Ans Mat

ANSYS Mechanical APDL Material Reference
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Table of Contents
1. Introduction to Material Models ............................................................................................................. 1 1.1. Material Models for Displacement Applications ................................................................................. 1 1.2. Material Models for Temperature Applications ................................................................................... 2 1.3. Material Models for Electromagnetic Applications ............................................................................. 2 1.4. Material Models for Coupled Applications ......................................................................................... 3 1.5. Material Parameters .......................................................................................................................... 3 2. Material Model Element Support ........................................................................................................... 5 3. Material Models .................................................................................................................................... 13 3.1. Understanding Material Data Tables ................................................................................................ 13 3.2. Experimental Data .......................................................................................................................... 14 3.3. Linear Material Properties ............................................................................................................... 14 3.3.1. Defining Linear Material Properties ......................................................................................... 15 3.3.2. Stress-Strain Relationships ...................................................................................................... 17 3.3.3. Anisotropic Elasticity .............................................................................................................. 18 3.3.4. Damping ............................................................................................................................... 18 3.3.5. Thermal Expansion ................................................................................................................. 19 3.3.6. Emissivity ............................................................................................................................... 20 3.3.7. Specific Heat .......................................................................................................................... 20 3.3.8. Film Coefficients ..................................................................................................................... 21 3.3.9. Temperature Dependency ...................................................................................................... 21 3.3.10. How Material Properties Are Evaluated ................................................................................. 21 3.4. Rate-Independent Plasticity ............................................................................................................ 21 3.4.1. Understanding the Plasticity Models ....................................................................................... 22 3.4.1.1. Nomenclature ............................................................................................................... 23 3.4.1.2. Strain Decomposition .................................................................................................... 24 3.4.1.3.Yield Criterion ................................................................................................................ 24 3.4.1.4. Flow Rule ...................................................................................................................... 25 3.4.1.5. Hardening ..................................................................................................................... 26 3.4.1.6. Large Deformation ........................................................................................................ 27 3.4.1.7. Output .......................................................................................................................... 27 3.4.1.8. Resources ...................................................................................................................... 28 3.4.2. Isotropic Hardening ............................................................................................................... 30 3.4.2.1. Yield Criteria and Plastic Potentials ................................................................................. 30 3.4.2.1.1. Von Mises Yield Criterion ....................................................................................... 30 3.4.2.1.2. Hill Yield Criterion ................................................................................................. 31 3.4.2.2. General Isotropic Hardening Classes .............................................................................. 33 3.4.2.2.1. Bilinear Isotropic Hardening .................................................................................. 33 3.4.2.2.1.1. Defining the Bilinear Isotropic Hardening Model ........................................... 34 3.4.2.2.2. Multilinear Isotropic Hardening ............................................................................. 34 3.4.2.2.2.1. Defining the Multilinear Isotropic Hardening Model ...................................... 35 3.4.2.2.3. Nonlinear Isotropic Hardening .............................................................................. 36 3.4.2.2.3.1. Power Law Nonlinear Isotropic Hardening .................................................... 36 3.4.2.2.3.2. Voce Law Nonlinear Isotropic Hardening ....................................................... 37 3.4.3. Kinematic Hardening ............................................................................................................. 38 3.4.3.1. Yield Criteria and Plastic Potentials ................................................................................. 38 3.4.3.2. General Kinematic Hardening Classes ............................................................................ 39 3.4.3.2.1. Bilinear Kinematic Hardening ................................................................................ 39 3.4.3.2.1.1. Defining the Bilinear Kinematic Hardening Model ......................................... 40 3.4.3.2.2. Multilinear Kinematic Hardening ........................................................................... 40 3.4.3.2.2.1. Defining the Multilinear Kinematic Hardening Model .................................... 42
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Material Reference 3.4.3.2.3. Nonlinear Kinematic Hardening ............................................................................ 43 3.4.3.2.3.1. Defining the Nonlinear Kinematic Hardening Model ..................................... 43 3.4.4. Generalized Hill ...................................................................................................................... 44 3.4.4.1. Defining the Generalized Hill Model ............................................................................... 46 3.4.5. Drucker-Prager ....................................................................................................................... 47 3.4.5.1. Classic Drucker-Prager ................................................................................................... 47 3.4.5.1.1. Defining the Classic Drucker-Prager Model ............................................................ 47 3.4.5.2. Extended Drucker-Prager (EDP) ...................................................................................... 48 3.4.5.2.1. EDP Yield Criteria Forms ........................................................................................ 48 3.4.5.2.1.1. Linear Form .................................................................................................. 48 3.4.5.2.1.2. Power Law Form ........................................................................................... 48 3.4.5.2.1.3. Hyperbolic Form .......................................................................................... 49 3.4.5.2.2. EDP Plastic Flow Potentials .................................................................................... 50 3.4.5.2.2.1. Linear Form .................................................................................................. 50 3.4.5.2.2.2. Power Law Form ........................................................................................... 51 3.4.5.2.2.3. Hyperbolic Form .......................................................................................... 51 3.4.5.2.3. Plastic Strain Increments for Flow Potentials .......................................................... 52 3.4.5.2.4. Example EDP Material Model Definitions ............................................................... 52 3.4.5.3. Extended Drucker-Prager Cap ........................................................................................ 53 3.4.5.3.1. Defining the EDP Cap Yield Criterion and Hardening .............................................. 55 3.4.5.3.2. Defining the EDP Cap Plastic Potential ................................................................... 56 3.4.5.3.3. Example EDP Cap Material Model Definition .......................................................... 56 3.4.6. Gurson ................................................................................................................................... 57 3.4.6.1. Void Volume Fraction ..................................................................................................... 57 3.4.6.2. Hardening ..................................................................................................................... 59 3.4.6.3. Defining the Gurson Material Model .............................................................................. 60 3.4.6.3.1. Defining the Gurson Base Model ........................................................................... 60 3.4.6.3.2. Defining Stress- or Strain-Controlled Nucleation .................................................... 60 3.4.6.3.3. Defining the Void Coalescence Behavior ............................................................... 61 3.4.6.3.4. Example Gurson Model Definition ......................................................................... 61 3.4.7. Cast Iron ................................................................................................................................ 62 3.4.7.1. Defining the Cast Iron Material Model ............................................................................ 64 3.5. Rate-Dependent Plasticity (Viscoplasticity) ...................................................................................... 64 3.5.1. Perzyna and Peirce Options .................................................................................................... 65 3.5.2. Exponential Visco-Hardening (EVH) Option ............................................................................. 65 3.5.3. Anand Option ........................................................................................................................ 66 3.5.4. Defining Rate-Dependent Plasticity (Viscoplasticity) ............................................................... 67 3.5.5. Creep ..................................................................................................................................... 67 3.5.5.1. Implicit Creep Equations ................................................................................................ 68 3.5.5.2. Explicit Creep Equations ................................................................................................ 70 3.5.5.2.1. Primary Explicit Creep Equation for C6 = 0 ............................................................. 71 3.5.5.2.2. Primary Explicit Creep Equation for C6 = 1 ............................................................. 71 3.5.5.2.3. Primary Explicit Creep Equation for C6 = 2 ............................................................. 71 3.5.5.2.4. Primary Explicit Creep Equation for C6 = 9 ............................................................. 71 3.5.5.2.4.1. Double Exponential Creep Equation (C4 = 0) ................................................. 71 3.5.5.2.4.2. Rational Polynomial Creep Equation with Metric Units (C4 = 1) ...................... 72 3.5.5.2.4.3. Rational Polynomial Creep Equation with English Units (C4 = 2) .................... 72 3.5.5.2.5. Primary Explicit Creep Equation for C6 = 10 ........................................................... 73 3.5.5.2.5.1. Double Exponential Creep Equation (C4 = 0) ................................................. 73 3.5.5.2.5.2. Rational Polynomial Creep Equation with Metric Units (C4 = 1) ...................... 73 3.5.5.2.5.3. Rational Polynomial Creep Equation with English Units (C4 = 2) .................... 73 3.5.5.2.6. Primary Explicit Creep Equation for C6 = 11 ........................................................... 73
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Material Reference 3.5.5.2.6.1. Modified Rational Polynomial Creep Equation (C4 = 0) .................................. 73 3.5.5.2.6.2. Rational Polynomial Creep Equation with Metric Units (C4 = 1) ...................... 74 3.5.5.2.6.3. Rational Polynomial Creep Equation with English Units (C4 = 2) .................... 74 3.5.5.2.7. Primary Explicit Creep Equation for C6 = 12 ........................................................... 74 3.5.5.2.8. Primary Explicit Creep Equation for C6 Equals 13 ................................................... 75 3.5.5.2.9. Primary Explicit Creep Equation for C6 = 14 ........................................................... 76 3.5.5.2.10. Primary Explicit Creep Equation for C6 = 15 ......................................................... 76 3.5.5.2.11. Primary Explicit Creep Equation for C6 = 100 ....................................................... 77 3.5.5.2.12. Secondary Explicit Creep Equation for C12 = 0 ..................................................... 77 3.5.5.2.13. Secondary Explicit Creep Equation for C12 = 1 ..................................................... 77 3.5.5.2.14. Irradiation Induced Explicit Creep Equation for C66 = 5 ........................................ 77 3.6. Hyperelasticity ................................................................................................................................ 77 3.6.1. Arruda-Boyce Hyperelasticity .................................................................................................. 78 3.6.2. Blatz-Ko Foam Hyperelasticity ................................................................................................. 79 3.6.3. Extended Tube Hyperelasticity ............................................................................................... 79 3.6.4. Gent Hyperelasticity ............................................................................................................... 80 3.6.5. Mooney-Rivlin Hyperelasticity ................................................................................................ 80 3.6.6. Neo-Hookean Hyperelasticity ................................................................................................. 82 3.6.7. Ogden Hyperelasticity ............................................................................................................ 82 3.6.8. Ogden Compressible Foam Hyperelasticity ............................................................................. 83 3.6.9. Polynomial Form Hyperelasticity ............................................................................................. 84 3.6.10. Response Function Hyperelasticity ....................................................................................... 85 3.6.11. Yeoh Hyperelasticity ............................................................................................................. 86 3.6.12. Special Hyperelasticity .......................................................................................................... 87 3.6.12.1. Anisotropic Hyperelasticity .......................................................................................... 87 3.6.12.2. Bergstrom-Boyce Material ............................................................................................ 88 3.6.12.3. Mullins Effect ............................................................................................................... 89 3.6.12.4. User-Defined Hyperelastic Material .............................................................................. 90 3.7. Viscoelasticity ................................................................................................................................. 90 3.7.1. Viscoelastic Formulation ......................................................................................................... 91 3.7.1.1. Small Deformation ......................................................................................................... 91 3.7.1.2. Small Strain with Large Deformation .............................................................................. 93 3.7.1.3. Large Deformation ........................................................................................................ 93 3.7.2. Time-Temperature Superposition ........................................................................................... 94 3.7.2.1. Williams-Landel-Ferry Shift Function .............................................................................. 94 3.7.2.2. Tool-Narayanaswamy Shift Function ............................................................................... 95 3.7.2.3. User-Defined Shift Function ........................................................................................... 96 3.7.3. Harmonic Viscoelasticity ......................................................................................................... 96 3.7.3.1. Prony Series Complex Modulus ...................................................................................... 97 3.7.3.2. Experimental Data Complex Modulus ............................................................................ 97 3.7.3.3. Frequency-Temperature Superposition .......................................................................... 99 3.7.3.4. Stress ............................................................................................................................ 99 3.8. Microplane ..................................................................................................................................... 99 3.8.1. Microplane Modeling ........................................................................................................... 100 3.8.1.1. Discretization .............................................................................................................. 101 3.8.2. Material Models with Degradation and Damage .................................................................... 102 3.8.3. Material Parameters Definition and Example Input ................................................................ 104 3.8.4. Learning More About Microplane Material Modeling ............................................................. 105 3.9. Porous Media ................................................................................................................................ 105 3.9.1. Coupled Pore-Fluid Diffusion and Structural Model of Porous Media ...................................... 105 3.9.2. Johnson-Champoux-Allard Equivalent Fluid Model of a Porous Media ................................... 106 3.10. Electricity and Magnetism ........................................................................................................... 106
Material Reference 3.10.1. Piezoelectricity ................................................................................................................... 107 3.10.2. Piezoresistivity ................................................................................................................... 108 3.10.3. Magnetism ......................................................................................................................... 109 3.10.4. High-Frequency Electromagnetism ..................................................................................... 109 3.10.4.1. 3-D High-Frequency Material Properties ..................................................................... 109 3.10.4.1.1. Conductivity, Permittivity, and Permeability Matrices .......................................... 109 3.10.4.1.2. B-H Nonlinear Material Permeability Matrix ........................................................ 110 3.10.4.1.3. Anisotropic Electric and Magnetic Loss Tangents ............................................... 112 3.10.4.1.4. Frequency-Dependent Lossy Dielectric .............................................................. 112 3.10.4.2. 2-D High-Frequency Material Properties ..................................................................... 114 3.10.5. Anisotropic Electric Permittivity .......................................................................................... 115 3.11. Gasket ........................................................................................................................................ 116 3.12. Swelling ...................................................................................................................................... 117 3.13. Shape Memory Alloy (SMA) ......................................................................................................... 119 3.13.1. Shape Memory Alloy Model for Superelasticity .................................................................... 120 3.13.1.1. Constitutive Model for Superelasticity ........................................................................ 120 3.13.1.2. Material Parameters for the Superelastic SMA Material Model ..................................... 123 3.13.2. Shape Memory Material Model with Shape Memory Effect .................................................. 124 3.13.2.1. The Constitutive Model for Shape Memory Effect ........................................................ 124 3.13.2.2. Material Parameters for the Shape Memory Effect Option ........................................... 127 3.13.3. Element Support for SMA ................................................................................................... 127 3.13.4. Learning More About Shape Memory Alloy ......................................................................... 127 3.14. MPC184 Joint .............................................................................................................................. 128 3.14.1. Linear Elastic Stiffness and Damping Behavior ..................................................................... 128 3.14.2. Nonlinear Elastic Stiffness and Damping Behavior ............................................................... 129 3.14.2.1. Specifying a Function Describing Nonlinear Stiffness Behavior .................................... 130 3.14.3. Frictional Behavior .............................................................................................................. 131 3.15. Contact Friction .......................................................................................................................... 133 3.15.1. Isotropic Friction ................................................................................................................ 133 3.15.2. Orthotropic Friction ............................................................................................................ 134 3.15.3. Redefining Friction Between Load Steps ............................................................................. 134 3.15.4. User-Defined Friction .......................................................................................................... 135 3.16. Cohesive Zone ............................................................................................................................ 135 3.16.1. Exponential Cohesive Zone Material for Interface Elements ................................................. 136 3.16.2. Bilinear Cohesive Zone Material for Interface Elements ........................................................ 136 3.16.3. Cohesive Zone Material for Contact Elements ...................................................................... 137 3.17. Fluids .......................................................................................................................................... 138 3.18. User-Defined Material Model ....................................................................................................... 140 3.18.1. Using State Variables with UserMat .................................................................................... 140 3.19. Material Strength Limits .............................................................................................................. 140 3.20. Material Damage ........................................................................................................................ 142 3.20.1. Damage Initiation Criteria ................................................................................................... 143 3.20.2. Damage Evolution Law ....................................................................................................... 144 4. Explicit Dynamics Materials ................................................................................................................ 145 5. Material Curve Fitting ......................................................................................................................... 147 5.1. Hyperelastic Material Curve Fitting ................................................................................................ 147 5.1.1. Understanding the Hyperelastic Material Curve-Fitting Process ............................................. 147 5.1.2. Step 1. Prepare Experimental Data ........................................................................................ 148 5.1.3. Step 2. Input the Experimental Data ...................................................................................... 149 5.1.3.1. Batch ........................................................................................................................... 149 5.1.3.2. GUI .............................................................................................................................. 150 5.1.4. Step 3. Select a Material Model Option .................................................................................. 150
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Material Reference 5.1.4.1. Batch Method .............................................................................................................. 151 5.1.4.2. GUI Method ................................................................................................................. 151 5.1.5. Step 4. Initialize the Coefficients ............................................................................................ 151 5.1.5.1. Batch ........................................................................................................................... 152 5.1.5.2. GUI .............................................................................................................................. 152 5.1.6. Step 5. Specify Control Parameters and Solve ........................................................................ 152 5.1.6.1. Batch ........................................................................................................................... 153 5.1.6.2. GUI .............................................................................................................................. 153 5.1.7. Step 6. Plot Your Experimental Data and Analyze ................................................................... 153 5.1.7.1. GUI .............................................................................................................................. 153 5.1.7.2. Review/Verify .............................................................................................................. 154 5.1.8. Step 7. Write Data to the TB Command .................................................................................. 154 5.1.8.1. Batch ........................................................................................................................... 154 5.1.8.2. GUI .............................................................................................................................. 154 5.2. Viscoelastic Material Curve Fitting ................................................................................................. 154 5.2.1. Understanding the Viscoelastic Material Curve-Fitting Process .............................................. 155 5.2.2. Step 1. Prepare Experimental Data ........................................................................................ 155 5.2.3. Step 2. Input the Data ........................................................................................................... 156 5.2.3.1. Batch ........................................................................................................................... 156 5.2.3.2. GUI .............................................................................................................................. 157 5.2.4. Step 3. Select a Material Model Option .................................................................................. 157 5.2.4.1. Batch Method .............................................................................................................. 157 5.2.4.2. GUI Method ................................................................................................................. 158 5.2.5. Step 4. Initialize the Coefficients ............................................................................................ 158 5.2.5.1. Batch Method ............................................................................................................. 159 5.2.5.2. GUI Method ................................................................................................................. 160 5.2.6. Step 5. Specify Control Parameters and Solve ........................................................................ 160 5.2.6.1. Temperature-Dependent Solutions Using the Shift Function ......................................... 160 5.2.6.2. Temperature-Dependent Solutions Without the Shift Function ..................................... 161 5.2.6.3. Batch Method .............................................................................................................. 162 5.2.6.4. GUI Method ................................................................................................................. 163 5.2.7. Step 6. Plot the Experimental Data and Analyze ..................................................................... 163 5.2.7.1. Analyze Your Curves for Proper Fit ................................................................................ 163 5.2.8. Step 7. Write Data to the TB Command .................................................................................. 163 5.2.8.1. Batch Method .............................................................................................................. 164 5.2.8.2. GUI Method ................................................................................................................. 164 5.3. Creep Material Curve Fitting .......................................................................................................... 164 5.3.1. Understanding the Creep Material Curve-Fitting Process ....................................................... 164 5.3.2. Step 1. Prepare Experimental Data ........................................................................................ 165 5.3.3. Step 2. Input the Experimental Data ...................................................................................... 167 5.3.3.1. Batch Method .............................................................................................................. 167 5.3.3.2. GUI Method ................................................................................................................. 167 5.3.4. Step 3. Select a Material Model Option .................................................................................. 167 5.3.4.1. Batch Method .............................................................................................................. 167 5.3.4.2. GUI Method ................................................................................................................. 168 5.3.5. Step 4. Initialize the Coefficients ............................................................................................ 168 5.3.5.1. Batch Method .............................................................................................................. 169 5.3.5.2. GUI Method ................................................................................................................. 170 5.3.6. Step 5. Specify Control Parameters and Solve ........................................................................ 170 5.3.6.1. Batch Method .............................................................................................................. 170 5.3.6.2. GUI Method ................................................................................................................. 170 5.3.7. Step 6. Plot the Experimental Data and Analyze ..................................................................... 170
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Material Reference 5.3.7.1. GUI Method ................................................................................................................. 171 5.3.7.2. Analyze Your Curves for Proper Fit ................................................................................ 171 5.3.8. Step 7. Write Data to the TB Command .................................................................................. 171 5.3.8.1. Batch Method .............................................................................................................. 171 5.3.8.2. GUI Method ................................................................................................................. 171 5.3.9. Tips For Curve Fitting Creep Models ...................................................................................... 172 5.4. Chaboche Material Curve Fitting ................................................................................................... 173 5.4.1. Understanding the Chaboche Material Curve-Fitting Process ................................................ 173 5.4.2. Step 1. Prepare Experimental Data ........................................................................................ 174 5.4.3. Step 2. Input the Experimental Data ...................................................................................... 175 5.4.3.1. Batch Method .............................................................................................................. 175 5.4.3.2. GUI Method ................................................................................................................. 176 5.4.4. Step 3. Select a Material Model Option .................................................................................. 176 5.4.4.1. Batch Method .............................................................................................................. 176 5.4.4.2. GUI Method ................................................................................................................. 176 5.4.5. Step 4. Initialize the Coefficients ............................................................................................ 177 5.4.5.1. Including Isotropic Hardening Models with Chaboche Kinematic Hardening ................. 177 5.4.5.2. General Process for Initializing MISO Option Coefficients .............................................. 177 5.4.5.2.1. Batch Method ..................................................................................................... 178 5.4.5.2.2. GUI Method ........................................................................................................ 179 5.4.6. Step 5. Specify Control Parameters and Solve ........................................................................ 179 5.4.6.1. Temperature-Dependent Solutions .............................................................................. 179 5.4.6.2. Batch Method .............................................................................................................. 179 5.4.6.3. GUI Method ................................................................................................................. 180 5.4.7. Step 6. Plot the Experimental Data and Analyze ..................................................................... 180 5.4.7.1. Analyzing Your Curves for Proper Fit ............................................................................. 180 5.4.8. Step 7. Write Data to the TB Command .................................................................................. 181 6. Material Model Combinations ............................................................................................................ 183 7. Understanding Field Variables ............................................................................................................ 187 7.1. User-Defined Field Variables .......................................................................................................... 187 7.2. Data Processing ............................................................................................................................ 188 7.3. Example: One-Dimensional Interpolation ....................................................................................... 189 7.4. Example: Two-Dimensional Interpolation ....................................................................................... 189 7.5. Example: Multi-Dimensional Interpolation ..................................................................................... 190 8. GUI-Inaccessible Material Properties .................................................................................................. 193
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List of Figures
3.1. Stress-Strain Curve for an Elastic-Plastic Material .................................................................................... 22 3.2.Yield Surface in Principal Stress Space .................................................................................................... 25 3.3. Plastic Strain Flow Rule .......................................................................................................................... 25 3.4. Isotropic Hardening of the Yield Surface ................................................................................................ 26 3.5. Kinematic Hardening of the Yield Surface .............................................................................................. 27 3.6.Yield Surface for von Mises Yield Criterion .............................................................................................. 31 3.7. Stress vs. Total Strain for Bilinear Isotropic Hardening ............................................................................. 34 3.8. Stress vs. Total Strain for Multilinear Isotropic Hardening ........................................................................ 35 3.9. Stress vs. Plastic Strain for Voce Hardening ............................................................................................. 37 3.10. Stress vs. Total Strain for Bilinear Kinematic Hardening ......................................................................... 39 3.11. Stress vs. Total Strain for Multilinear Kinematic Hardening .................................................................... 41 3.12. Power Law Criterion in the Meridian Plane ........................................................................................... 49 3.13. Hyperbolic and Linear Criterion in the Meridian Plane .......................................................................... 50 3.14. Yield Surface for the Cap Criterion ....................................................................................................... 54 3.15. Growth, Nucleation, and Coalescence of Voids at Microscopic Scale ...................................................... 58 3.16. Cast Iron Yield Surfaces for Compression and Tension .......................................................................... 63 3.17. Generalized Maxwell Solid in One Dimension ...................................................................................... 91 3.18. Sphere Discretization by 42 Microplanes ............................................................................................ 102 3.19. Damage Parameter d Depending on the Equivalent Strain Energy ...................................................... 103 3.20. Stress-strain Behavior at Uniaxial Tension ........................................................................................... 104 3.21. Pseudoelasticity (PE) and Shape Memory Effect (SME) ........................................................................ 119 3.22. Typical Superelasticity Behavior ......................................................................................................... 120 3.23. Idealized Stress-Strain Diagram of Superelastic Behavior .................................................................... 122 3.24. Admissible Paths for Elastic Behavior and Phase Transformations ....................................................... 126
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List of Tables
3.1. Linear Material Property Descriptions .................................................................................................... 15 3.2. Implicit Creep Equations ....................................................................................................................... 68 3.3. Superelastic Option Constants ............................................................................................................ 123 3.4. Shape Memory Effect Option Constants .............................................................................................. 127 5.1. Experimental Details for Case 1 and 2 Models and Blatz-Ko .................................................................. 148 5.2. Experimental Details for Case 3 Models ............................................................................................... 148 5.3. Hyperelastic Curve-Fitting Model Types ............................................................................................... 150 5.4. Viscoelastic Data Types and Abbreviations ........................................................................................... 155 5.5. Creep Data Types and Abbreviations ................................................................................................... 165 5.6. Creep Model and Data/Type Attribute ................................................................................................. 166 5.7. Creep Models and Abbreviations ......................................................................................................... 168 6.1. Material Model Combination Possibilities ............................................................................................ 183
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Chapter 1: Introduction to Material Models

Material models (also called constitutive models), are the mathematical representation of a material's response to an applied load. Typical model classes include the relationships between stress-strain, heat flux-temperature gradient, voltage-strain, and current-voltage, but also include more general behaviors such as friction and bonding, and response due to changes in the physical environment such as thermal expansion and swelling. This reference provides information about material model behavior and application, including details about the load-response relationship and the necessary information required to use the material models in an analysis. The models are grouped based on the degrees of freedom that, directly or indirectly, give the loading function that serves as the input for the material model. The following related introductory topics are available: 1.1. Material Models for Displacement Applications 1.2. Material Models for Temperature Applications 1.3. Material Models for Electromagnetic Applications 1.4. Material Models for Coupled Applications 1.5. Material Parameters
1.1. Material Models for Displacement Applications

For analyses that include displacement degrees of freedom, the input is a function of deformation such as strain or displacement, and the response is given force-like quantities such as stress or normal and tangential forces. The following general material types are available: Type Linear elastic Behavior Application
The response is the stresses that are directly Many metals are linear proportional to the strains and the material will elastic at room temperature fully recover the original shape when unloaded. when the strains are small. For isotropic materials, the relationship is given by Hooke's law and this relationship can be generalized to define anisotropic behavior. The deformation of the material includes a permanent, or plastic, component that will not return to the original configuration if the load is removed and evolves in response to the deformation history. These materials also typically have an elastic behavior so that the combined deformation includes a part that is recoverable upon unloading. The behavior of these models is defined by a strain-energy potential, which is the energy stored in the material due to strain. The mathPlastic deformation is observed in many materials such as metals, alloys, soils, rocks, concrete, and ceramics.
Plastic and elastic-plastic
Hyperelastic
Hyperelastic models are often used for materials that undergo large elastic de-
Introduction to Material Models Type Behavior ematical formulation is convenient for largedeformation analyses. Rate effects and time dependency This is a general behavior in which the response of the material depends on the rate of deformation, and thus also the time. Examples include viscoelasticity, viscoplasticity, creep and damping. Application formation, such as polymers and biological materials. Metal alloys that show significant creep deformation under elevated temperature, rate-dependent metal forming applications, polymers which typically get stiffer for increased deformation rate, and structures that damp out high frequency waves under dynamic loading. Radiation environments, bonded materials with thermal strain mismatch, and soils that absorb water.
Expansion and swelling
Materials often respond to changes in the physical environment and this response affects the structural behavior. Examples include thermal expansion in which changes in material volume depend on changes in temperature and swelling behaviors that depend on hygroscopic effects or neutron flux.
Interaction
These models produce a response based on the Gasket and joint materials interaction of structures. and also models of bonded and separating surfaces along interfaces or material cleavage. An elastic constitutive model with an internal phase transformation. The phase transformation depends on the stress and temperature that cause an internal transformation strain.
Shape memory alloy
1.2. Material Models for Temperature Applications

For analyses that include temperature as a degree of freedom, the material model for conduction gives a heat flux due to the gradient of temperature and also interaction between bodies due to radiative heat transfer that is dependent on surface temperature differences.
1.3. Material Models for Electromagnetic Applications

Material models for use in analyses with electromagnetic degrees of freedom include: Type Magnetic Conductivity Permittivity and Permeability Description Gives the magnetomotive force in response to the magnetic flux. For electric and magnetic current conductivities that model the relationship between the respective field and its flux. Gives the energy storage in a material in response to an electromagnetic field.
Material Parameters Type Loss Description Represents the energy lost in electromagnetic and dielectric materials in response to changes in electromagnetic fields.
1.4. Material Models for Coupled Applications

Some models are valid in analyses that couple the thermal, electromagnetic, and displacement degrees of freedom. Although the models are coupled, they remain distinct and give the same load-response behavior. However, the piezoelectric and piezoresistive materials are electromechanical coupled models that give a strain in response to a voltage and also produce a voltage in response to straining.
1.5. Material Parameters

Because a material model represents a mathematical relationship between response and load, it requires input parameters so that the model matches the material behavior. In some cases, the parameters can be a function of physical field quantities such as temperature, frequency or time or interaction quantities such as normal pressure, relative distance, or relative velocity. Matching the model to the actual behavior can be challenging; therefore, some built-in curve-fitting methods are available that use minimization to select a set of parameters that give a close fit to measured material behavior. The curve-fitting methods help you to select material parameters for creep, hyperelastic, viscoelastic, and some plastic models.
Chapter 2: Material Model Element Support

Following is a list of available material models and the elements that support each material. Material models are specified via the TB,Lab command, where Lab represents the material model label (shortcut name). For a list of elements and the material models they support (Lab value), see Element Support for Material Models in the Element Reference. Label (Lab) AHYPER Material Model Anisotropic hyperelasticity Elements SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, SHELL208, SHELL209, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290
ANEL
Anisotropic elasticity SOLID5, PLANE13, SOLID98, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Also, explicit dynamic elements SOLID164, SOLID168
ANISO
Anisotropic plasticity SOLID65 Nonlinear legacy elements only
BB
Bergstrom-Boyce
PLANE182, PLANE183, SHELL181, SOLID185, SOLID186, SOLID187, SOLSH190, SHELL208, SHELL209, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 SOLID5, PLANE13, PLANE53, SOLID96, SOLID97, SOLID98, PLANE233, SOLID236, SOLID237 von Mises plasticity: SOLID65, LINK180, SHELL181, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SHELL281, PIPE288, PIPE289, ELBOW290, PLANE182, PLANE183 Also , explicit dynamic elements LINK160, BEAM161, PLANE162, SHELL163, SOLID164, SOLID168 Hill plasticity: SOLID65, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOL-
BH BISO
Magnetic Bilinear isotropic hardening
Material Model Element Support Label (Lab) Material Model Elements ID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 BKIN Bilinear kinematic hardening von Mises plasticity: SOLID65, LINK180, SHELL181, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SHELL281, PIPE288, PIPE289, ELBOW290, PLANE182, PLANE183 Also , explicit dynamic elements LINK160, BEAM161, PLANE162, SHELL163, SOLID164 Hill plasticity: SOLID65, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 CAST Cast iron PLANE182 (not applicable for plane stress), PLANE183 (not applicable for plane stress), SOLID185, SOLID186, SOLID187, SOLSH190, PLANE223, SOLID226, SOLID227, SOLID272, SOLID273, SOLID285, PIPE288, PIPE289 SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, SOLSH190, SHELL208, SHELL209, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 PLANE182, SOLID185 von Mises or Hill plasticity: LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 CNDE CNDM Anisotropic electric current conductivity HF119, HF120
CDM
Mullins effect
CGCR
Crack growth fracture criterion
CHABOCHE Chaboche nonlinear kinematic hardening
Anisotropic magnet- HF119, HF120 ic current conductivity Composite damage Explicit dynamic elements PLANE162, SHELL163, SOLID164, SOLID168
COMP
Label (Lab) CONCR
Material Model Concrete or concrete damage
Elements SOLID65 Concrete damage model using explicit dynamic elements SOLID164, SOLID168
CREEP
Creep
Implicit creep with von Mises or Hill potential: SOLID65, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SOLID285, SHELL281, PIPE288, PIPE289, ELBOW290
CTE
Coefficient of thermal expansion
LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, CPT212, CPT213, CPT215, CPT216, CPT217, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 CONTA171, CONTA172, CONTA173, CONTA174, CONTA175, CONTA176, CONTA177, INTER202, INTER203, INTER204, INTER205 COMBI165 Progressive damage evolution (MPDG option): LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Continuum damage mechanics (CDM option): SHELL181, PLANE182 (plane stress option), PLANE183 (plane stress option), SHELL208, SHELL209, SHELL281, PIPE288 (thin pipe formulation), PIPE289 (thin pipe formulation), ELBOW290
CZM
Cohesive zone
DISCRETE DMGE
Explicit springdamper (discrete) Damage evolution law
DMGI
Damage initiation criteria
LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 SOLID65 HF118, HF119, HF120, PLANE223, SOLID226, SOLID227
DP DPER
Drucker-Prager plasticity Anisotropic electric permittivity
Material Model Element Support Label (Lab) EDP Material Model Extended DruckerPrager Elements PLANE182 (not applicable for plane stress), PLANE183 (not applicable for plane stress), SOLID185, SOLID186, SOLID187, SOLSH190, PLANE223, SOLID226, SOLID227, SOLID272, SOLID273, SOLID285, PIPE288, PIPE289 LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, CPT212, CPT213, CPT215, CPT216, CPT217, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Explicit dynamic elements only Explicit dynamic elements BEAM161, PLANE162, SOLID164, SOLID168 Used only with other material models FLUID116
ELASTIC
Elasticity
EOS EVISC EXPE FCON FCLI FLUID FOAM FRIC
Equation of state Viscoelasticity Experimental data Fluid conductance data
Failure criteria mater- All structural elements ial strength limits Fluid Foam Coefficient of friction HSFLD241, HSFLD242 Explicit dynamic elements PLANE162, SOLID164, SOLID168 CONTA171, CONTA172, CONTA173, CONTA174, CONTA175, CONTA176, CONTA177, CONTA178 Orthotropic friction (TB,FRIC,,,,ORTHO) is not applicable to the 2-D contact elements CONTA171 and CONTA172, nor to CONTA178.
GASKET GCAP GURSON
Gasket Geological cap Gurson pressure-dependent plasticity
INTER192, INTER193, INTER194, INTER195 Explicit dynamic elements SOLID164, SOLID168 PLANE182 (not applicable for plane stress), PLANE183 (not applicable for plane stress), SOLID185, SOLID186, SOLID187, SOLSH190, PLANE223, SOLID226, SOLID227, SOLID272, SOLID273, SOLID285 HF119, HF120 FLUID116 LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Explicit dynamic elements PLANE162, SOLID164, SOLID168 SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, SOLSH190, CPT212, CPT213, CPT215, CPT216, CPT217,
HFFDLD HFLM HILL
Frequency-dependent lossy dielectric Film coefficient data Hill anisotropy
HONEY HYPER
Honeycomb Hyperelasticity
Label (Lab)
Material Model
Elements SHELL208, SHELL209, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290,
INTER JOIN
User-defined contact CONTA171, CONTA172, CONTA173, CONTA174, CONTA175, interaction CONTA176, CONTA177, CONTA178 Joint (linear and MPC184 nonlinear elastic stiffness, linear and nonlinear damping, and frictional behavior) Multilinear kinematic hardening von Mises plasticity: SOLID65, PLANE13, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SHELL281, PIPE288, PIPE289, ELBOW290 Hill plasticity: SOLID65, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, ELBOW290
KINH
LSEM
Anisotropic electric and magnetic loss tangents Multilinear elasticity
HF119, HF120
MELAS
SOLID65 Nonlinear legacy elements only
MISO
Multilinear isotropic hardening
von Mises plasticity: SOLID65, LINK180, SHELL181,PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Hill plasticity: SOLID65, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, RE-
Material Model Element Support Label (Lab) Material Model Elements INF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, ELBOW290 MKIN Multilinear kinematic hardening von Mises plasticity: LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Hill plasticity: LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, ELBOW290 MOONEY MPLANE Mooney-Rivlin hyper- Explicit dynamic elements PLANE162, SHELL163, SOLID164, elasticity SOLID168 Microplane PLANE182, PLANE183, SOLID185, SOLID186, SOLID187 Can be used with reinforcing elements REINF263, REINF264 and REINF265 to model reinforced concrete. MUR NLISO Anisotropic relative permeability Voce isotropic hardening law HF118, HF119, HF120 von Mises or Hill plasticity: LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 PERF Johnson-ChampouxAllard Equivalent Fluid Model of a Porous Media Piezoelectric matrix Plasticity FLUID30, FLUID220, FLUID221
PIEZ PLASTIC
SOLID5, PLANE13, SOLID98, PLANE223, SOLID226, SOLID227 LINK180, SHELL181, PIPE288, PIPE289, ELBOW290, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285 Explicit dynamic elements LINK160, BEAM161, PLANE162, SHELL163, SOLID164, SOLID168
PLAW
Plasticity laws
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Label (Lab) PM
Material Model Coupled Pore-Fluid Diffusion and Structural Model of Porous Media Prony series constants for viscoelastic materials
Elements CPT212, CPT213, CPT215, CPT216, CPT217
PRONY
LINK180, SHELL181, PIPE288, PIPE289, ELBOW290, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285 PLANE223, SOLID226, SOLID227
PZRS RATE
Piezoresistivity
Rate-dependent LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLplasticity (viscoplasti- ID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, REINF263, REcity) INF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Anand unified plasticity option: SHELL181 (except plane stress), PLANE182 (except plane stress), PLANE183 (except plane stress), SOLID185, SOLID186, SOLID187, SOLSH190, SOLID272, SOLID273, SOLID285, PIPE288, PIPE289
SDAMP
Material structural damping
SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, SOLID272, SOLID273, SHELL281, SOLID285, ELBOW290 LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, PLANE223, SOLID226, SOLID227, PIPE288, PIPE289, ELBOW290, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285 PLANE182, PLANE183, PLANE223 (with plane strain or axisymmetric stress states), SOLID185, SOLID186, SOLID187, SOLSH190, SOLID226, SOLID227, SOLID272, SOLID273, SOLID285
SHIFT
Shift function for viscoelastic materials
SMA
Shape memory alloy
STATE
State variables (user- SOLID65, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, defined) SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Also, user-defined plasticity or viscoplasticity: PLANE183
SWELL
Swelling
SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290
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Material Model Element Support Label (Lab) UNIAXIAL Material Model Uniaxial stress-strain relation User-defined Elements PLANE182, PLANE183, PLANE223 (not applicable for plane stress), SOLID185, SOLID186, SOLID187, SOLSH190, SOLID226, SOLID227, SOLID272, SOLID273, SOLID285 SOLID65, LINK180, SHELL181, PLANE182, PLANE183, SOLID185, SOLID186, SOLID187, BEAM188, BEAM189, SOLSH190, SHELL208, SHELL209, REINF263, REINF264, REINF265, SOLID272, SOLID273, SHELL281, SOLID285, PIPE288, PIPE289, ELBOW290 Also, user-defined plasticity or viscoplasticity: PLANE183
USER
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Chapter 3: Material Models

This document describes all material models available for implicit analysis, including information about material data table input (TB). The following material model topics are available: 3.1. Understanding Material Data Tables 3.2. Experimental Data 3.3. Linear Material Properties 3.4. Rate-Independent Plasticity 3.5. Rate-Dependent Plasticity (Viscoplasticity) 3.6. Hyperelasticity 3.7. Viscoelasticity 3.8. Microplane 3.9. Porous Media 3.10. Electricity and Magnetism 3.11. Gasket 3.12. Swelling 3.13. Shape Memory Alloy (SMA) 3.14. MPC184 Joint 3.15. Contact Friction 3.16. Cohesive Zone 3.17. Fluids 3.18. User-Defined Material Model 3.19. Material Strength Limits 3.20. Material Damage For a list of the elements that support each material model, see Material Model Element Support (p. 5). For related information, see Nonlinear Structural Analysis in the Structural Analysis Guide. For information about explicit dynamics material models, including detailed data table input, see Material Models in the ANSYS LS-DYNA User's Guide.
3.1. Understanding Material Data Tables

A material data table is a series of constants that are interpreted when they are used. Data tables are always associated with a material number and are most often used to define nonlinear material data (stress-strain curves, creep constants, swelling constants, and magnetization curves). Other material properties are described in Linear Material Properties (p. 14). For some element types, the data table is used for special element input data other than material properties. The form of the data table (referred to as the TB table) depends upon the data being defined: Where the form is peculiar to only one element type, the table is described with the element in Element Library. If the form applies to more than one element, it is described here and referenced in the element description.
13
Material Models
3.2. Experimental Data

The experimental data table option (TB,EXPE) allows you to input experimental data. The data is used with other material models. Initiate the experimental data table, then specify the appropriate experimental data type (TBOPT), as follows: UNIAXIAL - Uniaxial experimental data BIAXIAL - Equibiaxial experimental data SHEAR - Pure shear experimental data, also known as planar tension VOLUMETRIC - Volumetric experimental data SSHEAR - Simple shear experimental data UNITENSION - Uniaxial tension experimental data UNICOMPRESSION - Uniaxial compression experimental data GMODULUS - Shear modulus experimental data KMODULUS - Bulk modulus experimental data EMODULUS - Tensile modulus experimental data NUXY - Poisson's ratio experimental data Enter the data (TBPT) for each data point. Each data point entered consists of the independent variable followed by one or more dependent variables. The specific definition of the input points depends on the requirements of the material model associated with the experimental data. You can also define experimental data as a function of field variables. Field-dependent data are entered by preceding a set of experimental data (TBFIELD) to define the value of the field.
3.3. Linear Material Properties

Material properties (which may be functions of temperature) are described as linear properties because typical non-thermal analyses with these properties require only a single iteration. Conversely, if properties needed for a thermal analysis (such as KXX) are temperature-dependent, the problem is nonlinear. Properties such as stress-strain data are described as nonlinear properties because an analysis with these properties requires an iterative solution. Linear material properties that are required for an element, but which are not defined, use default values. (The exception is that EX and KXX must be input with a nonzero value where applicable.) Any additional material properties are ignored. The X, Y, and Z portions of the material property labels refer to the element coordinate system. In general, if a material is isotropic, only the X and possibly the XY term is input. The following topics concerning linear material properties are available: 3.3.1. Defining Linear Material Properties 3.3.2. Stress-Strain Relationships 3.3.3. Anisotropic Elasticity 3.3.4. Damping 3.3.5.Thermal Expansion 3.3.6. Emissivity 3.3.7. Specific Heat 3.3.8. Film Coefficients 3.3.9.Temperature Dependency
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Linear Material Properties 3.3.10. How Material Properties Are Evaluated
3.3.1. Defining Linear Material Properties

The linear material properties used by the element type are listed under "Material Properties" in the input table for each element type. The following table describes all available linear material properties, defined via the Lab value on the MP command: Table 3.1: Linear Material Property Descriptions MP, Lab Value EX EY EZ PRXY PRYZ PRXZ NUXY NUYZ NUXZ GXY GYZ GXZ ALPX ALPY ALPZ CTEX CTEY CTEZ THSX THSY THSZ REFT MU ALPD BETD Temp None None None Strain Strain/Temp Strain/Temp Force/Area None Force/Area Units Description Elastic modulus, element x direction Elastic modulus, element y direction Elastic modulus, element z direction Major Poisson's ratio, x-y plane Major Poisson's ratio, y-z plane Major Poisson's ratio, x-z plane Minor Poisson's ratio, x-y plane Minor Poisson's ratio, y-z plane Minor Poisson's ratio, x-z plane Shear modulus, x-y plane Shear modulus, y-z plane Shear modulus, x-z plane Secant coefficient of thermal expansion, element x direction Secant coefficient of thermal expansion, element y direction Secant coefficient of thermal expansion, element z direction Instantaneous coefficient of thermal expansion, element x direction Instantaneous coefficient of thermal expansion, element y direction Instantaneous coefficient of thermal expansion, element z direction Thermal strain, element x direction Thermal strain, element y direction Thermal strain, element z direction Reference temperature (as a property) (see also TREF) Coefficient of friction (or, for FLUID29 and FLUID30 elements, boundary admittance) Mass matrix multiplier for damping (also see ALPHAD) Stiffness matrix multiplier for damping (also see BETAD)
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Material Models MP, Lab Value DMPR DENS KXX KYY KZZ C ENTH HF EMIS Heat/Mass*Temp Heat/Vol Heat / (Time*Area*Temp) None Heat/Time QRATE None Force*Time/ Length2 Length/Time Heat*Length/ (Time*Area*Temp) None Mass/Vol
Units
Description Constant material damping coefficient Mass density Thermal conductivity, element x direction Thermal conductivity, element y direction Thermal conductivity, element z direction Specific heat Enthalpy ( DENS*C d(Temp)) Convection (or film) coefficient Emissivity Heat generation rate for thermal mass element MASS71 Fraction of plastic work converted to heat (Taylor-Quinney coefficient) for coupled-field elements PLANE223, SOLID226, and SOLID227 Viscosity Sonic velocity (FLUID29, FLUID30, FLUID129, and FLUID130 elements only) Magnetic relative permeability, element x direction Magnetic relative permeability, element y direction Magnetic relative permeability, element z direction Magnetic coercive force, element x direction
VISC SONC MURX MURY MURZ MGXX MGYY MGZZ RSVX RSVY RSVZ PERX PERY PERZ LSST SBKX SBKY SBKZ DXX DYY DZZ CREF CSAT
None
Current/Length
Magnetic coercive force, element y direction Magnetic coercive force, element z direction Electrical resistivity, element x direction
Resistance*Area/Length
Electrical resistivity, element y direction Electrical resistivity, element z direction Electric relative permittivity, element x direction
None None Voltage/Temp
Electric relative permittivity, element y direction Electric relative permittivity, element z direction Dielectric loss tangent Seebeck coefficient, element x direction Seebeck coefficient, element y direction Seebeck coefficient, element z direction Diffusion coefficient, element x direction
Length /Time Mass/Length Mass/Length

3 3
Diffusion coefficient, element y direction Diffusion coefficient, element z direction Saturated concentration Reference concentration
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Linear Material Properties MP, Lab Value BETX BETY BETZ Length /Mass
3
Units
Description Coefficient of diffusion expansion, element x direction Coefficient of diffusion expansion, element y direction Coefficient of diffusion expansion, element z direction
3.3.2. Stress-Strain Relationships

Structural material properties must be input as an isotropic, orthotropic, or anisotropic material. If the material is isotropic: Young's modulus (EX) must be input. Poisson's ratio (PRXY or NUXY) defaults to 0.3. If a zero value is desired, input PRXY or NUXY with a zero or blank value. Poisson's ratio should not be 0.5 nor -1.0. The shear modulus (GXY) defaults to EX/(2(1+NUXY)). If GXY is input, it must match EX/(2 (1+NUXY)). The sole purpose for inputting GXY is to ensure consistency with the other two properties. If the material is orthotropic: EX, EY, EZ, (PRXY, PRYZ, PRXZ, or NUXY, NUYZ, NUXZ), GXY, GYZ, and GXZ must all be input if the element type uses the material property. There are no defaults. For example, if only EX and EY are input (with different values) to a plane stress element, The program generates an error message indicating that the material is orthotropic and that GXY and NUXY are also needed. Poisson's ratio may be input in either major (PRXY, PRYZ, PRXZ) or minor (NUXY, NUYZ, NUXZ) form, but not both for a particular material. The major form is converted to the minor form during the solve operation (SOLVE). Solution output is in terms of the minor form, regardless of how the data was input. If zero values are desired, input the labels with a zero (or blank) value. For axisymmetric analyses, the X, Y, and Z labels refer to the radial (R), axial (Z), and hoop () directions, respectively. Orthotropic properties given in the R, Z, system should be input as follows: EX = ER, EY = EZ, and EZ = E . An additional transformation is required for Poisson's ratios. If the given R, Z, properties are column-normalized (see the Mechanical APDL Theory Reference), NUXY = NURZ, NUYZ = NUZ = (ET/EZ) *NU Z, and NUXZ = NUR . If the given R, Z, properties are row-normalized, NUXY = (EZ/ER)*NURZ, NUYZ = (E /EZ)*NUZ = NU Z, and NUXZ = (E /ER)*NUR . For all other orthotropic material properties (including ALPX, ALPY, and ALPZ), the X, Y, and Z part of the label (as in KXX, KYY, and KZZ) refers to the direction (in the element coordinate system) in which that particular property acts. The Y and Z directions of the properties default to the X direction (for example, KYY and KZZ default to KXX) to reduce the amount of input required. If the material is anisotropic: See Anisotropic Elasticity (p. 18).
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Material Models
3.3.3. Anisotropic Elasticity

Anisotropic elastic capability (TB,ANEL) is available with current-technology plane and solid elements. Input the elastic coefficient matrix [D] either by specifying the stiffness constants (EX, EY, etc.) with MP commands, or by specifying the terms of the matrix with data table commands as described below. The matrix should be symmetric and positive definite (requiring all determinants to be positive). The full 6 x 6 elastic coefficient matrix [D] relates terms ordered x, y, z, xy, yz, xz via 21 constants as shown below.
11 21 31 41 51 61 22 32 42 52 62 33 43 53 63 Symmetric 44 54 64 55 65
66
For 2-D problems, a 4 x 4 matrix relates terms ordered x, y, z, xy via 10 constants (D11, D21, D22, D31, D32, D33, D41, D42, D43, D44). Note, the order of the vector is expected as {x, y, z, xy, yz, xz}, whereas for some published materials the order is given as {x, y, z, yz, xz, xy}. This difference requires the "D" matrix terms to be converted to the expected format. The "D" matrix can be defined in either "stiffness" form (with units of Force/Area operating on the strain vector) or in "compliance" form (with units of the inverse of Force/Area operating on the stress vector), whichever is more convenient. Select a form using TBOPT on the TB command. Both forms use the same data table input as described below. Enter the constants of the elastic coefficient matrix in the data table via the TB family of commands. Initialize the constant table with TB,ANEL. Define the temperature with TBTEMP, followed by up to 21 constants input with TBDATA commands. The matrix may be input in either stiffness or flexibility form, based on the TBOPT value. For the coupled-field elements, temperature- dependent matrix terms are not allowed. You can define up to six temperature-dependent sets of constants (NTEMP = 6 max on the TB command) in this manner. Matrix terms are linearly interpolated between temperature points. The constants (C1-C21) entered on TBDATA (6 per command) are: Constant C1-C6 C7-C12 C13-C18 C19-C21 Meaning Terms D11, D21, D31, D41, D51, D61 Terms D22, D32, D42, D52, D62, D33 Terms D43, D53, D63, D44, D54, D64 Terms D55, D65, D66
For a list of the elements that support this material model, see Material Model Element Support (p. 5).
3.3.4. Damping
Material dependent mass and stiffness damping (MP,ALPD and MP,BETD) is an additional method of including damping for dynamic analyses and is useful when different parts of the model have different damping values. ALPD and BETD are not assumed to be temperature dependent and are always evaluated at T = 0.0. Special-purpose elements, such as MATRIX27 and FLUID29, generally do not require damping. However,
18
Linear Material Properties if material property ALPD and BETD are specified for these elements, the value will be used to create the damping matrix at solution time. Constant material damping coefficient (DMPR) is a material-dependent structural damping coefficient that is constant with respect to the excitation frequency in harmonic analysis and is useful when different parts of the model have different damping values (see Damping Matrices in the Mechanical APDL Theory Reference). DMPR is not temperature dependent and is always evaluated at T = 0.0. See Damping Matrices in the Mechanical APDL Theory Reference for more details about the damping formulation. See Damping in the Structural Analysis Guide for more information about DMPR.
3.3.5. Thermal Expansion

The uniform temperature does not default to REFT (but does default to TREF on the TREF command). The effects of thermal expansion can be accounted for in three different (and mutually exclusive) ways: Secant coefficient of thermal expansion (ALPX, ALPY, ALPZ via the MP command) Instantaneous coefficient of thermal expansion (CTEX, CTEY, CTEZ via the MP or TB,CTE command) Thermal strain (THSX, THSY, THSZ via the MP command) When you use ALPX to enter values for the secant coefficient of thermal expansion (se), the program interprets those values as secant or mean values, taken with respect to some common datum or definition temperature. For example, suppose you measured thermal strains in a test laboratory, starting at 23C, and took readings at 200, 400, 600, 800, and 1000. When you plot this strain-temperature data, you could input this directly via THSX. The slopes of the secants to the strain-temperature curve would be the mean (or secant) values of the coefficient of thermal expansion, defined with respect to the common temperature of 23 (To). You can also input the instantaneous coefficient of thermal expansion (in, using CTEX). The slopes of the tangents to this curve represent the instantaneous values. Hence, the figure below shows how the alternate ways of inputting coefficients of thermal expansion relate to each other. th in se
To
Tn
The program calculates structural thermal strain as follows: th = se(T) * (T - TREF)
19
Material Models where: T = element evaluation temperature TREF = temperature at which zero thermal strains exist (TREF command or REFT ) se(T) = secant coefficient of thermal expansion, with respect to a definition temperature (in this case, same as TREF) (ALPX ) If the material property data is in terms of instantaneous values of , then the program will convert those instantaneous values into secant values as follows:
T T se n = o in n
where: Tn = temperature at which an se value is being evaluated To = definition temperature at which the se values are defined (in this case, same as TREF) in(T) = instantaneous coefficient of thermal expansion at temperature T (CTEX ) If the material property data is in terms of thermal strain, the program will convert those strains into secant values of coefficients of thermal expansion as follows: = where: ith(T) = thermal strain at temperature T (THSX) If necessary, the data is shifted so that the thermal strain is zero when Tn = Tref. If a data point at Tref exists, a discontinuity in se may be generated at Tn = Tref. This can be avoided by ensuring that the slopes of ith on both sides of Tref match. If the se values are based upon a definition temperature other than TREF, then you need to convert those values to TREF (MPAMOD). Also see the Mechanical APDL Theory Reference. th rf
3.3.6. Emissivity
EMIS defaults to 1.0 if not defined. However, if EMIS is set to zero or blank, EMIS is taken to be 0.0.
3.3.7. Specific Heat

You can input specific heat effects using either the C (specific heat) or ENTH (enthalpy) property. Enthalpy has units of heat/volume and is the integral of C x DENS over temperature. If both C and ENTH are specified, the program uses ENTH. ENTH should be used only in a transient thermal analysis. For phasechange problems, you must input ENTH as a function of temperature using the MP family of commands (MP, MPTEMP, MPTGEN, and MPDATA).
20
Rate-Independent Plasticity
3.3.8. Film Coefficients

Film coefficients are evaluated as described via the SF command. See the Mechanical APDL Theory Reference for additional details. Property evaluation at element temperatures beyond the supplied tabular range assumes a constant property at the extreme range value. An exception occurs for the ENTH property, which continues along the last supplied slope.
3.3.9. Temperature Dependency

Temperature-dependent properties may be input in tabular form (value vs. temperature [MP]) or as a fourth-order polynomial (value = f(temperature) [MPTEMP and MPDATA]). If input as a polynomial, the program evaluates the dependencies at discrete temperature points during PREP7 preprocessing and then converts the properties to tabular form. The tabular properties are then available to the elements.
3.3.10. How Material Properties Are Evaluated

Material properties are evaluated at or near the centroid of the element or at each of the integration points, as follows: Heat-transfer elements: All properties are evaluated at the centroid (except for the specific heat or enthalpy, which is evaluated at the integration points). Structural elements: All properties are evaluated at the integration points. All other elements: All properties are evaluated at the centroid. If the temperature of the centroid or integration point falls below or rises above the defined temperature range of tabular data, ANSYS assumes the defined extreme minimum or maximum value, respectively, for the material property outside the defined range.
3.4. Rate-Independent Plasticity

Plasticity is used to model materials subjected to loading beyond their elastic limit. As shown in the following figure, metals and other materials such as soils often have an initial elastic region in which the deformation is proportional to the load, but beyond the elastic limit a nonrecoverable plastic strain develops:
21
Material Models Figure 3.1: Stress-Strain Curve for an Elastic-Plastic Material
Unloading recovers the elastic portion of the total strain, and if the load is completely removed, a permanent deformation due to the plastic strain remains in the material. Evolution of the plastic strain depends on the load history such as temperature, stress, and strain rate, as well as internal variables such as yield strength, back stress, and damage. To simulate elastic-plastic material behavior, several constitutive models for plasticity are provided. The models range from simple to complex. The choice of constitutive model generally depends on the experimental data available to fit the material constants. The following rate-independent plasticity material model topics are available: 3.4.1. Understanding the Plasticity Models 3.4.2. Isotropic Hardening 3.4.3. Kinematic Hardening 3.4.4. Generalized Hill 3.4.5. Drucker-Prager 3.4.6. Gurson 3.4.7. Cast Iron
3.4.1. Understanding the Plasticity Models

The constitutive models for elastic-plastic behavior start with a decomposition of the total strain into elastic and plastic parts and separate constitutive models are used for each. The essential characteristics of the plastic constitutive models are: The yield criterion that defines the material state at the transition from elastic to elastic-plastic behavior. The flow rule that determines the increment in plastic strain from the increment in load. The hardening rule that gives the evolution in the yield criterion during plastic deformation. The following topics concerning plasticity theory and behavior are available to help you to further understand the plasticity material models: 3.4.1.1. Nomenclature
22
Rate-Independent Plasticity 3.4.1.2. Strain Decomposition 3.4.1.3.Yield Criterion 3.4.1.4. Flow Rule 3.4.1.5. Hardening 3.4.1.6. Large Deformation 3.4.1.7. Output 3.4.1.8. Resources
3.4.1.1. Nomenclature
Following are the common symbols used in the rate-independent plasticity theory documentation: Symbol Definition Identity tensor el p 0 ^ Strain Elastic strain Plastic strain Plastic strain components Effective plastic strain Accumulated equivalent plastic strain L,M,N Stress Stress components Principal stresses Stress minus backstress Yield stress Anisotropic yield stress in direction i Initial yield stress Initial yield stress in direction i Equivalent plastic stress Von Mises effective stress f + x Hill yield surface directional yield ratio Generalize Hill yield surface coefficients Generalized Hill constant Generalized Hill tensile and compressive yield strength Plastic work Uniaxial plastic work Drucker-Prager yield surface constant Drucker-Prager plastic potential constant Mohr-Coulomb cohesion Mohr-Coulomb internal friction angle Mohr-Coulomb flow internal friction angle
23
Symbol y ij
Definition Anisotropic directional yield strength Young's Modulus
T
F,G,H,
Elasto-Plastic tangent Elasto-Plastic tangent in direction i Plastic tangent Plastic tangent in direction i Hill yield surface coefficients
User input strain-stress data point
Material Models Symbol Definition Magnitude of plastic strain increment Effective stress function Yield function Plastic potential Translation of yield surface (backstress) Set of material internal variables Symbol Definition Extended Drucker-Prager yield surface pressure sensitivity Extended Drucker-Prager plastic potential pressure sensitivity Extended Drucker-Prager power law yield exponent Extended Drucker-Prager power law plastic potential exponent Extended Drucker-Prager hyperbolic yield constant Extended Drucker-Prager hypobolic plastic potential constant
( )
3.4.1.2. Strain Decomposition

From Figure 3.1: Stress-Strain Curve for an Elastic-Plastic Material (p. 22), a monotonic loading to gives a total strain . The total strain is additively decomposed into elastic and plastic parts: = el + pl
The stress is proportional to the elastic strain :
=
and the evolution of plastic strain is a result of the plasticity model.
For a general model of plasticity that includes arbitrary load paths, the flow theory of plasticity decomposes the incremental strain tensor into elastic and plastic strain increments: = + The increment in stress is then proportional to the increment in elastic strain, and the plastic constitutive model gives the incremental plastic strain as a function of the material state and load increment.
3.4.1.3. Yield Criterion

The yield criterion is a scalar function of the stress and internal variables and is given by the general function: = (3.1) where represents a set of history dependent scalar and tensor internal variables. Equation 3.1 (p. 24) is a general function representing the specific form of the yield criterion for each of the plasticity models. The function is a surface in stress space and an example, plotted in principal stress space, as shown in this figure:
24
Rate-Independent Plasticity Figure 3.2: Yield Surface in Principal Stress Space
Stress states inside the yield surface are given by < and result in elastic deformation. The material yields when the stress state reaches the yield surface and further loading causes plastic deformation. Stresses outside the yield surface do not exist and the plastic strain and shape of the yield surface evolve to maintain stresses either inside or on the yield surface.
3.4.1.4. Flow Rule

The evolution of plastic strain is determined by the flow rule: =
pl
is the plastic potential.
where
is the magnitude of the plastic strain increment and
When the plastic potential is the yield surface from Equation 3.1 (p. 24), the plastic strain increment is normal to the yield surface and the model has an associated flow rule, as shown in this figure: Figure 3.3: Plastic Strain Flow Rule
25
Material Models These flow rules are typically used to model metals and give a plastic strain increment that is proportional to the stress increment. If the plastic potential is not proportional to the yield surface, the model has a non-associated flow rule, typically used to model soils and granular materials that plastically deform due to internal frictional sliding. For non-associated flow rules, the plastic strain increment is not in the same direction as the stress increment. Non-associated flow rules result in an unsymmetric material stiffness tensor. Unsymmetric analysis can be set via the NROPT command. For a plastic potential that is similar to the yield surface, the plastic strain direction is not significantly different from the yield surface normal, and the degree of asymmetry in the material stiffness is small. In this case, a symmetric analysis can be used, and a symmetric material stiffness tensor will be formed without significantly affecting the convergence of the solution.
3.4.1.5. Hardening
The yield criterion for many materials depends on the history of loading and evolution of plastic strain. The change in the yield criterion due to loading is called hardening and is defined by the hardening rule. Hardening behavior results in an increase in yield stress upon further loading from a state on the yield surface so that for a plastically deforming material, an increase in stress is accompanied by an increase in plastic strain. Two common types of hardening rules are isotropic and kinematic hardening. For isotropic hardening, the yield surface given by Equation 3.1 (p. 24) has the form: y = where
( )
is a scalar function of stress and
is the yield stress.
1 2 Plastic loading from to increases the yield stress and results in uniform increase in the size of the yield surface, as shown in this figure: Figure 3.4: Isotropic Hardening of the Yield Surface
This type of hardening can model the behavior of materials under monotonic loading and elastic unloading, but often does not give good results for structures that experience plastic deformation after a load reversal from a plastic state. For kinematic hardening, the yield surface has the form:
26 Release 14.5 - SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.
Rate-Independent Plasticity = where is the back stress tensor. As shown in the following figure, the back stress tensor is the center (or origin) of the yield surface, 1 2 and plastic loading from to results in a change in the back stress and therefore a shift in the yield surface: Figure 3.5: Kinematic Hardening of the Yield Surface
Kinematic hardening is observed in cyclic loading of metals. It can be used to model behavior such as the Bauschinger effect, where the compressive yield strength reduces in response to tensile yielding. It can also be used to model plastic ratcheting, which is the buildup of plastic strain during cyclic loading. Many materials exhibit both isotropic and kinematic hardening behavior, and these hardening rules can be used together to give the combined hardening model. Other hardening behaviors include changes in the shape of the yield surface in which the hardening rule affects only a local region of the yield surface, and softening behavior in which the yield stress decreases with plastic loading.
3.4.1.6. Large Deformation

The plasticity constitutive models are applicable in both small-deformation and large-deformation analyses. For small deformation, the formulation uses engineering stress and strain. For large deformation (NLGEOM,ON), the constitutive models are formulated with the Cauchy stress and engineering strain.
3.4.1.7. Output
Output quantities specific to the plastic constitutive models are available for use in the POST1 database postprocessor (/POST1) and in the POST26 time-history results postprocessor (/POST26). The equivalent stress (label SEPL) is the current value of the yield stress evaluated from the hardening model.
27
Material Models The accumulated plastic strain (label EPEQ) is a path-dependent summation of the plastic strain rate over the history of the deformation: pl = where is the magnitude of the plastic strain rate.
The stress ratio (label SRAT) is the ratio of the elastic trial stress to the current yield stress and is an indicator of plastic deformation during an increment. If the stress ratio is: >1 A plastic deformation occurred during the increment. <1 An elastic deformation occurred during the increment. 1 The stress state is on the yield surface. Alternatively, the output quantities can have specialized meanings specific to the given material model. For example: For the kinematic hardening models, the equivalent stress is determined by evaluating the kinematic hardening rule for stress vs. plastic strain using the accumulated plastic strain. The stress ratio for multilinear kinematic hardening uses the trial stress and yield stress of the first subvolume. For the generalized Hill model, the equivalent plastic strain is given by:
1 2 2 +1 + +1 + x x = x
and the equivalent stress is:

e = + +
3.4.1.8. Resources
The following list of resources offers more information about plasticity: 1. Hill, R. The Mathematical Theory of Plasticity. New York: Oxford University Press, 1983. 2. Prager, W. The Theory of Plasticity: A Survey of Recent Achievements. Proceedings of the Institution of Mechanical Engineers. 169.1 (1955): 41-57. 3. Besseling, J. F. A Theory of Elastic, Plastic, and Creep Deformations of an Initially Isotropic Material Showing Anisotropic Strain-Hardening, Creep Recovery, and Secondary Creep. ASME Journal of Applied Mechanics. 25 (1958): 529-536. 4. Owen, D. R. J., A.Prakash, O. C. Zienkiewicz. Finite Element Analysis of Non-Linear Composite Materials by Use of Overlay Systems. Computers and Structures. 4.6 (1974): 1251-1267.
Rate-Independent Plasticity 5. Rice, J. R. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. Constitutive Equations in Plasticity. Ed. A. Argon. Cambridge, MA: MIT Press, 1975. 23-79. 6. Chaboche, J. L. Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity. International Journal of Plasticity. 5.3 (1989): 247-302. 7. Chaboche, J. L. On Some Modifications of Kinematic Hardening to Improve the Description of Ratchetting Effects. International Journal of Plasticity. 7.7 (1991): 661-678. 8. Shih, C. F., D. Lee. Further Developments in Anisotropic Plasticity. Journal of Engineering Materials and Technology. 100.3 (1978): 294-302. 9. Valliappan, S., P. Boonlaulohr, I. K. Lee. Non-Linear Analysis for Anisotropic Materials. International Journal for Numerical Methods in Engineering. 10.3 (1976): 597-606. 10. Drucker, D. C., W. Prager. Soil Mechanics and Plastic Analysis or Limit Design. Quarterly of Applied Mathematics. 10.2 (1952): 157-165. 11. Sandler, I. S, F. L. DiMaggio, G. Y. Baladi. A Generalized Cap Model for Geological Materials. Journal of the Geotechnical Engineering Division. 102.7 (1976): 683-699. 12. Schwer, L. E., Y. D. Murray. A Three-Invariant Smooth Cap Model with Mixed Hardening. International Journal for Numerical and Analytical Methods in Geomechanics. 18.10 (1994): 657-688. 13. Foster, C., R. Regueiro, A. Fossum, R. Borja. Implicit Numerical Integration of a Three-Invariant, Isotropic/Kinematic Hardening Cap Plasticity Model for Geomaterials. Computer Methods in Applied Mechanics and Engineering. 194.50-52 (2005): 5109-5138. 14. Pelessone, D. A Modified Formulation of the Cap Model. Technical Report GA-C19579. San Diego: Gulf Atomics, 1989. 15. Fossum, A.F., J. T. Fredrich. Cap Plasticity Models and Compactive and Dilatant Pre-Failure Deformation. Pacific Rocks 2000: Rock Around the Rim. Proceedings of the Fourth North American Rock Mechanics Symposium. Eds. J. Girard, M. Liebman, C. Breeds, T. Doe, A. A. Balkema. Rotterdam, 2000: 1169-1176. 16. Gurson, A. L. Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I--Yield Criteria and Flow Rules for Porous Ductile Media. Journal of Engineering Materials and Technology. 99.1 (1977): 2-15. 17. Needleman, A. V. Tvergaard. An Analysis of Ductile Rupture in Notched Bars. Journal of the Mechanics and Physics of Solids. 32.6 (1984): 461-490. 18. Arndt, S., B. Svendsen, D. Klingbeil. Modellierung der Eigenspannungen an der Rispitze mit einem Schdigungsmodell. Technische Mechanik. 4.17 (1997): 323-332. 19. Besson, J., C. Guillemer-Neel. An Extension of the Green and Gurson Models to Kinematic Hardening. Mechanics of Materials. 35.1-2 (2003): 1-18. 20. Hjelm, H. E. Yield Surface for Grey Cast Iron Under Biaxial Stress. Journal of Engineering Materials and Technology. 116.2 (1994): 148-154. 21. Chen, W. F., D. J. Han. Plasticity for Structural Engineers. New York: Springer-Verlag, 1988.
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Material Models
3.4.2. Isotropic Hardening

During plastic deformation, isotropic hardening causes a uniform increase in the size of the yield surface and results in an increase in the yield stress. The yield criterion has the form: y = is a scalar function of stress and is the yield stress that evolves as a function of the set of material internal variables . This type of hardening can model the behavior of materials under monotonic loading and elastic unloading, but often does not give good results for structures that experience additional plastic deformation after a load reversal from a plastic state. where Three general classes of isotropic hardening models are available: bilinear, multilinear, and nonlinear. Each of the hardening models assumes a von Mises yield criterion, unless an anisotropic Hill yield criterion is defined, and includes an associated flow rule. Isotropic hardening can also be combined with kinematic hardening and the Extended Drucker-Prager and Gurson models to provide an evolution of the yield stress. For more information, see Material Model Combinations (p. 183). The following topics related to the isotropic hardening material model are available: 3.4.2.1.Yield Criteria and Plastic Potentials 3.4.2.2. General Isotropic Hardening Classes
( )
3.4.2.1. Yield Criteria and Plastic Potentials

Hardening models assume a von Mises yield criterion, unless an anisotropic Hill yield criterion is defined. 3.4.2.1.1. Von Mises Yield Criterion 3.4.2.1.2. Hill Yield Criterion
3.4.2.1.1. Von Mises Yield Criterion

The von Mises yield criterion is commonly used in plasticity models for a wide range of materials. It is a good first approximation for metals, polymers, and saturated geological materials. The criterion is isotropic and independent of hydrostatic pressure, which can limit its applicability to microstructured materials and materials that exhibit plastic dilatation. The von Mises yield criterion is: = e = where = is the von Mises effective stress, also known as the von Mises equivalent stress,
(3.2)
( )2
and
is the yield strength and corresponds to the yield in uniaxial stress loading.
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Rate-Independent Plasticity In principal stress space, the yield surface is a cylinder with the axis along the hydrostatic line 1 = 2 = 3 and gives a yield criterion that is independent of the hydrostatic stress, as shown in the following figure: Figure 3.6: Yield Surface for von Mises Yield Criterion
For an associated flow rule, the plastic potential is the yield criterion in Equation 3.2 (p. 30) and the plastic strain increment is proportional to the deviatoric stress pl =
3.4.2.1.2. Hill Yield Criterion

The Hill yield criterion [1] is an anisotropic criterion that depends on the orientation of the stress relative to the axis of anisotropy. It can be used to model materials in which the microstructure influences the macroscopic behavior of the material such as forged metals and composites. In a coordinate system that is aligned with the anisotropy coordinate system, the Hill yield criterion given in stress components is: y = + +
+
+
= y
(3.3)
The coefficients in this yield criterion are functions of the ratio of the scalar yield stress parameter and the yield stress in each of the six stress components:
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Material Models = = = = = = 2 2 2 11 33 11 + 2 2 2 33 11 22 + 2 2 2 11 22 33 2 23 + 2 13 2 12
where the directional yield stress ratios are the user input parameters and are related to the isotropic yield stress parameter by:
y = y y = y = = = = y
where i is the yield stress in the direction indicated by the value of subscript i. The stress directions are in the anisotropy coordinate system which is aligned with the element coordinate system (ESYS). The isotropic yield stress is entered in the constants for the hardening model. The Hill yield criterion defines a surface in six-dimensional stress space and the flow direction is normal to the surface. The plastic strain increments in the anisotropy coordinate system are:
32
pl 11 =
( 1 2 ) + ( 1 3 )
pl 22 = pl 33 = pl 23 = pl 31 =
pl 12 = [
( 2 3 ) + ( 2 1 ) ( 3 1 ) + ( 3 2 )
23 31 12 ]
The Hill surface, used with a hardening model, replaces the default von Mises yield surface. After defining the material data table (TB,HILL), input the required constants (TBDATA): Constant C1 C2 C3 C4 C5 C6 Meaning R11 R22 R33 R12 R23 R13 Property Yield stress ratio in X direction Yield stress ratio in Y direction Yield stress ratio in Z direction Yield stress ratio in XY direction Yield stress ratio in YZ direction Yield stress ratio in XZ direction
The constants can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP). Example 3.1: Hill Surface
/prep7 MP,EX,1,20.0E5 MP,NUXY,1,0.3 ! ELASTIC CONSTANTS
TB,HILL,1 ! HILL TABLE TBDATA,1,1.0,1.1,0.9,0.85,0.9,0.80
3.4.2.2. General Isotropic Hardening Classes

Support is available for these general classes of isotropic hardening: 3.4.2.2.1. Bilinear Isotropic Hardening 3.4.2.2.2. Multilinear Isotropic Hardening 3.4.2.2.3. Nonlinear Isotropic Hardening
3.4.2.2.1. Bilinear Isotropic Hardening

Bilinear isotropic hardening is described by a bilinear effective stress versus effective strain curve. The initial slope of the curve is the elastic modulus of the material. Beyond the user-specified initial yield stress 0 , plastic strain develops and stress-vs.-total-strain continues along a line with slope defined by the user-specified tangent modulus than the elastic modulus.
T . The tangent modulus cannot be less than zero or greater
33
Material Models Figure 3.7: Stress vs. Total Strain for Bilinear Isotropic Hardening
3.4.2.2.1.1. Defining the Bilinear Isotropic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,BISO), input the following constants (TBDATA): Constant C1 C2 Meaning 0
T
Property Yield stress Tangent modulus
The constants can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP). Example 3.2: Bilinear Isotropic Hardening
/prep7 MPTEMP,1,0,500 MPDATA,EX,1,,14E6,12e6 MPDATA,PRXY,1,,0.3,0.3 TB,BISO,1,2 TBTEMP,0.0 TBDATA,1,44E3,1.2E6 TBTEMP,500 TBDATA,1,29.33E3,0.8E6 ! Define temperatures for Young's modulus
! ! ! ! !
Activate a data table Temperature = 0.0 Yield = 44,000; Tangent modulus = 1.2E6 Temperature = 500 Yield = 29,330; Tangent modulus = 0.8E6
3.4.2.2.2. Multilinear Isotropic Hardening

The behavior of multilinear isotropic hardening is similar to bilinear isotropic hardening except that a multilinear stress versus total or plastic strain curve is used instead of a bilinear curve.
34
Rate-Independent Plasticity The multilinear hardening behavior is described by a piece-wise linear stress-total strain curve, starting at the origin and defined by sets of positive stress and strain values, as shown in this figure: Figure 3.8: Stress vs. Total Strain for Multilinear Isotropic Hardening
The first stress-strain point corresponds to the yield stress. Subsequent points define the elastic plastic response of the material.
3.4.2.2.2.1. Defining the Multilinear Isotropic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,PLASTIC,,,,MISO), input the following constants (TBDATA): Constant X Y Meaning
i
Property Strain value Stress value
The stress-plastic strain data points are entered into the table via the TBPT command. Temperature-dependent data can be defined (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP). Interpolation between temperatures occurs via stress-vs.-plastic-strain. Example 3.3: Multilinear Hardening with Plastic Strain
/prep7 MPTEMP,1,0,500 ! Define temperature-dependent EX, MPDATA,EX,1,,14.665E6,12.423e6 MPDATA,PRXY,1,,0.3 TB,PLASTIC,1,2,5,MISO TBTEMP,0.0 TBPT,DEFI,0,29.33E3 TBPT,DEFI,1.59E-3,50E3 ! Activate TB,PLASTIC data table ! Temperature = 0.0 ! Plastic strain, stress at temperature = 0
35
Material Models
TBPT,DEFI,3.25E-3,55E3 TBPT,DEFI,5.91E-3,60E3 TBPT,DEFI,1.06E-2,65E3 TBTEMP,500 TBPT,DEFI,0,27.33E3 TBPT,DEFI,2.02E-3,37E3 TBPT,DEFI,3.76E-3,40.3E3 TBPT,DEFI,6.48E-3,43.7E3 TBPT,DEFI,1.12E-2,47E3
! Temperature = 500 ! Plastic strain, stress at temperature = 500
3.4.2.2.3. Nonlinear Isotropic Hardening

Power law and Voce equations are available to model nonlinear isotropic hardening. 3.4.2.2.3.1. Power Law Nonlinear Isotropic Hardening 3.4.2.2.3.2. Voce Law Nonlinear Isotropic Hardening
3.4.2.2.3.1. Power Law Nonlinear Isotropic Hardening

The power law equation has a user-defined initial yield stress stress y is given by solving the following equation: Y Y 3G pl = +
N
and exponent N. The current yield
where G is the shear modulus determined from the user defined elastic constants and is the accumulated equivalent plastic strain.
Defining the Power Law Nonlinear Isotropic Hardening Model For the power law hardening model, define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,NLISO,,,,POWER), input the following constants (TBDATA): Constant C1 C2 Meaning N Property Initial yield stress Exponent
The exponent N must be positive and less than 1. Temperature-dependent data can be defined (NTEMP on the TB command), with temperatures specified for the subsequent set of constants (TBTEMP). Example 3.4: Power Law Nonlinear Isotropic Hardening
/prep7 TB,NLISO,1,2,,POWER TBTEMP,100 TBDATA,1,275,0.1 TBTEMP,200 TBDATA,1,275,0.1
! Define first temperature ! Define second temperature
36
3.4.2.2.3.2. Voce Law Nonlinear Isotropic Hardening

The Voce hardening model is similar to bilinear isotropic hardening, with an exponential saturation hardening term added to the linear term, as shown in this figure: Figure 3.9: Stress vs. Plastic Strain for Voce Hardening
The evolution of the yield stress for this model is specified by the following equation: Y = 0 + 0 pl +
pl
)
, the
where the user-defined parameters include
difference between the saturation stress and the , the hardening parameter that governs
initial yield stress, , the slope of the saturation stress and, the rate of saturation of the exponential term.
Defining the Voce Law Nonlinear Isotropic Hardening Model Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,NLISO,,,,VOCE), input the following constants (TBDATA): Constant C1 C2 C3 Meaning
Property Initial yield stress Linear coefficient Exponential coefficient
37
Material Models Constant C4 Meaning Property Exponential saturation parameter
The constants can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the subsequent set of constants (TBTEMP). Example 3.5: Voce Nonlinear Isotropic Hardening
/PREP7 TB,NLISO,1,2,,VOCE TBTEMP,40 TBDATA,1,280,7e3,155,7e2 TBTEMP,60 TBDATA,1,250,5e3,120,3e2 ! ! ! ! ! Activate NLISO data table Define first temperature Constants at first temperature Define second temperature Constants at second temperature
3.4.3. Kinematic Hardening

During plastic deformation, kinematic hardening causes a shift in the yield surface in stress space. In uniaxial tension, plastic deformation causes the tensile yield stress to increase and the magnitude of the compressive yield stress to decrease. This type of hardening can model the behavior of materials under either monotonic or cyclic loading and can be used to model phenomena such as the Bauschinger effect and plastic ratcheting. The yield criterion has the form: y = where is a scalar function of the relative stress and is the yield stress. The relative stress is: =
(3.4)
where the backstress is the shift in the position of the yield surface in stress space and evolves during plastic deformation. Three general classes of kinematic hardening models are available: bilinear, multilinear, and nonlinear. Each of the hardening models assumes a von Mises yield criterion, unless an anisotropic Hill yield criterion is defined, and includes an associated flow rule. Kinematic hardening can also be combined with isotropic hardening and the Gurson model to provide an evolution of the yield stress. For more information, see Material Model Combinations (p. 183). The following topics related to the kinematic hardening material model are available: 3.4.3.1.Yield Criteria and Plastic Potentials 3.4.3.2. General Kinematic Hardening Classes
3.4.3.1. Yield Criteria and Plastic Potentials

Kinematic hardening uses the von Mises yield criterion with an associated flow rule unless a Hill yield surface is defined. If a Hill yield surface is defined, the kinematic hardening model uses it for both the yield criterion and as the plastic potential to give an associated flow rule.
38
Rate-Independent Plasticity For more information about von Mises and Hill yield surfaces, see Yield Criteria and Plastic Potentials (p. 30).
3.4.3.2. General Kinematic Hardening Classes

Support is available for these general classes of kinematic hardening: 3.4.3.2.1. Bilinear Kinematic Hardening 3.4.3.2.2. Multilinear Kinematic Hardening 3.4.3.2.3. Nonlinear Kinematic Hardening
3.4.3.2.1. Bilinear Kinematic Hardening

The backstress tensor for bilinear kinematic hardening evolves so that the effective stress versus effective strain curve is bilinear. The initial slope of the curve is the elastic modulus of the material and beyond the user specified initial yield stress 0 , plastic strain develops and the back stress evolves so that stress versus total strain continues along a line with slope defined by the user specified tangent modulus This tangent modulus cannot be less than zero or greater than the elastic modulus.
T.
For uniaxial tension followed by uniaxial compression, the magnitude of the compressive yield stress decreases as the tensile yield stress increases so that the magnitude of the elastic range is always , as shown in this figure: Figure 3.10: Stress vs. Total Strain for Bilinear Kinematic Hardening
sh The backstress is proportional to the shift strain :
where G is the elastic shear modulus and the shift strain is numerically integrated from the incremental shift strain which is proportional to the incremental plastic strain:
Material Models sh = where = T T pl
and is Young's Modulus and is the user-defined tangent modulus [2]. The incremental plastic strain is defined by the associated flow rule for the von Mises or Hill potential given in Yield Criteria and Plastic Potentials (p. 30) with the stress given by the relative stress .
3.4.3.2.1.1. Defining the Bilinear Kinematic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,BKIN), input the following constants (TBDATA): Constant C1 C2 Meaning 0 Property Initial yield stress Tangent modulus
The constants can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP). This model can be used with the TB commands TBOPT option: For TBOPT 1, no stress relaxation occurs with an increase in temperature. This option is not recommended for non-isothermal problems. For TBOPT = 1, Rice's hardening rule [5] is applied, which takes stress relaxation with temperature increase into account. Example 3.6: Bilinear Kinematic Hardening
/prep7 MPTEMP,1,0,500 ! Define temperatures for Young's modulus MPDATA,EX,1,,14E6,12e6 MPDATA,PRXY,1,,0.3,0.3 TB,BKIN,1,2,2,1 ! Activate a data table with TBOPT=1 ! stress relaxation with temperature TBTEMP,0.0 ! Temperature = 0.0 TBDATA,1,44E3,1.2E6 ! Yield = 44,000; Tangent modulus = 1.2E6 TBTEMP,500 ! Temperature = 500 TBDATA,1,29.33E3,0.8E6 ! Yield = 29,330; Tangent modulus = 0.8E6
3.4.3.2.2. Multilinear Kinematic Hardening

The backstress tensor for multilinear kinematic hardening evolves so that the effective stress versus effective strain curve is multilinear with each of the linear segments defined by a set of user input stressstrain points, as shown in this figure:
40
Rate-Independent Plasticity Figure 3.11: Stress vs. Total Strain for Multilinear Kinematic Hardening
The model formulation is the sublayer or overlay model of Besselling [3] and Owen, Prakash and Zienkiewicz [4] in which the material is assumed to be composed of a number of sublayers or subvolumes, all subjected to the same total strain. The number of subvolumes is the same as the number of input stress-strain points, and the overall behavior is weighted for each subvolume where the weight is given by: k= Tk k 1 i i =1
Tk
where
is the tangent modulus for segment of the stress-strain curve.
The behavior of each subvolume is elastic-perfectly plastic, with the uniaxial yield stress for each subvolume given by: y =
( (
+ )
( ) )
where
) is the input stress-strain point for subvolume k.
The default yield surface is the von Mises surface, and each subvolume yields at an equivalent stress equal to the subvolume uniaxial yield stress. A Hill yield criterion can be used in which each subvolume yields according to the Hill criterion with the subvolume uniaxial yield as the isotropic yield stress and the subvolume anisotropic yield condition determined by the Hill surface.
41
Material Models The subvolumes undergo kinematic hardening with an associated flow rule and the plastic strain increment for each subvolume is the same as that for bilinear kinematic hardening. The total plastic strain is given by: pl =
Nsv i =1
pl i
where
is the number of subvolumes and
is the subvolume plastic strain increment.
3.4.3.2.2.1. Defining the Multilinear Kinematic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. To specify the hardening behavior, define the material data table (TB) and input the constants as stress vs. total strain points or as stress vs. plastic strain points. Constant P1 P2 Meaning Property Strain value Stress value
The constants can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP). When entering temperature-dependent stress-strain points, the set of data at each temperature must have the same number of points. Thermal softening for the multilinear kinematic hardening model is the same as that for bilinear kinematic hardening with Rice's hardening rule [5].
Entering Stress vs. Total Strain Points

After defining the material data table (TB,KINH,,,,0), enter the stress-strain points (TBPT). The slope of the first segment must correspond to the elastic modulus and no segment slope can be larger than the slope of the previous segment. Example 3.7: Multilinear Kinematic Hardening with Stress vs. Total Strain
/prep7 TB,KINH,1,2,3 TBTEMP,20.0 TBPT,,0.001,1.0 TBPT,,0.1012,1.2 TBPT,,0.2013,1.3 TBTEMP,40.0 TBPT,,0.008,0.9 TBPT,,0.09088,1.0 TBPT,,0.12926,1.05 ! ! ! ! ! ! ! ! ! Activate a data table Temperature = 20.0 Strain = 0.001, Stress = 1.0 Strain = 0.1012, Stress = 1.2 Strain = 0.2013, Stress = 1.3 Temperature = 40.0 Strain = 0.008, Stress = 0.9 Strain = 0.09088, Stress = 1.0 Strain = 0.12926, Stress = 1.05
Entering Stress vs. Plastic Strain Points

After defining the material data table (TB,KINH,,,,PLASTIC or TB,PLASTIC,,,,KINH), enter the stress-strain points (TBPT). No segment slope can be larger than the slope of the previous segment.
42
Rate-Independent Plasticity Example 3.8: Multilinear Kinematic Hardening with Stress vs. Plastic Strain
/prep7 TB,PLASTIC,1,2,3,KINH TBTEMP,20.0 TBPT,,0.0,1.0 TBPT,,0.1,1.2 TBPT,,0.2,1.3 TBTEMP,40.0 TBPT,,0.0,0.9 TBPT,,0.0900,1.0 TBPT,,0.129,1.05 ! ! ! ! ! ! ! ! ! Activate a data table Temperature = 20.0 Plastic Strain = 0.0000, Plastic Strain = 0.1000, Plastic Strain = 0.2000, Temperature = 40.0 Plastic Strain = 0.0000, Plastic Strain = 0.0900, Plastic Strain = 0.1290,
Stress = 1.0 Stress = 1.2 Stress = 1.3 Stress = 0.9 Stress = 1.0 Stress = 1.05
3.4.3.2.3. Nonlinear Kinematic Hardening

The nonlinear kinematic hardening model is a rate-independent version of the kinematic hardening model proposed by Chaboche [6][7]. The model allows the superposition of several independent backstress tensors and can be combined with any of the available isotropic hardening models. It can be useful in modeling cyclic plastic behavior such as cyclic hardening or softening and ratcheting or shakedown. The model uses an associated flow rule with either the default von Mises yield criterion or the Hill yield criterion if it is defined. The relative stress given by Equation 3.1 (p. 24) is used to evaluate the yield function, and the backstress tensor is given by the superposition of a number of evolving kinematic backstress tensors: =
i n
=1
where n is the number kinematic models to be superposed. The evolution of each backstress model in the superposition is given by the kinematic hardening rule: =
+
pl pl
is the plastic strain rate, where and are user-input material parameters, of the plastic strain rate, and is the temperature.

is the magnitude
3.4.3.2.3.1. Defining the Nonlinear Kinematic Hardening Model

Define the material data table (TB,CHABOCHE) and input the following constants (TBDATA): Constant C1 C2 C3 C4 Meaning
0
Property Initial yield stress Material constant for first kinematic model Material constant for first kinematic model Material constant for second kinematic model
43
Material Models Constant C5 ... C(2n) C(1+2n) Meaning 2 ...

n
Property Material constant for second kinematic model ... Material constant for last kinematic model Material constant for last kinematic model
Temperature-dependent data can be defined (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP). Set the TB commands NPTS value equal to n, the number of superimposed kinematic hardening models. Example 3.9: Nonlinear Kinematic Hardening
/prep7 TB,CHABOCHE,1,1,3 ! Activate Chaboche data table with ! 3 models to be superposed ! Define Chaboche material data TBDATA,1,18.8 ! C1 - Initial yield stress TBDATA,2,5174000,4607500 ! C2,C3 - Chaboche constants for 1st model TBDATA,4,17155,1040 ! C4,C5 - Chaboche constants for 2nd model TBDATA,6,895.18,9 ! C6,C7 - Chaboche constants for 3rd model
3.4.4. Generalized Hill

The generalized Hill plasticity model is an anisotropic model that accounts for different yield strength in tension than in compression. The anisotropic and asymmetric tension-compression behavior can be useful in modeling textured metals with low crystallographic symmetries. Examples of such materials include titanium and zirconium alloys, and composite materials with oriented microstructures such as natural and processed wood products and fiber-matrix composites. The model includes a yield surface that is a specialization of the Hill yield surface [8], an anisotropic work-hardening rule [9], and an associated flow rule. In a coordinate system that is aligned with the anisotropy coordinate system, the generalized Hill yield criterion given in stress components is:
+ + 33 33 + 44 1 + 55 3 + 66 31 + 11 11 111 + + 3 33 y = 111 + 3 33 + 3133 11
(3.5)
where the coefficients jj are functions of the parameter and the tensile and compressive yield stress: = = + where + and are the user-defined magnitudes of the tension and compression yield strength, respectively. The subscripts on the tension and compression yield stresses correspond to the Voigt notation coordinate directions
(3.6)
}{
}.
44
Rate-Independent Plasticity From the assumption of incompressible plastic deformation, the mixed subscript coefficients are given by:
12
= = =
( ( (
11
+ +
22
33
) ) )
j
13
11
22
33
(3.7)
23
11
22
33
The strength differential coefficients
are: (3.8)
( +
+
which, for incompressibility, must satisfy:
Setting the coefficient = +

y
, then:
and the coefficients in the yield criterion from Equation 3.5 (p. 44) are determined from Equation 3.6 (p. 44) through Equation 3.8 (p. 45), and the user-input tension and compression yield stresses. Due to the incompressibility assumption, + + + + + = + + +

(3.9)
and for a closed yield surface,
)<
(3.10)
Equation 3.9 (p. 45) and Equation 3.10 (p. 45) must be satisfied throughout the evolution of yield stresses that result from plastic deformation. The program checks these conditions through 20 percent equivalent plastic strain, but you must ensure that conditions are satisfied if the deformation exceeds that range. A bilinear anisotropic work hardening rule is used to evolve the individual components of tension and compression yield stresses. For a general state of deformation with a bilinear hardening law, the plastic work is: =
pl
where stress
is the effective stress at initial yield, and
is the current effective yield stress. For uniaxial
, the plastic strain is:
45
Material Models x 0 x pl x =
pl x
and the plastic work is: = ( + ) =

2 2 ( )
where the plastic slope is the slope of the stress versus plastic strain. The uniaxial plastic work is equivalent to the effective plastic work if: j =

j + j
1/
(3.11)

where the plastic tangent

is related to the user input tangent modulus
T
by:

Equation 3.11 (p. 46) is then the isotropic hardening evolution equation for the tensile and compressive yield stress components.
3.4.4.1. Defining the Generalized Hill Model

Define the isotropic or anisotropic elastic behavior via MP commands. Specify the Generalized Hill material parameters by defining the material data table (TB,ANISO) and entering the following input (TBDATA): Constant C1-C3 C4-C6 C7-C9 C10-C12 C13-C15 C16-C18 Meaning +

Property Tensile yield stresses in the material x, y, and z directions Tangent moduli of tension in material x, y, and z directions Compressive yield stresses in the material x, y, and z directions Tangent moduli of compression in material x, y, and z directions Shear yield stresses in the material xy, yz, and xz directions Tangent moduli in material xy, yz, and xz directions
= = = = = =

Temperature-dependent parameters are not allowed. Example 3.10: Generalized Hill

! Define Generalized Hill model /prep7
46
! Define elastic material properties mp,ex,1,210 mp,nuxy,1,0.3 ! Define anisotropic material properties tb,aniso,1 tbdata,1,0.33,0.33,0.495 ! Tensile yield stress (x,y & z) tbdata,4,0.21,0.21,0.315 ! Tangent moduli (tensile) tbdata,7,0.33,0.33,0.495 ! Compressive yield stress (x,y & z) tbdata,10,0.21,0.21,0.315 ! Tangent moduli (compressive) tbdata,13,0.1905,0.1905,0.1905 ! Shear yield stress (xy,yz,xz) tbdata,16,0.105,0.07,0.07 ! Tangent moduli (shear)
3.4.5. Drucker-Prager
The following topics related to Drucker-Prager plasticity are available: 3.4.5.1. Classic Drucker-Prager 3.4.5.2. Extended Drucker-Prager (EDP) 3.4.5.3. Extended Drucker-Prager Cap
3.4.5.1. Classic Drucker-Prager

The classic Drucker-Prager model [10] is applicable to granular (frictional) material such as soils, rock, and concrete and uses the outer cone approximation to the Mohr-Coulomb law. The input consists of only three constants: Cohesion value (> 0) Angle of internal friction Dilatancy angle The amount of dilatancy (the increase in material volume due to yielding) can be controlled via the dilatancy angle. If the dilatancy angle is equal to the friction angle, the flow rule is associative. If the dilatancy angle is zero (or less than the friction angle), there is no (or less of an) increase in material volume when yielding and the flow rule is non-associated. For more information about this material model, see Classic Drucker-Prager Model in the Mechanical APDL Theory Reference.
3.4.5.1.1. Defining the Classic Drucker-Prager Model

Define the isotropic or anisotropic elastic behavior via MP commands. Define the material data table (TB,DP) and define up to three constants (TBDATA), as follows: Constant C1 C2 C3 Meaning Force/Area Angle (in degrees) Angle (in degrees) Property Cohesion value Internal friction Dilatancy
Temperature-dependent parameters are not allowed. Example 3.11: Classic Drucker-Prager

MP,EX,1,5000 MP,NUXY,1,0.27
47
Material Models
TB,DP,1 TBDATA,1,2.9,32,0
! Cohesion = 2.9 (use consistent units), ! Angle of internal friction = 32 degrees, ! Dilatancy angle = 0 degrees
3.4.5.2. Extended Drucker-Prager (EDP)

The extended Drucker-Prager (EDP) material model includes three yield criteria and corresponding flow potentials similar to those of the classic Drucker-Prager model commonly used for geomaterials with internal cohesion and friction. The yield functions can also be combined with an isotropic or kinematic hardening rule to evolve the yield stress during plastic deformation. The model is defined via one of the three yield criteria combined with any of the three flow potentials and an optional hardening model. The following topics related to defining the EDP material model are available: 3.4.5.2.1. EDP Yield Criteria Forms 3.4.5.2.2. EDP Plastic Flow Potentials 3.4.5.2.3. Plastic Strain Increments for Flow Potentials 3.4.5.2.4. Example EDP Material Model Definitions
3.4.5.2.1. EDP Yield Criteria Forms

The EDP yield criteria include the following forms: 3.4.5.2.1.1. Linear Form 3.4.5.2.1.2. Power Law Form 3.4.5.2.1.3. Hyperbolic Form
3.4.5.2.1.1. Linear Form

The EDP linear yield criterion form is: y = e + y =
where the user-defined parameters are the pressure sensitivity and the uniaxial yield stress . Defining the EDP Linear Yield Criterion After initializing the extended Drucker-Prager linear yield criterion (TB,EDP,,,,LYFUN), enter the following constants (TBDATA): Constant C1 C2 Meaning Property Pressure sensitivity Uniaxial yield stress
The constants can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP).
3.4.5.2.1.2. Power Law Form

The EDP power law yield criteria form is:
48
Rate-Independent Plasticity y = b e + y = , pressure sensitivity , and uniaxial yield stress are the user-defined para-
where the exponent meters.
Figure 3.12: Power Law Criterion in the Meridian Plane
Defining the EDP Power Law Yield Criterion After initializing the extended Drucker-Prager power law yield criterion (TB,EDP,,,,PYFUN), enter the following constants (TBDATA): Constant C1 C2 C3 Meaning Property Pressure sensitivity Exponent Uniaxial yield stress
3.4.5.2.1.3. Hyperbolic Form

The EDP hyperbolic yield criteria form is: =
2
+ 2 +
where the constant meters.
, pressure sensitivity , and uniaxial yield stress are the user-defined para-
49
Material Models In the following figure, the hyperbolic yield criterion is plotted and compared to the linear yield criterion shown in the dashed line: Figure 3.13: Hyperbolic and Linear Criterion in the Meridian Plane
Defining the EDP Hyperbolic Yield Criterion After initializing the extended Drucker-Prager hyperbolic yield criterion (TB,EDP,,,,HYFUN), enter the following constants (TBDATA): Constant C1 C2 C3 y Meaning Property Pressure sensitivity Material parameter Uniaxial yield stress
3.4.5.2.2. EDP Plastic Flow Potentials

Three EDP flow potentials correspond in form to each of the yield criteria. However, the user-defined parameters for the flow potentials are independent of those for the yield criteria, and any potential can be combined with any yield criterion. The EDP plastic flow potentials include the following forms: 3.4.5.2.2.1. Linear Form 3.4.5.2.2.2. Power Law Form 3.4.5.2.2.3. Hyperbolic Form
3.4.5.2.2.1. Linear Form

The linear form of the plastic flow potential is:
50
Rate-Independent Plasticity y = e +
where is the flow potential pressure sensitivity. Defining the Linear Plastic Flow Potential After initializing the material data table (TB,EDP,,,,LFPOT), enter the following constant (TBDATA): Constant C1 Meaning Property Pressure sensitivity
The material behavior can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP).
3.4.5.2.2.2. Power Law Form

The power law form of the plastic flow potential is: = b +
where the exponent
and the pressure sensitivity are user-defined parameters.
Defining the Linear Plastic Flow Potential After initializing the material data table (TB,EDP,,,,PFPOT), enter the following constants (TBDATA): Constant C1 C2 Meaning Property Pressure sensitivity Exponent
3.4.5.2.2.3. Hyperbolic Form

The hyperbolic form of the plastic flow potential is: =
2
+ 2 +
where the pressure sensitivity the constant Defining the Linear Plastic Flow Potential
are user-defined parameters.
After initializing the material data table (TB,EDP,,,,HFPOT), enter the following constants (TBDATA):
51
Material Models Constant C1 C2 Meaning Property Pressure sensitivity Material parameter
3.4.5.2.3. Plastic Strain Increments for Flow Potentials

The plastic strain increment corresponding to each of the plastic flow potentials is: pl = pl = pl = where = +
b 1 +
+ 2+ 2
is the deviatoric stress:
( )
The dilatation for each of the flow potentials is:
( ) =
Associated flow is obtained if the plastic potential form and parameters are set equal to the yield criterion.
3.4.5.2.4. Example EDP Material Model Definitions

The following examples show how to define the EDP material model using various yield criteria and flow potentials: Example 3.12: EDP -- Linear Yield Criterion and Flow Potential
/prep7 !!! Define linear elasticity constants mp,ex,1,2.1e4 mp,nuxy,1,0.45 ! Extended Drucker-Prager Material Model Definition ! Linear Yield Function tb,edp,1,1,2,LYFUN tbdata,1,2.2526,7.894657 ! Linear Plastic Flow Potential tb,edp,1,1,2,LFPOT tbdata,1,0.566206,7.894657
52
Rate-Independent Plasticity Example 3.13: EDP -- Power Law Yield Criterion and Flow Potential
/prep7 !!! Define linear elasticity constants mp,ex,1,2.1e4 mp,nuxy,1,0.45 ! Extended Drucker-Prager Material Model Definition ! Power Law Yield Function tb,edp,1,1,3,PYFUN tbdata,1,8.33,1.5 ! Power Law Plastic Flow Potential tb,edp,1,1,2,PFPOT tbdata,1,8.33,1.5
Example 3.14: EDP -- Hyperbolic Yield Criterion and Flow Potential

/prep7 !!! Define linear elasticity constants mp,ex,1,2.1e4 mp,nuxy,1,0.45 ! Extended Drucker-Prager Material Model Definition ! Hyperbolic Yield Function tb,edp,1,1,3,HYFUN tbdata,1,1.0,1.75,7.89 ! Hyperbolic Plastic Flow Potential tb,edp,1,1,2,HFPOT tbdata,1,1.0,1.75
3.4.5.3. Extended Drucker-Prager Cap

The EDP Cap material model has a yield criterion similar to the other extended Drucker-Prager yield criteria with the addition of two cap surfaces that truncate the yield surface in tension and compression regions [11]. The model formulation follows that of Schwer and Murray [12] and Foster et al [13] and the numerical formulation is modified from the work of Pelessone [14]. The criterion is a function of the three stress invariants 1 , 2 , and 3 , given by:
= = =
( ) ( )
is the deviatoric stress.
where
Three functions define the surfaces that make up the yield criterion. The shear envelope function is given by:
s ( c ) = c
53
Material Models c
where ,
is the cohesion related yield parameter and is a user defined material parameter along with , and . This function reduces to the Drucker-Prager criterion for = . For positive values of
s(
1 , the shear failure envelope is evaluated at = 0, which gives the constant value
) =
The compaction function is itself a function of the shear envelope function and is given by:
( 0 ) =
0 0 ) ( 0 )
where is the Heaviside step function, is a user-input material parameter, and intersection of the compaction surface with the shear envelope, given by:
=
+ (
)
defines the
= , as where is the user-defined value of at the intersection of the compaction cap with shown in the following figure: Figure 3.14: Yield Surface for the Cap Criterion
< The compaction function defines the material yield surface when . The expansion function is a function of the shear envelope function and is given by: t ( ) = ( ) t ( ) where

is a user-input material parameter. The expansion function defines the material yield surface
> = ( ) . The expansion cap function reaches peak value at . when These functions define the yield criterion, given by:
) = ( ) ( ) ( )
(
3
(3.12)
where is the Lode angle function. The Lode angle is given by:
54
1
( 2 3 ) =
3 3/ 2 2
and the Lode angle function is: = +
( ) + (
( ))
where is a user-defined material parameter, a ratio of the extension strength to compression strength in triaxial loading. Two methods of isotropic hardening can be used to evolve the yield criterion due to plastic deformation. Hardening of the compaction cap is due to evolution of 0 , which is the intersection of the cap surface = with shown in Figure 3.14: Yield Surface for the Cap Criterion (p. 54). This value evolves due to p plastic volume strain v , and the relationship is given by [15]: =
c
((
c c ( i)
)(
i)
,
, and . The restriction
where

is the initial value of
, and the user-defined parameters are
is enforced so that the material does not soften.
Evolution of the yield surface at the intersection of the shear envelope with the expansion cap occurs by combining the cap model with an isotropic hardening model to evolve the value of . The bilinear, multilinear, or nonlinear isotropic hardening function can be used, and the yield stress from the isotropic hardening model must be consistent with the value of calculated from the cap material parameters = . given by The following topics related to defining the EDP Cap material model are available: 3.4.5.3.1. Defining the EDP Cap Yield Criterion and Hardening 3.4.5.3.2. Defining the EDP Cap Plastic Potential 3.4.5.3.3. Example EDP Cap Material Model Definition
3.4.5.3.1. Defining the EDP Cap Yield Criterion and Hardening

After initializing the material data table (TB,EDP,,,,CFYUN), enter the following constants (TBDATA): Constant C1 C2 C3 Material
t
Property Compaction cap parameter Expansion cap parameter Compaction cap yield pressure
55
Material Models Constant C4 C5 C6 C7 C8 C9 C10 C11

1 2
Material c
Property Cohesion yield parameter Shear envelope exponent Shear envelope exponential coefficient Shear envelope linear coefficient Ratio of extension to compression strength Limiting value of volumetric plastic strain Hardening parameter Hardening parameter
The yield criterion and hardening behavior can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP).
3.4.5.3.2. Defining the EDP Cap Plastic Potential

After initializing the material data table (TB,EDP,,,,CFPOT), enter the following constants (TBDATA): Constant C1 C2 C3 C4 Material
t i
Property Compaction cap parameter Expansion cap parameter Shear envelope exponent Shear envelope linear coefficient
The plastic flow potential can be defined as a function of temperature (NTEMP on the TB command), with temperatures specified for the table entries (TBTEMP). If the plastic flow potential is not defined, the yield surface is used as the flow potential, resulting in an associated flow model.
3.4.5.3.3. Example EDP Cap Material Model Definition

The following example input shows how to define an EDP Cap model by defining the yield criterion, hardening, and plastic flow potential: Example 3.15: EDP Cap Model Material Constant Input
/prep7 ! Define linear elasticity constants mp,ex ,1,14e3 mp,nuxy,1,0.0 ! Cap yield function
56
tb,edp ,1,1,,cyfun tbdata,1,2 tbdata,2,1.5 tbdata,3,-80 tbdata,4,10 tbdata,5,0.001 tbdata,6,2 tbdata,7,0.05 tbdata,8,0.9
! Rc Rt Xi SIGMA B ! A ! ALPHA ! PSI ! ! ! !
! Define hardening for cap-compaction portion tbdata,9,0.6 ! W1c tbdata,10,3.0/1000 ! D1c tbdata,11,0.0 ! D2c ! Cap plastic flow potential function tb,edp ,1,1,,cfpot tbdata,1,2 ! RC tbdata,2,1.5 ! RT tbdata,3,0.001 ! B tbdata,4,0.05 ! ALPHA
3.4.6. Gurson
The Gurson model is used to represent plasticity and damage in ductile porous metals [16][17]. When plasticity and damage occur, ductile metal undergoes a process of void growth, nucleation, and coalescence. The model incorporates these microscopic material behaviors into macroscopic plasticity behavior based on changes in the void volume fraction, also known as porosity, and pressure. A porosity increase corresponds to an increase in material damage, resulting in a diminished load-carrying capacity. The yield criterion and flow potential for the Gurson model is:
y = e + y
2 y
+ 3 *2 =
where
is the von Mises equivalent stress,
is the yield stress, , , and are user-input
Tvergaard-Needleman constants, and
is the modified void volume fraction.
The following additional Gurson model topics are available: 3.4.6.1. Void Volume Fraction 3.4.6.2. Hardening 3.4.6.3. Defining the Gurson Material Model
3.4.6.1. Void Volume Fraction

The following figure shows the phenomena of voids at the microscopic scale that are incorporated into the Gurson model:
57
Material Models Figure 3.15: Growth, Nucleation, and Coalescence of Voids at Microscopic Scale
(a): Existing voids grow when the solid matrix is in a hydrostatic-tension state. The solid matrix is assumed to be incompressible in plasticity so that any material volume growth is due to the void volume expansion. (b): Void nucleation occurs, where new voids are created during plastic deformation due to debonding of the inclusion-matrix or particle-matrix interface, or from the fracture of the inclusions or particles themselves. (c): Voids coalesce. In this process, the isolated voids establish connections. Although coalescence may not discernibly affect the void volume, the load-carrying capacity of the material begins to decay more rapidly at this stage. The void volume fraction is the ratio of void volume to the total volume. A volume fraction of 0 indicates no voids and the yield criterion reduces to the von Mises criterion. A volume fraction of 1 indicates all the material is void. The initial void volume fraction, 0 , is a user-defined parameter, and the rate of change of void volume fraction is a combination of the rate of growth and the rate of nucleation: = growth + nucleation
From the assumption of isochoric plasticity and conservation of mass, the rate of change of void volume fraction due to growth is proportional to the rate of volumetric plastic strain:
= (
p )
Void nucleation is controlled by either the plastic strain or the stress, and is assumed to follow a normal distribution of statistics. In the case of strain-controlled nucleation, the distribution is described by the mean strain, N , and deviation,
. The void nucleation rate due to strain control is given by:
58

nucleation
where
is the maximum void fraction for nucleated voids,
is the effective plastic strain, and the
rate of effective plastic strain, , is determined by equating the microscopic plastic work to the macroscopic plastic work:
In the case of stress-controlled nucleation, void nucleation is determined by the distribution of maximum + normal stress on the interfaces between inclusions and the matrix, equal to . Stress-controlled nucleation takes into account the effect of triaxial loading on the rate of void nucleation. The voidnucleation rate for stress control is given by:

+

where distribution of stress is described by the mean stress, The modified void volume fraction,
*
and deviation,
, is used to model the loss of material load carrying capacity as!
sociated with void coalescence. When the current void volume fraction reaches a critical value , the material load carrying capacity decreases rapidly due to coalescence. When the void volume fraction reaches , the load-carrying capacity of the material is lost completely. The modified void volume fraction is given by:
F

" "
"
"
>
"
"
3.4.6.2. Hardening
The Gurson model can be combined with one of the isotropic hardening models to incorporate isotropic hardening of the yield stress in the Gurson yield criterion. To combine the Gurson model with Chaboche kinematic hardening, the yield criterion is modified to:
( )
&
'
&
+
)
( +

) &
() 3
)=
59
Material Models where e is the von Mises equivalent modified relative stress, and y is the modified yield stress which are functions of the modified backstress given by: ff =
where is the kinematic hardening backstress. Then, the modified relative stress is: = and the modified yield stress is: = +
( )
( )
3.4.6.3. Defining the Gurson Material Model

The Gurson material model requires material parameters for the base model combined with parameters for either strain-controlled or stress-controlled nucleation. Additional input is required to define the void coalescence behavior. 3.4.6.3.1. Defining the Gurson Base Model 3.4.6.3.2. Defining Stress- or Strain-Controlled Nucleation 3.4.6.3.3. Defining the Void Coalescence Behavior 3.4.6.3.4. Example Gurson Model Definition
3.4.6.3.1. Defining the Gurson Base Model

To define the Gurson base model, initialize the material data table (TB,GURSON,,,,BASE), then input the following constants (TBDATA): Constant C1 C2 C3 C4 C5 Meaning
0 1 2
Property Initial yield strength Initial porosity First Tvergaard-Needleman constant Second Tvergaard-Needleman constant Third Tvergaard-Needleman constant
3.4.6.3.2. Defining Stress- or Strain-Controlled Nucleation

The Gurson base model is combined with either stress- or strain-controlled nucleation. To define stress-controlled nucleation, initialize the material data table (TB,GURSON,,,,SSNU), then input the following constants (TBDATA):
60
Rate-Independent Plasticity Constant C1 C2 C3 Meaning Property Nucleation porosity Mean stress Stress standard deviation
To define strain-controlled nucleation, initialize the material data table (TB,GURSON,,,,SNNU), then input the following constants (TBDATA): Constant C1 C2 C3 Meaning Property Nucleation porosity Mean strain Strain standard deviation
3.4.6.3.3. Defining the Void Coalescence Behavior

Define the void coalescence behavior after defining the Gurson base model and either the stress- or strain-controlled nucleation behavior. Initialize the material data table (TB,GURSON,,,,COAL), then input the following constants (TBDATA): Constant C1 C2 Meaning Property Critical porosity Failure porosity
c F
3.4.6.3.4. Example Gurson Model Definition

Following is an example Gurson plasticity material model definition: Example 3.16: Gurson Model with Isotropic and Kinematic Hardening
/prep7 !!! Define linear elasticity constants mp,ex,1,207.4E3 ! Young modulus (MPa) mp,nuxy,1,0.3 ! Poisson's ratio !!! Define parameters related to Gurson model with !!! the option of strain-controlled nucleation with !!! coalescence f_0=1E-8 q1=1.5 q2=1.0 q3=2.25 f_c=0.15 f_F=0.25 f_N=0.04 s_N=0.1 strain_N=0.3 sigma_Y=755 power_N=0.1 ! ! ! ! ! ! ! ! ! ! ! initial porosity first Tvergaard constant second Tvergaard constant third Tvergaard constant = q1^2 critical porosity failure porosity nucleation porosity standard deviation of mean strain mean strain initial yielding strength (MPa) power value for nonlinear isotropic
61
Material Models
! hardening power law !base model tb,gurson,1,,5,base tbdata,1,sigma_Y,f_0,q1,q2,q3 ! Strain-controlled nucleation tb,gurson,1,,3,snnu tbdata,1,f_N,strain_N,s_N ! Coalescence tb,gurson,1,,2,coal tbdata,1,f_c,f_F ! Power law isotropic hardening tb,nliso,1,,2,POWER tbdata,1,sigma_Y,power_N
3.4.7. Cast Iron

The cast iron plasticity model is used to model gray cast iron. The microstructure of gray cast iron is a two-phase material with graphite flakes embedded in a steel matrix [20]. The microstructure leads to different behavior in tension and compression. In tension, cracks form due to the graphite flakes and the material is brittle with low strength. In compression, the graphite flakes behave as incompressible media that transmit stress and the steel matrix governs the overall behavior. The model is isotropic elastic with the same elastic behavior in tension and compression. The yield strength and isotropic hardening behavior may be different in tension and in compression. Different yield criteria and plastic flow potentials are used for tension and compression. A composite yield surface is used to model different yield behavior in tension and compression. The tension behavior is pressure-dependent and the Rankine maximum stress criterion is used:
t
( ty ) =

( ) e +
( ) ty =
where
is the uniaxial tension yield stress, and is the Lode angle:

1
( 2 3 ) =
3 3/ 2 2
where and are the stress invariants:

= =
( )
is the deviatoric stress.
where
In compression, the pressure-independent von Mises yield criterion is used:

c
( c ) = c =
62
Rate-Independent Plasticity The following figure shows the yield surfaces in compression and tension for perfectly plastic behavior: Figure 3.16: Cast Iron Yield Surfaces for Compression and Tension
The yield surfaces are plotted in the meridian plane in which the ordinate and abscissa are von Mises equivalent stress and pressure, respectively. The evolution of the yield stress in tension and compression follows the user input piecewise linear stress-strain curves for compression and tension. The tension yield stress evolves as a function of the pl equivalent uniaxial plastic strain, t . The evolution of the equivalent uniaxial plastic strain is defined by equating the uniaxial plastic work increment to the total plastic work increment: = y
The compression yield stress evolves as a function of the equivalent plastic strain, , which is calculated from the increment in plastic strain determined by consistency with the yield criterion and the flow potential.
The plastic flow potential is defined by the von Mises yield criterion in compression and results in an associated flow rule. The flow potential in compression is: c = e c
( ) <
In tension, the Rankine cap yield surface is replaced by an ellipsoidal surface defined by:
( )
2
+ 2 =
( )
where
is a constant function of the user-defined plastic Poisson's ratio, :
63
Material Models The plastic Poisson's ratio determines the amount of volumetric expansion during tensile plastic deformation. For = , there is no plastic volume change, and the von Mises flow potential is used. The tensile flow potential gives a nonassociated flow model and results in an unsymmetric material stiffness tensor.
pl
3.4.7.1. Defining the Cast Iron Material Model

Define the isotropic elastic behavior (MP), then define the material data table (TB,CAST,,,,ISOTROPIC). Enter the tensile multilinear hardening stress-strain points into a data table (TB,UNIAXIAL,,,,TENSION). Do the same for the compressive multilinear hardening stress-strain points (TB,UNIAXIAL,,,,COMPRESSION). Enter tension and compression stress-strain points into their respective tables (TBPT), with the compression points entered as positive values. The plastic Poisson's ratio and stress-strain points can be defined as a function of temperature (NTEMP value on the TB command), with individual temperatures specified for the table entries (TBTEMP). Example 3.17: Cast Iron
/prep7 mp, ex, 1,14.773E6 mp,nuxy, 1,0.2273 ! Define cast iron model TB,CAST,1,,,ISOTROPIC TBDATA,1,0.04 TB,UNIAXIAL,1,1,5,TENSION TBTEMP,10 TBPT,,0.550E-03,0.813E+04 TBPT,,0.100E-02,0.131E+05 TBPT,,0.250E-02,0.241E+05 TBPT,,0.350E-02,0.288E+05 TBPT,,0.450E-02,0.322E+05 TB,UNIAXIAL,1,1,5,COMPRESSION TBTEMP,10 TBPT,,0.203E-02,0.300E+05 TBPT,,0.500E-02,0.500E+05 TBPT,,0.800E-02,0.581E+05 TBPT,,0.110E-01,0.656E+05 TBPT,,0.140E-01,0.700E+05
3.5. Rate-Dependent Plasticity (Viscoplasticity)

The following topics related to rate-dependent plasticity are available: 3.5.1. Perzyna and Peirce Options 3.5.2. Exponential Visco-Hardening (EVH) Option 3.5.3. Anand Option 3.5.4. Defining Rate-Dependent Plasticity (Viscoplasticity) 3.5.5. Creep For further information about rate-dependent plastic (viscoplastic) material options, see Rate-Dependent Plasticity in the Mechanical APDL Theory Reference and Viscoplasticity in the Structural Analysis Guide.
64
Rate-Dependent Plasticity (Viscoplasticity)
3.5.1. Perzyna and Peirce Options

To simulate viscoplasticity, combine the RATE option with the BISO, MISO, or NLISO options. To simulate anisotropic viscoplasticity, combine the RATE and HILL options with the BISO, MISO, or NLISO options. The Perzyna option has the following form: ^ pl = o
1/ m
The Peirce option has the following form:

=
is the equivalent plastic strain rate, m is the strain rate In both cases, is the material yield stress, hardening parameter, is the material viscosity parameter, and o is the static yield stress of material. o is a function of some hardening parameter and can be defined by isotropic plasticity (for example,
approaches zero, the solution approaches TB,BISO). As approaches , or m approaches zero, or the static (rate-independent) solution. When m is very small, the Peirce model has less difficulty converging as compared to the Perzyna model. For details, see Rate-Dependent Plasticity in the Mechanical APDL Theory Reference. The two material constants for the Perzyna and Peirce models (defined by the TBDATA) are: Constant C1 C2 Meaning m - Material strain rate hardening parameter - Material viscosity parameter
Specify the PERZYNA model (TBOPT = PERZYNA) as follows: TB,RATE,,,2,PERZYNA Specify the PEIRCE model (TBOPT = PEIRCE) as follows: TB,RATE,,,2,PEIRCE
3.5.2. Exponential Visco-Hardening (EVH) Option

This option has the following form: = =

where
+ +
b
The EVH option can be combined with nonlinear (Chaboche) kinematic hardening using von Mises or Hill yield criterion.
65
Material Models The six material constants in the EVH option are input (TBDATA) in the order shown: Constant C1 C2 C3 C4 C5 C6 Meaning Ko = Material strain hardening parameter Ro = Material strain hardening parameter R = Material strain hardening parameter b = Material strain hardening parameter m = Material strain rate hardening parameter K = Material viscosity parameter
Specify this option (TBOPT = EVH) as follows: TB,RATE,,,6,EVH
3.5.3. Anand Option

The Anand option offers a unified plasticity model requiring no combination with other material models. Details for the Anand option appear in Anand Viscoplasticity Option in the Mechanical APDL Theory Reference. This option requires nine material constants input via the data table command (TBDATA) in the order shown: Constant C1 Meaning s0 Property Initial value of deformation resistance Q = Activation energy C2 C3 C4 C5 C6 C7 C8 C9 Q/R A xi m h0
^
Units Stress Energy / Volume Energy / Volume temperature 1 / Time Dimensionless Dimensionless Stress Stress Dimensionless Dimensionless
R = Universal gas constant Pre-exponential factor Stress multiplier Strain rate sensitivity of stress Hardening / softening constant Coefficient for deformation resistance saturation value Strain rate sensitivity of saturation (deformation resistance) value Strain rate sensitivity of hardening or softening
n a
Specify this option (TBOPT = ANAND) as follows: TB,RATE,,,9,ANAND
66
Rate-Dependent Plasticity (Viscoplasticity) The Anand model supports plane strain, axisymmetric and full three-dimensional element behavior.
3.5.4. Defining Rate-Dependent Plasticity (Viscoplasticity)

Following is the general process for specifying rate-dependent plasticity (viscoplasticity): 1. Initialize the data table via TB,RATE and specify the model option (TBOPT) as described above. 2. Define the temperature (TBTEMP) for each set of data. 3. Define material constants (TBDATA) for each set of data. You can define up to nine material constants via TBDATA commands, but only six constants per command. For the Anand model, therefore, you must issue the TBDATA command more than once. The data table command (TBDATA) must be defined for the same material number to specify the static hardening behavior of the materials (rate-independent and isotropic).
3.5.5. Creep
The creep strain rate, cr , can be a function of stress, strain, temperature, and neutron flux level. Libraries of creep strain rate equations are included under the Implicit Creep Equations (p. 68) and Explicit Creep Equations (p. 70) sections. Enter the constants shown in these equations using TB,CREEP and TBDATA as described below. These equations (expressed in incremental form) are characteristic of materials being used in creep design applications (see the Mechanical APDL Theory Reference for details). For a list of the elements that support creep behavior, see Material Model Element Support (p. 5). Three types of creep equations are available: Primary creep Secondary creep Irradiation induced creep You can define the combined effects of more than one type of creep using the implicit equations specified by TBOPT = 11 or 12, the explicit equations, or a user-defined creep equation. The program analyzes creep using the implicit and the explicit time-integration method. The implicit method is robust, fast, accurate, and recommended for general use, especially with problems involving large creep strain and large deformation. It has provisions for including temperature-dependent constants. The program can model pure creep, creep with isotropic hardening plasticity, and creep with kinematic hardening plasticity, using both von Mises and Hill potentials. See Material Model Combinations (p. 183) for further information. Because the creep and plasticity are modeled simultaneously (no superposition), the implicit method is more accurate and efficient than the explicit method. Temperature dependency can also be incorporated by the Arrhenius function. (See the Mechanical APDL Theory Reference.) The explicit method is useful for cases involving very small time steps, such as in transient analyses. There are no provisions for temperature-dependent constants, nor simultaneous modeling of creep with any other material models such as plasticity. However, there is temperature dependency using the Arrhenius function, and you can combine explicit creep with other plasticity options using non-simultaneous modeling (superposition). In these cases, the program first performs the plastic analysis, then the creep calculation.
67
Material Models The terms implicit and explicit, as applied to creep, have no relationship to explicit dynamics, or any elements referred to as explicit elements.
3.5.5.1. Implicit Creep Equations

Enter an implicit creep equation using TBOPT within the TB command. Enter the value of TBOPT corresponding to the equation, as shown in Table 3.2: Implicit Creep Equations (p. 68). Table 3.2: Implicit Creep Equations Creep Model Name (TBOPT) 1 2 3 4 Strain Hardening Time Hardening Generalized Exponential Generalized Graham Generalized Blackburn Modified Time Hardening Modified Strain Hardening Generalized Garofalo Exponential form Norton Combined Time Hardening cr = 1C2cr C3 C4/ T =
=
t = 5 , = + + 6 7 = = ! +
% 8
Equation
Type
C1>0 C1>0
Primary Primary
C1>0, Primary C5>0 C1>0 Primary
= "
&
= $ +
'
C1>0, C3>0, Primary C6>0 C1>0 C1>0 C1>0 C1>0 C1>0 Primary Primary Secondary Secondary Secondary
6 7 8 9 10
() = ,*. *0 +, *9 : + ;< =
B =F D
+ ;< =G B?@=G +BA =E ? >

JN JO P K
HI = L
QR = V Y SW SX Y U Z[ = ^\_ \` a ] bd = gej ek +g el o f + i em pq = v
en o f h
11
Primary C1>0, + SecC5>0 ondary
12
Rational polynomial
p p = +
uy uz
s + C2>0
u
s = w
u
s{ u| = x
s~~u~ = v}
Primary + Secondary
68
Rate-Dependent Plasticity (Viscoplasticity) Creep Model Name (TBOPT) Generalized Time Hardening --cr =
r
Equation
Type
C 6 / T
13
= 1 + 2 2 + 3 3 = 4 + 5 User Creep ---
Primary
100 where:
---
cr = equivalent creep strain = change in equivalent creep strain with respect to time = equivalent stress T = temperature (absolute). The offset temperature (from TOFFST), is internally added to all temperatures for convenience. C1 through C12 = constants defined by the TBDATA command t = time at end of substep e = natural logarithm base You can define the user creep option by setting TBOPT = 100, and using TB,STATE to specify the number of state variables for the user creep subroutine. See the Guide to User-Programmable Features for more information. The RATE command is necessary to activate implicit creep for specific elements (see the RATE command description for details). The RATE command has no effect for explicit creep. For temperature-dependent constants, define the temperature using TBTEMP for each set of data. Then, define constants C1 through Cm using TBDATA (where m is the number of constants, and depends on the creep model you choose). The following example shows how you would define the implicit creep model represented by TBOPT = 1 at two temperature points.
TB,CREEP,1,,,1 TBTEMP,100 TBDATA,1,c11,c12,c13,c14 TBTEMP,200 TBDATA,1,c21,c22,c23,c24 !Activate creep data table, specify creep model 1 !Define first temperature !Creep constants c11, c12, c13, c14 at first temp. !Define second temperature !Creep constants c21, c22, c23, c24 at second temp.
Coefficients are linearly interpolated for temperatures that fall between user defined TBTEMP values. For some creep models, where the change in coefficients spans several orders of magnitude, this linear interpolation might introduce inaccuracies in solution results. Use enough curves to accurately capture the temperature dependency. Also, consider using the curve fitting subroutine to calculate a temperature dependent coefficient that includes the Arrhenius term. When a temperature is outside the range of defined temperature values, the program uses the coefficients defined for the constant temperature. For a list of elements that can be used with this material option, see Material Model Element Support (p. 5). See Creep in the Structural Analysis Guide for more information on this material option.
Material Models
3.5.5.2. Explicit Creep Equations

Enter an explicit creep equation by setting TBOPT = 0 (or leaving it blank) within the TB command, then specifying the constants associated with the creep equations using the TBDATA command. Specify primary creep with constant C6. Primary Explicit Creep Equation for C6 = 0 (p. 71), through Primary Explicit Creep Equation for C6 = 100 (p. 77), show the available equations. You select an equation with the appropriate value of C6 (0 to 15). If C1 0, or if T + Toffset 0, no primary creep is computed. Specify secondary creep with constant C12. Secondary Explicit Creep Equation for C12 = 0 (p. 77) and Secondary Explicit Creep Equation for C12 = 1 (p. 77) show the available equations. You select an equation with the appropriate value of C12 (0 or 1). If C7 0, or if T + Toffset 0, no secondary creep is computed. Also, primary creep equations C6 = 9, 10, 11, 13, 14, and 15 bypass any secondary creep equations since secondary effects are included in the primary part. Specify irradiation induced creep with constant C66. Irradiation Induced Explicit Creep Equation for C66 = 5 (p. 77) shows the single equation currently available; select it with C66 = 5. This equation can be used in conjunction with equations C6 = 0 to 11. The constants should be entered into the data table as indicated by their subscripts. If C55 0 and C61 0, or if T + Toffset 0, no irradiation induced creep is computed. A linear stepping function is used to calculate the change in the creep strain within a time step ( cr = ( cr )(t)). The creep strain rate is evaluated at the condition corresponding to the beginning of the time interval and is assumed to remain constant over the time interval. If the time step is less than 1.0e6, then no creep strain increment is computed. Primary equivalent stresses and strains are used to evaluate the creep strain rate. For highly nonlinear creep strain vs. time curves, use a small time step if you are using the explicit creep algorithm. A creep time step optimization procedure is available for automatically increasing the time step whenever possible. A nonlinear stepping function (based on an exponential decay) is also available (C11 = 1) but should be used with caution since it can underestimate the total creep strain where primary stresses dominate. This function is available only for creep equations C6 = 0, 1 and 2. Temperatures used in the creep equations should be based on an absolute scale (TOFFST). Use the BF or BFE commands to enter temperature and fluence values. The input fluence (t) includes the integrated effect of time and time explicitly input is not used in the fluence calculation. Also, for the usual case of a constant flux (), the fluence should be linearly ramp changed. Temperature dependent creep constants are not permitted for explicit creep. You can incorporate other creep options by setting C6 = 100. See the Guide to User-Programmable Features for more information. The following example shows how you would use the explicit creep equation defined by C6 = 1.
TB,CREEP,1 TBDATA,1,c1,c2,c3,c4,,1 !Activate creep data table !Creep constants c1, c2, c3, c4 for equation C6=1
The explicit creep constants that you enter with the TBDATA are:
70
Rate-Dependent Plasticity (Viscoplasticity) Constant C1-CN Meaning Constants C1, C2, C3, etc. (as defined in Primary Explicit Creep Equation for C6 = 0 (p. 71) to Irradiation Induced Explicit Creep Equation for C66 = 5 (p. 77)) These are obtained by curve fitting test results for your material to the equation you choose. Exceptions are defined below.
3.5.5.2.1. Primary Explicit Creep Equation for C6 = 0

cr = 1C2cr C3 C4/ T where: = change in equivalent strain with respect to time = equivalent stress T = temperature (absolute). The offset temperature (from TOFFST) is internally added to all temperatures for convenience. t = time at end of substep e = natural logarithm base

=

=
t where: = 5

Annealed 304 Stainless Steel: =
3.5.5.2.4.1. Double Exponential Creep Equation (C4 = 0)

To use the following Double Exponential creep equation to calculate = x where: x = 0 for C2
s +
m +
71
Material Models x = G + H for C2 < C3 C2 = 6000 psi (default), C3 = 25000 psi (default) s, r, m , G, and H = functions of temperature and stress as described in the reference. This double exponential equation is valid for Annealed 304 Stainless Steel over a temperature range from 800 to 1100F. The equation, known as the Blackburn creep equation when C1 = 1, is described completely in the High Alloy Steels. The first two terms describe the primary creep strain and the last term describes the secondary creep strain. To use this equation, input a nonzero value for C1, C6 = 9.0, and C7 = 0.0. Temperatures should be in R (or F with Toffset = 460.0). Conversion to K for the built-in property tables is done internally. If the temperature is below the valid range, no creep is computed. Time should be in hours and stress in psi. The valid stress range is 6,000 - 25,000 psi.
3.5.5.2.4.2. Rational Polynomial Creep Equation with Metric Units (C4 = 1)

To use the following standard Rational Polynomial creep equation (with metric units) to calculate c, enter C4 = 1.0: c + +
where: c = limiting value of primary creep strain p = primary creep time factor = secondary (minimum) creep strain rate This standard rational polynomial creep equation is valid for Annealed 304 SS over a temperature range from 427C to 704C. The equation is described completely in the High Alloy Steels. The first term describes the primary creep strain. The last term describes the secondary creep strain. The average "lot constant" is used to calculate . To use this equation, input C1 = 1.0, C4 = 1.0, C6 = 9.0, and C7 = 0.0. Temperature must be in C and Toffset must be 273 (because of the built-in property tables). If the temperature is below the valid range, no creep is computed. Also, time must be in hours and stress in Megapascals (MPa). Various hardening rules governing the rate of change of creep strain during load reversal may be selected with the C5 value: 0.0 - time hardening, 1.0 - total creep strain hardening, 2.0 - primary creep strain hardening. These options are available only with the standard rational polynomial creep equation.
3.5.5.2.4.3. Rational Polynomial Creep Equation with English Units (C4 = 2)

To use the above standard Rational Polynomial creep equation (with English units), enter C4 = 2.0. This standard rational polynomial equation is the same as described above except that temperature must be in F, Toffset must be 460, and stress must be in psi. The equivalent valid temperature range is 800 - 1300F.
72
Rate-Dependent Plasticity (Viscoplasticity)

Annealed 316 Stainless Steel: cr = 1 c
3.5.5.2.5.1. Double Exponential Creep Equation (C4 = 0)

To use the same form of the Double Exponential creep equation as described for Annealed 304 SS (C6 = 9.0, C4 = 0.0) in Primary Explicit Creep Equation for C6 = 9 (p. 71) to calculate c, enter C4 = 0.0. This equation, also described in High Alloy Steels, differs from the Annealed 304 SS equation in that the built-in property tables are for Annealed 316 SS, the valid stress range is 4000 - 30,000 psi, C2 defaults to 4000 psi, C3 defaults to 30,000 psi, and the equation is called with C6 = 10.0 instead of C6 = 9.0.

To use the same form of the standard Rational Polynomial creep equation with metric units as described for Annealed 304 SS (C6 = 9.0, C4 = 1.0) in Primary Explicit Creep Equation for C6 = 9 (p. 71), enter C4 = 1.0. This standard rational polynomial equation, also described in High Alloy Steels, differs from the Annealed 304 SS equation in that the built-in property tables are for Annealed 316 SS, the valid temperature range is 482 - 704C, and the equation is called with C6 = 10.0 instead of C6 = 9.0. The hardening rules for load reversal described for the C6 = 9.0 standard Rational Polynomial creep equation are also available. The average "lot constant" from High Alloy Steels is used in the calculation of m .

To use the previous standard Rational Polynomial creep equation with English units, enter C4 = 2.0. This standard rational polynomial equation is the same as described above except that the temperatures must be in F, Toffset must be 460, and the stress must be in psi (with a valid range from 0.0 to 24220 psi). The equivalent valid temperature range is 900 - 1300F.

Annealed 2 1/4 Cr - 1 Mo Low Alloy Steel: =
3.5.5.2.6.1. Modified Rational Polynomial Creep Equation (C4 = 0)

To use the following Modified Rational Polynomial creep equation to calculate c, enter C4 = 0.0: = +
73
Material Models A, B, and m are functions of temperature and stress as described in the reference. This modified rational polynomial equation is valid for Annealed 2 1/4 Cr -1 Mo Low Alloy steel over a temperature range of 700 - 1100F. The equation is described completely in the Low Alloy Steels. The first term describes the primary creep strain and the last term describes the secondary creep strain. No modification is made for plastic strains. To use this equation, input C1 = 1.0, C6 = 11.0, and C7 = 0.0. Temperatures must be in R (or F with Toffset = 460.0). Conversion to K for the built-in property tables is done internally. If the temperature is below the valid range, no creep is computed. Time should be in hours and stress in psi. Valid stress range is 1000 - 65,000 psi.

To use the following standard Rational Polynomial creep equation (with metric units) to calculate c, enter C4 = 1.0: c + +
where: c = limiting value of primary creep strain p = primary creep time factor = secondary (minimum) creep strain rate This standard rational polynomial creep equation is valid for Annealed 2 1/4 Cr - 1 Mo Low Alloy Steel over a temperature range from 371C to 593C. The equation is described completely in the Low Alloy Steels. The first term describes the primary creep strain and the last term describes the secondary creep strain. No tertiary creep strain is calculated. Only Type I (and not Type II) creep is supported. No modification is made for plastic strains. To use this equation, input C1 = 1.0, C4 = 1.0, C6 = 11.0, and C7 = 0.0. Temperatures must be in C and Toffset must be 273 (because of the built-in property tables). If the temperature is below the valid range, no creep is computed. Also, time must be in hours and stress in Megapascals (MPa). The hardening rules for load reversal described for the C6 = 9.0 standard Rational Polynomial creep equation are also available.

To use the above standard Rational Polynomial creep equation with English units, enter C4 = 2.0. This standard rational polynomial equation is the same as described above except that temperatures must be in F, Toffset must be 460, and stress must be in psi. The equivalent valid temperature range is 700 - 1100F.

r =
N (M1)
74
Rate-Dependent Plasticity (Viscoplasticity) where: C1 = Scaling constant M, N, K = Function of temperature (determined by linear interpolation within table) as follows: Constant C5 Meaning Number of temperature values to describe M, N, or K function (2 minimum, 6 maximum) First absolute temperature value Second absolute temperature value ... C48 + C5 C48 + C5 + 1 ... C48 + 2C5 C48 + 2C5 ... C48 + 2C5 C48 + 2C5 + 1 ... C48 + 3C5 C48 + 3C5 + 1 ... This power function creep law having temperature dependent coefficients is similar to Equation C6 = 1.0 except with C1 = f1(T), C2 = f2(T), C3 = f3(T), and C4 = 0. Temperatures must not be input in decreasing order. C5th N value First K value C5th M value First N value C5th M value C5th M value C5th absolute temperature value First M value
C49 C50
3.5.5.2.8. Primary Explicit Creep Equation for C6 Equals 13

Sterling Power Function: cr = where: acc = creep strain accumulated to this time (calculated by the program). Internally set to 1 x 10-5 at the first substep with nonzero time to prevent division by zero. A = C1/T B = C2/T + C3 acc
B acc
( 3 A + 2B + C )
75
Material Models C = C4/T + C5 This equation is often referred to as the Sterling Power Function creep equation. Constant C7 should be 0.0. Constant C1 should not be 0.0, unless no creep is to be calculated.

cr = where: c = cpt/(1+pt) + m ln c = -1.350 - 5620/T - 50.6 x 10-6 + 1.918 ln (/1000) ln p = 31.0 - 67310/T + 330.6 x 10-6 - 1885.0 x 10-12 2 ln = 43.69 - 106400/T + 294.0 x 10-6 + 2.596 ln (/1000) This creep law is valid for Annealed 316 SS over a temperature range from 800F to 1300F. The equation is similar to that given for C6 = 10.0 and is also described in High Alloy Steels. To use equation, input C1 = 1.0 and C6 = 14.0. Temperatures should be in R (or F with Toffset = 460). Time should be in hours. Constants are only valid for English units (pounds and inches). Valid temperature range: 800 - 1300F. Maximum stress allowed for ec calculation: 45,000 psi; minimum stress: 0.0 psi. If T + Toffset < 1160, no creep is computed. 1 c

General Material Rational Polynomial: = where: + = = = +
0
C C 2 3 4 8 9
7
( ust not be negative)
This rational polynomial creep equation is a generalized form of the standard rational polynomial equations given as C6 = 9.0, 10.0, and 11.0 (C4 = 1.0 and 2.0). This equation reduces to the standard equations for isothermal cases. The hardening rules for load reversal described for the C6 = 9.0 standard Rational Polynomial creep equation are also available.
76
Hyperelasticity

A user-defined creep equation is used. See the Guide to User-Programmable Features for more information.
3.5.5.2.12. Secondary Explicit Creep Equation for C12 = 0

cr = 7 / C8 C10 / T where: = equivalent stress T = temperature (absolute). The offset temperature (from TOFFST), is internally added to all temperatures for convenience. t = time e = natural logarithm base
3.5.5.2.13. Secondary Explicit Creep Equation for C12 = 1

=
3.5.5.2.14. Irradiation Induced Explicit Creep Equation for C66 = 5

t . +
= 55 6 where: B = FG + C63 =
9
+
= 2 = equivalent stress T = temperature (absolute). The offset temperature (from TOFFST) is internally added to all temperatures for convenience. t0.5 = neutron fluence (input on BF or BFE command) e = natural logarithm base t = time This irradiation induced creep equation is valid for 20% Cold Worked 316 SS over a temperature range from 700 to 1300F. Constants 56, 57, 58 and 62 must be positive if the B term is included. See Creep in the Structural Analysis Guide for more information on this material option.
3.6. Hyperelasticity
Hyperelastic material behavior is supported by current-technology shell, plane, and solid elements. For a list of elements that can be used with hyperelastic material models, see Material Model Element Support (p. 5). You can specify options to describe the hyperelastic material behavior for these elements.
77
Material Models Hyperelasticity options are available via the TBOPT argument on the TB,HYPER command. Several forms of strain energy potentials describe the hyperelasticity of materials. These are based on either strain invariants or principal stretches. The behavior of materials is assumed to be incompressible or nearly incompressible. The following hyperelastic material model topics are available: 3.6.1. Arruda-Boyce Hyperelasticity 3.6.2. Blatz-Ko Foam Hyperelasticity 3.6.3. Extended Tube Hyperelasticity 3.6.4. Gent Hyperelasticity 3.6.5. Mooney-Rivlin Hyperelasticity 3.6.6. Neo-Hookean Hyperelasticity 3.6.7. Ogden Hyperelasticity 3.6.8. Ogden Compressible Foam Hyperelasticity 3.6.9. Polynomial Form Hyperelasticity 3.6.10. Response Function Hyperelasticity 3.6.11.Yeoh Hyperelasticity 3.6.12. Special Hyperelasticity For information about other hyperelastic material models, see Special Hyperelasticity (p. 87).
3.6.1. Arruda-Boyce Hyperelasticity

The TB,HYPER,,,,BOYCE option uses the Arruda-Boyce form of strain energy potential given by: = +
1
+
4 1
2 L
2 1
4 L 5 1
3 1
6 L
8 L
where: W = strain energy per unit reference volume

= first deviatoric strain invariant
J = determinant of the elastic deformation gradient F = initial shear modulus of materials L = limiting network stretch d = material incompressibility parameter The initial bulk modulus is defined as: = As L approaches infinity, the option becomes equivalent to the Neo-Hookean option. The constants , L and d are defined by C1, C2, and C3 using the TBDATA command.
78
Hyperelasticity For a list of elements that can be used with this material option, see Material Model Element Support (p. 5). See Arruda-Boyce Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.2. Blatz-Ko Foam Hyperelasticity

The TB,HYPER,,,,BLATZ option uses the Blatz-Ko form of strain energy potential given by: = 2 + 3
3
where: W = strain energy per unit reference volume = initial strain shear modulus I2 and I3= second and third strain invariants The initial bulk modulus k is defined as: =
The model has only one constant and is defined by C1 using the TBDATA command. See Blatz-Ko Foam Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.3. Extended Tube Hyperelasticity

The extended tube model is available as a hyperelastic material option (TB,HYPER). The model simulates filler-reinforced elastomers and other rubber-like materials, supports material curve-fitting, and is available in all current-technology continuum, shell, and pipe elements. Five material constants are needed for the extended-tube model: TBOPT C1 C2 C3 C4 C5 Constants Gc Ge d Purpose Crosslinked network modulus Constraint network modulus Empirical parameter (0 1) Extensibility parameter Incompressibility parameter
Following the material data table command (TB), specify the material constant values via the TBDATA command , as shown in this example:
79
Material Models
TB,HYPER,1,,5,ETUBE ! Hyperelastic material, 1 temperature, ! 5 material constants, and the extended tube option TBDATA,1,0.25, 0.8,1.0,0.5,1.0e-5 ! Five material constant values (C1 through C5)
For more information, see the documentation for the TB,HYPER command, and Extended Tube Model in the Mechanical APDL Theory Reference.
3.6.4. Gent Hyperelasticity

The TB,HYPER,,,,GENT option uses the Gent form of strain energy potential given by: =
1 m
where: W = strain energy per unit reference volume = initial shear modulus of material
= li iting value of = rs d
rc sr r
J = determinant of the elastic deformation gradient F d = material incompressibility parameter The initial bulk modulus K is defined as: = As Jm approaches infinity, the option becomes equivalent to the Neo-Hookean option. The constants , Jm, and d are defined by C1, C2, and C3 using the TBDATA command. For a list of elements that can be used with this material option, see Material Model Element Support (p. 5). See Gent Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.5. Mooney-Rivlin Hyperelasticity

The Mooney-Rivlin model applies to current-technology shell, beam, solid, and plane elements. The TB,HYPER,,,,MOONEY option allows you to define 2, 3, 5, or 9 parameter Mooney-Rivlin models using NPTS = 2, 3, 5, or 9, respectively. For NPTS = 2 (2 parameter Mooney-Rivlin option, which is also the default), the form of the strain energy potential is: =
where:
80
Hyperelasticity W = strain energy potential

1 = first deviatoric strain invariant 2 =

c10, c01 = material constants characterizing the deviatoric deformation of the material d = material incompressibility parameter The initial shear modulus is defined as: =
0 + 0
and the initial bulk modulus is defined as: = where: d = (1 - 2*) / (C10 + C01) The constants c10, c01, and d are defined by C1, C2, and C3 using the TBDATA command. For NPTS = 3 (3 parameter Mooney-Rivlin option, which is also the default), the form of the strain energy potential is: = + +

The constants c10, c01, c11; and d are defined by C1, C2, C3, and C4 using the TBDATA command. For NPTS = 5 (5 parameter Mooney-Rivlin option), the form of the strain energy potential is: = + +

+
+
+
The constants c10, c01, c20, c11, c02, and d are material constants defined by C1, C2, C3, C4, C5, and C6 using the TBDATA command. For NPTS = 9 (9 parameter Mooney-Rivlin option), the form of the strain energy potential is: = + +

+

+
3 3+
+ +
+ 3
+ 3
The constants c10, c01, c20, c11, c02, c30, c21, c12, c03, and d are material constants defined by C1, C2, C3, C4, C5, C6, C7, C8, C9, and C10 using the TBDATA command.
81
Material Models See Mooney-Rivlin Hyperelastic Option (TB,HYPER) in the Structural Analysis Guide for more information on this material option.
3.6.6. Neo-Hookean Hyperelasticity

The option TB,HYPER,,,,NEO uses the Neo-Hookean form of strain energy potential, which is given by: =
1
where: W = strain energy per unit reference volume

= first deviatoric strain invariant
= initial shear modulus of the material d = material incompressibility parameter. J = determinant of the elastic deformation gradient F The initial bulk modulus is defined by: = The constants and d are defined via the TBDATA command. See Neo-Hookean Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.7. Ogden Hyperelasticity

The TB,HYPER,,,,OGDEN option uses the Ogden form of strain energy potential. The Ogden form is based on the principal stretches of the left Cauchy-Green tensor. The strain energy potential is:
N = + + 3 =
k = k
where: W = strain energy potential p ( = ,,) =

l
h
,

p = p = principal stretches of the left Cauchy-Green tensor J = determinant of the elastic deformation gradient N, p, p and dp = material constants In general there is no limitation on the value of N. (See the TB command.) A higher value of N can provide a better fit to the exact solution. It may however cause numerical difficulties in fitting the material constants. For this reason, very high values of N are not recommended. The initial shear modulus is defined by:

82
Hyperelasticity
i i
i =1
The initial bulk modulus K is defined by: =
For N = 1 and 1 = 2, the Ogden option is equivalent to the Neo-Hookean option. For N = 2, 1 = 2, and 2 = -2, the Ogden option is equivalent to the 2 parameter Mooney-Rivlin option. The constants p, p and dp are defined using the TBDATA command in the following order: For N (NPTS) = 1: 1, 1, d1 For N (NPTS) = 2: 1, 1, 2, 2, d1, d2 For N (NPTS) = 3: 1, 1, 2, 2, 3, 3, d1, d2, d3 For N (NPTS) = k: 1, 1, 2, 2, ..., k, k, d1, d2, ..., dk See Ogden Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.8. Ogden Compressible Foam Hyperelasticity

The TB,HYPER,,,,FOAM option uses the Ogden form of strain energy potential for highly compressible elastomeric foam material. The strain energy potential is based on the principal stretches of the left Cauchy-Green tensor and is given by:
= / 3 + 2 + 3 =
where: W = strain energy potential p (=,,) = dev

ator
c r
nc
al stretch J = determinant of the elastic deformation gradient N, i, i and k = material constants For this material option, the volumetric and deviatoric terms are tightly coupled. Hence, this model is meant to simulate highly compressible elastomers.
83
Material Models In general there is no limitation on the value of N. (See the TB command.) A higher value of N can provide a better fit to the exact solution. It may however cause numerical difficulties in fitting the material constants. For this reason, very high values of N are not recommended. The initial shear modulus is defined by:
= =
i
i 1
and the initial bulk modulus K is defined by: = =

For N = 1, 1 = 2, 1 = -, and 1 = 0.5, the Ogden foam option is equivalent to the Blatz-Ko option. The constants i, i and i are defined using the TBDATA command in the following order: For N (NPTS) = 1: 1, 1, 1 For N (NPTS) = 2: 1, 1, 2, 2, 1, 2 For N (NPTS) = 3: 1, 1, 2, 2, 3, 3, 1, 2, 3 For N (NPTS) = k: 1, 1, 2, 2, ..., k, k, 1, 2, ..., k See Ogden Compressible Foam Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.9. Polynomial Form Hyperelasticity

The TB,HYPER,,,,POLY option allows you to define a polynomial form of strain energy potential. The form of the strain energy potential for the Polynomial option is given by: =

+ j =
+
k
2k
where: W = strain energy potential
frst devatorc stran nvarant
84
Hyperelasticity
2 = second deviatoric strain invariant
J = determinant of the elastic deformation gradient F N, cij, and d = material constants In general there is no limitation on the value of N. (See the TB command.) A higher value of N can provide a better fit to the exact solution. It may however cause a numerical difficulty in fitting the material constants, and it also requests enough data to cover the whole range of deformation for which you may be interested. For these reasons, a very high value of N is not recommended. The initial shear modulus is defined by: =
10 + 01
and the initial bulk modulus is defined as: =
For N = 1 and c01 = 0, the polynomial form option is equivalent to the Neo-Hookean option. For N = 1, it is equivalent to the 2 parameter Mooney-Rivlin option. For N = 2, it is equivalent to the 5 parameter Mooney-Rivlin option, and for N = 3, it is equivalent to the 9 parameter Mooney-Rivlin option. The constants cij and d are defined using the TBDATA command in the following order: For N (NPTS) = 1: c10, c01, d1 For N (NPTS) = 2: c10, c01, c20, c11, c02, d1, d2 For N (NPTS) = 3: c10, c01, c20, c11, c02, c30, c21, c12, c03, d1, d2, d3 For N (NPTS) = k: c10, c01, c20, c11, c02, c30, c21, c12, c03, ..., ck0, c(k-1)1, ..., c0k, d1, d2, ..., dk See Polynomial Form Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.10. Response Function Hyperelasticity

The response function option for hyperelastic material constants (TB,HYPER,,,,RESPONSE) uses experimental data (TB,EXPE) to determine the constitutive response functions. The response functions (first derivatives of the hyperelastic potential) are used to determine hyperelastic constitutive behavior of the material. In general, the stiffness matrix requires derivatives of the response functions (second derivatives of the hyperelastic potential).
85
Material Models The method for determining the derivatives is ill-conditioned near the zero stress-strain point; therefore, a deformation limit is used, below which the stiffness matrix is calculated with only the response functions. The deformation measure is = I1 - 3, where I1 is the first invariant of the Cauchy-Green deformation tensor. The stiffness matrix is then calculated with only the response functions if < C1, where C1 is the material constant deformation limit (default 1 x 10-5). The remaining material parameters are for the volumetric strain energy potential, given by =
k =1 k
)2k
where N is the NPTS value (TB,HYPER,,,,RESPONSE) and dk represents the material constants incompressibility parameters (default 0.0) and J is the volume ratio. Use of experimental volumetric data requires NPTS = 0. Incompressible behavior results if all dk = 0 or NPTS = 0 with no experimental volumetric data.
3.6.11. Yeoh Hyperelasticity

The TB,HYPER,,,,YEOH option follows a reduced polynomial form of strain energy potential by Yeoh. The form of the strain energy potential for the Yeoh option is given by: = i0
i =
i+ =
where: W = strain energy potential
= frst devatorc stran nvarant

J = determinant of the elastic deformation gradient F N, ci0, and dk = material constants In general there is no limitation on the value of N. (See the TB command.) A higher value of N can provide a better fit to the exact solution. It may however cause a numerical difficulty in fitting the material constants, and it also requests enough data to cover the whole range of deformation for which you may be interested. For these reasons, a very high value of N is not recommended. The initial shear modulus is defined by: =
and the initial bulk modulus K is defined as: =
For N = 1 the Yeoh form option is equivalent to the Neo-Hookean option.
86
Hyperelasticity The constants ci0 and dk are defined using the TBDATA command in the following order: For N (NPTS) = 1: c10, d1 For N (NPTS) = 2: c10, c20, d1, d2 For N (NPTS) = 3: c10, c20, c30, d1, d2, d3 For N (NPTS) = k: c10, c20, c30, ..., ck0, d1, d2, ..., dk See Yeoh Hyperelastic Option in the Structural Analysis Guide for more information on this material option.
3.6.12. Special Hyperelasticity

The following hyperelastic material models have their own Lab value on the TB command (and are not simply TBOPT hyperelasticity options on the TB,HYPER command): 3.6.12.1. Anisotropic Hyperelasticity 3.6.12.2. Bergstrom-Boyce Material 3.6.12.3. Mullins Effect 3.6.12.4. User-Defined Hyperelastic Material
3.6.12.1. Anisotropic Hyperelasticity

The anisotropic hyperelasticity material model (TB,AHYPER) is available with current-technology shell, plane, and solid elements. Anisotropic hyperelasticity is a potential-based-function with parameters to define the volumetric part, the isochoric part and the material directions. Two strain energy potentials, as forms of polynomial or exponential function, are available for characterizing the isochoric part of strain energy potential. You can use anisotropic hyperelasticity to model elastomers with reinforcements, and for biomedical materials such as muscles or arteries. The strain energy potential for anisotropic hyperelasticity is given by: =
v
The volumetric strain energy is given by:
The polynomial-function-based strain energy potential is given by:
87
Material Models
i+ 3 j+ 6 k j 2 k 4 i =1 j =1 k =2 6 6 6 6 + l 5 l + m 6 m + n 7 n + o 8 o l= 2 m=2 n=2 o =2
= i 1
The exponential-function-based strain energy potential is given by:

(
) = ( ) + ( ) = =

Use TB,AHYPER,,TBOPT to define the isochoric part, material directions and the volumetric part. Only one TB table can be defined for each option. You can either define polynomial or exponential strain energy potential. TBOPT POLY EXP Constants C1 to C31 C1 to C10 Purpose Anisotropic strain energy potential Exponential anisotropic strain energy potential Material direction constants Material direction constants Volumetric potential Input Format
TB,AHYPER,,,POLY TBDATA,,A1,A2,A3,B1.... TB,AHYPER,,,EXPO TBDATA,1,A1,A2,A3,B1,B2,B3 TBDATA,7,C1,C2,E1,E2 TB,AHYPER,,,AVEC TBDATA,,A1,A2,A3 TB,AHYPER,,,BVEC TBDATA,,B1,B2, B3 TB,AHYPER,,,PVOL TBDATA,,D
AVEC BVEC PVOL
C1 to C3 C1 to C3 C1
You can enter temperature-dependent data for anisotropic hyperelastic material via the TBTEMP command. For the first temperature curve, issue TB, AHYPER,,,TBOPT, then input the first temperature (TBTEMP). The subsequent TBDATA command inputs the data. The program interpolates the temperature data to the material points automatically using linear interpolation. When the temperature is out of the specified range, the closest temperature point is used. For more information, see the TB command, and Anisotropic Hyperelasticity in the Mechanical APDL Theory Reference.
3.6.12.2. Bergstrom-Boyce Material

The Bergstrom Boyce option (TB,BB) is a phenomenological-based, highly nonlinear, rate-dependent material model for simulation of elastomer materials. The model assumes inelastic response only for shear distortional behavior defined by an isochoric strain energy potential, while the response to volumetric deformations is still purely elastic and characterized by a volumetric strain energy potential. This model requires seven material constants input for the isochoric (TBOPT = ISO) option and one material constant for the volumetric potential (TBOPT = PVOL) option. Issue the TBDATA data table command to input the constant values in the order shown:
Hyperelasticity Isochoric TB,BB,,,,ISO Constant C1 C2 C3 C4 C5 0 Meaning 0 N0 1 N1 Property Initial shear modulus for Part A Pa Units
( Alock )2, where lock is the Dimensionlimiting chain stretch less Initial shear modulus for Part B ( Block )2 Material constant Pa Dimensionless s-1(Pa)-m
m base c m Material constant Material constant Optional material constant Dimensionless Dimensionless Dimensionless
C6 C7 C8
The default optional material constant is = 1 x 10-5. However, if TBNPT > 7 or TBNPT is unspecified, the table value is used instead. If the table value is zero or exceeds 1 x 10-3, the default constant value is used. Volumetric Potential TB,BB,,,,PVOL Constant C1 Meaning d Property 1 / K, where K is the bulk modulus Units 1 / Pa
For more information, see: The BB argument and associated specifications in the TB command documentation Bergstrom-Boyce Hyperviscoelastic Material Model in the Structural Analysis Guide Bergstrom-Boyce in the Mechanical APDL Theory Reference
3.6.12.3. Mullins Effect

The Mullins effect is a modification to the nearly- and fully-incompressible isotropic hyperelastic constitutive models (all TB,HYPER options with the exception of TBOPT = BLATZ or TBOPT = FOAM) and is used with those models. The data table is initiated via the following command: TB,CDM,MAT,NTEMPS,NPTS,TBOPT
89
Material Models The material constants for each valid TBOPT value follow: Modified Ogden-Roxburgh Pseudo-Elastic TBOPT = PSE2 Constant C1 C2 C3 Meaning r m Property Damage variable parameter Damage variable parameter Damage variable parameter
For more information, see: The CDM argument and associated specifications in the TB command documentation Mullins Effect Material Model in the Structural Analysis Guide Mullins Effect in the Mechanical APDL Theory Reference.
3.6.12.4. User-Defined Hyperelastic Material

You can define a strain energy potential by using the option TB,HYPER,,,,USER. This allows you to provide a subroutine USERHYPER to define the derivatives of the strain energy potential with respect to the strain invariants. Refer to the Guide to User-Programmable Features for a detailed description on writing a user hyperelasticity subroutine. See User-Defined Hyperelastic Option (TB,HYPER,,,,USER) in the Structural Analysis Guide for more information on this material option.
3.7. Viscoelasticity
Viscoelastic materials are characterized by a combination of elastic behavior, which stores energy during deformation, and viscous behavior, which dissipates energy during deformation. The elastic behavior is rate-independent and represents the recoverable deformation due to mechanical loading. The viscous behavior is rate-dependent and represents dissipative mechanisms within the material. A wide range of materials (such as polymers, glassy materials, soils, biologic tissue, and textiles) exhibit viscoelastic behavior. Following are descriptions of the viscoelastic constitutive models, which include both small- and largedeformation formulations. Also presented is time-temperature superposition for thermorheologically simple materials and a harmonic domain viscoelastic model. 3.7.1. Viscoelastic Formulation 3.7.2.Time-Temperature Superposition 3.7.3. Harmonic Viscoelasticity For additional information, see Viscoelasticity in the Structural Analysis Guide.
90
Viscoelasticity
3.7.1. Viscoelastic Formulation

The following formulation topics for viscoelasticity are available: 3.7.1.1. Small Deformation 3.7.1.2. Small Strain with Large Deformation 3.7.1.3. Large Deformation
3.7.1.1. Small Deformation

The following figure shows a one dimensional representation of a generalized Maxwell solid. It consists of a spring element in parallel with a number of spring and dashpot Maxwell elements. Figure 3.17: Generalized Maxwell Solid in One Dimension
The spring stiffnesses are i, the dashpot viscosities are i , and the relaxation time is defined as the ratio of viscosity to stiffness, i = i / i. In three dimensions, the constitutive model for a generalized Maxwell model is given by: =
(3.13)
where: = Cauchy stress e = deviatoric strain = volumetric strain = past time I = identity tensor and G(t) and K(t) are the Prony series shear and bulk-relaxation moduli, respectively: n G = G + i i =1 K = K + = G i K (3.14)
(3.15)
91
Material Models where: G0, K0 = relaxation moduli at t = 0 nG, nK = number of Prony terms iG, iK = relative moduli iG, iK = relaxation time For use in the incremental finite element procedure, the solution for Equation 3.13 (p. 91) at t1 = t0 + t is:
i 1 = i( 0)
t + G G t 0 i i K + K
1 G i K
(3.16)
= ( )
(3.17)
where si and pi are the deviatoric and pressure components, respectively, of the Cauchy stress for each Maxwell element. The midpoint rule is used to approximate the integrals:

= ( )
(3.18)
= ( )
+
(
(3.19)
The model requires input of the parameters in Equation 3.14 (p. 91) and Equation 3.15 (p. 91). The relaxation moduli at t = 0 are obtained from the elasticity parameters input using the MP command or via an elastic data table (TB,ELASTIC). The Prony series relative moduli and relaxation times are input via a Prony data table (TB,PRONY), and separate data tables are necessary for specifying the bulk and shear Prony parameters. For the shear Prony data table, TBOPT = SHEAR, NPTS = nG, and the constants in the data table follow this pattern: Table Location 1 2 ... 2(NPTS - 1) 2(NPTS) Constant 1G 1G ... nGG nGG
For the bulk Prony data table, TBOPT = BULK, NPTS = nK, and the constants in the data table follow this pattern: Table Location 1
92
Constant 1K
Viscoelasticity Table Location 2 ... 2(NPTS - 1) 2(NPTS) Constant 1K ... nKK nKK
3.7.1.2. Small Strain with Large Deformation

This model is used when the large-deflection effects are active (NLGEOM,ON). To account for large displacement, the model is formulated in the co-rotated configuration using the co-rotated deviatoric stress = RTsR, where R is the rotation obtained from the polar decomposition of the deformation gradient. The pressure component of the Cauchy stress does not need to account for the material rotation and uses the same formulation as the small-deformation model. The deviatoric stress update is then expressed as:
i
( )
0
G i

G i
(3.20)
where R = R(t1)RT(t0) is the incremental rotation. Parameter input for this model resembles the input requirements for the small-deformation viscoelastic model.
3.7.1.3. Large Deformation

The large-strain viscoelastic constitutive model is a modification of the model proposed by Simo. Modifications are included for viscoelastic volumetric response and the use of time-temperature superposition. The linear structure of the formulation is provided by the generalized Maxwell model. Extension to large-deformation requires only a hyperelastic model for the springs in the Maxwell elements. Hyperelasticity is defined by a strain energy potential where, for isotropic materials: = where: = right Cauchy-Green deformation tensor = isochoric part of C = determinant of the deformation gradient The second Piola-Kirchhoff stress in the Maxwell element springs is then: =
(3.21)
(3.22)
and the large-deformation stress update for the Maxwell element stresses is given by:
93
Material Models + iG G i K + K G i K t
( 1) =
( 0)
(3.23)
( ) = ( )
(3.24)
where:

= =
deviatoric component of Si pressure component of Si
An anisotropic hyperelastic model can also be used for Equation 3.21 (p. 93) , in which case the form of the Maxwell element stress updates are unchanged. This model requires the Prony series parameters to be input via the Prony data table (as described in Small Deformation (p. 91)). The hyperelastic parameters for this model are input via a hyperelastic data table (TB,HYPER). For more information, see Hyperelasticity (p. 77).
3.7.2. Time-Temperature Superposition

For thermorheologically simple materials, the influence on the material behavior due to changing temperature is the same as that due to changing time. For these materials, a rate-dependent material response, P (a function of temperature and time), can be reduced to: = (3.25) r where: T = current temperature Tr = constant reference temperature = shifted time given by = t / A(T), where A(T) = shift function. The constitutive equations are solved in the shifted time scale. This method has the potential to reduce the experimental effort required to determine the material parameters but requires the determination of the shift function. The following forms of the shift function are available: 3.7.2.1. Williams-Landel-Ferry Shift Function 3.7.2.2.Tool-Narayanaswamy Shift Function 3.7.2.3. User-Defined Shift Function
3.7.2.1. Williams-Landel-Ferry Shift Function

The Williams-Landel-Ferry shift function has the form:
)=
) 2 +( )
(3.26)
where C1 and C2 are material parameters. (The shift function is often given in the literature with the opposite sign.) The parameters are input via a shift function data table (TB,SHIFT).
94
Viscoelasticity For the Williams-Landel-Ferry shift function, TBOPT = WLF, and the required input constants are: Table Location 1 2 3 Constant Tr C1 C2
3.7.2.2. Tool-Narayanaswamy Shift Function

Two forms of the Tool-Narayanaswamy shift function are available, one of which includes a fictive temperature. The first form is given by:
)=
(3.27)
where
is the scaled activation energy.
The parameters are input in a shift function data table (TB,SHIFT). For the Tool-Narayanaswamy shift function, TBOPT = TN, and the required input constants are: Table Location 1 2 Constant Tr
The second form of the Tool-Narayanaswamy shift function includes an evolving fictive temperature. The fictive temperature is used to model material processes that contain an intrinsic equilibrium temperature that is different from the ambient temperature of the material. The shift function is given by:
(
where:
)=
(3.28)
X = weight parameter TF = fictive temperature. The evolving fictive temperature is given by:
i =1
fi
fi
(3.29)
where: nf = number of partial fictive temperatures Cfi = fictive temperature relaxation coefficient
95
Material Models Tfi = partial fictive temperature The evolution of the partial fictive temperature is given by:
fi
fi
0 fi
+
0 F
0 F
fi +
(3.30)
where: fictive temperature relaxation time 0 (superscript) = values from the previous time step The fictive temperature model modifies the volumetric thermal strain model and gives an incremental thermal strain as: T = g + l
(3.31)
where g and l are the glass and liquid coefficients, respectively, of thermal expansion given by: = + 1 + 2 = + +
2
+ 3
+ 4
(3.32) (3.33)
The parameters are input in a shift function data table (TB,SHIFT). For the Tool-Narayanaswamy with fictive temperature shift function, TBOPT = FICT, NPTS = nf, and the required input constants are: Table Location 1 2 3 4 to 3(NPTS + 1) 3(NPTS + 1) + 1 to 3(NPTS + 1) + 5 3(NPTS + 1) + 6 to 3(NPTS + 1) + 10 Constant Tr H/R X Tf1, Cf1, f1, Tf2, Cf2, f2, ..., Tfn, Cfn, fn g0, g1, g2, g3, g4 l0, l1, l2, l3, l4
3.7.2.3. User-Defined Shift Function

Other shift functions can be accommodated via the user-provided subroutine UsrShift, described in the Programmer's Reference. Given the input parameters, the routine must evolve the internal state variables, then return the current and half-step shifted time.
3.7.3. Harmonic Viscoelasticity

For use in harmonic analyses, the generalized Maxwell model can be used to provide a constitutive model in the harmonic domain.
96
Viscoelasticity Assuming that the strain varies harmonically and that all transient effects have subsided, Equation 3.13 (p. 91) has the form: = where: = deviatoric and volumetric components of strain

+ +
(3.34)
= storage and loss shear moduli = storage and loss bulk moduli
= frequency and phase angle Comparing Equation 3.34 (p. 97) to the harmonic equation of motion, the material stiffness is due to the storage moduli and the material damping matrix is due to the loss moduli divided by the frequency. The following additional topics for harmonic viscoelasticity are available: 3.7.3.1. Prony Series Complex Modulus 3.7.3.2. Experimental Data Complex Modulus 3.7.3.3. Frequency-Temperature Superposition 3.7.3.4. Stress
3.7.3.1. Prony Series Complex Modulus

The storage and loss moduli are related to the Prony parameters by:
n G G 2 = 0 iG i i G 2 i = 1 + i K = K + =
n G G = 0 i i G 2 i =1 + i
(3.35)
K K
K K = K = +
(3.36)
Input of the Prony series parameters for a viscoelastic material in harmonic analyses follows the input method for viscoelasticity in the time domain detailed above.
3.7.3.2. Experimental Data Complex Modulus

Storage and loss moduli can also be input as piecewise linear functions of frequency on a data table for experimental data. Isotropic elastic moduli can be input for the complex shear, bulk and tensile modulus as well as the complex Poisson's ratio. The points for the experimental data table (input via the TBPT command) have frequency as the independent variable, and the dependent variables are the real component, imaginary component, and tan(). If the imaginary component is empty or zero for the data point, the tan() value is used to determine it; otherwise tan() is not used. Input complex shear modulus on an experimental data table (TB,EXPE) with TBOPT = GMODULUS. The data points are defined by:
97
Material Models Position 1 2 3 4 = Value

Input complex bulk modulus on an experimental data table (TB,EXPE) with TBOPT = KMODULUS. The data points are defined by: Position 1 2 3 4 = Value

Input complex tensile modulus on an experimental data table (TB,EXPE) with TBOPT = EMODULUS. The data points are defined by: Position 1 2 3 4 = Value

Input complex Poisson's ratio on an experimental data table (TB,EXPE) with TBOPT = NUXY. The data points are defined by: Position 1 2 3 4 = Value
98
Microplane Using experimental data to define the complex constitutive model requires elastic constants (defined via MP or by an elastic data table [TB,ELASTIC]). The elastic constants are unused if two sets of complex modulus experimental data are defined. This model also requires an empty Prony data table (TB,PRONY) with TBOPT = EXPERIMENTAL. Two elastic constants are required to define the complex constitutive model. If only one set of experimental data for a complex modulus is defined, the Poisson's ratio (defined via MP or by elastic data table) is used as the second elastic constant.
3.7.3.3. Frequency-Temperature Superposition

For thermorheologically simple materials in the frequency domain, frequency-temperature superposition is analogous to using time-temperature superposition to shift inverse frequency. The Williams-LandelFerry and Tool-Narayanaswamy (without fictive temperature) shift functions can be used in the frequency domain, and the material parameter input follows the shift table input described in Time-Temperature Superposition (p. 94). Frequency-temperature superposition can be used with either the Prony series complex modulus or any of the experimental data for complex moduli or Poisson's ratio.
3.7.3.4. Stress
The magnitude of the real and imaginary stress components are obtained from expanding Equation 3.34 (p. 97) and using the storage and loss moduli from either the Prony series parameters or the experimental data: = = where: Re() = real stress magnitude Im() = imaginary stress magnitude
( (
+ +
) ( ) + ( +
) )
(3.37) (3.38)
3.8. Microplane
The microplane model (TB,MPLANE) is based on research by Bazant and Gambarova [1][2] in which the material behavior is modeled through uniaxial stress-strain laws on various planes. Directional-dependent stiffness degradation is modeled through uniaxial damage laws on individual potential failure planes, leading to a macroscopic anisotropic damage formulation. The model is well suited for simulating engineering materials consisting of various aggregate compositions with differing properties (for example, concrete modeling, in which rock and sand are embedded in a weak matrix of cements). The microplane model cannot be combined with other material models. The following topics concerning the microplane material model are available: 3.8.1. Microplane Modeling 3.8.2. Material Models with Degradation and Damage 3.8.3. Material Parameters Definition and Example Input
99
Material Models 3.8.4. Learning More About Microplane Material Modeling Also see Material Model Element Support (p. 5) for microplane.
3.8.1. Microplane Modeling

Microplane theory is summarized in three primary steps. 1. Apply a kinematic constraint to relate the macroscopic strain tensors to their microplane counterparts. 2. Define the constitutive laws on the microplane levels, where unidirectional constitutive equations (such as stress and strain components) are applied on each microplane. 3. Relate the homogenization process on the material point level to derive the overall material response. (Homogenization is based on the principle of energy equivalence.) The microplane material model formulation is based on the assumption that microscopic free energy mic on the microplane level exists and that the integral of mic over all microplanes is equivalent to a macroscopic free Helmholtz energy mac [3], expressed as: mac =
mic
The factor
results from the integration of the sphere of unit radius with respect to the area .
The strains and stresses at microplanes are additively decomposed into volumetric and deviatoric parts, respectively, based on the volumetric-deviatoric (V-D) split. The strain split is expressed as: = D + v The scalar microplane volumetric strain v results from: = = 1
where V is the second-order volumetric projection tensor and 1 the second-order identity tensor. The deviatoric microplane strain vector D is calculated as: = : = 1 1 = de
where is the fourth-order identity tensor and the vector n describes the normal on the microsphere (microplane). The macroscopic strain is expressed as:
100
Microplane = ( v +
T
D )
The stresses can then be derived as = mic = ( +
where v and D are the scalar volumetric stress and the deviatoric stress tensor on the microsphere, and = = .
Assume isotropic elasticity: = and =

=
=
where Kmic and Gmic are microplane elasticity parameters and can be interpreted as a sort of microplane bulk and shear modulus. The integrals of the macroscopic strain (p. 100) equation and the derived stresses (p. 101) equation are solved via numerical integration: =

( ) =1
Np
where wi is the weight factor.
3.8.1.1. Discretization
Discretization is the transfer from the microsphere to microplanes which describe the approximate form of the sphere. Forty-two microplanes are used for the numerical integration. Due to the symmetry of the microplanes (where every other plane has the same normal direction), 21 microplanes are considered and summarized.[3] The following figure illustrates the discretization process:
101
Material Models Figure 3.18: Sphere Discretization by 42 Microplanes
3.8.2. Material Models with Degradation and Damage

To account for material degradation and damage, the microscopic free-energy function is modified to include a damage parameter, yielding: mic ( v D
mic ) = (
mic ) mic ( v D )
mic
The damage parameter d
{ is the normalized damage variable

T )
}.
The stresses are derived by: = = ( +
where
) = (
and
) = (
) .
The damage status of a material is described by the equivalent-strain-based damage function , where mic is the equivalent strain energy, which characterizes the damage evolution law and is defined as: a = =
22 01 + 1 1 + 2 2
where I1 is the first invariant of the strain tensor , J2 is the second invariant of the deviatoric part of the strain tensor , and k0, k1, and k2 are material parameters that characterize the form of damage function. The equivalent strain function (p. 102) implies the Mises-Hencky-Huber criterion for k0 = k1 = 0, and k2 = 1, and the Drucker-Prager-criterion for k0 > 0, k1 = 0, and k2 = 1. The damage evolution is modeled by the following function:
=
o +
( ( o ) )
102
Microplane where mic defines the maximal degradation, mic determines the rate of damage evolution, and characterizes the equivalent strain energy on which the material damaging starts (damage starting boundary). The following figure shows the evolution of the damage variable d as a function of equivalent strain energy mic for the implemented exponential damage model: Figure 3.19: Damage Parameter d Depending on the Equivalent Strain Energy mic 0
This figure shows the stress-strain behavior for uniaxial tension:
103
Material Models Figure 3.20: Stress-strain Behavior at Uniaxial Tension
3.8.3. Material Parameters Definition and Example Input

mic The material parameters in the model are: E, , k0, k1, k2, 0 , and mic. E is Youngs modulus and is Poissons ratio. Both are microplane elastic properties and are defined via the MP command. The parameters k0, k1, k2, , and mic are defined via the TB command (TB,MPLANE). The command syntax is: TB,MPLAN,MAT,NTEMP,NPTS,TBOPT TBDATA,1,C1,C2,C3,C4,C5,C6 The following table describes the material constants: Constant C1 C2 C3 C4 C5 C6 Meaning k0 k1 k2 mic
mic
Property Damage function constant Damage function constant Damage function constant Critical equivalent-strain-energy density Maximum damage parameter Scale for rate of damage
104
Porous Media Example 3.18: Microplane Material Constant Input

Define elastic properties of material MP,EX,1,60000.0 MP,NUXY,1,0.36 Define microplane model properties TB,MPLANE,1,,6 TBDATA,1,0,0,1,0.1,0.1,0.1
3.8.4. Learning More About Microplane Material Modeling

The following list of resources offers more information about microplane material modeling: 1. Bazant, Z. P., P.G. Gambarova . Crack Shear in Concrete: Crack Band Microplane Model. Journal of Structural Engineering . 110 (1984): 2015-2036. 2. Bazant, Z. P., B. H. Oh. Microplane Model for Progressive Fracture of Concrete and Rock. Journal for Engineering Mechanics . 111 (1985): 559-582. 3. Leukart, M., E. Ramm. A Comparison of Damage Models Formulated on Different Material Scales. Computational Materials Science. 28.3-4 (2003): 749-762.
3.9. Porous Media

3.9.1. Coupled Pore-Fluid Diffusion and Structural Model of Porous Media
Issue the TB,PM command to define material model constants for a porous medium. Fluid permeability (PERM) and Biot coefficient (BIOT) options are available. Material constants for TBOPT = PERM Constant C1 C2 C3 Meaning kx ky kz Property Permeability coefficient Permeability coefficient Permeability coefficient Units -
Material constants for TBOPT = BIOT Constant C1 C2 Meaning km Property Biot coefficient Biot modulus Units Dimensionless Defaults to zero.
For more information, see: The PM argument and associated specifications in the TB command documentation Pore-Fluid-Diffusion-Structural Analysis in the Coupled-Field Analysis Guide Porous Media Flow in the Mechanical APDL Theory Reference
105
Material Models
3.9.2. Johnson-Champoux-Allard Equivalent Fluid Model of a Porous Media

Issue the TB,PERF command in an acoustic full harmonic analysis to define a Johnson-Champoux-Allard equivalent fluid model of a porous medium. The effective density is given by: = 0 + 0 where: 0 = density of fluid = fluid resistivity = porosity = tortuosity = viscous characteristic length = dynamic viscosity The effective bulk modulus is given by: = + rt where: = specific heat ratio P0 = static reference pressure Prt = Prandtl number ' = thermal characteristic length The constants (C1-C5) entered on TBDATA are: Constant C1 C2 C3 C4 C5 Meaning Fluid resistivity (N s/m4) Porosity (defaults to 1) Tortuosity (defaults to 1) Viscous characteristic length (m) Thermal characteristic length (m) 2 0 + 2 22
1/ 2
(3.39)
rt
(3.40)
Additional material parameters are input with the MP and R commands. For more information, see Equivalent Fluid of Perforated Materials in the Mechanical APDL Theory Reference.
3.10. Electricity and Magnetism

The following material model topics related to electricity and magnetism are available: 3.10.1. Piezoelectricity 3.10.2. Piezoresistivity
Electricity and Magnetism 3.10.3. Magnetism 3.10.4. High-Frequency Electromagnetism 3.10.5. Anisotropic Electric Permittivity
3.10.1. Piezoelectricity
Piezoelectric capability (TB,PIEZ) is available with the coupled-field elements. (See Material Model Element Support (p. 5) for piezoelectricity.) Material properties required for the piezoelectric effects include the dielectric (relative permittivity) constants, the elastic coefficient matrix, and the piezoelectric matrix. Input the dielectric constants either by specifying orthotropic dielectric permittivity (PERX, PERY, PERZ) on the MP command or by specifying the terms of the anisotropic permittivity matrix [] on the TB,DPER command. The values input on the MP command will be interpreted as permittivity at constant strain [S]. Using TB,DPER, you can specify either permittivity at constant strain [S] (TBOPT = 0), or permittivity at constant stress [T] (TBOPT = 1). Input the elastic coefficient matrix [c] either by specifying the stiffness constants (EX, EY, etc.) with MP commands, or by specifying the terms of the anisotropic elasticity matrix with TB commands as described in Anisotropy. You can define the piezoelectric matrix in [e] form (piezoelectric stress matrix) or in [d] form (piezoelectric strain matrix). The [e] matrix is typically associated with the input of the anisotropic elasticity in the form of the stiffness matrix [c], and the permittivity at constant strain [S]. The [d] matrix is associated with the input of compliance matrix [s] and permittivity at constant stress [T]. Select the appropriate matrix form for your analysis using the TB,PIEZ command. The full 6 x 3 piezoelectric matrix relates terms x, y, z, xy, yz, xz to x, y, z via 18 constants as shown:
11 21 31 41 51 61 12 22 32 42 52 62 13 33 43 53 63 23
For 2-D problems, a 4 x 2 matrix relates terms ordered x, y, z, xy via 8 constants (e11, e12, e21, e22, e31, e32, e41, e42). The order of the vector is expected as {x, y, z, xy, yz, xz}, whereas for some published materials the order is given as {x, y, z, yz, xz, xy}. This difference requires the piezoelectric matrix terms to be converted to the expected format. You can define up to 18 constants (C1-C18) with TBDATA commands (6 per command): Constant C1-C6 C7-C12 C13-C18 Meaning Terms e11, e12, e13, e21, e22, e23 Terms e31, e32, e33, e41, e42, e43 Terms e51, e52, e53, e61, e62, e63
107
Material Models See Piezoelectric Analysis in the Coupled-Field Analysis Guide for more information on this material model.
3.10.2. Piezoresistivity
Elements with piezoresistive capabilities use the TB,PZRS command to calculate the change in electric resistivity produced by elastic stress or strain. Material properties required to model piezoresistive materials are electrical resistivity, the elastic coefficient matrix, and the piezoresistive matrix. You can define the piezoresistive matrix either in the form of piezoresistive stress matrix [] (TBOPT = 0) or piezoresistive strain matrix [m] (TBOPT = 1). The piezoresistive stress matrix [] uses stress to calculate the change in electric resistivity due to piezoresistive effect, while the piezoresistive strain matrix [m] (TBOPT = 1) uses strain to calculate the change in electric resistivity. See Piezoresistivity in the Mechanical APDL Theory Reference for more information. The full 6x6 piezoresistive matrix relates the x, y, z, xy, yz, xz terms of stress to the x, y, z, xy, yz, xz terms of electric resistivity via 36 constants: 11 21 31 41 51 61 Constant C1-C6 C7-C12 C13-C18 C19-C24 C25-C30 C31-C36 12 22 32 42 52 62 13 23 33 43 53 63 14 24 34 44 54 64 15 25 35 45 55 65 16 26 36 46 56 66
Meaning Terms 11, 12, 13, 14, 15, 16 Terms 21, 22, 23, 24, 25, 26 Terms 31, 32, 33, 34, 35, 36 Terms 41, 42, 43, 44, 45, 46 Terms 51, 52, 53, 54, 55, 56 Terms 61, 62, 63, 64, 65, 66
For 2-D problems, a 4x4 matrix relates terms ordered x, y, z, xy via 16 constants. Constant C1-C4 C7-C10 C13-C16 C19-C22 Meaning Terms 11, 12, 13, 14 Terms 21, 22, 23, 24 Terms 31, 32, 33, 34 Terms 41, 42, 43, 44
The order of the vector is expected as {x, y, z, xy, yz, xz}, whereas for some published materials the order is given as {x, y, z, yz, xz, xy}. This difference requires the piezoresistive matrix terms to be converted to the expected format.
108
Electricity and Magnetism See Piezoresistive Analysis in the Coupled-Field Analysis Guide for more information on this material model.
3.10.3. Magnetism
Elements with magnetic capability use the TB table to input points characterizing B-H curves. Temperature-dependent curves cannot be input. Initialize the curves with the TB,BH command. Use TBPT commands to define up to 500 points (H, B). The constants (X, Y) entered on TBPT (two per command) are: Constant X Y Meaning Magnetomotive force/length Flux/Area Property Magnetic field intensity (H) Corresponding magnetic flux density (B)
Specify the system of units (MKS or user defined) with EMUNIT, which also determines the value of the permeability of free space. This value is used with the relative permeability property values (MP) to establish absolute permeability values. The defaults (also obtained for Lab = MKS) are MKS units and free-space permeability of 4 E-7 Henries/meter. You can specify Lab = MUZRO to define any system of units, then input free-space permeability. For more information about this material option, see Additional Guidelines for Defining Regional Material Properties and Real Constants in the Low-Frequency Electromagnetic Analysis Guide
3.10.4. High-Frequency Electromagnetism

Specify high-frequency electromagnetic material properties using the TB command. The following topics provide specific information about the various high-frequency material model (Lab) options available: 3.10.4.1. 3-D High-Frequency Material Properties 3.10.4.2. 2-D High-Frequency Material Properties
3.10.4.1. 3-D High-Frequency Material Properties

High-frequency elements use the TB table to specify material properties such as anisotropic electric current conductivity (mhos/m)(1/m), anisotropic magnetic current conductivity (ohms/m), anisotropic relative permittivity, anisotropic relative permeability, anisotropic electric and magnetic loss tangents, and frequency-dependent lossy dielectric. For a list of the elements that support 3-D high-frequency material properties, see Material Model Element Support (p. 5).
3.10.4.1.1. Conductivity, Permittivity, and Permeability Matrices

For electric current conductivity (TB,CNDE), magnetic current conductivity (CNDM), relative permittivity (DPER), and relative permeability (TB,MUR,MAT,,,TBOPT with TBOPT = 0 input a permeability matrix), a 3 x 3 matrix relates terms ordered by x, y, and z.
109
Material Models = 23 33
11 21 31
12 22 32
13
The constants (C1-C9) entered on the TBDATA command are: Constant C1-C9 Meaning X11, X22, X33, X12, X23, X13, X21, X32, X31
If Xij is 0 where i and j are indexes, then Xji must also be zero. For TB, DPER and TB,MUR the diagonal elements cannot be zero.
3.10.4.1.2. B-H Nonlinear Material Permeability Matrix

TB,MUR,MAT,,,TBOPT with TBOPT = 1 or 2 generates a B-H nonlinear material permeability matrix with a uniform or nonuniform dc magnetic field, respectively. A nonuniform dc magnetic field can be the solution of a magneto static analysis. The matrix is give by the following equations for a static magnetic field (Ho) in the x, y, and z directions, respectively. r = = = where: = + = m 0 + 0 + " # +
& s $ !
(Fo H in x-diection)
y

z
% = % ' = * = +
110
Electricity and Magnetism = 0 and is the gyromagnetic ratio 0 is the electron gyromagnetic ratio g is the Lande factor Ho is the static magnetic field in the z, y, or x direction Ms is the saturation magnetization introduced by Ho H is the resonance line width is the working angular frequency The constants (C1-C7) entered on TBDATA are: Constant C1 C2 C3 C4 C5 Meaning Saturation magnetization 4Ms (Gauss) (no default). Lande g-factor (1.8 to 2.5, defaults to 2.0). Resonance line width H (Oe) (defaults to 0). Internal dc magnetic field Ho (Oe) (no default). Direction of Ho. 0 - z direction (default) 1 - y direction (default) 2 - x direction (default) C6 Sign of off-diagonal element of permeability matrix.
C6 ,
r r = r (default)
<
= C7

Symmetry key for permeability matrix. 0 antisymmetric matrix (default)
111
Material Models Constant Meaning 1 symmetric matrix
3.10.4.1.3. Anisotropic Electric and Magnetic Loss Tangents

You can specify anisotropic electric and magnetic loss tangent matrices using TB,LSEM. The anisotropic electric loss tangent matrix is defined as: ,1 r [ ] = 0 where: = = = 1 3
,2 r
,3 r
The anisotropic magnetic loss tangent matrix is defined as: [ ] = where:

=
=
= m m
The constants (C1-C6) entered on TBDATA are: Constant C1 C2 C3 C4 C5 C6 Meaning Electric loss tangent in the X direction, tan1 Electric loss tangent in the Y direction, tan2 Electric loss tangent in the Z direction, tan3 Magnetic loss tangent in the X direction, tanm1 Magnetic loss tangent in the Y direction, tanm2 Magnetic loss tangent in the Z direction, tanm3
3.10.4.1.4. Frequency-Dependent Lossy Dielectric

You can specify a frequency-dependent lossy dielectric using TB,HFFDLD. In this Debye relaxation polarization model, the complex permittivity is given by:
112
Electricity and Magnetism r,complex = r + where: = relaxation time = optical permittivity rs = static permittivity = working angular frequency The frequency-dependent lossy dielectric is characterized by a dielectric constant (r) and a loss tangent (tan) at two frequencies. In terms of Maxwell's equations, the real parts of the dielectric constant and the conductivity are given by: = + + 2 rs r +
= 0 + where:
0 +
0 = DC conductivity 0 = free space permittivity r and are determined by the four parameters: rs, 0, , and . Experimentally measured values of the dielectric constant and loss tangent are usually available at two frequencies: r1 and tan1 at a lower frequency 1 of approximately 1 MHz r2 and tan2 at a higher frequency 2 between 1 and 2 GHz The lower frequency data is considered static or DC values. Accordingly, the static permittivity and DC conductivity are given by: rs = r1 0 = 10r1tan1 If is known from experimental measurements, the Debye's model can then be completely defined by calculating the relaxation time by: =
where: =
and =
113
Material Models Knowing , the material characteristics can be determined over the entire frequency range. If r is unknown, the following equations (based on the higher frequency data) can be solved simultaneously for and . = 0 s 2 0
+
= where: 2 = 20r2tan2
The constants (C1-C8) entered on TBDATA are: Constant C1 C2 C3 C4 C5 Meaning Lower frequency (f1) at which measured data is considered static or DC values. Higher frequency (f2) at which measured data is available. Relative permittivity at lower frequency (r1). Relative permittivity at lower frequency (r2). Relative permittivity at optical frequency ( ). Input if known. Calculated if it is not known. DC conductivity (0). It does not have to be defined if the loss tangent at lower frequency is defined. Loss tangent at lower frequency (tan1). It does not have to be defined if the DC conductivity is defined. Loss tangent at higher frequency (tan2).
C6 C7 C8
3.10.4.2. 2-D High-Frequency Material Properties

The high-frequency 2-D modal analysis element HF118 uses the TB table to specify the following material properties: TB,DPER anisotropic relative permittivity TB,MUR,MAT,,,TBOPT with TBOPT = 0 anisotropic relative permeability The element must lie in an X-Y plane and a 3 x 3 matrix relates terms ordered by x, y, and z.
114
Electricity and Magnetism xx = zz
yy
The constants (C1-C9) entered on the TBDATA command are: Constant C1-C9 Meaning Xxx, Xyy, Xzz, Xxy, 0, 0, Xyx, 0, 0
Note
HF118 can not use the TB table to specify the following: Anisotropic electric current conductivity (TB,CNDE) Anisotropic magnetic current conductivity (TB,CNDM) B-H nonlinear material permeability matrix with a uniform or nonuniform dc magnetic field (TB,MUR,MAT,,,TBOPT with TBOPT = 1 or 2).
3.10.5. Anisotropic Electric Permittivity

Elements with piezoelectric capabilities use the TB,DPER command to specify anisotropic relative electric permittivity. You can define electric permittivity at constant strain [S] (TBOPT = 0) or constant stress [T] (TBOPT = 1) The program converts matrix [T] to [S] using piezoelectric strain and stress matrices. The full 3x3 electric permittivity matrix relates x, y, z components of electric field to the x, y, z components of electric flux density via 6 constants:
11 12 13 22 23 33 sm
Constant C1-C6
Meaning 11, 22, 33, 12, 23, 13
For 2-D problems, a 2x2 matrix relates terms ordered x, y via 3 constants (11 22 12): Constant C1, C2, C4 Meaning 11, 22, 12
115
Material Models
3.11. Gasket
The gasket model (TB,GASKET) allows you to simulate gasket joints with the interface elements. The gasket material is usually under compression and is highly nonlinear. The material also exhibits quite complicated unloading behavior when compression is released. You can define some general parameters including the initial gap, stable stiffness for numerical stabilization, and stress cap for a gasket in tension. You can also directly input data for the experimentally measured complex pressure closure curves for the gaskets. Sub-options are also available to define gasket unloading behavior including linear and nonlinear unloading. Linear unloading simplifies the input by defining the starting closure at the compression curves and the slope. Nonlinear unloading option allows you to directly input unloading curves to more accurately model the gasket unloading behavior. When no unloading curves are defined, the material behavior follows the compression curve while it is unloaded. Enter the general parameters and the pressure closure behavior data via the TBOPT option on the TB,GASKET command. Input the material data (TBDATA or TBPT) as shown in the following table: Gasket Data Type TBOPT Constants Meaning Initial gap (default = 0, meaning there is no initial gap). Stable stiffness (default = 0, meaning there is no stable stiffness. [1] Maximum tension stress allowed when the gasket material is in tension (default = 0, meaning there is no tension stress in the gasket material). Closure value. Pressure value.
TB,GASKET,,,2,COMP TBPT,,X1,Y1 TBPT,,X2,Y2
Input Format
C1
C2 General parameters PARA
TB,GASKET,,,,PARA TBDATA,1,C1,C2,C3
C3
Compression load closure curve Linear unloading data Nonlinear unloading data [2] Transverse shear
Xi COMP Yi
Xi LUNL Yi Xi NUNL TSS Yi XY, XZ
Closure value on compression curve where unloading started. Unloading slope value. Closure value. Pressure value. Transverse shear values
TB,GASKET,,,2,LUNL TBPT,,X1,Y1 TBPT,,X2,Y2
TB,GASKET,,,2,NUNL TBPT,,X1,Y1 TBPT,,X2,Y2 TB,GASKET,,,2,TSS TBDATA,1,TSSXY,TSSXZ
1. Stable stiffness is used for numerical stabilization such as the case when the gasket is opened up and thus no stiffness is contributed to the element nodes, which in turn may cause numerical difficulty.
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Swelling 2. Multiple curves may be required to define the complex nonlinear unloading behavior of a gasket material. When there are several nonlinear unloading curves defined, the program requires that the starting point of each unloading curve be on the compression curve to ensure the gasket unloading behavior is correctly simulated. Though it is not a requirement that the temperature dependency of unloading data be the same as the compression data, when there is a missing temperature, the program uses linear interpolation to obtain the material data of the missing temperature. This may result in a mismatch between the compression data and the unloading data. Therefore, it is generally recommended that the number of temperatures and temperature points be the same for each unloading curve and compression curve. When using the material GUI to enter data for the nonlinear unloading curves, an indicator at the top of the dialog box states the number of the unloading curve whose data is currently displayed along with the total number of unloading curves defined for the particular material (example: Curve number 2/5). To enter data for the multiple unloading curves, type the data for the first unloading curve, then click on the Add Curve button and type the data for the second curve. Repeat this procedure for entering data for the remaining curves. Click the Del Curve button if you want to remove the curve whose data is currently displayed. Click the > button to view the data for the next curve in the sequence, or click the < button to view the data for the previous curve in the sequence. To insert a curve at a particular location in the sequence, click on the > or < buttons to move to the curve before the insertion location point and click on the Add Curve button. For example, if the data for Curve number 2/5 is currently displayed and you click on the Add Curve button, the dialog box changes to allow you to enter data for Curve number 3/6. You can define a total of 100 nonlinear unloading curves per material. You can enter temperature-dependent data (TBTEMP) for any of the gasket data types. For the first temperature curve, issue TB,GASKET,,,,TBOPT, then input the first temperature using TBTEMP, followed by the data using either TBDATA or TBPT depending on the value of TBOPT as shown in the table. The program automatically interpolates the temperature data to the material points using linear interpolation. When the temperature is out of the specified range, the closest temperature point is used. For more information, see Gasket Material in the Mechanical APDL Theory Reference. For a detailed description of the gasket joint simulation capability, see Gasket Joints Simulation in the Structural Analysis Guide.
3.12. Swelling
Swelling (TB,SWELL) is a material enlargement (volume expansion) caused by neutron bombardment or other effects (such as moisture). The swelling strain rate is generally nonlinear and is a function of factors such as temperature, time, neutron flux level, stress, and moisture content. Irradiation-induced swelling and creep apply to metal alloys that are exposed to nuclear radiation. However, the swelling equations and the fluence input may be completely unrelated to nuclear swelling. You can also model other types of swelling behavior, such as moisture-induced volume expansion. Swelling strain is modeled using additive decomposition of strains, expressed as: = el + pl + cr + sw
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Material Models where is the total mechanical strain, el is the elastic strain, pl is the plastic strain, and sw is the swelling strain. You can combine swelling strain with other material models such as plasticity and creep; however, you can use swelling with any hyperelasticity or anisotropic hyperelasticity material model. Irradiation-induced swelling is generally accompanied by irradiation creep for metals and composites, such as silicon carbide (SiC). The irradiation-induced swelling strain rate may depend on temperature, time, fluence (the flux x time), and stress, such as: sw sw ( t )
where t is time, T is the temperature, t is the fluence, and is the stress. Temperatures used in the swelling equations should be based on an absolute scale (TOFFST). Specify temperature and fluence values via the BF or BFE command. The following options for modeling swelling are available: Linear swelling defines swelling strain rate as a function of fluence rate, expressed as:
where C is the swelling constant, which may depend on temperature. Exponential swelling defines swelling strain as a function of fluence, expressed as:
1+ 2
+ 3
A user-defined swelling option is available if you wish to create your own swelling function. For more information, see userswstrain in the Guide to User-Programmable Features. Swelling equations are material-specific and are empirical in nature. For highly nonlinear swelling strain vs. fluence curves, it is good practice to use a small fluence step for better accuracy and solution stability. If time is changing, a constant flux requires a linearly changing fluence (because the swelling model uses fluence [t] rather than flux []). Initialize the swelling table (TB,SWELL) with the desired data table option (TBOPT), as follows: Swelling Model Options (TB,SWELL,,TBOPT) Option (TBOPT) LINE EXPT USER Constant C1 C1, C2, C3, C4 C1, ..., Cn Description Linear swelling Exponential swelling User-defined Constant Value Input TBDATA,1,C1 TBDATA,1,C1,C2,C3,C4 TBDATA,1,C1,C2,
Issue the TBDATA command to enter the swelling table constants (up to six per command), as shown in the table. For a list of the elements that you can use with the swelling model, see Material Model Element Support (p. 5) For more information about this material model, see Swelling in the Structural Analysis Guide.
Shape Memory Alloy (SMA)
3.13. Shape Memory Alloy (SMA)

A shape memory alloy (SMA) is a metallic alloy that remembers its original shape. Upon loading and unloading cycles, an SMA can undergo large deformation without showing residual strains (pseudoelasticity effect, also often called superelasticity), and can recover its original shape through thermal cycles (the shape memory effect). Such distinct material behavior is due to the material microstructure in which there exists two different crystallographic structures, one characterized by austenite (A), and another one by martensite (M). Austenite is the crystallographically more-ordered phase, and martensite is the crystallographically lessordered phase. The key characteristic of an SMA is the occurrence of a martensitic phase transformation. Typically, the austenite is stable at high temperatures and low stress, while the martensite is stable at low temperatures and high stress. The reversible martensitic phase transformation results in unique effects: the pseudoelasticity (PE) and the shape memory effect (SME). As shown by (a) in the following figure, whenever L is positive, the specimen recovers its original shape completely and returns to a stress-free configuration (PE). Figure 3.21: Pseudoelasticity (PE) and Shape Memory Effect (SME)
(a) PE -- High Temperature (b) SME -- Low Temperature As shown by (b) in the figure, when L is negative, residual strains (E and E') can be observed after unloading into a stress-free configuration. If the material is heated, then eventually L becomes positive; however, the admissible configuration under a stress-free state points to A. The material therefore undergoes an inverse transformation process (SME). Nitinol A typical shape memory alloy is Nitinol, a nickel titanium (Ni-Ti) alloy discovered in the 1960s at the U.S. Naval Ordnance Laboratory (NOL). The acronym NiTi-NOL (or Nitinol) has since been commonly used when referring to Ni-Ti-based shape memory alloys. Two SMA material model options (accessed via TB,SMA) are available, one for simulating superelastic behavior and the other for simulating the shape memory effect behavior of shape memory alloys.
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Material Models The material option for superelasticity is based on Auricchio et al. [1] in which the material undergoes large-deformation without showing permanent deformation under isothermal conditions, as shown by (a) in Figure 3.21: Pseudoelasticity (PE) and Shape Memory Effect (SME) (p. 119). The material option for the shape memory effect is based on the 3-D thermomechanical model for stress-induced solid phase transformations [2] [3] [4]. The following shape memory alloy topics are available: 3.13.1. Shape Memory Alloy Model for Superelasticity 3.13.2. Shape Memory Material Model with Shape Memory Effect 3.13.3. Element Support for SMA 3.13.4. Learning More About Shape Memory Alloy
3.13.1. Shape Memory Alloy Model for Superelasticity

The following topics are available for the SMA superelasticity option: 3.13.1.1. Constitutive Model for Superelasticity 3.13.1.2. Material Parameters for the Superelastic SMA Material Model
3.13.1.1. Constitutive Model for Superelasticity

From a macroscopic perspective, the phase-transformation mechanisms involved in superelastic behavior are: 1. Austenite to martensite (A->S) 2. Martensite to austenite (S->A) 3. Martensite reorientation (S->S) Figure 3.22: Typical Superelasticity Behavior
Two of the phase transformations are considered here: A->S and S->A. The material is composed of two phases, the austenite (A) and the martensite (S). Two internal variables, the martensite fraction (S) and the austenite fraction (A), are introduced. One of them is a dependent variable, and they are assumed to satisfy the relation expressed as: S + A =
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Shape Memory Alloy (SMA) The independent internal variable chosen here is S. The material behavior is assumed to be isotropic. The pressure dependency of the phase transformation is modeled by introducing the Drucker-Prager loading function, as follows: = +
= = =
where is the material parameter, is the stress, and 1 is the identity tensor. The evolution of the martensite fraction, S, is then defined as follows: AS = S SA S where:
AS f
transormation tr ransormation
SA f
+ = = +
where
d
are the material parameters shown in the following figure:
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Material Models Figure 3.23: Idealized Stress-Strain Diagram of Superelastic Behavior
AS AS s SA s SA

< < f i > otherwie

< < <

+ = and
are the material parameters shown in Figure 3.23: Idealized Stress-Strain Diagram where of Superelastic Behavior (p. 122). The material parameter characterizes the material response in tension and compression. If tensile and compressive behaviors are the same, then = 0. For a uniaxial tension-compression test, can be related to the initial value of austenite to martensite phase transformation in tension and compression !"# ( c , respectively) as:
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Shape Memory Alloy (SMA)

AS AS c t AS AS c + t
The stress-strain relation is: = r
where D is the elastic stiffness tensor, is the transformation strain tensor, and is the material parameter shown in Figure 3.23: Idealized Stress-Strain Diagram of Superelastic Behavior (p. 122).
3.13.1.2. Material Parameters for the Superelastic SMA Material Model

To model the superelastic behavior of shape memory alloys, initialize the data table using the TB,SMA command's SUPE option. Define the elastic behavior in the austenite state (MP). The superelastic SMA option is described by six constants that define the stress-strain behavior in loading and unloading for the uniaxial stress-state. For each data set, define the temperature (TBTEMP), then define constants C1 through C6 (TBDATA). You can define up to 99 sets of temperature-dependent constants in this manner. Table 3.3: Superelastic Option Constants Constant C1 C2 C3 C4 C5 C6 Meaning
s
Property Starting stress value for the forward phase transformation Final stress value for the forward phase transformation Starting stress value for the reverse phase transformation Final stress value for the reverse phase transformation Maximum residual strain Parameter measuring the difference between material responses in tension and compression

Example 3.19: Defining Elastic Properties of the Austenite Phase

MP,EX,1,60000.0 MP,NUXY,1,0.36 Define SMA material properties TB,SMA,1,,,SUPE TBDATA,1, 520, 600, 300, 200, 0.07, 0.0
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Material Models
3.13.2. Shape Memory Material Model with Shape Memory Effect

The following topics concerning SMA and the shape memory effect are available: 3.13.2.1.The Constitutive Model for Shape Memory Effect 3.13.2.2. Material Parameters for the Shape Memory Effect Option
3.13.2.1. The Constitutive Model for Shape Memory Effect

The shape memory effect was based on a 3-D thermomechanical model for stress-induced solid phase transformations that was presented in [1] [2] [3]. Within the framework of classical irreversible thermodynamics, the model is able to reproduce all of the primary features relative to shape memory materials in a 3-D stress state. The free energy potential is set to: where: D = material elastic stiffness tensor = total strain
tr
( )
tr
( ) + ( )
tr M
' tr
2 ' tr
( )
' tr
= total transformation strain = deviatoric transformation strain
M(T) = a positive and monotonically increasing function of the temperature as (T - T0)+ in which + is the positive part of the argument (also known as Maxwell stress). = material parameter T = temperature T0 = temperature below which no twinned martensite is observed h = material parameter related to the hardening of the material during the phase transformation
( ) = indicator function introduced to satisfy the constraint on the transformation norm [1]

in which

( ) =

from which we have =

where Xtr is defined as the transformation stress. Stresses, strains, and the transformation strains are then related as follows:
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Shape Memory Alloy (SMA) =
( )
tr
Splitting the stress into deviatoric and volumetric components, we have = =
where S is the deviatoric stress and p is the volumetric stress (also called hydrostatic pressure) The transformation stress is given as follows: = M (
)+
' ' + '
where is defined by =
< L = L
where
is a maximum transformation strain.
Numerous experimental tests show an asymmetric behavior of SMA in tension and compression, and suggest describing SMA as an isotropic material with a Prager-Lode-type limit surface. Accordingly, the following yield function is assumed: = +
3 2
where Xtr is the transformation stress, J2 and J3 are the second and third invariants of transformation stress, m is a material parameter related to Lode dependency, and R is the elastic domain radius. J2 and J3 are defined as follows: =
The evolution of transformation strain is defined as: =
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Material Models where is an internal variable and is called as transformation strain multiplier. and F(Xtr) must satisfy the classical Kuhn-Tucker conditions, as follows:
tr
)=
which also reduces the problem to a constrained optimization problem. The elastic properties of austenite and martensite phase differ. During the transformation phase, the elastic stiffness tensor of material varies with the deformation. The elastic stiffness tensor L is therefore assumed to be a function of the transformation strain = ' L
, defined as:
)+
where DA is the elastic stiffness tensor of austenite phase, and DS is the elastic stiffness tensor of martensite phase. The Poissons ratio of the austenite phase is assumed to be the same as the martensite phase. When the material is in its austenite phase, D = DA, and when the material undergoes full transformation (martensite phase), D = DS. The following figure illustrates a number of the mechanical model features: Figure 3.24: Admissible Paths for Elastic Behavior and Phase Transformations
The austenite phase is associated with the horizontal region abcd. Mixtures of phases are related to the surface cdef. The martensite phase is represented by the horizontal region efgh. Point c corresponds to the nucleation of the martensite phase. Phase transformations take place only along line cf, where
3 2
. Saturated phase transformations are represented by paths on line fg. The horizontal region efgh contains elastic processes except, of course, those on line fg.
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Shape Memory Alloy (SMA) A backward Euler integration scheme is used to solve the stress update and the consistent tangent stiffness matrix required by the finite element solution for obtaining a robust nonlinear solution.
3.13.2.2. Material Parameters for the Shape Memory Effect Option

To model the shape memory effect behavior of shape memory alloys, initialize the data table using the TB,SMA commands MEFF option. Define the elastic behavior in the austenite state (MP). The shape memory effect option is described by seven constants that define the stress-strain behavior of material in loading and unloading cycles for the uniaxial stress-state and thermal loading. For each data set, define the temperature (TBTEMP), then define constants C1 through C7 (TBDATA). You can define up to 99 sets of temperature-dependent constants in this manner. Table 3.4: Shape Memory Effect Option Constants Constant C1 C2 C3 C4 C5 C6 C7 Meaning h To R
L
Property Hardening parameter Reference temperature Elastic limit Temperature scaling parameter Maximum transformation strain Martensite modulus Lode (p. 125) dependency parameter
Em m
Example 3.20: Defining Shape Memory Effect Properties of the Austenite Phase
MP,EX,1,60000.0 MP,NUXY,1,0.36 Define SMA material properties TB,SMA,1,,,MEFF TBDATA,1,1000, 223, 50, 2.1, 0.04, 45000 TBDATA,7,0.05
3.13.3. Element Support for SMA

Support for SMA material models is available only with current-technology plane, solid, and solid-shell elements where 3-D stress states are applicable (including 3-D solid elements, solid-shell elements, 2D plane strain, axisymmetric elements, and solid pipe elements).
3.13.4. Learning More About Shape Memory Alloy

A considerable body of literature exists concerning shape memory alloy material models. The following list of resources offers a wealth of information but is by no means exhaustive: 1. Auricchio, F. A Robust Integration-Algorithm for a Finite-Strain Shape-Memory-Alloy. International Journal of Plasticity. 17 (2001): 971-990.
127
Material Models 2. Souza, A. C., E. N. Mamiya, N. Zouain. Three-Dimensional Model for Solids Undergoing Stress-Induced Phase Transformations. European Journal of Mechanics-A/Solids . 17 (1998): 789-806. 3. Auricchio, F., R. L. Taylor, J. Lubliner. Shape-Memory Alloys: Macromodeling and Numerical Simulations of the Superelastic Behavior. Computational Methods in Applied Mechanical Engineering. 146, 1 (1997): 281-312. 4. Auricchio, F., L. Petrini. Improvements and Algorithmical Considerations on a Recent Three-Dimensional Model Describing Stress-Induced Solid Phase Transformations. International Journal for Numerical Methods in Engineering. 55 (2005): 1255-1284. 5. Auricchio, F., D. Fugazza, R. DesRoches. Numerical and Experimental Evaluation of the Damping Properties of Shape-Memory Alloys. Journal of Engineering Materials and Technology. 128:3 (2006): 312-319.
3.14. MPC184 Joint

The TB,JOIN option allows you to impose linear and nonlinear elastic stiffness and damping behavior or Coulomb friction behavior on the available components of relative motion of an MPC184 joint element. The stiffness and damping behaviors described here apply to all joint elements except the weld, orient, and spherical joints. The Coulomb friction behavior described here applies only to the revolute, slot, and translational joints. The TB command may be repeated with the same material ID number to specify both the stiffness and damping behavior. The following joint material models are available: 3.14.1. Linear Elastic Stiffness and Damping Behavior 3.14.2. Nonlinear Elastic Stiffness and Damping Behavior 3.14.3. Frictional Behavior
3.14.1. Linear Elastic Stiffness and Damping Behavior

Input the linear stiffness or damping behavior for the relevant components of relative motion of a joint element by specifying the terms as part of a 6 x 6 matrix with data table commands as described below. The 6 x 6 matrix for linear stiffness or damping behavior is as follows:
11 21 31 41 51 61 22 32 42 52 62 33 43 53 63 44 54 64 55 65
66
Enter the stiffness or damping coefficient of the matrix in the data table with TB set of commands. Initialize the constant table with TB,JOIN,,,STIF (for stiffness behavior) or TB,JOIN,,,DAMP (for damping behavior). Define the temperature with TBTEMP, followed by the relevant constants input with TBDATA commands. Matrix terms are linearly interpolated between temperature points. Based on the joint type, the relevant constant specification is as follows:
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MPC184 Joint Joint Element x-axis Revolute joint z-axis Revolute joint Universal joint Slot joint Point-in-plane joint Translational joint x-axis Cylindrical joint z-axis Cylindrical joint x-axis Planar joint z-axis Planar joint General joint Screw joint Constant C16 C21 C16, C18, C21 C1 C7, C8, C12 C1 C1, C4, C16 C12, C15, C21 C7, C8, C9, C12, C13, C16 C1, C2, C6, C7, C11, C21 Use appropriate entries based on unconstrained degrees of freedom. C12, C15, C21 Meaning Term D44 Term D66 Terms D44, D64, D66 Term D11 Terms D22, D32, D33 Term D11 Terms D11, D41, D44 Terms D33, D63, D66 Terms D22, D32, D42, D33, D43, D44 Terms D11, D21, D61, D22, D62, D66 --Terms D33, D63, D66
The following example shows how you would define the uncoupled linear elastic stiffness behavior for a universal joint at the two available components of relative motion, with two temperature points:
TB,JOIN,1,2,,STIF ! Activate JOIN material model with linear elastic stiffness TBTEMP,100.0 ! Define first temperature TBDATA,16,D44 ! Define constant D44 in the local ROTX direction TBDATA,21,D66 ! Define constant D66 in the local ROTZ direction TBTEMP,200.0 ! Define second temperature TBDATA,16,D44 ! Define constant D44 in the local ROTX direction. TBDATA,21,D66 ! Define constant D66 in the local ROTZ direction.
3.14.2. Nonlinear Elastic Stiffness and Damping Behavior

You can specify nonlinear elastic stiffness as a displacement (rotation) versus force (moment) curve using the TB,JOIN command with a suitable TBOPT setting. Use the TBPT command to specify the data points or specify the name of a function that defines the curve on the TB command. (Use the Function Tool to generate the specified function.) The values may be temperature-dependent. You can specify nonlinear damping behavior in a similar manner by supplying velocity versus damping force (or moment). The appropriate TBOPT labels for each joint element type are shown in the following tables. For a description of each TBOPT label, see Joint Element Specifications (JOINT) in the TB command documentation. Nonlinear Stiffness Behavior Joint Element x-axis Revolute joint z-axis Revolute joint Universal joint Slot joint Point-in-plane joint TBOPT on TB command JNSA, JNS4 JNSA, JNS6 JNSA, JNS4, and JNS6 JNSA and JNS1 JNSA, JNS2, and JNS3
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Material Models Nonlinear Stiffness Behavior Joint Element Translational joint x-axis Cylindrical joint z-axis Cylindrical joint x-axis Planar joint z-axis Planar joint General joint Screw joint TBOPT on TB command JNSA and JNS1 JNSA, JNS1, and JNS4 JNSA, JNS3, and JNS6 JNSA, JNS2, JNS3, and JNS4 JNSA, JNS1, JNS2, and JNS6 Use appropriate entries based on unconstrained degrees of freedom JNSA, JNS3, and JNS6 Nonlinear Damping Behavior Joint Element x-axis Revolute joint z-axis Revolute joint Universal joint Slot joint Point-in-plane joint Translational joint x-axis Cylindrical joint z-axis Cylindrical joint x-axis Planar joint z-axis Planar joint General joint Screw joint TBOPT on TB command JNDA, JND4 JNDA, JND6 JNDA, JND4, and JND6 JNDA and JND1 JNDA, JND2, and JND3 JNDA and JND1 JNDA, JND1, and JND4 JNDA, JND3, and JND6 JNDA, JND2, JND3, and JND4 JNDA, JND1, JND2, and JND6 Use appropriate entries based on unconstrained degrees of freedom JNDA, JND3, and JND6
The following example illustrates the specification of nonlinear stiffness behavior for a revolute joint that has only one available component of relative motion (the rotation around the axis of revolution). Two temperature points are specified.
TB,JOIN,1,2,2,JNS4 TBTEMP,100. TBPT,,rotation_value_1,moment_value_1 TBPT,,rotation_value_2,moment_value_2 TBTEMP,200.0 TBPT,,rotation_value_1,moment_value_1 TBPT,,rotation_value_2,moment_value_2
3.14.2.1. Specifying a Function Describing Nonlinear Stiffness Behavior

When specifying a function that describes the nonlinear stiffness behavior, the Function Tool allows the force to be defined as a function of temperature and relative displacement; the two independent variables are named as TEMP and DJU. Similarly, when specifying a function that describes the nonlinear damping behavior, the Function Tool allows the damping force to be defined as a function of temperature and relative velocity; the two independent variables are identified as TEMP and DJV. Example
130
Consider a function where the damping force varies with temperature and relative velocity:
MPC184 Joint F = (-0.005 * Temperature + 0.25) * Relative Velocity Define the function using the Function Editor, then retrieve and load it using the Function Loader. (The editor and the loader are both components of the Function Tool.) Assuming a function name of dampfunc, you can then use the TB command to define the joint material:
TB, JOIN, 1, , , JND4, , %dampfunc%
For more information about the Function Tool utility, see Using the Function Tool in the Basic Analysis Guide.
3.14.3. Frictional Behavior

Frictional behavior along the unrestrained components of relative motion influences the overall behavior of the Joints. You can model Coulomb friction for joint elements via the TB,JOIN command with an appropriate TBOPT label. The joint frictional behavior can be specified only for the following joints: Revolute joint, Slot joint, and Translational joint. The friction parameters are described below. Coulomb Friction Coefficient Specification There are three options for defining the Coulomb friction coefficient. Define a single value of the Coulomb friction coefficient by specifying TBOPT = MUSx, where the value of x depends on the joint under consideration. Use the TBDATA command to specify the value of the friction coefficient. Define the Coulomb friction coefficient as a function of the sliding velocity. Use TBOPT = MUSx (as stated above) and use the TBPT command to specify the data values. Use the exponential law for friction behavior. Specify TBOPT = EXPx, where the value of x depends on the joint under consideration, and use the TBDATA command to specify the values required for the exponential law. In this case, the TBDATA command format is: TBDATA, s, d, c where s is the coefficient of friction in the static regime, d is the coefficient of friction in the dynamic regime, and c is the decay coefficient. Maximum or Critical Force/Moment The maximum allowable value of critical force/moment can be specified using TBOPT = TMXx, where x depends on the joint under consideration. Elastic Slip The elastic slip can be specified by setting TBOPT = SLx, where x depends on the joint under consideration. If the stick-stiffness value is not specified, then this value along with the critical force/moment is used to determine the stick-stiffness. If the elastic slip is not specified, then a default value is computed for stick-stiffness calculations if necessary.
131
Material Models Stick-Stiffness A stick-stiffness value can be specified for controlling the behavior in the stick regime when friction behavior is specified. Use TBOPT = SKx, where x depends on the joint under consideration. If the stick-stiffness value is not specified, then the following procedure is adopted: If both maximum force/moment and elastic slip are specified, then the stick-stiffness is calculated from these values. If only maximum force/moment is specified, then a default elastic slip is computed and then the stickstiffness is calculated. If only the elastic slip is specified, then the stick-stiffness value is computed based on the current normal force/moment (Friction Coefficient * Normal Force or Moment/elastic-slip). Interference Fit Force/Moment If the forces that are generated during a joint assembly have to be modeled, the interference fit force/moment can be specified using TBOPT = FIx, where x depends on the joint under consideration. This force/moment will contribute to the normal force/moment in friction calculations. The appropriate TBOPT labels (TB command) for each joint element type are shown in the table below: TBOPT Labels for Elements Supporting Coulomb Friction Friction Parameter Static Friction Exponential Friction Law Max. Allowable Shear Force/Moment Elastic Slip Interference Fit Force/Moment Stick-Stiffness x-axis Revolute Joint MUS4 EXP4 TMX4 z-axis Revolute Joint MUS6 EXP6 TMX6 Slot Joint MUS1 EXP1 TMX1 Translational Joint MUS1 EXP1 TMX1
SL4 FI4 SK4
SL6 FI6 SK6
SL1 FI1 SK1
SL1 FI1 SK1
The following examples illustrate how to specify Coulomb friction parameters for various scenarios. Example 1 Specifying a single value of coefficient of friction and other friction parameters for an xaxis revolute joint.
TB, JOIN, 1, , , MUS4 TBDATA, 1, 0.1 TB, JOIN, 1, , , SK4 TBDATA, 1, 3.0E4 TB, JOIN, 1, , , FI4 TBDATA, 1, 10000.00 ! ! ! ! ! ! Label Value Label Value Label Value for friction coefficient of coefficient of friction for stick-stiffness for stick-stiffness for interference fit force for interference fit force
Example 2 Specifying temperature dependent friction coefficient and other friction parameters for a z-axis revolution joint.
TB, JOIN, 1,2 , 1, MUS6 TBTEMP, 10 ! 2 temp points, 2 data points and label for friction coefficient ! 1st temperature
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Contact Friction
TBDATA, 1, 0.15 TBTEMP, 20 TBDATA, 1, 0.1 ! TB, JOIN, 1, , , SK4 TBDATA, 1, 3.0E4 TB, JOIN, 1, , , FI4 TBDATA, 1, 10000.00 ! Value of coefficient of friction ! 2nd temperature ! Value of coefficient of friction ! ! ! ! Label Value Label Value for for for for stick-stiffness stick-stiffness interference fit force interference fit force
Example 3 joint.
Specifying the exponential law for friction and other friction parameters for a z-axis revolute
! Label for friction coefficient ! Static friction coeff, dynamic friction coeff, decay constant ! Label for stick-stiffness ! Value for stick-stiffness
TB, JOIN, 1, , , EXP6 TBDATA, 1, 0.4, 0.2, 0.5 ! TB, JOIN, 1, , , SK6 TBDATA, 1, 3.0E4
Example 4
Specifying friction as a function of sliding velocity for a slot joint.

! ! ! ! ! ! ! ! Label for friction coefficient Sliding velocity, coefficient of friction Sliding velocity, coefficient of friction Sliding velocity, coefficient of friction Label Value Label Value for max allowable frictional force for max allowable frictional force for elastic slip of elastic slip
TB, JOIN, 1, , 3, MUS1 TBPT, , 1.0, 0.15 TBPT, , 5.0, 0.10 TBPT, , 10.0, 0.09 ! TB, JOIN, 1, , , TMX1 TBDATA, 1, 3.0E4 TB, JOIN, 1, , , SL1 TBDATA, 1, 0.04
3.15. Contact Friction

Contact friction (TB,FRIC) is a material property used with current-technology contact elements. It can be specified either through the coefficient of friction (MU) for isotropic or orthotropic friction models or as user defined friction properties.
3.15.1. Isotropic Friction

Isotropic friction is applicable to 2-D and 3-D contact and is available for all contact elements. Use the TB,FRIC command with TBOPT = ISO to define isotropic friction, and specify the coefficient of friction MU on the TBDATA command. This is the recommended method for defining isotropic friction. To define a coefficient of friction that is dependent on temperature, time, normal pressure, sliding distance, or sliding relative velocity, use the TBFIELD command. Suitable combinations of up to two fields can be used to define dependency, for example, temperature and sliding distance as shown below:
TB,FRIC,1,,,ISO TBFIELD,TEMP,100.0 TBFIELD,SLDI,0.1 TBDATA,1,MU TBFIELD,SLDI,0.5 TBDATA,1,MU TBFIELD,TEMP,200.0 TBFIELD,SLDI,0.2 TBDATA,1,MU TBFIELD,SLDI,0.7 TBDATA,1,MU ! Activate isotropic friction model ! Define first value of temperature ! Define first value of sliding distance ! Define coefficient of friction ! Define second value of sliding distance ! Define coefficient of friction ! Define second value of temperature ! Define first value of sliding distance ! Define coefficient of friction ! Define second value of sliding distance ! Define coefficient of friction
See Understanding Field Variables (p. 187) for more information on the interpolation scheme used for field-dependent material properties defined using TBFIELD. To define a coefficient of friction that is dependent on temperature only, use the TBTEMP command as shown below:
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Material Models
TB,FRIC,1,2,,ISO TBTEMP,100.0 TBDATA,1,MU TBTEMP,200.0 TBDATA,1,MU ! ! ! ! ! Activate isotropic friction model Define first temperature Define coefficient of friction at temp 100.0 Define second temperature Define coefficient of friction at temp 200.0
Alternatively, you can use MU on the MP command to specify the isotropic friction. Use the MPTEMP command to define MU as a function of temperature. See Linear Material Properties (p. 14) for details. Note that if the coefficient of friction is defined as a function of temperature, the program always uses the contact surface temperature as the primary variable (not the average temperature from the contact and target surfaces).
3.15.2. Orthotropic Friction

The orthotropic friction model uses two different coefficients of friction in two principal directions (see Frictional Model in the Mechanical APDL Theory Reference for details). It is applicable only to 3-D contact and is available for current-technology contact elements. Issue the TB,FRIC command with TBOPT = ORTHO to define orthotropic friction, and specify the coefficients of friction, MU1 and MU2, on the TBDATA command. To define a coefficient of friction that is dependent on temperature, time, normal pressure, sliding distance, or sliding relative velocity, use the TBFIELD command. Suitable combinations of up to two fields can be used to define dependency, for example, sliding relative velocity and normal pressure as shown below:
TB,FRIC,1,,,ORTHO TBFIELD,SLRV,10.0 TBFIELD,NPRE,200.0 TBDATA,1,MU1,MU2 TBFIELD,NPRE,250.0 TBDATA,1,MU1,MU2 TBFIELD,SLRV,20.0 TBFIELD,NPRE,150.0 TBDATA,1,MU1,MU2 TBFIELD,NPRE,300.0 TBDATA,1,MU1,MU2 ! Activate orthotropic friction model ! Define first value of sliding relative velocity ! Define first value of normal pressure ! Define coefficients of friction ! Define second value of normal pressure ! Define coefficients of friction ! Define second value of sliding relative velocity ! Define first value of normal pressure ! Define coefficients of friction ! Define second value of normal pressure ! Define coefficients of friction
See Understanding Field Variables (p. 187) for more information on the interpolation scheme used for field-dependent material properties defined using TBFIELD. To define a coefficient of friction that is dependent on temperature only, use the TBTEMP command as shown below:
TB,FRIC,1,2,,ORTHO TBTEMP,100.0 TBDATA,1,MU1,MU2 TBTEMP,200.0 TBDATA,1,MU1,MU2 ! ! ! ! ! Activate orthotropic friction model Define first temperature Define coefficients of friction at temp 100.0 Define second temperature Define coefficients of friction at temp 200.0
Note that if the coefficient of friction is defined as a function of temperature, the program always uses the contact surface temperature as the primary variable (not the average temperature from the contact and target surfaces).
3.15.3. Redefining Friction Between Load Steps

If the friction behavior changes between initial loading and secondary loading (for example, during cyclic loading of seabed pipelines), you can reissue the TB,FRIC command between load steps to define new values for the coefficient of friction. This is true for both temperature-dependent friction (isotropic
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Cohesive Zone or orthotropic) defined via the TBTEMP command and field-dependent friction (isotropic or orthotropic) defined via the TBFIELD command. The following example shows the latter case:
TB,FRIC,1,,,ORTHO !Activate orthotropic friction model TBFIELD,SLDI,0. !Define initial curve for coefficient of friction TBDATA,1,0.0,0.0 TBFIELD,SLDI,0.25 TBDATA,1,0.0,1.25 TBFIELD,SLDI,0.5 TBDATA,1,0.0,1.0 TBFIELD,SLDI,20. TBDATA,1,0.0,1.1 /SOLUTION !* LOAD STEP 1 ... TIME,1 SOLVE TB,FRIC,1,,,ORTHO TBFIELD,SLDI,0. TBDATA,1,0.0,20.0 TBFIELD,SLDI,1.1 TBFIELD,SLDI,20.25 TBDATA,1,0.0,0.0 TBFIELD,SLDI,20.5 TBDATA,1,0.0,0.8 TBFIELD,SLDI,21 TBDATA,1,0.0,0.7 TBFIELD,SLDI,35 TBDATA,1,0.0,0.75 !* LOAD STEP 2 ... TIME,2 SOLVE !Activate orthotropic friction model !Define secondary curve for coefficient of friction
3.15.4. User-Defined Friction

As an alternative to the program-supplied friction models, you can define your own friction model with the user programmable friction subroutine, USERFRIC. The frictional stresses can be defined as a function of variables such as slip increments, sliding rate, temperature, and other arguments passed into the subroutine. You can specify a number of properties or constants associated with your friction model, and you can introduce extra solution-dependent state variables that can be updated and used within the subroutine. User-defined friction is applicable to 2-D and 3-D contact elements. To specify user-defined friction, use the TB,FRIC command with TBOPT = USER and specify the friction properties on the TBDATA command, as shown below. Also, use the USERFRIC subroutine to program the friction model.
TB,FRIC,1,,2,USER TBDATA,1,PROP1,PROP2 ! Activate user defined friction model; NPTS = 2 ! Define friction properties
Field variables specified with the TBFIELD command are not available for TB,FRIC,,,,USER. For detailed information on using the USERFRIC subroutine, see Writing Your Own Friction Law (USERFRIC) in the Contact Technology Guide.
3.16. Cohesive Zone

Cohesive zone materials can be used with interface elements (INTERnnn) and contact elements (CONTAnnn), as described here: 3.16.1. Exponential Cohesive Zone Material for Interface Elements
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Material Models 3.16.2. Bilinear Cohesive Zone Material for Interface Elements 3.16.3. Cohesive Zone Material for Contact Elements For more detailed information about cohesive zone materials, see Cohesive Zone Material (CZM) Model in the Mechanical APDL Theory Reference.
3.16.1. Exponential Cohesive Zone Material for Interface Elements

The interface elements allow exponential cohesive zone materials to be used for simulating interface delamination and other fracture phenomena. To define exponential material behavior, issue the TB,CZM,,,,EXPO command, then specify the following material constants via the TBDATA command: Constant C1 C2 C3 Meaning max n t Property Maximum normal traction at the interface Normal separation across the interface where the maximum normal traction is attained Shear separation where the maximum shear traction is attained
To define a temperature dependent material, use the TBTEMP command as shown below:
TB,CZM,1,2,,EXPO TBTEMP,100.0 TBDATA,1, max, n, t TBTEMP,200.0 TBDATA,1, max, n, t ! ! ! ! ! Activate exponential material model Define first temperature Define material constants at temp 100.0 Define second temperature Define material constants at temp 200.0
3.16.2. Bilinear Cohesive Zone Material for Interface Elements

The interface elements allow bilinear cohesive zone materials to be used for simulating interface delamination and other fracture phenomena. To define bilinear material behavior, issue the TB,CZM,,,,BILI command, then specify the following material constants via the TBDATA command: Constant C1 C2 C3 C4 C5 C6 Meaning max
c n
Property Maximum normal traction Normal displacement jump at the completion of debonding Maximum tangential traction Tangential displacement jump at the completion of debonding Ratio of
*
max
t
to
, or ratio of
to
Non-dimensional weighting parameter
To define a temperature-dependent material, issue the TBTEMP command as shown in the following example input fragment:
TB,CZM,1,2,,BILI ! Activate bilinear CZM material model ! ! Define first temperature !
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Cohesive Zone
TBTEMP,100.0 ! ! Define Mode I dominated material constants at temp 100.0: ! c !TBDATA,1, max, n ,- max, t , ! ! Define second temperature ! TBTEMP,200.0 TBTEMP,200.0 ! ! Define Mode I dominated material constants at temp 200.0: TBDATA,1, , ,max
, , max
Debonding Interface Modes Three modes of interface debonding comprise bilinear CZM law: Case Mode I Dominated Mode II Dominated Mixed-Mode Input on the TBDATA command as follows: C1, C2, C3, C4, C5 (where C3 = -max) C1, C2, C3, C4, C5 (where C1 = -max) C1, C2, C3, C4, C5, C6 (where C1 = max and C3 = max)
3.16.3. Cohesive Zone Material for Contact Elements

To model interface delamination, also known as debonding, the contact elements support a cohesive zone material model with bilinear behavior. This model allows two ways to specify material data. Bilinear Material Behavior with Tractions and Separation Distances To define bilinear material behavior with tractions and separation distances, issue the TB,CZM,,,,CBDD comman, then specify the following material constants via the TBDATA command: Constant C1 C2 C3 C4 C5 C6 Meaning max

Property Maximum normal contact stress Contact gap at the completion of debonding Maximum equivalent tangential contact stress Tangential slip at the completion of debonding Artificial damping coefficient Flag for tangential slip under compressive normal contact stress
max

TB,CZM,1,2,,CBDD TBTEMP,100.0 ! Activate bilinear material model with tractions ! and separation distances ! Define first temperature ! Define material constants at temp 100.0

TBDATA,1, max, , max, , ,
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Material Models
TBTEMP,200.0 ! Define second temperature ! Define material constants at temp 200.0
c TBDATA,1, max, n , max, t , ,
Bilinear Material Behavior with Tractions and Critical Fracture Energies Use the TB,CZM command with TBOPT = CBDE to define bilinear material behavior with tractions and critical fracture energies, and specify the following material constants using the TBDATA command. Constant C1 C2 C3 C4 C5 C6 Meaning max Gcn max Gct Property Maximum normal contact stress Critical fracture energy for normal separation Maximum equivalent tangential contact stress Critical fracture energy for tangential slip Artificial damping coefficient Fag for tangential slip under compressive normal contact stress
TB,CZM,1,2,,CBDE TBTEMP,100.0 TBDATA,1, max,Gcn, max,Gct, , TBTEMP,200.0 TBDATA,1, max,Gcn, max,Gct, , ! ! ! ! ! ! Activate bilinear material model with tractions and facture energies Define first temperature Define material constants at temp 100.0 Define second temperature Define material constants at temp 200.0
3.17. Fluids
Fluid material models can be used with hydrostatic fluid elements to model compressible fluids. For theoretical background on these materials, see Fluid Material Models in the Mechanical APDL Theory Reference. For more information on using these fluid material models with the hydrostatic fluid elements, see Modeling Hydrostatic Fluids in the Structural Analysis Guide. There are three ways to define material data for compressible fluids: liquid, gas, or pressure-volume data. Liquid Use the TB,FLUID command with TBOPT = LIQUID to define material behavior for a liquid, and specify the following material constants using the TBDATA command: Constant C1 C2 C3 Meaning K 0f Property Bulk modulus Coefficient of thermal expansion Initial density
You can define a temperature dependent liquid material with up to 20 temperatures (NTEMP = 20 max on the TB command) by using the TBTEMP command, as shown in the example below:
Fluids
TB,FLUID,1,2,,LIQUID TBTEMP,100.0 TBDATA,1,K, , 0f TBTEMP,200.0 TBDATA,1,K, , 0f ! ! ! ! ! Activate liquid material model Define first temperature Define material constants at temp 100.0 Define second temperature Define material constants at temp 200.0
When specifying temperature dependent density values for a liquid, keep in mind that the current density (f) for hydrostatic fluid elements is computed at each iteration as a function of pressure change (P), bulk modulus (K), coefficient of thermal expansion (), and temperature change (T). A reference temperature may be input using the TREF or MP,REFT command. For details on how the current density is calculated, refer to Liquid in the Mechanical APDL Theory Reference. Gas Use the TB,FLUID command with TBOPT = GAS to define material behavior for a gas, and specify the following material constant using the TBDATA command: Constant C1 Meaning 0f Property Initial density
You can define a temperature dependent gas material with up to 20 temperatures (NTEMP = 20 max on the TB command) by using the TBTEMP command, as shown in the example below:
TB,FLUID,1,2,,GAS TBTEMP,100.0 TBDATA,1, 0f TBTEMP,200.0 TBDATA,1, 0f ! ! ! ! ! Activate gas material model Define first temperature Define material constants at temp 100.0 Define second temperature Define material constants at temp 200.0
When specifying temperature dependent density values for a gas, keep in mind that the current density (f ) for hydrostatic fluid elements is computed at each iteration based on the Ideal Gas Law. For details on how the current density is calculated, refer to Gas in the Mechanical APDL Theory Reference. To use the Ideal Gas Law, you also need to define a reference pressure (input as real constant PREF) and a reference temperature (input with the TREF or MP,REFT command) with temperature offset (input with the TOFFST command). Pressure-Volume Data Use the TB,FLUID command with TBOPT = PVDATA to define compressible fluid behavior in terms of a pressure-volume curve. You can specify up to 20 temperature-dependent pressure-volume curves (NTEMP = 20 max on the TB command). The temperature for the first curve is input with TBTEMP, followed by TBPT commands for up to 100 pressure-volume data points. The data points (X, Y) entered on TBPT are: Constant X Y Meaning Pressure value Corresponding volume value
The pressure-volume data point must be defined in terms of total pressure and total volume of the fluid in the containing vessel.
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Material Models
3.18. User-Defined Material Model

The User-Defined material option (TB,USER) describes input parameters for defining your own material model via the UserMat subroutine, which defines any material type except incompressible materials. The subroutine supports current-technology elements only. For more information, see Subroutine UserMat (Creating Your Own Material Model) in the Guide to User-Programmable Features. Input for the user-defined option is determined by constants which you define. The TB,USER command initializes the constant table. The constants are defined via TBDATA commands (six per command). The number of constants can be any combination of the number of temperatures (NTEMP) and the number of data points per temperature (NPTS), to a maximum of NTEMP x NPTS = 1000. Example 3.21: Input for a User-Defined Material
TB,USER,1,2,4 ! ! ! ! ! ! ! ! ! ! Set material 1 as a user-defined material with two temperatures and four data points at each temperature point. First temperature. Four material constants for first temperature. Second temperature. Four material constants for second temperature.
TBTEMP,1.0 TBDATA,1,19e5,0.3,1e3,100, TBTEMP,2.0 TBDATA,1,21e5,0.3,2e3,100,
3.18.1. Using State Variables with UserMat

The UserMat subroutine supports state variables. To use them, initialize the constant table (TB,STATE), then define the constants (TBDATA). Example 3.22: Initializing the Values of State Variables for a User-Defined Material
TB,STATE,1,,4, TBDATA,1,C1,C2,C3,C4, ! Define material 1, which ! has four state variables. ! Initialize the four state variables.
You can define a maximum of 1000 state variables (NPTS = 1000).
3.19. Material Strength Limits

Material strength limits represent maximum stresses or strains that a material can sustain. This data table defines the strength limits and other related constants required for computing failure criteria (FC) index of a material under various loading conditions. Specify a TBOPT value on the TB,FCLI command to correspond to the stress limits (TBOPT = 1) or strain limits (TBOPT = 2). The following table lists the coefficient values that are addressed for the available TBOPT values: Strength Limit Constants C1 TBOPT = 1 Stress Limits (NPTS = 16) TBOPT = 2 Strain Limits (NPTS = 9)
XTEN -- Allowable tensile stress in material X-direction (must be positive)
XTEN -- Allowable tensile strain in material X-direction (must be positive)
C2
XCMP -- Allowable compressive XCMP -- Allowable compressive stress in material X-direction strain in material X-direction (de(default to the negative of XTEN) fault to the negative of XTEN)
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Material Strength Limits Strength Limit Constants C3 TBOPT = 1 Stress Limits (NPTS = 16) TBOPT = 2 Strain Limits (NPTS = 9)
YTEN -- Allowable tensile stress in material Y-direction (must be positive)
YTEN -- Allowable tensile strain in material Y-direction (must be positive)
C4
YCMP -- Allowable compressive YCMP -- Allowable compressive stress in material Y-direction strain in material Y-direction (de(default to the negative of YTEN) fault to the negative of YTEN) ZTEN -- Allowable tensile stress in material Z-direction (must be positive) ZTEN -- Allowable tensile strain in material Z-direction (must be positive)
C5
C6
ZCMP -- Allowable compressive ZCMP -- Allowable compressive stress in material Z-direction strain in material Z-direction (de(default to the negative of ZTEN) fault to the negative of ZTEN) XY -- Allowable XY shear stress (must be positive) YZ -- Allowable YZ shear stress (must be positive) XZ -- Allowable XZ shear stress (must be positive) XYCP -- XY coupling coefficient for Tsai-Wu strength index (default = -1.0) YZCP -- YZ coupling coefficient for Tsai-Wu failure index (default = -1.0) XZCP -- XZ coupling coefficient for Tsai-Wu failure index (default = -1.0) XZIT -- XZ tensile inclination parameter for Puck failure index (default = 0.0) XZIC -- XZ compressive inclination parameter for Puck failure index (default = 0.0) YZIT -- YZ tensile inclination parameter for Puck failure index (default = 0.0) YZIC -- YZ compressive inclination parameter for Puck failure index (default = 0.0) G1G2 -- Fracture toughness ratio between GI (mode I) and GII (mode II) XY -- Allowable XY shear strain (must be positive) YZ -- Allowable YZ shear strain (must be positive) XZ -- Allowable XZ shear strain (must be positive) --
C7 C8 C9 C10
C11
--
C12
--
C13
--
C14
--
C15
--
C16
--
C17
--
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Material Models Strength Limit Constants C18 C19 C20 TBOPT = 1 Stress Limits (NPTS = 16) TBOPT = 2 Strain Limits (NPTS = 9)
ETAL -- Longitudinal friction coefficient ETAT -- Transverse friction coefficient ALP0 -- Fracture angle under pure transverse compression
----
To determine physical failure criteria in unidirectional fiber-reinforced composite materials, including Puck and Hashin, LaRc03, and LaRc04 criteria, always define the reinforced fiber direction as the material X direction. The following table summarizes the applicable strength-limit constants for each failure criterion: Strength Limit Constants C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 Max Strain Criterion Y Y Y Y Y Y Y Y Y -----------Max Stress Criterion Y Y Y Y Y Y Y Y Y -----------Tsai-Wu Strength Ratio Y Y Y Y Y Y Y Y Y Y Y Y --------Puck Criterion Y Y Y Y --Y -----Y Y Y Y ----Hashin Criterion Y Y Y Y --Y Y -------------
LaRc03/04 UserCriterion Defined Y Y Y Y --Y ---------Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y
3.20. Material Damage

The following topics concerning material damage initiation and evolution are available:
Material Damage 3.20.1. Damage Initiation Criteria 3.20.2. Damage Evolution Law
3.20.1. Damage Initiation Criteria

This data table defines the criteria type for determining the onset of material damage under loading. Specify the TBOPT value on the TB,DMGI command to correspond to failure criteria (TBOPT = 1 or FCRT). The following table shows the coefficient values addressed for the available TBOPT values: Constants C1 TBOPT = 1 (or FCRT) NPTS = 4 Failure criteria type [1] for the tensile fiber failure mode Failure criteria type [1] for the compressive fiber failure mode Failure criteria type [1] for the tensile matrix failure mode Failure criteria type [1] for the compressive matrix failure mode 1. The available failure criteria are as follows: 1 -- Maximum strain 2 -- Maximum stress 3 -- Puck 4 -- Hashin 5 -- LaRc03 6 -- LaRc04 11 -- User-defined #1 ... 19 -- User-defined #9 To complete the material damage definition, it is also necessary to specify a compatible material damage evolution law (TB,DMGE). Without a damage evolution law, the damage-initiation criteria have no effect on the material. The following table summarizes the compatible damage evolution laws with specific damage-initiation criteria: Compatible TB,DMGE,,,,TBOPT Option 1 or MPDG TB,DGMI,TBOPT option 1 or FCRT Y
C2
C3
C4
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Material Models
3.20.2. Damage Evolution Law

This data table defines the material damage evolution law (or the way a material degrades) following the initiation of damage (TB,DMGE). Specify the TBOPT value on the TB,DMGE command to correspond to the instant stiffness reduction (TBOPT = 1 or MPDG). The following table shows the coefficient values addressed for the available TBOPT values: Constants C1 TBOPT = 1 or MPDG NPTS = 4 Tensile fiber stiffness reduction [1] Compressive fiber stiffness reduction [1] Tensile matrix stiffness reduction [1] Compressive matrix stiffness reduction [1] 1. 0 -- no damage, 1 -- complete damage To complete the material damage definition, it is also necessary to specify a compatible material damage initiation criterion (TB,DMGI). Without a damage initiation criterion, the damage evolution law has no effect on the material. The following table summarizes the compatible damage-initiation criteria with specific damage evolution laws: Compatible TB,DMGI,,,,TBOPT Option 1 or FCRT TB,DMGE,TBOPT option 1 or MPDG Y
C2
C3
C4
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Chapter 4: Explicit Dynamics Materials

Material properties used in explicit dynamic analyses (ANSYS LS-DYNA User's Guide program) differ somewhat from those used in implicit analyses (described in Linear Material Properties (p. 14) and Material Models (p. 13).) Most explicit dynamics material models require data table input. A data table is a series of constants that are interpreted when they are used. Data tables are always associated with a material number and are most often used to define nonlinear material data (that is, stress-strain curves). The form of the data table (referred to as the TB table) depends on the material model being defined. For a complete description of all explicit dynamics material models, including detailed data table input, see Material Models in the ANSYS LS-DYNA User's Guide.
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Chapter 5: Material Curve Fitting

Material curve fitting allows you to derive coefficients from experimental data that you provide for your material. Curve fitting involves comparing your experimental data to certain preexisting nonlinear material models. With this capability, you compare experimental data versus program-calculated data for different nonlinear models; based on those comparisons, you can determine the best material model to use during solution. Curve fitting is based on the data table configurations outlined in the TB command. The data manipulations and constructions are performed via the TBFT command. The following material curve-fitting topics are available: 5.1. Hyperelastic Material Curve Fitting 5.2. Viscoelastic Material Curve Fitting 5.3. Creep Material Curve Fitting 5.4. Chaboche Material Curve Fitting
5.1. Hyperelastic Material Curve Fitting

Hyperelastic curve fitting is a tool for estimating your material constants by inputting your experimental data and comparing it to the hyperelastic material models. Your stress-strain curves can be converted to any of the supported hyperelastic models, including Mooney-Rivlin, Ogden, Neo-Hookean, Polynomial, Arruda-Boyce, Gent, and Yeoh. Compressible hyperelastic Ogden hyper-foam and Blatz-Ko models are also supported. You perform curve fitting either interactively (GUI) or via batch commands. You input your experimental data, choose a model from one of nine supplied hyperelastic models, perform a regression analysis, graphically view the curve-fitting results, compare the fits to the experimental data, and write the fitted coefficients to the database as nonlinear data table commands for the subsequent finite element analyses. Hyperelastic models can define three types of behavior: purely incompressible, nearly incompressible, and compressible. Hyperelastic curve fitting is based on the HYPER option of the TB command.
5.1.1. Understanding the Hyperelastic Material Curve-Fitting Process

The steps for hyperelastic curve fitting are defined as follows: 1 Step 1. Prepare Experimental Data (p. 148) 2 Step 2. Input the Experimental Data (p. 149) 3 Step 3. Select a Material Model Option (p. 150) The experimental data must be a plain text file delimited by a space or a comma. The experimental data can be read in by browsing to the file location (GUI) or by specifying the filename and path (batch) on the command line. The material options for the applicable curve-fitting regimen are defined in the TB command. Nine hyperelastic models are supported. After you pick a model, you can still change to another model if an ideal fit is not realized.
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Material Curve Fitting 4 Step 4. Initialize the Coefficients (p. 151) Depending on the model, hyperelastic curve fitting can be a linear or nonlinear regression process. The hyperelastic material models, along with the associated process for each are listed in Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150). You will specify the type of error norm to be used to generate the curve fit. You review and verify the results by comparing the experimental data and the regression errors. If the results you obtain are not acceptable, repeat steps 3 to 5 to perform a new curve-fitting solution. Write your curve-fitting results to the database in the TB command table format.
5 Step 5. Specify Control Parameters and Solve (p. 152) 6 Step 6. Plot Your Experimental Data and Analyze (p. 153)
7 Step 7. Write Data to the TB Command (p. 154)
5.1.2. Step 1. Prepare Experimental Data

Curve fitting requires experimental test data. Your hyperelastic curve-fitting data needs to be a comma or space delimited file, referencing your stress vs. strain values. Hyperelastic curve fitting supports three main behaviors: Case 1 - Totally Incompressible Models (see Table 5.1: Experimental Details for Case 1 and 2 Models and Blatz-Ko (p. 148)) Case 2 - Nearly Incompressible Models (see Table 5.1: Experimental Details for Case 1 and 2 Models and Blatz-Ko (p. 148)) Case 3 - Compressible Models (see Table 5.2: Experimental Details for Case 3 Models (p. 148)) The types of data required for each of these cases is defined in the tables below. Table 5.1: Experimental Details for Case 1 and 2 Models and Blatz-Ko Experimental Type Uniaxial Test Biaxial Test Planar/Shear Test Column 1 Engineering Strain Engineering Strain Engineering Strain (in loading direction) Engineering Shear Strain Volume Ratio (J) Column 2 Engineering Stress Engineering Stress Engineering Stress Column 3
Simple Shear Test
Engineering Shear Stress Hydrostatic Pressure
(Optional) Engineering Normal Stress (normal to the edge of shear stress)
Volumetric Test
Table 5.2: Experimental Details for Case 3 Models Experiment Type Uniaxial Test Column 1 Engineering Strain Column 2 Lateral Direction Engineering Strain Column 3 Engineering Stress
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Hyperelastic Material Curve Fitting Experiment Type Biaxial Test Shear Test Simple Shear Test Column 1 Engineering Strain Engineering Strain (in loading direction) Column 2 Engineering Strain (in thickness direction) Engineering Strain (in thickness direciton) Column 3 Engineering Stress Engineering Stress (Optional) Engineering Normal Stress (normal to the edge of shear stress)
Engineering Shear Strain Engineering Shear Stress
Volumetric Test
Volume Ratio (J)
Hydrostatic Pressure
J is the ratio of current volume to the original volume. All stresses output in POST1/POST26 are true stresses and logarithmic strains.
5.1.3. Step 2. Input the Experimental Data

Use the EADD option on the TBFT command to input your data files. First, designate whether they are uniaxial, biaxial, shear, simple shear, or volumetric, and then designate the location in the Option2, Option3, and Option4 fields. All of the stress values will be engineering stress, except for the volumetric option (true stress). The file should be a simple, delimited set of stress and strain values similar to the following:
0.9703 0.9412 0.9127 0.8847 60.00 118.2 175.2 231.1
For temperature-dependent curve fitting, specify your temperature values at the top of the experimental data using the
/temp,value
line. This header format specifies the attribute (temp) and its value (100). An example of a typical data input using these attributes follows:
/temp,100 0.9703 60.00 0.9412 118.2 0.9127 175.2 0.8847 231.1
Adding this header introduces a temperature attribute of 100 degrees. You can add additional data sets at other temperatures, in additional files. One file can have data at only one temperature. For compressible materials, the curve-fitting tool's default behavior is to solve only for stress as a function of strain and lateral strain. To force the curve-fitting tool to also fit experimental lateral strain data to generate the coefficients for the Ogden compressible foam model, add the line /USEL,1 near the top of the experimental data file. This option is valid for uniaxial, biaxial and shear test data.
5.1.3.1. Batch
TBFT,EADD,ID,Option1,Option2,Option3,Option4 ! experimental data input
where:
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Material Curve Fitting ID = Index corresponding to the material number Option1 = UNIA, BIAX, SHEA, SSHE or VOLU Option2 = name of file containing experimental data Option3 = file name extension Option4 = file directory
5.1.3.2. GUI
The Material Properties GUI provides an input field where you can type in the filename of your data file, and also include the appropriate path. You can also browse to a file in a specified location. Separate input is performed for each Option1 value (UNIA, BIAX, SHEA, etc.).
5.1.4. Step 3. Select a Material Model Option

Use the FADD option on the TBFT command to define your constitutive model. Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) below lists the models that are available for hyperelastic curve fitting. When volumetric data is supplied, a compressible or nearly incompressible model is implied, and only the designated options will be available. When no volumetric data is supplied, the model is understood to be incompressible, and those model options will be available. Supplying zero coefficients for the volumetric data field will also denote an incompressible model. Table 5.3: Hyperelastic Curve-Fitting Model Types Model Name Mooney-Rivlin Polynomial Yeoh Neo-Hookean Ogden Arruda-Boyce Gent Blatz-Ko Ogden Hyperfoam Abbreviation moon poly yeoh neoh ogde boyc gent blat foam Order/Options 2, 3, 5, 9 1 to N 1 to N 1 to N 1 to N No. of Coefficients [1] 2/3/5/9+1 see below [2] N+N 1+1 2*N+N 2+1 2+1 1 2*N+N Linear/Nonlinear Fitting Linear Linear Linear Linear Nonlinear Nonlinear Nonlinear Nonlinear Nonlinear
1. The number of coefficients is usually the sum of the number of deviatoric coefficients and the number of volumetric coefficients. 2. The number of coefficients for a polynomial will be dependent on the polynomial order N. =

Number of Coefficients
=1
Blatz-Ko and Ogden hyper-foam are compressible models. For Ogden hyper-foam, the experimental data you supply will require additional fields. For more information about the hyperelastic models available for curve fitting, see Hyperelasticity in the Material Reference.
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Hyperelastic Material Curve Fitting
5.1.4.1. Batch Method

TBFT,FADD,ID,HYPER,Option2,Option3 ! material model options input
where: ID = Index corresponding to the material number Option2 = model name, as specified in Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) Option3 = order or number of coefficients, where applicable. Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) specifies the number and type of coefficient(s) necessary for each hyperelastic model type.
5.1.4.2. GUI Method

A pull-down selection menu in the material GUI allows you to pick the applicable material model option. The options displayed will be dependent on the format of your experimental data file.
5.1.5. Step 4. Initialize the Coefficients

Depending on the model you choose, hyperelastic curve fitting can be a linear or a nonlinear regression process. In both cases, the initial coefficients you supply will determine how accurate and efficient your curve fit will be. The initial values of the coefficients generally come from experience, and also from studying the function that defines the model you are attempting to compare/fit your data to. For most hyperelastic models, 1 or -1 is a good starting point. However, coefficient values can vary greatly depending on the model chosen. The Gent model, for instance, provides good fit with initial coefficient values as high as 1000. You can also fix (hold constant) your coefficients (Option4 = FIX on the TBFT command). You specify a value for a coefficient and keep it unchanged, while allowing the other coefficients to be operated on. You can then release the coefficient you fixed and operate on it after the others are optimized. By default, all of the coefficients are free to vary. The capability to fix coefficients applies only to nonlinear curve fits (as listed in Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150)). For temperature-dependent experimental data, enable temperature-dependency and specify a reference temperature before solving for the coefficients. You can set the reference temperature only to values specified using the /temp,value header line in the experimental data. You can also specify tref = all and initiate multiple solves to evaluate coefficients at all available discrete temperature values. In this case, for data at three temperatures (t1, t2, and t3), a single TBFT,SOLVE entry will initiate three different solve operations at those three discrete temperatures. With temperature dependency on and the reference temperature set to a particular value, a TBFT,SOLVE command solves for coefficients only at that temperature. To solve for coefficients at other temperatures, you set the reference temperature to each desired discrete temperature value and solve again. You can initialize the coefficients before or after turning temperature dependency on. If the coefficients are initialized before turning temperature dependency on, the specified coefficients will become the initial coefficients for all future solves for that particular model. These coefficients are, however, overridden when temperature dependency is turned on and another set of initial values are specified at discrete temperature values. The curve-fitting tool looks for the initial coefficients at a particular temperature. If no coefficients are specified at a discrete temperature, the initial coefficients set before temperature dependency was turned on are used.
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Material Curve Fitting
5.1.5.1. Batch
TBFT,SET,ID,HYPER,Option2,Option3,Option4,Option5 ! initialize coefficients
where: ID = Index corresponding to the material number Option2 = model name (see Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for the available models) Option3 = order, if applicable Option4 = index of coefficient Option5 = value of the coefficient To set Option4 for a reference temperature, or for temperature dependency:
TBFT,SET,ID,HYPER,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = model name. See Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for the available models. Option3 = order, if applicable Option4 = tdep or tref Option5 = If Option4 = tdep, Option5 = 1 turns on temperature dependency, and Option5 = 0 turns it off. If Option4 = tref, Option5 will be either all, or the reference temperature.
TBFT,FIX,ID,HYPER,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = model name See Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for the available models. Option3 = (Blank--not applicable) Option4 = index of coefficient Option5 = 1 for fixed and 0 to vary
5.1.5.2. GUI
The GUI automatically updates your coefficient tables depending on the model you pick. You can modify individual coefficients to initialize them at values you believe are more appropriate.
5.1.6. Step 5. Specify Control Parameters and Solve

Depending on the model, your hyperelastic curve fitting will be either a linear or nonlinear regression process (see Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for details). Your error norms can be either normalized or non-normalized. Normalized error norms (the default regression option) generally give better results than the non-normalized error norms, since normalized error gives equal weight to all of your data points. The solution control parameters of a nonlinear regression include: 1. Number of iterations 2. Residual tolerance
Hyperelastic Material Curve Fitting 3. Coefficient change tolerance The solution stops when both the residual tolerance and the coefficient change tolerance of your error norm are met, or if the number of iterations criteria is met. When you use nonlinear regression, you must initialize your coefficients.
5.1.6.1. Batch
TBFT,SOLVE,ID,HYPER,Option2,Option3,Option4, ..., Option7 ! set control parameters and solve
where: ID = Index corresponding to the material number Option2 = model name. See Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for the models available. Option3 = order or number of your coefficients. See Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for possible values. Option4 = curve-fitting procedure: 0 = non-normalized least squares, 1 = normalized least squares Option5 = maximum number of iterations Option6 = tolerance of residual changes Option7 = tolerance of coefficient changes Other solution parameters are available. See the TBFT command for details.
5.1.6.2. GUI
The GUI lets you specify all of your control parameters (error norm, solution control parameters, and the solver options) interactively. You select the appropriate options from the provided menus, and solve to generate the coefficients. You can change the parameters and repeat the solution as necessary to ensure an accurate result. The unused options are disabled whenever necessary.
5.1.7. Step 6. Plot Your Experimental Data and Analyze

After you initiate Solve, the coefficient tables will contain the fitted coefficients and also the residual errors. You can then plot your data and visually interpret the results. The first column is always your Xaxis, with each additional column plotted separately as a function of column one. You should try to reserve column one for the variable that you would like to see vary in the plot. For example, if your data contains time, temperature, stress, and creep strain, and you wish to see the creep strain vary as a function of time at different temperatures and stresses in the plot, you add your experimental data using multiple TBFT,EADD commands (or the corresponding GUI operations) by splitting the file into multiple experimental files as prescribed earlier, one for each combination of temperature and stress.
5.1.7.1. GUI
Use the GRAPH button to plot the data. Your plots will show columns 2 and above as separate curves, plotted as a function of column 1. The data in column 1 is always the X-axis. By default, all the experiments are plotted in the GUI window. To view specific data and its corresponding fitting result, you can click the right mouse button (RMB) on the specific data set, and pick a desired option to view the results. Other RMB plotting utilities can be found for different data fields in the curve-fitting GUI window. Use RMB functions to Zoom, Fit, Save Plot to File, View/Hide Legend and View/Hide Grid. Two or more fitted functions can also be compared in the same plot. For example, you can view Mooney2 Uniaxial
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Material Curve Fitting and Mooney9 Uniaxial plots directly on top of each other, provided you have already solved for both of these functions. RMB also allows you to set the number of points used to generate the plot, and also change the minimum X value and the maximum X value in a plot. You can also hide a particular curve within a graph, turn the legend and/or axis displays on and off, and switch the scales between log scale and regular scale. Use the middle-mouse button to eliminate a specific curve and clarify or refine the remaining curve.
5.1.7.2. Review/Verify
The two factors you consider in determining results acceptability are visual fit and the error norm/residual values. When you plot the curve, the error norm/residual values are printed in the curve-fitting GUI window. Error norm values help you determine the quality of curve fitting and whether to accept the results, but are not always the best indicator of a valid curve fit. Plotting the curves and visually assessing the result is usually the best indication. If the results are unacceptable, you may want to go back to step 3 and solve again, either by picking a different model, increasing the order, or redefining your initial values of the coefficients or other control parameters. You can continue to use your original experimental data, repeating step 3 through step 7 until you get an acceptable solution.
5.1.8. Step 7. Write Data to the TB Command

After you are satisfied with your curve-fitting results, you can write the curve-fitting data to the database using TBFT,FSET. Both the GUI and the command line convert the coefficients to the appropriate form before writing to TB tables. The program stores the data as part of the material property set for use in subsequent analyses.
5.1.8.1. Batch
TBFT,FSET,ID,HYPER,Option2,Option3 ! write data to TB
where: ID = Index corresponding to the material number Option2 = model name. See Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for the models available. Option3 = order or number of your coefficients. See Table 5.3: Hyperelastic Curve-Fitting Model Types (p. 150) (above) for possible values.
5.1.8.2. GUI
Once you complete the process and update your material data properties with the representative curve data, you are returned to the material properties dialog. The curve data can now be accessed for the full range of material behavior.
5.2. Viscoelastic Material Curve Fitting

Viscoelastic material curve fitting determines your material constants by relating your experimental data to the Prony series expansion for both the shear and bulk modulus of the hypoviscoelastic material option. Curve fitting is performed either interactively or via batch commands. You input the experimental data, define the order of Prony series expansion, perform nonlinear regression, view the curve-fitting results graphically, compare the experimental data, and write the fitted coefficients to the database as nonlinear data table commands for the subsequent finite elements analyses. You can fit shear modulus and/or
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Viscoelastic Material Curve Fitting bulk modulus and/or shift functions, along with discrete temperature dependencies for multiple data sets.
5.2.1. Understanding the Viscoelastic Material Curve-Fitting Process

The steps for viscoelastic curve fitting are defined as follows: 1 Step 1. Prepare Experimental Data (p. 155) 2 Step 2. Input the Data (p. 156) 3 Step 3. Select a Material Model Option (p. 157) 4 Step 4. Initialize the Coefficients (p. 158) 5 Step 5. Specify Control Parameters and Solve (p. 160) 6 Step 6. Plot the Experimental Data and Analyze (p. 163) The experimental data must be a plain text file delimited by a space or a comma. The experimental data can be read in from the GUI or via batch commands, as a plain text file. This includes Prony series expansion of shear and/or bulk moduli as well as shift function. The supported shift functions include WLF and TN. Viscoelastic curve fitting is a nonlinear regression; the initial value of your coefficients is very important for a successful solution. Specify the error norm to be used, the solution control parameters, and perform the nonlinear regression. Graphically view the curve-fitting results. Review and verify the results by comparing them with the experimental data and the regression errors. If any factor is not acceptable, repeat steps 3 to 7 to obtain a new curve-fitting solution. Write your curve-fitting results in TB command format to the database.
7 Step 7. Write Data to the TB Command (p. 163)

Curve fitting requires experimental test data. To use curve fitting with viscoelastic materials, the experimental data must be shear modulus and/or bulk modulus as a function of time and temperature. The experimental data is named sdec or bdec to distinguish it from other data types, such as hyperelastic or creep. This also makes documenting your analyses more convenient. Your viscoelastic test data must be a plain text file with headers to define the test data. The data file should be in table form, delimited by a space or a comma. The header defines the test data type and the temperature for your test data. For viscoelastic curve fitting with multiple temperatures, you can evaluate coefficients at each discrete temperature point and write it as a temperature-dependent Prony data table, or you can use the Williams-Landau-Ferry (WLF) or Tool-Narayanaswamy (TN) shift functions to account for the temperature dependency. (See Shift Functions in the Mechanical APDL Theory Reference.) A separate data file must be provided for each discrete temperature. The viscoelastic test data can be any of 4 data types, see Table 5.4: Viscoelastic Data Types and Abbreviations (p. 155). Table 5.4: Viscoelastic Data Types and Abbreviations Time Shear modulus Bulk modulus Temperature time smod bmod temp
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Material Curve Fitting The headers are used to describe the data types that characterize the test data columns or attributes of the data. The following listing contains the appropriate headers, followed by the delimited data:
/temp,100 ! define temperature attribute 0.01 2992.53 1 2978.514207 2 2965.45541 4 2942.293214 6 2922.530649 8 2905.612202 10 2891.073456 20 2842.506984 40 2798.142793 60 2772.383729 80 2750.631843 100 2730.398114 200 2643.125432 400 2517.475394 600 2431.262053 800 2366.580897 1000 2313.955396 2000 2117.922594 4000 1833.734397 6000 1627.199197 8000 1470.6806 10000 1347.264527 20000 964.0141125 40000 586.1405449 60000 392.186777 80000 277.2706253 100000 202.0025278 200000 46.87056342 400000 2.669209118 600000 0.156653269 800000 0.0137224 1000000 0.005591539
5.2.3. Step 2. Input the Data

You use the EADD option of the TBFT command to input your data files. The experimental data must be read in from a plain text file. The experimental data must be prepared as discussed in the previous section, and include both the header information and the formatted test data. Each file is viewed as a data set, and can be the complete set of experimental test data or a part of an experimental test data. You can include several data sets, including tests performed at different temperatures. Although different data sets can have the same/or different temperature, each file can have only one temperature. Multiple temperature data sets must be input with multiple files. Two types of data may be required for viscoelastic curve fitting, either shear modulus vs. time and/or bulk modulus vs. time. The data can also be a function of temperature, which can then be accounted for by either WLF or TN shift function. You can use the GUI or batch command to input your experimental data.
5.2.3.1. Batch
TBFT,EADD,ID,Option1,Option2,Option3,Option4 ! input data
where: ID = index corresponding to the material number Option1 = experimental data type, either sdec or bdec
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Viscoelastic Material Curve Fitting Option2 = name of file containing experimental data Option3 = file name extension Option4 = file directory The sdec coefficient refers to the shear modulus as a function of time. The bdec coefficient refers to the bulk modulus as a function of time.
5.2.3.2. GUI
Click on the Add Dataset button and type the filename into the area provided. You can also browse to a file in a specified location. Separate input is performed for each data type (Option1 = sdec, or bdec)

The TBFT command provides the curve-fitting tools for viscoelastic material modeling. You represent your viscoelastic material behavior by a set of Prony-series expansions of shear and/or bulk moduli to characterize the shear and the bulk deformation of the material. You can also use the shift functions to characterize the material's temperature dependency. First you define a case name to associate the set of coefficients for the Prony expansions with the shift functions that characterize the material behavior. You can use the case name to define several different options that characterize the same test data, and then to compare the curve-fitting results. To define the material model, you must first define a case name, and then specify the order of shear and bulk moduli and the type of the shift function(s), if required. You need to create an additional case to define different shear order, bulk order or shift options. Once you create a case, the number of shear terms, bulk terms, or shift options cannot be changed.

You define a viscoelastic material with the Prony series expansion by creating a case and setting the order of shear modulus, bulk modulus, and shift options. You create the case with the TBFT,FCASE command. The first line will include FCASE,ID,NEW. Then you specify the number of shear terms, bulk terms, and the shift function. The case is actually created only after the option is issued. The following syntax examples and argument descriptions illustrate the relationship of these activities:
TBFT,FCASE,ID,NEW,Option2,Option3
where: ID = Index corresponding to the material number Option2 = PVHE (refers to Prony Viscohypoelastic) Option3 = Case name
TBFT,FADD,ID,CATEGORY,Option2,Option3
where: ID = Index corresponding to the material number CATEGORY = VISCO Option2 = pshear or pbulk or shift Option3 = dependent on Option2 as follows: Option2 = pshear or bulk, Option3 = NONE, or 1 to N
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Material Curve Fitting Option2 = shift, Option3 = NONE or TN or WLF

TBFT,FCASE,ID,FINI
5.2.4.2. GUI Method

You can use the GUI to interactively navigate the tree structure of the curve-fitting window. Each of the shear, bulk and shift options can be selected, and you can fill in the appropriate case name in a text box field. As you choose the options, the coefficient table is automatically created.

The initial values chosen for your coefficients determines the success of your viscoelastic curve-fitting operations. A complete model has (2*NG+1) + (2* Nk +1) + NS number of the coefficients, where NG is the order of the Prony series expansion of the shear modulus, Nk is the order of the Prony series expansion of the bulk modulus, and NS is the number of coefficients of the shift function. (NS = 2 for the TN option, and NS = 3 for the WLF option.) The coefficients are ordered as shear terms first, then the bulk terms, and then the shift function. The coefficients are ordered as A0G, A1G, 1G, A2G, 2G, AnG, and nG for shear modulus, and A0K, A1K, 1K, A2K, 2K, AnK, and nK for bulk modulus. A shift function must be used together with your shear and/or bulk modulus for temperature-dependent experimental data. The default coefficient is set to 1, but it is good practice to redefine the initial values before solving. When initializing your coefficients, set AnKs to 1 and nKs to time values that are equally distributed in the log scale, spanning the data range from minimum to maximum time. For example, consider the shear-decay versus time-data file. If the time values vary from 1 to 10000, and if you use third-order Prony, logical guesses for 1G , 2G and 3G that span this range could be 1G = 1, 2G = 100, and 3G = 10,000 (also (1), (10) and (10,000) or (1), (1,000) and (10,000), respectively). The equations below describe the relationship between the Prony Coefficient (n) and the corresponding coefficient generated in curve fitting (An). K and G are the shear modulus and bulk modulus at t = 0. This was done to keep all nG and nK values used in the TB tables positive. nK = (AnK * AnK) / K nG = (AnG * AnG) / G A good guess for the WLF or TN parameter is the reference temperature you used during your partial solve for shear and bulk. The index of the reference/base temperatures is the sum of NumShear + NumBulk + 1.
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Viscoelastic Material Curve Fitting You can also fix (hold constant) your coefficients. You specify a value for a coefficient and keep it unchanged, while allowing the other coefficients to be operated on. You can then release the fixed coefficient later if desired. By default, all of the coefficients are free to vary. You estimate coefficients for temperature-dependent data either by using the shift function or by setting the temperature-dependency flag and setting a reference temperature before solving for the coefficients. You can set the reference temperature only to values specified using the /temp,value header line in the experimental data. You can also specify tref = all and initiate multiple solves to evaluate coefficients at all available discrete temperature values. In this case, for data at three temperatures (t1, t2, and t3), a single TBFT,SOLVE command initiates three separate solve operations at those three discrete temperature values, and generate data at three corresponding discrete temperatures. With temperature dependency specified and the reference temperature set to a particular value, a TBFT,SOLVE command solves for coefficients only at that discrete temperature. To solve for coefficients at other temperatures, set the reference temperature to each of the desired discrete temperature values and solve again. You can initialize the coefficients before or after activating temperature dependency. If the coefficients initialize before setting temperature dependency, the specified coefficients become the initial coefficients for all future solves for that particular model. These coefficients are, however, overridden when temperature dependency is active and another set of values is specified at a discrete temperature value. The curve-fitting tool looks for the initial coefficients at a particular temperature. If no coefficients are specified at discrete temperature values, the initial coefficients set before temperature dependency was activated are used.

TBFT,SET,ID,CASE,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = Index of coefficient Option5 = Value of coefficient Example 5.1: Initializing Coefficients Using the Batch Method
TBFT,SET,1, myvisco1,,1,1.2 ! Initialize the first coefficient to 1.2 TBFT,SET,1, myvisco1,,2,1.5 ! Initialize the second coefficient to 1.5
Use the TBFT,FIX command to fix a coefficient to a value set by the TBFT, SET command or to release a previously fixed coefficient. By default, coefficients are not fixed.
TBFT,FIX,ID,CASE,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = Index of coefficient Option5 = 1 to fix, 0 to vary
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Material Curve Fitting Example 5.2: Fixing Coefficient Values

TBFT,FIX,1, myvisco1,,1,1.! Fix the first coefficient to a value set via TBFT,SET TBFT,FIX,1, myvisco1,,2,1 ! Fix the second coefficient to a value set via TBFT,SET
Temperature dependency uses Option4, and references data files you entered with the appropriate temp header.
TBFT,SET,ID,VISCO,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = Model name. See Table 5.4: Viscoelastic Data Types and Abbreviations (p. 155) (above) for the models available. Option3 = (Blank--not applicable) Option4 = tdep or tref Option5 = If Option 4 = tdep, then 1 turns temperature dependency on and 0 turns it off. If Option 4 = tref, this value will be a specific temperature, or ALL.
5.2.5.2. GUI Method

The coefficients table is automatically updated in the viscoelastic curve-fitting GUI window when the order of shear modulus and/or bulk modulus and/or shift function are defined. Specify values for your coefficients in the coefficients table in the curve-fitting GUI window, and check the appropriate boxes to fix them or allow them to vary.

Viscoelastic curve fitting is a nonlinear regression process. You can use either normalized or non-normalized error norm for the regression. The normalized error norm is the default regression option for calculating the error. This error norm generally gives better results than the non-normalized error norm, since the normalized error gives equal weight to all data points. The solution control parameters for nonlinear regression include number of iterations, residual tolerance, and coefficient change tolerance. The solution stops when both residual tolerance of error norm and coefficient change tolerance is met or if the number of iterations criteria is met. The coefficients are updated when the solution is completed. In general it is very difficult to directly solve a complete case including coefficients of the shear modulus, the bulk modulus and the shift function. Three solver options including shear modulus only, bulk modulus, and shift function (or all) are provided to allow you to solve only Prony coefficients of the shear modulus, Prony coefficients of the bulk modulus, and coefficients of the shift function. In many cases, however, the coefficients of shift function can't be solved until the shear or bulk modulus are solved. It is normal for a solution to not converge at first, but to stop when the maximum iterations criteria is reached. At that point, you should examine the curve-fitting results and the solution history before proceeding any further. You can then adjust parameters and resolve the problem whenever it is necessary.
5.2.6.1. Temperature-Dependent Solutions Using the Shift Function

For viscoelastic curve fitting using the shift function, follow these three steps to perform the regression: 1. Solve the shear coefficients (if there are any): Set the partial solve option using TBFT, SET,,,,,COMP, PSHEAR.
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Viscoelastic Material Curve Fitting Set the reference temperature at which your partial solution will be performed using TBFT, SET,,,,,TREF, TX. Only data at temperature TX will be used to estimate shear coefficients Solve 2. Solve the bulk coefficients (if there are any): Set the partial solve option using TBFT, SET,,,,,COMP, PBULK. Set the reference temperature at which your partial solution will be performed using TBFT, SET,,,,,TREF, TX. The reference temperature should be the same for both shear and bulk. Only data at temperature TX will be used to estimate shear coefficients TBFT,SOLVE 3. Solve the shift function (or all) coefficients: Set the partial solve option using TBFT, SET,,,,,COMP, PVHE. TREF is not used when solving for all parameters. All temperature data is used to estimate the coefficients. TBFT,SOLVE For GUI operations, when only the shear and bulk buttons are checked, only your shear coefficients are solved. To solve for both shear and bulk, you must check all three buttons.
5.2.6.2. Temperature-Dependent Solutions Without the Shift Function

If you do not use the shift function, discrete data must be supplied to account for temperature dependency. Follow these steps to perform the regression: 1. Set the temperature-dependency flag using the command TBFT, SET,,,,,TDEP,1. 2. Solve the shear coefficients (if there are any): Set the partial solve option using TBFT, SET,,,,,COMP, PSHEAR. Set the reference temperature at which your partial solution will be performed using TBFT, SET,,,,,TREF, TX. Only data at temperature TX will be used to estimate shear coefficients Initialize the coefficients. TBFT,SOLVE Repeat the above steps for all temperatures. 3. Solve the bulk coefficients (if there are any): Set the partial solve option using TBFT, SET,,,,,COMP, PBULK. Set the reference temperature at which your partial solution will be performed using TBFT, SET,,,,,TREF, TX. Only data at temperature TX will be used to estimate the shear coefficients. Initialize the coefficients.
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Material Curve Fitting TBFT,SOLVE Repeat the above steps for all temperatures. Alternatively, you can solve for both shear and bulk data at the same time. 1. Set the temperature dependency flag using the command TBFT, SET,,,,,TDEP,1. 2. Solve for ALL coefficients. Set the partial solve option using TBFT, SET,,,,,COMP, PVHE. Set the reference temperature at which your partial solution will be performed using TBFT, SET,,,,,TREF, TX. Only data at temperature TX will be used to estimate shear and bulk coefficients Initialize the coefficients. TBFT,SOLVE Repeat the above steps for all of the desired temperatures.

Solution option command is:
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = comp Option5 = pshea (for Shear only) or pbulk (for bulk only) The SOLVE option allows you to specify procedure types, tolerances, and number of iterations.
TBFT,SOLVE,ID,CASE,Option2,Option3,Option4, ..., Option7
where: ID = Index corresponding to the material number Option2 = visco function name (See Table 5.4: Viscoelastic Data Types and Abbreviations (p. 155).) Option3 = (Blank--not applicable) Option4 = Curve-fitting procedure: 0 = non-normalized least squares, 1 = normalized least squares (default) Option5 = Maximum number of iterations Option6 = Tolerance of residual changes Option7 = Tolerance of coefficient changes Other solving parameters are available. See the TBFT command for details.
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Viscoelastic Material Curve Fitting
5.2.6.4. GUI Method

The GUI provides access for you to choose your error norm, solution control parameters, and solver options. Once you complete these specifications and SOLVE, you can go back and modify your parameters as necessary to obtain a good curve fit.
5.2.7. Step 6. Plot the Experimental Data and Analyze

The best way to verify a good fit between your experimental data and the provided curves is to plot your curves and visually inspect them. The Graph button provides direct means to plot the data.
5.2.7.1. Analyze Your Curves for Proper Fit

All of your data is plotted as a function of column 1 for the X-axis. Columns 2 and above are each plotted in a separate graph, in a separate window, with the corresponding fitted data as a function of column 1. By default, all the shear data sets and/or the bulk data sets as well as the corresponding fitting results are plotted in two separated graphs in a GUI window. To view a specific data and its corresponding fitting result, you click the right mouse button (RMB) on the specific data set, and pick a desired option. You can also use the right mouse button to turn the legend and/or axis on and off. The scales can be also switched between log scale and regular scale. With the middle button, you can eliminate certain curves from each window's display in order to see the remaining data more clearly. Right mouse button (RMB) functions allow you to Zoom, Fit, Save Plot to File, View/Hide Legend, and View/Hide Grid. Two or more fitted functions can also be compared in the same plot. RMB also allows you to see the number of points used to generate the plot, and also change the minimum X value and the maximum X value in a plot. You can use the middle-mouse button (context sensitive) to hide a particular curve within a graph. Reviewing your curve-fitting result graphically is the only way to ensure a good fit. After plotting the curve-fitting results, you can then review multiple plots and also verify the error norm/residual value that is printed in the curve-fitting GUI window. This information helps you determine the quality of a curve fit and decide whether to accept the results. If not, you may want to go back to step 3, solve again by changing the order of the Prony series, redefining certain initial values of the coefficients, and possibly other control parameters. You can continue to use your original data, repeating step 3 through step 6 until you are satisfied with the solution.

After a successful curve fitting, the last step is to write the curve-fitting data as a Prony data table (the TB,PRONY command) to the database. The GUI or the command line converts the coefficients to the appropriate form before writing TB commands. The curve-fitting tool calculates coefficients that differ from those shown in the TB,PRONY table. The following equations are used to convert the calculated coefficients to the TB,PRONY format: Curve-Fitting Equation: =
2 0
+
i
2 i
t / i
=1N
Prony Equation:
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=
= +
t/ 0 i
i and i (i = 1 to N) are the Prony coefficients (entered via the TB,PRONY) command. A0 is not entered in the Prony table. Conversion Procedure:
=
2 N

TBFT,FSET,ID,Option2,Option3
where: Option2 = Case Option3 = Case name
5.2.8.2. GUI Method

Click the Write to Database button and the fitted coefficients are automatically written to the material database according Prony data table. Please note that the coefficients you see in the curve-fitting module are different from those in the TB tables. The values remain the same, but the values are different. The values shown are the square of the values derived during curve fitting. They are also normalized to make at time 0 = 1.
5.3. Creep Material Curve Fitting

Use curve fitting to determine your creep material behavior. Thirteen creep models are available, along with the tools to generate and fit derived coefficients to your experimental data. Interactive (GUI) or command line (batch) input is available. The following topics concerning creep material curve fitting are available: 5.3.1. Understanding the Creep Material Curve-Fitting Process 5.3.2. Step 1. Prepare Experimental Data 5.3.3. Step 2. Input the Experimental Data 5.3.4. Step 3. Select a Material Model Option 5.3.5. Step 4. Initialize the Coefficients 5.3.6. Step 5. Specify Control Parameters and Solve 5.3.7. Step 6. Plot the Experimental Data and Analyze 5.3.8. Step 7. Write Data to the TB Command 5.3.9.Tips For Curve Fitting Creep Models
5.3.1. Understanding the Creep Material Curve-Fitting Process

The general process for creep curve fitting is defined as follows:
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Creep Material Curve Fitting Step Detailed Information Found Here 1 Step 1. Prepare Experimental Data (p. 165) Step 2. Input the Experimental Data (p. 167) Step 3. Select a Material Model Option (p. 167) Comments The experimental data must be a plain text file with headers to describe the data types and attributes. The test data must be delimited by a space or a comma. The experimental data can be read into the program by browsing to the file location in the GUI or by specifying the location on the command line. The material options for the applicable curve-fitting regimen are defined via the TB command. Support is offered for 13 implicit creep models. After you choose a model, you can still switch to another model if an ideal fit is not realized. Creep curve fitting is a nonlinear regression; the initial values of the coefficients to be determined can be very important for a successful solution. Choose the error norm to be used for an acceptable curve fit. Review and verify the results by comparing them with the experimental data and the regression errors. If they are unacceptable, repeat steps 3 through 5 to obtain a new curve-fitting solution. Write the curve-fitting results in the TB command format to the database.
Step 4. Initialize the Coefficients (p. 168) Step 5. Specify Control Parameters and Solve (p. 170) Step 6. Plot the Experimental Data and Analyze (p. 170)
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Step 7. Write Data to the TB Command (p. 171)

You need to provide accurate experimental test data in order to achieve valid curve-fitting results. For creep analyses, you use either the creep strain value or creep strain rate, derived as a function of time, temperature, stress, and/or creep strain. The type of data you need to provide will depend on the creep model you choose. The experimental data is named "creep" to distinguish it from other types of data, such as uniaxial, tension, biaxial, etc. The creep data must be a plain text file that contains the headers and the test data in table form, delimited by a space or a comma. The header is used to describe the data types that characterize the test data columns or attributes of the data. Five different creep data types are available: Table 5.5: Creep Data Types and Abbreviations Time Equivalent Creep Strain Equivalent Creep Strain Rate Equivalent Stress Temperature time creq dcreq seqv temp
The header format to define each column's data type is /n, abbr, where n is the index of the data column in the file, and abbr is the abbreviation for the type of data in the column, as described in Table 5.5: Creep Data Types and Abbreviations (p. 165).
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Material Curve Fitting Following is a typical data input file:

/1,seqv ! indicates first column is stress /2,creq ! indicates second column is creep strain /3,temp ! indicates third column is temperature /4,dcreq ! indicates fourth column is creep strain rate 4000 0.00215869 100 0.000203055 4000 0.00406109 100 0.000181314 4000 0.00664691 100 0.000165303 4000 0.0102068 100 0.000152217 4000 0.0151416 100 0.000140946
When a particular column is unchanged over the loading history, you can define it as an attribute. For instance, in the above example, the stress and temperature are constant throughout the range. You define this data as an attribute. The header format to define a data attribute is /attr, value, where attr is the data-type abbreviation, and value is the value of the attribute. The constant stress and temperature values above can be written into the file header, as follows:
/seqv,4000 ! indicate this creep has a constant stress of 4000 /temp,100 ! indicate this creep data is at a constant temperature of 100 /1,creq ! indicate first column is creep strain /2,dcreq ! indicate second column is creep strain rate 0.00215869 0.000203055 0.00406109 0.000181314 0.00664691 0.000165303 0.0102068 0.000152217 0.0151416 0.000140946 0.0220102 0.000130945
Thirteen model types are available for creep curve fitting. The model you choose determines the experimental data required for the curve-fitting process. The following table describes the creep data required to perform curve fitting for each model type. For strain hardening and modified strain hardening, you must input both creep strain and creep strain rate in the experimental data. Table 5.6: Creep Model and Data/Type Attribute Creep Model Strain Hardening Time Hardening Generalized Exponential Generalized Graham Generalized Blackburn Modified Time Hardening Modified Strain Hardening Generalized Garofalo Exponential Form Norton Combined Time Hardening Prim+Sec Rational Polynomial Generalized Time Hardening x x x x x x x x x x x x creq x dcreq x x x x x x x x x x time seqv x x x x x x x x x x x x x x x x x x x x temp x x x x
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Creep Material Curve Fitting

The experimental data must be read in from a plain text file. Prepare this file according to the previous section, including both header information and formatted test data. The header portion is required for creep analyses. Each file is viewed as a data set, and can be a complete set of experimental test data or a part of a series of files of experimental test data. You can include several data sets, such as tests performed at different loading conditions, strain ranges, and temperatures. Input your experimental data using either the batch method or the GUI method.

The EADD argument on the TBFT command is used to identify and specify the location of your data files. The command syntax is:
TBFT,EADD,ID,Option1,Option2,Option3,Option4
where: ID = Index corresponding to the material number Option1 = creep Option2 = name of file containing experimental data Option3 = file name extension Option4 = file directory
5.3.3.2. GUI Method

In interactive mode, you can input experimental data by typing the filename (with the appropriate path if the file is not in the default directory) into the appropriate area. You can also browse to a file in a particular location. Click on Add Data Set to add additional data sets for creep curve fitting. There is no restriction on the number of data sets you can add.

The models available for creep curve fitting are defined in Table 5.7: Creep Models and Abbreviations (p. 168). Select the one that best satisfies your requirements, and use the creep model abbreviation from the table in subsequent fitting operations. It is helpful to view the formula before solving, as you can better determine the initial coefficients to use and the format of your experimental data. See Table 5.7: Creep Models and Abbreviations (p. 168) to determine a starting point for the initial creep model coefficients.

TBFT,FADD,ID,CREEP,Option2,Option3
where: ID = Index corresponding to the material number Option2 = creep model abbreviation. See Table 5.7: Creep Models and Abbreviations (p. 168). Option3 = not used for creep curve fitting.
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Material Curve Fitting The following table describes the creep models available and their abbreviated names for Option2 (above). Table 5.7: Creep Models and Abbreviations Model Number 1 2 3 4 5 6 7 8 9 10 11 12 13 Name Strain Hardening Time Hardening Generalized Exponential Generalized Graham Generalized Blackburn Modified Time Hardening Modified Strain Hardening Generalized Garofalo Exponential Form Norton Prim+Sec Time Hardening Prim+Sec Rational Polynomial Generalized Time Hardening Fitting Name/Option2 shar thar gexp ggra gbla mtha msha ggar expo nort psth psrp gtha
The experimental data must be consistent with the creep model you choose. See Table 5.6: Creep Model and Data/Type Attribute (p. 166) for the data types required for each creep model.
5.3.4.2. GUI Method

You can pick the appropriate model option from a menu in the data entry area. All of the options and constraints listed for batch input apply.

Creep curve fitting is a nonlinear regression process. A successful curve fit depends greatly on the initial coefficient values; certain variances will prevent your curve fit from converging. When a solution doesn't converge, you can adjust the initial value of specific coefficients and rerun the problem (you do not need to input the data again). In general, the more parameters a model has, the more difficult it is to get the solution to converge. Models with many parameters will sometimes converge more easily if you fix (hold constant) your coefficients. You specify a value for a coefficient and keep it unchanged, while allowing the other coefficients to be operated on. You can then release the fixed coefficients to obtain a solution. By default, all of the coefficients are free to vary. Although some creep models implicitly address temperature dependency, other models lack this capability. For both types of models, you can account for temperature dependency by generating coefficients at discrete temperature values. Solve for separate coefficients at each of your desired discrete temperatures. If the equation implicitly supports temperature dependency already, that portion is eliminated by appropriately fixing the coef-
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Creep Material Curve Fitting ficient to a specific value. For example, if you have e(-C*T), then C is set to 0 to eliminate this term. To do this, you activate temperature dependency by setting the tdep parameter to 1. For temperature-dependent experimental data, you enable temperature-dependency and specify a reference temperature before solving for the coefficients. You can set the reference temperature only to values specified using the /temp,value header line in the experimental data. You can also specify tref = all and initiate multiple solves to evaluate coefficients at all available discrete temperature values. In this case, for data at three temperatures (t1, t2, and t3), a single TBFT,SOLVE entry will initiate three different solve operations at those three discrete temperatures. With temperature dependency on and the reference temperature set to a particular value, a TBFT,SOLVE command solves for coefficients only at that temperature. To solve for coefficients at other temperatures, you set the reference temperature to each of the desired discrete temperature values and solve again. You can initialize the coefficients before or after turning temperature dependency on. If the coefficients are initialized before turning temperature dependency on, the specified coefficients will become the initial coefficients for all future solves for that particular model. These coefficients are, however, overridden when temperature dependency is turned on and another set of initial values is specified at discrete temperatures. The curve-fitting tool looks for the initial coefficients at a particular temperature. If no coefficients are specified at discrete temperatures, the initial coefficients set before temperature dependency was turned on are used.

Define your coefficient values and temperature dependency using the SET option of the TBFT command, as follows:
TBFT,SET,ID,CREEP,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = creep model name Option3 = order, if applicable Option4 = index of coefficient Option5 = value of coefficient To modify the coefficients, use the FIX option of the TBFT command.
TBFT,FIX,ID,CREEP,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = creep model name Option3 = order, if applicable Option4 = index of coefficient Option5 = 0 - variable, 1 - fixed
TBFT,SET,ID,CREEP,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = creep model name Option3 = order, if applicable
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Material Curve Fitting Option4 = tdep or tref Option5 = For Option4 = tdep, 1 to activate, 0 to deactivate. For Option4 = tdep, a specific temp value, or all.
5.3.5.2. GUI Method

The models listed in Table 5.7: Creep Models and Abbreviations (p. 168) are available directly from the GUI. When you select a model from the available options, the program automatically configures the coefficients for the model. You can then make modifications, including initializing and/or fixing certain coefficients.

Your error norm, your maximum number iterations allowed, and your error tolerance will affect the accuracy of your results. There are two available error norms available for the regression. The default error norm calculation option is normalized curve fitting. It usually yields better results than non-normalized curve fitting, since normalized fitting gives equal weight to all data points when minimizing the error norms. Other available solve criteria are number of iterations, residual tolerance, and coefficient change tolerance. The solution stops when both residual tolerance and coefficient change tolerance is met or if the number of iterations criteria is met. The coefficients are updated after every iteration during the solve process.

The batch command is:
TBFT,SOLVE,ID,CREEP,Option2,Option3,Option4, ..., Option7
where: ID = Index corresponding to the material number Option2 = creep function name (See Table 5.7: Creep Models and Abbreviations (p. 168)) Option3 = ((Blank--not applicable for creep models) Option4 = Error norm: 0 = non-normalized, 1 = normalized (default) Option5 = Maximum number of iterations Option6 = Tolerance of residual changes Option7 = Tolerance of coefficient changes Other solving parameters are available. See the TBFT command for details.
5.3.6.2. GUI Method

After you fill in the last set of coefficient values, the analysis moves automatically to the solution phase. Each of the options specified in the command line description is presented as a pull down menu or fill in box, and each option must be specified before Solve will begin. The coefficients are updated in the GUI after the solution is complete.

After you initiate Solve, the coefficient tables will contain the fitted coefficients and also the residual errors. You can then plot your data and visually interpret the results. Column one is your X- axis, with each additional column plotted separately as a function of column one.
Creep Material Curve Fitting You should reserve column one for the variable that you would like to see vary in the plot. For example, if your data contains time, temperature, stress and creep strain, you may wish to see the creep strain vary as a function of time, at different temperatures and stresses in the plot. Add your experimental data using multiple TBFT,EADD commands (or the corresponding GUI method). Split the file into multiple experimental fields as prescribed earlier, one for each combination of temperature and stress. Right mouse button (RMB) functions allow you to Zoom, Fit, Save Plot to File, View/Hide Legend and View/Hide Grid. Two or more fitted functions can also be compared in the same plot. For example, you can view Mooney2 Uniaxial and Mooney9 Uniaxial plots right on top of each other, provided both of these function are already solved for. RMB also allows you to see the number of points used to generate the plot, and also change the Minimum X Value and the Maximum X Value in a plot. You can use the middle-mouse button (context sensitive) to hide a particular curve within a graph.
5.3.7.1. GUI Method

You can simultaneously display many data sets for each function plotted. Each window of your display can be used to display each one of the data sets you are plotting against column one.
5.3.7.2. Analyze Your Curves for Proper Fit

Use the GUI to graphically review the curve-fitting result and ensure a good fit throughout the range of data. Use the plotted curve-fitting results both to determine the degree of fit at various locations and to verify the error norm/residual value. You can then determine the quality of a curve fitting and decide whether to accept the results. If the results are unacceptable, you may want to go back to step 3, and then solve again with a new model, redefining certain initial values of the coefficients, and also modifying some of the other control parameters. You can continue to use your experimental data, repeating step 3 through step 6 until you obtain a satisfactory solution.

After you are satisfied with your curve-fitting results, you can write the curve-fitting data to the database using TBFT, FSET. The GUI or the command line converts the coefficients to the appropriate form before writing to TB tables. The data is stored as part of the material property set for use in subsequent analyses.

TBFT,FSET,ID,CREEP,Option2,Option3
where: ID = Index corresponding to the material number Option2 = Creep model abbreviation Option3 = (Blank--not applicable)
5.3.8.2. GUI Method

When you complete the process, click the Write to Database button to write the fitted coefficients of your creep model as a creep data table in the material database. You are then returned to the material properties dialog. The curve data can now be accessed for the full range of material behavior.
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5.3.9. Tips For Curve Fitting Creep Models

The following tips can help to ensure successful curve fitting. They are only suggestions, and do not represent a singular method or strategy. Also, following them does not guarantee a solution. Refer to Creep for additional details about each implicit creep model. Strain Hardening Strain hardening has 4 coefficients, with C4 dedicated to temperature dependency. If you do not have temperature-dependent data, set C4 to zero. If you have difficulty solving temperature-dependent data, use experimental data for only one temperature and fix C4 to zero. Solve, and then add data for your other temperatures. Then release C4 and solve for all coefficients. You can also solve for just C4 by fixing the C1, C2 and C3 values. Time hardening has 4 coefficients, with C4 dedicated to temperature dependency. If you do not have temperature-dependent data, set C4 to zero. If you have difficulty solving temperature-dependent data, use experimental data for only one temperature and fix C4 to zero. Solve, and then add data for your other temperatures. Then release C4 and solve for all coefficients. You can also solve for just C4 by fixing the C1, C2 and C3 values. Generalized exponential has 5 coefficients, with C4 dedicated to temperature dependency. If you do not have temperature-dependent data, set C4 to zero. Use a low value for C5 (e.g., 1e-3) to avoid floating-point overflows in the exponential term. If you have difficulty solving for temperature-dependent data, use experimental data for only one temperature, fix C4 to zero, then solve. Then add data for your other temperatures, release C4 and solve for all coefficients. You can also solve for just C4 by fixing C1, C2 and C3. Generalized Graham has 8 coefficients. You use C8 for temperature dependency. If you do not have temperature-dependent data, set C8 to zero. If you have difficulty solving temperature-dependent data, use experimental data for only one temperature, fix C8 to zero, and solve. Then add data for other temperatures, release C8 and solve for the remaining coefficients individually. Generalized Blackburn has 7 coefficients. It is advisable to look at the exponential terms and try to keep them from floating-point overflows. To keep eC2 within floating-point range, make sure the initial value of C2 is such that C2 is close to 1. Similarly try to keep /C4 and C7 close to 1. Modified time hardening has 4 coefficients. C4 is for temperature dependency. If you do not have temperature-dependent data, fix C4 to zero. Modified Strain Hardening has 3 coefficients. This model can be
C C2 3 + complex for curve fitting. To keep the 1 term from going negative, C1 is replaced with C12 but converted to the proper form before beings written to the database.
Time Hardening
Generalized Exponential
Generalized Graham
Generalized Blackburn
Modified Time Hardening Modified Strain Hardening
Generalized Garofalo
Generalized Garofalo has 4 coefficients, with C4 dedicated to temperature dependency. If you do not have temperature-dependent
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Chaboche Material Curve Fitting data, set C4 to zero. To keep the Sinh term within floating-point range, keep c2 close to one when you initialize the coefficients. Exponential Form Exponential form has 3 coefficients, with C3 dedicated to temperature dependency. If you do not have temperature-dependent data, set C3 to zero. To keep eC2 within floating-point range, keep /C2 close to one. Norton model has 3 coefficients, with C3 dedicated to temperature dependency. If you do not have temperature-dependent data, set C3 to zero. Time hardening has 7 coefficients. This is a complex model. Here it is advisable to solve for temperature independent data first and then introduce temperature related data. Rational polynomial is a very complex model for curve fitting, with 10 coefficients. If you find it hard to fit this data, it is advisable that you split the experimental data into primary creep data and secondary creep data. Primary creep data is the initial part of the curve that covers the nonlinearity in the strain rate. Fit only the secondary data by fixing C1 to 1 and then set all other coefficients except C2, C3 and C4 to zero. Use a low value of C3 to keep 10C3 within floating-point range. Coefficients C5 to C10 in curve fitting refers to coefficients C7 to C12 in the implicit creep equation (rational polynomial). Then add the primary creep data, release all coefficients, and solve. Generalized time hardening has 6 coefficients. Set C6 to zero if you have temperature independent data. When initializing coefficients set C5 close to 1 to avoid floating-point overflows.
Norton
Prim+Sec Time Hardening Prim+Sec Rational Polynomial
Generalized Time Hardening
5.4. Chaboche Material Curve Fitting

Chaboche material curve fitting determines your material constants by relating your experimental data to the Chaboche nonlinear kinematic hardening model. Curve fitting is performed either interactively or via batch commands. You can fit uniaxial plastic strain vs. stress data, along with discrete temperature dependencies for multiple data sets. The following topics concerning Chaboche material curve fitting are available: 5.4.1. Understanding the Chaboche Material Curve-Fitting Process 5.4.2. Step 1. Prepare Experimental Data 5.4.3. Step 2. Input the Experimental Data 5.4.4. Step 3. Select a Material Model Option 5.4.5. Step 4. Initialize the Coefficients 5.4.6. Step 5. Specify Control Parameters and Solve 5.4.7. Step 6. Plot the Experimental Data and Analyze 5.4.8. Step 7. Write Data to the TB Command
5.4.1. Understanding the Chaboche Material Curve-Fitting Process

Chaboche material curve fitting determines your material constants by relating your experimental data to the Chaboche nonlinear kinematic hardening model. Isotropic hardening can also be modeled by including a supported isotropic hardening model with the kinematic hardening model in the curve-fitting process.
Material Curve Fitting Following is the general process for using curve fitting to determine the coefficients for the Chaboche material model: Step Detailed Information Found Here 1 Step 1. Prepare Experimental Data (p. 174) Step 2. Input the Experimental Data (p. 175) Step 3. Select a Material Model Option (p. 176) Step 4. Initialize the Coefficients (p. 177) Step 5. Specify Control Parameters and Solve (p. 179) Step 6. Plot the Experimental Data and Analyze (p. 180) Comments The experimental data must be a plain text file with headers to describe the data types and attributes. The test data must be delimited by a space or a comma. The experimental data can be read into the program by browsing to the file location in the GUI or by specifying the location on the command line. The material options for the applicable curve-fitting regimen are defined (TBFT). This step includes selecting the kinematic hardening model. Chaboche curve fitting is a nonlinear regression; the initial values of the coefficients to be determined is important for a successful solution. Specify the error norm to be used, the solution control parameters, and perform the nonlinear regression. Review and verify the results by comparing them with the experimental data and the regression errors. If any factor is unacceptable, repeat steps 3 through 5 to obtain a new curve-fitting solution. Write the curve-fitting results in the TB command format to the database.
5 6
Step 7. Write Data to the TB Command (p. 181)

Curve fitting requires experimental test data. To use curve fitting with plasticity, the only experimental data supported is uniaxial test data. Uniaxial test data has two columns, plastic strain and true stress. Experimental data for plasticity is path dependent. Your uniaxial test data must be a plain text file with headers to define the test data. The data file should be in table format, delimited by spaces or commas. Headers can be used to describe the data types that characterize the test data columns or additional attributes of the data. For Chaboche curve fitting with multiple temperatures, you can evaluate coefficients at each discrete temperature point and write it as a temperature-dependent Chaboche data table. A separate data file is necessary for each discrete temperature. Issue this command at the top of the experimental data file to specify the temperature for the experiment:
/temp,TempValue
where TempValue is your specified temperature. Following is a typical data input file:
/temp,100 ! define temperature attribute 4.57E-06 2.43E+02 4.89E-04 2.67E+02 1.01E-03 2.83E+02 1.55E-03 2.94E+02
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Chaboche Material Curve Fitting

2.11E-03 3.02E+02 2.68E-03 3.09E+02 3.26E-03 3.14E+02 3.84E-03 3.18E+02 4.42E-03 3.22E+02 4.42E-03 7.78E+01 4.41E-03 -1.65E+02 3.54E-03 -2.31E+02 2.51E-03 -2.66E+02 1.40E-03 -2.84E+02 2.49E-04 -2.95E+02 -9.11E-04 -3.03E+02 -2.08E-03 -3.10E+02 -3.24E-03 -3.17E+02 -4.41E-03 -3.24E+02 -4.41E-03 -7.95E+01 -4.40E-03 1.63E+02 -3.53E-03 2.30E+02 -2.50E-03 2.65E+02 -1.39E-03 2.83E+02 -2.44E-04 2.94E+02 9.16E-04 3.02E+02 2.08E-03 3.09E+02 3.25E-03 3.16E+02 4.41E-03 3.23E+02 4.41E-03 7.87E+01 4.41E-03 -1.64E+02 3.53E-03 -2.31E+02 2.51E-03 -2.66E+02 1.40E-03 -2.84E+02 2.48E-04 -2.95E+02
Uniaxial test data can include loading, unloading, and cyclic loading. For plasticity, experimental data is path-dependent and the stress-strain behavior depends on the history of the loading and/or unloading.

The experimental data must be read in from a plain text file (TBFT,EADD). Prepare this file as described in Step 1. Prepare Experimental Data. Each file is viewed as a data set, and can be a complete set of experimental test data or a part of a series of files of experimental test data. You can include several data sets, such as tests performed at different stress levels and/or temperatures, when you perform creep curve fitting. Input your experimental data using either the batch method or the GUI method.

Issue this command to identify and specify the location of a data file:
TBFT,EADD,ID,Option1,Option2,Option3,Option4
where: ID = Index corresponding to the material number Option1 = Experimental data type UNIA (uniaxial test data) Option2 = Experimental data file name Option3 = File name extension Option4 = File directory
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5.4.3.2. GUI Method

Click on Add Data Set and enter the experimental data file name in the field provided. You can also browse to a file in a specified location. Separate input is performed for each data type (Option1 = SDEC or BDEC) /

The TBFT command provides the curve-fitting tools for Chaboche material modeling. To define the material model, you must specify a case name, the order of Chaboche kinematic model, and finally the isotropic hardening option if needed. After you create a case, the number of Chaboche terms or the isotropic hardening options cannot be changed without deleting the case. Additional cases with different case names can be created to model Chaboche kinematic hardening with different orders.

Define a Chaboche material model by defining a case (TBFT,FCASE), then specifying the order of kinematic hardening model. The case is created only after the TBFT,FCASE,ID,FINISH command executes. The following syntax example and argument descriptions illustrates a complete case definition:
TBFT,FCASE,ID,NEW,Option2,Option3 ! define case
where: ID = Index corresponding to the material number Option2 = CPLA Option3 = Your specified case name
TBFT,FADD,ID,Category,Option2,Option3 ! specify kinematic hardening model
where: ID = Index corresponding to the material number Category = PLAS Option2 = CHABOCHE or BISO or MISO or VOCE Option3 = Dependent on Option2, as follows: When Option2 = CHABOCHE, Option3 = 1 to N When Option2 = MISO, Option3 = 1 to Niso When Option2 = BISO or VOCE, Option3 is not used
TBFT,FCASE,ID,FINISH ! create case
5.4.4.2. GUI Method

Interactively navigate the tree structure of the curve-fitting window. Select the order of the Chaboche model, select the isotropic hardening option if needed, then specify the appropriate case name in the text box field. As you select the options, the coefficient table is created automatically.
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The initial values chosen for your coefficients (TBFT,SET) determines the success of the curve-fitting operation. The number of parameter depends on the order of the Chaboche model. For a Chaboche model of order N, there are 2*N+1 coefficients. Coefficients 1 through 2*N are Chaboche parameters, and coefficient 2*N+1 is the yield stress. The odd coefficients in the Chaboche model refer to the slope of the curve and the even coefficients are decay function parameters. The number of terms depend on the complexity of the curve. Evenly distributed estimates of the slope over the range of the curve can be used as initial guesses for the curve-fitting process. Different terms can dominate at different parts of the curve. Initial guesses for the decay parameters can typically be chosen one or two orders less than odd coefficients (slope).
5.4.5.1. Including Isotropic Hardening Models with Chaboche Kinematic Hardening

When isotropic hardening models are included with Chaboche kinematic hardening, the number of coefficients is 2*N+Niso, where Niso is 2 for the BKIN option, 2*Ni for the MISO option (where Ni is the number of MISO terms), and 4 for the VOCE NLISO option. The index of the isotropic hardening coefficients start from 2*N+1. Initial yield stress is generally estimated from separate experiments outside of curve fitting and is generally fixed in the curve-fitting process. For the BKIN and VOCE options, the first isotropic hardening coefficient refers to yield stress.
5.4.5.2. General Process for Initializing MISO Option Coefficients

The MISO option used here is same as the TB,PLAS,,,MISO option. It has Ni coefficients, and the second isotropic hardening coefficient refers to the yield stress. Odd isotropic coefficients are plastic strain values and the even coefficients are stress values. The actual index of the coefficient is 2*N+Niso, where N is the order of the Chaboche model. If the experiment has a maximum accumulated plastic strain of pl,max, set the odd coefficients to values equally distributed from 0 to pl,max. These values are then fixed (TBFT,FIX) before solving (TBFT,SOLVE). If you require greater accuracy at certain strain ranges, you can distribute the strain values unevenly as you wish. To fix (hold constant) your coefficients (TBFT,FIX), specify a value for a coefficient and keep it unchanged, while allowing the other coefficients to be operated on. You can then release the fixed coefficient later if desired. By default, all of the coefficients are free to vary. Estimate coefficients for temperature-dependent data by setting the temperature-dependency flag and setting a reference temperature before solving for the coefficients. You can set the reference temperature only to values specified via the /temp,TempValue header line in the experimental data. You can also specify tref = all and initiate multiple solves to evaluate coefficients at all available discrete temperature values. In this case, for data at three temperatures (t1, t2, and t3), a single TBFT,SOLVE command initiates three separate solve operations at those three discrete temperature values, and generates data at three corresponding discrete temperatures. With temperature dependency specified and the reference temperature set to a specific value, a TBFT,SOLVE command solves for coefficients only at that discrete temperature. To solve for coefficients
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Material Curve Fitting at other temperatures, set the reference temperature to each of the desired discrete temperature values and solve again. You can initialize the coefficients before or after activating temperature dependency. If the coefficients initialize before setting temperature dependency, the specified coefficients become the initial coefficients for all future solves for that particular model. These coefficients are, however, overridden when temperature dependency is active and another set of values is specified at a discrete temperature value. The curve-fitting tool looks for the initial coefficients at a particular temperature. If no coefficients are specified at discrete temperature values, the initial coefficients set before temperature dependency was activated are used.
5.4.5.2.1. Batch Method

The following syntax example and argument descriptions illustrate coefficient initialization:
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = Index of coefficient Option5 = Value of coefficient Example 5.3: Initialize Coefficients
TBFT,SET,1,case1,,1,1.2 ! Initialize the first coefficient to 1.2 TBFT,SET,1,case1,,2,1.5 ! Initialize the second coefficient to 1.5
By default, coefficients are not fixed. To fix a coefficient to a value (p. 177) set via the TBFT,SET command, or to release a previously fixed coefficient, issue the TBFT,FIX command.
TBFT,FIX,ID,CASE,Option2,Option3,Option4,Option5
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = Index of coefficient Option5 = 1 to fix, 0 to vary (default) Temperature dependency uses Option4 and references your specified data files with the appropriate temp header:
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = tdep or tref Option5 = If Option4 = tdep, then 1 activates temperature dependency 0 deactivates it. If Option4 = tref, this value is either a specific temperature or all temperatures (ALL (p. 177)).
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Chaboche Material Curve Fitting Example 5.4: Fixing Coefficient Values

TBFT,FIX,1, case1,,1,1 ! Fix the first coefficient to a value set via TBFT,SET TBFT,FIX,1, case1,,2,1 ! Fix the second coefficient to a value set via TBFT,SET
5.4.5.2.2. GUI Method

The coefficients table is updated automatically in the Chaboche curve-fitting window when the order of kinematic hardening model is specified. Specify values for your coefficients in the coefficients table in the curve-fitting GUI window, and check the appropriate boxes to fix them or allow them to vary.

Chaboche curve fitting is a nonlinear regression process. Only the non-normalized error norm is available for the regression. The solution control parameters for nonlinear regression include number of iterations, residual tolerance, and coefficient change tolerance. The solution stops when both residual tolerance of the error norm and coefficient change tolerance is met, or if the number of iterations criteria is met. The coefficients are updated when the solution is completed.
5.4.6.1. Temperature-Dependent Solutions

Separate coefficients at each temperate must be calculated to account for temperature dependency. Perform the regression as follows: 1. Set the temperature-dependency flag (TBFT,SET,,,,,TDEP,1). 2. Solve for all (tref = ALL) coefficients. a. Set the reference temperature at which your partial solution should be performed (TBFT,SET,,,,,TREF,TX). Only data at temperature TX is used to estimate the combined plasticity coefficients. b. Initialize the coefficients. c. Solve (TBFT,SOLVE). Repeat the regression process for all desired temperatures.

The following syntax examples and argument descriptions illustrate how to set control parameters and solve:
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = tdep Option5 = 1 to activate temperature dependency, 0 to deactivate (default)
where:
Material Curve Fitting ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = tref Option5 = Valid temperature values found in the experimental data The SOLVE option allows you to specify procedure types, tolerances, and the number of iterations:
TBFT,SOLVE,ID,CASE,Option2,Option3,Option4, ... , Option7
where: ID = Index corresponding to the material number Option2 = Case name Option3 = (Blank--not applicable) Option4 = Curve-fitting procedure: 0 = non-normalized least squares Option5 = Maximum number of iterations Option6 = Residual change tolerance Option7 = Coefficient change tolerance Other parameters for solving are available. See the TBFT command for more information.
5.4.6.3. GUI Method

The GUI allows you to choose your error norm, solution control parameters, and solver options. After you complete these specifications and solve, you can go back and modify your parameters as necessary to obtain a good curve fit.

The best method for ensuring a good fit between your experimental data and the provided curves is to plot your curves and visually inspect them via the GUI. The Graph button provides a direct means for plotting the data.
5.4.7.1. Analyzing Your Curves for Proper Fit

All of your data is plotted as a function of column 1 (X axis). Column 2 (Y axis) and the corresponding fitted data are plotted as a function of column 1. Two or more fitted functions can be compared in the same plot. Take advantage of the right-mouse-button (RMB) functions to zoom, fit, save your plot to a file, view or hide objects, toggle between log scale and regular scale, and so on. With the middle mouse button, you can eliminate specific curves from each window's display in order to view the remaining data more clearly. After plotting the curve-fitting results, you can then review multiple plots and also verify the error norm/residual value displayed in the curve-fitting GUI window. This information helps you to determine the quality of a curve fit and decide whether or not to accept the results. If the curve-fitting results are unsatisfactory, you may want to go back to Step 3. Select a Material Model Option and solve again by changing the order of the Chaboche model or other options, redefining certain initial values of the coefficients, and possibly redefining other control parameters. You can continue to use your original data, repeating step 3 through step 6 until you are satisfied with the solution.
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After a successful curve fitting, the last step is to write the curve-fitting data as a Chaboche data table (TB,CHABOCHE) to the database. The program converts the coefficients to the appropriate form before writing to the database.
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Chapter 6: Material Model Combinations

You can combine some material models to simulate various material behaviors. The following table presents the model options you can combine along with the associated TB command labels, and links to sample input listings located under Material Model Combinations in the Structural Analysis Guide. If a material model does not appear in the table, it cannot be combined with another material model. Table 6.1: Material Model Combination Possibilities Model Plasticity Plasticity Plasticity Plasticity Plasticity Plasticity Viscoplasticity Viscoplasticity Viscoplasticity Viscoplasticity Viscoplasticity Viscoplasticity Viscoplasticity Viscoplasticity Gurson Plasticity With ... Combined Hardening Combined Hardening Combined Hardening Combined Hardening Combined Hardening Combined Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Combined Hardening Combined Hardening Combined Hardening Combined Hardening Isotropic Hardening Combination Type Bilinear Multilinear Multilinear Nonlinear Multilinear Multilinear Bilinear Multilinear Multilinear Nonlinear Nonlinear Nonlinear Nonlinear Nonlinear Bilinear Command, Label TB,BISO + TB,CHAB TB,MISO + TB,CHAB TB, PLAS,,,,MISO + TB,CHAB TB,NLISO + TB,CHAB TB,PLAS,,,,MISO + TB,EDP TB,MISO+ TB,EDP TB,PLAS,,,,BISO + TB,EDP TB,MISO + TB,RATE TB,PLAS,,,,MISO + TB,RATE TB,NLISO + TB,RATE TB,CHAB+ TB,RATE+TB,BISO TB,CHAB+ TB,RATE+TB,MISO TB,CHAB+ TB,RATE+TB,PLASTIC TB,CHAB+ TB,RATE+TB,NLISO TB,GURS + TB,BISO Link to Example BISO and CHAB Example MISO and CHAB Example PLAS (Multilinear Isotropic Hardening) and CHAB Example NLISO and CHAB Example PLAS (MISO) and EDP Example MISO and EDP Example BISO and RATE Example MISO and RATE Example PLAS (MISO) and RATE Example NLISO and RATE Example CHAB and RATE and BISO Example CHAB and RATE and MISO Example CHAB and RATE and PLASTIC Example CHAB and RATE and NLISO Example GURSON and BISO Example
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Material Model Combinations Model Gurson Plasticity Gurson Plasticity Gurson Plasticity Plasticity and Creep (Implicit) Plasticity and Creep (Implicit) Plasticity and Creep (Implicit) Plasticity and Creep (Implicit) Plasticity and Creep (Implicit) Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity Anisotropic Plasticity With ... Isotropic Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Kinematic Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening Kinematic Hardening Kinematic Hardening Kinematic Hardening Kinematic Hardening Kinematic Hardening Combined Hardening Combined Hardening Combination Type Multilinear Multilinear Nonlinear Bilinear Multilinear Multilinear Nonlinear Bilinear Bilinear Multilinear Multilinear Nonlinear Bilinear Multilinear Multilinear Multilinear Chaboche Bilinear Isotropic and Chaboche Multilinear Isotropic and Chaboche Command, Label TB,GURS + TB,MISO TB,GURS + TB,PLAS,,, MISO TB,GURS + TB,NLISO TB,BISO + TB,CREEP TB,MISO + TB,CREEP TBPLAS,,,,MISO + TB,CREEP TB,NLISO + TB,CREEP TB,BKIN + TB,CREEP TB,HILL + TB,BISO TB,HILL + TB,MISO TB,HILL + TBPLAS,,,,MISO TB,HILL + TB,NLSIO TB,HILL + TB,BKIN TB,HILL + TB,MKIN TB,HILL + TB,/ KINH TB,HILL + TBPLAS,,,, KINH TB,HILL + TB,CHAB TB,HILL + TB,BISO + TB,CHAB TB,HILL + TB,MISO + TB,CHAB Link to Example GURSON and MISO Example GURSON and PLAS (MISO) Example GURSON and NLISO Example BISO and CREEP Example MISO and CREEP Example PLAS (MISO) and CREEP Example NLISO and CREEP Example BKIN and CREEP Example HILL and BISO Example HILL and MISO Example HILL and PLAS (MISO) Example HILL and NLISO Example HILL and BKIN Example HILL and MKIN Example HILL and KINH Example HILL and PLAS (KINH) Example HILL and CHAB Example HILL and BISO and CHAB Example HILL and MISO and CHAB Example
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Model
With ...
Combination Type Multilinear Isotropic and Chaboche Nonlinear Isotropic and Chaboche Bilinear Multilinear Nonlinear --Bilinear
Command, Label TB,HILL + TB,PLAS,,,,MISO + TB,CHAB TB,HILL + TB,NLISO + TB,CHAB TB,HILL + TB,RATE + TB,BISO TB,HILL + TB,RATE + TB,MISO TB,HILL + TB,RATE + TB,NLISO TB,HILL + TB,CREEP TB,HILL + TB,CREEP + TB,BISO TB,HILL + TB,CREEP + TB,MISO TB,HILL + TB,CREEP + TB,MISO TB,HILL + TB,CREEP + TB,NLISO TB,HILL +
Link to Example HILL and PLAS (MISO) and CHAB Example
Anisotropic Plasticity
Combined Hardening
Anisotropic Plasticity Anisotropic Viscoplasticity Anisotropic Viscoplasticity Anisotropic Viscoplasticity Anisotropic Creep (Implicit)
Combined Hardening Isotropic Hardening Isotropic Hardening Isotropic Hardening
HILL and NLISO and CHAB Example
HILL and RATE and BISO Example HILL and RATE and MISO Example HILL and RATE and NLISO Example HILL and CREEP Example HILL and CREEP and BISO Example HILL and CREEP and MISO Example HILL and CREEP and PLAS (MISO) Example HILL and CREEP and NLISO Example HILL and CREEP and BKIN Example AHYPER and PRONY (Anisotropic Hyperelasticity and Viscoelasticity) Example HYPER and PRONY (Hyperelasticity and Viscoelasticity) Example EDP and CREEP and PLAS (MISO) Example
Anisotropic Isotropic Creep and PlastiHardening city (Implicit) Anisotropic Isotropic Creep and PlastiHardening city (Implicit) Anisotropic Isotropic Creep and PlastiHardening city (Implicit) Anisotropic Isotropic Creep and PlastiHardening city (Implicit) Anisotropic Kinematic Creep and PlastiHardening city (Implicit) Anisotropic Hyperelasticity and Viscoelasticity Hyperelasticity and Viscoelasticity
Multilinear
Multilinear
Nonlinear
Bilinear
Finite Strain Nonlinear Visco-AnisoAnisotropic tropic HypereElasticity lasticity Finite Strain Visco-Hyperelasticity Nonlinear Isotropic Elasticity Bilinear, Multilinear, or Nonlinear
TB,AHYPER + TB,PRONY
TB,HYPER + TB,PRONY TB,EDP + TB,CREEP + TB,BISO or TB,MISO or TB,NLISO
Extended Drucker-Prager (EDP) Isotropic and Creep (Impli- Hardening cit)
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Material Model Combinations Model Cap and Creep (Implicit) With ... Isotropic Hardening Combination Type Bilinear, Multilinear, or Nonlinear Command, Label TB,EDP + TB,CREEP + TB,BISO or TB,MISO or TB,NLISO Link to Example CAP and CREEP and PLAS (MISO) Example
Following are cross-reference links to other sections in the documentation that provide descriptions of the individual material model options represented in the table above. Bilinear Isotropic Hardening (TB,BISO) Bilinear Kinematic Hardening (TB,BKIN) Chaboche Nonlinear Kinematic Hardening (TB,CHAB) Creep (Implicit) (TB,CREEP) -- Also see Creep in the Structural Analysis Guide. Hill Anisotropy (TB,HILL] Multilinear Isotropic Hardening (TB,MISO) Multilinear Kinematic Hardening (TB,MKIN or KINH) Nonlinear Isotropic Hardening (TB,NLISO). Rate-Dependent Plasticity (TB,RATE) Hyperelasticity (TB,HYPER) Anisotropic Hyperelasticity (TB, AHYPER) -- Also see Anisotropic Hyperelasticity in the Mechanical APDL Theory Reference. Viscoelasticity
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Chapter 7: Understanding Field Variables

When you define field-dependent material properties (TBFIELD and related TB commands), the program uses linear interpolation between the data points you provide to determine specific material property values. To do so, the program: 1. Creates a grid using your defined field data values. 2. Assumes that you have defined curve-based data and automatically provides the missing grid data points. 3. Performs linear interpolation on this populated grid to find material property values. The following related topics are available: 7.1. User-Defined Field Variables 7.2. Data Processing 7.3. Example: One-Dimensional Interpolation 7.4. Example: Two-Dimensional Interpolation 7.5. Example: Multi-Dimensional Interpolation
7.1. User-Defined Field Variables

For elastic (TB,ELASTIC) and thermal expansion (TB,CTE) material models, you can define your own field variables (TBFIELD). Valid user-defined field variables are UF01 through UF09. Use your field variables with initial state loading to define a field of material properties. When you define a field variable as node-based field data over the finite element model (INISTATE), the solver reads the initial-state data at the nodes and determines the correct value of the material property necessary for performing the finite element calculations. The following example input shows how to define Youngs Modulus as a function of global Y using the elastic material model and the node-based initial state capability: Example 7.1: Defining Youngs Modulus as Function of Global Y
TB,ELASTIC,1 TBFIELD,UF01,0.0 TBDATA,1,1e6,0.3 TBFIELD,UF01,1.0 TBDATA,1,1e7,0.3 ! Define a UF01 field over the finite element model INISTATE,SET,NODE,1 INISTATE,SET,DTYP,UF01 *GET,NumNodes,node,,num,max *DO,iI,1,NumNodes iExis=NSEL(iI) *IF,iExis,eq,1,then yval=NY(iI) INISTATE,DEFINE,iI,,,,yval *ENDIF *ENDDO
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Understanding Field Variables
7.2. Data Processing

Consider a case where isotropic friction (TB,FRIC) field data is dependent on both temperature and sliding distance. Assume the following command input:
TB,FRIC,1,2, ,ISO TBFIELD,TEMP,100.0 TBFIELD,SLDI,0.1 TBDATA,1,0.3 TBFIELD,SLDI,0.5 TBDATA,1,0.5 TBFIELD,TEMP,200.0 TBFIELD,SLDI,0.2 TBDATA,1,0.2 TBFIELD,SLDI,0.7 TBDATA,1,0.1
Listing the data shows a user-defined 4x2 grid:

(FRIC) Table For Material 1
Data for Isotropic friction TEMPERATURE SLIDE DIST FRICTION DIR. 1 TEMPERATURE SLIDE DIST FRICTION DIR. 1 TEMPERATURE SLIDE DIST FRICTION DIR. 1 TEMPERATURE SLIDE DIST FRICTION DIR. 1 = 100.00 = 0.10000 FRICTION COEFF. 0.30000 = 100.00 = 0.50000 FRICTION COEFF. 0.50000 = 200.00 = 0.20000 FRICTION COEFF. 0.20000 = 200.00 = 0.70000 FRICTION COEFF. 0.10000
A tabular format represents the data in the 4x2 grid as shown: Sliding Distance Temperature 100 200 0.1 0.2 0.5 0.7 0.3 0.2 0.5 0.1
When defining tabular data, the first specified field variable forms the rows of the table. The subsequent variables form the columns. In this example, Temperature is the first defined field variable. In this case, the user defined only four out of a possible eight grid locations. To populate the interpolation search space, the program fills the missing grid points in each row from left to right. If the first or subsequent grid locations of a row are not defined, the program uses the first
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Example: Two-Dimensional Interpolation defined value within the row to backfill the grid. The program then fills any undefined locations within the grid by linearly interpolating between defined points in each row. If the last value(s) along a row are not defined, the program gives them the last previously defined value within that row. Therefore, based on the defined field-dependent friction values, the program generates the following grid automatically (where values in italics represent those provided by the program): Sliding Distance Temperature 100 200 0.1 0.2 0.5 0.7 0.3 0.35 0.5 0.5 0.2 0.2 0.14 0.1
7.3. Example: One-Dimensional Interpolation

To demonstrate the interpolation of data in a sparsely defined grid, consider the results of a 2-D interpolation at a temperature of 100 and a sliding distance of 0.40. In this case, the program performs only a 1-D interpolation because the defined temperature value (100) lies directly on the defined grid field. For this case, the program obtains a friction coefficient value of 0.45 based on the following calculations: = 0 + 1 0 where = Equation (2) Equation (1)
Substituting the tabular values x = 0.4, y0 = 0.35 x0 = 0.2, y1 = 0.5 x1 = 0.5
Substituting these values into Equation (2): = = Equation (3)
and solving for the interpolated values using Equation (1), we obtain = + = Equation (4)
7.4. Example: Two-Dimensional Interpolation

Consider the case where a true 2-D interpolation is required at a temperature of 180 and a sliding distance of 0.40. The program performs three different linear interpolations to determine the property value
189
Understanding Field Variables within the grid. When performing two-dimensional interpolation, the program always interpolates first along the two relevant rows of the grid (Temperature in this case), then between the rows. In this example, the program performs the first interpolation at a temperature of 100 and a sliding distance of 0.4, yielding the result of 0.45 (as shown in Equation (4)). The program performs the second interpolation for a temperature of 200 and a sliding distance of 0.4. In this case, we find that x = 0.4, y0 = 0.2 x0 = 0.2, y1 = 0.14 x1 = 0.5
Finally, the program performs a third interpolation between the temperature value of 100 and 200 at a sliding distance of 0.4. t = 180, y0 = 0.45 t0 = 100, y1 = 0.16 t1 = 200
7.5. Example: Multi-Dimensional Interpolation

The program uses a radial-basis interpolation function to perform interpolation when more than two field variables are input. You can input a random collection of data points where the material property was experimentally evaluated. It is not necessary to input data in any specific grid format. It is your responsibility to provide enough data to cover the region that you intend to interpolate your experimental data. Nonlinear interpolation can produce results slightly different from that of a linear interpolation function. Because the data is a random sampling, the algorithm creates an n-dimensional rectangular bounding box, and the queries outside of the bounding box are projected to fall on the surface or edge of the
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Example: Multi-Dimensional Interpolation bounding box. The projection is done by calculating a normal to each axis until it finds a position within the bounding box. In the following figure, the stars represent the data points input that you provide, and the blue stars represent queries outside of the bounding box:
The figure shows how the data points are projected first in one dimension and then in the second dimension. This method was extrapolated and implemented for multiple dimensions. Implementation of the radial basis function is expressed as: =
i N
=1
( =
1
j,i
where N is the number of data points and O is the number of free variables (or the order of the interpolation). Input data is (xj,1, xj,2, , zj) where j varies from 1 to the N. The unknown values are ai (where i varies from 1 to N ) and c. The equation is evaluated for all data points provided in the input to calculate the ai and c values. Reference For further information about multidimensional field-variable interpolation, consult this reference: 1. Amidror, Isaac. Scattered Data Interpolation Methods for Electronic Imaging Systems: A Survey . Journal of Electronic Imaging 11:2 (2002: 157-176.
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Chapter 8: GUI-Inaccessible Material Properties

The following material properties are not available via the material property menus of the interactive GUI. You can specify them from the command line or input file. Subsequent graphic display and postprocessing is still displayed. Material Property Anisotropic hyperelasticity Bergstrom-Boyce Mullins effect Crack growth fracture criterion Coefficient of thermal expansion Cohesive zone Damage evolution law Damage initiation criteria Extended Drucker-Prager Experimental data Failure criteria material strength limits Fluid Coefficient of friction Gurson pressure-dependent plasticity Porous media TB Command Lab Value AHYPER BB CDM CGCR CTE CZM DMGE DMGI EDP EXPE FCLI FLUID FRIC GURSON PM
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ANSYS Mechanical APDL Material Reference

Release 14.5 October 2012

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U.S. Government Rights

Chapter 1: Introduction to Material Models

1.1. Material Models for Displacement Applications

Plastic and elastic-plastic

Expansion and swelling

Shape memory alloy

1.2. Material Models for Temperature Applications

1.3. Material Models for Electromagnetic Applications

1.4. Material Models for Coupled Applications

1.5. Material Parameters

Chapter 2: Material Model Element Support

Anisotropic plasticity SOLID65 Nonlinear legacy elements only

Magnetic Bilinear isotropic hardening

Crack growth fracture criterion

CHABOCHE Chaboche nonlinear kinematic hardening

Label (Lab) CONCR

Material Model Concrete or concrete damage

Coefficient of thermal expansion

Explicit springdamper (discrete) Damage evolution law

Damage initiation criteria

Drucker-Prager plasticity Anisotropic electric permittivity

EOS EVISC EXPE FCON FCLI FLUID FOAM FRIC

Equation of state Viscoelasticity Experimental data Fluid conductance data

GASKET GCAP GURSON

Gasket Geological cap Gurson pressure-dependent plasticity

HFFDLD HFLM HILL

Frequency-dependent lossy dielectric Film coefficient data Hill anisotropy

Anisotropic electric and magnetic loss tangents Multilinear elasticity

SOLID65 Nonlinear legacy elements only

Multilinear isotropic hardening

Elements CPT212, CPT213, CPT215, CPT216, CPT217

Material structural damping

Shift function for viscoelastic materials

Shape memory alloy

Chapter 3: Material Models

3.1. Understanding Material Data Tables

3.2. Experimental Data

3.3. Linear Material Properties

Linear Material Properties 3.3.10. How Material Properties Are Evaluated

3.3.1. Defining Linear Material Properties

None None Voltage/Temp

Length /Time Mass/Length Mass/Length

3.3.2. Stress-Strain Relationships

3.3.3. Anisotropic Elasticity

3.3.5. Thermal Expansion

The program calculates structural thermal strain as follows: th = se(T) * (T - TREF)

3.3.7. Specific Heat

3.3.8. Film Coefficients

3.3.9. Temperature Dependency

3.3.10. How Material Properties Are Evaluated

3.4. Rate-Independent Plasticity

Material Models Figure 3.1: Stress-Strain Curve for an Elastic-Plastic Material

3.4.1. Understanding the Plasticity Models

Definition Anisotropic directional yield strength Young's Modulus