Theoretical analysis of strain-and stress-based forming limit diagrams

Original Article Theoretical analysis of strain- and stress-based forming limit diagrams J Strain Analysis 48(3) 177–188 Ó IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0309324712468524 sdj.sagepub.com Surajit Kumar Paul Abstract Forming limit diagram is extensively used in the analysis of sheet metal forming to define the limit of deformation of materials without necking or fracture. This article contributes in construction of strain- and stress-based forming limit diagrams by theoretical analysis of several available instability criteria, including bifurcation analysis of diffuse and through-thickness neck formation, shear stress-based criteria onset of through-thickness necking, and so on, and comparative studies among different available theories are done with earlier published work on forming limit in different literatures. The scope of Tresca criterion in forming limit diagram is also investigated in this article. It is found that apart from stress-based forming limit diagram, conventional failure criterion equivalent failure strain versus triaxiality (h) plot is also strain path independent. Keywords Strain- and stress-based forming limit diagram, equivalent failure strain versus triaxiality plot, Hill–Tresca criterion, Hill– Bressan–Williams criterion, Hill–Swift criterion, Storen–Rice criterion Date received: 21 June 2012; accepted: 26 October 2012 Introduction Sheet metal formability can be defined as the ability of metal to deform without necking or fracture into desired shape. Every sheet metal can be deformed without failing only up to a certain limit, which is normally known as forming limit curve (FLC). FLC is generally governed by localized necking, which eventually leads to the ductile fracture. FLC can be represented as a curve of the major strain (e1) at the onset of localized necking for all values of the minor strain (e2), and the full graph is named as forming limit diagram (FLD). A schematic of FLD is illustrated in Figure 1. The FLC can be split into two branches: ‘‘left branch’’ and ‘‘right branch.’’ Keeler and Backhofen1 have first introduced the ‘‘right branch’’ of FLC, which is valid for positive major and minor strains. Goodwin2 has completed the FLC by introducing the ‘‘left branch’’ of FLC, which is applicable for positive major and negative minor strains. FLC is solely applicable for proportional strain path. Therefore, to construct FLC, different ratios of major and minor strains are chosen in proportional strain paths. The ‘‘left side’’ represents strain paths with strain ratios (a = e2/e1) that vary from uniaxial tension (a = 20.5) to plane strain (a = 0). On the ‘‘right branch,’’ the strain ratios differ from plane strain to full biaxial (a = 1) stretching. Usually, FLD is determined by using one of the following two types of test method: (a) Marciniak et al.’s3 in-plane test, where a sheet metal sample is strained by a flat-bottomed cylindrical punch and it creates a frictionless in-plane deformation of the sheet, and (b) Nakazima et al.’s4 out-of-plane test (dome), which uses a hemispherical punch. As this test deformation is not frictionless, lubricants are used. The necessary strain paths can be obtained by using different sample widths. The strain levels in the stamped part should be located below the FLC everywhere in order to avoid necking of the material. By offsetting the FLC (generally 10%), a safety margin is normally introduced. Usually by evaluating how close the strain state is to the FLC, the risk of necking/failure is determined. Rejection of the formed part not only depends on necking but also depends on other problems such as excessive thinning, wrinkling, or insufficient stretch. Therefore, it is not sufficient to design a component that can be manufactured in forming route by considering only the risk of failure through necking. Apart from Research and Development, Tata Steel Limited, Jamshedpur, India Corresponding author: Surajit Kumar Paul, R&D, Tata Steel Limited, Jamshedpur 831001, India. Email: paulsurajit@yahoo.co.in; surajit.paul@tatasteel.com 178 Figure 1. A schematic plot of forming limit diagram. Journal of Strain Analysis 48(3) Figure 3. A schematic plot of the forming limit diagram and other failure limits. FLC: forming limit curve. Figure 2. Schematic representation of forming limit diagram indicating safe forming region. necking, all other failure conditions are normally also evaluated by studying the strain levels. The details of different zones of a FLD are constructed from the FLC, as shown in Figure 2. Wrinkling may occur when the minor stress in the sheet is compressive. Wrinkling of the flange areas can be suppressed by the blank holder. However, wrinkling may also occur in unsupported regions or regions in contact with only one tool. During actual sheet metal forming operations, deformation limit is not only constrained by the necking limit, that is FLC, but also constrained by the shear fracture and thinning fracture limit that is depicted in Figure 3.5 The procedure for determining FLCs may differ concerning the geometry of the forming punch used, the specimen dimensions, the number of specimen geometries considered, the friction conditions during testing, the technique used for the evaluation of strain states, and the type of curve-fitting used. Due to the bending effects during forming, a small deviation from linear strain paths may occur. The determination of the experimental FLC is a quite lengthy procedure, which is a real problem in the practical forming process design. Nowadays, in the competitive market scenario of the automotive industry, there is a strong demand to reduce product design cycle time, which means that there is not enough time to determine experimental FLC. Therefore, a profound demand from the automotive industry comes to generate the FLC from different theoretical and empirical formulas. The large scatter in experimental results while determining a FLC is another significant drawback. The most commonly used failure criterion for sheet metal forming applications is still FLC despite of its all the drawbacks. This is due to its simplicity, excellent performance, and for its historical reasons in many cases. Although the FLC method is widely used tool in sheet metal forming process, it is valid only for proportional loading conditions, where the ratio of major and minor plastic strains remains constant throughout the forming process.6 Prediction by finite element method (FEM) and experimental measurements have shown that in most first draw forming operations are almost proportional, hence the path-dependent limitations of the FLC are often not considered. The issue of path dependency of FLC cannot be ignored in the analysis of secondary forming processes like redrawing and flanging dies where strain paths are not the same for the first and second forming processes. In such cases, conventional FLC is not valid, major and minor strains lie far below the FLC found to neck while in other areas strains far above the conventional FLC found to be safe from necking. The path dependence of the relation between the stress and strain rates in the incremental plasticity is the sole cause of the path dependency in the strain-based FLC. The limitations of the strain-based FLC are addressed by Kleemola and Pelkkikangas7 for the analysis of flanging operations followed by a draw forming operation for different metals and alloys. They proposed an alternative solution that is known as stress-based FLC to overcome the path dependency of strain-based FLC. Arrieux et al.8 also proposed a similar stress-based criterion for all secondary forming operations. Stoughton6 also proposed the stress-based FLC in all forming operations, including the first draw die, in order to get a robust measure of forming severity. 179 Paul The FLC can be achieved before the onset of local necking. The commonly used necking models are the Swift diffuse criterion9 and the Hill local criterion.10 The Swift diffuse criterion was derived by assuming that there is a maximum loading force and the Hill localization criterion is obtained by assuming that there is a maximum principal stress. Storen and Rice11 used a bifurcation analysis to describe the behavior of the FLC that arises from a vertex on the yield surface by imposing force equilibrium between the necked and nonnecked region of the metal. Another well-known methodology is the geometric imperfection model proposed by Marciniak and Kuczynski,3,12 which is referred as M–K model. FLC generated by the M–K method depends on the size and shape of geometrical imperfection. Therefore, exact FLC cannot be obtained from the M–K method, but a band can be obtained or calibrate it with at least one experimental point.13 All these models are extensively used to generate theoretical FLC. An alternative to the above-mentioned prediction methods is proposed by Bressan and Williams;14 a novel shear failure theory was used to calculate the maximum shear stress to estimate the onset of local necking. However, only the right-hand side of FLC can be generated by Bressan–Williams (BW) model, as they prescribed in their original article. All these models support stressbased FLC. Since the stress-based FLC and the strainbased FLC are identical under proportional loading conditions, these models can equally be viewed to explain the strain-based FLC under that condition. In this study, Swift diffuse necking criterion,9 Hill local necking criterion,10 Storen and Rice11 bifurcation analysis, maximum shear stress-based Tresca failure criterion,15 and Bressan and Williams14 shear failure criterion are investigated. where s1 and s2 are the major and minor principal true stresses, and plane stress (s3 =0) condition is adopted. seq is the equivalent true stress. Associated flow rule can be described as dep = dl df d s ð2Þ where dl is the proportionality constant, and for uniaxial condition, it is equal to the effective strain rate; dep is the plastic strain rate. The ratio of the minor true strain rate (de2) to the major true strain rate (de1) is defined by the parameter a= de2 de1 ð3Þ Similarly, the ratio of the minor true stress (s2) to the major true stress (s1) is defined by the parameter b= s2 s1 ð4Þ This flow rule (equation (2)) leads to a relation between a and b that can be expressed as a= b= 2b  1 2b ð5Þ 1 + 2a a+2 ð6Þ The rate of change of the yield function can be defined as df ∂f ∂f dl = ds1 + ds2 p de ∂s1 ∂s2 ð7Þ The most commonly used representation of stress– strain relation is the power law Plasticity relations Predictive capabilities of the existing models are validated by FLD data available in the literatures. The material property details are given in Table 1. All the mathematical formulations described in this article are derived from isotropic plasticity theory. Von Mises yield function and associated flow rule are used for all analyses. Von Mises yield function can be defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f = seq = s21 + s22  s1 s2 ð1Þ seq = Keneq ð8Þ where K and n are material constants, K is power law coefficient, and n is power law exponent. By rearranging equations (1) and (4), equivalent stress (seq) can be written as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi seq = s1 1  b + b2 ð9Þ Similarly, equivalent plastic strain increment (deeq) can be written as Table 1. Material constants details. Serial no. Material Reference 1 2 3 4 5 DP600 TRIP 600 Steel Fep04 AISI 1012 low carbon steel AA3105-U Al alloy Uthaisangsuk et al.16 Uthaisangsuk et al.16 Brunet and Morestin17 Nurcheshmeh and Green18 Aghaie-Khafri and Mahmudi19 K (MPa) 950 1090 522 238 236 n Uniform elongation Thickness (mm) 0.16 0.21 0.237 0.35 0.26 0.16 0.26 — — 0.15 1.0 1.0 — — 1.0 180 Journal of Strain Analysis 48(3) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 deeq = pffiffiffi de1 1 + a + a2 3 ð10Þ Triaxiality (h) is the ratio of mean stress (sm) and equivalent stress (seq), and it can be expressed as h= sm 1+a = pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi seq 3 1 + a + a2 ð11Þ where mean stress (sm) is sm = 1 ðs1 + s2 + s3 Þ 3 ð12Þ All those relationships are used in next section to derive the FLC by various criteria. FLD prediction methods FLC-zero (FLC0) is the most important parameter in the FLC. It is the major principal strain at the onset of necking at plane strain condition, that is, the minor principal strain is equal to zero. Experimentally, FLC0 can be determined form tensile test with specimen geometry of large sample width compared to its length. The parameter FLC0 is often also defined from empirical formula or makes approximation from similar metals. An empirical formula used to determine FLC0, in a wide range of steels, was proposed by Keeler and Brazier20     13:3 + 14:13t n for n40:21 FLC0 = ln 1 + 100 0:21 ð13Þ where t is the sheet thickness measured in millimeters, and n is the strain hardening exponent in power law expression (equation (8)). In equation (13), FLC0 is represented in true strain. Actually, stress-based FLC can be directly constructed from Swift, Hill, Storen and Rice, and Bressan and Williams models. By imposing proportional straining condition, strain-based FLCs can be obtained from those stress-based FLCs. In this study, proportional straining condition is assumed in the beginning, and for simplicity, strain-based FLCs are first shown and then mapped in the stress space. Swift diffuse necking criterion For sheet metal forming, Swift9 proposed a diffuse neck criterion, which described as diffuse neck starts when the load reaches a maximum along both principal directions. The diffuse necking condition can be expressed by the constraints in the form of equation (14)  ds1 = s1 de1 ð14Þ ds2 = s2 de2 From equations (2), (7), and (14), we can obtain the following set of points defining swift instability (for details see Stoughton and Zhu’s21 study) Figure 4. FLC from Swift diffusive necking theory for different strain hardening exponents.  e1 e2  0 B B =B @  2nð2  bÞ 1  b + b2  1 C 2 3 4  3b  3b  + 4b 2 C C 2nð1  2bÞ 1  b + b A ð15Þ 4  3b  3b2 + 4b3 Equivalent strain can be obtained from equation (10) and equivalent stress from equation (8). Stress-based FLC can be generated from equations (4) and (9). In equation (15), power law exponent n is equal to uniform elongation eUL; therefore, eUL can be used instead of n. FLC generated from Swift diffusive necking theory is portrayed in Figure 4, for different n values. Hill local necking criterion Hill10 assumed that a local neck will form with an angle c to the direction of the major principal stress. The orientation angle of the neck with respect of loading axes may be pffiffiffiffiffiffiffi expressed as c = tan1 ( a). Due to the square root in the expression, the instability described in equation (16) has physical significance only when a is negative. The reduction in thickness and the effect of strain hardening balance each other exactly when local neck formed. This means that the fractions within the material reach a state where traction increments equal to zero, and from there, the following criterion for local necking can be derived ds1 = s 1 ð 1 + aÞ de1 ð16Þ For proportional straining, the familiar strain-based Hill’s expression appears e1 = n 1+a ð17Þ In equation (17), power law exponent n is equal to uniform elongation eUL; therefore, eUL can be used instead of n. The set of points defining Hill’s instability can be expressed as 181 Paul  e1 e2  0 eUL 1 +aA = @ 1eUL a 1+a for b \ 0 ð18Þ The equivalent strain at local necking can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2eUL 1 + a + a2 pffiffiffi ð19Þ eeq = 3ð 1 + aÞ Equivalent stress for power law expression can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!n 2eUL 1 + a + a2 pffiffiffi ð20Þ seq = K 3 ð 1 + aÞ From equation (20), a path-independent stress-based FLC can be derived as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!n 1 Kð2 + aÞ 2eUL 1 +a + a2 pffiffiffi C   B pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 C B 3 1 +a + a 3ð1+ aÞ s1 C forb \0 ! =B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p n C B s2 2eUL 1 +a + a2 A @ Kð1+ 2aÞ pffiffiffi pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð1+ aÞ 3 1 +a + a2 0 ð21Þ A similar derivation has been shown by Stoughton and Zhu21 and Alsos et al.22 Using power law expression, the stress-based FLC can be constructed as !n 1 Kð2 + aÞ 3a2 + nð2 + aÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C   B pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi B 3 1 + a + a ð1 + 2aÞ 3ð1 + a + a2 Þ C s1 !n C =B C B s2 3a2 + nð2 + aÞ2 A @ Kð1 + 2aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 + a + a2 ð1 + 2aÞ 3ð1 + a + a2 Þ 0 ð24Þ Hill–Swift criterion Poor shape of FLC is observed in the negative region of a (Figure 1) by Swift theory. Hill’s criterion is applicable for negative value of a. Limitation of Hill–Swift (HS) theory can be eliminated by the combination of the two. In HS theory, Hill criterion is applied for a negative value of a, and Swift criterion is applied for a positive value of a. Figure 5 shows the effect of power law exponent (n) in FLC by HS theory. Storen–Rice bifurcation criterion Storen and Rice11 introduced a bifurcation that allows a neck to form in-plane strain. They introduce a vertex on the yield surface whose discontinuity grows until a plane strain plastic strain increment is possible. Thus, they got two sets of solutions, those two solutions for FLD in two directions of sheets (rolling and transverse directions). Using one solution, FLC can be constructed as (Stoughton and Zhu’s21 study) 0 1 3a2 + nð2 + aÞ2 C   B B 2ð1 + 2aÞð1 + a + a2 Þ C e1 C ð22Þ =B B a 3a2 + nð2 + aÞ2 C e2 A @ 2ð1 + 2aÞð1 + a + a2 Þ The equivalent strain from e1 can be expressed as 2 eeq = Figure 5. FLC from the Hill–Swift theory for different strain hardening exponents. 3a2 + nð2 + aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 + 2aÞ 3ð1 + a + a2 Þ ð23Þ FLCs generated from Storen-Rice (SR) theory for different values of power law exponent (n) are shown in Figure 6. This figure shows that the FLC shapes become distorted for high value of n. BW shear instability criterion An interesting way of predicting stability of sheets was proposed by Bressan and Williams14 and applied to the problem of ship grounding by Alsos et al.22 To derive BW instability criterion, they make three basic assumptions, which are as follows: (a) shear instability is initiated in the direction through the thickness at which the material element experiences no change of length, (b) instability starts when local shear stress exceeds a critical value, and (c) elastic components of strains are neglected during calculation. The orientation of the plane of critical shear stress can be predicted by cos 2u =  a 2+a ð25Þ where u represents the direction of the localization band predicted by the BW criterion. And critical shear stress (tcr) is predicted by t cr = s1 sin 2u 2 ð26Þ By combining equations (25) and (26), we can obtain the BW criterion 182 Journal of Strain Analysis 48(3) Hill–Bressan–Williams criterion The mathematical expression of the BW criterion is valid for both positive and negative values of minor strain. The initial BW criterion was used for the positive quadrant of the FLD. Therefore, the Hill and BW criterion have been combined into one criterion in order to cover the full range of a, from now on referred to as the Hill–Bressan–Williams (HBW) criterion. In the HBW criterion, Hill criterion is applied for the negative value of a, and the BW criterion is applied for the positive value of a. Hill–Tresca criterion Figure 6. FLC from Storen–Rice theory for different strain hardening exponents. 2t cr s1 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a 2 1  2+ a ð27Þ A similar derivation is given by Alsos et al.22 Bressan and Williams initially suggested to conduct the material constant calibration from uniaxial tensile test or biaxial tensile test. The material constant for BW model can be calibrated from plane strain, a = 022 condition, or simply from Hill’s analysis. The BW criterion can be calibrated by considering that the Hill expression and the BW criterion will be the same at plane strain condition. From equation (21) (s1 vs ^e1 expression), the critical BW shear stress takes the following form  n 1 2n t cr = pffiffiffi K pffiffiffi 3 3 ð28Þ In equation (28), 2n is equal to uniform elongation 2eUL; therefore, 2eUL can be used instead of 2n. Then, rearranging equations (27) and (28), we can obtain n 2e pUL ffiffi 2 3 s1 = pffiffiffi K qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a 2 3 1 ð29Þ 2+a Therefore, the set of points for stress-based FLC can be expressed as  s1 s2   n 1 1 2 + a 2eUL pffiffiffi B pffiffi3ffi K pffiffiffiffiffiffiffiffiffiffiffiffi 1+a 3  C C =B @ 1 2a + 1 2eUL n A pffiffiffi K pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3 3 1+a 0 ð30Þ With the help of equations (3) and (8)–(10), conversion from stress-based FLC to strain-based FLC can be made. The deformation comes from slip mechanism, which can be described as, the slip will occur on certain preferred combinations of crystallographic planes when shear stress reaches a critical limit. Furthermore, experimental observation reveals that the direction of maximum shear stress is close to the failure planes.14,23 Therefore, it can be assumed that the instability may occur before any visual signs of local necking. Accordingly, a shear stress-based instability criterion can be used to calculate the point of local necking. Therefore, Tresca’s maximum shear stress-based theory15 is used in this investigation to predict necking of sheet materials. In this investigation, Hill’s criterion is used for the negative strain ratios, that is, drawing operation, and Tresca’s criterion is used for the positive strain ratios, that is, stretching operation. According to the Tresca criterion, instability can occur when maximum shear stress exceeds a critical value at any location. Distance from Tresca’s hexagon center to top side in the first quadrant can be defined as s1 = str ð31Þ The parameter str can be determined from plane strain condition. According to the Hill criterion, equivalent strain can be determined when necking starts inplane strain condition from equation (19) by putting a = 0. Therefore, equivalent strain at plane strain condition becomes 2 eeqtr = pffiffiffi eUL 3 ð32Þ where n = eUL for power law relationship. Therefore, equivalent stress will be  n 2 seqtr = K pffiffiffi eUL 3 ð33Þ The Tresca parameter (str) will be  n 2 2 str = pffiffiffi K pffiffiffi eUL 3 3 ð34Þ 183 Paul Figure 7. TRIP 600:16 (a) strain-based FLC, (b) stress-based FLC, and (c) equivalent failure strain versus triaxiality (h) plot. Therefore, for stretching (0 4 a 4 1), stress-based FLC can be represented as 0 1 n   p2ffiffi K p2ffiffi eUL s1 3 3 ð35Þ =@ nA s2 1 + 2a p2ffiffi p2ffiffi eUL K 2+a 3 3 Similarly to the BW criterion, conversion from stress-based to strain-based FLC can be made with the help of equations (3) and (8)–(10). Discussion There are several types of failure criteria available to define the limit of sheet metal deformation. They are (a) strain-based FLC or FLC or eFLC,1,2,20 (b) stressbased FLC or forming stress limit curve (FSLC) or sFLC,6–8 and (c) extended stress-based FLC (XSFLC).24 Apart from these three, one more criteria that is widely used to define failure limit in general is also shown; it is (d) equivalent failure strain versus triaxiality plot.25,26 Stress-based FLC and XSFLC are similar types of representation for ordinary forming operations; therefore, to avoid repetition instead of two, only stress-based FLC is discussed in latter part of this section. The effect of prestraining on subsequent strainbased FLC can be determined by the following two assumptions:6 (a) in the second loading stage, the material does not yield plastically until the effective stress rises to the same level of stress attained at the end of the prestrain loading stage, that is, pure isotropic hardening is considered. (b) Material fails after prestraining at the exactly same equivalent failure strain/stress as it is in the as-received condition. This assumptions work well in the absence of shear stresses, as this is the case in the usual FLD experiments, where the principal axes are aligned with the coordinate axis. For prestraining, the power law equation (equation (8)) becomes  n seq = K ePSeq + eeq ð36Þ where ePSeq is the equivalent strain during prestraining and seq and eeq are the equivalent stress and strain, respectively, in the second loading stage. With the help of equations (9), (10), and (36), FLC after prestraining can be constructed. TRIP 600 steel In this subsection, experimental FLC of TRIP 600 steel represented by Uthaisangsuk et al.16 is compared with FLC derived theoretically from different criteria. The material constants are tabulated in Table 1. Strain-based FLC, stress-based FLC, and equivalent failure strain versus triaxiality (h) plots are shown in 184 Journal of Strain Analysis 48(3) Figure 8. DP 600:16 (a) strain-based FLC, (b) stress-based FLC, and (c) equivalent failure strain versus triaxiality (h) plot. Figure 7 for TRIP 600 steel. TRIP 600 has clearly higher uniform elongation than its strain hardening exponent, because the deformation mechanism is influenced by the phase transformation that occurred during straining.16 Therefore, during theoretical FLC derivation by different criteria, n is equal to uniform elongation (eUL) of the material used. The HBW criterion shows relatively good agreement with experimental data. DP 600 steel Experimental FLC of DP 600 steel presented by Uthaisangsuk et al.16 is used in this study to compare the FLC drawn from various criteria. In Figure 8, strain-based FLC, stress-based FLC, and equivalent failure strain versus triaxiality (h) plots are depicted. DP 600 steel microstructure contains soft ferrite matrix and hard martensite distributed in soft matrix. During deformation, strain partitioning takes place among soft and hard phases.16,27,28 The presence of different phases makes damage initiation complex in multiphase steels, in single-phase steel damage, that is, necking initiates from initiation and coalescence of voids, but in DP steel damage also initiates from cracking of hard phase, decohesion of hard and soft phases, and so on.16 Complexity in deformation and damage is also reflected in FLC. The HBW criterion predicts FLC better than other models. The same is also reflected in stress-based FLC and equivalent failure strain versus triaxiality (h) plot. AA 3105 Al alloy Aghaie-Khafri and Mahmudi19 generated experimental FLC of AA 3105 Al alloy, which has been used in this study. The uniform elongation (eUL) of this Al alloy is lower than the strain hardening exponent (n). As uniform elongation is showing good correlation with FLC0; therefore, n equal to uniform elongation (eUL) is used for this material. None of the models are able to predict the FLC of AA 3105 Al alloy; only the SR criterion predicts relatively well in the range of 0 4 a 4 1 but over predict in the range of 20.5 4 a 4 0, and the Hill criterion shows relatively well result in the range of 20.5 4 a 4 0, which is shown in Figure 9. AISI 1012 low carbon steel Nurcheshmeh and Green18 generated experimental FLC of base material and of 10% tensile prestraining for AISI 1012 low carbon steel. Their experimentally generated data are used to examine the different models predicting capabilities. FLC predictions by different criteria are shown in Figure 10(a) for base material and in Figure 10(b) for FLC after 10% tensile Paul 185 Figure 9. AA 3105 Al alloy:19 (a) strain-based FLC, (b) stress-based FLC, and (c) equivalent failure strain versus triaxiality (h) plot. Figure 10. AISI 1012 low carbon steel:18 (a) strain-based FLC for base material, (b) strain-based FLC after material uniaxial tensile prestrained to 10%, (c) stress-based FLC, and (d) equivalent failure strain versus triaxiality (h) plot. 186 Journal of Strain Analysis 48(3) Figure 11. Steel Fep04:17 (a) strain-based FLC for base material, (b) strain-based FLC after material uniaxial tensile prestrained to 10%, (c) strain-based FLC after material equibiaxial tensile prestrained to 15%, (d) stress-based FLC, and (e) equivalent failure strain versus triaxiality (h) plot. prestraining. For prestraining condition, path dependency of strain-based FLC is well predicted by theoretical isotropic plasticity formulation. Therefore, it can be said that the path dependency of strain-based FLC arises from path-dependent stress–strain plasticity theories. For both FLCs, base material and material after 10% uniaxial tensile prestraining are comparatively well predicted by the HBW criterion. Experimental stressbased FLC can be generated by transforming FLC from strain space to stress space by J2 plasticity theory. It can be observed from Figure 10(c) that experimental stress-based FLC for both base material and material after 10% tensile prestraining is almost the same. Similarly, in Figure 10(d), also experimental equivalent failure strain versus triaxiality (h) plot for both base material and material after 10% tensile prestraining is almost the same. Therefore, from Figure 10(c) and (d), it can be observed that both stress-based FLC and equivalent failure strain versus triaxiality (h) plot are strain path independent. 187 Paul Steel Fep04 1. 17 Brunet and Morestin investigated the effect of nonlinear strain paths in FLC for steel Fep04. The specimens were first prestrained in uniaxial and equibiaxial tension; thereafter, FLCs were generated. In Brunet and Morestin’s experiments, uniaxial tensile prestraining was done at 10% strain level and equibiaxial tensile prestraining was conducted at 15% strain level. Strainbased FLC predictions by different theories are shown in Figure 11(a) for base Fep04 steel sheet, Figure 11(b) for Fep04 steel sheet after 10% uniaxial tensile prestraining, and Figure 11(c) for Fep04 steel sheet after 15% equibiaxial tensile prestraining. The HBW criterion predicts well in base material as well as in uniaxial prestrained material. Similarly, like AISI 1012 low carbon steel, strain-based FLC for uniaxial and equibiaxial tensile prestraining conditions generated from theoretical isotropic plasticity formulation for steel Fep04 is also capable to capture path dependency of strain-based FLC. This path dependency of strain-based FLC solely arises from path-dependent stress–strain plasticity theories, which is observed in this study. The HBW criterion predicts well is also reflected on stress-based FLC in Figure 11(d) and equivalent failure strain versus triaxiality (h) plot in Figure 11(e). Experimental pure, uniaxial and biaxial prestrained FLCs are converted into stress space from strain space and are shown in Figure 11(d). Few experimental points in biaxial prestrained FLC are excluded during conversion from strain space to stress space, because principal strains in the second loading operation is small, and there is a possibility of error during transformation. From Figure 11(d), it can be observed that experimental stress-based FLC for the base material, materials after uniaxial and biaxial prestraining is almost the same. These results have supported the earlier finding that stress-based FLC is strain path independent. Equivalent failure strains are calculated from pure, uniaxial, and biaxial prestrained experimental FLC and are plotted with triaxiality (h) in Figure 11(e). It can be observed from Figure 11(e) that equivalent failure strain versus triaxiality (h) plot for pure, uniaxial, and biaxial prestrained conditions is almost the same, that is, equivalent failure strain versus triaxiality (h) plot is also strain path independent. Concluding remarks Theoretical formulations of strain- and stress-based FLD by different instability criteria are presented in this study; they are the HS criterion, the SR criterion, the Hill–Tresca criterion, and the HBW criterion. Comparative studies are also done among those FLDs generated from different criteria with the help of experimental data available in the literatures, TRIP 600 steel, DP600 steel, AA 3105 Al alloy, AISI 1012 low carbon steel, and steel Fep04. From this study, the following conclusions can be made: 2. 3. 4. This investigation supports the statement by presenting data of AISI 1012 low carbon steel and Fep04 steel that the stress-based FLD is strain path independent. 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