Finite Element Procedures For Nonlinear Analysis Lagrangian and Eulerian Formulations of 1D
Finite Element Procedures For Nonlinear Analysis Lagrangian and Eulerian Formulations of 1D
Finite Element Procedures For Nonlinear Analysis Lagrangian and Eulerian Formulations of 1D
Topics
Introduction and overview
Governing equations: total Lagrangian formulation
Weak form: total Lagrangian formulation
Finite element discretizations: total Lagrangian Formulation
Element and Global matrices: total Lagrangian formulation
Governing equations: Updated Lagrangian formulation
Weak form: updated Lagrangian formulation
SESSION TITLE:
Finite element Procedures for Nonlinear Analysis:
Lagrangian and Eulerian Formulations of 1D Continua
There are many more measures of strain but this is the most
convenient for this presentation.
•Stress measures:
The stress measure used in the TLF is not the physical stress (also
known as Cauchy stress). Let the total force across a given cross
section be denoted by T and assume that the stress is constant in the
cross-section.
The Cauchy stress is given by
= T/A ---------------------------------------------(7)
Note that this measure of stress refers to the current area A. In the
TLF, we will use the nominal stress. The nominal stress will be
denoted by P and is given by
P = T/Ao ---------------------------------------------(8)
This is equivalent to the definition of the engineering stress.
However, no definition of the engineering stress is available in
multi-dimensional stress states.
Comparing (7) and (8):
or in rate form
P(X, t) S PF F (X, t ), F(X, t ) , t t (18)
t
Here SPF and StPF are functionals of the deformation history. The
superscripts are appended to the constitutive functionals to
indicate which measures of stress and strain they relate.
The stress is assumed to be a continuous function of strain. As
indicated in equation (17), the stress can depend on both F and
F and on other state variables, such as temperature, etc.
The stress can also depend on the history of deformation, as in
elastic-plastic materials; This is indicated in (17) and (18) by
letting the constitutive functions depend on deformation for all
time prior to t.
The constitutive equation for a solid is customarily expressed in
material coordinates because it depends on the history of
deformation at a material point.
Examples of constitutive equations are
1. Linear Elastic Material
Total form: P(X,t) = EPF(X,t) = EPF (F(X,t)-1)
Rate form: P(X,t) = EPF (X,t) = EPF F(X,t)
2. Linear viscoelastic material:
P(X,t) = EPF [F(X,t)-1) +F(X,t)]
or P = EPF (+)
For small deformations, the material parameter EPF corresponds to
Young’s modulus; the constant determines the magnitude of
damping.
• Momentum equation in terms of displacements:
A single governing equation can be obtained by substituting the
relevant constitutive equation (17) or (18), into the momentum
equation (12) and expressing the strain measure in terms of the
displacement by (6). For the total form of the constitutive
equation, the resulting equation is
A Pu
0 ,x , u ,X ,X ρ0 A0 b ρ0 A0u (22)
• Boundary conditions:
A boundary is called displacement boundary and denoted by u if
the displacement is prescribed. The prescribed value is surface
traction on t. The boundary conditions are
u = ŭ on u
noP = txo on t
Initial conditions are expressed in terms of displacement and
velocity.
u(X,0) = uo(x) for X[Xa,Xb]
ŭ (X,0) = o(X) for X[ Xa,Xb]
If the body is initially undeformed and at rest, the initial conditions
can be written
u(X,0) = 0
ŭ (X,0) = 0
WEAK FORM FOR TOTAL LAGRANGIAN
FORMULATION:
The momentum equation cannot be discretized directly in the FEM.
In order to discretize this equation, a weak form, often called a
variational form is needed. The principle of virtual work, or weak
form, which will be developed next, is equivalent to the momentum
equation and the surface traction on boundary conditions.
Collectively, the latter is called the classical strong form.
To develop the weak form, we require the trial functions u(X,t) to
satisfy all displacement boundary conditions and to be smooth
enough so that all derivatives in the momentum equation are well
defined. So also are the test functions u(X)!
The weak form is obtained by taking the product of the momentum
equation with the test function and integrating over the domain.
Smoothness of test and trial functions:
Kinematic admissibility: The weak form is integrable if the test and
trial functions are Co.
In the weak form for the TLF, all integrals are over material
domains, i.e., the reference configuration.
Each of the terms in the weak form represents a VIRTUAL WORK
due to the virtual displacement u. The test function u(X) is often
called a virtual displacement to indicate that it is not the actual
displacement. The virtual work of the body forces b(X,t) and the
prescribed tractions tox is called the virtual external work since it
results from the external (applied) loads.
The external work is
Xb
δW ext δu ρ δu 0
o A 0 b dX A 0 t x
Xa Γt
The virtual internal work, which arises from the stresses in the
material can be written as
Xb
δF PA 0dX
Xa
The term ρ0A0u can be considered as a body force which acts in the
direction opposite to the acceleration i.e., a D’Alembert force. We
will define the corresponding virtual work by Wkin and call it the
virtual inertial work or virtual kinetic work.
Xb
δW kin δu ρ 0 A 0 u dX
Xa
• Principle of Virtual Work:
δW δu, u δW δW int ext
δW kin
0
This in fact is the required WEAK FORM of the momentum
equation, the traction boundary conditions, and the stress jump
conditions. The equivalence of the strong and weak forms for the
momentum equation is called the principle of virtual work.
A key step in the development of the weak form is the integration
by parts. This eliminates the derivatives on the stress , without
this step stress would have to be Co and u would have the C1!
Furthermore, the traction boundary conditions would have to be
imposed on the trial functions. It is therefore convenient to
integrate by parts and reduce the smoothness requirements on the
stress and hence the trial displacements.
FINITE ELEMENT DISCRETIZATION IN TLF:
•The discrete equations for a finite element are obtained from the
principle of virtual work by using finite element interpolants for the
trial and test functions.
•The finite element trial function:
nN
u(X, t) Σ N I (X) u I (t)
I 1
•The test functions (or virtual displacements)
nN
δu(X) Σ N I (X) δu I
I 1
• Nodal Forces:
δW int δu T f int
δW ext δu T f ext
δW kin δu T f kin
fint are the internal nodal forces (corresponds to the stresses in the
material)
fext are the external nodal forces (corresponds to externally applied
loads)
fkin are the inertial nodal forces (corresponds to the inertia)
•Nodal forces are always defined so that they are conjugate to the
nodal displacement u in the sense of work so that the scalar product
of an increment of nodal displacement with the nodal forces gives an
increment of work.
• Semi-discrete equations
Ma = fext – fint or f = Ma Where f= fext – fint
The above is the semi discrete momentum equation, which is
called the equation of motion. These equations are called semi-
discrete because they are discrete in space but continuous in
time.
The equations of motion are a system of nN-1 second-order
ordinary differential equations. The independent variable is the
time t.These equations are easily remembered as f = Ma
Newton’s second law of motion.
The mass matrix infinite element discretiztion is often not
diagonal, so the equations of motion differ from the Newton’s
second law of motion in that a force at node I can generate
accelerations at node J if MIJ 0.
A diagonal approximation to the mass matrix is often used. In that
case, the discrete equations of motion are identical to Newton’s
equations for a system of particles inter connected by deformable
element.
ELEMENT AND GLOBAL MATRICES
•In FEA programs, the nodal forces (internal and external) and
mass matrix are usually computed on an element level.
•The element nodal forces are combined into the global matrix by
an operation called scatter or vector assembly.
•The mass matrix and other square matrices are similarly
combined from the element level to the global level by an
operation called matrix assembly.
•The element nodal displacements are extracted from the global
matrix by an operation called gather.
•The connectivity matrix is a Boolean matrix, i.e., it consists of the
integers 0 and 1.
•The total internal energy is the sum of the element internal
energies.
TWO-NODE LINEAR DISPLACEMENT ELEMENT
•Consider a two-node ROD element as shown in Figure 2.4.
•Displacement field:
u1 (t)
uX, t X 2 X1 , X X1
1
l0 u 2 (t)
•Strain measure:
u1 (t)
ε(X, t) u ,X 1 1
1
l0 u 2 (t)
The Bo matrix is
Bo = 1/lo [-1, +1]
• Nodal Internal Forces
1
X2
1
f int
B P dΩ 0
Γ
PA 0 dx
1
e 0
Ωe
l
X1 0
0
int
f1 1
A 0P
f 2 e 1
M e ρ 0 N N dΩ 0
T
Ω e0
ρ 0 A 0l0 2 1
1 2
6
THREE-NODE QUADRATIC DISPLACEMENT ELEMENT
•Consider the three-node element of length lo and cross section
area Ao shown in Figure 2.7.
•The mapping between the material coordinates X and the element
(natural) coordinates ξ is
X1
1
Xξ Nξ X e ξξ 1,1 ξ , ξξ 1 X 2
2 1
2 2
X 3
ξ 1,1
•The displacement field is given by the same shape functions
(Interpolants)
u1 t
u ξ, t Nξ u e t Nξ u 2 t
u t
3
•Strain measures
ε B0 u e
B0
1
2ξ 1, 4ξ 2ξ 1
2X ,ξ
•Internal nodal forces
int
f1
f eint B0 P dΩ 0 f 2
T
Ω e0 f
3 e
•The external nodal forces
1
ξ ξ 1
1 2
1 1 ξ2
ρ 0 bA 0 X ,ξ dξ
1
2 ξ ξ 1
f eext
1
ξ ξ 1
1 2
1 ξ 2 A 0 t 0x
1 1
2 ξ ξ 1
Γ et
•Element mass matrix
1
ξ ξ 1
1 2
2 1
M e 1 ξ ξ ξ 1, 1 ξ , ξ ξ 1
2 1
1 1 2 2
2 ξ ξ 1
ρ 0 A 0 X ,ξ dξ
4 2 1
ρ0A 0L0
Me 2 16 2
30
1 2 4
1 0 0
ρ0A0L0
Me
diag
0 4 0
6
0 0 1