The Finite Element Method for the Analysis of Non-Linear and ...

The Finite Element Method for the Analysis of 

Non-Linear and Dynamic Systems 

Prof. Dr. Eleni Chatzi 

Lecture 7 - 5 November, 2010 

Institute of Structural Engineering Method of Finite Elements II 1

Constitutive Relations 

Previously we examined the kinematic equations formulation 

(displacement, strain displacement relations) 

The next step is to determine appropriate constitutive relationships 

of the form: 

ex. linear analysis ⇒ σ = Eε 

σ = f(ε) 

When dealing with incremental analysis this is written in tensor 

form for time t: 

t 

σ = t C ijrst ɛ rs 


Constitutive Relations 

It is necessary that kinematic and constitutive relations are 

appropriate (ex. The Second Piola-Kirchhoff stress tensor is to 

be used with the Green Lagrange strain tensor). 

Ultimately, the position of the observer frame should not affect 

the constitutive relations of a material (material frame 

indifference or objectivity of the material response). 


Overview of Material Descriptions 

We can discriminate amongst the following major classes of 

material behavior 

Elastic, linear or nonlinear 

Hyperelastic 

Hypoelastic 

Elastoplastic 

Creep 

Viscoplastic 


Solution Flowchart 

General Solution process in incremental nonlinear FE 

Known Solution at t: 

Stresses 

t σ, strains t ε, 

Internal material parameters t κ 

Known Quantities at iterations i- 1 : 

Nodal Displacements at first Iteration: 

and hence 

Element strains 

t+Δt 

ε i−1 

t+Δt 

U i−1 

Calculate at t+ Δt: 

t+Δt 

Stresses σ i−1 

Repat till Convergence 

Tangent stress strain matrix C i−1 

Internal material parameters 

t+Δt 

κ i−1 

• Elastic Analysis: directly obtain 

t+Δt 

t+Δt 

σ i−1 , C i−1 from ε i−1 

• Inelastic Analysis: Integrate to get 

t+Δt 

σ i−1 

t t+Δti−1 

= σ + ∫ dσ 

t 

Calculate: 

Incremental Displacement Vector ΔU i: 

t+Δt 

K i−1 ΔU i = 

Then, 

t+Δt 

R 

− 

t+Δt 

F i−1 

t+Δt t+Δt 

U i = U i−1 + 

ΔU i 


Notation 

Main Stress - Strain pairs: 

Material Nonlinearity (small deformations) 

Engineering Stress σ 

Engineering Strain ε 

TL formulation (large deformations) 

2nd Piola-Kirchhoff Stress S 

Green-Lagrange Strain ɛ 

UL formulation (large deformations) 

Cauchy Stress τ 

Almansi Strain ɛ A 


Elastic Material 

For an elastic material the stress is a function of strain only The 

stress path is the same both in loading and unloading 

Linear Elastic 

The elasticity (constitutive) 

tensor components, C ijrs are 

constant 

Nonlinear Elastic 

The elasticity (constitutive) 

tensor components, C ijrs are 

a function of strain 

Example: Almost all materials under small stress 

σ 

ε 



For the case of an elastic material we already saw that the TL 

Formulation (used for large deformation analysis) yields: 

t 

0S ij = t t 

0C ijrs 0ɛ rs 

The elasticity tensor for 3D stress conditions is defined as: 

t 

C ijrs = λδ ij δrs + µ(δ ir δjs + δ is δjr) 

where λ and µ are the Lamé constants and δ ij is the Kronecker delta, 

Eν 

λ = 

(ν)(1 − 2ν) , µ = E 

2(1 + ν) 

{ 0 i ≠ j 

δ ij = 

1 i = j 



Important Note 

“The 2nd Piola-Kirchhoff (PK2) stress and Green-Lagrange strain 

tensor components are invariant to rigid body motions.” 

For problems with small strains we can take advantage of this 

observation and use any constitutive relationship that has been 

developed for engineering stress and strain measures by just 

substituting with the PK2 stress and Green-Lagrange strain 

This observation can be extended to all problems with large 

deformations but small strain conditions such as the elastic or 

elastoplastic buckling problem and the collapse analysis of slender 

structures. 



UL Formulation 

We now write 

t 

τ ij = t tC ∗ ijrs 

t tɛ A rs 

where the elasticity tensor components of the UL constitutive matrix 

C ∗ , are related to the one of the TL formulation C, through the 

following relationship: 

t 

tC ∗ ijrs = 

t 

ρ 

0 

ρ 

t 

0x i,m 

t 

0x j,n 

t 

0C mnpq 

t 

0x r,p 

t 

0x s,q 

Also the Almansi strain tensor is related to the Green-Lagrange one 

through 

t 

tɛ A = 0 t 

X T 0ɛ t 0 t 

X 

Therefore the two formulations are equivalent and interchangeable. 


Example 

Consider the four node element shown below. Examine the effect of using two 

different stress measures and the same constitutive matrix on the Cauchy stresses. 

2 

1 

3 4 

A. TL formulation Using the PK2 stress measure and the Green-Lagrange strain 

tensor we have: 

t 

0S ij = t t 

0C ijrs 0 ɛ rs (1) 


Example 

In order to define the required stresses and strains we need to determine the deformation 

gradient t 0S, which in 2D is written as: 

⎡ 

∂ t x 1 ∂ t ⎤ 

x 1 

t 

0 X = ⎢ ∂ 0 x 1 ∂ 0 x 2 

⎣ ∂ t x 2 ∂ t ⎥ 

x 2 ⎦ (2) 

∂ 0 x 1 ∂ 0 x 2 

The coordinates of a random point within the element are given as the weighted sum of 

the nodal coordinates, where the weights are the shape functions. However, the shape 

functions are written with respect to the r, s system (isoparametric representation) as: 

N 1 = 1 4 (1 + r)(1 + s) N 2 = 1 (1 − r)(1 + s) 

4 

N 3 = 1 4 (1 − r)(1 − s) N 4 = 1 (3) 

4 (1 + r)(1 − s) 

From the given figure however we see that the 0 x 1 , 0 x 2 system is related to the r, s 

system through: 

0 x 1 = r + 1 

0 x 2 = s + 1 ⇒ 

N 1 = 1 4 0 x 1 0 x 2 N 2 = 1 4 (2 − 0 x 1 )( 0 x 2 ) 

(4) 

N 3 = 1 4 (2 − 0 x 1 )(2 − 0 x 2 ) N 4 = 1 4 0 x 1 (2 − 0 x 2 ) 


Example 

Therefore, we ultimately have: 

∂ t x i 

∂ 0 x j 

= 

The nodal coordinates at time t are: 

4∑ 

k=1 

( ) ∂Nk t x k 

∂ 0 i (5) 

x j 

( t x 1 

1 

, t x 1 2) = (2 + 4t, 2) ( t x 2 1, t x 2 2) = (4t, 2) 

( t x 3 1, t x 3 2) = (0, 0) ( t x 4 1, t x 4 2) = (2, 0) 

(6) 

By substituting (4), (5) and (6) into Eqn (2) we obtain: 

[ ] 

t 1 2t 

0X = 

0 1 

From Lecture 4 we know that the Green-Langrange strain tensor will then be: 

t 

0ɛ = 1 2 (t 0X T t 0X − I) ⇒ 

[ ] 

t 0 t 

0ɛ = 

t 2t 2 


Example 

Also, for plane strain strain conditions the constitutive tensor is: 

⎡ 

1 ν 0 

C = 

E 

⎢ ν 1 0 

1 − ν 2 ⎣ 

1 − ν 

0 0 

2 

⎤ 

⎡ 

E=5000,ν=0.3 

⎥ 

⎢ 

⎦ → ⎣ 

6731 2885 0 

2885 6731 0 

0 0 1923 

Now, using the assumption of small strain we can use the above constitutive 

tensor for the relationship between the PK2 stress and the Green-Lagrange strain. 

Hence from Eqn (1): 

⎡ 

⎢ 

⎣ 

S 11 

S 22 

S 12 

⎤ 

⎡ 

⎥ ⎢ 

⎦ = ⎣ 

5770t 2 

13462t 2 

3846t 

⎤ 

⎥ 

⎦ 

⎤ 

⎥ 

⎦ 


Example 

Then, the Cauchy stress at time t can be obtained from the PK2 

stress as (Lecture 4): 

t ρ t 

0 ρ 

t τ = 0X t 0S t 0X T ⇒ 

⎡ ⎤ ⎡ 

τ 11 21000t 2 + 54000t 4 ⎤ 

⎢ ⎥ ⎢ 

⎣ τ 22 ⎦ = ⎣ 13000t 2 ⎥ 

⎦ 

τ 12 3800t + 27000t 3 


Example 

B. Jaumann stress rate formulation 

This formulation uses the following constitutive relationship: 

t˜τ ij = t C ijrs t D rs (7) 

where the velocity strain tensor t D is computed using the velocity gradient L (see 

Lecture 4) : 

[ ] 

0 2 

L = ẊX −1 = 

0 0 

L can be decomposed to s symmetric part D = D T (the velocity strain tensor) and a 

skew symmetric part W = −W T (the spin tensor): 

L = D + W 

[ 

0 1 

D = 

1 0 

which in this case yields 

] [ ] 

0 1 

, W = 

−1 0 

Now we use the same constitutive matrix C and Eqn (7) to obtain the Jaumann stress at 

time t: 

⎡ ⎤ ⎡ ⎤ 

˜τ 11 0 

⎢ ⎥ ⎢ ⎥ 

⎣ ˜τ 22 ⎦ = ⎣ 0 ⎦ 

˜τ 12 

3846 


Example 

The Jaumann stress is connected to the Cauchy stress through: 

˜τ ij = ˙τ ij + τ ip W pj + τ jp W pi 

which from the above yields the follwoing formula for the Cauchy components τ ij 

⎡ 

⎢ 

⎣ 

˙τ 11 

˙τ 22 

˙τ 12 

⎤ ⎡ 

⎥ ⎢ 

⎦ = ⎣ 

2τ 12 

⎤ 

−2τ 12 

⎥ 

⎦ 

The above system of ordinary differential equations can be solved to finally get: 

⎡ 

⎢ 

⎣ 

τ 11 

τ 22 

τ 12 

⎤ ⎡ 

⎥ ⎢ 

⎦ = ⎣ 

1900(1 − cos2t) 

−1900(1 − cos2t) 

1900sin2t 

⎤ 

⎥ 

⎦ 

The results from methods A and B are rather close for small values of the deformation 

measure t but grow quite different as t gets larger than 0.1, indicating that the same C 

can no longer be used. 


Hyperelastic Material 

Hyperelastic (rubberlike) materials 

exhibit an incompressible response, 

path independence and no energy 

dissipation. 

The stress is now calculated through 

the strain energy functional W 

t 

0S ij = ∂W 

∂ t 0ɛ ij 

Figure: Stress-strain curves for various 

hyperelastic material models. 


Hyperelastic Material 

Hyperelastic Material Models 

Saint Venant-Kirchhoff model 

W (ɛ) = λ 2 [tr(ɛ)]2 + µtr(ɛ 2 ) 

and the second Piola-Kirchhoff stress can be derived as 

S = λ[tr(ɛ)]I + 2µɛ 

λ, µ are the Lamé constants 

Mooney-Rivlin model 

W (ɛ) = C 1 (I 1 − 3) + C 2 (I 2 − 3) 

where C1 and C2 are empirically determined material constants and 

I 1 = tr(C) = C 11 + C 22 + C 33 

where C is the Cauchy-Green deformation tensor (see Lecture 4) and 

I 2 = 1 2 [(I 1) 2 − tr(C) 2 ] 


Hypoelastic Material 

In this case, stress increments are calculated from strain increments 

dσ ij = C ijrs dɛ rs 

The material moduli C ijrs are defined as functions of 

stress 

strain 

fracture criteria 

loading and unloading parameters 

maximum strains reached and so on 

Example Concrete models 

(*Hypoelasticity is not an elastic type of behavior in the sense that it 

does not exhibit path independence) 


Inelasticity 

Elastoplasticity, Creep and Viscoplasticity are types of Inelastic 

behavior 

Elastic behavior ⇒ stresses can be directly calculated from the strain 

Inelastic behavior ⇒ the stress at time t depends on the stress strain 

history 

In the incremental analysis of inelastic response we had three main scenarios 

Small displacements-rotations / small strains ⇒ use linear elastic 

solution, engineering stress and strain measures 

Large displacements-rotations / small strains ⇒ use TL formulation 

by substituting the appropriate stress - strain measures (PK2, 

Green-Lagrange) in the place of the engineering stress and strain 

measures 

Large displacements-rotations / large strains ⇒ use either TL or UL 

formulation, more complex constitutive laws 


Elastoplasticity 

In this formulation we encounter a linearly elastic behavior until yield 

and usually a gardening post yield behavior 

Examples Metals, soild and Rocks when subjected to high stresses 



The strain and stress increments are given by: 

dɛ rs = dɛ E rs + dɛ P rs 

dσ ij = C E ijrs (dɛ rs − dɛ P rs) 

where C E ijrs are the components of the elastic constitutive tensor and dɛ rs, dɛ E rs, 

dɛ P rs are the components of the total strain increment. 

To calculate the plastic strains we use the following three properties: 

Yield Function f y(σ, ɛ P ) 

f y < 0 ⇒ Elastic behavior 

f y = 0 ⇒ Plastic behavior 

f y > 0 ⇒ Inadmissible 

Flow rule 

The yield function is used in the flow rule in order to obtain the plastic 

strain increments 

λ is a scalar to be determined 

dɛ P ij = λ ∂fy 

∂σ ij 

Hardening rule 

This specifies how the yield function is modified during plastic flow 



Example: Von Mises yield criterion (in 3D): 

f y = 0 ⇒ (σ 11 − σ 22) 2 + (σ 22 − σ 33) 2 + (σ 11 − σ 33) 2 + 6(σ 2 12 + σ 2 23 + σ 2 31) − 2σ 2 y = 0 



Isotropic & Kinematic hardening Rules 

In the case of isotropic hardening, the yield surface expands 

uniformly. 

In the case of kinematic hardening, the size of the yield surface 

remains unchanged and the center location of the yield surface 

is shifted. 



Response for cyclic loading 

Isotropic hardening: the yield stress is higher as the cyclic loading progresses 

Kinematic hardening: the difference between unloading stress and new yield 

stress in the opposite direction of loading is constant and equal to 2σ y. 

Figure: isotropic hardening in tension 

Figure: kinematic hardening 


Thermoelastoplasticity and Creep 

This behavior exhibits the time effect of increasing strains under constant 

loads or decreasing stress under constant deformations (relaxation) 

Typical examples of such behavior are metals at high temperatures 

The thermal strain(ɛ = α∆T ) and the creep strain now enter the 

formulation of the stress strain relationships. 

Creep 

Creep is the tendency of a solid material to slowly move or deform 

permanently under constant stresses. Creep tests measure the strain 

response due to a constant stress. The classical creep curve represents the 

evolution of strain as a function of time in a material subjected to uniaxial 

stress at a constant temperature. The creep test, for instance, is performed 

by applying a constant force/stress and analyzing the strain response of the 

system. In general, this curve usually shows three phases or periods of 

behavior. 


Creep 

1. A primary creep stage, also known as transient creep, is the starting stage 

during which hardening of the material leads to a decrease in the rate of flow 

which is initially very high. (0 ≤ ε ≤ ε 1). 

2. The secondary creep stage, also known as the steady state, is where the strain 

rate is constant. (ε 1 ≤ ε ≤ ε 2). 

3. A tertiary creep phase in which there is an increase in the strain rate up to the 

fracture strain. (ε 2 ≤ ε ≤ ε R). 


Relaxation 

A relaxation test is defined as the stress response due to a constant strain for a 

period of time. In viscoplastic materials, relaxation tests demonstrate the stress 

relaxation in uniaxial loading at a constant strain. The decompositon of strain 

rate is dε 

dt = dεe 

dt + dεvp 

dt 

The elastic part of the strain rate is given by dεe 

dt = dσ 

E−1 

dt 

For the flat region of the strain-time curve, the total strain rate is zero. 

Hence we have, dεvp dσ 

−1 

= −E 

dt 

dt 

Therefore the relaxation curve can be used to determine rate of viscoplastic strain 

and hence the viscosity of the dashpot in a 1D viscoplastic material model. 


Viscoplasticity 

Viscoplasticity describes the rate-dependent inelastic behavior of 

solids. Rate-dependence in this context means that the deformation 

of the material depends on the rate at which loads are applied[1]. 

The inelastic behavior that is the subject of viscoplasticity is plastic 

deformation which means that the material undergoes unrecoverable 

deformations when a load level is reached. Rate-dependent plasticity 

is important for transient plasticity calculations. 

The main difference between rate-independent plastic and 

viscoplastic material models is that the latter exhibit not only 

permanent deformations after the application of loads but continue 

to undergo a creep flow as a function of time under the influence of 

the applied load. 

Typical examples of such behavior are Polymers and Metals 



The elastic response of viscoplastic materials can be represented in one-dimension 

by Hookean spring elements. Rate-dependence can be represented by nonlinear 

dashpot elements. 

Plasticity can be accounted for by 

adding sliding frictional elements. In the 

figure E is the modulus of elasticity, λ is 

the viscosity parameter and N is a 

power-law type parameter that 

represents non-linear dashpot 

σ = λ dɛ 1/N 

. The sliding element can 

dt 

have a yield stress (σy) that is strain 

rate dependent, or even constant, as 

shown in Figure (c). 



Stress-strain response of a 

viscoplastic material at different 

strain rates. The dotted lines 

show the response if the 

strain-rate is held constant. The 

blue line shows the response when 

the strain rate is changed 

suddenly. 


NL FE Special Considerations - The Contact Problem 

Difficult non linear behavior = contact between two or more 

bodies 

Contacts = From frictionless in small displacement to friction in 

general large strain conditions 

Nonlinearity of the analysis is not only geometric and material 

but also contact conditions 


Contact Conditions 

Usual term 

Contribution of contact 

forces 

Consider N bodies that are in contact at 

time t: 

t S c is the complete area of contact 

t f c i : Components of the contact 

tractions t f S i : components of the known 

externally applied traction. 

Then the virtual work for the N bodies at time t is give by: 


Contact Conditions 

Usual term 

Cont 

force 

We denote the two bodies as I and J. Each body is supported such that 

without contact no rigid motion is possible. 


Definitions used in Contact Analysis 

Let t f IJ : vector of contact surface traction on body I due to contact with 

body J then t f IJ = − t f JI . Hence, the virtual work due to the contact 

traction can be written: 

∫ 

δu I 

S IJ i fi IJ dS IJ + 

∫ 

δu J 

S IJ i fi JI dS JI = 

∫ 

δu IJ 

S IJ i fi 

IJ dS IJ 

where 

where 

δu IJ 

i = δu I i − δu J i 

We call the pair of surfaces S IJ , S IJ a contact surface pair. 

S IJ is named the ”contactor surface” and 

S JI is named the ”target surface” 

Hence, the right hand side of the previous eqn represents the virtual work 

of the contact tractions over the relative virtual displacements δu I i , δuJ i 


Definitions used in Contact Analysis 


Contact Analysis-Interface Conditions 

Normal Conditions 





Tangential Conditions 

Coulomb’s Law of friction States: 

Definitions Coulomb’s law of friction states : 


Contact Analysis 

Complexities 

In the previous we consider pseudo-static contact conditions 

In dynamic analysis: 

Body forces include inertial force effects and the kinematic interface 

conditions must be satisfied at all instances of time, requiring 

displacement, velocity and acceleration compatibility between the 

contacting bodies. 

The time integrations schemes (ex. trapezoidal rule) do not 

automatically satisfy compatibility which therefore has to be imposed 

separately on the step by step solution 

Various algorithms have been proposed to solve contact problems in 

Finite Element analysis. 


Contact Analysis-Solution Approach 

The Constraint Function Method 

Let w be a function of λ and g such that the solutions of w(g, λ) = 0 

satisfy the Normal conditions 

Let v be a function of τ and ˙u such that the solutions v( ˙u, τ) = 0 satisfy 

the Tangential Conditions. Then, the contact conditions are given by: 

w(g, λ) = 0 v( ˙u, τ) = 0 

These can now be imposed on the principle of virtual work using either a 

penalty approach or a Lagrange Multiplier method. Variables λ and τ can 

be considered Lagrange multipliers, and so we consider their variations δλ, 

δτ. By multiplying by the variations and integrating in the domain we 

obtain the constraint equation: 

∫ 

S IJ [δλ w(g, λ) + δτ u( ˙u, τ)]dS IJ = 0 

The governing equations of motion in this case are now both the principle 

of virtual work and the constraint equation

The Finite Element Method for the Analysis of Non-Linear and ...

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