The Finite Element Method for the Analysis of Non-Linear ......Figure 1.1: Uniaxial Response of an...

The Finite Element Method for the Analysis ofNon-Linear and Dynamic Systems

Prof. Dr. Eleni Chatzi

Lecture ST1 - 19 November, 2015

Institute of Structural Engineering Method of Finite Elements II 1

Constitutive Relations

Overview so far

Material Nonlinearity

Large Displacements

Dynamic Analysis

Special Topics

Material Laws

The Contact problem

Fracture & Special Formulations (XFEM, SBFEM)



Previously we examined the kinematic equations formulation(displacement, strain displacement relations)

The next step is to determine appropriate constitutive relationshipsof the form:

σ = f(ε)

ex. linear analysis in 1D ⇒ σ = Eε

When dealing with higher dimensions & incremental analysis, thisis written in tensor form for time t:

tσ = tCijrstεrs



It is necessary that kinematic and constitutive relations areappropriate. Previously we saw that in the large displacementformulation appropriate tensors need to be defined.e.g. TL Formulation ⇒ Second Piola-Kirchhoff stresstensor, Green Lagrange strain tensor).

Therefore a problem involving large strains should also becombined with a material law that admits large strains.

However, we might be examining a problem of large displacements with

small strains. In that case we can still use the material laws defined for

classic engineering stress and strain measures (for small displacements) but

this time combined with the SP-K stress and G-L strain tensors.


Notation

Main Stress - Strain pairs:

Material Nonlinearity (small deformations)Engineering (or Nominal) Stress σEngineering Strain ε

TL formulation (large deformations)2nd Piola-Kirchhoff Stress SGreen-Lagrange Strain ε

UL formulation (large deformations)Cauchy (or True) Stress τAlmansi Strain εA

Note that: τ =L

L0σ, ε =

L− L0

L0


Solution Flowchart

General Solution process in incremental nonlinear FE

Known Solution at t:

Stresses 𝝈𝑡 , strains 𝜺𝑡 ,

Internal material parameters 𝜿𝑡

Calculate at t+Δt:

Stresses 𝝈𝒊−𝟏𝑡+Δ𝑡

Tangent stress strain matrix 𝑪𝒊−𝟏

Internal material parameters 𝜿𝒊−𝟏𝑡+Δ𝑡

• Elastic Analysis: directly obtain

𝝈𝒊−𝟏𝑡+Δ𝑡 ,𝑪𝒊−𝟏 from 𝜺𝒊−𝟏𝑡+Δ𝑡

• Inelastic Analysis: Integrate to get

𝝈𝒊−𝟏𝑡+Δ𝑡 = 𝝈 + ∫ 𝒅𝝈𝑡+Δ𝑡𝒊−𝟏𝒕

𝑡

𝑼𝒊 = 𝑼𝒊−𝟏 +𝑡+Δ𝑡𝑡+Δ𝑡 Δ𝑼i

Calculate:

Incremental Displacement Vector Δ𝑼i:

𝑲𝒊−𝟏𝑡+Δ𝑡 Δ𝑼i = 𝑹𝑡+Δ𝑡 − 𝑭𝒊−𝟏𝑡+Δ𝑡

Then,

Known Quantities at iterations i- 1 :

Nodal Displacements at first Iteration: 𝑼𝒊−𝟏𝑡+Δ𝑡

and hence

Element strains 𝜺𝒊−𝟏𝑡+Δ𝑡

Repa

t til

l Con

verg

ence


Overview of Material Descriptions

We can discriminate amongst the following major classes ofmaterial behavior

Elastic, linear or nonlinear

Hyperelastic

Hypoelastic

Elastoplastic

Creep

Viscoplastic


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 ElasticThe elasticity (constitutive)tensor components, Cijrs areconstant

Nonlinear ElasticThe elasticity (constitutive)tensor components, Cijrs area function of strain

𝝈

𝜺

Example: Almost all materials under small stress


Elastic Material

(a) (b)

s s

e e

Figure 1.1: Uniaxial Response of an Elastic Material: (a) Linear, (b) Nonlinear

1.2.1 Cauchy Elastic Models

Some nonlinear elastic models are based on the general premise that for an elastic materialthe current state of Cauchy stress is a function of the current state of strain, and not of thehistory of strain. That is, σij = Fij (εkl), where Fij is some nonlinear function. Once thiscondition is satisfied, however, the inverse relation does not necessarily exist. Materials thatsatisfy this fundamental requirement of an elastic body are called Cauchy elastic materials.

More specifically, the general functional form relationship for Cauchy elastic models iswritten in indicial form as

σij = α0δij + α1εij + α2εimεmj + α3εimεmnεnj + · · · (1.1)

or, in direct formσ = α0I + α1ε+ α2ε

2 + α3ε3 + · · · (1.2)

where I is the second-order identity tensor. The αi (i = 1, 2, 3, · · · ) represent model parame-ters whose values are determined from the results of laboratory experiments. Further detailspertaining to such parameter determination are given in the following example.

.Example 1.1: First-Order Cauchy Elastic Model

A first order Cauchy model corresponds to the case of isotropic linear elasticity. As such,only the first-order terms in equation (1.1) are retained, giving

σij = α0δij + α1εij (1.3)

In order to determine the values of the parameters α0 and α1, consider two simple lab-oratory experiments. First consider simple shear deformation. Since there is no volumetricstrain, and since only one nonzero component of strain exists (say ε12 = ε21), it follows that

σ12 = α1ε12 = α1γ122

(1.4)

Solving for α1 gives

2


Elastic Material

For the case of an elastic material we already saw that the TLFormulation (used for large deformation analysis) yields:

t0Sij = t

0Cijrst0εrs

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

tCijrs = λδijδrs+ µ(δirδjs+ δisδjr)

where λ and µ are the Lame constants and δij is the Kronecker delta,

λ =Eν

(1 + ν)(1− 2ν), µ =

E

2(1 + ν)(homogeneous isotropic material)

δij =

{0 i 6= j1 i = j


Elastic Material

Important Note“The 2nd Piola-Kirchhoff (PK2) stress and Green-Lagrange straintensor components are invariant to rigid body motions.”

For problems with small strains we can take advantage of thisobservation and use any constitutive relationship that has beendeveloped for engineering stress and strain measures by justsubstituting with the PK2 stress and Green-Lagrange strain

This observation can be extended to all problems with largedeformations but small strain conditions such as the elastic orelastoplastic buckling problem and the collapse analysis of slenderstructures.


Hyperelastic Material

Hyperelastic (rubberlike) materialsexhibit an incompressible response(volume preserving), pathindependence and no energydissipation.

The stress is now calculated throughthe strain energy functional W

t0Sij =

∂W

∂t0εij

Figure: Stress-strain curves for varioushyperelastic material models.


Hyperelastic Material

Hyperelastic Material Models

Saint Venant-Kirchhoff model

W (ε) =λ

2[tr(ε)]2 + µtr(ε2)

and the second Piola-Kirchhoff stress can be derived as

S = λ[tr(ε)]I + 2µε

λ, µ are the Lame constants

Mooney-Rivlin model

W (ε) = C1(I1 − 3) + C2(I2 − 3)

where C1 and C2 are empirically determined material constants and

I1 = tr(C) = C11 + C22 + C33

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

I2 =1

2[(I1)2 − tr(C)2]


Inelasticity

Elastoplasticity, Creep and Viscoplasticity are types of Inelasticbehavior

Elastic behavior ⇒ stresses can be directly calculated from the strain

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

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

Small displacements-rotations / small strains ⇒ use linear elasticsolution, engineering stress and strain measures

Large displacements-rotations / small strains ⇒ use TL formulationby substituting the appropriate stress - strain measures (PK2,Green-Lagrange) in the place of the engineering stress and strainmeasures

Large displacements-rotations / large strains ⇒ use either TL or ULformulation, more complex constitutive laws


Elastoplasticity

In this formulation we encounter a linearly elastic behavior until yieldand usually a hardening post yield behavior

Examples: Metals, soild and Rocks when subjected to high stresses.Institute of Structural Engineering Method of Finite Elements II 15

Elastoplasticity

The strain and stress increments are given by:

dεrs = dεE

rs + dεP

rs dσij = CE

ijrs (dεrs − dεP

rs)

where CE

ijrs are the components of the elastic constitutive tensor and dεrs, dεE

rs,

dεP

rs are the components of the total strain increment.To calculate the plastic strains we use the following three properties:

Yield Function fy(σ, εP)

fy < 0 ⇒ Elastic behaviorfy >= 0 ⇒ Plastic or elastic behavior depending on the loading condition

Flow ruleThe yield function is used in the flow rule in order to obtain the plasticstrain increments

dεP

ij = λ∂fy∂σij

λ is a scalar to be determined

Hardening ruleThis specifies how the yield function is modified during the progression ofloading.


Elastoplasticity

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

fy = 0⇒ (σ11 − σ22)2 + (σ22 − σ33)

2 + (σ11 − σ33)2 + 6(σ2

12 + σ223 + σ2

31)− 2σ2y = 0


Elastoplasticity

Isotropic & Kinematic hardening Rules

In the case of isotropic hardening, the yield surface expandsuniformly.

In the case of kinematic hardening, the size of the yield surfaceremains unchanged and the center location of the yield surfaceis shifted. (Bauschinger effect)


Elastoplasticity

Cyclic Loading

PLASTICITY. Flow rule for kinematic hardening

Obviously, for a reversed loading process like the one in the cyclic loading diagram of Fig. 1, the

isotropic hardening will lead to a cyclic test behaviour according to the solid line OABCDE of Fig. 2

(in which the length of line segment BC is the same as that of line segment AB). It is, however, a well-

established fact that in most materials there is a Bauschinger effect, by which a reversed loading will

E

t

D

C

B

A

s

s

O

Fig. 1 Cyclic loading

D’

C’

A,E

B,D

C

O

Fig. 2 diagram of uniaxial cyclic test.

Isotropic hardening

Kinematic hardening

Hardening

PLASTICITY. Flow rule for kinematic hardening

Obviously, for a reversed loading process like the one in the cyclic loading diagram of Fig. 1, the

isotropic hardening will lead to a cyclic test behaviour according to the solid line OABCDE of Fig. 2

(in which the length of line segment BC is the same as that of line segment AB). It is, however, a well-

established fact that in most materials there is a Bauschinger effect, by which a reversed loading will

E

t

D

C

B

A

s

s

O

Fig. 1 Cyclic loading

D’

C’

A,E

B,D

C

O

Fig. 2 diagram of uniaxial cyclic test.

Isotropic hardening

Kinematic hardening


Thermoelastoplasticity and Creep

This behavior exhibits time effect of increasing strains under constant loadsor decreasing stress under constant deformations (relaxation)Typical examples of such behavior are metals at high temperatures

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

formulation of the stress strain relationships.Institute of Structural Engineering Method of Finite Elements II 20

Viscoplasticity

Viscoplasticity describes the rate-dependent inelastic behavior ofsolids. Rate-dependence in this context means that the deformationof the material depends on the rate at which loads are applied. Theinelastic behavior that is the subject of viscoplasticity is plasticdeformation which means that the material undergoes unrecoverabledeformations when a load level is reached. Rate-dependent plasticityis important for transient plasticity calculations.

The main difference between rate-independent plastic andviscoplastic material models is that the latter exhibit not onlypermanent deformations after the application of loads but continueto undergo a creep flow as a function of time under the influence ofthe applied load.

Typical examples of such behavior are Polymers and Metals


Viscoplasticity


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