the finite element method for the analysis of non-linear ......figure 1.1: uniaxial response of an...
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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)
Institute of Structural Engineering Method of Finite Elements II 2
Constitutive Relations
Overview so far
Material Nonlinearity
Large Displacements
Dynamic Analysis
Special Topics
Material Laws
The Contact problem
Fracture & Special Formulations (XFEM, SBFEM)
Institute of Structural Engineering Method of Finite Elements II 2
Constitutive Relations
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
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Constitutive Relations
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.
Institute of Structural Engineering Method of Finite Elements II 4
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
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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
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Overview of Material Descriptions
We can discriminate amongst the following major classes ofmaterial behavior
Elastic, linear or nonlinear
Hyperelastic
Hypoelastic
Elastoplastic
Creep
Viscoplastic
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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
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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
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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
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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.
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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.
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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]
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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
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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.
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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
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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)
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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
Institute of Structural Engineering Method of Finite Elements II 19
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
Institute of Structural Engineering Method of Finite Elements II 21
Viscoplasticity
Institute of Structural Engineering Method of Finite Elements II 22