the finite element method for the analysis of non … and dynamic systems: computational plasticity...
TRANSCRIPT
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The Finite Element Method for the Analysis ofNon-Linear and Dynamic Systems: Computational
Plasticity Part II
Prof. Dr. Eleni ChatziDr. Giuseppe Abbiati, Dr. Konstantinos Agathos
Lecture 3 - 5 October, 2017
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Learning Goals
To recall the basics of linear elasticity and the importance ofVoigt notation for representing tensors.
To understand basic rate-independent plasticity modelsformulated in terms of stress and strain fields.
To derive displacement-based finite elements based on suchconstitutive models.
References:
Ren de Borst, Mike A. Crisfield, Joris J. C. Remmers, Clemens V.Verhoosel, Nonlinear Finite Element Analysis of Solids andStructures, 2nd Edition, Wiley, 2012.
Example: Forming of a metal profile
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Lumped vs. Continuous Plasticity Models
Lumped parameter model:
Finite dimensional stateexpressed in terms of thescalar r
Described by a set ofOrdinary DifferentialEquations (ODE)
Continuous parameter model:
Infinite dimensional stateexpressed in terms of thefield σ (x)
Described by a set of PartialDifferential Equations (PDE)
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Voigt Notation
Stresses and strains are second order tensors related by a fourthorder tensor describing the elastic properties of the continuum.
σij = Deijklεkl
i , j , k, l → 1, 2, 3↓
σ6×1
= [De ]6×6ε6×1
However, in order to facilitate the implementation of computerprograms -when possible- it is more convenient to work with vectorsand matrices. A clear description of Voigt notation is reported in:
Belytschko, T., Wing Kam L., Brian M., and Khalil E.. Nonlinearfinite elements for continua and structures, Appendix 1, John wiley& sons, 2013.
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Voigt Notation
Graphical representation of the Cauchy stress tensor.
σ =
σxx σxy σxzσyy σyz
sym σzz
→
σxxσyyσzzσyzσxzσxy
= σ
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Voigt Notation
Graphical representation of the Green-Lagrange (small) strain tensor.
ε =
εxx εxy εxzεyy εyz
sym εzz
εxx =
∂u
∂x, εxy =
γxy2
=1
2
(∂u
∂y+∂v
∂x
)εyy =
∂v
∂y, εxz =
γxz2
=1
2
(∂u
∂z+∂w
∂x
)εzz =
∂w
∂z, εyz =
γyz2
=1
2
(∂v
∂z+∂w
∂y
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Voigt Notation
Graphical representation of the Green-Lagrange (small) strain tensor.
ε =
εxx εxy εxzεyy εyz
sym εzz
→
εxxεyyεzz
2εyz2εxz2εxy
=
εxxεyyεzzγyzγxzγxy
= ε
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Voigt Notation
Cauchy stress tensor. Green-Lagrange (small) strain tensor.
δw int =3∑
i=1
3∑j=1
δεijσij = δεijσij = δε : σ = δεTσ
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Voigt Notation
Cauchy stress tensor. Green-Lagrange (small) strain tensor.
δw int =3∑
i=1
3∑j=1
δεijσij = δεijσij = δε : σ = δεTσ
Principle of virtual displacement !!!
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Voigt Notation
Isotropic elastic compliance from tensor:
εij = C eijklσkl or ε = Ce : σ
to Voigt notation:
ε = [Ce ] σ
εxxεyyεzzγyzγxzγxy
=1
E
1 −ν −ν 0 0 0−ν 1 −ν 0 0 0−ν −ν 1 0 0 00 0 0 2 (1 + ν) 0 00 0 0 0 2 (1 + ν) 00 0 0 0 0 2 (1 + ν)
σxxσyyσzzσyzσxzσxy
E : Young modulus, ν : Poisson ratio.
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Voigt Notation
Isotropic elastic stiffness from tensor:
σij = Deijklεkl or σ = De : ε
to Voigt notation:
σ = [De ] ε
σxxσyyσzzσyzσxzσxy
=E
(1 + ν) (1− 2ν)
1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 00 0 0 1−2ν
2 0 00 0 0 0 1−2ν
2 00 0 0 0 0 1−2ν
2
εxxεyyεzzγyzγxzγxy
E : Young modulus, ν : Poisson ratio.
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From Lumped to Continuous Plasticity Models
Lumped plasticity modelr, u, Ke
Continuous plasticity modelσ, ε, [De ]
Elastic regimeif f (r) < 0
↓r = Ke u
if f (σ) < 0
↓σ = [De ] ε
Elastoplastic regimeif f (r) = 0
↓r = Ke (u− up)
f = 0
with up = λm
if f (σ) = 0
↓σ = [De ] (ε − εp)f = 0
with εp = λm
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From Lumped to Continuous Plasticity Models
Lumped plasticity modelr, u, Ke
Continuous plasticity modelσ, ε, [De ]
if f (r) = 0
↓r = Ke (u− up)
f = 0
with up = λm
if f (σ) = 0
↓σ = [De ] (ε − εp)f = 0
with εp = λm
Yield criterion : this is a scalar function that determines theboundary of the elastic domain.
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From Lumped to Continuous Plasticity Models
Lumped plasticity modelr, u, Ke
Continuous plasticity modelσ, ε, [De ]
if f (r) = 0
↓r = Ke (u− up)
f = 0
with up = λm
if f (σ) = 0
↓σ = [De ] (ε − εp)f = 0
with εp = λm
Flow rule : this is a vector function that determines the direction ofthe plastic strain flow.
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From Lumped to Continuous Plasticity Models
Lumped plasticity modelr, u, Ke
Continuous plasticity modelσ, ε, [De ]
if f (r) = 0
↓r = Ke (u− up)
f = 0
with up = λ∂f
∂r
if f (σ) = 0
↓σ = [De ] (ε − εp)f = 0
with εp = λ∂f
∂σ
In the case of associated plasticity, the same function f defines bothyield criterion and flow rule i.e. the plastic displacement/strain flow
is co-linear with the yielding surface normal.
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Invariants of the Stress Tensor
Invariants of stress tensor σ are used to formulate yielding criteria.
σ =
σxx σxy σxzσyy σyz
sym σzz
↓
det (σ − λI) = det
σxx − λ σxy σxzσyy − λ σyz
sym σzz − λ
↓
λ3 − I1λ2 − I2λ− I3 = 0
where I1, I2 and I3 are the invariants of the stress tensor andλ = σ11, σ22, σ33 are the eigenvalues of the stress tensor alsocalled principal stresses.
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Invariants of the Stress Tensor
Invariants of stress tensor σ are used to formulate yielding criteria.
λ3 − I1λ2 − I2λ− I3 = 0
with,
I1 = σxx + σyy + σzz
I2 = σ2xy + σ2
yz + σ2zx − σxxσyy − σyyσzz − σzzσxx
I3 = σxxσyyσzz + 2σxyσyzσzx − σxxσ2yz − σyyσ2
zx − σzzσ2xy
↓
Ψ =1
2σT [Ce ] σ =
1
2E
(I 21 + 2I2 (1 + ν)
)where Ψ is the elastic energy potential.
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Invariants of the Deviatoric Stress Tensor
Invariants of deviatoric stress tensor s are used to formulate yieldingcriteria.
σ =
σxx σxy σxzσyy σyz
sym σzz
↓
p =σxx + σyy + σzz
3↓
s = σ − pI =
σxx − p σxy σxzσyy − p σyz
sym σzz − p
=
sxx sxy sxzsyy syz
sym szz
where p is the hydrostatic pressure.
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Invariants of the Deviatoric Stress Tensor
Invariants of deviatoric stress tensor s are used to formulate yieldingcriteria.
s =
sxx sxy sxzsyy syz
sym szz
↓
det (s− λI) = det
sxx − λ sxy sxzsyy − λ syz
sym szz − λ
↓
λ3 − J1λ2 − J2λ− J3 = 0
where J1, J2 and J3 are the invariants of the deviatoric stress tensor.
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Invariants of the Deviatoric Stress Tensor
Invariants of deviatoric stress tensor s are used to formulate yieldingcriteria.
λ3 − J1λ2 − J2λ− J3 = 0
with,
J1 = sxx + syy + szz
J2 = s2xy + s2
yz + s2zx − sxxsyy − syy szz − szzsxx
J3 = sxxsyy szz + 2sxy syzszx − sxxs2yz − syy s
2zx − szzs
2xy
↓
Ψd =1
2sT [Ce ] s =
1
2E
(J2
1 + 2J2 (1 + ν))
where Ψd is the deviatoric elastic energy potential.
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Invariants of the Deviatoric Stress Tensor
Invariants of deviatoric stress tensor s are used to formulate yieldingcriteria.
λ3 − J1λ2 − J2λ− J3 = 0
with,
J1 = sxx + syy + szz = 0
J2 = s2xy + s2
yz + s2zx − sxxsyy − syy szz − szzsxx =
I 21
3+ I2
J3 = sxxsyy szz + 2sxy syzszx − sxxs2yz − syy s
2zx − szzs
2xy
↓
Ψd =1
2sT [Ce ] s =
J2 (1 + ν)
E=
(I 21
3+ I2
)(1 + ν)
E
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Von Mises Yield Function
The J2 invariant of the deviatoric stress tensor is used to define theVon Mises yield function:
fVM (σ) = q − σ = 0
where σ is the pure uniaxial yielding stress and,
q =√
3J2 =
=
√(σxx − σyy )2 + (σyy − σzz)2 + (σzz − σxx)2
2+ 3σ2
xy + 3σ2xz + 3σ2
yz
=
√(σ11 − σ22)2 + (σ22 − σ33)2 + (σ33 − σ11)2
2
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Von Mises Yield Function
The J2 invariant of the deviatoric stress tensor is used to define theVon Mises yield function:
fVM (σ) = q − σ = 0
where σ is the pure uniaxial yielding stress and,
q =√
3J2 =
√3
2σTPσ
with,
P =
2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0
0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2
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Drucker-Prager Yield Function
The J2 invariant of the deviatoric stress tensor is used to define theDrucker-Prager yield function that accounts for hydrostatic pressuredependency:
fDP (σ) = q + αp − k = 0
where α and k are material parameters and,
q =√
3J2 =
√3
2σTPσ, p = πTσ
with,
P =
2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0
0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2
, π =
1/31/31/3
000
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Tresca Yield Function
The Tresca yield function reads,
fTR (σ) =
σ11−σ222 − τmax = 0
σ22−σ112 − τmax = 0
σ11−σ332 − τmax = 0
σ33−σ112 − τmax = 0
σ22−σ332 − τmax = 0
σ33−σ222 − τmax = 0
where τmax = σ/2 is used to approximate the Von Mises yieldfunction.
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Coulomb Yield Function
The Coulomb yield function reads,
fCL (σ) =
σ11−σ222 + σ11+σ22
2 sin (ϕ)− c · cos (ϕ) = 0σ22−σ11
2 + σ11+σ222 sin (ϕ)− c · cos (ϕ) = 0
σ11−σ332 + σ11+σ33
2 sin (ϕ)− c · cos (ϕ) = 0σ33−σ11
2 + σ11+σ332 sin (ϕ)− c · cos (ϕ) = 0
σ22−σ332 + σ22+σ33
2 sin (ϕ)− c · cos (ϕ) = 0σ33−σ22
2 + σ22+σ332 sin (ϕ)− c · cos (ϕ) = 0
where α = 6sin(ϕ)3−sin(ϕ) and k = 6c·cos(ϕ)
3−sin(ϕ) are used to approximate theDrucker-Prager yield function.
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Continuous Plasticity Problem
Stress-strain response of an elastic perfectly-plastic material.
Let’s imagine to turn this into a computer program:
1: function [σj+1] = material (εj+1)2: ...3: end
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Return Mapping Algorithm: (σ,ε) vs. (r,u)
The return mapping algorithm if form of residual minimizationproblem is reported for a generic continuous plasticity model:
σj+1, ∆λj+1 :
εσ = σj+1 − σe+ Dem∆λj+1
εf = f (σj+1)
For the sake of comparison, the return mapping algorithm is reportedalso for a generic lumped plasticity model (e.g. spring-slider):
rj+1, ∆λj+1 :
εr = rj+1 − re + Dem∆λj+1
εf = f (rj+1)
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Return Mapping Algorithm: (σ,ε) vs. (r,u)
The corresponding Newton-Raphson algorithm is reported for ageneric continuous plasticity model:[
σk+1j+1
∆λk+1j+1
]=
[σkj+1
∆λkj+1
]−[∂εσ∂σ
∂εσ∂∆λ
∂εf∂σ
∂εf∂∆λ
]−1 [εkσεkf
]The Newton-Raphson algorithm is reported also for a generic lumpedplasticity model (e.g. spring-slider):[
rk+1j+1
∆λk+1j+1
]=
[rkj+1
∆λkj+1
]−[∂εr∂r
∂εr∂∆λ
∂εf∂r
∂εf∂∆λ
]−1 [εkrεkf
]
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Von Mises Plasticity with Associated Flow Rule
The gradient of the Von Mises yield surface is function of σ:
fVM (σ) =
√3
2σTPσ − σ = 0
↓
nVM = mVM =∂fVM∂σ
=3Pσ
2√
32σTPσ
where σ is the pure uniaxial yielding stress and,
P =
2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0
0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2
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Drucker-Prager Plasticity with Associated Flow Rule
The gradient of the Drucker-Prager yield surface is function of σ:
fDP (σ) =
√3
2σTPσ+ απTσ − k
↓
nDP = mDP =∂fVM∂σ
=3Pσ
2√
32σTPσ
+ απ
where α and k are material parameters and,
P =
2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0
0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2
,π =
1/31/31/3
000
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Return Mapping Algorithm with Curved Yield Surfaces
In order to guarantee convergence of the return mapping algorithmwhen the yield surface is curved, the strain increment has to besmall.
e.g. spring-slider return mapping.
re = rj + Ke∆uj+1
e.g. Von Mises return mapping.
σe = σj + [De ] ∆εj+1
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Return Mapping Algorithm
A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models, is reported in form ofresidual minimization problem:
σj+1, ∆λj+1 :
εσ = σj+1 − σe+ [De ]m (σj+1) ∆λj+1 = 0
εf = f (σj+1) = 0
↓[σk+1
j+1
∆λk+1j+1
]=
[σkj+1
∆λkj+1
]−[∂εσ∂σ
∂εσ∂∆λ
∂εf∂σ
∂εf∂∆λ
]−1 [εkσεkf
]
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Return Mapping Algorithm
A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models, is reported in form ofresidual minimization problem:
σj+1, ∆λj+1 :
εσ = σj+1 − σe+ [De ]m (σj+1) ∆λj+1 = 0
εf = f (σj+1) = 0
↓[σk+1
j+1
∆λk+1j+1
]=
[σkj+1
∆λkj+1
]−[I + [De ] ∂m∂σ∆λkj+1 [De ]m
∂f∂σ 0
]−1 [εkσεkf
]where m and f and their partial derivatives are functions of σkj+1.
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Consistent Tangent Stiffness
A formulation of the consistent tangent operator, which iscompatible with both Von Mises and Drucker-Prager plasticitymodels, is reported:
σj+1, ∆λj+1 :
εσ = σj+1 − σe+ [De ]m (σj+1) ∆λj+1 = 0
εf = f (σj+1) = 0
↓[σk+1
j+1
∆λk+1j+1
]=
[σkj+1
∆λkj+1
]−
[∂σ∂εσ
∂σ∂εf
∂∆λ∂εσ
∂∆λ∂εf
] [εkσεkf
]↓
[D]j+1 =∂σj+1
∂εj+1= −
∂σj+1
∂εσ
∂εσ∂εj+1
with,
∂ (∆εj+1) = ∂ (εj+1 − εj) = ∂εj+1 −*constant
∂εj = ∂εj+1
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Hardening Behaviour
So far we assumed that the yield function f depends only on thestress tensor σ and material parameters are constant. However,this is almost never the case:
Cyclic loading in metals: Bauschinger effect.
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Hardening Behaviour
We can identify two complementary hardening phenomena:
Isotropic hardening: expansion ofthe yield surface.
f = f (σ, κ)
κ is a scalar variable.
Kinematic hardening: translationof the yield surface.
f = f (σ, α)
α is a tensor variable.
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Isotropic Hardening
The Von Mises yield function modified by the linear isotropichardening rule reads,
fVM (σ) = q (σ)− (σ0 + hκ)
where the evolution of κ, which accounts for the expansion of theyield surface, reads,
κ = λp (σ, κ)→ κ =
∫κdt
with σ0 is the initial yield strength, h is the hardening modulus andp (σ, κ) is a scalar function depending on the hardeninghypothesis. It is noteworthy that the gradient of the yield functiondoes not depend on the isotropic hardening variable κ in this case:
∂fVM∂σ
=3Pσ
2√
32σTPσ
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Isotropic Hardening
These are some examples of isotropic hardening hypothesis:
κ :
σTεp = λ
(σTm
), work-hardening√
23εpTQεp = λ
√23m
TQm, strain-hardening
−3πT εp = −λ(3πTm
), volumetric-hardening
with,
Q =
2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0
0 0 0 1/2 0 00 0 0 0 1/2 00 0 0 0 0 1/2
, π =
1/31/31/3
000
, εp = mλ
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Return Mapping Algorithm with Isotropic Hardening
A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with isotropic hardening,is reported in form of residual minimization problem:
σj+1, κj+1, ∆λj+1 :
εσ = σj+1 − σe+ [De ]m (σj+1, κj+1) ∆λj+1
εκ = κj+1 − κj −∆λj+1p (σj+1, κj+1)
εf = f (σj+1, κj+1)
↓σk+1j+1
κk+1j+1
∆λk+1j+1
=
σkj+1
κkj+1
∆λkj+1
−∂εσ∂σ ∂εσ
∂κ∂εσ∂∆λ
∂εκ∂σ
∂εκ∂κ
∂εκ∂∆λ
∂εf∂σ
∂εf∂κ
∂εf∂∆λ
−1 εkσεkκεkf
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Return Mapping Algorithm with Isotropic Hardening
A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with isotropic hardening,is reported in form of residual minimization problem:
σj+1, κj+1, ∆λj+1 :
εσ = σj+1 − σe+ [De ]m (σj+1, κj+1) ∆λj+1
εκ = κj+1 − κj −∆λj+1p (σj+1, κj+1)
εf = f (σj+1, κj+1)
↓σk+1j+1
κk+1j+1
∆λk+1j+1
=
σkj+1
κkj+1
∆λkj+1
−I + [De ] ∂m∂σ∆λkj+1 [De ] ∂m∂κ∆λkj+1 [De ]m
− ∂p∂σ∆λkj+1 1− ∂p
∂κ∆λkj+1 −p∂f∂σ
∂f∂κ 0
−1 εkσεkκεkf
where m, p and f and their partial derivatives are functions ofσkj+1 and κkj+1.
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Kinematic Hardening
The Von Mises yield function modified by the Ziegler kinematichardening rule reads,
fVM (σ) = q (σ − α)− σ
where the evolution of α, which represents the position of thecentroid of the yield function, reads
α = λa (σ − α)→ α =
∫αdt
where a is a material parameter. It is noteworthy that the gradientof the yield function depends on the hardening variable α in thiscase:
∂fVM∂σ
=3 (Pσ − α)
2√
32 (Pσ − α)T P (Pσ − α)
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Return Mapping Algorithm with Kinematic Hardening
A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with kinematichardening, is reported in form of residual minimization problem:
σj+1, αj+1, ∆λj+1 :
εσ = σj+1 − σe+ [De ]m (σj+1, αj+1) ∆λj+1
εα = αj+1 −αj −∆λj+1a (σj+1 − αj+1)
εf = f (σj+1, αj+1)
↓σk+1j+1
αk+1j+1
∆λk+1j+1
=
σkj+1
αkj+1
∆λkj+1
− ∂εσ∂σ ∂εσ
∂α∂εσ∂∆λ
∂εα∂σ
∂εα∂α
∂εα∂∆λ
∂εf∂σ
∂εf∂α
∂εf∂∆λ
−1 εkσεkαεkf
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Return Mapping Algorithm with Kinematic Hardening
A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with kinematichardening, is reported in form of residual minimization problem:
σj+1, αj+1, ∆λj+1 :
εσ = σj+1 − σe+ [De ]m (σj+1, αj+1) ∆λj+1
εα = αj+1 −αj −∆λj+1a (σj+1 − αj+1)
εf = f (σj+1, αj+1)
↓σk+1j+1
αk+1j+1
∆λk+1j+1
=
σkj+1
αkj+1
∆λkj+1
−I + [De ] ∂m∂σ∆λkj+1 [De ] ∂m∂α∆λkj+1 [De ]m
−a∆λkj+1 1 + a∆λkj+1 −a (σj+1 − αj+1)∂f∂σ
∂f∂α 0
−1 εkσεkαεkf
where m, p and f and their partial derivatives are functions ofσkj+1 and αk
j+1.
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Return Mapping Algorithm (σ,ε): Code Template
1: ∆εj+1 ← εj+1 − εj2: σe ← σj + [De ] ∆εj+1
3: if f (σe) ≥ 0 then4: σj+1 ← σe5: ∆λj+1 ← 06: εr ← σj+1 − σe + [De ]m∆λj+1
7: εf ← f (σj+1)8: repeat
9:
[σj+1
∆λj+1
]←[σj+1
∆λj+1
]−
[∂εr∂σ
∂εr∂∆λ
∂εf∂σ
∂εf∂∆λ
]−1 [εrεf
]10: εr ← σj+1 − σe + [De ]m∆λj+1
11: εf ← f (σj+1)12: until ‖ε‖ >= Tol
13: [D]j+1 ← −∂σ∂εr
∂εr∂ε
14: else if f (σe) < 0 then15: σj+1 ← σe16: [D]j+1 ← [De ]17: end if
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Finite Element Discretization: from (σ,ε) to (r,u)
So far we derived a procedure for calculating the punctual stressresponse given a punctual strain increment for a generic plasticityconstitutive model ...
... however we want to formulate finite (length, area or volume)elements that relate nodal forces to nodal displacements.
The principle of virtual displacements facilitates their derivation:
r (uj) = f (tj)∫ΩδεTσjdΩ =
∫ΩδuTpvolj dΩ +
∫ΓδuTpsurj ·
−→dΓ
σj : stress state generated by volume pvol and surface psur
loads up to tj .
δu and δε : compatible variations of displacement u andstrain ε fields.
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Finite Element Discretization: from (σ,ε) to (r,u)
The restoring force is calculated according to the principle of virtualdisplacement for a bar element:
r (uj) =
∫ L
0δεTσjdx
The two nodal displacements completely characterize displacementand strain fields within the element:
Shape functionsn1 (x) = 1− x
L
n2 (x) = xL
Shape functions’ derivativesb1 (x) = dn1
dx = − 1L
b2 (x) = dn2dx = 1
L
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Finite Element Discretization: from (σ,ε) to (r,u)
The restoring force is calculated according to the principle of virtualdisplacement for a bar element:
r (uj) =
∫ L
0δεTσjdx
The two nodal displacements completely characterize displacementand strain fields within the element:
Displacement field Shape functions
u (x) = u1n1 (x) + u2n2 (x) =[n1 (x) n2 (x)
] [u1
u2
]= N (x)u
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Finite Element Discretization: from (σ,ε) to (r,u)
The restoring force is calculated according to the principle of virtualdisplacement for a bar element:
r (uj) =
∫ L
0δεTσjdx
The two nodal displacements completely characterize displacementand strain fields within the element:
Strain field Shape functions’ derivatives
ε (x) = u1b1 (x) + u2b2 (x) =[b1 (x) b2 (x)
] [u1
u2
]= B (x)u
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Finite Element Discretization: Restoring Force
The restoring force is calculated according to the principle of virtualdisplacement by integrating the stress field:
uj → interp.→ εj → ret. mapping→ σj
↓
r (uj) =
∫ L
0δεTσjdx
r (uj) =
∫ L
0
[b1 (x)b2 (x)
]σj (x) dx
r (uj) ≈∑m
ωm
[b1 (xm)b2 (xm)
]σj (xm)
Interpolation works exactly like for linear finite elements. The returnmapping algorithm is formulated for the specific plasticity model.
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Finite Element Discretization: Consistent Tangent Stiffness
The consistent tangent stiffness is calculated according to theprinciple of virtual displacement:
uj → interp.→ εj → ret. mapping→ [D]j
↓
∂rj∂uj
= Kj =
∫ L
0δεT
∂σj∂εj
∂εj∂uj
dx
Kj =
∫ L
0
[b1 (x)b2 (x)
][D]j (x)
[b1 (x) b2 (x)
]dx
Kj ≈∑m
ωm
[b1 (xm)b2 (xm)
][D]j (xm)
[b1 (xm) b2 (xm)
]Interpolation works exactly like for linear finite elements. Theconsistent tangent stiffness is formulated for the specific plasticitymodel.
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Nonlinear Static Analysis (r,u)
We derived a procedure for calculating the force response of a singleelement given a displacement trial ...
... but we want to solve the static displacement response of a model,which combines several elements, subjected to an external loadhistory.
The corresponding balance equation reads,
uj : r (uj)− f (tj) = 0
where,
uj : global displacement vector
r (uj) : global restoring force vector
f (tj) : global external load vector
at time step j-th.
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Nonlinear Static Analysis (r,u): Code Template
1: for j = 1 to J do2: uj ← uj−1
3: for i = 1 to I do4: ri,j ← elementForce (Ziuj)5: rj ← rj + ZT
i ri,j6: end for7: εr ← rj − f (tj)8: repeat9: for i = 1 to I do
10: Ki,j ← elementStiff (Ziuj)11: Kj ← Kj + ZT
i Ki,jZi
12: end for13: uj ← uj −K−1
j εr14: for i = 1 to I do15: ri,j ← elementForce (Ziuj)16: rj ← rj + ZT
i ri,j17: end for18: εr ← rj − f (tj)19: until ‖εr‖ >= Tol20: end for
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