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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 4b, October, 2019
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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:
de Borst, R., Crisfield, M. A., Remmers, J. J. C., Verhoosel, C.V., Nonlinear finite element analysis of solids and structures,2nd Edition, Wiley, 2012.
de Souza Neto, E. A., Peric, D., Owen, D. R., Computationalmethods for plasticity: theory and applications, John Wiley &Sons, 2011.
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Lumped vs. Continuous Plasticity Models
Lumped model:
The restoring force r is ascalar
Described by a set ofOrdinary DifferentialEquations (ODE)
Continuous model:
The stress σ is a 2nd ordertensor
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}↓
{σ}6×1
= [De ]6×6{ε}6×1
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
)Institute of Structural Engineering Method of Finite Elements II 6
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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. Cauchy (small) strain tensor.
δw int =3∑
i=1
3∑j=1
δεijσij = δεijσij = δε : σ = {δε}T{σ}
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Voigt Notation
Cauchy stress tensor. Cauchy (small) strain tensor.
δw int =3∑
i=1
3∑j=1
δεijσij = δεijσij = δε : σ = {δε}T{σ}
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 model{σ}, {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 model{σ}, {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 model{σ}, {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 model{σ}, {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
2{σ}T [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
2{s}T [Ce ] {s} =
1
2E
(J2
1 + 2J2 (1 + ν))
where Ψd is the deviatoric elastic energy potential.Institute of Structural Engineering Method of Finite Elements II 16
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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 = 0
J2 =(σxx − σyy )2 + (σyy − σzz)2 + (σzz − σxx)2
6+ σ2
xy + σ2xz + σ2
yz
↓
Ψd =1
2{s}T [Ce ] {s} =
J2 (1 + ν)
E
where Ψd is the deviatoric elastic energy potential.Institute of Structural Engineering Method of Finite Elements II 16
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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) = σVM − σ = 0
with,
σVM =√
3J2
=
√(σxx − σyy )2 + (σyy − σzz)2 + (σzz − σxx)2
2+ 3σ2
xy + 3σ2xz + 3σ2
yz
σ is the uniaxial yielding stress.
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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) = σVM − σ = 0
with,
σVM =√
3J2 =
√3
2{σ}TP{σ}
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
σ is the uniaxial yielding stress.
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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) = σDP − σ = 0
with,
σDP =
√3
2{σ}TP{σ}+ απT{σ}
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
σ is the uniaxial yielding stress and and α accounts for the effect ofhydrostatic pressure.
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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 σ = 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 [{σ}j+1] = material ({ε}j+1)2: ...3: end
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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} = {σ}j + [De ] {∆εj+1}
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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:
{{σ}j+1, ∆λj+1} :
{εσ = {σ}j+1 − {σe}+ Dem∆λj+1
εf = f ({σ}j+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:[
{σ}k+1j+1
∆λk+1j+1
]=
[{σ}kj+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
2{σ}TP{σ} − σ = 0
↓
mVM =∂fVM∂{σ}
=3P{σ}
2√
32{σ}TP{σ}
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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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:
{{σ}j+1, ∆λj+1} :
{εσ = {σ}j+1 − {σe}+ [De ]m ({σ}j+1) ∆λj+1 = 0
εf = f ({σ}j+1) = 0
↓[{σ}k+1
j+1
∆λk+1j+1
]=
[{σ}kj+1
∆λkj+1
]−[∂εσ∂σ
∂εσ∂∆λ
∂εf∂σ
∂εf∂∆λ
]−1 [εkσεkf
]k indicates a generic Newton iteration of the solution of a singlesolution step j .
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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:
{{σ}j+1, ∆λj+1} :
{εσ = {σ}j+1 − {σe}+ [De ]m ({σ}j+1) ∆λj+1 = 0
εf = f ({σ}j+1) = 0
↓[{σ}k+1
j+1
∆λk+1j+1
]=
[{σ}kj+1
∆λkj+1
]−[I + [De ] ∂m∂σ∆λkj+1 [De ]m
∂f∂σ 0
]−1 [εkσεkf
]k indicates a generic Newton iteration of the solution of a singlesolution step j .
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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:
{{σ}j+1, ∆λj+1} :
{εσ = {σ}j+1 − {σe}+ [De ]m ({σ}j+1) ∆λj+1 = 0
εf = f ({σ}j+1) = 0
↓[∂σ∂εσ
∂σ∂εf
∂∆λ∂εσ
∂∆λ∂εf
]→ [D]j+1 =
∂{σ}j+1
∂{ε}j+1= −
∂{σ}j+1
∂εσ
∂εσ∂{ε}j+1
with,
∂ (∆{ε}j+1) = ∂ ({ε}j+1 − {ε}j) = ∂{ε}j+1 −����*constant
∂{ε}j = ∂{ε}j+1
The Jacobian is evaluated numerically and then inverted.
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Return Mapping Algorithm ({σ},{ε}): Code Template
1: ∆{ε}j+1 ← {ε}j+1 − {ε}j2: {σ}e ← {σ}j + [De ] ∆{ε}j+1
3: if f ({σ}e) ≥ 0 then4: {σ}j+1 ← {σ}e5: ∆λj+1 ← 06: εr ← {σ}j+1 − {σ}e + [De ]m∆λj+1
7: εf ← f ({σ}j+1)8: repeat
9:
[{σ}j+1
∆λj+1
]←[{σ}j+1
∆λj+1
]−
[∂εr∂{σ}
∂εr∂∆λ
∂εf∂{σ}
∂εf∂∆λ
]−1 [εrεf
]10: εr ← {σ}j+1 − {σ}e + [De ]m∆λj+1
11: εf ← f ({σ}j+1)12: until ‖ε‖ >= Tol
13: [D]j+1 ← −∂{σ}∂εr
∂εr∂{ε}
14: else if f ({σ}e) < 0 then15: {σ}j+1 ← {σ}e16: [D]j+1 ← [De ]17: end if
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Hardening Behaviour
The yield function f evolves after plastic deformation:
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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Hardening Behaviour
Cyclic loading in metals (Bauschinger effect):
An increase in tensile yield strength occurs at the expense ofcompressive yield strength.
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Isotropic Hardening
The Von Mises yield function modified by the linear isotropichardening rule reads,
fVM ({σ}) =
√3
2{σ}TP{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}
=3P{σ}
2√
32{σ}TP{σ}
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Isotropic Hardening
These are some examples of isotropic hardening hypothesis:
κ :
{σ}T{εp} = λ
({σ}Tm
), work-hardening√
23{εp}TQ{εp} = 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:
{{σ}j+1, κj+1, ∆λj+1} :
εσ = {σ}j+1 − {σe}+ [De ]m ({σ}j+1, κj+1) ∆λj+1
εκ = κj+1 − κj −∆λj+1p ({σ}j+1, κj+1)
εf = f ({σ}j+1, κj+1)
↓{σ}k+1j+1
κk+1j+1
∆λk+1j+1
=
{σ}kj+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
k indicates a generic iteration within the analysis step j .
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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:
{{σ}j+1, κj+1, ∆λj+1} :
εσ = {σ}j+1 − {σe}+ [De ]m ({σ}j+1, κj+1) ∆λj+1
εκ = κj+1 − κj −∆λj+1p ({σ}j+1, κj+1)
εf = f ({σ}j+1, κj+1)
↓{σ}k+1j+1
κk+1j+1
∆λk+1j+1
=
{σ}kj+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
k indicates a generic iteration within the analysis step j .
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Kinematic Hardening
The Von Mises yield function modified by the Ziegler kinematichardening rule reads,
fVM ({σ}) =
√3
2({σ}T − {α}T )P ({σ} − {α})− σ
where the evolution of {α}, which represents the position of thecentroid of the yield function, reads
{α} = λa ({σ} − {α})→ {α} =
∫{α}dt
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}
=3P ({σ} − {α})
2√
32 (P{σ} − {α})T P (P{σ} − {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:
{{σ}j+1, {α}j+1, ∆λj+1} :εσ = {σ}j+1 − {σe}+ [De ]m ({σ}j+1, {α}j+1) ∆λj+1
εα = {α}j+1 − {α}j −∆λj+1a ({σ}j+1 − {α}j+1)
εf = f ({σ}j+1, {α}j+1){σ}k+1j+1
{α}k+1j+1
∆λk+1j+1
=
{σ}kj+1
{α}kj+1
∆λkj+1
− ∂εσ∂σ ∂εσ
∂α∂εσ∂∆λ
∂εα∂σ
∂εα∂α
∂εα∂∆λ
∂εf∂σ
∂εf∂α
∂εf∂∆λ
−1 εkσεkαεkf
k indicates a generic iteration within the analysis step j .
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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:
{{σ}j+1, {α}j+1, ∆λj+1} :εσ = {σ}j+1 − {σe}+ [De ]m ({σ}j+1, {α}j+1) ∆λj+1
εα = {α}j+1 − {α}j −∆λj+1a ({σ}j+1 − {α}j+1)
εf = f ({σ}j+1, {α}j+1){σ}k+1j+1
{α}k+1j+1
∆λk+1j+1
=
{σ}kj+1
{α}kj+1
∆λkj+1
−I + [De ] ∂m∂σ∆λkj+1 [De ] ∂m∂α∆λkj+1 [De ]m
−a∆λkj+1 1 + a∆λkj+1 −a ({σ}j+1 − {α}j+1)∂f∂σ
∂f∂α 0
−1 εkσεkαεkf
k indicates a generic iteration within the analysis step j .
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Finite Element Discretization: from ({σ},{ε}) to (fint ,u)
In order to compute element nodal forces from stresses we apply theweak form:
∫δεTσdV︸ ︷︷ ︸Fint
=
∫δuT
bdV ‘ +
∫δuT
tdA︸ ︷︷ ︸Fext
Let’s consider an incremental decomposition of the strains
ε = εi + ∆ε
where δεi = 0 as variation of a known quantity.Then the stress is written as
σ = σi + ∆σ = σi +∂σ
∂ε︸︷︷︸D
∆ε
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Finite Element Discretization: from ({σ},{ε}) to (fint ,u)
In order to compute element nodal forces from stresses we apply theweak form:
∫δεTσdV︸ ︷︷ ︸Fint
=
∫δuT
bdV ‘ +
∫δuT
tdA︸ ︷︷ ︸Fext
Substituting the previous in the weak form gives∫δεTσdV =
∫δ(εi + ∆ε)T (σi + ∆σ)dV
=
∫(δ∆εTσi + δ∆εT∆σ)dV =
∫δ∆εTσidV︸ ︷︷ ︸
Fint
+
∫δ∆εTD∆εdV
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Finite Element Discretization: from ({σ},{ε}) to (fint ,u)
Using the discretization of the FE formulation for displacements
U = NUn
where N are the FE shape functions, and Un is the nodaldisplacement vector, with
∆U = N∆Un, δU = NδUn, δ∆U = Nδ∆Un
For a bar element N =[N1 N2
]=[1− x/L x/L
]
Displacement field Shape functions
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Finite Element Discretization: from ({σ},{ε}) to (fint ,u)
Discretization of strains
ε = BUn
where B is the strain-displacement matrix containing the shapefunction derivatives.
For a bar element ε =[−1 1
]︸ ︷︷ ︸B
[U1
U2
]︸ ︷︷ ︸Un
Strain field Shape functions derivatives
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Finite Element Discretization: from ({σ},{ε}) to (fint ,u)
Discretization of strains
ε = BUn
with
∆ε = B∆Un, δε = NδUn, δ∆ε = Nδ∆Un
f iint =
∫δ∆εTσidV = δ∆UnT
∫BTσidV
For the bar element this means:
f iint =[δ∆U1 δ∆U2
] ∫ [−11
]σ11dV∫
δ∆εTD∆εdV = δ∆UnT∫
BTDBdV︸ ︷︷ ︸KT
∆Un
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