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
Institute of Structural Engineering Method of Finite Elements II 1
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.
Institute of Structural Engineering Method of Finite Elements II 2
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)
Institute of Structural Engineering Method of Finite Elements II 3
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.
Institute of Structural Engineering Method of Finite Elements II 4
Voigt Notation
Graphical representation of the Cauchy stress tensor.
σ =
σxx σxy σxzσyy σyz
sym σzz
→
σxxσyyσzzσyzσxzσxy
= {σ}
Institute of Structural Engineering Method of Finite Elements II 5
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
Voigt Notation
Graphical representation of the Green-Lagrange (small) strain tensor.
ε =
εxx εxy εxzεyy εyz
sym εzz
→
εxxεyyεzz
2εyz2εxz2εxy
=
εxxεyyεzzγyzγxzγxy
= {ε}
Institute of Structural Engineering Method of Finite Elements II 6
Voigt Notation
Cauchy stress tensor. Cauchy (small) strain tensor.
δw int =3∑
i=1
3∑j=1
δεijσij = δεijσij = δε : σ = {δε}T{σ}
Institute of Structural Engineering Method of Finite Elements II 7
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 !!!
Institute of Structural Engineering Method of Finite Elements II 7
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.
Institute of Structural Engineering Method of Finite Elements II 8
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.
Institute of Structural Engineering Method of Finite Elements II 9
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
Institute of Structural Engineering Method of Finite Elements II 10
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.
Institute of Structural Engineering Method of Finite Elements II 11
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.
Institute of Structural Engineering Method of Finite Elements II 11
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.
Institute of Structural Engineering Method of Finite Elements II 11
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.
Institute of Structural Engineering Method of Finite Elements II 12
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.
Institute of Structural Engineering Method of Finite Elements II 13
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.
Institute of Structural Engineering Method of Finite Elements II 14
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.
Institute of Structural Engineering Method of Finite Elements II 15
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
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
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.
Institute of Structural Engineering Method of Finite Elements II 17
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.
Institute of Structural Engineering Method of Finite Elements II 17
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.
Institute of Structural Engineering Method of Finite Elements II 18
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.
Institute of Structural Engineering Method of Finite Elements II 19
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.
Institute of Structural Engineering Method of Finite Elements II 20
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
Institute of Structural Engineering Method of Finite Elements II 21
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}
Institute of Structural Engineering Method of Finite Elements II 22
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)
Institute of Structural Engineering Method of Finite Elements II 23
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
]
Institute of Structural Engineering Method of Finite Elements II 24
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
Institute of Structural Engineering Method of Finite Elements II 25
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 .
Institute of Structural Engineering Method of Finite Elements II 26
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 .
Institute of Structural Engineering Method of Finite Elements II 26
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.
Institute of Structural Engineering Method of Finite Elements II 27
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
Institute of Structural Engineering Method of Finite Elements II 28
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.
Institute of Structural Engineering Method of Finite Elements II 29
Hardening Behaviour
Cyclic loading in metals (Bauschinger effect):
An increase in tensile yield strength occurs at the expense ofcompressive yield strength.
Institute of Structural Engineering Method of Finite Elements II 30
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}
Institute of Structural Engineering Method of Finite Elements II 31
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
Institute of Structural Engineering Method of Finite Elements II 32
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 .
Institute of Structural Engineering Method of Finite Elements II 33
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 .
Institute of Structural Engineering Method of Finite Elements II 33
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})
Institute of Structural Engineering Method of Finite Elements II 34
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 .
Institute of Structural Engineering Method of Finite Elements II 35
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 .
Institute of Structural Engineering Method of Finite Elements II 35
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
Institute of Structural Engineering Method of Finite Elements II 36
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
Institute of Structural Engineering Method of Finite Elements II 36
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
Institute of Structural Engineering Method of Finite Elements II 37
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
Institute of Structural Engineering Method of Finite Elements II 38
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
Institute of Structural Engineering Method of Finite Elements II 39