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
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:
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
Institute of Structural Engineering Method of Finite Elements II 2
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)
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↓
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.
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. Green-Lagrange (small) strain tensor.
δw int =3∑
i=1
3∑j=1
δεijσij = δεijσij = δε : σ = δεTσ
Institute of Structural Engineering Method of Finite Elements II 7
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 !!!
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 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
Institute of Structural Engineering Method of Finite Elements II 10
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.
Institute of Structural Engineering Method of Finite Elements II 11
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.
Institute of Structural Engineering Method of Finite Elements II 11
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.
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
2σT [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
2sT [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 = 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
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) = 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
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) = 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
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) = 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
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 k = 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 [σj+1] = material (εj+1)2: ...3: end
Institute of Structural Engineering Method of Finite Elements II 21
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)
Institute of Structural Engineering Method of Finite Elements II 22
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
]
Institute of Structural Engineering Method of Finite Elements II 23
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
Institute of Structural Engineering Method of Finite Elements II 24
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
Institute of Structural Engineering Method of Finite Elements II 25
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
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:
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
]
Institute of Structural Engineering Method of Finite Elements II 27
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.
Institute of Structural Engineering Method of Finite Elements II 27
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
Institute of Structural Engineering Method of Finite Elements II 28
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.
Institute of Structural Engineering Method of Finite Elements II 29
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.
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) = 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
Institute of Structural Engineering Method of Finite Elements II 31
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
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:
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
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:
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.
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) = 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)
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:
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
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:
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.
Institute of Structural Engineering Method of Finite Elements II 35
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
Institute of Structural Engineering Method of Finite Elements II 36
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.
Institute of Structural Engineering Method of Finite Elements II 37
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
Institute of Structural Engineering Method of Finite Elements II 38
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
Institute of Structural Engineering Method of Finite Elements II 38
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
Institute of Structural Engineering Method of Finite Elements II 38
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.
Institute of Structural Engineering Method of Finite Elements II 39
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.
Institute of Structural Engineering Method of Finite Elements II 40
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.
Institute of Structural Engineering Method of Finite Elements II 41
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
Institute of Structural Engineering Method of Finite Elements II 42
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