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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

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.


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)


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.


Voigt Notation

Graphical representation of the Cauchy stress tensor.

σ =

σxx σxy σxzσyy σyz

sym σzz

→

σxxσyyσzzσyzσxzσxy

= {σ}


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}


Voigt Notation

Cauchy stress tensor. Cauchy (small) strain tensor.

δw int =3∑

i=1

3∑j=1

δεijσij = δεijσij = δε : σ = {δε}T{σ}


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 !!!


Voigt Notation

Isotropic elastic compliance from tensor:

εij = C eijklσkl or ε = Ce : σ

to Voigt notation:

{ε} = [Ce ] {σ}


=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)


E : Young modulus, ν : Poisson ratio.


Voigt Notation

Isotropic elastic stiffness from tensor:

σij = Deijklεkl or σ = De : ε

to Voigt notation:

{σ} = [De ] {ε}


=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


E : Young modulus, ν : Poisson ratio.


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





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.





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.





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.


Invariants of the Stress Tensor

Invariants of stress tensor σ are used to formulate yielding criteria.

σ =


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.


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.


Invariants of the Deviatoric Stress Tensor

Invariants of deviatoric stress tensor s are used to formulate yieldingcriteria.

σ =


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.




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.




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



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.


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.


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.


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.


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.


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


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}


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)


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

]


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


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 .


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 .


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.


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


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.


Hardening Behaviour

Cyclic loading in metals (Bauschinger effect):

An increase in tensile yield strength occurs at the expense ofcompressive yield strength.


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}


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


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 .


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



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})


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



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



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



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



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



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



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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