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### Transcript of ETH Z£¼rich - Homepage | ETH Z£¼rich - The Finite Element ... ... methods...

• The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Computational

Plasticity Part II

Prof. Dr. Eleni Chatzi Dr. 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 of Voigt notation for representing tensors.

To understand basic rate-independent plasticity models formulated in terms of stress and strain fields.

To derive displacement-based finite elements based on such constitutive 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., Computational methods 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 a scalar

Described by a set of Ordinary Differential Equations (ODE)

Continuous model:

The stress σ is a 2nd order tensor

Described by a set of Partial Differential Equations (PDE)

Institute of Structural Engineering Method of Finite Elements II 3

• Voigt Notation

Stresses and strains are second order tensors related by a fourth order tensor describing the elastic properties of the continuum.

σij = D e ijkl�kl

i , j , k, l → {1, 2, 3} ↓

{σ} 6×1

= [De ] 6×6 {�} 6×1

However, in order to facilitate the implementation of computer programs -when possible- it is more convenient to work with vectors and matrices. A clear description of Voigt notation is reported in:

Belytschko, T., Wing Kam L., Brian M., and Khalil E.. Nonlinear finite 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

→ 

σxx σyy σzz σyz σxz σxy

 = {σ}

Institute of Structural Engineering Method of Finite Elements II 5

• Voigt Notation

Graphical representation of the Green-Lagrange (small) strain tensor.

� =

 �xx �xy �xz�yy �yz sym �zz

 �xx =

∂u

∂x , �xy =

γxy 2

= 1

2

( ∂u

∂y + ∂v

∂x

) �yy =

∂v

∂y , �xz =

γxz 2

= 1

2

( ∂u

∂z + ∂w

∂x

) �zz =

∂w

∂z , �yz =

γyz 2

= 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.

� =

 �xx �xy �xz�yy �yz sym �zz

→ 

�xx �yy �zz

2�yz 2�xz 2�xy

 = 

�xx �yy �zz γyz γxz γxy

 = {�}

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

δ�ijσij = δ�ijσij = δ� : σ = {δ�}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

δ�ijσij = δ�ijσij = δ� : σ = {δ�}T{σ}

Principle of virtual displacement !!!

Institute of Structural Engineering Method of Finite Elements II 7

• Voigt Notation

Isotropic elastic compliance from tensor:

�ij = C e ijklσkl or � = C

e : σ

to Voigt notation:

{�} = [Ce ] {σ}



�xx �yy �zz γyz γxz γxy

 = 1

E



1 −ν −ν 0 0 0 −ν 1 −ν 0 0 0 −ν −ν 1 0 0 0 0 0 0 2 (1 + ν) 0 0 0 0 0 0 2 (1 + ν) 0 0 0 0 0 0 2 (1 + ν)





σxx σyy σzz σyz σxz σxy

 E : Young modulus, ν : Poisson ratio.

Institute of Structural Engineering Method of Finite Elements II 8

• Voigt Notation

Isotropic elastic stiffness from tensor:

σij = D e ijkl�kl or σ = D

e : �

to Voigt notation:

{σ} = [De ] {�}

 σxx σyy σzz σyz σxz σxy

 = E

(1 + ν) (1− 2ν)

 1− ν ν ν 0 0 0 ν 1− ν ν 0 0 0 ν ν 1− ν 0 0 0 0 0 0 1−2ν2 0 0 0 0 0 0 1−2ν2 0 0 0 0 0 0 1−2ν2



 �xx �yy �zz γyz γxz γxy

 E : Young modulus, ν : Poisson ratio.

Institute of Structural Engineering Method of Finite Elements II 9

• From Lumped to Continuous Plasticity Models

Lumped plasticity model r, u, Ke

Continuous plasticity model {σ}, {�}, [De ]

Elastic regime if f (r) < 0

↓ ṙ = Ke u̇

if f ({σ}) < 0 ↓

{σ̇} = [De ] {�̇}

Elastoplastic regime if f (r) = 0

↓{ ṙ = Ke (u̇− u̇p) ḟ = 0

with u̇p = λ̇m

if f ({σ}) = 0 ↓{

{σ̇} = [De ] ({�̇} − {�̇p}) ḟ = 0

with {�̇p} = λ̇m Institute of Structural Engineering Method of Finite Elements II 10

• From Lumped to Continuous Plasticity Models

Lumped plasticity model r, u, Ke

Continuous plasticity model {σ}, {�}, [De ]

if f (r) = 0

↓{ ṙ = Ke (u̇− u̇p) ḟ = 0

with u̇p = λ̇m

if f ({σ}) = 0 ↓{

{σ̇} = [De ] ({�̇} − {�̇p}) ḟ = 0

with {�̇p} = λ̇m

Yield criterion : this is a scalar function that determines the boundary of the elastic domain.

Institute of Structural Engineering Method of Finite Elements II 11

• From Lumped to Continuous Plasticity Models

Lumped plasticity model r, u, Ke

Continuous plasticity model {σ}, {�}, [De ]

if f (r) = 0

↓{ ṙ = Ke (u̇− u̇p) ḟ = 0

with u̇p = λ̇m

if f ({σ}) = 0 ↓{

{σ̇} = [De ] ({�̇} − {�̇p}) ḟ = 0

with {�̇p} = λ̇m

Flow rule : this is a vector function that determines the direction of the plastic strain flow.

Institute of Structural Engineering Method of Finite Elements II 11

• From Lumped to Continuous Plasticity Models

Lumped plasticity model r, u, Ke

Continuous plasticity model {σ}, {�}, [De ]

if f (r) = 0

↓{ ṙ = Ke (u̇− u̇p) ḟ = 0

with u̇p = λ̇ ∂f

∂r

if f ({σ}) = 0 ↓{

{σ̇} = [De ] ({�̇} − {�̇p}) ḟ = 0

with {�̇p} = λ̇ ∂f

∂{σ}

In the case of associated plasticity, the same function f defines both yield 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 and λ = {σ11, σ22, σ33} are the eigenvalues of the stress tensor also called 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 = σ 2 xy + σ

2 yz + σ

2 zx − σxxσyy − σyyσzz − σzzσxx

I3 = σxxσyyσzz + 2σxyσyzσzx − σxxσ2yz − σyyσ2zx − 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 yielding criteria.

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 Structura