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  • 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