Finite Element Method Elasto-Plastic Materials using the ... Finite Element Method Matthias M£¼ller

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  • 1

    1M Müller – Elasto-Plastic FEM

    ETH Zurich

    Interactive Simulation of Elasto-Plastic Materials using the Finite Element Method

    Matthias Müller Seminar – Wintersemester 02/03

    Movie

    2M Müller – Elasto-Plastic FEM

    ETH Zurich Outline

    FEM vs. Mass-Spring Stiffness

    The Stiffness Matrix Static/Dynamic Deformation

    Continuum Mechanics and FEM Strain and Stress Tensors Continuous PDE’s FEM Discretization

    Plasticity Plastic Strain Update Rules

    Fracture Principal Stresses Crack Computation

    3M Müller – Elasto-Plastic FEM

    ETH Zurich Mass-Spring vs. FEM

    1. Discretization of an object into mass points

    2. Representation of forces between mass points with springs

    3. Computation of the dynamics

    deformable mass-spring system

    1. Discretization of an object into elements (tetrahedra)

    2. Discretization of continuous energy equations into algebraic equations for forces acting at vertices

    3. Computation of the dynamics

    deformable FEM system

    4M Müller – Elasto-Plastic FEM

    ETH Zurich Pros and Cons of FEM

    Pros: • No individual spring constants needed

    (only 2 known material parameters E,ν) • No inversion problems

    (inverted tetrahedra produce forces) • Stress and strain tensors allow

    • fracture and • plasticity simulations

    Cons: • (Pre-)compute stiffness matrix • Store stiffness matrix (3x3) per edge • Store original and actual positions of vertices

  • 2

    5M Müller – Elasto-Plastic FEM

    ETH Zurich Outline

    FEM vs. Mass-Spring Stiffness

    The Stiffness Matrix Static/Dynamic Deformation

    Continuum Mechanics and FEM Strain and Stress Tensors Continuous PDE’s FEM Discretization

    Plasticity Plastic Strain Update Rules

    Fracture Principal Stresses Crack Computation

    6M Müller – Elasto-Plastic FEM

    ETH Zurich One-dimensional Spring

    f

    ∆x

    f = k · ∆x

    7M Müller – Elasto-Plastic FEM

    ETH Zurich Three-dimensional Object

    Finite Element Mesh • 903 tetrahedra • 393 vertices • 3 x 393 = 1179 dof.

    ∆x1x ∆x1y ∆x1z ... ∆xnx ∆xny ∆xnz

    ∆x =

    f1x f1y f1z ... fn

    x

    fny fnz

    f =

    8M Müller – Elasto-Plastic FEM

    ETH Zurich Three-dimensional Object

    f

    ∆x

    fel = K · ∆x (Stiffness Matix K ∈ R3nx3n) fel = F(∆x) (Function F : R3n→R3n)

  • 3

    9M Müller – Elasto-Plastic FEM

    ETH Zurich Static Deformation

    f

    ∆xfext = fel = K · ∆x ∆x = K-1 · fext

    fext = fel = F(∆x) ∆x = F-1(fext)

    Solve linear system (Conjugate Gradients)

    Solve non-linear system (Newton-Raphson - generalized Newton-Method)

    10M Müller – Elasto-Plastic FEM

    ETH Zurich Dynamic Deformation

    Mx´´ + Cx´ + K∆x = fext • Coupled system of 3n linear ODEs • Explicit integration: No solver needed • Implicit integration: Linear solver per time step

    Mx´´ + Cx´ + F(∆x) = fext • Coupled system of 3n non-linear ODEs • Explicit integration: No solver needed • Implicit integration: Linearize at every time step: K = dF/dx

    11M Müller – Elasto-Plastic FEM

    ETH Zurich Outline

    FEM vs. Mass-Spring Stiffness

    The Stiffness Matrix Static/Dynamic Deformation

    Continuum Mechanics and FEM Strain and Stress Tensors Continuous PDE’s FEM Discretization

    Plasticity Plastic Strain Update Rules

    Fracture Principal Stresses Crack Computation

    12M Müller – Elasto-Plastic FEM

    ETH Zurich Continuous Elasticity 1-d

    fn/A = E ∆l/l

    fn A

    ∆l l

    strain ε [1] (Dehnung)

    Elasticity (Young‘s) Modulus [N/m2]

    Metal: ~1011 N/m2

    Soft material: ~106 N/m2

    stress σ [N/m2] (Normal-Spannung)

  • 4

    13M Müller – Elasto-Plastic FEM

    ETH Zurich Continuous Elasticity 3-d

    Deformation:

    x y

    z

    • Continuous 3-d vector field u: R3 → R3

    • Defined within undeformed object

    u(x) u(x,y,z) v(x,y,z) w(x,y,z)

    x+u(x)x

    14M Müller – Elasto-Plastic FEM

    ETH Zurich Linear 3-d Strain

    normal strain in x - direction:

    x x+dx

    u(x) u(x+dx)

    dx

    dx u(x+dx)-u(x) εx = u(x+dx)–u(x)

    dx = ∂/∂xu = ux

    shear strain:

    x x+dx

    v(x) v(x+dx)

    dx

    v(x+dx)-v(x) γxy = v(x+dx)–v(x)

    dx = ∂/∂xv = vx

    15M Müller – Elasto-Plastic FEM

    ETH Zurich Linear 3-d Strain

    Linear strain tensor:

      

      

    ++ ++ ++

    =   

      

     ==

    zzyzx

    yzyyx

    xzxyx

    zzyzx

    yzyyx

    xzxyx

    wvwuw wvvuv wuvuu

    zyx εγγ γεγ γγε

    ),,(εε

      

      

           

           

    ∂∂∂∂ ∂∂∂∂

    ∂∂∂∂ ∂∂

    ∂∂ ∂∂

    =

           

           

    = w v u

    xz yz

    xy z

    y x

    zx

    yz

    xy

    z

    y

    x

    /0/ //0 0// /00 0/0 00/

    γ γ γ ε ε ε

    ε

    Symmetric, 3x3 matrix → 6 vector:

    16M Müller – Elasto-Plastic FEM

    ETH Zurich Non-Linear 3-d Strain

    Green-Saint-Venant strain tensor:

    ( )   

      

      

      

    +∂∂ +∂∂ +∂∂

      

      

    +∂∂ +∂∂ +∂∂

      

      

    +∂∂ +∂∂ +∂∂

    == )(/ )(/ )(/

    , )(/ )(/ )(/

    , )(/ )(/ )(/

    ,, wzz vyz uxz

    wzy vyy uxy

    wzx vyx uxx

    zyx JJJJ

    • Transformation we use: x → x + u(x): • Use Jacobian of transformation:

    IJJε −= TGreen

    • Interpretation:

    x y

    z x

    x + u(x)

    Jx

    Jy Jz

  • 5

    17M Müller – Elasto-Plastic FEM

    ETH Zurich 3-d Stress

    An f

    A fσ

    ⋅ == dA d

    d d

    Stress is force per (oriented) area:

    A

    zyzxz

    yzyxy

    xzxyx

    AdA d nnσf ⋅

      

      

    =⋅= σττ τστ ττσ

      

      

    xz

    xy

    x

    τ τ σ

      

      

    yz

    y

    yx

    τ σ τ

      

      

    z

    zy

    zx

    σ τ τ

    nA f

    A

    18M Müller – Elasto-Plastic FEM

    ETH Zurich Constitutive Relation (isotropic)

    The stress tensor is symmetric, 3x3 matrix → 6 vector:

    Only two scalar parameters: E: Young’s modulus, ν: Poisson ratio

           

           

           

           

    − −

    =

           

           

    zx

    yz

    xy

    z

    y

    x

    zx

    yz

    xy

    z

    y

    x

    G G

    G ccc

    ccc ccc

    γ γ γ ε ε ε

    ννν ννν ννν

    τ τ τ σ σ σ

    0

    )1( 0)1(

    )1(

    )1(2 , )21)(1( ννν +

    = −+

    = EGEc

    E εσ =Hooke‘s law:

    19M Müller – Elasto-Plastic FEM

    ETH Zurich Putting it all together

    Given u(x) we can compute • strain ε(x) and • stress σ(x) = E ε(x) at every point x within the object.

    → Find u(x) such that corresponding stresses σ(x) are in balance with external forces f(x) everywhere within object:

    0

    0

    0

    ,,,

    ,,,

    ,,,

    =+++

    =+++

    =+++

    zzzyzyxzx

    yzyzyyxyx

    xzxzyxyxx

    fσττ fτστ fττσ • Strong formulation

    • Coupled system of partial differential equations!

    20M Müller – Elasto-Plastic FEM

    ETH Zurich Energy Formulation

    • Energy U is a scalar • U at point x is given by „displacement × force“:

    Eεεσε TTelastic 2 1

    2 1 ==U

    • The total Energy of the deformed body:

    ∫  

       ⋅−=

    body

    T body fu2

    1 dVU Eεε

    • Given f, E we can compute Ubody for any u(x) • Find u(x) such that Ubody is a minimum (δU = 0)

  • 6

    21M Müller – Elasto-Plastic FEM

    ETH Zurich Outline

    FEM vs. Mass-