Tutorial 5: Hyper-elasticity

Problem statement

using Gridap
using LineSearches: BackTracking

Material parameters

const λ = 100.0
const μ = 1.0

Deformation Gradient

F(∇u) = one(∇u) + ∇u'

J(F) = sqrt(det(C(F)))

#Green strain

#E(F) = 0.5*( F'*F - one(F) )

dE(∇du,∇u) = 0.5*( ∇du⋅F(∇u) + (∇du⋅F(∇u))' )

Right Cauchy-green deformation tensor

C(F) = (F')⋅F

Constitutive law (Neo hookean)

function S(∇u)
  Cinv = inv(C(F(∇u)))
  μ*(one(∇u)-Cinv) + λ*log(J(F(∇u)))*Cinv
end

function dS(∇du,∇u)
  Cinv = inv(C(F(∇u)))
  _dE = dE(∇du,∇u)
  λ*(Cinv⊙_dE)*Cinv + 2*(μ-λ*log(J(F(∇u))))*Cinv⋅_dE⋅(Cinv')
end

Cauchy stress tensor

σ(∇u) = (1.0/J(F(∇u)))*F(∇u)⋅S(∇u)⋅(F(∇u))'

Model

domain = (0,1,0,1)
partition = (20,20)
model = CartesianDiscreteModel(domain,partition)

Define new boundaries

labels = get_face_labeling(model)
add_tag_from_tags!(labels,"diri_0",[1,3,7])
add_tag_from_tags!(labels,"diri_1",[2,4,8])

Setup integration

degree = 2
Ω = Triangulation(model)
dΩ = Measure(Ω,degree)

Weak form

res(u,v) = ∫( (dE∘(∇(v),∇(u))) ⊙ (S∘∇(u)) )*dΩ

jac_mat(u,du,v) =  ∫( (dE∘(∇(v),∇(u))) ⊙ (dS∘(∇(du),∇(u))) )*dΩ

jac_geo(u,du,v) = ∫( ∇(v) ⊙ ( (S∘∇(u))⋅∇(du) ) )*dΩ

jac(u,du,v) = jac_mat(u,v,du) + jac_geo(u,v,du)

Construct the FEspace

reffe = ReferenceFE(lagrangian,VectorValue{2,Float64},1)
V = TestFESpace(model,reffe,conformity=:H1,dirichlet_tags = ["diri_0", "diri_1"])

Setup non-linear solver

nls = NLSolver(
  show_trace=true,
  method=:newton,
  linesearch=BackTracking())

solver = FESolver(nls)

function run(x0,disp_x,step,nsteps,cache)

  g0 = VectorValue(0.0,0.0)
  g1 = VectorValue(disp_x,0.0)
  U = TrialFESpace(V,[g0,g1])

  #FE problem
  op = FEOperator(res,jac,U,V)

  println("\n+++ Solving for disp_x $disp_x in step $step of $nsteps +++\n")

  uh = FEFunction(U,x0)

  uh, cache = solve!(uh,solver,op,cache)

  writevtk(Ω,"results_$(lpad(step,3,'0'))",cellfields=["uh"=>uh,"sigma"=>σ∘∇(uh)])

  return get_free_dof_values(uh), cache

end

function runs()

 disp_max = 0.75
 disp_inc = 0.02
 nsteps = ceil(Int,abs(disp_max)/disp_inc)

 x0 = zeros(Float64,num_free_dofs(V))

 cache = nothing
 for step in 1:nsteps
   disp_x = step * disp_max / nsteps
   x0, cache = run(x0,disp_x,step,nsteps,cache)
 end

end

#Do the work!
runs()

Picture of the last load step

Extension to 3D

Extending this tutorial to the 3D case is straightforward. It is left as an exercise.

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