Von Mises plasticity¶

function MaterialState()
    return MaterialState(
        zero(SymmetricTensor{2, 3}),
        zero(SymmetricTensor{2, 3}),
        0.0
    )
end

function compute_stress_tangent(ϵ::SymmetricTensor{2, 3}, material::J2Plasticity, state::MaterialState)
    # unpack some material parameters
    G = material.G
    H = material.H

    # We use (•)ᵗ to denote *trial*-values
    σᵗ = material.Dᵉ ⊡ (ϵ - state.ϵᵖ) # trial-stress
    sᵗ = dev(σᵗ)         # deviatoric part of trial-stress
    J₂ = 0.5 * sᵗ ⊡ sᵗ   # second invariant of sᵗ
    σᵗₑ = sqrt(3.0 * J₂) # effective trial-stress (von Mises stress)
    σʸ = material.σ₀ + H * state.k # Previous yield limit

    φᵗ = σᵗₑ - σʸ # Trial-value of the yield surface

    if φᵗ < 0.0 # elastic loading
        return σᵗ, material.Dᵉ, MaterialState(state.ϵᵖ, σᵗ, state.k)
    else # plastic loading
        h = H + 3G
        μ = φᵗ / h # plastic multiplier

        c1 = 1 - 3G * μ / σᵗₑ
        s = c1 * sᵗ           # updated deviatoric stress
        σ = s + vol(σᵗ)       # updated stress

        # Compute algorithmic tangent stiffness $D = \frac{\Delta \sigma }{\Delta \epsilon}$
        κ = H * (state.k + μ) # drag stress
        σₑ = material.σ₀ + κ  # updated yield surface

        δ(i, j) = i == j ? 1.0 : 0.0
        Isymdev(i, j, k, l) = 0.5 * (δ(i, k) * δ(j, l) + δ(i, l) * δ(j, k)) - 1.0 / 3.0 * δ(i, j) * δ(k, l)
        Q(i, j, k, l) = Isymdev(i, j, k, l) - 3.0 / (2.0 * σₑ^2) * s[i, j] * s[k, l]
        b = (3G * μ / σₑ) / (1.0 + 3G * μ / σₑ)

        Dtemp(i, j, k, l) = -2G * b * Q(i, j, k, l) - 9G^2 / (h * σₑ^2) * s[i, j] * s[k, l]
        D = material.Dᵉ + SymmetricTensor{4, 3}(Dtemp)

        # Return new state
        Δϵᵖ = 3 / 2 * μ / σₑ * s # plastic strain
        ϵᵖ = state.ϵᵖ + Δϵᵖ      # plastic strain
        k = state.k + μ          # hardening variable
        return σ, D, MaterialState(ϵᵖ, σ, k)
    end
end

function create_dofhandler(grid, interpolation)
    dh = DofHandler(grid)
    add!(dh, :u, interpolation) # add a displacement field with 3 components
    close!(dh)
    return dh
end

function doassemble!(
        K::SparseMatrixCSC, r::Vector, cellvalues::CellValues, dh::DofHandler,
        material::J2Plasticity, u, states, states_old
    )
    assembler = start_assemble(K, r)
    nu = getnbasefunctions(cellvalues)
    re = zeros(nu)     # element residual vector
    ke = zeros(nu, nu) # element tangent matrix

    for (i, cell) in enumerate(CellIterator(dh))
        fill!(ke, 0)
        fill!(re, 0)
        eldofs = celldofs(cell)
        ue = u[eldofs]
        state = @view states[:, i]
        state_old = @view states_old[:, i]
        assemble_cell!(ke, re, cell, cellvalues, material, ue, state, state_old)
        assemble!(assembler, eldofs, ke, re)
    end
    return K, r
end

function assemble_cell!(Ke, re, cell, cellvalues, material, ue, state, state_old)
    n_basefuncs = getnbasefunctions(cellvalues)
    reinit!(cellvalues, cell)

    for q_point in 1:getnquadpoints(cellvalues)
        # For each integration point, compute stress and material stiffness
        ϵ = function_symmetric_gradient(cellvalues, q_point, ue) # Total strain
        σ, D, state[q_point] = compute_stress_tangent(ϵ, material, state_old[q_point])

        dΩ = getdetJdV(cellvalues, q_point)
        for i in 1:n_basefuncs
            δϵ = shape_symmetric_gradient(cellvalues, q_point, i)
            re[i] += (δϵ ⊡ σ) * dΩ # add internal force to residual
            for j in 1:i # loop only over lower half
                Δϵ = shape_symmetric_gradient(cellvalues, q_point, j)
                Ke[i, j] += δϵ ⊡ D ⊡ Δϵ * dΩ
            end
        end
    end
    symmetrize_lower!(Ke)
    return
end

function symmetrize_lower!(K)
    for i in 1:size(K, 1)
        for j in (i + 1):size(K, 1)
            K[i, j] = K[j, i]
        end
    end
    return
end;

function doassemble_neumann!(r, dh, facetset, facetvalues, t)
    n_basefuncs = getnbasefunctions(facetvalues)
    re = zeros(n_basefuncs)                      # element residual vector
    for fc in FacetIterator(dh, facetset)
        # Add traction as a negative contribution to the element residual `re`:
        reinit!(facetvalues, fc)
        fill!(re, 0)
        for q_point in 1:getnquadpoints(facetvalues)
            dΓ = getdetJdV(facetvalues, q_point)
            for i in 1:n_basefuncs
                δu = shape_value(facetvalues, q_point, i)
                re[i] -= (δu ⋅ t) * dΓ
            end
        end
        assemble!(r, celldofs(fc), re)
    end
    return r
end

function solve()
    # Define material parameters
    E = 200.0e9  # [Pa]
    H = E / 20   # [Pa]
    ν = 0.3      # [-]
    σ₀ = 200.0e6 # [Pa]
    material = J2Plasticity(E, ν, σ₀, H)

    L = 10.0 # beam length [m]
    w = 1.0  # beam width [m]
    h = 1.0  # beam height[m]
    n_timesteps = 10
    u_max = zeros(n_timesteps)
    traction_magnitude = 1.0e7 * range(0.5, 1.0, length = n_timesteps)

    # Create geometry, dofs and boundary conditions
    n = 2
    nels = (10n, n, 2n) # number of elements in each spatial direction
    P1 = Vec((0.0, 0.0, 0.0))  # start point for geometry
    P2 = Vec((L, w, h))        # end point for geometry
    grid = generate_grid(Tetrahedron, nels, P1, P2)
    interpolation = Lagrange{RefTetrahedron, 1}()^3

    dh = create_dofhandler(grid, interpolation) # JuaFEM helper function
    dbcs = create_bc(dh, grid) # create Dirichlet boundary-conditions

    cellvalues, facetvalues = create_values(interpolation)

    # Pre-allocate solution vectors, etc.
    n_dofs = ndofs(dh)  # total number of dofs
    u = zeros(n_dofs)   # solution vector
    Δu = zeros(n_dofs)  # displacement correction
    r = zeros(n_dofs)   # residual
    K = allocate_matrix(dh) # tangent stiffness matrix

    # Create material states. One array for each cell, where each element is an array of material-
    # states - one for each integration point
    nqp = getnquadpoints(cellvalues)
    states = [MaterialState() for _ in 1:nqp, _ in 1:getncells(grid)]
    states_old = [MaterialState() for _ in 1:nqp, _ in 1:getncells(grid)]

    # Newton-Raphson loop
    NEWTON_TOL = 1 # 1 N
    print("\n Starting Netwon iterations:\n")

    for timestep in 1:n_timesteps
        t = timestep # actual time (used for evaluating d-bndc)
        traction = Vec((0.0, 0.0, traction_magnitude[timestep]))
        newton_itr = -1
        print("\n Time step @time = $timestep:\n")
        update!(dbcs, t) # evaluates the D-bndc at time t
        apply!(u, dbcs)  # set the prescribed values in the solution vector

        while true
            newton_itr += 1
            if newton_itr > 8
                error("Reached maximum Newton iterations, aborting")
                break
            end
            # Tangent and residual contribution from the cells (volume integral)
            doassemble!(K, r, cellvalues, dh, material, u, states, states_old)
            # Residual contribution from the Neumann boundary (surface integral)
            doassemble_neumann!(r, dh, getfacetset(grid, "right"), facetvalues, traction)
            norm_r = norm(r[Ferrite.free_dofs(dbcs)])

            print("Iteration: $newton_itr \tresidual: $(@sprintf("%.8f", norm_r))\n")
            if norm_r < NEWTON_TOL
                break
            end

            apply_zero!(K, r, dbcs)
            Δu = Symmetric(K) \ r
            u -= Δu
        end

        # Update the old states with the converged values for next timestep
        states_old .= states

        u_max[timestep] = maximum(abs, u) # maximum displacement in current timestep
    end

    # ## Postprocessing
    # Only a vtu-file corresponding to the last time-step is exported.
    #
    # The following is a quick (and dirty) way of extracting average cell data for export.
    mises_values = zeros(getncells(grid))
    κ_values = zeros(getncells(grid))
    for (el, cell_states) in enumerate(eachcol(states))
        for state in cell_states
            mises_values[el] += vonMises(state.σ)
            κ_values[el] += state.k * material.H
        end
        mises_values[el] /= length(cell_states) # average von Mises stress
        κ_values[el] /= length(cell_states)     # average drag stress
    end
    VTKGridFile("plasticity", dh) do vtk
        write_solution(vtk, dh, u) # displacement field
        write_cell_data(vtk, mises_values, "von Mises [Pa]")
        write_cell_data(vtk, κ_values, "Drag stress [Pa]")
    end

    return u_max, traction_magnitude
end

u_max, traction_magnitude = solve();

using Plots
plot(
    vcat(0.0, u_max),                # add the origin as a point
    vcat(0.0, traction_magnitude),
    linewidth = 2,
    title = "Traction-displacement",
    label = nothing,
    markershape = :auto
)
ylabel!("Traction [Pa]")
xlabel!("Maximum deflection [m]")