{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# Von Mises plasticity\n",
    "\n",
    "![Shows the von Mises stress distribution in a cantilever beam.](plasticity.png)\n",
    "\n",
    "*Figure 1.* A coarse mesh solution of a cantilever beam subjected to a load\n",
    "causing plastic deformations. The initial yield limit is 200 MPa but due to\n",
    "hardening it increases up to approximately 240 MPa."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Introduction\n",
    "\n",
    "This example illustrates the use of a nonlinear material model in Ferrite.\n",
    "The particular model is von Mises plasticity (also know as J₂-plasticity) with\n",
    "isotropic hardening. The model is fully 3D, meaning that no assumptions like *plane stress*\n",
    "or *plane strain* are introduced.\n",
    "\n",
    "Also note that the theory of the model is not described here, instead one is\n",
    "referred to standard textbooks on material modeling.\n",
    "\n",
    "To illustrate the use of the plasticity model, we setup and solve a FE-problem\n",
    "consisting of a cantilever beam loaded at its free end. But first, we shortly\n",
    "describe the parts of the implementation dealing with the material modeling."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Material modeling\n",
    "This section describes the `struct`s and methods used to implement the material\n",
    "model"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Material parameters and state variables\n",
    "\n",
    "Start by loading some necessary packages"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "using Ferrite, Tensors, SparseArrays, LinearAlgebra, Printf"
   ],
   "metadata": {},
   "execution_count": 1
  },
  {
   "cell_type": "markdown",
   "source": [
    "We define a J₂-plasticity-material, containing material parameters and the elastic\n",
    "stiffness Dᵉ (since it is constant)"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "struct J2Plasticity{T, S <: SymmetricTensor{4, 3, T}}\n",
    "    G::T  # Shear modulus\n",
    "    K::T  # Bulk modulus\n",
    "    σ₀::T # Initial yield limit\n",
    "    H::T  # Hardening modulus\n",
    "    Dᵉ::S # Elastic stiffness tensor\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 2
  },
  {
   "cell_type": "markdown",
   "source": [
    "Next, we define a constructor for the material instance."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "function J2Plasticity(E, ν, σ₀, H)\n",
    "    δ(i, j) = i == j ? 1.0 : 0.0 # helper function\n",
    "    G = E / 2(1 + ν)\n",
    "    K = E / 3(1 - 2ν)\n",
    "\n",
    "    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)\n",
    "    temp(i, j, k, l) = 2.0G * (0.5 * (δ(i, k) * δ(j, l) + δ(i, l) * δ(j, k)) + ν / (1.0 - 2.0ν) * δ(i, j) * δ(k, l))\n",
    "    Dᵉ = SymmetricTensor{4, 3}(temp)\n",
    "    return J2Plasticity(G, K, σ₀, H, Dᵉ)\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 3
  },
  {
   "cell_type": "markdown",
   "source": [
    "Define a `struct` to store the material state for a Gauss point."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "struct MaterialState{T, S <: SecondOrderTensor{3, T}}\n",
    "    # Store \"converged\" values\n",
    "    ϵᵖ::S # plastic strain\n",
    "    σ::S # stress\n",
    "    k::T # hardening variable\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 4
  },
  {
   "cell_type": "markdown",
   "source": [
    "Constructor for initializing a material state. Every quantity is set to zero."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Main.var\"##325\".MaterialState"
     },
     "metadata": {},
     "execution_count": 5
    }
   ],
   "cell_type": "code",
   "source": [
    "function MaterialState()\n",
    "    return MaterialState(\n",
    "        zero(SymmetricTensor{2, 3}),\n",
    "        zero(SymmetricTensor{2, 3}),\n",
    "        0.0\n",
    "    )\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 5
  },
  {
   "cell_type": "markdown",
   "source": [
    "For later use, during the post-processing step, we define a function to\n",
    "compute the von Mises effective stress."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "function vonMises(σ)\n",
    "    s = dev(σ)\n",
    "    return sqrt(3.0 / 2.0 * s ⊡ s)\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 6
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Constitutive driver\n",
    "\n",
    "This is the actual method which computes the stress and material tangent\n",
    "stiffness in a given integration point.\n",
    "Input is the current strain and the material state from the previous timestep."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "compute_stress_tangent (generic function with 1 method)"
     },
     "metadata": {},
     "execution_count": 7
    }
   ],
   "cell_type": "code",
   "source": [
    "function compute_stress_tangent(ϵ::SymmetricTensor{2, 3}, material::J2Plasticity, state::MaterialState)\n",
    "    # unpack some material parameters\n",
    "    G = material.G\n",
    "    H = material.H\n",
    "\n",
    "    # We use (•)ᵗ to denote *trial*-values\n",
    "    σᵗ = material.Dᵉ ⊡ (ϵ - state.ϵᵖ) # trial-stress\n",
    "    sᵗ = dev(σᵗ)         # deviatoric part of trial-stress\n",
    "    J₂ = 0.5 * sᵗ ⊡ sᵗ   # second invariant of sᵗ\n",
    "    σᵗₑ = sqrt(3.0 * J₂) # effective trial-stress (von Mises stress)\n",
    "    σʸ = material.σ₀ + H * state.k # Previous yield limit\n",
    "\n",
    "    φᵗ = σᵗₑ - σʸ # Trial-value of the yield surface\n",
    "\n",
    "    if φᵗ < 0.0 # elastic loading\n",
    "        return σᵗ, material.Dᵉ, MaterialState(state.ϵᵖ, σᵗ, state.k)\n",
    "    else # plastic loading\n",
    "        h = H + 3G\n",
    "        μ = φᵗ / h # plastic multiplier\n",
    "\n",
    "        c1 = 1 - 3G * μ / σᵗₑ\n",
    "        s = c1 * sᵗ           # updated deviatoric stress\n",
    "        σ = s + vol(σᵗ)       # updated stress\n",
    "\n",
    "        # Compute algorithmic tangent stiffness $D = \\frac{\\Delta \\sigma }{\\Delta \\epsilon}$\n",
    "        κ = H * (state.k + μ) # drag stress\n",
    "        σₑ = material.σ₀ + κ  # updated yield surface\n",
    "\n",
    "        δ(i, j) = i == j ? 1.0 : 0.0\n",
    "        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)\n",
    "        Q(i, j, k, l) = Isymdev(i, j, k, l) - 3.0 / (2.0 * σₑ^2) * s[i, j] * s[k, l]\n",
    "        b = (3G * μ / σₑ) / (1.0 + 3G * μ / σₑ)\n",
    "\n",
    "        Dtemp(i, j, k, l) = -2G * b * Q(i, j, k, l) - 9G^2 / (h * σₑ^2) * s[i, j] * s[k, l]\n",
    "        D = material.Dᵉ + SymmetricTensor{4, 3}(Dtemp)\n",
    "\n",
    "        # Return new state\n",
    "        Δϵᵖ = 3 / 2 * μ / σₑ * s # plastic strain\n",
    "        ϵᵖ = state.ϵᵖ + Δϵᵖ      # plastic strain\n",
    "        k = state.k + μ          # hardening variable\n",
    "        return σ, D, MaterialState(ϵᵖ, σ, k)\n",
    "    end\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 7
  },
  {
   "cell_type": "markdown",
   "source": [
    "## FE-problem\n",
    "What follows are methods for assembling and and solving the FE-problem."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "function create_values(interpolation)\n",
    "    # setup quadrature rules\n",
    "    qr = QuadratureRule{RefTetrahedron}(2)\n",
    "    facet_qr = FacetQuadratureRule{RefTetrahedron}(3)\n",
    "\n",
    "    # cell and facetvalues for u\n",
    "    cellvalues_u = CellValues(qr, interpolation)\n",
    "    facetvalues_u = FacetValues(facet_qr, interpolation)\n",
    "\n",
    "    return cellvalues_u, facetvalues_u\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 8
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Add degrees of freedom"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "create_dofhandler (generic function with 1 method)"
     },
     "metadata": {},
     "execution_count": 9
    }
   ],
   "cell_type": "code",
   "source": [
    "function create_dofhandler(grid, interpolation)\n",
    "    dh = DofHandler(grid)\n",
    "    add!(dh, :u, interpolation) # add a displacement field with 3 components\n",
    "    close!(dh)\n",
    "    return dh\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 9
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Boundary conditions"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "function create_bc(dh, grid)\n",
    "    dbcs = ConstraintHandler(dh)\n",
    "    # Clamped on the left side\n",
    "    dofs = [1, 2, 3]\n",
    "    dbc = Dirichlet(:u, getfacetset(grid, \"left\"), (x, t) -> [0.0, 0.0, 0.0], dofs)\n",
    "    add!(dbcs, dbc)\n",
    "    close!(dbcs)\n",
    "    return dbcs\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 10
  },
  {
   "cell_type": "markdown",
   "source": [
    "### Assembling of element contributions\n",
    "\n",
    "* Residual vector `r`\n",
    "* Tangent stiffness `K`"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "doassemble! (generic function with 1 method)"
     },
     "metadata": {},
     "execution_count": 11
    }
   ],
   "cell_type": "code",
   "source": [
    "function doassemble!(\n",
    "        K::SparseMatrixCSC, r::Vector, cellvalues::CellValues, dh::DofHandler,\n",
    "        material::J2Plasticity, u, states, states_old\n",
    "    )\n",
    "    assembler = start_assemble(K, r)\n",
    "    nu = getnbasefunctions(cellvalues)\n",
    "    re = zeros(nu)     # element residual vector\n",
    "    ke = zeros(nu, nu) # element tangent matrix\n",
    "\n",
    "    for (i, cell) in enumerate(CellIterator(dh))\n",
    "        fill!(ke, 0)\n",
    "        fill!(re, 0)\n",
    "        eldofs = celldofs(cell)\n",
    "        ue = u[eldofs]\n",
    "        state = @view states[:, i]\n",
    "        state_old = @view states_old[:, i]\n",
    "        assemble_cell!(ke, re, cell, cellvalues, material, ue, state, state_old)\n",
    "        assemble!(assembler, eldofs, ke, re)\n",
    "    end\n",
    "    return K, r\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 11
  },
  {
   "cell_type": "markdown",
   "source": [
    "Compute element contribution to the residual and the tangent."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "assemble_cell! (generic function with 1 method)"
     },
     "metadata": {},
     "execution_count": 12
    }
   ],
   "cell_type": "code",
   "source": [
    "function assemble_cell!(Ke, re, cell, cellvalues, material, ue, state, state_old)\n",
    "    n_basefuncs = getnbasefunctions(cellvalues)\n",
    "    reinit!(cellvalues, cell)\n",
    "\n",
    "    for q_point in 1:getnquadpoints(cellvalues)\n",
    "        # For each integration point, compute stress and material stiffness\n",
    "        ϵ = function_symmetric_gradient(cellvalues, q_point, ue) # Total strain\n",
    "        σ, D, state[q_point] = compute_stress_tangent(ϵ, material, state_old[q_point])\n",
    "\n",
    "        dΩ = getdetJdV(cellvalues, q_point)\n",
    "        for i in 1:n_basefuncs\n",
    "            δϵ = shape_symmetric_gradient(cellvalues, q_point, i)\n",
    "            re[i] += (δϵ ⊡ σ) * dΩ # add internal force to residual\n",
    "            for j in 1:i # loop only over lower half\n",
    "                Δϵ = shape_symmetric_gradient(cellvalues, q_point, j)\n",
    "                Ke[i, j] += δϵ ⊡ D ⊡ Δϵ * dΩ\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    symmetrize_lower!(Ke)\n",
    "    return\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 12
  },
  {
   "cell_type": "markdown",
   "source": [
    "Helper function to symmetrize the material tangent"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "doassemble_neumann! (generic function with 1 method)"
     },
     "metadata": {},
     "execution_count": 13
    }
   ],
   "cell_type": "code",
   "source": [
    "function symmetrize_lower!(K)\n",
    "    for i in 1:size(K, 1)\n",
    "        for j in (i + 1):size(K, 1)\n",
    "            K[i, j] = K[j, i]\n",
    "        end\n",
    "    end\n",
    "    return\n",
    "end;\n",
    "\n",
    "function doassemble_neumann!(r, dh, facetset, facetvalues, t)\n",
    "    n_basefuncs = getnbasefunctions(facetvalues)\n",
    "    re = zeros(n_basefuncs)                      # element residual vector\n",
    "    for fc in FacetIterator(dh, facetset)\n",
    "        # Add traction as a negative contribution to the element residual `re`:\n",
    "        reinit!(facetvalues, fc)\n",
    "        fill!(re, 0)\n",
    "        for q_point in 1:getnquadpoints(facetvalues)\n",
    "            dΓ = getdetJdV(facetvalues, q_point)\n",
    "            for i in 1:n_basefuncs\n",
    "                δu = shape_value(facetvalues, q_point, i)\n",
    "                re[i] -= (δu ⋅ t) * dΓ\n",
    "            end\n",
    "        end\n",
    "        assemble!(r, celldofs(fc), re)\n",
    "    end\n",
    "    return r\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 13
  },
  {
   "cell_type": "markdown",
   "source": [
    "Define a function which solves the FE-problem."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "solve (generic function with 1 method)"
     },
     "metadata": {},
     "execution_count": 14
    }
   ],
   "cell_type": "code",
   "source": [
    "function solve()\n",
    "    # Define material parameters\n",
    "    E = 200.0e9  # [Pa]\n",
    "    H = E / 20   # [Pa]\n",
    "    ν = 0.3      # [-]\n",
    "    σ₀ = 200.0e6 # [Pa]\n",
    "    material = J2Plasticity(E, ν, σ₀, H)\n",
    "\n",
    "    L = 10.0 # beam length [m]\n",
    "    w = 1.0  # beam width [m]\n",
    "    h = 1.0  # beam height[m]\n",
    "    n_timesteps = 10\n",
    "    u_max = zeros(n_timesteps)\n",
    "    traction_magnitude = 1.0e7 * range(0.5, 1.0, length = n_timesteps)\n",
    "\n",
    "    # Create geometry, dofs and boundary conditions\n",
    "    n = 2\n",
    "    nels = (10n, n, 2n) # number of elements in each spatial direction\n",
    "    P1 = Vec((0.0, 0.0, 0.0))  # start point for geometry\n",
    "    P2 = Vec((L, w, h))        # end point for geometry\n",
    "    grid = generate_grid(Tetrahedron, nels, P1, P2)\n",
    "    interpolation = Lagrange{RefTetrahedron, 1}()^3\n",
    "\n",
    "    dh = create_dofhandler(grid, interpolation) # JuaFEM helper function\n",
    "    dbcs = create_bc(dh, grid) # create Dirichlet boundary-conditions\n",
    "\n",
    "    cellvalues, facetvalues = create_values(interpolation)\n",
    "\n",
    "    # Pre-allocate solution vectors, etc.\n",
    "    n_dofs = ndofs(dh)  # total number of dofs\n",
    "    u = zeros(n_dofs)   # solution vector\n",
    "    Δu = zeros(n_dofs)  # displacement correction\n",
    "    r = zeros(n_dofs)   # residual\n",
    "    K = allocate_matrix(dh) # tangent stiffness matrix\n",
    "\n",
    "    # Create material states. One array for each cell, where each element is an array of material-\n",
    "    # states - one for each integration point\n",
    "    nqp = getnquadpoints(cellvalues)\n",
    "    states = [MaterialState() for _ in 1:nqp, _ in 1:getncells(grid)]\n",
    "    states_old = [MaterialState() for _ in 1:nqp, _ in 1:getncells(grid)]\n",
    "\n",
    "    # Newton-Raphson loop\n",
    "    NEWTON_TOL = 1 # 1 N\n",
    "    print(\"\\n Starting Netwon iterations:\\n\")\n",
    "\n",
    "    for timestep in 1:n_timesteps\n",
    "        t = timestep # actual time (used for evaluating d-bndc)\n",
    "        traction = Vec((0.0, 0.0, traction_magnitude[timestep]))\n",
    "        newton_itr = -1\n",
    "        print(\"\\n Time step @time = $timestep:\\n\")\n",
    "        update!(dbcs, t) # evaluates the D-bndc at time t\n",
    "        apply!(u, dbcs)  # set the prescribed values in the solution vector\n",
    "\n",
    "        while true\n",
    "            newton_itr += 1\n",
    "            if newton_itr > 8\n",
    "                error(\"Reached maximum Newton iterations, aborting\")\n",
    "                break\n",
    "            end\n",
    "            # Tangent and residual contribution from the cells (volume integral)\n",
    "            doassemble!(K, r, cellvalues, dh, material, u, states, states_old)\n",
    "            # Residual contribution from the Neumann boundary (surface integral)\n",
    "            doassemble_neumann!(r, dh, getfacetset(grid, \"right\"), facetvalues, traction)\n",
    "            norm_r = norm(r[Ferrite.free_dofs(dbcs)])\n",
    "\n",
    "            print(\"Iteration: $newton_itr \\tresidual: $(@sprintf(\"%.8f\", norm_r))\\n\")\n",
    "            if norm_r < NEWTON_TOL\n",
    "                break\n",
    "            end\n",
    "\n",
    "            apply_zero!(K, r, dbcs)\n",
    "            Δu = Symmetric(K) \\ r\n",
    "            u -= Δu\n",
    "        end\n",
    "\n",
    "        # Update the old states with the converged values for next timestep\n",
    "        states_old .= states\n",
    "\n",
    "        u_max[timestep] = maximum(abs, u) # maximum displacement in current timestep\n",
    "    end\n",
    "\n",
    "    # ## Postprocessing\n",
    "    # Only a vtu-file corresponding to the last time-step is exported.\n",
    "    #\n",
    "    # The following is a quick (and dirty) way of extracting average cell data for export.\n",
    "    mises_values = zeros(getncells(grid))\n",
    "    κ_values = zeros(getncells(grid))\n",
    "    for (el, cell_states) in enumerate(eachcol(states))\n",
    "        for state in cell_states\n",
    "            mises_values[el] += vonMises(state.σ)\n",
    "            κ_values[el] += state.k * material.H\n",
    "        end\n",
    "        mises_values[el] /= length(cell_states) # average von Mises stress\n",
    "        κ_values[el] /= length(cell_states)     # average drag stress\n",
    "    end\n",
    "    VTKGridFile(\"plasticity\", dh) do vtk\n",
    "        write_solution(vtk, dh, u) # displacement field\n",
    "        write_cell_data(vtk, mises_values, \"von Mises [Pa]\")\n",
    "        write_cell_data(vtk, κ_values, \"Drag stress [Pa]\")\n",
    "    end\n",
    "\n",
    "    return u_max, traction_magnitude\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 14
  },
  {
   "cell_type": "markdown",
   "source": [
    "Solve the FE-problem and for each time-step extract maximum displacement and\n",
    "the corresponding traction load. Also compute the limit-traction-load"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " Starting Netwon iterations:\n",
      "\n",
      " Time step @time = 1:\n",
      "Iteration: 0 \tresidual: 1435838.41167605\n",
      "Iteration: 1 \tresidual: 118655.22430368\n",
      "Iteration: 2 \tresidual: 59.50456058\n",
      "Iteration: 3 \tresidual: 0.00002560\n",
      "\n",
      " Time step @time = 2:\n",
      "Iteration: 0 \tresidual: 159537.60129725\n",
      "Iteration: 1 \tresidual: 1706974.26597926\n",
      "Iteration: 2 \tresidual: 97346.48157049\n",
      "Iteration: 3 \tresidual: 37.17532011\n",
      "Iteration: 4 \tresidual: 0.00001524\n",
      "\n",
      " Time step @time = 3:\n",
      "Iteration: 0 \tresidual: 159537.60129701\n",
      "Iteration: 1 \tresidual: 3033614.51718249\n",
      "Iteration: 2 \tresidual: 183762.82986491\n",
      "Iteration: 3 \tresidual: 187.23777242\n",
      "Iteration: 4 \tresidual: 0.00023135\n",
      "\n",
      " Time step @time = 4:\n",
      "Iteration: 0 \tresidual: 159537.60129742\n",
      "Iteration: 1 \tresidual: 3668226.41190261\n",
      "Iteration: 2 \tresidual: 85645.15221552\n",
      "Iteration: 3 \tresidual: 33.39133787\n",
      "Iteration: 4 \tresidual: 0.00002312\n",
      "\n",
      " Time step @time = 5:\n",
      "Iteration: 0 \tresidual: 159537.60129764\n",
      "Iteration: 1 \tresidual: 4942707.09611024\n",
      "Iteration: 2 \tresidual: 822244.81049667\n",
      "Iteration: 3 \tresidual: 2806.49948363\n",
      "Iteration: 4 \tresidual: 0.04196782\n",
      "\n",
      " Time step @time = 6:\n",
      "Iteration: 0 \tresidual: 159537.60129723\n",
      "Iteration: 1 \tresidual: 6350622.82330476\n",
      "Iteration: 2 \tresidual: 1433617.64531907\n",
      "Iteration: 3 \tresidual: 11917.22662334\n",
      "Iteration: 4 \tresidual: 0.96519065\n",
      "\n",
      " Time step @time = 7:\n",
      "Iteration: 0 \tresidual: 159537.60130042\n",
      "Iteration: 1 \tresidual: 7442093.81842929\n",
      "Iteration: 2 \tresidual: 2293366.32653456\n",
      "Iteration: 3 \tresidual: 27806.00144416\n",
      "Iteration: 4 \tresidual: 4.78936691\n",
      "Iteration: 5 \tresidual: 0.00002337\n",
      "\n",
      " Time step @time = 8:\n",
      "Iteration: 0 \tresidual: 159537.60129787\n",
      "Iteration: 1 \tresidual: 7898429.46749798\n",
      "Iteration: 2 \tresidual: 2166408.36902476\n",
      "Iteration: 3 \tresidual: 19078.14976325\n",
      "Iteration: 4 \tresidual: 2.12913739\n",
      "Iteration: 5 \tresidual: 0.00003700\n",
      "\n",
      " Time step @time = 9:\n",
      "Iteration: 0 \tresidual: 159537.60129718\n",
      "Iteration: 1 \tresidual: 9113096.78354406\n",
      "Iteration: 2 \tresidual: 1942261.17847130\n",
      "Iteration: 3 \tresidual: 14972.09948485\n",
      "Iteration: 4 \tresidual: 1.53213288\n",
      "Iteration: 5 \tresidual: 0.00003588\n",
      "\n",
      " Time step @time = 10:\n",
      "Iteration: 0 \tresidual: 159537.60129681\n",
      "Iteration: 1 \tresidual: 9810716.23843057\n",
      "Iteration: 2 \tresidual: 1947382.98912119\n",
      "Iteration: 3 \tresidual: 34190.85171497\n",
      "Iteration: 4 \tresidual: 4.44141634\n",
      "Iteration: 5 \tresidual: 0.00005322\n"
     ]
    }
   ],
   "cell_type": "code",
   "source": [
    "u_max, traction_magnitude = solve();"
   ],
   "metadata": {},
   "execution_count": 15
  },
  {
   "cell_type": "markdown",
   "source": [
    "Finally we plot the load-displacement curve."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "Plot{Plots.GRBackend() n=1}",
      "image/png": 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      ]
     },
     "metadata": {},
     "execution_count": 16
    }
   ],
   "cell_type": "code",
   "source": [
    "using Plots\n",
    "plot(\n",
    "    vcat(0.0, u_max),                # add the origin as a point\n",
    "    vcat(0.0, traction_magnitude),\n",
    "    linewidth = 2,\n",
    "    title = \"Traction-displacement\",\n",
    "    label = nothing,\n",
    "    markershape = :auto\n",
    ")\n",
    "ylabel!(\"Traction [Pa]\")\n",
    "xlabel!(\"Maximum deflection [m]\")"
   ],
   "metadata": {},
   "execution_count": 16
  },
  {
   "cell_type": "markdown",
   "source": [
    "*Figure 2.* Load-displacement-curve for the beam, showing a clear decrease\n",
    "in stiffness as more material starts to yield."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "---\n",
    "\n",
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