finite_strain_elastoplasticity.ipynb 8.86 KB
 Jeremy BLEYER committed Apr 03, 2020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Finite-strain elastoplasticity within the logarithmic strain framework\n", "\n", "This demo is dedicated to the resolution of a finite-strain elastoplastic problem using the logarithmic strain framework proposed in (Miehe et al., 2002). \n", "\n", "## Logarithmic strains \n", "\n", "This framework expresses constitutive relations between the Hencky strain measure $\\boldsymbol{H} = \\dfrac{1}{2}\\log (\\boldsymbol{F}^T\\cdot\\boldsymbol{F})$ and its dual stress measure $\\boldsymbol{T}$. This approach makes it possible to extend classical small strain constitutive relations to a finite-strain setting. In particular, the total (Hencky) strain can be split **additively** into many contributions (elastic, plastic, thermal, swelling, etc.). Its trace is also linked with the volume change $J=\\exp(\\operatorname{tr}(\\boldsymbol{H}))$. As a result, the deformation gradient $\\boldsymbol{F}$ is used for expressing the Hencky strain $\\boldsymbol{H}$, a small-strain constitutive law is then written for the $(\\boldsymbol{H},\\boldsymbol{T})$-pair and the dual stress $\\boldsymbol{T}$ is then post-processed to an appropriate stress measure such as the Cauchy stress $\\boldsymbol{\\sigma}$ or Piola-Kirchhoff stresses.\n", "\n", "## MFront implementation\n", "\n", " TODO:\n", "* Write the gallery example comments and refer to it?\n", "\n", "* or use standard brick with comments here ?\n", " \n", "\n", "\n", "## FEniCS implementation\n", "\n", "We define a box mesh representing half of a beam oriented along the $x$-direction. The beam will be fully clamped on its left side and symmetry conditions will be imposed on its right extremity. The loading consists of a uniform self-weight." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "from dolfin import *\n", "import mfront_wrapper as mf\n", "import numpy as np\n", "import ufl\n", "\n", "length, width, height = 1., 0.04, 0.1\n", "nx, ny, nz = 30, 5, 10\n", "mesh = BoxMesh(Point(0, -width/2, -height/2.), Point(length, width/2, height/2.), nx, ny, nz)\n", "\n", "V = VectorFunctionSpace(mesh, \"CG\", 2)\n", "u = Function(V, name=\"Displacement\")\n", "\n", "def left(x, on_boundary):\n", " return near(x[0], 0) and on_boundary\n", "def right(x, on_boundary):\n", " return near(x[0], length) and on_boundary\n", "\n", "bc = [DirichletBC(V, Constant((0.,)*3), left),\n", " DirichletBC(V.sub(0), Constant(0.), right)]\n", "\n", "selfweight = Expression((\"0\", \"0\", \"-t*qmax\"), t=0., qmax = 50e6, degree=0)\n", "\n", "file_results = XDMFFile(\"results/finite_strain_plasticity.xdmf\")\n", "file_results.parameters[\"flush_output\"] = True\n", "file_results.parameters[\"functions_share_mesh\"] = True" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The MFrontNonlinearMaterial instance is loaded from the MFront LogarithmicStrainPlasticity behaviour. This behaviour is a finite-strain behaviour (material.finite_strain=True) which relies on a kinematic description using the total deformation gradient $\\boldsymbol{F}$. By default, a MFront behaviour always returns the Cauchy stress as the stress measure after integration. However, the stress variable dual to the deformation gradient is the first Piloa-Kirchhoff (PK1) stress. An internal option of the MGIS interface is therefore used in the finite-strain context to return the PK1 stress as the \"flux\" associated to the \"gradient\" $\\boldsymbol{F}$. Both quantities are non-symmetric tensors, aranged as a 9-dimensional vector in 3D following [MFront conventions on tensors](http://tfel.sourceforge.net/tensors.html)." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "StandardFiniteStrainBehaviour\n", "F_CAUCHY\n", "['DeformationGradient'] [9]\n", "['FirstPiolaKirchhoffStress'] [9]\n" ] } ], "source": [ "material = mf.MFrontNonlinearMaterial(\"../materials/src/libBehaviour.so\",\n", " \"LogarithmicStrainPlasticity\")\n", "print(material.behaviour.getBehaviourType())\n", "print(material.behaviour.getKinematic())\n", "print(material.get_gradient_names(), material.get_gradient_sizes())\n", "print(material.get_flux_names(), material.get_flux_sizes())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The MFrontNonlinearProblem instance must therefore register the deformation gradient as Identity(3)+grad(u). This again done automatically since \"DeformationGradient\" is a predefined gradient. The following message will be shown upon calling solve:\n", "\n", "Automatic registration of 'DeformationGradient' as I + (grad(Displacement)).\n", "\n", "The loading is then defined and, as for the [small-strain elastoplasticity example](small_strain_elastoplasticity.ipynb), state variables include the ElasticStrain and EquivalentPlasticStrain since the same behaviour is used as in the small-strain case with the only difference that the total strain is now given by the Hencky strain measure. In particular, the ElasticStrain is still a symmetric tensor (vector of dimension 6). Note that it has not been explicitly defined as a state variable in the MFront behaviour file since this is done automatically when using the IsotropicPlasticMisesFlow parser.\n", "\n", "Finally, we setup a few parameters of the Newton non-linear solver." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(6,)\n" ] } ], "source": [ "problem = mf.MFrontNonlinearProblem(u, material, bcs=bc)\n", "problem.set_loading(dot(selfweight, u)*dx)\n", "\n", "p = problem.get_state_variable(\"EquivalentPlasticStrain\")\n", "epsel = problem.get_state_variable(\"ElasticStrain\")\n", "print(ufl.shape(epsel))\n", "\n", "prm = problem.solver.parameters\n", "prm[\"absolute_tolerance\"] = 1e-6\n", "prm[\"relative_tolerance\"] = 1e-6\n", "prm[\"linear_solver\"] = \"mumps\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "During the load incrementation, we monitor the evolution of the vertical downwards displacement at the middle of the right extremity.\n", "\n", "This simulation is a bit heavy to run so we suggest running it in parallel:\n", "bash\n", "mpirun -np 16 python3\n", "" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Increment 1\n", "Automatic registration of 'DeformationGradient' as I + (grad(Displacement)).\n", "\n", "Automatic registration of 'Temperature' as a Constant value = 293.15.\n", "\n", "Increment 2\n", "Increment 3\n", "Increment 4\n" ] } ], "source": [ "P0 = FunctionSpace(mesh, \"DG\", 0)\n", "p_avg = Function(P0, name=\"Plastic strain\")\n", "\n", "Nincr = 30\n", "load_steps = np.linspace(0., 1., Nincr+1)\n", "results = np.zeros((Nincr+1, 3))\n", "for (i, t) in enumerate(load_steps[1:]):\n", " selfweight.t = t\n", " print(\"Increment \", i+1)\n", " problem.solve(u.vector())\n", "\n", " results[i+1, 0] = -u(length, 0, 0)[2]\n", " results[i+1, 1] = t\n", "\n", " file_results.write(u, t)\n", " p_avg.assign(project(p, P0))\n", " file_results.write(p_avg, t)\n", "\n", "plt.figure()\n", "plt.plot(results[:, 0], results[:, 1], \"-o\")\n", "plt.xlabel(r\"Displacement\")\n", "plt.ylabel(\"Load\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The load-displacement curve exhibits a classical elastoplastic behaviour rapidly followed by a stiffening behaviour due to membrane catenary effects. " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.9" } }, "nbformat": 4, "nbformat_minor": 4 }