02_kinematics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Kinematics\n",
    "\n",
    "Created by: Dominik Klein\n",
    "\n",
    "Contact: dominik.klein@stud.tu-darmstadt.de\n",
    "\n",
    "Date of creation: 2020-04-04\n",
    "\n",
    "Last update: 2020-07-01"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Information:**\n",
    "Run the following cell to define all commands for this notebook. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "Jupyter.notebook.execute_cells([23])"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "Jupyter.notebook.execute_cells([5])"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "Jupyter.notebook.execute_cells([15])"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "%run -i ./code/02_kinematics/main.py"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Description\n",
    "\n",
    "In this notebook, an introduction into deformation kinematic is presented. Beside some simple strain measures, the polar and volumetric-deviatoric decomposition of the deformation gradient as well as it's principal values and invariants are shown. \n",
    "\n",
    "This notebook provides interactive content, which can be activated in the cell below. By default, the notebook content is set to static."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Static:**\n",
    "Run the following cell for static content (default). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "Jupyter.notebook.execute_cells([10])"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "notebook_static()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Interactive:**\n",
    "Run the following cell for interactive content. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "notebook_interactive()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Deformation Gradient"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A body (solid continuum) in 3-dimensional space is given by a domain $\\Omega\\subset\\mathbb{R}^3$ as a collection of material points $\\boldsymbol{p}$. The body before and after deformation is represented by two different configurations. The reference (initial) configuration describes the undeformed body with material points $\\boldsymbol{p}$ with coordinates $\\boldsymbol{X}\\in\\Omega_0\\subset\\mathbb{R}^3$. A change of shape of the body (deformation) yields the current (deformed) configuration with material points $\\boldsymbol{p}'$ with coordinates $\\boldsymbol{x}\\in\\Omega\\subset\\mathbb{R}^3$.\n",
    "\n",
    "<img src=\"files/code/02_kinematics/configurations.JPG\" alt=\"Drawing\" style=\"width: 500px;\"/>\n",
    "\n",
    "Both configurations are connected by the deformation field \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{\\phi}:\\Omega_0\\rightarrow\\Omega,\\;\\boldsymbol{x}=\\boldsymbol{\\phi}\\left(\\boldsymbol{X},t\\right)\n",
    "\\end{equation}\n",
    "\n",
    "with the corresponding displacement field \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{u}:\\Omega_0\\rightarrow\\mathbb{R}^3,\\;\\boldsymbol{u}\\left(\\boldsymbol{X},t\\right)=\\boldsymbol{\\phi}\\left(\\boldsymbol{X},t\\right)-\\boldsymbol{X}=\\boldsymbol{x}-\\boldsymbol{X}\n",
    "\\end{equation}\n",
    "\n",
    "Capital letters denote the reference configuration, and smallercase letters denote the deformed configuration. A first order Taylor series approximation of the mapping $\\boldsymbol{\\phi}$\n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{x}+d\\boldsymbol{x}=\\boldsymbol{\\phi}\\left(\\boldsymbol{X}+d\\boldsymbol{X}\\right)\\approx\\boldsymbol{\\phi}\\left(\\boldsymbol{X}\\right)+\\frac{d\\boldsymbol{\\phi}\\left(\\boldsymbol{X}\\right)}{d\\boldsymbol{X}}d\\boldsymbol{X}=\\boldsymbol{x}+\\nabla_{\\boldsymbol{X}}\\boldsymbol{\\phi}\\left(\\boldsymbol{X}\\right)d\\boldsymbol{X}\n",
    "\\end{equation}\n",
    "\n",
    "gives an approximation of the deformation of line elements $d\\boldsymbol{x}$ by\n",
    "\n",
    "\\begin{equation}\n",
    "d\\boldsymbol{x}\\approx\\nabla_{\\boldsymbol{X}}\\boldsymbol{\\phi}=\\left(\\boldsymbol{X}\\right)d\\boldsymbol{X}\n",
    "\\end{equation}\n",
    "\n",
    "The second order tensor $\\boldsymbol{F}:=\\nabla_{\\boldsymbol{X}}\\boldsymbol{\\phi}\\left(\\boldsymbol{X}\\right)=\\nabla_\\boldsymbol{X}\\boldsymbol{x}\\left(\\boldsymbol{X}\\right)=\\boldsymbol{I}+\\nabla_\\boldsymbol{X}\\boldsymbol{u}\\left(\\boldsymbol{X}\\right)$ defines the deformation gradient. The deformation gradient is the linearization of the mapping between reference and current deformation.\n",
    "\n",
    "The gradient operator corresponds to the componentwise derivative of the deformed vector $\\boldsymbol{x}$ with respect to the undeformed vector $\\boldsymbol{X}$\n",
    "\n",
    "\\begin{equation}\n",
    " F_{ij}=x_{i,j}=   \\frac{\\partial x_i}{\\partial X_j} =\n",
    " \\begin{bmatrix}\n",
    "   \\frac{\\partial x_1}{\\partial X_1} &\n",
    "   \\frac{\\partial x_1}{\\partial X_2} & \\frac{\\partial x_1}{\\partial X_3} \\\\\n",
    "   \\frac{\\partial x_2}{\\partial X_1} &\n",
    "   \\frac{\\partial x_2}{\\partial X_2} & \\frac{\\partial x_2}{\\partial X_3} \\\\\n",
    "      \\frac{\\partial x_3}{\\partial X_1} &\n",
    "   \\frac{\\partial x_3}{\\partial X_2} & \\frac{\\partial x_3}{\\partial X_3} \\\\\n",
    "   \\end{bmatrix}\n",
    "   \\end{equation}\n",
    "\n",
    "In the following we will restrict to plane stress states.\n",
    "\n",
    "The most simple deformation is stretching along the axes directions. Elongation in the 1-direction by 100% and elongation in the 2-direction by 60% is given by the deformation field\n",
    "\n",
    "\\begin{equation}\n",
    "x_1=2.0\\cdot X_1 + 0.0 \\cdot X_2\n",
    "\\\\\n",
    "x_2=0.0 \\cdot X_1 + 1.6 \\cdot X_2\n",
    "\\end{equation}\n",
    "\n",
    "which gives the deformation gradient\n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{F}=\n",
    " \\begin{bmatrix}\n",
    "   2 & 0 \\\\\n",
    "   0 & 1.6\n",
    "   \\end{bmatrix}\n",
    "   \\end{equation}\n",
    "   \n",
    "## Polar Decomposition\n",
    "\n",
    "The deformation gradient can be uniquely composed into a part which rotates the body, and a part which changes the shape of the body. This multiplicative decomposition is called polar decomposition, which results in \n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{F}=\\boldsymbol{R}\\cdot\\boldsymbol{U}=\\boldsymbol{v}\\cdot\\boldsymbol{R}\n",
    "\\end{equation}\n",
    "\n",
    "The symbols $\\boldsymbol{U}:=\\sqrt{\\boldsymbol{F}^T\\boldsymbol{F}}$ and $\\boldsymbol{v}:=\\sqrt{\\boldsymbol{F}\\boldsymbol{F}^T}$ denote the right and left stretch tensors, respectively. Both are symmetric positive definite.The symbol $\\boldsymbol{R}:=\\boldsymbol{F}\\cdot\\boldsymbol{U}^{-1}$ denotes the unimodular, orthogonal rotation tensor, i.e. $\\boldsymbol{R}^T\\cdot\\boldsymbol{R}=\\boldsymbol{R}\\cdot\\boldsymbol{R}^T=\\boldsymbol{I}$ and $\\text{det}\\left(\\boldsymbol{R}\\right)=1$\n",
    ".\n",
    "## Volume Change\n",
    "\n",
    "The volumetric strain is given by the determinant of the deformation gradient $J:=\\text{det}\\left(\\boldsymbol{F}\\right)>0$. At volume preserving deformations which we know from (nearly) incompressible materials, the Jacobian determinant equals one.\n",
    "\n",
    "The deformation gradient can further be composed into a part which changes the volume, and a part which changes the shape of body while preserving the volume, which is called deviatoric deformation.\n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{F}=\\boldsymbol{R}\\cdot\\boldsymbol{U}=\\boldsymbol{R}\\cdot\\boldsymbol{U}_{dev}\\cdot\\boldsymbol{U}_{vol}\n",
    "\\end{equation}\n",
    "\n",
    "The deviatoric stretch tensor is unimodular, i.e. $\\text{det}\\left(\\boldsymbol{U}_{dev}\\right)=1$. The volumetric stretch tensor can be expressed with the scaled second order unit tensor $\\boldsymbol{U}_{vol}=\\epsilon_{vol}\\cdot\\boldsymbol{I}$\n",
    "\n",
    "\n",
    "The following examples are restricted to plane strain deformation states.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Deformation Gradient\n",
      "\n",
      "F =\n",
      "[[1.100 0.300 0.000]\n",
      " [0.000 1.100 0.000]\n",
      " [0.000 0.000 1.000]]\n",
      "\n",
      "\n",
      "Rotation Tensor\n",
      "\n",
      "R =\n",
      "[[0.991 0.135 0.000]\n",
      " [-0.135 0.991 0.000]\n",
      " [0.000 0.000 1.000]]\n",
      "\n",
      "\n",
      "Right Stretch Tensor\n",
      "\n",
      "U =\n",
      "[[1.090 0.149 0.000]\n",
      " [0.149 1.130 0.000]\n",
      " [0.000 0.000 1.000]]\n",
      "\n",
      "\n",
      "Volume Ratio\n",
      "\n",
      "J =\n",
      "   1.210\n"
     ]
    },
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "Jupyter.notebook.execute_cells([13, 18, 20])"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "F_decomposition()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Strain Measures"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At deformation, the length of the infinitesimal line element $d\\boldsymbol{X}$ is changed. The change of length between the infinitesimal line element in the reference configuration $d\\boldsymbol{X}$ and the length of the line element $d\\boldsymbol{x}$ in the deformed configuration leads to\n",
    "\n",
    "\\begin{equation}\n",
    "||d\\boldsymbol{x}||^2-||d\\boldsymbol{X}||^2=d\\boldsymbol{x}\\cdot d\\boldsymbol{x}-d\\boldsymbol{X}\\cdot d\\boldsymbol{X}\n",
    "\\\\\n",
    "=\\left(\\boldsymbol{F}d\\boldsymbol{X}\\right)^T\\left(\\boldsymbol{F}d\\boldsymbol{X}\\right)-d\\boldsymbol{X}^Td\\boldsymbol{X}=d\\boldsymbol{X}^T\\left(\\boldsymbol{F}^T\\boldsymbol{F}-\\boldsymbol{I}\\right)d\\boldsymbol{X}\n",
    "\\end{equation}\n",
    "\n",
    "With the right Cauchy-Green strain tensor\n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{C}=\\boldsymbol{F}^T\\boldsymbol{F} = \\boldsymbol{U}^T\\boldsymbol{U}\n",
    ".\\end{equation}\n",
    "\n",
    "Therefore the so called Green-Lagrange strain tensor\n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{E}=\\frac{1}{2}\\left(\\boldsymbol{F}^T\\boldsymbol{F}-\\boldsymbol{I}\\right)\n",
    "\\end{equation}\n",
    "\n",
    "is a measure for the change of length of an infinitesimal line element with respect to the material configuration.\n",
    "For small deformations, the Green-Lagrange tensor can be approximated by it's linearization. The linearized strain tensor for small deformations is called infinitesimal strain tensor, and given by\n",
    "\n",
    "\\begin{equation}\n",
    "\\boldsymbol{E}_{lin}=\\boldsymbol{\\epsilon}=\\frac{1}{2}\\left(\\boldsymbol{F}^T+\\boldsymbol{F}\\right)-\\boldsymbol{I}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Green-Lagrange Strain Tensor\n",
      "E =\n",
      "[[0.105 0.165 0.000]\n",
      " [0.165 0.150 0.000]\n",
      " [0.000 0.000 0.000]]\n",
      "\n",
      "\n",
      "Linearized Strain Tensor\n",
      "E_lin =\n",
      "[[0.100 0.150 0.000]\n",
      " [0.150 0.100 0.000]\n",
      " [0.000 0.000 0.000]]\n"
     ]
    }
   ],
   "source": [
    "strain_measures(F11,F22,F12,F21)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the graphic below, the Green-Lagrange Strain Tensor and the Linearized Strain Tensor for uniaxial tension are compared. The 11-component of the strain tensors is shown."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "strain_measures_lin_plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Eigenvalues and  Invariants"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The eigenvalue problem for the right Cauchy-Green tensor $\\boldsymbol{C}$\n",
    "\n",
    "\\begin{equation}\n",
    "\\left(\\boldsymbol{C}-\\lambda_i^2\\boldsymbol{I}\\right)\\boldsymbol{N}_i=\\boldsymbol{0},\\;\\;\\;\\left(i=1,2,3;\\;\\text{no summation}\\right)\n",
    "\\end{equation}\n",
    "\n",
    "yields the three positive and real eigenvalues $\\lambda_i^2$ with their corresponding orthogonal and normalized eigenvectors $\\boldsymbol{N}_i$. The square roots of the eigenvalues give the principal stretches $\\lambda_i$, with the eigenvectors $\\boldsymbol{N}_i$ as principal directions. Rotated into the principal system, the Right Cauchy-Green tensor consists only of main diagonal elements. The tension entries become extremal, while the shear entries vanish.\n",
    "\n",
    "In the graphic below, the principal stretches for the plane strain deformation defined in the beginning of the notebook are shown."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Principal Stretches\n",
      "[0.960 1.260 1.000]\n",
      "\n",
      "\n",
      "Eigenvectors (ordered in columns)\n",
      "[[-0.753 -0.658 0.000]\n",
      " [0.658 -0.753 0.000]\n",
      " [0.000 0.000 1.000]]\n"
     ]
    }
   ],
   "source": [
    "eigenvalues(F11,F22,F12,F21)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The invariants of a tensor are scalar quantities which remain unchanged under change of observer. For example, the Euclidean norm of a first order tensor, which corresponds to the length of a vector, is invariant under rotation or translation:\n",
    "\n",
    "\\begin{equation}\n",
    "I\\left(\\boldsymbol{a}\\right)=\\sqrt{a_i^2}\n",
    "\\end{equation}\n",
    "\n",
    "A second order tensor $\\boldsymbol{A}$ with eigenvalues $A_i$ has three independent invariants.\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{aligned}\n",
    "I_1\\left(\\boldsymbol{A}\\right)&=\\text{tr}\\boldsymbol{A}\n",
    "\\\\\n",
    "&=A_1+A_2+A_3\n",
    "\\\\\n",
    "I_2\\left(\\boldsymbol{A}\\right)&=\\frac{1}{2}\\left(\\left(\\text{tr}\\boldsymbol{A}\\right)^2-\\text{tr}\\left(\\boldsymbol{A}^2\\right)\\right)\n",
    "\\\\\n",
    "&=A_1A_2+A_1A_3+A_2A_3\n",
    "\\\\\n",
    "I_3\\left(\\boldsymbol{A}\\right)&=\\text{det}\\boldsymbol{A}\n",
    "\\\\\n",
    "&=A_1A_2A_3\n",
    "\\\\\n",
    "\\end{aligned}\n",
    "\\end{equation}\n",
    "\n",
    "which can be derived from the Caley-Hamilton theorem, see for example Holzapfel (2000).\n",
    "The invariants of strain and deformation tensors can be used as a scalar measure for the state of deformation. For example, the third invariant of the deformation gradient corresponds to the volumetric deformation ratio.\n",
    "\n",
    "With the three invariants, the deviatoric state of deformation can be further examined. The determinant of the deviatoric Right Cauchy-Green tensor equals one, since it contains no volumetric deformation. Two independent principal stretches remain, which can be uniquely determined when the first two invariants are given.\n",
    "\n",
    "In the graph below, the invariant plane for the first and second invariant of the Right Cauchy-Green tensor is displayed. It can be shown that every deformation lies between the \"Uniaxial Tension\" and the \"Equibiaxial Tension\" prototype deformations, with \"Shear\" between them, see e.g. Baaser (2013). With the first two invariants of an arbitrary deviatoric deformation it can be examined which kind of deformation is occuring. This can be important in the design process of mechanical components, e.g. with shear buffers made of rubber (is the state of deformation in the buffer actually a shear deformation?)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "deformation_state(F11,F22,F12,F21)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Further Information"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Holzapfel, G. (2000). Nonlinear solid mechanics: a continuum approach for engineering.\n",
    "\n",
    "Destrade, M. et al. (2012). Simple shear is not so simple.\n",
    "\n",
    "Baaser, H. et al. (2013). Reformulation of strain invariants at incompressibility."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Magic functions for interactive graphs\n",
    "def interactive_plot(x):\n",
    "    if x:\n",
    "        %matplotlib notebook\n",
    "    else:\n",
    "        %matplotlib inline"
   ]
  }
 ],
 "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": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}