{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Homework Assignment 1\n",
    "=====================\n",
    "\n",
    "\n",
    "**Note**:  This is a computable IPython notebook who's source code can be downloaded [here](https://raw.githubusercontent.com/johnfoster-pge-utexas/PGE383-AdvGeomechanics/master/files/assignment1_solution.ipynb).\n",
    "\n",
    "## Problem 1\n",
    " \n",
    "The motion of a certain continuous medium is defined by the equations\n",
    " \n",
    "\\begin{align}\n",
    "  x_1 &= \\frac{1}{2} \\left( X_1 + X_2 \\right) e^t + \\frac{1}{2} \\left( X_1-X_2 \\right) e^{-t}, \\notag \\\\\n",
    "  x_2 &= \\frac{1}{2} \\left( X_1 + X_2 \\right) e^t - \\frac{1}{2} \\left( X_1-X_2 \\right) e^{-t}, \\notag \\\\\n",
    "  x_3 &= X_3 \\notag\n",
    "\\end{align}\n",
    "\n",
    "1. Compute the following\n",
    "\n",
    "   1. The Green-Lagrange strain tensor $E$\n",
    "   \n",
    "   1. The linear (small) strain tensor $\\varepsilon$\n",
    "\n",
    "   Plot the $11$, $22$, and $12$ components of $E$ and $\\varepsilon$ on the same figure\n",
    "   from time $t=0$ to $t=0.05$.\n",
    "   \n",
    "\n",
    "1. Compute the following\n",
    "\n",
    "   1. The rate-of-deformation tensor $D$\n",
    "   \n",
    "   1. The rate-of-change of the small strain tensor $\\dot{\\varepsilon} = \\frac{d\\varepsilon}{dt}$\n",
    "\n",
    "   Plot the $11$, $22$, and $12$ components of $D$ and $\\dot{\\varepsilon}$ on the same figure\n",
    "   from time $t=0$ to $t=0.05$.\n",
    "        \n",
    "        \n",
    "**Solution**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This cell loads some important packages from sympy, numpy, and matplotlib that I will use the perform calculations and display the results neatly.  In order to run this notebook, you will have to have these packages installed in your Python distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from sympy.matrices import *\n",
    "import sympy.mpmath\n",
    "from sympy.utilities.lambdify import lambdify\n",
    "init_printing()\n",
    "\n",
    "import numpy\n",
    "\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "#plt.style.available\n",
    "#plt.style.use('bmh')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining which variables will be \"symbolic\" in nature, i.e., they will not take on numerical values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t, X1, X2, X3 = symbols('t, X_1, X_2, X_3')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This defines the deformation mapping as in the problem statement."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x1 = Rational(1, 2) * (X1 + X2) * exp(t) + Rational(1, 2) * (X1 - X2) * exp(-t)\n",
    "x2 = Rational(1, 2) * (X1 + X2) * exp(t) - Rational(1, 2) * (X1 - X2) * exp(-t)\n",
    "x3 = X3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we compute the deformation gradient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAABLCAMAAAACojjaAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cosh{\\left (t \\right )} & \\sinh{\\left (t \\right )} & 0\\\\\\sinh{\\left (t \\right )} & \\cosh{\\left (t \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡cosh(t)  sinh(t)  0⎤\n",
       "⎢                   ⎥\n",
       "⎢sinh(t)  cosh(t)  0⎥\n",
       "⎢                   ⎥\n",
       "⎣   0        0     1⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = simplify(Matrix([[diff(x1, X1), diff(x1, X2), diff(x1, X3)],\n",
    "            [diff(x2, X1), diff(x2, X2), diff(x2, X3)],\n",
    "            [diff(x3, X1), diff(x3, X2), diff(x3, X3)]])); F"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the Green-Lagrange strain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPoAAABOCAMAAAA+aftsAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n/5JH/h3mH+y4YuqXGcGFAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\sinh^{2}{\\left (t \\right )} & \\frac{1}{2} \\sinh{\\left (2 t \\right )} & 0\\\\\\frac{1}{2} \\sinh{\\left (2 t \\right )} & \\sinh^{2}{\\left (t \\right )} & 0\\\\0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    2      sinh(2⋅t)   ⎤\n",
       "⎢sinh (t)   ─────────  0⎥\n",
       "⎢               2       ⎥\n",
       "⎢                       ⎥\n",
       "⎢sinh(2⋅t)      2       ⎥\n",
       "⎢─────────  sinh (t)   0⎥\n",
       "⎢    2                  ⎥\n",
       "⎢                       ⎥\n",
       "⎣    0          0      0⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E = Rational(1, 2) * (F.T * F - eye(3)); simplify(E)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can immediatelly evaluate the linear \"small\" strain as well, directly with the deformation gradient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARAAAABLCAMAAACP6iinAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cosh{\\left (t \\right )} - 1 & \\sinh{\\left (t \\right )} & 0\\\\\\sinh{\\left (t \\right )} & \\cosh{\\left (t \\right )} - 1 & 0\\\\0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡cosh(t) - 1    sinh(t)    0⎤\n",
       "⎢                           ⎥\n",
       "⎢  sinh(t)    cosh(t) - 1  0⎥\n",
       "⎢                           ⎥\n",
       "⎣     0            0       0⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon = simplify(Rational(1,2) * (F.T + F) - eye(3)); simplify(epsilon)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This turns our symbolic output into actual functions of $t$ that we can evaluate and plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "E11_function = lambdify(t, E[0,0], \"numpy\")\n",
    "epsilon11_function = lambdify(t, epsilon[0,0], \"numpy\")\n",
    "\n",
    "E22_function = lambdify(t, E[1,1], \"numpy\")\n",
    "epsilon22_function = lambdify(t, epsilon[1,1], \"numpy\")\n",
    "\n",
    "E12_function = lambdify(t, E[0,1], \"numpy\")\n",
    "epsilon12_function = lambdify(t, epsilon[0,1], \"numpy\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evaluating the functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t0 = numpy.linspace(0.0,0.05)\n",
    "\n",
    "E11 = E11_function(t0)\n",
    "epsilon11 = epsilon11_function(t0)\n",
    "\n",
    "E22 = E22_function(t0)\n",
    "epsilon22 = epsilon22_function(t0)\n",
    "\n",
    "E12 = E12_function(t0)\n",
    "epsilon12 = epsilon12_function(t0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZgAAAEYCAYAAABhi+CNAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1093617d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(1)\n",
    "ax = fig.add_subplot(111,xlabel='$t$', ylabel='$\\epsilon$')\n",
    "ax.plot(t0, E22,'k-', t0, epsilon22, 'k--',\n",
    "         t0, E12,'b-', t0, epsilon12, 'b--');\n",
    "plt.grid()\n",
    "ax.annotate('$E_{11}, E_{22}$', xy=(0.04, 0.001),  xycoords='data',\n",
    "            xytext=(0.03, 0.01), textcoords='data',\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc,angleA=0,armA=20,angleB=80,armB=10,rad=10\"),\n",
    "            fontsize='15'\n",
    "           )\n",
    "ax.annotate('$\\epsilon_{11}, \\epsilon_{22}$', xy=(0.05, 0.0008),  xycoords='data',\n",
    "            xytext=(0.042, 0.011), textcoords='data',\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc,angleA=0,armA=20,angleB=80,armB=10,rad=10\"),\n",
    "            fontsize='15'\n",
    "           )\n",
    "ax.annotate('$E_{12}, \\epsilon_{12}$', xy=(0.03, 0.03),  xycoords='data',\n",
    "            xytext=(0.035, 0.02), textcoords='data',\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc,angleA=90,armA=20,angleB=-80,armB=10,rad=10\"),\n",
    "            fontsize='15'\n",
    "           );\n",
    "ax.xaxis.label.set_size(20)\n",
    "ax.yaxis.label.set_size(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now compute the rate-of-deformation tensor, i.e. the symmetric part of the velocity gradient"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nBUsqL0RiPGE+T4rlBUuM4Qdc7inlXZCnqwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  1  0⎤\n",
       "⎢       ⎥\n",
       "⎢1  0  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  0  0⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fdot = F.diff(t);\n",
    "\n",
    "L = expand(Fdot * F.inv());\n",
    "\n",
    "D = simplify(Rational(1,2) * (L.T + L)); D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the linear \"small\" strain-rate, $\\dot{\\epsilon}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAABLCAMAAAACojjaAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\sinh{\\left (t \\right )} & \\cosh{\\left (t \\right )} & 0\\\\\\cosh{\\left (t \\right )} & \\sinh{\\left (t \\right )} & 0\\\\0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡sinh(t)  cosh(t)  0⎤\n",
       "⎢                   ⎥\n",
       "⎢cosh(t)  sinh(t)  0⎥\n",
       "⎢                   ⎥\n",
       "⎣   0        0     0⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "epsilon_dot = epsilon.diff(t); epsilon_dot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we will turn the symbolic small strain components into functions that we can evaluate in time. It's not necassary to perform this operation on the rate-of-deformation tensor because it has constant components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "epsilon11_dot_function = lambdify(t, epsilon_dot[0,0], \"numpy\")\n",
    "\n",
    "epsilon22_dot_function = lambdify(t, epsilon_dot[1,1], \"numpy\")\n",
    "\n",
    "epsilon12_dot_function = lambdify(t, epsilon_dot[0,1], \"numpy\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evaluating the functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "D11 = numpy.zeros_like(t0)\n",
    "epsilon11_dot = epsilon11_dot_function(t0)\n",
    "\n",
    "D22 = D11\n",
    "epsilon22_dot = epsilon22_dot_function(t0)\n",
    "\n",
    "D12 = numpy.ones_like(t0)\n",
    "epsilon12_dot = epsilon12_dot_function(t0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZIAAAEYCAYAAAB2qXBEAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nwzrT9oIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1093f6790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(2)\n",
    "ax = fig.add_subplot(111,xlabel='$t$', ylabel='$\\dot{\\epsilon}$')\n",
    "ax.plot(t0, D11,'k-', t0, epsilon11_dot, 'k--',\n",
    "         t0, D12,'b-', t0, epsilon12_dot, 'b--');\n",
    "plt.grid()\n",
    "ax.annotate('$D_{11}, D_{22}$', xy=(0.04, 0.0),  xycoords='data',\n",
    "            xytext=(0.025, 0.15), textcoords='data',\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc,angleA=0,armA=20,angleB=80,armB=10,rad=10\"),\n",
    "            fontsize='15'\n",
    "           )\n",
    "ax.annotate('$\\dot{\\epsilon}_{11}, \\dot{\\epsilon}_{22}$', xy=(0.05, 0.03),  xycoords='data',\n",
    "            xytext=(0.04, 0.15), textcoords='data',\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc,angleA=0,armA=20,angleB=80,armB=10,rad=10\"),\n",
    "            fontsize='15'\n",
    "           )\n",
    "ax.annotate('$D_{12}, \\dot{\\epsilon}_{12}$', xy=(0.04, 1.0),  xycoords='data',\n",
    "            xytext=(0.035, 0.75), textcoords='data',\n",
    "            arrowprops=dict(arrowstyle=\"->\", connectionstyle=\"arc,angleA=90,armA=20,angleB=-80,armB=10,rad=10\"),\n",
    "            fontsize='15'\n",
    "           );\n",
    "ax.xaxis.label.set_size(20)\n",
    "ax.yaxis.label.set_size(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2\n",
    "\n",
    "\n",
    "Given the following stress tensor\n",
    "$$\n",
    "\\sigma =\n",
    "\\begin{bmatrix}\n",
    "  36 & 27 & 0 \\\\\n",
    "  27 & -36 & 0 \\\\\n",
    "  0  & 0  & 18 \n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Find:\n",
    "\n",
    "1. the components of the traction vector acting on a plane with unit normal\n",
    "   vector $\\hat{n}^T = \\left[ 2/3, -2/3, 1/3 \\right]$\n",
    "\n",
    "1. the magnitude of the traction vector found in (a)\n",
    "\n",
    "1. its component in the direction of the normal\n",
    "\n",
    "1. a. the angle between the traction vector and the normal\n",
    "\n",
    "**Solution**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining the stress tensor and normal vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sigma = Matrix([[36, 27, 0],[27, -36, 0],[0, 0, 18]])\n",
    "n = Matrix([Rational(2, 3), Rational(-2, 3), Rational(1, 3)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The traction vector is then $\\vec{t} = \\sigma^{T} \\hat{n}$ according to the Cauchy stress equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACUAAABLCAMAAADAvr4bAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n0BxNNyqMcgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}6\\\\42\\\\6\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡6 ⎤\n",
       "⎢  ⎥\n",
       "⎢42⎥\n",
       "⎢  ⎥\n",
       "⎣6 ⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sigma.T * n; t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing the magnitude"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "42.8485705712571"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.mpmath.mp.pretty = True\n",
    "magnitude = sympy.mpmath.norm(t,2); magnitude"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the projection in the direction of the normal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAAZBAMAAAC86AERAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAu90iEM0ymauJRO92VEnJ\nOWQAAACrSURBVCgVYxBSMmBAB1xKmgwK6IJgvgUx4uaFYgwMYIIBWT1LA8O8BWCCAUWcw4GBewOY\nQBXnVmDgeAYmUMVZnwDFwQSqONCBTE+gBLK9QKF1E6AEmrgiUJgBRIDEWYKUgEA1gYGBXQAoBCZQ\n1Z8ACjOACRRx3gIGYwYwgeIvhnQGhsMQAkWcWXNmhQKYQHU/67t37xTABKo4yFIYQLEXJgikB05c\nCWv60QAAIbIqR7bpCM0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-22\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[-22]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.T * n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The angle between the normal and the traction vector (in radians)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$2.10998044394962$$"
      ],
      "text/plain": [
       "2.10998044394962"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acos(((t / magnitude).T * n)[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or in degree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$120.892974293453$$"
      ],
      "text/plain": [
       "120.892974293453"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_ * 180. / numpy.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3\n",
    "\n",
    "Given the following stress tensor\n",
    "$$\n",
    "\\sigma =\n",
    "\\begin{bmatrix}\n",
    "   18 & 0 & 24 \\\\\n",
    "   0 & -50 & 0 \\\\\n",
    "   24  & 0  & 32\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "    \n",
    "Find:\n",
    "\n",
    "1. the principle stresses $\\sigma_{I}, \\sigma_{II}, \\sigma_{III}$\n",
    "\n",
    "1. the three invariants $I_{1}, I_{2}, I_{3}$\n",
    "\n",
    "1. the deviatoric stress\n",
    "\n",
    "1. the two nonzero invariants of the deviatoric stress, i.e. $J_{2}, J_{3}$\n",
    "\n",
    "**Solution**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining the stress tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sigma = Matrix([[18, 0, 24],[0, -50, 0],[24, 0, 32]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we use sympy to diagonalize (or find the eigenvalues, they are shown on the diagonal of the matrix.  We then define $\\sigma_I > \\sigma_{II} > \\sigma_{III}$ accordingly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAABLCAMAAABqfXinAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nJ4TqG0mnMdrmk+pNAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-50 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 50\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-50  0  0 ⎤\n",
       "⎢          ⎥\n",
       "⎢ 0   0  0 ⎥\n",
       "⎢          ⎥\n",
       "⎣ 0   0  50⎦"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_, D = sigma.diagonalize();\n",
    "sigma1 = D[2,2];sigma2 = D[1,1]; sigma3 = D[0,0]; D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first, second, and third invariants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB5wID+0cG/gsMfN8Z5BUY+L4wzDdg\nYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxylQEA9FISlFfRJtkAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I1 = sigma1 + sigma2 + sigma3; I1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAOBAMAAAB5gJkQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92\nVGZ6zyUAAAAA80lEQVQYGU3Qv0oDMQCA8S/XBqsWCU4+gHtBkY7eG1gRnIuDUpcWHHS0FVwrgmKh\noG9QUDt1uRssBZebXPsAgh2EYruc+WPjZfhI8ksIBP6GPD4PCeL8KaI2DF0WxgbLc1bTNCJIxI6L\nxyYMCB7e4BNuXTy+QDssmOUFjJWNx5Ja4A/0EhuPerKrCl/DUEw1xiZRBldmLCk5lzPYuzO5yWCu\nYRZnWVzfNmNTb5/Yg/V9fakXm0R2w6Z4CS14fNLPjRObf/xAVp6hrsowcfGYr1KsVGGLdzhy8XjQ\nub/ikLU+uUS8ungspek3snatEN2R/niTX7UxUXc+k3IOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$-2500$$"
      ],
      "text/plain": [
       "-2500"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I2 = sigma1 * sigma2 + sigma1 * sigma3 + sigma2 * sigma3; I2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB5wID+0cG/gsMfN8Z5BUY+L4wzDdg\nYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxylQEA9FISlFfRJtkAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I3 = sigma1 * sigma2 * sigma3; I3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The deviatoric stress"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIgAAABLCAMAAACcGD96AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}18 & 0 & 24\\\\0 & -50 & 0\\\\24 & 0 & 32\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡18   0   24⎤\n",
       "⎢           ⎥\n",
       "⎢0   -50  0 ⎥\n",
       "⎢           ⎥\n",
       "⎣24   0   32⎦"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sij = sigma - 1. / 3. * I1 * eye(3); Sij"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we perform the same procedure on the deviatoric stress and compute the invariants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAOBAMAAAB5gJkQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92\nVGZ6zyUAAAAA80lEQVQYGU3Qv0oDMQCA8S/XBqsWCU4+gHtBkY7eG1gRnIuDUpcWHHS0FVwrgmKh\noG9QUDt1uRssBZebXPsAgh2EYruc+WPjZfhI8ksIBP6GPD4PCeL8KaI2DF0WxgbLc1bTNCJIxI6L\nxyYMCB7e4BNuXTy+QDssmOUFjJWNx5Ja4A/0EhuPerKrCl/DUEw1xiZRBldmLCk5lzPYuzO5yWCu\nYRZnWVzfNmNTb5/Yg/V9fakXm0R2w6Z4CS14fNLPjRObf/xAVp6hrsowcfGYr1KsVGGLdzhy8XjQ\nub/ikLU+uUS8ungspek3snatEN2R/niTX7UxUXc+k3IOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$-2500$$"
      ],
      "text/plain": [
       "-2500"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_, D = Sij.diagonalize();\n",
    "Sij1 = D[2,2]; Sij2 = D[1,1]; Sij3 = D[0,0]; \n",
    "\n",
    "J2 = Sij1 * Sij2 + Sij1 * Sij3 + Sij2 * Sij3; J2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB5wID+0cG/gsMfN8Z5BUY+L4wzDdg\nYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxylQEA9FISlFfRJtkAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J3 = Sij1 * Sij2 * Sij3; J3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 4\n",
    "\n",
    "Show that\n",
    "$$\n",
    " \\frac{\\partial J_2}{\\partial \\sigma_{ij}} = S_{ij}\n",
    "$$\n",
    "where $J_2$ is the second invariant of the deviatoric stress tensor, $S_{ij}$.\n",
    "    \n",
    "\n",
    "**Solution**\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\partial J_2}{\\partial \\sigma_{ij}} &= \\frac{\\partial}{\\partial \\sigma_{ij}}\\left(J_2\\right) \\\\\n",
    "                                          &= \\frac{\\partial}{\\partial \\sigma_{ij}}\\left(\\frac{1}{2}S_{kl}S_{kl}\\right) \\\\\n",
    "                                          &= \\frac{1}{2} \\left(\\frac{\\partial}{\\partial \\sigma_{ij}}\\left(S_{kl}\\right)S_{kl} + S_{kl}\\frac{\\partial}{\\partial \\sigma_{ij}}\\left(S_{kl}\\right) \\right) \\\\   \n",
    "                                          &= S_{kl}\\frac{\\partial}{\\partial \\sigma_{ij}}\\left(S_{kl}\\right) \\\\\n",
    "                                          &= S_{kl}\\frac{\\partial}{\\partial \\sigma_{ij}}\\left(\\sigma_{kl} - \\frac{1}{3} \\sigma_{mm}\\delta_{kl}\\right) \\\\\n",
    "                                          &= S_{kl}\\left(\\delta_{il}\\delta_{kj} - \\frac{1}{3} \\delta_{im}\\delta_{jm}\\delta_{kl}\\right) \\\\\n",
    "                                          &= S_{kl}\\left(\\delta_{il}\\delta_{kj} - \\frac{1}{3} \\delta_{ij}\\delta_{kl}\\right) \\\\\n",
    "                                          &= S_{ij} - \\frac{1}{3} S_{kk} \\delta_{ij} \\\\\n",
    "                                          &= S_{ij}\n",
    "\\end{align}        \n",
    "\n",
    "because\n",
    "\n",
    "$$\n",
    "S_{kk} =0\n",
    "$$\n",
    "\n",
    "by definition of $S$ being a deviatoric tensor."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5\n",
    " \n",
    "For each of the following stress states (values not given are zero), plot the three Mohr's circles and\n",
    "determine the maximum shear stress.\n",
    "    \n",
    "1. Uniaxial tension $\\sigma_{11} = 40$\n",
    "\n",
    "1. Biaxial stress $\\sigma_{11} = -10, \\sigma_{22} = 30$\n",
    "\n",
    "1. Hydrostatic tension of magnitude 100 psi\n",
    "    \n",
    "1. $\\sigma_{11} = -60, \\sigma_{22} = 100, \\sigma_{33} = 40$\n",
    "\n",
    "1. $\\sigma_{11} = 10, \\sigma_{22} = 40, \\sigma_{21} = \\sigma_{12} = 20$\n",
    "\n",
    "\n",
    "**Solution**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZIAAAEkCAYAAAAFEbeSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109724450>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111,xlabel='$\\sigma_n$', ylabel='$\\sigma_t$',title='Uniaxial tension, $\\sigma_{11}=40$')\n",
    "ax.xaxis.label.set_size(20)\n",
    "ax.yaxis.label.set_size(20)\n",
    "ax.axhline(0, color='black', lw=2)\n",
    "ax.axvline(0, color='black', lw=2)\n",
    "ax.annotate('$\\sigma_{II},\\sigma_{III}$', xy=(0.04, 1.0),  xycoords='data', fontsize='15');\n",
    "ax.annotate('$\\sigma_{I}$', xy=(40, 1.0),  xycoords='data', fontsize='15');\n",
    "circ = plt.Circle((0.5*(40-0), 0), radius=20, fill=False, color='b')\n",
    "ax.add_patch(circ)\n",
    "plt.axis('equal')\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZIAAAEkCAYAAAAFEbeSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n4ziOUy3ckDiO4zjVwg2J4ziOUy3+HxbpszsAIcEYAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109768b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111,xlabel='$\\sigma_n$', ylabel='$\\sigma_t$',title='Biaxial stress, $\\sigma_{11}=-10, \\sigma_{22}=30$')\n",
    "ax.xaxis.label.set_size(20)\n",
    "ax.yaxis.label.set_size(20)\n",
    "ax.axhline(0, color='black', lw=2)\n",
    "ax.axvline(0, color='black', lw=2)\n",
    "ax.annotate('$\\sigma_{III}$', xy=(-10, 1.0),  xycoords='data', fontsize='15');\n",
    "ax.annotate('$\\sigma_{II}$', xy=(0.04, 1.0),  xycoords='data', fontsize='15');\n",
    "ax.annotate('$\\sigma_{I}$', xy=(30, 1.0),  xycoords='data', fontsize='15');\n",
    "circ1 = plt.Circle((0.5*(30+(-10)), 0), radius=20, fill=False, color='b')\n",
    "circ2 = plt.Circle((0.5*(30+(0)), 0), radius=15, fill=False, color='b')\n",
    "circ3 = plt.Circle((-5, 0), radius=5, fill=False, color='b')\n",
    "ax.add_patch(circ1)\n",
    "ax.add_patch(circ2)\n",
    "ax.add_patch(circ3)\n",
    "plt.axis('equal')\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**For the hydostatic tension case, there are no Mohr's circles to draw because $\\sigma_I = \\sigma_{II} =  \\sigma_{III}$**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYwAAAEjCAYAAAAhczZxAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109530710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111,xlabel='$\\sigma_n$', ylabel='$\\sigma_t$',\n",
    "                     title='$\\sigma_{11} = -60, \\sigma_{22} = 100, \\sigma_{33} = 40$')\n",
    "ax.xaxis.label.set_size(20)\n",
    "ax.yaxis.label.set_size(20)\n",
    "ax.axhline(0, color='black', lw=2)\n",
    "ax.axvline(0, color='black', lw=2)\n",
    "ax.annotate('$\\sigma_{III}$', xy=(-60, 1.0),  xycoords='data', fontsize='15');\n",
    "ax.annotate('$\\sigma_{II}$', xy=(40, 1.0),  xycoords='data', fontsize='15');\n",
    "ax.annotate('$\\sigma_{I}$', xy=(100, 1.0),  xycoords='data', fontsize='15');\n",
    "circ1 = plt.Circle((0.5*(100+(-60)), 0), radius=80, fill=False, color='b')\n",
    "circ2 = plt.Circle((70, 0), radius=30, fill=False, color='b')\n",
    "circ3 = plt.Circle((-10, 0), radius=50, fill=False, color='b')\n",
    "ax.add_patch(circ1)\n",
    "ax.add_patch(circ2)\n",
    "ax.add_patch(circ3)\n",
    "plt.axis('equal')\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAABLCAMAAABTGsn8AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 50\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  0  0 ⎤\n",
       "⎢        ⎥\n",
       "⎢0  0  0 ⎥\n",
       "⎢        ⎥\n",
       "⎣0  0  50⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sigma = Matrix([[10,20,0],[20,40,0],[0,0,0]])\n",
    "_, D = sigma.diagonalize(); D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZIAAAEjCAYAAAAYFIcqAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nj9YSYQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1096e7050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111,xlabel='$\\sigma_n$', ylabel='$\\sigma_t$',\n",
    "                     title='$\\sigma_{11} = 10, \\sigma_{22} = 40, \\sigma_{21} = \\sigma_{12} = 20$')\n",
    "ax.xaxis.label.set_size(20)\n",
    "ax.yaxis.label.set_size(20)\n",
    "ax.axhline(0, color='black', lw=2)\n",
    "ax.axvline(0, color='black', lw=2)\n",
    "ax.annotate('$\\sigma_{II},\\sigma_{III}$', xy=(0.04, 1.0),  xycoords='data', fontsize='15');\n",
    "ax.annotate('$\\sigma_{I}$', xy=(50, 1.0),  xycoords='data', fontsize='15');\n",
    "circ = plt.Circle((0.5*(50-0), 0), radius=25, fill=False, color='b')\n",
    "ax.add_patch(circ)\n",
    "plt.axis('equal')\n",
    "plt.grid()\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}