Elasticity.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drawing Mohr's circles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  50,  300,    0],\n",
       "       [ 300, -400,    0],\n",
       "       [   0,    0,    0]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Given a certain stress tensor\n",
    "σ = np.array([[50, 300, 0], [300, -400, 0], [0, 0, 0]])\n",
    "σ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n = 100000\n",
    "\n",
    "σn = np.zeros([n, 1]) # array of normal stress component\n",
    "τ = np.zeros([n, 1]) # array of shear stress component\n",
    "\n",
    "for i in range(n):\n",
    "    \n",
    "    # Generate normal vectors to n random planes\n",
    "    θ = np.random.uniform(0, 2*np.pi)\n",
    "    ϕ = np.random.uniform(0, np.pi)\n",
    "    η = np.array([np.sin(ϕ)*np.cos(θ), np.sin(ϕ)*np.sin(θ), np.cos(ϕ)])\n",
    "    \n",
    "    # Cauchy's Law\n",
    "    T = σ.dot(η)\n",
    "    \n",
    "    # Stress components\n",
    "    σn[i, 0] = η.dot(T)\n",
    "    τ[i, 0] = np.sqrt(np.array([(T**2).sum() - σn[i, 0]**2]).astype(np.float32))[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fb8efabafd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot circles\n",
    "plt.plot(σn, τ, 'c.')\n",
    "plt.axis('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Checking how the maximum and minimum values of the normal stress component match the principal normal stresses."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-549.99030444725634, 199.97520159297252)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Principal normal stresses (not including σz), numerically\n",
    "σn.min(), σn.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-550.0, 200.0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "σx = σ[0, 0]\n",
    "σy = σ[1, 1]\n",
    "τxy = σ[0, 1]\n",
    "\n",
    "# Center and radius of the external circle\n",
    "C = (σx + σy)/2\n",
    "R = np.sqrt(((σx - σy)/2)**2 + τxy**2)\n",
    "\n",
    "# Principal normal stresses (not including σz), graphically\n",
    "C - R, C + R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 200., -550.,    0.])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Principal normal stresses, analitically\n",
    "np.linalg.eigvals(σ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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