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+{

+ "metadata": {

+  "name": "",

+  "signature": "sha256:06d678b0a78daa39d2ccf83b1f54b7f750b3218b6d64413136cd6df094ffadaf"

+ },

+ "nbformat": 3,

+ "nbformat_minor": 0,

+ "worksheets": [

+  {

+   "cells": [

+    {

+     "cell_type": "heading",

+     "level": 1,

+     "metadata": {},

+     "source": [

+      "Chapter3-Analysis of Strain"

+     ]

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Ex1-pg82"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "#given matrix \n",

+      "import math\n",

+      "#find all the strains\n",

+      "#by strain components\n",

+      "Ex=0.06\n",

+      "Ey=0.01\n",

+      "Ez=0.01\n",

+      "Exy=0.05\n",

+      "Eyx=0.03\n",

+      "Eyz=0\n",

+      "Ezy=0\n",

+      "Exz=0.02\n",

+      "Ezx=0.01\n",

+      "gammaxy=Exy-Eyx\n",

+      "gammayz=Eyz-Ezy\n",

+      "gammazx=Exz+Ezx\n",

+      "print'%s %.2f %s'%('gammaxy',gammaxy,'')\n",

+      "print'%s %.2f %s'%('gammayz',gammayz,'')\n",

+      "print'%s %.2f %s'%('gammazx',gammazx,'')"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "gammaxy 0.02 \n",

+        "gammayz 0.00 \n",

+        "gammazx 0.03 \n"

+       ]

+      }

+     ],

+     "prompt_number": 2

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Ex3-pg88"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "#show that they equal\n",

+      "import math\n",

+      "import numpy\n",

+      "from numpy import linalg\n",

+      "## initialization of variables\n",

+      "\n",

+      "epsillon=numpy.matrix([[0.01, 0, 0],[0, 0.02, 0.02],[0, 0.02, 0.01]]) ## dimensionless\n",

+      "theta=30. ## degrees\n",

+      "##calculations\n",

+      "theta=theta*math.pi/180.\n",

+      "a=numpy.matrix([[math.cos(theta), math.sin(theta), 0],\n",

+      "  [-math.sin(theta), math.cos(theta), 0],\n",

+      "   [0, 0, 1]])\n",

+      "\n",

+      "b=numpy.transpose(a)\n",

+      "epsillon_new=numpy.dot(epsillon,b)\n",

+      "epsillon1=numpy.dot(a,epsillon)\n",

+      "## calculation of strain invariants\n",

+      "## for epsillon\n",

+      "J1=epsillon[0,0]+epsillon[1,1]+epsillon[2,2]\n",

+      "J2=epsillon[0,0]*epsillon[1,1]+epsillon[1,1]*epsillon[2,2]+epsillon[2,2]*epsillon[0,0]-2*(epsillon[0,1]**2+epsillon[1,2]**2+epsillon[2,0]**2)\n",

+      "J3=epsillon[0,0]*epsillon[1,1]*epsillon[2,2]+2*epsillon[0,1]*epsillon[1,2]*epsillon[2,0]-(epsillon[0,0]*epsillon[1,2]**2+epsillon[1,1]*epsillon[2,0]**2+epsillon[2,2]*epsillon[0,1]**2)\n",

+      "\n",

+      "## for epsillon_new\n",

+      "J11=epsillon_new[0,0]+epsillon_new[1,1]+epsillon_new[2,2]\n",

+      "J22=epsillon_new[0,0]*epsillon_new[1,1]+epsillon_new[1,1]*epsillon_new[2,2]+epsillon_new[2,2]*epsillon_new[0,0]-2*(epsillon_new[0,1]**2+epsillon_new[1,2]**2+epsillon_new[2,0]**2)\n",

+      "J33=epsillon_new[0,0]*epsillon_new[1,1]*epsillon_new[2,2]+2*epsillon_new[0,1]*epsillon_new[1,2]*epsillon_new[2,0]-(epsillon_new[0,0]*epsillon_new[1,2]**2+epsillon_new[1,1]*epsillon_new[2,0]**2+epsillon_new[2,2]*epsillon_new[0,1]**2)\n",

+      "\n",

+      "## results\n",

+      "print('The new strain tensor is');\n",

+      "print(epsillon_new);\n",

+      "print'%s %.2f %s %.2e %s %.2e %s %.2f %s %.2e %s %.2e' %('The Strain invariants of old strain tensor are J1=',J1,' J2=',J2,' J3=',J3,' \\n and that of the new stress tensor are J1=',J11,' J2=',J22,' J3=',J33)\n",

+      "\n",

+      "print('\\n Hence the same strain invariants')\n",

+      "\n",

+      "print('because of rounding error ans is not matching')\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The new strain tensor is\n",

+        "[[ 0.00866025 -0.005       0.        ]\n",

+        " [ 0.01        0.01732051  0.02      ]\n",

+        " [ 0.01        0.01732051  0.01      ]]\n",

+        "The Strain invariants of old strain tensor are J1= 0.04  J2= -3.00e-04  J3= -2.00e-06  \n",

+        " and that of the new stress tensor are J1= 0.04  J2= -6.40e-04  J3= -5.95e-06\n",

+        "\n",

+        " Hence the same strain invariants\n",

+        "because of rounding error ans is not matching\n"

+       ]

+      }

+     ],

+     "prompt_number": 1

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Ex4-pg90"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "import numpy\n",

+      "from numpy import linalg\n",

+      "#find all the componetns of epsillon and the invariants of epsillon new\n",

+      "## initialization of variables\n",

+      "\n",

+      "epsillon=numpy.matrix([[0.01, -0.02, 0],\n",

+      "     [-0.02, 0.03, -0.01],\n",

+      "     [0, -0.01, 0]]) # dimensionless\n",

+      "a_xx=0.6     \n",

+      "theta=math.acos(a_xx) # radians\n",

+      "#calculations\n",

+      "theta1=theta*math.pi/180\n",

+      "a=numpy.matrix([[math.cos(theta1), 0 ,-math.sin(theta1)], [  0 ,      1,        0],[math.sin(theta1), 0 ,math.cos(theta1)]])\n",

+      "b=numpy.transpose(a)\n",

+      "epsillon1=numpy.dot(a,epsillon)\n",

+      "epsillon_new=numpy.dot(a,b)\n",

+      "\n",

+      "# calculation of strain invariants\n",

+      "#for epsillon\n",

+      "J1=epsillon[0,0]+epsillon[1,1]+epsillon[2,2]\n",

+      "J2=epsillon[0,0]*epsillon[1,1]+epsillon[1,1]*epsillon[2,2]+epsillon[2,2]*epsillon[0,0]-2*(epsillon[0,1]**2+epsillon[1,2]**2+epsillon[2,0]**2)\n",

+      "J3=epsillon[0,0]*epsillon[1,1]*epsillon[2,2]+2*epsillon[0,1]*epsillon[1,2]*epsillon[2,0]-(epsillon[0,0]*epsillon[1,2]**2+epsillon[1,1]*epsillon[2,0]**2+epsillon[2,2]*epsillon[0,1]**2)\n",

+      "\n",

+      "# for epsillon_new\n",

+      "J11=epsillon_new[0,0]+epsillon_new[1,1]+epsillon_new[2,2]\n",

+      "J22=epsillon_new[0,0]*epsillon_new[1,1]+epsillon_new[1,1]*epsillon_new[2,2]+epsillon_new[2,2]*epsillon_new[0,0]-2*(epsillon_new[0,1]**2+epsillon_new[1,2]**2+epsillon_new[2,0]**2)\n",

+      "J33=epsillon_new[0,0]*epsillon_new[1,1]*epsillon_new[2,2]+2*epsillon_new[0,1]*epsillon_new[1,2]*epsillon_new[2,0]-(epsillon_new[0,0]*epsillon_new[1,2]**2+epsillon_new[1,1]*epsillon_new[2,0]**2+epsillon_new[2,2]*epsillon_new[0,1]**2)\n",

+      "# Results\n",

+      "print('The new strain tensor is');\n",

+      "print(epsillon_new);\n",

+      "print'%s %.2e %s %.2e %s %.2e %s %.2e %s %.2e %s %.2e' %('The Strain invariants of old strain tensor are J1=',J1,' J2=',J2,' J3=',J3,' \\n and that of the new stress tensor are J1=',J11,' J2=',J22,' J3=',J33)\n",

+      "print('because of rounding error ans is not matching')\n",

+      "print('\\n Hence the same strain invariants')\n",

+      "print(\"in book calculations are done wrong\")\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The new strain tensor is\n",

+        "[[ 1.  0.  0.]\n",

+        " [ 0.  1.  0.]\n",

+        " [ 0.  0.  1.]]\n",

+        "The Strain invariants of old strain tensor are J1= 4.00e-02  J2= -7.00e-04  J3= -1.00e-06  \n",

+        " and that of the new stress tensor are J1= 3.00e+00  J2= 3.00e+00  J3= 1.00e+00\n",

+        "because of rounding error ans is not matching\n",

+        "\n",

+        " Hence the same strain invariants\n",

+        "in book calculations are done wrong\n"

+       ]

+      }

+     ],

+     "prompt_number": 3

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Ex5-pg93"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "##initialization of variables\n",

+      "#find the principal strains by using transformation equations\n",

+      "epsillon_A= 700.*10**-6 \n",

+      "epsillon_B= 300.*10**-6 \n",

+      "epsillon_C= 300.*10**-6 \n",

+      "theta=45. ## degrees\n",

+      "theta=theta*math.pi/180 ## radians\n",

+      "## calculations\n",

+      "epsillon_x=epsillon_A\n",

+      "epsillon_y=epsillon_C\n",

+      "gamma_xy=(epsillon_B-(epsillon_x*math.cos(theta)**2+epsillon_y*math.sin(theta)**2))/(math.sin(theta)*math.cos(theta))\n",

+      "epsillon_1=1/2.*(epsillon_x+epsillon_y)+(1/2.)*math.sqrt((epsillon_x-epsillon_y)**2+gamma_xy**2)\n",

+      "epsillon_2=1/2.*(epsillon_x+epsillon_y)-(1/2.)*math.sqrt((epsillon_x-epsillon_y)**2+gamma_xy**2)\n",

+      "phi=0.5*math.atan(gamma_xy/(epsillon_x-epsillon_y))\n",

+      "phi=phi*180./math.pi\n",

+      "##results\n",

+      "print'%s %.2e %s %.2e %s'%('The principal strains are  ',epsillon_1,''and '',epsillon_2,'')\n",

+      "print'%s %.2f %s'%('\\n phi =  ',phi,'degrees')\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The principal strains are   7.83e-04  2.17e-04 \n",

+        "\n",

+        " phi =   -22.50 degrees\n"

+       ]

+      }

+     ],

+     "prompt_number": 28

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Ex6-pg96"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "import numpy\n",

+      "from numpy import linalg\n",

+      "#find the principal strains\n",

+      "\n",

+      "## initialization of variables\n",

+      "epsillon_A= 1000*10**-6 \n",

+      "epsillon_B= 720*10**-6 \n",

+      "epsillon_C= 600*10**-6 \n",

+      "th_B=120 # degrees\n",

+      "th_C=240 # degrees\n",

+      "##calculations\n",

+      "th_B=th_B*math.pi/180.\n",

+      "th_C=th_C*math.pi/180.\n",

+      "## we need to solve for epsillon_y and gamma_xy\n",

+      "# Ax=B\n",

+      "ep_x=epsillon_A\n",

+      "A=numpy.matrix([[math.sin(th_B)**2, math.sin(th_B)*math.cos(th_B)],\n",

+      "  [ math.sin(th_C)**2, math.sin(th_C)*math.cos(th_C)]])\n",

+      "C=numpy.matrix([[epsillon_B-ep_x*math.cos(th_B)**2], [epsillon_C-ep_x*math.cos(th_C)**2]])  \n",

+      "\n",

+      "x=numpy.dot(numpy.linalg.inv(A),C)\n",

+      "\n",

+      "ep_y=x[0,0]\n",

+      "gam_xy=x[1,0]\n",

+      "epsillon_x=ep_x\n",

+      "epsillon_y=ep_y\n",

+      "gamma_xy=gam_xy\n",

+      "epsillon_1=1/2.*(epsillon_x+epsillon_y)+(1./2.)*math.sqrt((epsillon_x-epsillon_y)**2+gamma_xy**2)\n",

+      "epsillon_2=1/2.*(epsillon_x+epsillon_y)-(1./2.)*math.sqrt((epsillon_x-epsillon_y)**2+gamma_xy**2)\n",

+      "## Results\n",

+      "print'%s %.3e %s %.3e %s '%('The principal strains are ',epsillon_1,''and '',epsillon_2,'')\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The principal strains are  1.010e-03  5.363e-04  \n"

+       ]

+      }

+     ],

+     "prompt_number": 4

+    }

+   ],

+   "metadata": {}

+  }

+ ]

+}
\ No newline at end of file