{
"cells": [
 {
		   "cell_type": "markdown",
	   "metadata": {},
	   "source": [
       "# Chapter 3: ANALYSIS OF STRAINS"
	   ]
	},
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.3: Chapter3_Example_3.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"clc\n",
"// initialization of variables\n",
"clear\n",
"epsillon=[0.01 0 0\n",
"     0 0.02 0.02\n",
"     0 0.02 0.01] // dimensionless\n",
"theta=30 // degrees\n",
"//calculations\n",
"theta=theta*%pi/180\n",
"a=[cos(theta) sin(theta) 0\n",
"  -sin(theta) cos(theta) 0\n",
"   0 0 1]\n",
"b=a.'\n",
"epsillon_new=a*epsillon*b\n",
"// calculation of strain invariants\n",
"// for epsillon\n",
"J1=epsillon(1,1)+epsillon(2,2)+epsillon(3,3)\n",
"J2=epsillon(1,1)*epsillon(2,2)+epsillon(2,2)*epsillon(3,3)+epsillon(3,3)*epsillon(1,1)-2*(epsillon(1,2)^2+epsillon(2,3)^2+epsillon(3,1)^2)\n",
"J3=epsillon(1,1)*epsillon(2,2)*epsillon(3,3)+2*epsillon(1,2)*epsillon(2,3)*epsillon(3,1)-(epsillon(1,1)*epsillon(2,3)^2+epsillon(2,2)*epsillon(3,1)^2+epsillon(3,3)*epsillon(1,2)^2)\n",
"\n",
"// for epsillon_new\n",
"J11=epsillon_new(1,1)+epsillon_new(2,2)+epsillon_new(3,3)\n",
"J22=epsillon_new(1,1)*epsillon_new(2,2)+epsillon_new(2,2)*epsillon_new(3,3)+epsillon_new(3,3)*epsillon_new(1,1)-2*(epsillon_new(1,2)^2+epsillon_new(2,3)^2+epsillon_new(3,1)^2)\n",
"J33=epsillon_new(1,1)*epsillon_new(2,2)*epsillon_new(3,3)+2*epsillon_new(1,2)*epsillon_new(2,3)*epsillon_new(3,1)-(epsillon_new(1,1)*epsillon_new(2,3)^2+epsillon_new(2,2)*epsillon_new(3,1)^2+epsillon_new(3,3)*epsillon_new(1,2)^2)\n",
"\n",
"// results\n",
"printf('The new strain tensor is');\n",
"disp(epsillon_new);\n",
"printf('The strain invariants of old strain tensor are J1=%0.2f J2=%.e J3=%.e \n and that of the new strain tensor are J1=%0.2f J2=%.e J3=%.e',J1,J2,J3,J11,J22,J33)\n",
"printf('\n Hence the same strain invariants')\n",
"\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.4: Chapter3_Example_4.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"clc\n",
"// initialization of variables\n",
"clear\n",
"epsillon=[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=acos(a_xx) // radians\n",
"//calculations\n",
"// theta=theta*%pi/180\n",
"a=[cos(theta) 0 -sin(theta)\n",
"   0       1        0\n",
"   sin(theta) 0 cos(theta)]\n",
"b=a.'\n",
"epsillon_new=a*epsillon*b\n",
"\n",
"// calculation of strain invariants\n",
"// for epsillon\n",
"J1=epsillon(1,1)+epsillon(2,2)+epsillon(3,3)\n",
"\n",
"J2=epsillon(1,1)*epsillon(2,2)+epsillon(2,2)*epsillon(3,3)+epsillon(3,3)*epsillon(1,1)-2*(epsillon(1,2)^2+epsillon(2,3)^2+epsillon(3,1)^2)\n",
"\n",
"J3=epsillon(1,1)*epsillon(2,2)*epsillon(3,3)+2*epsillon(1,2)*epsillon(2,3)*epsillon(3,1)-(epsillon(1,1)*epsillon(2,3)^2+epsillon(2,2)*epsillon(3,1)^2+epsillon(3,3)*epsillon(1,2)^2)\n",
"\n",
"// for epsillon_new\n",
"J11=epsillon_new(1,1)+epsillon_new(2,2)+epsillon_new(3,3)\n",
"\n",
"J22=epsillon_new(1,1)*epsillon_new(2,2)+epsillon_new(2,2)*epsillon_new(3,3)+epsillon_new(3,3)*epsillon_new(1,1)-2*(epsillon_new(1,2)^2+epsillon_new(2,3)^2+epsillon_new(3,1)^2)\n",
"\n",
"J33=epsillon_new(1,1)*epsillon_new(2,2)*epsillon_new(3,3)+2*epsillon_new(1,2)*epsillon_new(2,3)*epsillon_new(3,1)-(epsillon_new(1,1)*epsillon_new(2,3)^2+epsillon_new(2,2)*epsillon_new(3,1)^2+epsillon_new(3,3)*epsillon_new(1,2)^2)\n",
"\n",
"// Results\n",
"printf('The new strain tensor is')\n",
"disp(epsillon_new)\n",
"printf('The strain invariants of old stress tensor are J1=%0.2f J2=%.e J3=%.e \n and that of the new stress tensor are J1=%0.2f J2=%.e J3=%.e',J1,J2,J3,J11,J22,J33)\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.5: Chapter3_Example_5.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"clc\n",
"// initialization of variables\n",
"clear\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*%pi/180 // radians\n",
"// calculations\n",
"epsillon_x=epsillon_A\n",
"epsillon_y=epsillon_C\n",
"gamma_xy=(epsillon_B-(epsillon_x*cos(theta)^2+epsillon_y*sin(theta)^2))/(sin(theta)*cos(theta))\n",
"epsillon_1=1/2*(epsillon_x+epsillon_y)+(1/2)*sqrt((epsillon_x-epsillon_y)^2+gamma_xy^2)\n",
"epsillon_2=1/2*(epsillon_x+epsillon_y)-(1/2)*sqrt((epsillon_x-epsillon_y)^2+gamma_xy^2)\n",
"phi=0.5*atan(gamma_xy/(epsillon_x-epsillon_y))\n",
"phi=phi*180/%pi\n",
"//results\n",
"printf('The principal strains are %.3e, %.3e',epsillon_1,epsillon_2)\n",
"printf('\n phi = %.2f degrees',phi)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 3.6: Chapter3_Example_6.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"clc\n",
"// initialization of variables\n",
"clear\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*%pi/180\n",
"th_C=th_C*%pi/180\n",
"// we need to solve for epsillon_y and gamma_xy\n",
"// Ax=B\n",
"ep_x=epsillon_A\n",
"A=[sin(th_B)^2 sin(th_B)*cos(th_B)\n",
"   sin(th_C)^2 sin(th_C)*cos(th_C)]\n",
"C=[epsillon_B-ep_x*cos(th_B)^2; epsillon_C-ep_x*cos(th_C)^2]   \n",
"x=inv(A)*C\n",
"ep_y=x(1,1)\n",
"gam_xy=x(2,1)\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)*sqrt((epsillon_x-epsillon_y)^2+gamma_xy^2)\n",
"epsillon_2=1/2*(epsillon_x+epsillon_y)-(1/2)*sqrt((epsillon_x-epsillon_y)^2+gamma_xy^2)\n",
"// Results\n",
"printf('The principal strains are %.3e, %.3e',epsillon_1,epsillon_2)"
   ]
   }
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			 "text": "MetaKernel Magics",
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