06_Vergleichsspannung.ipynb

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   "source": [
    "# Vergleichsspannung\n",
    "\n",
    "Dr. Zoufiné Lauer-Baré, FEM course\n",
    "\n",
    " $$\\mathbf{u}(x_1,x_2,x_3)= \\begin{pmatrix}u_1(x_1,x_2,x_3)\\\\u_2(x_1,x_2,x_3)\\\\u_3(x_1,x_2,x_3) \\end{pmatrix}$$\n",
    " \n",
    " $$\\mathbf{E} = \\frac{1}{2}\\left(\\nabla \\mathbf{u}+(\\nabla \\mathbf{u})^T\\right)$$\n",
    " \n",
    " $$\\mathbf{\\Sigma}  =2\\mu \\mathbf{E}+\\lambda \\text{tr}(\\mathbf{E})\\mathbf{I} $$"
   ]
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   "source": [
    "S. 182 in [Issler et. al, 2003]."
   ]
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   "source": [
    "$$\\mathbf{\\Sigma}  =\\begin{pmatrix}\\sigma_{11}&&\\sigma_{12}&&\\sigma_{13}\\\\\n",
    "                                   \\sigma_{21}&&\\sigma_{22}&&\\sigma_{23}\\\\\n",
    "                                   \\sigma_{31}&&\\sigma_{31}&&\\sigma_{33} \n",
    "                                 \\end{pmatrix} $$"
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   "source": [
    "Wie bekommxt man an einer Stelle $\\mathbf{x}\\in\\mathbb{R}^3$ einen Spannungswert, den man farbcodieren und mit anderen Spannungswerten (z.B. Streckgrenze) vergleichen kann? Z.B. durch die von Mises Vergleichsspannung:\n",
    "\n",
    "$$\\sigma_{v}=\\frac{1}{\\sqrt{2}}\\sqrt{(\\sigma_{11}-\\sigma_{22})^2+(\\sigma_{22}-\\sigma_{33})^2+(\\sigma_{33}-\\sigma_{11})^2+6\\cdot(\\sigma_{12}^2+\\sigma_{23}^2+\\sigma_{31}^2)}$$\n",
    "\n",
    "Sind die Hauptspannungen $\\sigma_{I},\\sigma_{II},\\sigma_{III}$ bekannt, dann ist die von Mises Vergleichsspannung:\n",
    "\n",
    "$$\\sigma_{v}=\\frac{1}{\\sqrt{2}}\\sqrt{(\\sigma_{I}-\\sigma_{II})^2+(\\sigma_{II}-\\sigma_{III})^2+(\\sigma_{III}-\\sigma_{I})^2}$$.\n",
    "\n",
    "Die die Hauptspannungen $\\sigma_{I},\\sigma_{II},\\sigma_{III}$ sind die Eigenwerte der Spannungsmatrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Im Folgenden ein Beispiel. Auf der Seite [sympy live](https://live.sympy.org/) können Sie die Eigenwerte von Matrizen symbolsich berechnen."
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
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   "outputs": [],
   "source": [
    "from sympy import *\n",
    "import numpy as np\n",
    "from numpy import sqrt\n",
    "from matplotlib import pylab as plt\n",
    "import plotly.express as px\n",
    "import plotly.graph_objects as go"
   ]
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}20 & 0 & 0\\\\0 & 30 & 40\\\\0 & 40 & -30\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[20,  0,   0],\n",
       "[ 0, 30,  40],\n",
       "[ 0, 40, -30]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "Sigma=Matrix(([20,0,0],[0,30,40],[0,40,-30]))\n",
    "Sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{20: 1, -50: 1, 50: 1}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sigma.eigenvals()"
   ]
  },
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   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigmaI=20\n",
    "sigmaII=-50 \n",
    "sigmaIII=50 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "summe=(sigmaI-sigmaII)**2+(sigmaII-sigmaIII)**2+(sigmaIII-sigmaI)**2\n",
    "sigma_v=(1/np.sqrt(2.))*np.sqrt(summe)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
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     "data": {
      "text/plain": [
       "88.88194417315589"
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     "execution_count": 6,
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    "sigma_v"
   ]
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "Spannung=Matrix(([-30,0,-60  ],[0,30,-60],[-60,-60,0]))"
   ]
  },
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   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-30 & 0 & -60\\\\0 & 30 & -60\\\\-60 & -60 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-30,   0, -60],\n",
       "[  0,  30, -60],\n",
       "[-60, -60,   0]])"
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     "execution_count": 8,
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   ],
   "source": [
    "Spannung"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-90: 1, 90: 1, 0: 1}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "Spannung.eigenvals()"
   ]
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   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigmaI=-90\n",
    "sigmaII=0\n",
    "sigmaIII=90"
   ]
  },
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       "155.88457268119893"
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     "execution_count": 11,
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   "source": [
    "summe=(sigmaI-sigmaII)**2+(sigmaII-sigmaIII)**2+(sigmaIII-sigmaI)**2\n",
    "sigma_v=(1/np.sqrt(2.))*np.sqrt(summe)\n",
    "sigma_v"
   ]
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   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}20 & 40 & -20\\\\40 & 20 & -20\\\\-20 & -20 & -10\\end{matrix}\\right]$"
      ],
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       "Matrix([\n",
       "[ 20,  40, -20],\n",
       "[ 40,  20, -20],\n",
       "[-20, -20, -10]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "Spannung2=Matrix(([20,40,-20  ],[40,20,-20],[-20,-20,-10]))\n",
    "Spannung2"
   ]
  },
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   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{70: 1, -20: 2}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
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   "source": [
    "Spannung2.eigenvals()"
   ]
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  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "90.0"
      ]
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     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "sigmaI=70\n",
    "sigmaII=-20\n",
    "sigmaIII=-20\n",
    "\n",
    "summe=(sigmaI-sigmaII)**2+(sigmaII-sigmaIII)**2+(sigmaIII-sigmaI)**2\n",
    "sigma_v=(1/np.sqrt(2.))*np.sqrt(summe)\n",
    "sigma_v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2 & 5 & 3\\\\5 & 1 & 4\\\\3 & 4 & 3\\end{matrix}\\right]$"
      ],
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       "Matrix([\n",
       "[2, 5, 3],\n",
       "[5, 1, 4],\n",
       "[3, 4, 3]])"
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     "execution_count": 15,
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   "source": [
    "Spannung3=Matrix(([2,5,3  ],[5,1,4],[3,4,3]))\n",
    "\n",
    "Spannung3"
   ]
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   "cell_type": "code",
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     "data": {
      "text/plain": [
       "{10: 1, -2 - sqrt(3): 1, -2 + sqrt(3): 1}"
      ]
     },
     "execution_count": 16,
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     "output_type": "execute_result"
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    "Spannung3.eigenvals()"
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   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
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    {
     "data": {
      "text/plain": [
       "[10, -2 - sqrt(3), -2 + sqrt(3)]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(Spannung3.eigenvals())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{PurePoly}{\\left( \\lambda^{3} - 6 \\lambda^{2} - 39 \\lambda - 10, \\lambda, domain=\\mathbb{Z} \\right)}$"
      ],
      "text/plain": [
       "PurePoly(lambda**3 - 6*lambda**2 - 39*lambda - 10, lambda, domain='ZZ')"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Spannung3.charpoly()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12.369316876852979"
      ]
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     "execution_count": 19,
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   "source": [
    "sigmaI=10\n",
    "sigmaII=-2 - np.sqrt(3)\n",
    "sigmaIII= -2 + np.sqrt(3)\n",
    "\n",
    "summe=(sigmaI-sigmaII)**2+(sigmaII-sigmaIII)**2+(sigmaIII-sigmaI)**2\n",
    "sigma_v=(1/np.sqrt(2.))*np.sqrt(summe)\n",
    "sigma_v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "lamda = symbols('lamda')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\lambda^{3} - 6 \\lambda^{2} - 39 \\lambda - 10$"
      ],
      "text/plain": [
       "lamda**3 - 6*lamda**2 - 39*lamda - 10"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Spannung3.charpoly(lamda).as_expr()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
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   "source": [
    "poly = lambdify(lamda, Spannung3.charpoly(lamda).as_expr())"
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",
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     },
     "metadata": {},
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    }
   ],
   "source": [
    "#plt.plot(np.linspace(-15, 15), poly(np.linspace(-15, 15)))\n",
    "fig=px.line(x=np.linspace(-15, 15),y=poly(np.linspace(-15, 15)))\n",
    "fig.add_trace(go.Scatter(x=np.asfarray(list(Spannung3.eigenvals())), \n",
    "                         y=poly(np.asfarray(list(Spannung3.eigenvals()))),\n",
    "                         mode='markers',))\n",
    "fig.show()"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "factor(Spannung3.charpoly(lamda).as_expr())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.asfarray(list(Spannung3.eigenvals()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly(np.asfarray(list(Spannung3.eigenvals())))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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