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{
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 "metadata": {},
 "outputs": [
 {
 "name": "stdout",
 "output_type": "stream",
 "text": [
 "Requirement already satisfied: numpy in /usr/local/python/3.12.1/lib/python3.12/site-packages (2.3.5)\n",
 "Requirement already satisfied: matplotlib in /usr/local/python/3.12.1/lib/python3.12/site-packages (3.10.8)\n",
 "Requirement already satisfied: contourpy>=1.0.1 in /usr/local/python/3.12.1/lib/python3.12/site-packages (from matplotlib) (1.3.3)\n",
 "Requirement already satisfied: cycler>=0.10 in /usr/local/python/3.12.1/lib/python3.12/site-packages (from matplotlib) (0.12.1)\n",
 "Requirement already satisfied: fonttools>=4.22.0 in /usr/local/python/3.12.1/lib/python3.12/site-packages (from matplotlib) (4.61.1)\n",
 "Requirement already satisfied: kiwisolver>=1.3.1 in /usr/local/python/3.12.1/lib/python3.12/site-packages (from matplotlib) (1.4.9)\n",
 "Requirement already satisfied: packaging>=20.0 in /home/codespace/.local/lib/python3.12/site-packages (from matplotlib) (25.0)\n",
 "Requirement already satisfied: pillow>=8 in /usr/local/python/3.12.1/lib/python3.12/site-packages (from matplotlib) (12.0.0)\n",
 "Requirement already satisfied: pyparsing>=3 in /usr/local/python/3.12.1/lib/python3.12/site-packages (from matplotlib) (3.2.5)\n",
 "Requirement already satisfied: python-dateutil>=2.7 in /home/codespace/.local/lib/python3.12/site-packages (from matplotlib) (2.9.0.post0)\n",
 "Requirement already satisfied: six>=1.5 in /home/codespace/.local/lib/python3.12/site-packages (from python-dateutil>=2.7->matplotlib) (1.17.0)\n"
 ]
 }
 ],
 "source": [
 "!pip install numpy matplotlib"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "5hzPDd4kzdfJ"
 },
 "outputs": [],
 "source": [
 "import numpy as np\n",
 "import matplotlib.pyplot as plt\n",
 "from math import *\n",
 "import cmath\n",
 "import scipy.special as sp"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "g790NNzfzezc"
 },
 "outputs": [],
 "source": [
 "def normale(theta,phi):\n",
 " # retourne le vecteur unitaire définit par (cos(theta)*sin(phi),sin(theta)*sin(phi),cos(phi))\n",
 " vN = np.array([cos(theta)*sin(phi),sin(theta)*sin(phi),cos(phi)])\n",
 " return vN.T"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "C6Z8nsTLzgUX"
 },
 "outputs": [],
 "source": [
 "def tens_to_mat(liste):\n",
 " if isinstance(liste,list):\n",
 " liste = np.array(liste)\n",
 " res = np.array([[liste[0],liste[3],liste[4]],\n",
 " [liste[3],liste[1],liste[5]],\n",
 " [liste[4],liste[5],liste[2]]])\n",
 " return res\n"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "t1mSr0W6zjfv"
 },
 "outputs": [],
 "source": [
 "def contTang(tens,vN):\n",
 " # calcul le vecteur contrainte tangentiell sur une facette de normale vN\n",
 " M = tens_to_mat(tens)# vecteur contrainte\n",
 " cont = M@vN# contrainte normale\n",
 " \n",
 " cN = cont@vN\n",
 " contT = cont-cN*vN\n",
 " return contT\n"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "JiyijqVBzl-C"
 },
 "outputs": [],
 "source": [
 "def hydro(tens):\n",
 " # cette fonction doit retourner la pression hydrostatique associée à ce tenseur (c'est pour un instant du cycle !)\n",
 " p = (tens[0]+tens[1]+tens[2])/3\n",
 " return p\n"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "a7AsZuRSznR4"
 },
 "outputs": [],
 "source": [
 "def genereTens(sigma1,omega,pasTemps,fin):\n",
 " tens = np.array([sigma1,0,0,0,0,0])\n",
 " for i in range(int(fin/pasTemps)):\n",
 " t = (i+1 )* pasTemps\n",
 " ligne = np.array([sigma1*cos(omega*t),0,0,0,0,0])\n",
 " tens = np.vstack((tens, ligne))\n",
 " # omega est la pulsation, vous pouvez choisir 2*pi par exemple\n",
 " # sigma1 est fixe, par exemple 100 MPa\n",
 " # cette fonction doit générer une matrice de 6 colonnes, chaque ligne étant le tenseur à un instant du cycle, et de la forme [sigma1*cos(omega*t),0,0,0,0,0]\n",
 " return tens\n"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "colab": {
 "base_uri": "https://localhost:8080/"
 },
 "collapsed": true,
 "id": "W-LUazbv2rzT",
 "outputId": "c0f130ef-b7f4-4225-dbfc-8b3558703bff"
 },
 "outputs": [
 {
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 ]
 },
 "execution_count": 7,
 "metadata": {},
 "output_type": "execute_result"
 }
 ],
 "source": [
 "genereTens(100,2*pi,0.01,1)"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "6Jq7YQXKzpfj"
 },
 "outputs": [],
 "source": [
 "def amplitudeTangMax(tens):\n",
 " # cette fonction doit retourner pour UN instant une liste de deux éléments : le premier élément est la valeur max_n (norme de contTang) et le deuxième les angles du plan associés\n",
 " # il faut balayer les facette !\n",
 " maxi = 0\n",
 " theta = 0\n",
 " planMax = [0,0]\n",
 " phi = 0\n",
 " pasTheta = pi/180\n",
 " pasPhi = pi/180\n",
 " vect_norm = normale(theta,phi)\n",
 " for i in range(180+1):\n",
 " theta = i*pasTheta\n",
 " for j in range(180+1):\n",
 " phi = j*pasPhi\n",
 " # on construit le vecteur normal\n",
 " vect_norm = normale(theta,phi)\n",
 " # on calcule la contrainte tangentielle\n",
 " contT = contTang(tens,vect_norm)\n",
 " # on calcule sa norme\n",
 " norme = np.linalg.norm(contT)\n",
 " # si elle est plus grande que maxi, elle devient maxi\n",
 " if norme > maxi:\n",
 " maxi = norme\n",
 " planMax = [theta,phi]\n",
 " #on actualise planMax\n",
 " # on retourne [maxi,planMax]\n",
 " return [maxi,planMax]\n",
 "\n",
 "\n"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "u60vUh2SF1QP"
 },
 "outputs": [],
 "source": []
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "w57rhdov7ajH"
 },
 "outputs": [],
 "source": [
 "def nuage(sigma1,omega,pasTemps,fin):\n",
 " \"\"\"\n",
 " Le but de la fonction est de tracer les contraintes tangentielles maximales en fonction de la \n",
 " pression hydrostatique\n",
 " \"\"\"\n",
 " points = np.array([0,0])\n",
 " tensTot = genereTens(sigma1,omega,pasTemps,fin)\n",
 " for t in range(int(fin/pasTemps)):\n",
 " tens = tensTot[t]\n",
 " cisMax ,_= amplitudeTangMax(tens)\n",
 " hydros = hydro(tens)\n",
 " ligne = np.array([cisMax,hydros])\n",
 " points = np.vstack((points, ligne))\n",
 " return points\n",
 "\n"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": null,
 "metadata": {
 "id": "7kamwWhNAcfE"
 },
 "outputs": [],
 "source": [
 "def traceNuage(points):\n",
 " plt.scatter(points[:, 0], points[:, 1])\n",
 " plt.xlabel(\"amplitude de cisaillement max\")\n",
 " plt.ylabel(\"pression hydrostatique\")\n",
 " plt.title(\"Nuage de points\")\n",
 " plt.show()"
 ]
 },
 {
 "cell_type": "code",
 "execution_count": 11,
 "metadata": {
 "colab": {
 "base_uri": "https://localhost:8080/",
 "height": 472
 },
 "id": "VYwe_MwBErb-",
 "outputId": "690674cc-88a8-446c-eb65-38e14cb7775a"
 },
 "outputs": [
 {
 "data": {
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jJSIiEi71K3b2RWZCGtLKcYH+lR4bPHaHXNgNLe6AODPP/+EXC7HixcvYG9vj/3798Pf3x9xcXGwtraGk5OTyvFubm5ITU2t8Jx5eXnIy8sTn2dnZ+sidCIiogpVVqispEndDgCxNqdkHx9jm4auLUaV+DRr1gxxcXHIysrC3r17MWrUKJw5c6Za5wwPD8fSpUu1FCEREZHmKmswWJymdTuAedTuqMuoa3yCgoLg6+uLYcOGoVevXnjy5InKqI+3tzdmzpyJWbNmlXuOskZ8PD09WeNDRER6UV6hslLJGVWa1PiYU2Jj0jU+SgqFAnl5eQgICICVlRVOnDgh7rt16xaSk5Mhl8srPIeNjY04RV75ICIi0gd1GwzmFyjE56zbqR6judUVFhaGvn37wsvLCzk5Ofjpp59w+vRpHD16FDKZDKGhoZg9ezacnZ3h6OiIadOmQS6XazSji4iISJ80KVQu3jSQdTtVZzSJT3p6OkJCQpCSkgKZTIbWrVvj6NGjeOONNwAA69atg1QqxdChQ5GXl4fg4GB89913Bo6aiIjMjbpFyoDmhcrFsW6naoy6xkcX2MeHiIiqqqwi5YpGYbadu4dlh25Uet6F/Zsb9TIR+mAWNT5EREQ1hbJIueStK2U35eJLSSiNlDcstURESVJJ0XGkHUx8iIiIqqkqRcoAC5UNgVeSiIiomjQpUi7JXBYHrSmMpriZiIiopqpOkTLAQmV9YuJDRERUjCazspSq0k25JGtLKQuY9YCJDxER0f+nydIRxY2UN8TyiBuVdlNmkbLhcQyNiIgIVZuVpcQiZePB7wAREZm9qs7KKo5FysaBt7qIiMjsVXXpiJJYpFzzMfEhIiKzV91ZWcWxSLlmY+JDREQmoyozsgDtzMoi48DEh4iITEJVZ2QBnJVlTnjTkYiIjF51ZmQBnJVlTvgdJCIio6aNGVkAZ2WZAyY+RERk1LQ5IwvgrCxTxxSWiIiMGmdkkSaY+BARkUHlFyiw7dw9LDp4FdvO3au0FqekkfKGpepySuKMLFLirS4iIjKY6kxBV1LOyNp8tvwCZ87IIiUmPkREZBDKKeglKaegA1A7+VEeVzKJkkqgURJFpk8iCEIltfDmJTs7GzKZDFlZWXB0dDR0OEREJim/QAG/hYcrbRh4c1lfjUZqqtq5mYyfup/fHPEhIiK90+YU9OI4I4sqwzSYiIj0TttT0InUxcSHiIj0jlPQyVCY+BARkd5xCjoZChMfIiLSSHX77gBcFJQMh8XNRESkNm303VHiFHQyBE5nL4HT2YmIylZe3x2lqq5gzinopA2czk5ERFqTX6DA1nPlJz1A0cjNnN5+GictnIJO+sSUmoiIKqVJ3x2imoyJDxERVYp9d8hUMPEhIqJKse8OmQomPkREVCn23SFTwcSHiMhEaaPfjhL77pCp4KwuIiITpM1+O0rsu0OmgH18SmAfHyIydrrqt6PEvjtUE7GPDxGRGdJlvx0l9t0hY8YUnYjIhLDfDlHFmPgQEZkQ9tshqhgTHyIiE8J+O0QVM5rEJzw8HK+99hocHBzg6uqKwYMH49atWyrHvHjxAlOmTEHdunVhb2+PoUOHIi0tzUARExHpH/vtEFXMaBKfM2fOYMqUKbh06RKOHz+Oly9fonfv3sjNzRWPmTVrFn799Vf88ssvOHPmDB4+fIghQ4YYMGoiosqx3w6R/hjtdPZHjx7B1dUVZ86cQdeuXZGVlQUXFxf89NNPePvttwEAN2/eRPPmzREZGYnXX39drfNyOjsR6VNZ/Xa00RdHV+clqqlMfjp7VlYWAMDZ2RkAEBMTg5cvXyIoKEg8xs/PD15eXhUmPnl5ecjLyxOfZ2dn6zBqIqL/Ka/fjkKAuL06zQbn9PZjvx2iEowy8VEoFJg5cyY6d+6Mli1bAgBSU1NhbW0NJycnlWPd3NyQmppa7rnCw8OxdOlSXYZLRFQK++0QGYZRpv5TpkzB1atXsXv37mqfKywsDFlZWeLj/v37WoiQiKhi7LdDZBhGN+IzdepU/Pbbbzh79iwaNGggbnd3d0d+fj4yMzNVRn3S0tLg7u5e7vlsbGxgY2Ojy5CJiEphvx0iwzCaER9BEDB16lTs378fJ0+ehI+P6qyFgIAAWFlZ4cSJE+K2W7duITk5GXK5XN/hEhFViP12iAzDaEZ8pkyZgp9++gkHDx6Eg4ODWLcjk8lga2sLmUyG0NBQzJ49G87OznB0dMS0adMgl8vVntFFRKQvI+UNsTziRoW3u9hvh0j7jGbEZ+PGjcjKykL37t1Rv3598bFnzx7xmHXr1mHAgAEYOnQounbtCnd3d+zbt8+AURMRlY39dogMw2j7+OgK+/gQUVnyCxQ6mRrOfjtE2qHu5zcTnxKY+BBRSbpOTnSVVBGZE5NvYEhEpA+6bDKoxH47RPrDPymIiMqhbpPB6qytRUT6xcSHi

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