J3_derivative_check_v2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Checking \n",
    "\n",
    "From wikipedia and from Kayenta\n",
    "$$\\frac{\\partial J_{3} }{\\partial \\sigma} = s^{2} - \\frac{2}{3} J_{2} 1$$\n",
    "\n",
    "I want to see if the values come out the same if I calcualte the derivatives of the determinant "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Varaibles that will be operated on\n",
    "# p: Mean stress\n",
    "# t_ij: Components of the stress tensor\n",
    "t11, t22, t33, t12, t13, t23, p= sp.symbols('t11 t22 t33 t12 t13 t23 p') \n",
    "\n",
    "s, J2, I = sp.symbols('s, J2, I')\n",
    "\n",
    "I = sp.Matrix([[1, 0, 0],\n",
    "               [0, 1, 0],\n",
    "               [0, 0, 1]]\n",
    "              )\n",
    "\n",
    "p = 1/3* (t11 + t22 + t33)\n",
    "\n",
    "# J3: Stress invariant (det (s) )\n",
    "J3=  (t11 - p) * (t22 - p) * (t33 - p) - (t11 - p) * t23**2 - (t22 - p) * t13**2 - (t33 - p) * t12**2 + 2 * t12 * t13 * t23\n",
    "\n",
    "stress_list = [t11, t22, t33, t12, t13, t23]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- t_{12}^{2} \\left(- \\frac{t_{11}}{3} - \\frac{t_{22}}{3} + \\frac{2 t_{33}}{3}\\right) + 2 t_{12} t_{13} t_{23} - t_{13}^{2} \\left(- \\frac{t_{11}}{3} + \\frac{2 t_{22}}{3} - \\frac{t_{33}}{3}\\right) - t_{23}^{2} \\left(\\frac{2 t_{11}}{3} - \\frac{t_{22}}{3} - \\frac{t_{33}}{3}\\right) + \\left(- \\frac{t_{11}}{3} - \\frac{t_{22}}{3} + \\frac{2 t_{33}}{3}\\right) \\left(- \\frac{t_{11}}{3} + \\frac{2 t_{22}}{3} - \\frac{t_{33}}{3}\\right) \\left(\\frac{2 t_{11}}{3} - \\frac{t_{22}}{3} - \\frac{t_{33}}{3}\\right)\n"
     ]
    }
   ],
   "source": [
    "print(sp.latex(sp.nsimplify(J3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$- t_{12}^{2} \\left(- \\frac{t_{11}}{3} - \\frac{t_{22}}{3} + \\frac{2 t_{33}}{3}\\right) + 2 t_{12} t_{13} t_{23} - t_{13}^{2} \\left(- \\frac{t_{11}}{3} + \\frac{2 t_{22}}{3} - \\frac{t_{33}}{3}\\right) - t_{23}^{2} \\left(\\frac{2 t_{11}}{3} - \\frac{t_{22}}{3} - \\frac{t_{33}}{3}\\right) + \\left(- \\frac{t_{11}}{3} - \\frac{t_{22}}{3} + \\frac{2 t_{33}}{3}\\right) \\left(- \\frac{t_{11}}{3} + \\frac{2 t_{22}}{3} - \\frac{t_{33}}{3}\\right) \\left(\\frac{2 t_{11}}{3} - \\frac{t_{22}}{3} - \\frac{t_{33}}{3}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Derivative wrt. t11\n",
      "\\frac{2 t_{11}^{2}}{9} - \\frac{2 t_{11} t_{22}}{9} - \\frac{2 t_{11} t_{33}}{9} + \\frac{t_{12}^{2}}{3} + \\frac{t_{13}^{2}}{3} - \\frac{t_{22}^{2}}{9} + \\frac{4 t_{22} t_{33}}{9} - \\frac{2 t_{23}^{2}}{3} - \\frac{t_{33}^{2}}{9}\n",
      "-----------------------------------------------------\n",
      "Derivative wrt. t22\n",
      "- \\frac{t_{11}^{2}}{9} - \\frac{2 t_{11} t_{22}}{9} + \\frac{4 t_{11} t_{33}}{9} + \\frac{t_{12}^{2}}{3} - \\frac{2 t_{13}^{2}}{3} + \\frac{2 t_{22}^{2}}{9} - \\frac{2 t_{22} t_{33}}{9} + \\frac{t_{23}^{2}}{3} - \\frac{t_{33}^{2}}{9}\n",
      "-----------------------------------------------------\n",
      "Derivative wrt. t33\n",
      "- \\frac{t_{11}^{2}}{9} + \\frac{4 t_{11} t_{22}}{9} - \\frac{2 t_{11} t_{33}}{9} - \\frac{2 t_{12}^{2}}{3} + \\frac{t_{13}^{2}}{3} - \\frac{t_{22}^{2}}{9} - \\frac{2 t_{22} t_{33}}{9} + \\frac{t_{23}^{2}}{3} + \\frac{2 t_{33}^{2}}{9}\n",
      "-----------------------------------------------------\n",
      "Derivative wrt. t12\n",
      "\\frac{2 t_{11} t_{12}}{3} + \\frac{2 t_{12} t_{22}}{3} - \\frac{4 t_{12} t_{33}}{3} + 2 t_{13} t_{23}\n",
      "-----------------------------------------------------\n",
      "Derivative wrt. t13\n",
      "\\frac{2 t_{11} t_{13}}{3} + 2 t_{12} t_{23} - \\frac{4 t_{13} t_{22}}{3} + \\frac{2 t_{13} t_{33}}{3}\n",
      "-----------------------------------------------------\n",
      "Derivative wrt. t23\n",
      "- \\frac{4 t_{11} t_{23}}{3} + 2 t_{12} t_{13} + \\frac{2 t_{22} t_{23}}{3} + \\frac{2 t_{23} t_{33}}{3}\n",
      "-----------------------------------------------------\n"
     ]
    }
   ],
   "source": [
    "Latex = True\n",
    "\n",
    "# Loop over the stress terms and calculate the derivatives\n",
    "for stress in stress_list:\n",
    "    # Calc the derivative\n",
    "    deriv = J3.diff(stress, evaluate = True)\n",
    "    deriv_simp = sp.nsimplify(deriv)\n",
    "    \n",
    "    # Print the result with a label and a newline\n",
    "    print(f\"Derivative wrt. {stress}\")\n",
    "\n",
    "    # Expand the terms and combine like terms\n",
    "    deriv_simp = sp.expand(deriv_simp)\n",
    "\n",
    "    # Switch for if the output should be in latex format\n",
    "    if Latex:\n",
    "        deriv_simp = sp.latex(deriv_simp)\n",
    "        deriv_simp\n",
    "        print(deriv_simp)\n",
    "    else:\n",
    "        sp.pprint(deriv_simp)\n",
    "    print(\"-----------------------------------------------------\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = sp.Matrix([[t11 - p, t12, t13],\n",
    "               [t12, t22 - p, t23],\n",
    "               [t13, t23, t33-p]]\n",
    "            )\n",
    "\n",
    "# J2 Stress invariant\n",
    "J2 = 1/2 * ((t11-p)**2 + (t22-p)**2 + (t33-p)**2 + 2*(t12**2 + t13**2 + t23**2))\n",
    "\n",
    "imech_ans = s *s - 2/3 * J2 * I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{2 t_{11}}{3} - \\frac{t_{22}}{3} - \\frac{t_{33}}{3} & t_{12} & t_{13}\\\\t_{12} & - \\frac{t_{11}}{3} + \\frac{2 t_{22}}{3} - \\frac{t_{33}}{3} & t_{23}\\\\t_{13} & t_{23} & - \\frac{t_{11}}{3} - \\frac{t_{22}}{3} + \\frac{2 t_{33}}{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2⋅t₁₁   t₂₂   t₃₃                                          ⎤\n",
       "⎢───── - ─── - ───          t₁₂                  t₁₃        ⎥\n",
       "⎢  3      3     3                                           ⎥\n",
       "⎢                                                           ⎥\n",
       "⎢                     t₁₁   2⋅t₂₂   t₃₃                     ⎥\n",
       "⎢       t₁₂         - ─── + ───── - ───          t₂₃        ⎥\n",
       "⎢                      3      3      3                      ⎥\n",
       "⎢                                                           ⎥\n",
       "⎢                                          t₁₁   t₂₂   2⋅t₃₃⎥\n",
       "⎢       t₁₃                 t₂₃          - ─── - ─── + ─────⎥\n",
       "⎣                                           3     3      3  ⎦"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.nsimplify(s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "imech_ans_simp =sp.nsimplify(imech_ans)\n",
    "imech_ans_simp = sp.expand(imech_ans_simp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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UNkAgKCDuve74YOewlSXLfz6dGMZGmt/W6W50GwKcKVx5y0RkQjLK9d8k8eORXOq7+juoq7Pe8fkKS+3bx8o0TDma8uG7Ose64+PhJFuZPVOFRpj7LfixrH/YDpRpmDmyfeTTTMOGmTzlrZAhqI0mEc3Bj28Q9Njc3fpnUbrFp/Yt27dyJHMUidFt3fFhzmUrsztVUIp7p9zvVPGvLfgqtruU4MNwrmHKW7mSV65lrn9yo5jGhTNXrMtsWFt2PeNT+5YtUzmSOQrF6Lnu+DBnsZWlG9Xt/ik2wHTv0Ai3PrLdI1nxFxwpbyvyO05auR6z4T8HR5xd5lt/1rFx6ekPlbvFp/ZtTTbsK0dhbiRJ722Diz+HrfwFKJwqRlT3m5MAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle \\frac{2 t_{11}^{2}}{9} - \\frac{2 t_{11} t_{22}}{9} - \\frac{2 t_{11} t_{33}}{9} + \\frac{t_{12}^{2}}{3} + \\frac{t_{13}^{2}}{3} - \\frac{t_{22}^{2}}{9} + \\frac{4 t_{22} t_{33}}{9} - \\frac{2 t_{23}^{2}}{3} - \\frac{t_{33}^{2}}{9}$"
      ],
      "text/plain": [
       "     2                              2      2      2                    2      2\n",
       "2⋅t₁₁    2⋅t₁₁⋅t₂₂   2⋅t₁₁⋅t₃₃   t₁₂    t₁₃    t₂₂    4⋅t₂₂⋅t₃₃   2⋅t₂₃    t₃₃ \n",
       "────── - ───────── - ───────── + ──── + ──── - ──── + ───────── - ────── - ────\n",
       "  9          9           9        3      3      9         9         3       9  "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imech_ans_simp[0,0]"
   ]
  },
  {
   "attachments": {
    "image.png": {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{t_{11} t_{12}}{3} + \\frac{t_{12} t_{22}}{3} - \\frac{2 t_{12} t_{33}}{3} + t_{13} t_{23}$"
      ],
      "text/plain": [
       "t₁₁⋅t₁₂   t₁₂⋅t₂₂   2⋅t₁₂⋅t₃₃          \n",
       "─────── + ─────── - ───────── + t₁₃⋅t₂₃\n",
       "   3         3          3              "
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imech_ans_simp[0,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASoAAAArCAYAAADPEcU+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJdUlEQVR4Ae1d7ZHVNhTdZbYAIB1AB7BUwNJBIBUEOkiGX7v/dqAD0kECHUAqCNABpALIdkDO8Uoe2U/Slf10Zfvt1YxXsj7ux5HetXQta49//PhxZMEQWAsCFxcX9yDL706eU8TfeY/8zy7PohuIwHFoqDAYbgODT7ieIP1VC49WfLTkz9FtoVsLHjkdtcqgF43UG8RPPA+kXyH9Gy6OyQ8+nzHu1cdrCx6hTpaOI3BrlH2Gew4WPsU0Qys+mjqkaLfQrQWPlH6a+TRKL0IGMBScXV3hehvmu3QLHFrwiKhmWSECY0PFJ9lXDA4ODM3Qio+mDinaLXRrwSOln2Y+jcIXN4sJ+XAmdRv5fIiGoQUOLXiEOlk6gsDYUHGgvIvUq53Vik9tuUvotdCtBY8SXWvXoUHKPSi51AtDCxxa8Ah1snQEgePz83N2BKfbHARMc7BwRvUPnmCvEVcJoNWETxVhJxJpoVsLHhPVblYdun8Bs3uIj1vg0IJHM/AOhFHvTEfn/Ayd3nIwaOrWio+mDinaLXRrwSOl3xL50PcB+PIFD9/89Q/OFji04LEEplvkeRIIzbV49BUwOoyzrWe4XiD9MGjTJaXyUf1ZfByPl46W91X8inzO/tYSWugW5SHhI5WvBcCIHHSiv4P8vZFydaI4sMzpGh2vE3GI8gANjj+uEPjSielHuC6R3/1+pHLUtTARgdBQEfgd/xRA5xONZQw0WIMglQ8qX9/M4oOmr8CrfyOE9Bvk8Ul7/5rsKv620C3KA9pL+EjlqwAwFML1MX1WT8N8l47igLrZ8Yq2U3CI8iANXJTrD8qCmNsn/sZ1h/cIUvl1LftbjEDnTAfQNEB8Mrwft0TZZ1x8mkX3VUnlIb19+IDOc7TnwPGBg4F+Cw7MxUML3XI8AICEj1S+OIahAND1Oe7vIuasZhByOKAsO15BqAiHHA/QuMT1ZyDUT0iHW3qk8qCpJUsQ6AwVKnYGAJ3Tb6hzHVVCY0qdffhwNvVxCrPGdVvoluMh4SOVN4YrzQ5jj/7S+4j7mRTSfCjxYcqQw+G6RvpvKQ5JHpCDxjB0k1BePji7IJX7ehaXI3Diqg7W4gCawHMGFXZGOdV0zdl8IFM3zQ5Ic8Bx+l1bxoDFpGQL3ZI8JHyk8kmaKlaGnJwhP0LsP6Px3Dgm/RhI4uArp+IJOIg8QItLvl9wcTe9l61nLZX3FS0hIuBnVHdRs1vaAdxuGYhYwwBU4QPZOJg5cHcc+6LGehVa6FbEQ8JHKteDKE8ZcnHGROc5N3fyx99fyOOLHP/ipAiHPLfOt5QbRyIPyEOXyGNc/LyHRmsQpPJBZbvJIuANFZ9eHBz0CzxzAGcbzizcmw9k42DmNPsh0n7gzhSnarMWuok8JHyk8qqITCdGHyn7l+NwfIV9LeIgsS7AoYiHG4OsSyc9Dd8gSOWDynaTRKBb+gFMzqY41VUN+/JBew5i7qfpZHX3R4ijjn5VZUbEnQyzMXS6ZHWTeEg0pPKRSs1vIV/RG1wJB0nwEhxSPJDPFce/uB4j7Vcd3oieunbJcrTzbSQxrTxAwPuogqx1JjEAaKS4JYE/Zv/kop+KT7NNhxq6STSk8k0DOEH4fXFA+ytc38EyfDjS8U5j9ZdUPkFUtaqQkcZW9ZSU2jxEQwWGNBA0COwMvnnhsusb4m4DnlSOukWhgA6B9QD3NNGOsu0VQMPrtuMQ3Yuwa9xINwkfqbyGqovTaIQ130a+BK9vTmFu+AxdEVL5bJwqjdVuvEMIGlytUJVH/wmNlrRboIvOp2OeRni8+3kL4puMNwiBGmMVNLgyOUNctNSeA29tHrfmCGFtDAFDYNMIcLaz8xVKZY2q8jipLJyRMwQMgRUigBkODQfdJHSf0J3zAHncClLtlBRNHmao0FMWDIFDRwBGhF+dfEBMNweXfbPfUKew0uRhS78U6pZvCBwmAjRQ2lskqvO4UTMqWPzOiRgZf9yFfITy2BtEftfVf3MWaXtjsoADlw08JYBxaXiKdto/jFJZNlMPmGmNVS4Bo/4p17+p43G4XGRbvilkmm86+6NtkA5DlIej/9JVJA2G/qgmlCd5nKDwYP5fFnTJHvqH8pghOkL+Xm/90P5GYAg9rzCwVD5bOiQMu59fwR/onByvKKs+VkHT+6dip6RwbyINDEPsQcRtSbmjbbqGOR6oIB2xk+RBQ5UEq+Nsf0QEDEMRIrGCYShCVKNCZ4iA9eCUFNxzEytnvVw98KEdC5ejzPHRNr44yQMVnoM+TxH2/GmYmEfHPvkneZx46hYbAobAwSMw8B3BONAocYe9uDR3hiQEiG1paMYhx4OzxORRTTkeZqjGMNu9IXC4CNAX2336A6PQLQMRR/1VKQhQP3u0DdoleaDt+MsPGq6do5piPG6lBLJ8Q8AQODgEik6EyGkNI5I92gZti3iADn1inJXt+DxjPGxGlesVKzMEDggBGADOprg02yuADn1aNEifEHNvVr90RFrkgTp8u8dlY/h95EAm1BnwUDNUThhO7Rj82wZ6/b0j7bpkHX+vIAavVYWNYbgq7HLCbBzX5mMVePH3W+XoGof9znFGoP89x0PFUDnFKIw3VEdIc5r3HjH31UxaF6OdaoA8qzOekImDYzMYqnZQReJbx3WJsQqe2aNtSrsHdDiT4v6wnaOaJB4qhgqCdKczgjkNkzdK3hhww5fPQ9JCAgHDMAHMntmGawRAZ0Q4sTjDtXOcE/JqHF0jHTWU5KFlqLhmHUxRncVEtoVCBAzDQqAmVjNcI4Dh90nfUvIQSpQTN16zA2j4/3sYpZHjoWKowJCzp4FQyOPSj4FTPwsCAoahANDMYsN1JnALN2uyPQGDg9NJevn5n0TGeykWhmAb7A1DnX4yXHVwrU1VZUblhcQg8N8P8QNGThs/+jKLyxAwDMtwmlrLcJ2K2LL1mx1FjIHBpR8P6lrdW79lu6Ccu2FYjtWUmobrFLSWqdvMUFE9DIj/EPG1+x2k6Wy3MBEBw3AiYIXVDddCoBaqpuKjQqfza2gu+8bBL/3os7KQQcAwzICzR5Hhugd4CzZVMVTQh/sluL2esycL8xAwDOfhJrUyXCWEVliu5Uznso7fAI2Xd6cOA7/5c4WQrEYkw1CnKwxXHVxVqWoZqp2NYzBadKbfxsUtCmMDpqrkRokbhjodZ7jq4KpKVc2ZDmNEPxS3xPvA73zW+lGyl3FVsWGo0x2Gqw6umlT/BwNehBmxk1ZrAAAAAElFTkSuQmCC",
      "text/latex": [
       "$\\displaystyle \\frac{t_{11} t_{12}}{3} + \\frac{t_{12} t_{22}}{3} - \\frac{2 t_{12} t_{33}}{3} + t_{13} t_{23}$"
      ],
      "text/plain": [
       "t₁₁⋅t₁₂   t₁₂⋅t₂₂   2⋅t₁₂⋅t₃₃          \n",
       "─────── + ─────── - ───────── + t₁₃⋅t₂₃\n",
       "   3         3          3              "
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imech_ans_simp[1, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{2 t_{11} t_{12}}{3} + \\frac{2 t_{12} t_{22}}{3} - \\frac{4 t_{12} t_{33}}{3} + 2 t_{13} t_{23}$"
      ],
      "text/plain": [
       "2⋅t₁₁⋅t₁₂   2⋅t₁₂⋅t₂₂   4⋅t₁₂⋅t₃₃            \n",
       "───────── + ───────── - ───────── + 2⋅t₁₃⋅t₂₃\n",
       "    3           3           3                "
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imech_ans_simp[0,1] + imech_ans_simp[1, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "VT_Research",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}