add small exercice related with the initial stress

tpv13/tpv13.ipynb

@@ -190,9 +190,61 @@
   },
   {
    "cell_type": "markdown",
+   "id": "5291f06b-f8c9-473f-9361-eddfa993b0c1",
    "metadata": {},
    "source": [
-    "# Rupture Speed Analysis\n",
+    "# Initial stress\n",
+    "\n",
+    "The loading in tpv13 is defined as a stress tensor, which is converted into fault tractions by SeisSol.\n",
+    "In the next cell, we compute analytically the normal and shear tractions, and compute the relative prestress ratio, a key parameter for rupture dynamics, characterizing the initial stress.\n",
+    "By modifying params with the nucleation static friction (mu_s=0.54), verify that this area is critically stressed (R>1)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sympy as sp\n",
+    "import numpy as np\n",
+    "\n",
+    "# Define symbolic variables\n",
+    "s3_to_s1, sigma_max, dip, mu_s, mu_d = sp.symbols(\"s3_to_s1 sigma_max dip mu_s mu_d\")\n",
+    "\n",
+    "# stress tensor\n",
+    "stress = sp.diag((1 + s3_to_s1) * sigma_max / 2, s3_to_s1 * sigma_max, sigma_max)\n",
+    "# normal and along dip unit vectors\n",
+    "u_n = sp.Matrix([0, -sp.sin(dip), sp.cos(dip)])\n",
+    "udip = sp.Matrix([0, sp.cos(dip), sp.sin(dip)])\n",
+    "\n",
+    "# tractions\n",
+    "traction = stress * u_n\n",
+    "\n",
+    "sigma_n = traction.dot(u_n)\n",
+    "print(\"sigma_n:\", sigma_n)\n",
+    "sigma_d = traction.dot(udip)\n",
+    "print(\"sigma_d:\", sigma_d)\n",
+    "\n",
+    "R = (sigma_d - mu_d * sigma_n) / ((mu_s - mu_d) * sigma_n)\n",
+    "print(\"relative prestress ratio R:\", R)\n",
+    "\n",
+    "params = {dip: 60 * np.pi / 180, s3_to_s1: 0.3495, mu_s: 0.7, mu_d: 0.1}\n",
+    "print(\n",
+    "    f\"relative prestress ration R evaluated with {params}:\",\n",
+    "    R.subs(params),\n",
+    ")\n",
+    "\n",
+    "print(\"sigma_n:\", sigma_n.subs(params))\n",
+    "print(\"sigma_d:\", sigma_d.subs(params))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "de67024b-6854-4fa3-9fe4-13562d828801",
+   "metadata": {},
+   "source": [
+    "# Rupture speed analysis\n",
     "\n",
     "We analyze the mechanism causing supershear in this setup and disable it.\n",
     "\n",