update of exercise 3.4

exercises/X0304_example_ductile_bond.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "73893c57-4e0f-4f4a-b626-e9450ecc9a47",
+   "metadata": {},
+   "source": [
+    "# Notebook supporting exercise X0304\n",
+    "\n",
+    "Explain the inverse relation between ductility interface and ductility of the structure\n",
+    "Hardening bond-slip law can be regarded as ductile phenomenon. In the first branch of the constitutive law (shear stress – slip) grows until it reaches the elastic limit. After that, the shear stress either remains constant or further slowly increases with an increasing slip in the respective material point. \n",
+    "\n",
+    "Figure\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ec0fc4d4-e7f9-446b-83b3-e38c196787bb",
+   "metadata": {},
+   "source": [
+    "Explain how does the interface ductility influence the structural ductility of the pullout response or even of a tensile test. Some experts say, that by increasing the stress-strain ductility (i.e. the hardening) of an interface at the level of a material point it is possible to increase the structural ductility, i.e. to obtain larger deformation at failure. However, this is not the case. Explain why. You can use this notebook to support your explanation by the stress profiles at fiber rupture of the pullout test"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d198c000-aaba-420f-bb4e-71d4fee6e737",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%matplotlib widget\n",
+    "import matplotlib.pylab as plt\n",
+    "import numpy as np\n",
+    "from bmcs_cross_section.pullout import PullOutModel1D"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "08fbd1b1-8f48-4189-b85f-08852c870bdc",
+   "metadata": {},
+   "source": [
+    "Constract a model with the trilinear bond-slip law"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "32539418-1076-44c5-bcfd-be07823b1193",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "po_trc = PullOutModel1D(n_e_x=100, w_max=2.3) # mm \n",
+    "po_trc.geometry.L_x = 500\n",
+    "po_trc.time_line.step = 0.05\n",
+    "po_trc.cross_section.trait_set(A_m=1543, A_f=16.7, P_b=10)\n",
+    "po_trc.material_model='trilinear'\n",
+    "po_trc.material_model_.trait_set(E_m=28000, E_f=170000, tau_1=5.5, tau_2=5.5, s_1=0.07, s_2=6)\n",
+    "po_trc.material_model_.interact()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c4c01ad7-1e7f-4f9a-bc5d-2d2137e4c1fc",
+   "metadata": {},
+   "source": [
+    "Show the response for `tau_2 = 5.5` MPa"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ccb47fb4-03fb-40bb-aaa1-988660e71109",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "po_trc.material_model_.tau_2 = 5.5 # MPa\n",
+    "po_trc.run()\n",
+    "po_trc.interact()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "96da92b1-dc4b-4dcf-aef6-b157bd2327d1",
+   "metadata": {},
+   "source": [
+    "Optically determine at which displacement does the fiber break. The ultimate force is given as the product of the strength and cross-sectional area"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8425a30b-da92-4094-9d44-1b0b68e4165a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "f_trc_t = 1300 # [MPa]\n",
+    "P_fu = po_trc.cross_section.A_f * f_trc_t # [N]\n",
+    "P_fu / 1000 # [kN]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51372c58-2a6d-4ae9-8093-022df704d4fc",
+   "metadata": {},
+   "source": [
+    "Now, how does the pull-out displacement change if you increase the hardening slope to `tau_2 = 10` MPa?\n",
+    "Provide an explanation of your answer before verifying using the following code"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0f0fe225-8cf6-4846-9d92-ba1f562e2faf",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# po_trc.material_model_.tau_2 = 10.0 # MPa\n",
+    "# po_trc.run()\n",
+    "# po_trc.interact()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "40119ceb-27f9-430f-9986-9ce39d8e2a87",
+   "metadata": {},
+   "source": [
+    "Finally, you can run the evaluation in a loop to see the effect even for softening bond-slip laws"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0f206477-247a-4a09-9877-10fa9269d32d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "tau_list = [0.0, 2, 3, 5.5, 8, 12, 16]\n",
+    "Pw_list = []\n",
+    "po_trc.w_max = 2.3\n",
+    "for tau_2 in tau_list:\n",
+    "    print('evaluating pullout curve for tau_2', tau_2)\n",
+    "    po_trc.material_model_.tau_2=tau_2\n",
+    "    po_trc.reset()\n",
+    "    po_trc.run()\n",
+    "    Pw_t = po_trc.history.get_Pw_t()\n",
+    "    Pw_list.append(Pw_t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f61a5d31-f96c-4d22-8273-89fa7d58f420",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "%matplotlib widget\n",
+    "fig, (ax, ax_w_fu) = plt.subplots(1,2, figsize=(10,4), tight_layout=True)\n",
+    "fig.canvas.header_visible = False\n",
+    "w_fu_list = []\n",
+    "for Pw_t, tau, color in zip(Pw_list, tau_list, ['red','green','blue','black','orange','gray', 'yellow']):\n",
+    "    P_range, w_unloaded, w_loaded = Pw_t\n",
+    "    ax.plot(w_loaded, P_range/1000, color=color, linestyle='solid', label=r'$\\tau_2$ = %d' % tau)\n",
+    "    ax.plot(w_unloaded, P_range/1000, color=color, linestyle='dashed')\n",
+    "    arg_fu = np.argmax( P_range > P_fu )\n",
+    "    w_fu = w_loaded[arg_fu] # mm\n",
+    "    w_fu_list.append(w_fu)\n",
+    "\n",
+    "print(tau_list, w_fu_list)\n",
+    "# Plotting\n",
+    "ax.set_xlabel(r'half of crack opening $w/2$ [mm]'); ax.set_ylabel(r'$P$ [kN]')\n",
+    "ax.legend()\n",
+    "ax_w_fu.plot(tau_list, w_fu_list, 'o-', color='blue', label=r'$P_\\mathrm{max}$')\n",
+    "ax_w_fu.set_xlabel(r'residual shear (ductility) $\\tau_2$ [MPa]')\n",
+    "ax_w_fu.set_ylabel(r'failure displacement $w_\\mathrm{fu}$ [mm]')\n",
+    "ax_w_fu.set_xlim(xmin=0); ax_w_fu.set_ylim(ymin=0)\n",
+    "ax.plot([0, 2.3], [P_fu/1000, P_fu/1000], linestyle='dashed', linewidth=1, color='red', label='breaking force' );\n",
+    "# ax_w_fu.plot([0, 350], [P_fu/1000, P_fu/1000], linestyle='dashed', linewidth=1, color='red', label='breaking force' );\n",
+    "ax_w_fu.legend();"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c0385e43-2b56-4bb7-9d1f-d4cca752fef5",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "bmcs_env2",
+   "language": "python",
+   "name": "bmcs_env2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

extras/newton_method.ipynb

@@ -331,9 +331,9 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "bmcs_env2",
    "language": "python",
-   "name": "python3"
+   "name": "bmcs_env2"
   },
   "language_info": {
    "codemirror_mode": {

tour3_nonlinear_bond/3_1_nonlinear_bond.ipynb

@@ -556,6 +556,7 @@
    "outputs": [],
    "source": [
     "%matplotlib widget\n",
+    "po.material_model='trilinear'\n",
     "po.interact()"
    ]
   },

tour3_nonlinear_bond/3_2_anchorage_length.ipynb

@@ -671,6 +671,15 @@
     "</div>"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<div style=\"background-color:lightgray;text-align:left\"> <img src=\"../icons/exercise.png\" alt=\"Run\" width=\"40\" height=\"40\">\n",
+    "    &nbsp; &nbsp; <a href=\"../exercises/X0304_example_ductile_bond.ipynb\"><b>Exercise X0304:</b></a> <b>Anchorage length</b> \n",
+    "</div>"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},