.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/analysis/directional_stiffness.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_analysis_directional_stiffness.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_analysis_directional_stiffness.py:


Effective properties of a cubic material
========================================

This tutorial studies the directional stiffness of a cubic material

.. GENERATED FROM PYTHON SOURCE LINES 7-15

.. code-block:: Python


    import numpy as np
    import matplotlib.pyplot as plt
    from matplotlib import cm, colors
    from simcoon import simmit as sim
    import os









.. GENERATED FROM PYTHON SOURCE LINES 16-18

Define a grid of directions in 3D space using spherical coordinates (θ, φ)
and compute the components of the unit vector n.

.. GENERATED FROM PYTHON SOURCE LINES 18-32

.. code-block:: Python


    phi = np.linspace(0, 2 * np.pi, 128)  # azimuthal angle in the xy-plane
    theta = np.linspace(0, np.pi, 128).reshape(128, 1)  # polar angle

    n_1 = np.sin(theta) * np.cos(phi)
    n_2 = np.sin(theta) * np.sin(phi)
    n_3 = np.cos(theta) * np.ones(128)

    n = (
        np.array([n_1 * n_1, n_2 * n_2, n_3 * n_3, n_1 * n_2, n_1 * n_3, n_2 * n_3])
        .transpose(1, 2, 0)
        .reshape(128, 128, 1, 6)
    )








.. GENERATED FROM PYTHON SOURCE LINES 33-34

Use Simcoon to obtain the cubic stiffness matrix L and its inverse (compliance M).

.. GENERATED FROM PYTHON SOURCE LINES 34-42

.. code-block:: Python


    C11 = 185000.0
    C12 = 158000.0
    C44 = 39700.0

    L = sim.L_cubic([C11, C12, C44], "Cii")
    M = np.linalg.inv(L)








.. GENERATED FROM PYTHON SOURCE LINES 43-44

Compute the directional stiffness E(n) = 1 / (nᵀ·M·n) for all directions.

.. GENERATED FROM PYTHON SOURCE LINES 44-52

.. code-block:: Python


    S = (n @ M @ n.reshape(128, 128, 6, 1)).reshape(128, 128)
    E = 1.0 / S

    x = E * n_1
    y = E * n_2
    z = E * n_3








.. GENERATED FROM PYTHON SOURCE LINES 53-54

Plot the directional stiffness as a 3D surface.

.. GENERATED FROM PYTHON SOURCE LINES 54-82

.. code-block:: Python


    fig = plt.figure(figsize=plt.figaspect(1))
    ax = fig.add_subplot(111, projection="3d")

    norm = colors.Normalize(vmin=np.min(E), vmax=np.max(E), clip=False)
    surf = ax.plot_surface(
        x,
        y,
        z,
        rstride=1,
        cstride=1,
        norm=norm,
        facecolors=cm.bone(norm(E)),
        linewidth=0,
        antialiased=False,
        shade=False,
    )

    ax.set_xlabel(r"$E_x$")
    ax.set_ylabel(r"$E_y$")
    ax.set_zlabel(r"$E_z$")

    scalarmap = cm.ScalarMappable(cmap=plt.cm.bone, norm=norm)
    cbar = fig.colorbar(scalarmap, ax=ax, shrink=0.7, pad=0.15)
    cbar.set_label(r"Directional stiffness $E$", rotation=270, labelpad=20)

    plt.tight_layout()
    plt.show()



.. image-sg:: /examples/analysis/images/sphx_glr_directional_stiffness_001.png
   :alt: directional stiffness
   :srcset: /examples/analysis/images/sphx_glr_directional_stiffness_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 2.433 seconds)


.. _sphx_glr_download_examples_analysis_directional_stiffness.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: directional_stiffness.ipynb <directional_stiffness.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: directional_stiffness.py <directional_stiffness.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: directional_stiffness.zip <directional_stiffness.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_