.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/plate_tectonics/plot_04_hertz_floating_plate.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_plate_tectonics_plot_04_hertz_floating_plate.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_plate_tectonics_plot_04_hertz_floating_plate.py:


Hertz's floating elastic plate (1884)
=====================================

Heinrich Hertz solved for the bending of a thin elastic plate floating on a
fluid, the problem of a load on lake ice. The same equation describes the
lithosphere floating on the mantle: rigidity grows as the cube of the elastic
thickness and the flexural length as its three-quarter power.

.. GENERATED FROM PYTHON SOURCE LINES 10-32



.. image-sg:: /api/gallery/plate_tectonics/images/sphx_glr_plot_04_hertz_floating_plate_001.png
   :alt: Same plate equation, from ice to lithosphere
   :srcset: /api/gallery/plate_tectonics/images/sphx_glr_plot_04_hertz_floating_plate_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    flexural length of 0.5 m lake ice: 14.4 m
    flexural length of a 30 km plate: 74 km






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.constants import GRAVITY
    from earthsciencekit.plate_tectonics import flexural_parameter, flexural_rigidity

    ice = flexural_parameter(flexural_rigidity(9e9, 0.5, 0.33), 1000, GRAVITY)
    thickness = np.logspace(3, 5, 50)
    rigidity = flexural_rigidity(7e10, thickness, 0.25)
    alpha = flexural_parameter(rigidity, 3300 - 1000, GRAVITY)
    assert np.isclose(np.polyfit(np.log(thickness), np.log(alpha), 1)[0], 0.75)
    assert np.isclose(np.polyfit(np.log(thickness), np.log(rigidity), 1)[0], 3)
    print(f"flexural length of 0.5 m lake ice: {float(ice):.1f} m")
    print(f"flexural length of a 30 km plate: {float(flexural_parameter(flexural_rigidity(7e10, 30e3, 0.25), 2300, GRAVITY)) / 1e3:.0f} km")

    fig, ax = plt.subplots()
    ax.loglog(thickness / 1e3, alpha / 1e3, label="lithosphere on mantle")
    ax.set(xlabel="elastic thickness [km]", ylabel="flexural length [km]", title="Same plate equation, from ice to lithosphere")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.123 seconds)


.. _sphx_glr_download_api_gallery_plate_tectonics_plot_04_hertz_floating_plate.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../lite/lab/index.html?path=api/gallery/plate_tectonics/plot_04_hertz_floating_plate.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_hertz_floating_plate.ipynb <plot_04_hertz_floating_plate.ipynb>`

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

      :download:`Download Python source code: plot_04_hertz_floating_plate.py <plot_04_hertz_floating_plate.py>`

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

      :download:`Download zipped: plot_04_hertz_floating_plate.zip <plot_04_hertz_floating_plate.zip>`


.. only:: html

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

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