.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/elasticity/plot_02_duhamel_thermal_stress.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_materials_elasticity_plot_02_duhamel_thermal_stress.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_materials_elasticity_plot_02_duhamel_thermal_stress.py:


Duhamel's thermal stress (1838)
===============================

Jean-Marie Duhamel (1838) added temperature to Navier's equations of
elasticity: a change :math:`\Delta T` produces a free strain
:math:`\alpha\,\Delta T` (:func:`~engineeringkit.materials.thermal_strain`)
that causes no stress if the body can expand, and a stress
:math:`-E\alpha\,\Delta T` (:func:`~engineeringkit.materials.constrained_thermal_stress`)
if it is held. The stress does not depend on the length of the bar, which is
why continuously welded rail must be laid at a chosen neutral temperature.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import constrained_thermal_stress, thermal_strain








.. GENERATED FROM PYTHON SOURCE LINES 21-27

Continuously welded rail
------------------------

Rail steel has :math:`E = 210` GPa and :math:`\alpha = 11.5\times10^{-6}`/K.
Laid stress-free at 25 °C, it is in compression on hot days and in
tension in winter. The total strain, free plus elastic, stays zero.

.. GENERATED FROM PYTHON SOURCE LINES 27-34

.. code-block:: Python


    E, alpha, neutral = 210e9, 11.5e-6, 25.0
    T = np.linspace(-20, 60, 81)
    stress = constrained_thermal_stress(E, alpha, T - neutral)
    print(f"at 60 C: {stress[-1] / 1e6:.1f} MPa; at -20 C: {stress[0] / 1e6:.1f} MPa")
    assert np.allclose(thermal_strain(alpha, T - neutral) + stress / E, 0)





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

 .. code-block:: none

    at 60 C: -84.5 MPa; at -20 C: 108.7 MPa




.. GENERATED FROM PYTHON SOURCE LINES 35-41

Jointed rail: a gap delays the stress
-------------------------------------

Bolted 18 m rails leave a gap of a few millimetres at each joint. The rail
expands freely until the gap closes, and only the excess strain is
restrained.

.. GENERATED FROM PYTHON SOURCE LINES 41-58

.. code-block:: Python


    length = 18.0
    fig, ax = plt.subplots(figsize=(6, 3.8))
    ax.plot(T, stress / 1e6, label="welded (no gap)")
    for gap in (0.003, 0.006):
        excess = np.maximum(thermal_strain(alpha, T - neutral) - gap / length, 0)
        jointed = -E * excess
        closes = neutral + gap / (alpha * length)
        ax.plot(T, jointed / 1e6, label=f"{1e3 * gap:.0f} mm gap, closes at {closes:.0f} C")
        assert np.all(jointed[T <= closes] == 0)
        assert np.allclose(jointed[T > closes], constrained_thermal_stress(E, alpha, T[T > closes] - closes))
    ax.axhline(0, color="0.6", lw=0.8)
    ax.set(xlabel="rail temperature [C]", ylabel="axial stress [MPa]", title="Thermal stress in rail")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/elasticity/images/sphx_glr_plot_02_duhamel_thermal_stress_001.png
   :alt: Thermal stress in rail
   :srcset: /api/gallery/materials/elasticity/images/sphx_glr_plot_02_duhamel_thermal_stress_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.034 seconds)


.. _sphx_glr_download_api_gallery_materials_elasticity_plot_02_duhamel_thermal_stress.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/materials/elasticity/plot_02_duhamel_thermal_stress.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_duhamel_thermal_stress.ipynb <plot_02_duhamel_thermal_stress.ipynb>`

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

      :download:`Download Python source code: plot_02_duhamel_thermal_stress.py <plot_02_duhamel_thermal_stress.py>`

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

      :download:`Download zipped: plot_02_duhamel_thermal_stress.zip <plot_02_duhamel_thermal_stress.zip>`


.. only:: html

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

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