.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/elasticity/plot_01_young_modulus.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_01_young_modulus.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_01_young_modulus.py:


Young's modulus (1807)
======================

Thomas Young's *Lectures on Natural Philosophy* (1807) gave a single number
for the stiffness of a material, its "modulus of elasticity". Young expressed
it as the height of a column of the material whose weight would compress it
to nothing if the law stayed linear; Navier (1826) restated it as the ratio
of stress to strain, :math:`E = \sigma/\varepsilon`, the form used today
(:func:`~engineeringkit.materials.hooke_stress`). The example reduces a
simulated tensile test from load and extension to stress and strain
(:func:`~engineeringkit.materials.normal_stress`,
:func:`~engineeringkit.materials.normal_strain`), fits the modulus with
:class:`~engineeringkit.materials.StressStrainCurve` and measures the energy
the bar absorbs.

.. GENERATED FROM PYTHON SOURCE LINES 19-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import StressStrainCurve, hooke_stress, normal_strain, normal_stress, resilience_modulus








.. GENERATED FROM PYTHON SOURCE LINES 25-31

From the testing machine to the stress-strain curve
---------------------------------------------------

A 10 mm steel bar with a 50 mm gauge length yields at 250 MPa and then
hardens linearly. The machine records force and extension; dividing by
the original area and length gives engineering stress and strain.

.. GENERATED FROM PYTHON SOURCE LINES 31-44

.. code-block:: Python


    E, sy, H = 200e9, 250e6, 1.2e9
    area, gauge = np.pi * 0.005**2, 0.050
    strain_in = np.concatenate((np.linspace(0, sy / E, 8), np.linspace(sy / E, 0.20, 200)[1:]))
    stress_in = np.where(strain_in <= sy / E, hooke_stress(E, strain_in), sy + H * (strain_in - sy / E))
    force, extension = stress_in * area, strain_in * gauge

    curve = StressStrainCurve(normal_strain(extension, gauge), normal_stress(force, area))
    summary = curve.summary()
    print(f"fitted E = {summary['young_modulus_Pa'] / 1e9:.1f} GPa, ultimate {summary['ultimate_strength_Pa'] / 1e6:.0f} MPa")
    print(f"toughness {summary['toughness_J_m3'] / 1e6:.1f} MJ/m^3, resilience {resilience_modulus(sy, E) / 1e3:.1f} kJ/m^3")
    assert np.isclose(summary["young_modulus_Pa"], E)





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

 .. code-block:: none

    fitted E = 200.0 GPa, ultimate 488 MPa
    toughness 73.5 MJ/m^3, resilience 156.2 kJ/m^3




.. GENERATED FROM PYTHON SOURCE LINES 45-52

Young's "height of the modulus"
-------------------------------

Young's own measure was a length, :math:`E/\rho g`. For steel it is
thousands of kilometres. The elastic part of the curve stores the
modulus of resilience :math:`\sigma_y^2/2E`, the area of the elastic
triangle, which is tiny next to the area under the whole curve.

.. GENERATED FROM PYTHON SOURCE LINES 52-68

.. code-block:: Python


    rho, g = 7850.0, 9.81
    print(f"height of the modulus of steel: {E / (rho * g) / 1e3:.0f} km")
    elastic = StressStrainCurve(curve.strain[:8], curve.stress[:8])
    assert np.isclose(elastic.toughness(), resilience_modulus(sy, E))

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9, 3.6))
    curve.plot(ax=ax1)
    ax1.set_title("tensile test")
    elastic.plot(ax=ax2)
    ax2.fill_between(elastic.strain, elastic.stress / 1e6, alpha=0.3, label="resilience")
    ax2.set_title("elastic region")
    ax2.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/elasticity/images/sphx_glr_plot_01_young_modulus_001.png
   :alt: tensile test, elastic region
   :srcset: /api/gallery/materials/elasticity/images/sphx_glr_plot_01_young_modulus_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    height of the modulus of steel: 2597 km





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

   **Total running time of the script:** (0 minutes 0.045 seconds)


.. _sphx_glr_download_api_gallery_materials_elasticity_plot_01_young_modulus.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_01_young_modulus.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_young_modulus.ipynb <plot_01_young_modulus.ipynb>`

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

      :download:`Download Python source code: plot_01_young_modulus.py <plot_01_young_modulus.py>`

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

      :download:`Download zipped: plot_01_young_modulus.zip <plot_01_young_modulus.zip>`


.. only:: html

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

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