.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/plasticity/plot_02_ramberg_osgood.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_plasticity_plot_02_ramberg_osgood.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_plasticity_plot_02_ramberg_osgood.py:


Ramberg and Osgood's stress-strain curve (1943)
===============================================

Aluminium alloys have no sharp yield point: their stress-strain curves bend
over smoothly, and aircraft designers of the Second World War needed them
in a form they could compute with. Walter Ramberg and William Osgood (NACA,
1943) described the whole curve with three numbers,
:math:`\varepsilon = \sigma/E + K(\sigma/E)^n`, usually written today with
the 0.2% offset yield strength,
:math:`\varepsilon = \sigma/E + 0.002(\sigma/\sigma_y)^n`
(:func:`~engineeringkit.materials.ramberg_osgood_strain`). The example
plots the family and recovers the offset yield strength from a sampled
curve.

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import StressStrainCurve, ramberg_osgood_strain








.. GENERATED FROM PYTHON SOURCE LINES 24-28

Ramberg and Osgood's form and the offset form
---------------------------------------------

The two forms are the same curve with :math:`K = 0.002(E/\sigma_y)^n`.

.. GENERATED FROM PYTHON SOURCE LINES 28-35

.. code-block:: Python


    E, sy = 72e9, 345e6
    stress = np.linspace(0, 1.15 * sy, 400)
    n = 15
    K = 0.002 * (E / sy) ** n
    assert np.allclose(ramberg_osgood_strain(stress, E, sy, n), stress / E + K * (stress / E) ** n)








.. GENERATED FROM PYTHON SOURCE LINES 36-43

The 0.2% offset yield strength
------------------------------

Draw the elastic line shifted by 0.002 strain; it crosses the curve at
:math:`\sigma_y`. :meth:`~engineeringkit.materials.StressStrainCurve.offset_yield_strength`
does this on sampled data. Smaller :math:`n` means more gradual yielding
and more hardening after it.

.. GENERATED FROM PYTHON SOURCE LINES 43-59

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 3.8))
    for exponent in (5, 10, 20, 50):
        curve = StressStrainCurve(ramberg_osgood_strain(stress, E, sy, exponent), stress)
        found = curve.offset_yield_strength()
        print(f"n = {exponent:2d}: offset yield {found / 1e6:.1f} MPa")
        assert np.isclose(found, sy, rtol=5e-3)
        curve.plot(ax=ax)
        ax.lines[-1].set_label(f"n = {exponent}")
    offset = np.linspace(0.002, 0.002 + 1.15 * sy / E, 2)
    ax.plot(offset, E * (offset - 0.002) / 1e6, "k:", lw=1, label="0.2% offset line")
    ax.set(xlim=(0, 0.012), title="Ramberg-Osgood curves, 2024-T3 aluminium")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/plasticity/images/sphx_glr_plot_02_ramberg_osgood_001.png
   :alt: Ramberg-Osgood curves, 2024-T3 aluminium
   :srcset: /api/gallery/materials/plasticity/images/sphx_glr_plot_02_ramberg_osgood_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    n =  5: offset yield 345.0 MPa
    n = 10: offset yield 345.0 MPa
    n = 20: offset yield 345.0 MPa
    n = 50: offset yield 345.0 MPa





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

   **Total running time of the script:** (0 minutes 0.038 seconds)


.. _sphx_glr_download_api_gallery_materials_plasticity_plot_02_ramberg_osgood.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/plasticity/plot_02_ramberg_osgood.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_ramberg_osgood.ipynb <plot_02_ramberg_osgood.ipynb>`

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

      :download:`Download Python source code: plot_02_ramberg_osgood.py <plot_02_ramberg_osgood.py>`

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

      :download:`Download zipped: plot_02_ramberg_osgood.zip <plot_02_ramberg_osgood.zip>`


.. only:: html

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

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