.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/composites/plot_02_halpin_tsai.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_composites_plot_02_halpin_tsai.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_composites_plot_02_halpin_tsai.py:


The Halpin-Tsai equations (1969)
================================

The Voigt and Reuss bounds are far apart for fibre composites, and the
exact elasticity solutions for fibre arrays were too complicated for
design. John Halpin and Stephen Tsai, at the US Air Force Materials
Laboratory, fitted those solutions with one interpolating formula,

.. math:: \frac{P}{P_m} = \frac{1 + \xi\eta V_f}{1 - \eta V_f}, \qquad
          \eta = \frac{P_f/P_m - 1}{P_f/P_m + \xi}

(:func:`~engineeringkit.materials.halpin_tsai`), in which a single
parameter :math:`\xi` measures how well the reinforcement carries load:
:math:`\xi = 0` is the Reuss bound, :math:`\xi \to \infty` the Voigt bound,
and :math:`\xi = 2` fits the transverse modulus of round fibres. The
example estimates the transverse and shear moduli of glass-epoxy and the
stiffness of short fibres.

.. GENERATED FROM PYTHON SOURCE LINES 22-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import halpin_tsai, rule_of_mixtures_E_longitudinal, rule_of_mixtures_E_transverse








.. GENERATED FROM PYTHON SOURCE LINES 28-35

Transverse and shear moduli
---------------------------

For E-glass (:math:`E_f = 72` GPa, :math:`G_f = 30` GPa) in epoxy
(:math:`E_m = 3.5` GPa, :math:`G_m = 1.3` GPa) at :math:`V_f = 0.6`,
Halpin-Tsai gives about 15 GPa across the fibres, well above the
Reuss value of 8 GPa that springs in series predict.

.. GENERATED FROM PYTHON SOURCE LINES 35-44

.. code-block:: Python


    Ef, Em, Gf, Gm, Vf = 72e9, 3.5e9, 30e9, 1.3e9, 0.6
    E2 = halpin_tsai(Ef, Em, Vf, 2.0)
    G12 = halpin_tsai(Gf, Gm, Vf, 1.0)
    reuss = rule_of_mixtures_E_transverse(Ef, Em, Vf)
    print(f"E2 = {E2 / 1e9:.1f} GPa (Reuss {reuss / 1e9:.1f}), G12 = {G12 / 1e9:.2f} GPa")
    assert reuss < E2 < rule_of_mixtures_E_longitudinal(Ef, Em, Vf)
    assert np.isclose(halpin_tsai(Ef, Em, Vf, 0.0), reuss)





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

 .. code-block:: none

    E2 = 14.9 GPa (Reuss 8.2), G12 = 4.48 GPa




.. GENERATED FROM PYTHON SOURCE LINES 45-51

One parameter between the bounds
--------------------------------

For aligned short fibres of aspect ratio :math:`\ell/d`, the longitudinal
modulus uses :math:`\xi = 2\ell/d`: short stubby fibres sit near the lower
bound, and long ones approach the Voigt rule of mixtures.

.. GENERATED FROM PYTHON SOURCE LINES 51-65

.. code-block:: Python


    fractions = np.linspace(0, 0.7, 100)
    fig, ax = plt.subplots(figsize=(6, 3.8))
    ax.plot(fractions, rule_of_mixtures_E_longitudinal(Ef, Em, fractions) / 1e9, "k--", label="Voigt")
    for aspect in (1, 10, 100):
        ax.plot(fractions, halpin_tsai(Ef, Em, fractions, 2 * aspect) / 1e9, label=f"l/d = {aspect}")
    ax.plot(fractions, rule_of_mixtures_E_transverse(Ef, Em, fractions) / 1e9, "k:", label="Reuss")
    ax.set(xlabel="fibre volume fraction", ylabel="modulus [GPa]", title="Halpin-Tsai for glass-epoxy")
    ax.legend()
    fig.tight_layout()
    short, long_ = halpin_tsai(Ef, Em, 0.5, 2.0), halpin_tsai(Ef, Em, 0.5, 200.0)
    assert short < long_ < rule_of_mixtures_E_longitudinal(Ef, Em, 0.5)

    plt.show()



.. image-sg:: /api/gallery/materials/composites/images/sphx_glr_plot_02_halpin_tsai_001.png
   :alt: Halpin-Tsai for glass-epoxy
   :srcset: /api/gallery/materials/composites/images/sphx_glr_plot_02_halpin_tsai_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.114 seconds)


.. _sphx_glr_download_api_gallery_materials_composites_plot_02_halpin_tsai.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/composites/plot_02_halpin_tsai.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_halpin_tsai.ipynb <plot_02_halpin_tsai.ipynb>`

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

      :download:`Download Python source code: plot_02_halpin_tsai.py <plot_02_halpin_tsai.py>`

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

      :download:`Download zipped: plot_02_halpin_tsai.zip <plot_02_halpin_tsai.zip>`


.. only:: html

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

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