.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/plasticity/plot_01_dislocations.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_01_dislocations.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_01_dislocations.py:


Taylor, Orowan and Polanyi: dislocations (1934)
===============================================

Jacob Frenkel (1926) estimated the shear stress needed to slide one plane of
a perfect crystal over the next as about :math:`G/2\pi`
(:func:`~engineeringkit.materials.ideal_shear_strength`). Single crystals of
copper and aluminium slip at stresses of order 1 MPa, thousands of times
less. In 1934 Geoffrey Taylor, Egon Orowan and Michael Polanyi independently
explained the gap with the dislocation, a line defect whose glide moves one
row of atoms at a time. The lattice resists it only with the Peierls stress
(Peierls 1940, Nabarro 1947), which falls exponentially as the slip planes
get farther apart (:func:`~engineeringkit.materials.peierls_stress`).

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import ideal_shear_strength, peierls_stress








.. GENERATED FROM PYTHON SOURCE LINES 23-30

The strength gap
----------------

Typical critical resolved shear stresses of annealed, high-purity single
crystals against Frenkel's estimate.
In a face-centred cubic metal the close-packed :math:`\{111\}` planes are
:math:`a_0/\sqrt3` apart and the Burgers vector is :math:`a_0/\sqrt2`.

.. GENERATED FROM PYTHON SOURCE LINES 30-40

.. code-block:: Python


    metals = {"copper": (48e9, 0.34, 0.361e-9, 0.65e6), "aluminium": (26e9, 0.35, 0.405e-9, 0.79e6)}
    for name, (G, nu, a0, crss) in metals.items():
        b, d = a0 / np.sqrt(2), a0 / np.sqrt(3)
        ideal = ideal_shear_strength(G, b, d)
        tau_p = peierls_stress(G, nu, b, d)
        print(f"{name:9s}: ideal {ideal / 1e9:5.2f} GPa, Peierls {tau_p / 1e6:5.1f} MPa, measured {crss / 1e6:.2f} MPa")
        print(f"           ideal / measured = {ideal / crss:,.0f}")
        assert ideal / crss > 5000 and tau_p < ideal / 100





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

 .. code-block:: none

    copper   : ideal  9.36 GPa, Peierls  61.2 MPa, measured 0.65 MPa
               ideal / measured = 14,394
    aluminium: ideal  5.07 GPa, Peierls  29.9 MPa, measured 0.79 MPa
               ideal / measured = 6,415




.. GENERATED FROM PYTHON SOURCE LINES 41-50

Wide planes slip easily
-----------------------

The Peierls-Nabarro estimate is crude, it still exceeds the measured
stress in FCC metals, whose dislocations split into partials and spread
further, but its trend is right: closely spaced planes with short Burgers
vectors resist slip far more than widely spaced close-packed ones. That is
why slip happens on close-packed planes in close-packed directions, and
why covalent crystals with narrow dislocations are brittle.

.. GENERATED FROM PYTHON SOURCE LINES 50-66

.. code-block:: Python


    G, nu = 48e9, 0.34
    ratio = np.linspace(0.5, 1.5, 101)
    tau = peierls_stress(G, nu, 1.0, ratio)
    assert np.all(np.diff(tau) < 0) and np.isclose(tau[0] / tau[-1], np.exp(2 * np.pi / (1 - nu)))

    fig, ax = plt.subplots(figsize=(6, 3.8))
    ax.semilogy(ratio, tau / G, label="Peierls-Nabarro")
    ax.axhline(1 / (2 * np.pi), color="k", ls="--", label="Frenkel, G / 2 pi")
    ax.axhline(0.65e6 / 48e9, color="C2", ls=":", label="copper, measured")
    ax.axvline(np.sqrt(2 / 3), color="0.6", lw=0.8, label="FCC {111}")
    ax.set(xlabel="plane spacing / Burgers vector", ylabel="shear stress / G", title="Why dislocations make metals soft")
    ax.legend(loc="lower left")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/plasticity/images/sphx_glr_plot_01_dislocations_001.png
   :alt: Why dislocations make metals soft
   :srcset: /api/gallery/materials/plasticity/images/sphx_glr_plot_01_dislocations_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.058 seconds)


.. _sphx_glr_download_api_gallery_materials_plasticity_plot_01_dislocations.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_01_dislocations.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_dislocations.ipynb <plot_01_dislocations.ipynb>`

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

      :download:`Download Python source code: plot_01_dislocations.py <plot_01_dislocations.py>`

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

      :download:`Download zipped: plot_01_dislocations.zip <plot_01_dislocations.zip>`


.. only:: html

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

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