.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/hardness/plot_01_brinell_hardness.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_hardness_plot_01_brinell_hardness.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_hardness_plot_01_brinell_hardness.py:


Brinell's hardness test (1900)
==============================

Johan August Brinell, chief engineer of the Fagersta ironworks in Sweden,
showed his ball test at the 1900 Paris Exposition. A 10 mm hardened steel
ball pressed into the part with 3000 kgf leaves a dent of diameter
:math:`d`; the load divided by the curved area of the dent is the
hardness number (:func:`~engineeringkit.materials.brinell_hardness`),

.. math:: \mathrm{HB} = \frac{2P}{\pi D\left(D - \sqrt{D^2 - d^2}\right)}.

Indentation is plastic flow under a constrained punch, so for steels HB
tracks the tensile strength, :math:`S_{ut} \approx 3.45\,\mathrm{HB}` MPa
(:func:`~engineeringkit.materials.brinell_tensile_strength`): a test that
leaves only a small mark estimates the strength of a finished part.

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import brinell_hardness, brinell_tensile_strength, vickers_hardness








.. GENERATED FROM PYTHON SOURCE LINES 26-33

From the dent to the number
---------------------------

A 3000 kgf load and a dent 4.0 mm across give about 229 HB, the hardness
of a normalised medium-carbon steel. The projected-area pressure,
:math:`4P/\pi d^2`, is a few per cent higher than HB because the
curved cap has more area than its shadow.

.. GENERATED FROM PYTHON SOURCE LINES 33-40

.. code-block:: Python


    P, D = 3000 * 9.80665, 10e-3
    HB = brinell_hardness(P, D, 4.0e-3)
    mean_pressure = 4 * (P / 9.80665) / (np.pi * 4.0**2)
    print(f"HB = {HB:.0f}, projected mean pressure {mean_pressure:.0f} kgf/mm^2, Sut ~ {brinell_tensile_strength(HB) / 1e6:.0f} MPa")
    assert 1 < mean_pressure / HB < 1.05





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

 .. code-block:: none

    HB = 229, projected mean pressure 239 kgf/mm^2, Sut ~ 789 MPa




.. GENERATED FROM PYTHON SOURCE LINES 41-48

Hardness against strength
-------------------------

Dent diameters in Brinell's valid range, :math:`0.24D` to :math:`0.6D`,
cover soft to hardened steels. The right-hand axis reads the strength
estimate. A Vickers diamond, whose dents are geometrically similar at any
load, gives a number close to HB on the same steel below about 400 HB.

.. GENERATED FROM PYTHON SOURCE LINES 48-65

.. code-block:: Python


    d = np.linspace(2.4e-3, 6.0e-3, 50)
    hb = brinell_hardness(P, D, d)
    sut = brinell_tensile_strength(hb)
    print(f"range {hb[-1]:.0f} to {hb[0]:.0f} HB, Sut {sut[-1] / 1e6:.0f} to {sut[0] / 1e6:.0f} MPa")
    hv = vickers_hardness(30 * 9.80665, 0.50e-3)
    print(f"Vickers, 30 kgf, 0.50 mm diagonal: {hv:.0f} HV")
    assert np.all(np.diff(hb) < 0) and np.allclose(sut / hb, 3.45e6)

    fig, ax = plt.subplots(figsize=(6, 3.8))
    ax.plot(1e3 * d, hb)
    ax.set(xlabel="dent diameter [mm]", ylabel="Brinell hardness HB", title="3000 kgf, 10 mm ball")
    ax.secondary_yaxis("right", functions=(lambda h: 3.45 * h, lambda s: s / 3.45)).set_ylabel("estimated Sut [MPa]")
    ax.grid(True, alpha=0.3)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/hardness/images/sphx_glr_plot_01_brinell_hardness_001.png
   :alt: 3000 kgf, 10 mm ball
   :srcset: /api/gallery/materials/hardness/images/sphx_glr_plot_01_brinell_hardness_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    range 95 to 653 HB, Sut 329 to 2254 MPa
    Vickers, 30 kgf, 0.50 mm diagonal: 223 HV





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

   **Total running time of the script:** (0 minutes 0.090 seconds)


.. _sphx_glr_download_api_gallery_materials_hardness_plot_01_brinell_hardness.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/hardness/plot_01_brinell_hardness.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_brinell_hardness.ipynb <plot_01_brinell_hardness.ipynb>`

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

      :download:`Download Python source code: plot_01_brinell_hardness.py <plot_01_brinell_hardness.py>`

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

      :download:`Download zipped: plot_01_brinell_hardness.zip <plot_01_brinell_hardness.zip>`


.. only:: html

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

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