.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/fatigue/plot_05_neuber_rule.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_fatigue_plot_05_neuber_rule.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_fatigue_plot_05_neuber_rule.py:


Neuber's rule for notches that yield (1961)
===========================================

A notch raises the elastic stress at its root by :math:`K_t`, but in a
ductile part the root yields long before the nominal stress does, and
:math:`K_t S` overstates the stress and understates the strain. Heinz
Neuber, at the Technical University of Munich, showed for a shear-loaded
notch that the stress and strain concentration factors keep their geometric
mean equal to :math:`K_t` after yield, :math:`K_\sigma K_\varepsilon = K_t^2`,
so

.. math:: \sigma\varepsilon = \frac{(K_t S)^2}{E}

(:func:`~engineeringkit.materials.neuber_notch_stress`). Intersected with
the material's stress-strain curve, the hyperbola gives the local strain
that strain-life fatigue needs. The example also builds the fatigue notch
factor from the notch sensitivity
(:func:`~engineeringkit.materials.peterson_notch_sensitivity`,
:func:`~engineeringkit.materials.fatigue_notch_factor`).

.. GENERATED FROM PYTHON SOURCE LINES 24-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import (
        elliptical_hole_concentration,
        fatigue_notch_factor,
        neuber_notch_stress,
        peterson_notch_sensitivity,
        ramberg_osgood_strain,
    )








.. GENERATED FROM PYTHON SOURCE LINES 36-42

Notch sensitivity
-----------------

A hole in a wide plate has :math:`K_t = 3` (Inglis). In fatigue a small
hole hurts less: with Peterson's material length :math:`\alpha = 0.25` mm,
a 1 mm radius gives :math:`q = 0.8` and :math:`K_f = 2.6`.

.. GENERATED FROM PYTHON SOURCE LINES 42-49

.. code-block:: Python


    Kt = elliptical_hole_concentration(1.0, 1.0)
    q = peterson_notch_sensitivity(1e-3, 0.25e-3)
    Kf = fatigue_notch_factor(Kt, q)
    print(f"Kt = {Kt:.1f}, q = {q:.2f}, Kf = {Kf:.2f}")
    assert np.isclose(Kf, 2.6)





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

 .. code-block:: none

    Kt = 3.0, q = 0.80, Kf = 2.60




.. GENERATED FROM PYTHON SOURCE LINES 50-57

The Neuber hyperbola
--------------------

A steel with :math:`E = 200` GPa and a Ramberg-Osgood curve through a
0.2% offset yield of 400 MPa (:math:`n = 8`) carries a nominal 250 MPa at
the hole. The elastic answer, 750 MPa, is far above yield; Neuber gives a
local stress near 440 MPa and a strain 1.7 times the elastic one.

.. GENERATED FROM PYTHON SOURCE LINES 57-77

.. code-block:: Python


    E, sy, n, S = 200e9, 400e6, 8.0, 250e6
    stress, strain = neuber_notch_stress(S, Kt, E, sy, n)
    print(f"Neuber: sigma = {stress / 1e6:.0f} MPa, strain = {strain:.4f}; elastic: {Kt * S / 1e6:.0f} MPa, {Kt * S / E:.4f}")
    assert np.isclose(stress * strain, (Kt * S) ** 2 / E) and stress < Kt * S and strain > Kt * S / E

    curve_stress = np.linspace(0, 700e6, 300)
    curve_strain = ramberg_osgood_strain(curve_stress, E, sy, n)
    hyper_strain = np.linspace(0.0015, 0.012, 200)
    fig, ax = plt.subplots(figsize=(6, 3.8))
    ax.plot(curve_strain, curve_stress / 1e6, label="Ramberg-Osgood curve")
    ax.plot(hyper_strain, (Kt * S) ** 2 / E / hyper_strain / 1e6, "--", label="Neuber hyperbola")
    ax.plot([0, Kt * S / E], [0, Kt * S / 1e6], ":", color="0.4", label="elastic line")
    ax.plot(strain, stress / 1e6, "ko", label="notch root")
    ax.plot(Kt * S / E, Kt * S / 1e6, "kx", label="elastic estimate")
    ax.set(xlim=(0, 0.012), ylim=(0, 800), xlabel="strain", ylabel="stress [MPa]", title="Neuber's rule at a hole")
    ax.legend(fontsize=8)
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/fatigue/images/sphx_glr_plot_05_neuber_rule_001.png
   :alt: Neuber's rule at a hole
   :srcset: /api/gallery/materials/fatigue/images/sphx_glr_plot_05_neuber_rule_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Neuber: sigma = 439 MPa, strain = 0.0064; elastic: 750 MPa, 0.0037





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

   **Total running time of the script:** (0 minutes 0.076 seconds)


.. _sphx_glr_download_api_gallery_materials_fatigue_plot_05_neuber_rule.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/fatigue/plot_05_neuber_rule.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_neuber_rule.ipynb <plot_05_neuber_rule.ipynb>`

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