.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/materials/creep/plot_01_norton_creep.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_creep_plot_01_norton_creep.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_creep_plot_01_norton_creep.py:


Norton's power-law creep (1929)
===============================

Steam turbines of the 1920s ran hot enough for their steel to stretch slowly
under constant load. Frederick Norton's *The Creep of Steel at High
Temperatures* (1929) found that the steady creep rate rises as a power of
stress, :math:`\dot\varepsilon = A\sigma^n`, with :math:`n` around 5. Its
temperature dependence is thermally activated,
:math:`e^{-Q/RT}`, with an activation energy close to that of
self-diffusion (Dorn, 1950s)
(:func:`~engineeringkit.materials.arrhenius_creep_rate`). The example
recovers :math:`n` and :math:`Q` from simulated creep tests.

.. GENERATED FROM PYTHON SOURCE LINES 17-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.materials import arrhenius_creep_rate
    from engineeringkit.materials.utils import power_law_fit








.. GENERATED FROM PYTHON SOURCE LINES 24-30

Creep tests at three temperatures
---------------------------------

Minimum creep rates of a ferritic steel with :math:`n = 5` and
:math:`Q = 280` kJ/mol, measured with 15% scatter at five stresses and
three temperatures.

.. GENERATED FROM PYTHON SOURCE LINES 30-37

.. code-block:: Python


    A, Q, R, n = 1.8e-32, 280e3, 8.314, 5.0
    stresses = np.geomspace(60e6, 200e6, 5)
    temperatures = np.array([850.0, 900.0, 950.0])
    rng = np.random.default_rng(11)
    rates = arrhenius_creep_rate(A, Q, R, temperatures[:, None], stresses[None, :], n) * rng.lognormal(0, 0.15, (3, 5))








.. GENERATED FROM PYTHON SOURCE LINES 38-44

The stress exponent and the activation energy
---------------------------------------------

At each temperature the slope of :math:`\log\dot\varepsilon` against
:math:`\log\sigma` is :math:`n`. At fixed stress the slope of
:math:`\ln\dot\varepsilon` against :math:`1/T` is :math:`-Q/R`.

.. GENERATED FROM PYTHON SOURCE LINES 44-62

.. code-block:: Python


    exponents = [power_law_fit(stresses, row)[1] for row in rates]
    slope = np.polyfit(1 / temperatures, np.log(rates).mean(axis=1), 1)[0]
    print(f"stress exponents {np.round(exponents, 2).tolist()}; activation energy {-slope * R / 1e3:.0f} kJ/mol")
    assert np.allclose(exponents, n, atol=0.4) and abs(-slope * R / Q - 1) < 0.05

    life = 0.01 / arrhenius_creep_rate(A, Q, R, 850.0, 100e6, n)
    print(f"time to 1% creep strain at 850 K and 100 MPa: {life / 3600:.0f} h")

    fig, ax = plt.subplots(figsize=(6, 3.8))
    for T, row in zip(temperatures, rates):
        ax.loglog(stresses / 1e6, row, "o", label=f"{T:.0f} K")
        ax.loglog(stresses / 1e6, arrhenius_creep_rate(A, Q, R, T, stresses, n), "k-", lw=0.8)
    ax.set(xlabel="stress [MPa]", ylabel="minimum creep rate [1/s]", title="Norton's law")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/materials/creep/images/sphx_glr_plot_01_norton_creep_001.png
   :alt: Norton's law
   :srcset: /api/gallery/materials/creep/images/sphx_glr_plot_01_norton_creep_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    stress exponents [4.87, 4.88, 4.8]; activation energy 279 kJ/mol
    time to 1% creep strain at 850 K and 100 MPa: 2487 h





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

   **Total running time of the script:** (0 minutes 0.061 seconds)


.. _sphx_glr_download_api_gallery_materials_creep_plot_01_norton_creep.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/creep/plot_01_norton_creep.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_norton_creep.ipynb <plot_01_norton_creep.ipynb>`

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

      :download:`Download Python source code: plot_01_norton_creep.py <plot_01_norton_creep.py>`

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

      :download:`Download zipped: plot_01_norton_creep.zip <plot_01_norton_creep.zip>`


.. only:: html

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

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