.. _sphx_glr_api_gallery_materials_elasticity:

Elasticity
----------

Young's modulus, thermal stress and the isotropic elastic constants.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Thomas Young&#x27;s Lectures on Natural Philosophy (1807) gave a single number for the stiffness of a material, its &quot;modulus of elasticity&quot;. Young expressed it as the height of a column of the material whose weight would compress it to nothing if the law stayed linear; Navier (1826) restated it as the ratio of stress to strain, E = \sigma/\varepsilon, the form used today (~engineeringkit.materials.hooke_stress). The example reduces a simulated tensile test from load and extension to stress and strain (~engineeringkit.materials.normal_stress, normal_strain), fits the modulus with StressStrainCurve and measures the energy the bar absorbs.">

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  .. image:: /api/gallery/materials/elasticity/images/thumb/sphx_glr_plot_01_young_modulus_thumb.png
    :alt:

  :doc:`/api/gallery/materials/elasticity/plot_01_young_modulus`

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      <div class="sphx-glr-thumbnail-title">Young's modulus (1807)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Jean-Marie Duhamel (1838) added temperature to Navier&#x27;s equations of elasticity: a change \Delta T produces a free strain \alpha\,\Delta T (~engineeringkit.materials.thermal_strain) that causes no stress if the body can expand, and a stress -E\alpha\,\Delta T (~engineeringkit.materials.constrained_thermal_stress) if it is held. The stress does not depend on the length of the bar, which is why continuously welded rail must be laid at a chosen neutral temperature.">

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  .. image:: /api/gallery/materials/elasticity/images/thumb/sphx_glr_plot_02_duhamel_thermal_stress_thumb.png
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  :doc:`/api/gallery/materials/elasticity/plot_02_duhamel_thermal_stress`

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      <div class="sphx-glr-thumbnail-title">Duhamel's thermal stress (1838)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Early elasticity theories built from molecular forces predicted a single elastic constant for an isotropic solid, with Poisson&#x27;s ratio fixed at 1/4. Measurements disagreed. Gabriel Lamé&#x27;s Leçons sur la théorie mathématique de l&#x27;élasticité (1852) wrote the isotropic law with two constants,">

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  .. image:: /api/gallery/materials/elasticity/images/thumb/sphx_glr_plot_03_lame_constants_thumb.png
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  :doc:`/api/gallery/materials/elasticity/plot_03_lame_constants`

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      <div class="sphx-glr-thumbnail-title">Lamé's elastic constants (1852)</div>
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.. toctree::
   :hidden:

   /api/gallery/materials/elasticity/plot_01_young_modulus
   /api/gallery/materials/elasticity/plot_02_duhamel_thermal_stress
   /api/gallery/materials/elasticity/plot_03_lame_constants