.. _sphx_glr_api_gallery_materials_yield:

Yield
-----

Multiaxial yield criteria for ductile metals.


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    <div class="sphx-glr-thumbcontainer" tooltip="Henri Tresca (1864) punched and extruded lead, copper and iron through dies at the Conservatoire des Arts et Métiers and found that metals flow like a viscous fluid once the largest shear stress reaches a constant. With principal stresses \sigma_1 \ge \sigma_2 \ge \sigma_3 the criterion is \sigma_1 - \sigma_3 = \sigma_y (~engineeringkit.materials.tresca_equivalent). It is the first criterion for yielding under combined stresses, and in the plane-stress principal plane it draws a hexagon. The example applies it to a shaft in bending and torsion and to biaxial stress states.">

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  .. image:: /api/gallery/materials/yield/images/thumb/sphx_glr_plot_01_tresca_criterion_thumb.png
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  :doc:`/api/gallery/materials/yield/plot_01_tresca_criterion`

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      <div class="sphx-glr-thumbnail-title">Tresca's maximum-shear yield criterion (1864)</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Tresca&#x27;s hexagon has corners, which make the equations of plasticity awkward. Richard von Mises (1913) proposed a smooth surface through the corners: yield when the second invariant of the deviatoric stress reaches \sigma_y^2/3, in plane stress \sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2 = \sigma_y^2 (~engineeringkit.materials.von_mises_plane_stress). Huber (1904) had proposed the same rule, and Hencky (1924) explained it as the elastic energy of distortion. Taylor and Quinney&#x27;s tension-torsion tests on thin tubes (1931) fell closer to von Mises than to Tresca.">

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  .. image:: /api/gallery/materials/yield/images/thumb/sphx_glr_plot_02_von_mises_criterion_thumb.png
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  :doc:`/api/gallery/materials/yield/plot_02_von_mises_criterion`

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      <div class="sphx-glr-thumbnail-title">Von Mises's yield criterion (1913)</div>
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.. toctree::
   :hidden:

   /api/gallery/materials/yield/plot_01_tresca_criterion
   /api/gallery/materials/yield/plot_02_von_mises_criterion