.. _appendix-common-simple-shear:

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A.2 Simple Shear
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An alternative, and often simpler to implement, shear problem is that of simple shear. With such a deformation field, only one shear stress component is non-zero (like the pure shear case). The difference arises in that given a simple shear loading the diagonal stresses are not necessarily zero. This state may be produced by a motion, :math:`\chi(X_i,t)` of the form :math:`\chi(X_i,t)=X_i+\gamma(t) X_2\delta_{i1}`. The resultant deformation gradient, :math:`F_{ij}`, takes the form,

.. math::

   F_{ij}=\delta_{ij}+\gamma\left(t\right)\delta_{i1}\delta_{j2}

and it is noted that this deformation is volume preserving (:math:`J=\det F_{ij}=1`). Numerically, such a deformation field results from applying a displacement in the :math:`x` direction along the :math:`y` surface.
