.. _appendix-common-uniaxial-stress:

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A.1 Uniaxial Stress
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In all likelihood, the most common test (experimentally or numerically) is that of uniaxial stress. Such a state may be produced via either stress or displacement control. Here, the latter case is discussed as displacement control can be essential when considering model responses that soften through damage or other mechanisms. To produce the uniaxial stress of interest, a displacement of the form :math:`u_1=\lambda(t)` is applied along the :math:`x_1` edge. In three dimensional finite element cases, it is also essential to leave the :math:`x_2` and :math:`x_3` surfaces with a traction free condition. With elastically isotropic materials, this produces a strain field of the form,

.. math::

   \varepsilon_{ij}=\left[\delta_{i1}\delta_{j1}-\nu\left(\delta_{i2}\delta_{j2}+\delta_{i3}\delta_{i3}\right)\right]\ln\left(1+\lambda\right),

which produces :math:`\sigma_{11}` as the only non-zero stress.

