.. _failure-wilkins:

*******
Wilkins
*******

The Wilkins failure model, proposed by Wilkins [:footcite:`Wilkins1980`] is implemented with the form:

.. math::
  :label: mat:eq:modularfailure_wilkins

   d = \frac{1}{d_{\text{crit}}} \int_0^{\bar{\varepsilon}^p}  w_1 w_2  d\bar{\varepsilon}^{p},

where :math:`w_1` represents a pressure-dependent term defined as,

.. math::

   w_1 = \left( \frac{1}{1 - \frac{p}{B}} \right) ^{\alpha},

with :math:`\alpha` and :math:`B` as fitting parameters and :math:`p` is mean hydrostatic stress, and :math:`w_2` represents a Lode-angle-related term defined as,

.. math::

   w_2 = (2 - A)^{\beta},

with :math:`\beta` as fitting parameter, and :math:`A` defined as a function of deviatoric principal stresses (:math:`s_1 \geq s_2 \geq s_3`),

.. math::

   A = \max \left( \frac{s_2}{s_3}, \frac{s_2}{s_1} \right) . 

The failure process initiates once the integral term reaches the critical failure parameter, such that :math:`d = 1`.

User Guide
==========

.. code-block:: sierrainput

   #
   # WILKINS Failure model definitions
   #
   WILKINS ALPHA    = <real>
   WILKINS BETA     = <real>
   WILKINS PRESSURE = <real>

.. raw::
   html

   <hr>

.. footbibliography::
