.. _failure-tearing:

Tearing Parameter
=================

The tearing parameter model, proposed by Wellman [:footcite:`int:ref:wellman`], is implemented with the form,

.. math::
   :label: mat:eq:modularfailure_tp

   d = \frac{1}{d_{\text{crit}}} \int_0^{\bar{\varepsilon}^p}  {\Big\langle \frac{{2\sigma_{\max}}}{{3\left( {\sigma_{\max}  - p } \right)}}\Big\rangle } ^m d\bar{\varepsilon}^{p},

where :math:`\sigma_{ij}` is the Cauchy stress tensor, :math:`p= \frac13 \sigma_{kk}` is the mean hydrostatic stress, :math:`\sigma_{max}` is the maximum principal stress, and :math:`\bar{\varepsilon}^{p}` is the equivalent plastic strain. The two parameters of the model as :math:`m`, a fit exponent, and the critical failure (tearing) parameter, :math:`d_{\text{crit}}`. The angle brackets :math:`\langle \cdot \rangle`, denoting Macaulay brackets,

.. math::

   \langle x \rangle = \begin{cases}
   0 \;\;\; \text{if } x \leq 0 \\
   x \;\;\; \text{if } x > 0
   \end{cases},

are used to ensure that the failure process occurs only with tensile stress states and prevent "damage healing". The failure process initiates once the integral term reaches the critical tearing parameter, such that :math:`d = 1`.

.. %and the corresponding stress decay occurs over a strain interval corresponding to the \emph{critical crack opening strain}, :math:`\varepsilon_{\text{ccos}}`.  
.. %Importantly, the :math:`\varepsilon_{\text{ccos}}` serves a dual role in that it may also be used to control 
.. %the energy dissipated during failure.  With respect to the latter point, careful selection of the 
.. %critical crack opening strain may be used to ensure consistent energy is dissipated through different meshes.  
.. %This decay process is isotropic and linear with the current damage value being equivalent to the 
.. %ratio of crack opening strain in the direction of the maximum principal stress to the critical value.

User Guide
----------

.. code-block:: sierrainput

   #
   # TEARING_PARAMETER Failure model definitions
   #
   TEARING PARAMETER EXPONENT = m

.. raw::
   html

   <hr>

.. footbibliography::
