.. _cohesive-int:

****************
Intrinsic Models
****************

.. _cohesive-int-MDGc:

Mixed-mode Dependent Toughness
==============================

The MDGc CZM (Mixed-mode Dependent Toughness Cohesive Zone Mode) has two elements. Mode I energy dissipation is defined by a trapezoidal traction-separation relationship that depends only on normal separation. Mode II (III) dissipation is generated by shear yielding that depends only on the tangential separation components. A perfect plasticity-like formulation is used to define shear yielding by relating effective shear traction to effective slip rate. Shear yielding occurs within the region where Mode I separation (softening) occurs and can also occur ahead of that region. The MDGc CZM was developed to model crack propagation along an epoxy/solid interface when there is small-scale crack-tip yielding and when the epoxy and solid materials can be idealized as linear elastic. Nevertheless, this model might be applicable to other types of interfaces, but the user needs to use care in doing so. The MDGc CZM is described in detail in reference [:footcite:`ReedyMDGc`]. Note that the current implementation of the MDGc CZM differs slightly from that described in reference [:footcite:`ReedyMDGc`] in that shear unloading occurs after Mode I separation is complete (i.e. the normal traction has dropped to zero). In the initial implementation described in reference [:footcite:`ReedyMDGc`], shear unloading commenced as soon as Mode I softening initiated. A clear preference for either option is not obvious and the current choice generates a smoother solution.

.. code-block:: sierrainput

   BEGIN PARAMETERS FOR MODEL MDGc
     PEAK NORMAL TRACTION = <real>
     NORMAL LENGTH SCALE = <real>
     TANGENTIAL LENGTH SCALE = <real> 
     LAMBDA_1 = <real>
     LAMBDA_2 = <real>
     PEAK SHEAR TRACTION = <real>
     LAMBDA_3 = <real>
     PENETRATION PENALTY = <real>
     UNLOAD TYPE = ELASTIC
   END [PARAMETERS FOR MODEL MDGc]

In the above command blocks:

- The maximum normal traction is specified by the ``PEAK NORMAL TRACTION`` command.
- The normal separation at which the normal traction falls to zero is prescribed by the ``NORMAL LENGTH SCALE`` command.
- The effective tangential separation over which plastic yield occurs before the interface fails in shear is prescribed by the ``TANGENTIAL LENGTH SCALE`` command. This should be large compared to ``NORMAL LENGTH SCALE``. A recommended value is 100.0.
- ``LAMBDA_1`` indicates the normalized separation at which the normal traction response flattens with an additional increase in normal separation. The initial Mode I loading slope :math:`K` equals the ``PEAK NORMAL TRACTION``/(``LAMBDA_1``:math:`\times` ``NORMAL LENGTH SCALE}``).
- ``LAMBDA_2`` indicates the normalized separation at which the normal traction begins to decrease with additional increase in normal separation. Setting ``LAMBDA_1=LAMBDA_2`` generates a triangular traction-separation relationship.
- The maximum shear traction is specified through the ``PEAK SHEAR TRACTION`` command.
- ``LAMBDA_3`` controls the rapidity with which the shear is released. The shear unloading slope, :math:`K_u`, equals the negative of the initial Mode I loading slope, :math:`K`, times the ratio of ``LAMBDA_1/LAMBDA_3``. One reasonable choice is ``LAMBDA_3=LAMBDA_1``.
- The ``PENETRATION PENALTY`` parameter multiplies the Mode I loading slope, :math:`K`, to provide an artificially increased penetration stiffness to help prevent interpenetration of cohesive surfaces when crack closure occurs. It is recommended that this parameter be set to zero (no penetration stiffness) and that Sierra/SM contact surfaces be used to prevent interpenetration.
- The only currently supported option for ``UNLOAD TYPE`` is ``ELASTIC``.

The state variables for this model are listed in :numref:`cse-int-tab-MDGc-StateVar`.

.. _cse-int-tab-MDGc-StateVar:

.. csv-table:: State Variables for MDGc CZM (:numref:`cohesive-int-MDGc`)
   :align: center
   :delim: &
   :header: Name, Description

   ``LAMBDA_MAX`` & Maximum lambda the model has experienced (lambda equals the normal separation divided by the ``NORMAL LENGTH SCALE``)
   ``TRACTION AT LAMBDA MAX`` & Traction at ``LAMBDA_MAX``

.. raw::
   html

   <hr>

.. footbibliography::
