.. _xfem-multiple_holes_plate:

*************************
Plate with Multiple Holes
*************************

Product: Sierra/SolidMechanics - Explicit Analysis

.. figure:: multiple_holes_plate/figs/multiHolePlate.png
   :align: center
   :width: 600px
	   
   Plate with multiple holes problem set-up.

* :download:`Mesh <multiple_holes_plate/multiHoles.g>`   
* :download:`Input Deck <multiple_holes_plate/multiHoles.i>`

.. Problem Description
.. ===================

.. .. _fig-HoleSetup:

.. .. figure:: ../../../../tests/adagio_rtest/ExampleProblemsManual/XfemExampleProblems/MultipleHolesPlate/figs/multiHolePlate.png
..    :align: center
..    :scale: 75 %

..    Plate with multiple holes problem setup.

.. The purpose of the following numerical examples is to display the X-FEM nucleation and branching capabilities for 2-D shell elements. Consider a plate with a multiple holes under uniform tensile loading in :numref:`fig-HoleSetup`. As the force increases as the top and bottom are pulled apart, a stress concentration forms at the sides of both large holes. Once the stress concentration reaches the fracture stress, a crack nucleates at these locations. The cracks grow in multiple different modes, and the cracks branch when their user specified branching criteria is reached.

.. Loading and Boundary Conditions
.. ===============================

.. A force is applied to the top and bottom of the plate at a linear rate. The nucleation method that was chosen for this problem is element based nucleation and the nucleation and branching failure parameter is maximum principal stress.  

.. Material Model
.. ==============

.. An elastic-plastic material model is used, and the material properties can be found in :numref:`tab-xfem_multiholes_mats` below. Metric units are used: displacement is in meters, mass is in kilograms, time is in seconds, force is in Newtons, and temperature is in Kelvin.

.. .. _tab-xfem_multiholes_mats:

.. .. csv-table:: Plate with Multiple Holes Materials
..    :align: center
..    :delim: &

..    Young's Modulus: & :math:`E` & :math:`210\times10^3`
..    Poisson's Ratio & :math:`\nu` & :math:`0.3`
..    Density & :math:`\rho` & :math:`0.0078`
..    Yield Stress & :math:`\sigma_{yield}` & :math:`360`
..    Hardening Modulus & :math:`H` & :math:`50\times 10^3`
..    Beta & :math:`\beta` & :math:`0.75`

.. Finite Element Model
.. ====================

.. The elements that were used for this simulation were Belytschko-Tsay shell elements. 

.. Feature Tested
.. ==============

.. X-FEM crack nucleation, piecewise-linear crack growth, and crack branching in shells under dynamic conditions.

.. Results and Discussion
.. ======================

.. As can be seen in :numref:`fig-nucleation`\ (a) below, the stress concentration just before crack nucleation is the highest around the 2 large holes.  Once crack nucleation is initiated, stress waves propagate throughout the geometry as shown in :numref:`fig-nucleation`\ (b). The cracks propagate in multiple different fashions, with other cracks nucleating in the geometry as well as crack branching occurring in :numref:`fig-branching`. See :ref:`Appendix A <input-multiple_holes_plate>` for the input deck.

.. .. %\begin{figure}[h]
.. .. %\centering
.. .. %\epsfig{figure=figs/twoHoleBeforeCrack.png,width=.8\textwidth}
.. .. %\caption{Multi holes before nucleation.}
.. .. %\label{fig:RB}
.. .. %\end{figure}

.. .. _fig-nucleation:

.. .. subfigure:: AB
..    :subcaptions: below
..    :align: center

..    .. image:: ../../../../tests/adagio_rtest/ExampleProblemsManual/XfemExampleProblems/MultipleHolesPlate/figs/twoHoleBeforeCrack.png
..       :alt: Multi holes before nucleation.
..       :scale: 40 %

..    .. image:: ../../../../tests/adagio_rtest/ExampleProblemsManual/XfemExampleProblems/MultipleHolesPlate/figs/twoHoleBeforeCrack.png
..       :alt: Stress waves after nucleation.
..       :scale: 40 %

.. .. %\begin{figure}[h]
.. .. %\centering
.. .. %\epsfig{figure=figs/stressWaves.png,width=.8\textwidth}
.. .. %\caption{Stress waves after nucleation.}
.. .. %\label{fig:RB}
.. .. %\end{figure}

.. .. %%\begin{figure}[h]
.. .. %%\centering
.. .. %%\epsfig{figure=figs/multiHoleSnapshots.png,width=.8\textwidth}
.. .. %%\caption{Plate with Multiple Holes Snapshots}
.. .. %%\label{fig:RB}
.. .. %%\end{figure}

.. .. _fig-branching:

.. .. figure:: ../../../../tests/adagio_rtest/ExampleProblemsManual/XfemExampleProblems/MultipleHolesPlate/figs/multiHoleEnd.png
..    :align: center
..    :scale: 60 %

..    Plate with multiple holes snapshots.
