Integrator::Fracture
====================

:bdg-link-secondary-line:`src/Integrator/Fracture.H <../../doxygen/Fracture_8H_source.html>`

This class implements second and fourth order phase field brittle fracture with near singular solver.
This class inherits from the base class :code:`Base/Mechanics.H`.
See `this link <https://doi.org/10.1007/s00466-023-02325-8>`__
for more information on this implementation.

The energy functional for a second order model is given by

.. math::

     \mathcal{L}= \int_\Omega \left[\left(g(c) + \eta \right)W_0^+ + W_0^-\right] dV + \int_\Omega G_c \left[ \frac{w(c)}{4\xi} + \xi |\nabla c|^2\right] dV

where :math:`g(c)` and :math:`w(c)` are interpolation functions specified by the user.
The fracture energy :math:`G_c` and crack length scale :math:`xi` are also read from input file.
The tension compression asymmetry is accounted by splitting the strain energy :math:`W_0` as

.. math::

   W_0^\pm = \frac{1}{2} \lambda (\operatorname{tr}\bm{\varepsilon}_\pm)^2 + \mu \operatorname{tr}\left(\bm{\varepsilon}_\pm^2\right), \quad \bm{\varepsilon_\pm} = \sum_{i=1}^d \left(\varepsilon_i \right)_\pm \hat{\bm{v}}_i\otimes\hat{\bm{v}}_i

where :math:`\lambda` and :math:`\mu` are material models for linear elastic isotropic material, :math:`\bm{\varepsilon}_\pm` are the positive and negative components of the strain tensor :math:`\bm{\varepsilon}` computed through eigenvalue decomposition.
The fourth order model adds a laplacian term to the free energy functional.
The code performs a staggered solve where it solves the elastic problem implicitly and crack problem explicitly.

Class methods:

#. :code:`Fracture()`:
   Basic constructor. Does nothing, and leaves all values initiated as NAN.
#. :code:`Fracture(IO::ParmParse &pp)`:
   Calls the parser.
#. :code:`static void Parse(Fracture &value, IO::ParmParse &pp)`
   Parses input file, ICs, BCs, and sets up multifabs appropriately
#. :code:`void Initialize(int lev) override`
   Calls IC and sets up the initial crack geometry.
#. :code:`virtual void UpdateModel(int a_step) override`
   Performs degradation by updating the :math:`\psi` field using :math:`g(c)`.
#. :code:`void TimeStepBegin(Set::Scalar a_time, int a_step) override`
   Solves the elastic problem, performs eigen value decomposition of strain, and computes the positive part of the strain energy.
#. :code:`void Advance(int a_lev, amrex::Real a_time, amrex::Real a_dt)`
   Advances the crack field by computing the variational derivative of energy functional with :math:`c`.
#. :code:`void TagCellsForRefinement(int lev, amrex::TagBoxArray &a_tags, Set::Scalar a_time, int a_ngrow) override`
   Refines the grid based on the crack field.
#. :code:`void Integrate(int amrlev, Set::Scalar time, int step, const amrex::MFIter &mfi, const amrex::Box &a_box) override`
   Performs spatial integration of the driving force to check for convergence of crack problem
#. :code:`void TimeStepComplete(Set::Scalar /*time*/, int /* iter*/)`
   Checks whether the solver should work on crack problem or elastic problem.

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         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-option">constant</span> <span class="input-reference-badge input-reference-badge-option">ellipse</span> <span class="input-reference-badge input-reference-badge-option">expression</span> <span class="input-reference-badge input-reference-badge-option">notch</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
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           </div>
           <div class="input-reference-executable">
             <strong>fracture-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>crack.ic</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-crack-model-2bb405294f">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../doxygen/Fracture_8H_source.html#l00147"><code>crack.model</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-meta">repeatable</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">This is needed for the specific crack model we are implementing</p>
       <details class="input-reference-resolved">
         <summary>Names by executable <span>4 executables, 4 uses</span></summary>
         <div class="input-reference-resolved-body">
           <div class="input-reference-executable">
             <strong>alamo-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>crack.model</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>crack.model</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>fracture-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>crack.model</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>fracture-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>crack.model</code>
             </div>
           </div>
         </div>
       </details>
     </article>
   </div>