Model::Solid::Linear::Hexagonal
===============================

:bdg-link-secondary-line:`src/Model/Solid/Linear/Hexagonal.H <../../../../doxygen/Linear_2Hexagonal_8H_source.html>`

This class implements elasticity for a linear material with hexagonal symmetry.
The elastic modulus tensor :math:`\mathbb{C}` has major and minor symmetry but not cubic symmetry,
so the symmetry class for this material is MajorMinor
The five elastic constants in the unrotated system are :math:`C_{11},C_{12},C_{13},C_{33},C_{44}`,
and the resulting elastic modulus tensor is

.. math::

   \mathbb{C} &= 
   \begin{bmatrix}
      C_{0000} & C_{0011} & C_{0022} & C_{0012} & C_{0020} & C_{0001} \\
      C_{1100} & C_{1111} & C_{1122} & C_{1112} & C_{1120} & C_{1101} \\
      C_{2200} & C_{2211} & C_{2222} & C_{2212} & C_{2220} & C_{2201} \\
      C_{1200} & C_{1211} & C_{1222} & C_{1212} & C_{1220} & C_{1201} \\
      C_{2000} & C_{2011} & C_{2022} & C_{2012} & C_{2020} & C_{2001} \\
      C_{0100} & C_{0111} & C_{0122} & C_{0112} & C_{0120} & C_{0101} 
   \end{bmatrix} \\
   &=
   \begin{bmatrix}  
      C_{11}  &   C_{12}  &   C_{12}  &          &                      &                     \\
      C_{12}  &   C_{11}  &   C_{13}  &          &                      &                     \\
      C_{12}  &   C_{13}  &   C_{33}  &          &                      &                     \\
              &           &           &  C_{44}  &                      &                     \\
              &           &           &          &   0.5(C_{11}-C_{12}) &                     \\
              &           &           &          &                      &  0.5(C_{11}-C_{12})  
   \end{bmatrix}

Rotations can be specified using Bunge euler angles to generate a rotation matrix :math:`\mathbf{R}`.
The rotated modulus tensor :math:`\mathbb{C}^{rot}` is then

.. math::
   
   \mathbb{C}^{rot}_{psqt} = \mathbb{C}_{ijkl}R_{pi}R_{qj}R_{sk}R_{tl}

.. container:: input-reference-math-loader

   :math:`\phantom{0}`

.. raw:: html

   <div class="input-reference-list">
     <article class="input-reference-item" id="input-c11-63801c28ec">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00171"><code>C11</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-required">required</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Elastic constant</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>mechanics.model0.C11</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.C11</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C11</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C11</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-c12-caa5f5664a">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00172"><code>C12</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-required">required</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Elastic constant</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>mechanics.model0.C12</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.C12</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C12</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C12</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-c13-ccf6078f78">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00173"><code>C13</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-required">required</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Elastic constant</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>mechanics.model0.C13</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.C13</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C13</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C13</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-c33-8441182a36">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00174"><code>C33</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-required">required</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Elastic constant</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>mechanics.model0.C33</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.C33</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C33</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C33</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-c44-27d1a1bc9e">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00175"><code>C44</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-required">required</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Elastic constant</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>mechanics.model0.C44</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.C44</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C44</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.C44</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-phi1-c738b25888">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00184"><code>phi1</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-default">0</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Bunge Euler angle <span class="math notranslate nohighlight">\(\phi_1\)</span></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>mechanics.model0.phi1</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.phi1</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.phi1</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.phi1</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-phi-f34f3fdf1b">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00185"><code>Phi</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-default">0</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Bunge Euler angle <span class="math notranslate nohighlight">\(\Phi\)</span></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>mechanics.model0.Phi</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.Phi</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.Phi</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.Phi</code>
             </div>
           </div>
         </div>
       </details>
     </article>
     <article class="input-reference-item" id="input-phi2-af356ca444">
       <header class="input-reference-item-header">
         <a class="input-reference-name" href="../../../../doxygen/Linear_2Hexagonal_8H_source.html#l00186"><code>phi2</code></a>
         <div class="input-reference-badges"><span class="input-reference-badge input-reference-badge-default">0</span> <span class="input-reference-badge input-reference-badge-meta">conditional</span></div>
       </header>
       <p class="input-reference-description">Bunge Euler angle <span class="math notranslate nohighlight">\(\phi_2\)</span></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>mechanics.model0.phi2</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>alamo-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>mechanics.model0.phi2</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-2d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.phi2</code>
             </div>
           </div>
           <div class="input-reference-executable">
             <strong>mechanics-3d-coverage-g++</strong>
             <div class="input-reference-paths">
               <code>model0.phi2</code>
             </div>
           </div>
         </div>
       </details>
     </article>
   </div>