.. _large_deformation-hyperelastic: large_deformation/hyperelastic.py ================================= **Description** Nearly incompressible Mooney-Rivlin hyperelastic material model. Large deformation is described using the total Lagrangian formulation. Models of this kind can be used to model e.g. rubber or some biological materials. Find :math:\ul{u} such that: .. math:: \intl{\Omega\suz}{} \left( \ull{S}\eff(\ul{u}) + K(J-1)\; J \ull{C}^{-1} \right) : \delta \ull{E}(\ul{v}) \difd{V} = 0 \;, \quad \forall \ul{v} \;, where .. list-table:: :widths: 20 80 * - :math:\ull{F} - deformation gradient :math:F_{ij} = \pdiff{x_i}{X_j} * - :math:J - :math:\det(F) * - :math:\ull{C} - right Cauchy-Green deformation tensor :math:C = F^T F * - :math:\ull{E}(\ul{u}) - Green strain tensor :math:E_{ij} = \frac{1}{2}(\pdiff{u_i}{X_j} + \pdiff{u_j}{X_i} + \pdiff{u_m}{X_i}\pdiff{u_m}{X_j}) * - :math:\ull{S}\eff(\ul{u}) - effective second Piola-Kirchhoff stress tensor The effective stress :math:\ull{S}\eff(\ul{u}) is given by: .. math:: \ull{S}\eff(\ul{u}) = \mu J^{-\frac{2}{3}}(\ull{I} - \frac{1}{3}\tr(\ull{C}) \ull{C}^{-1}) + \kappa J^{-\frac{4}{3}} (\tr(\ull{C}\ull{I} - \ull{C} - \frac{2}{6}((\tr{\ull{C}})^2 - \tr{(\ull{C}^2)})\ull{C}^{-1}) \;. .. image:: /../gallery/images/large_deformation-hyperelastic.png :download:source code  .. literalinclude:: /../examples/large_deformation/hyperelastic.py