Elasticity::StVenantKirchhoff Namespace Reference

Functions
template<int dim, typename number >
void	cell_residual (dealii::Vector< number > &result, const dealii::FEValuesBase< dim > &fe, const dealii::VectorSlice< const std::vector< std::vector< dealii::Tensor< 1, dim >>>> &input, double lambda=0., double mu=1.)
template<int dim>
void	cell_matrix (dealii::FullMatrix< double > &M, const dealii::FEValuesBase< dim > &fe, const dealii::VectorSlice< const std::vector< std::vector< dealii::Tensor< 1, dim >>>> &input, double lambda=0., double mu=1.)

Detailed Description

Integrators for linear stress-strain relation in the case of geometrically nonlinear elasticity, also known as St. Venant-Kirchhoff materials or as large displacement/small strain.

Function Documentation

template<int dim>

void Elasticity::StVenantKirchhoff::cell_matrix ( dealii::FullMatrix< double > &  M, const dealii::FEValuesBase< dim > &  fe, const dealii::VectorSlice< const std::vector< std::vector< dealii::Tensor< 1, dim >>>> &  input, double  lambda = 0., double  mu = 1.  )

inline

Gateaux derivative matrix for nonlinear Elasticity problem.

\[ \int_\Omega \left \{\big(I+ \nabla v \big) \, \big( \lambda \, \operatorname{tr}(\tfrac12 E) \, I + \mu E \big) \, \, + \, \, \big(I + \nabla u \big) \, \big(\lambda \, \operatorname{tr} (\tfrac12 (\,\nabla v + (\nabla v)^T + (\nabla u)^T \nabla v + (\nabla v)^T \nabla u \,)) \, I + \mu \, (\, \nabla v + (\nabla v)^T + (\nabla u)^T \nabla v + (\nabla v)^T \nabla u \,) \big) \right \} \, : \, \nabla v \, dx \]

where \(E = \tfrac12\bigl(\nabla u + (\nabla u)^T + (\nabla u)^T \nabla u \bigr)\).

template<int dim, typename number >

void Elasticity::StVenantKirchhoff::cell_residual ( dealii::Vector< number > &  result, const dealii::FEValuesBase< dim > &  fe, const dealii::VectorSlice< const std::vector< std::vector< dealii::Tensor< 1, dim >>>> &  input, double  lambda = 0., double  mu = 1.  )

inline

The residual for nonlinear Elasticity problem.

\[ \int_\Omega \bigl((I+F) \Sigma\bigr) \colon \nabla v \,dx \]

Author

Guido Kanschat