This class implements a constitutive formulation for an Neo-Hookean elastic solid in terms of the current configuration. More...

#include <nonlinear_neohooke_cc.h>

Public Member Functions
	NonlinearNeoHookeCurrentConfig (const std::vector< std::vector< RealGradient >> &dphi_in, GetPot &args, bool calculate_linearized_stiffness_in)

void	init_for_qp (VectorValue< Gradient > &grad_u, unsigned int qp)
	Initialize the class for the given displacement gradient at the specified quadrature point. More...

void	get_residual (DenseVector< Real > &residuum, unsigned int &i)
	Return the residual vector for the current state. More...

void	get_linearized_stiffness (DenseMatrix< Real > &stiffness, unsigned int &i, unsigned int &j)
	Return the stiffness matrix for the current state. More...

Public Attributes
bool	calculate_linearized_stiffness
	Flag to indicate if it is necessary to calculate values for stiffness matrix during initialization. More...

Private Member Functions
void	build_b_0_mat (int i, DenseMatrix< Real > &b_l_mat)

void	calculate_stress ()

void	calculate_tangent ()

Static Private Member Functions
static void	tensor_to_voigt (const RealTensor &tensor, DenseVector< Real > &vec)

Private Attributes
unsigned int	current_qp

const std::vector< std::vector< RealGradient > > &	dphi

DenseMatrix< Real >	C_mat

Real	E

Real	nu

RealTensor	F

RealTensor	S

RealTensor	tau

RealTensor	sigma

DenseMatrix< Real >	B_L

DenseMatrix< Real >	B_K

Detailed Description

This class implements a constitutive formulation for an Neo-Hookean elastic solid in terms of the current configuration.

This implementation is suitable for computing large deformations. See e.g. "Nonlinear finite element methods" (P. Wriggers, 2008, Springer) for details.

Definition at line 40 of file nonlinear_neohooke_cc.h.

Constructor & Destructor Documentation

◆ NonlinearNeoHookeCurrentConfig()

NonlinearNeoHookeCurrentConfig::NonlinearNeoHookeCurrentConfig	(	const std::vector< std::vector< RealGradient >> &	dphi_in,
		GetPot &	args,
		bool	calculate_linearized_stiffness_in
	)

inline

Definition at line 43 of file nonlinear_neohooke_cc.h.

                                                                          :
     calculate_linearized_stiffness(calculate_linearized_stiffness_in),
     dphi(dphi_in)
   {
     E = args("material/neohooke/e_modulus", 10000.0);
     nu = args("material/neohooke/nu", 0.3);
   }

Member Function Documentation

◆ build_b_0_mat()

void NonlinearNeoHookeCurrentConfig::build_b_0_mat	(	int	i,
		DenseMatrix< Real > &	b_l_mat
	)

private

Definition at line 141 of file nonlinear_neohooke_cc.C.

 {
   for (unsigned int ii = 0; ii < 3; ++ii)
     b_0_mat(ii, ii) = dphi[i][current_qp](ii);
  
   b_0_mat(0, 3) = dphi[i][current_qp](1);
   b_0_mat(1, 3) = dphi[i][current_qp](0);
   b_0_mat(1, 4) = dphi[i][current_qp](2);
   b_0_mat(2, 4) = dphi[i][current_qp](1);
   b_0_mat(0, 5) = dphi[i][current_qp](2);
   b_0_mat(2, 5) = dphi[i][current_qp](0);
 }

References current_qp, and dphi.

Referenced by get_linearized_stiffness(), and get_residual().

◆ calculate_stress()

void NonlinearNeoHookeCurrentConfig::calculate_stress ( )

private

Definition at line 73 of file nonlinear_neohooke_cc.C.

 {
   Real mu = E / (2.0 * (1.0 + nu));
   Real lambda = E * nu / ((1 + nu) * (1 - 2 * nu));
  
   Real detF = F.det();
   RealTensor Ft = F.transpose();
  
   RealTensor C = Ft * F;
   RealTensor b = F * Ft;
   RealTensor identity;
   identity(0, 0) = 1.0; identity(1, 1) = 1.0; identity(2, 2) = 1.0;
   RealTensor invC = C.inverse();
  
   S = 0.5 * lambda * (detF * detF - 1) * invC + mu * (identity - invC);
  
   tau = (F * S) * Ft;
   sigma = 1/detF * tau;
 }

References E, F, libMesh::TypeTensor< T >::inverse(), nu, libMesh::Real, S, sigma, and tau.

Referenced by init_for_qp().

◆ calculate_tangent()

void NonlinearNeoHookeCurrentConfig::calculate_tangent ( )

private

Definition at line 52 of file nonlinear_neohooke_cc.C.

 {
   Real mu = E / (2 * (1 + nu));
   Real lambda = E * nu / ((1 + nu) * (1 - 2 * nu));
  
   Real detF = F.det();
  
   C_mat.resize(6, 6);
   for (unsigned int i = 0; i < 3; ++i) {
     for (unsigned int j = 0; j < 3; ++j) {
       if (i == j) {
         C_mat(i, j) = 2 * mu + lambda;
         C_mat(i + 3, j + 3) = mu - 0.5 * lambda * (detF * detF - 1);
       } else {
         C_mat(i, j) = lambda * detF * detF;
       }
     }
   }
 }

References C_mat, E, F, nu, libMesh::Real, and libMesh::DenseMatrix< T >::resize().

Referenced by init_for_qp().

◆ get_linearized_stiffness()

void NonlinearNeoHookeCurrentConfig::get_linearized_stiffness	(	DenseMatrix< Real > &	stiffness,
		unsigned int &	i,
		unsigned int &	j
	)

Return the stiffness matrix for the current state.

Definition at line 118 of file nonlinear_neohooke_cc.C.

 {
   stiffness.resize(3, 3);
  
   Real G_IK = (sigma * dphi[i][current_qp]) * dphi[j][current_qp];
   stiffness(0, 0) += G_IK;
   stiffness(1, 1) += G_IK;
   stiffness(2, 2) += G_IK;
  
   B_L.resize(3, 6);
   this->build_b_0_mat(i, B_L);
   B_K.resize(3, 6);
   this->build_b_0_mat(j, B_K);
  
   B_L.right_multiply(C_mat);
   B_L.right_multiply_transpose(B_K);
   B_L *= 1/F.det();
  
   stiffness += B_L;
 }

References B_K, B_L, build_b_0_mat(), C_mat, current_qp, dphi, F, libMesh::Real, libMesh::DenseMatrix< T >::resize(), libMesh::DenseMatrix< T >::right_multiply(), libMesh::DenseMatrix< T >::right_multiply_transpose(), and sigma.

Referenced by SolidSystem::element_time_derivative().

◆ get_residual()

void NonlinearNeoHookeCurrentConfig::get_residual	(	DenseVector< Real > &	residuum,
		unsigned int &	i
	)

Return the residual vector for the current state.

Definition at line 93 of file nonlinear_neohooke_cc.C.

 {
   B_L.resize(3, 6);
   DenseVector<Real> sigma_voigt(6);
  
   this->build_b_0_mat(i, B_L);
  
   tensor_to_voigt(sigma, sigma_voigt);
  
   B_L.vector_mult(residuum, sigma_voigt);
 }

References B_L, build_b_0_mat(), libMesh::DenseMatrix< T >::resize(), sigma, tensor_to_voigt(), and libMesh::DenseMatrix< T >::vector_mult().

Referenced by SolidSystem::element_time_derivative().

◆ init_for_qp()

void NonlinearNeoHookeCurrentConfig::init_for_qp	(	VectorValue< Gradient > &	grad_u,
		unsigned int	qp
	)

Initialize the class for the given displacement gradient at the specified quadrature point.

Definition at line 25 of file nonlinear_neohooke_cc.C.

 {
   this->current_qp = qp;
   F.zero();
   S.zero();
  
   {
     RealTensor invF;
     invF.zero();
     for (unsigned int i = 0; i < 3; ++i)
       for (unsigned int j = 0; j < 3; ++j)
         invF(i, j) += libmesh_real(grad_u(i)(j));
  
     F.add(invF.inverse());
  
     libmesh_assert_greater (F.det(), -TOLERANCE);
   }
  
   if (this->calculate_linearized_stiffness)
     this->calculate_tangent();
  
   this->calculate_stress();
 }

References calculate_linearized_stiffness, calculate_stress(), calculate_tangent(), current_qp, F, libMesh::libmesh_real(), S, libMesh::TOLERANCE, and libMesh::TypeTensor< T >::zero().

Referenced by SolidSystem::element_time_derivative().

◆ tensor_to_voigt()

void NonlinearNeoHookeCurrentConfig::tensor_to_voigt	(	const RealTensor &	tensor,
		DenseVector< Real > &	vec
	)

staticprivate

Definition at line 106 of file nonlinear_neohooke_cc.C.

 {
   vec(0) = tensor(0, 0);
   vec(1) = tensor(1, 1);
   vec(2) = tensor(2, 2);
   vec(3) = tensor(0, 1);
   vec(4) = tensor(1, 2);
   vec(5) = tensor(0, 2);
  
 }