void HookesLaw::compute_stress_imp( unsigned int dim,
const libMesh::TensorValue<libMesh::Real>& g_contra,
const libMesh::TensorValue<libMesh::Real>& g_cov,
const libMesh::TensorValue<libMesh::Real>& /*G_contra*/,
const libMesh::TensorValue<libMesh::Real>& G_cov,
libMesh::TensorValue<libMesh::Real>& stress )
{
stress.zero();
for( unsigned int i = 0; i < dim; i++ )
{
for( unsigned int j = 0; j < dim; j++ )
{
for( unsigned int k = 0; k < dim; k++ )
{
for( unsigned int l = 0; l < dim; l++ )
{
libMesh::Real strain_kl = 0.5*(G_cov(k,l) - g_cov(k,l));
_C(i,j,k,l) = _lambda*g_contra(i,j)*g_contra(k,l) +
_mu*(g_contra(i,k)*g_contra(j,l) + g_contra(i,l)*g_contra(j,k));
stress(i,j) += _C(i,j,k,l)*strain_kl;
}
}
}
}
return;
}
void IncompressiblePlaneStressHyperelasticity<StrainEnergy>::compute_stress_imp( unsigned int /*dim*/,
const libMesh::TensorValue<libMesh::Real>& a_contra,
const libMesh::TensorValue<libMesh::Real>& a_cov,
const libMesh::TensorValue<libMesh::Real>& A_contra,
const libMesh::TensorValue<libMesh::Real>& A_cov,
libMesh::TensorValue<libMesh::Real>& stress )
{
// We're treating a_* and A_* as 2x2, but we're cheating to pick up lambda^2
libMesh::Real lambda_sq = A_cov(2,2);
libMesh::Real A_over_a = 1.0/lambda_sq; // We're incompressible
libMesh::Real I1, I2;
this->compute_I1_I2(a_contra,a_cov,A_contra,A_cov,lambda_sq,A_over_a,I1,I2);
libMesh::Real a_term, A_term;
this->compute_stress_terms( lambda_sq, A_over_a, I1, I2, a_term, A_term );
// Now compute stress
stress.zero();
for( unsigned int alpha = 0; alpha < 2; alpha++ )
{
for( unsigned int beta = 0; beta < 2; beta++ )
{
stress(alpha,beta) = a_contra(alpha,beta)*a_term + A_contra(alpha,beta)*A_term;
}
}
return;
}
void IncompressiblePlaneStressHyperelasticity<StrainEnergy>::compute_stress_deriv_terms( libMesh::Real lambda_sq, libMesh::Real A_over_a,
libMesh::Real I1, libMesh::Real I2,
const libMesh::TensorValue<libMesh::Real>& a_contra,
const libMesh::TensorValue<libMesh::Real>& A_contra,
libMesh::TensorValue<libMesh::Real>& daterm_dstrain,
libMesh::TensorValue<libMesh::Real>& dAterm_dstrain) const
{
daterm_dstrain.zero();
dAterm_dstrain.zero();
libMesh::Real dWdI1 = _W.dI1(I1,I2,1.0); // We're incompressible
libMesh::Real dWdI2 = _W.dI2(I1,I2,1.0);
// A = det(A_cov) = 1/det(A_contra)
//libMesh::Real A = 1.0/( A_contra(0,0)*A_contra(1,1) - A_contra(0,1)*A_contra(1,0) );
for( unsigned int alpha = 0; alpha < 2; alpha++ )
{
for( unsigned int beta = 0; beta < 2; beta++ )
{
// a_term = 2.0*(dWdI1 + dWdI2*lambda^2);
// => da_dstrain = 2.0*dWdI2*dlambda^2_dstrain
// dlambda_sq_dstrain = -2*lambda^2 A^{alpha,beta}
const libMesh::Real dlamsq_dstrain = -2.0*lambda_sq*A_contra(alpha,beta);
daterm_dstrain(alpha,beta) = 2.0*dWdI2*dlamsq_dstrain;
// A_term = 2.0*dWdI2*A_over_a + p;
// A = det(A_cov) ==> dA_dstrain = 2*A*A_contra(alpha,beta)
// p = -2.0*lambda_sq*( dWdI1 + dWdI2*(I1-lambda_sq) );
const libMesh::Real dI1_dstrain = 2.0*a_contra(alpha,beta) + dlamsq_dstrain;
const libMesh::Real dp_dstrain = -2.0*dlamsq_dstrain*( dWdI1 + dWdI2*(I1-lambda_sq) )
-2.0*lambda_sq*dWdI2*(dI1_dstrain - dlamsq_dstrain);
dAterm_dstrain(alpha,beta) = 2.0*dWdI2*A_over_a*(2.0*A_contra(alpha,beta)) + dp_dstrain;
}
}
return;
}
void Hyperelasticity<StrainEnergy>::compute_stress_imp( unsigned int dim,
const libMesh::TensorValue<libMesh::Real>& g_contra,
const libMesh::TensorValue<libMesh::Real>& g_cov,
const libMesh::TensorValue<libMesh::Real>& G_contra,
const libMesh::TensorValue<libMesh::Real>& G_cov,
libMesh::TensorValue<libMesh::Real>& stress )
{
stress.zero();
// Compute strain invariants
libMesh::Real I3 = (g_contra*G_cov).det();
libMesh::Real I1 = 0.0;
libMesh::Real I2 = 0.0;
for( unsigned int i = 0; i < dim; i++ )
{
for( unsigned int j = 0; j < dim; j++ )
{
I1 += g_contra(i,j)*G_cov(i,j);
I2 += G_contra(i,j)*g_cov(i,j);
}
}
I2 *= I3;
libMesh::Real dWdI1 = _W.dI1(I1,I2,I3);
libMesh::Real dWdI2 = _W.dI2(I1,I2,I3);
libMesh::Real dWdI3 = _W.dI3(I1,I2,I3);
// Now compute stress
for( unsigned int i = 0; i < dim; i++ )
{
for( unsigned int j = 0; j < dim; j++ )
{
for( unsigned int k = 0; k < dim; k++ )
{
for( unsigned int l = 0; l < dim; l++ )
{
stress(i,j) += 2.0*dWdI1*g_contra(i,j)
+ 2.0*dWdI2*(I1*g_contra(i,j) - g_contra(i,k)*g_contra(j,l)*G_cov(k,l))
+ 2.0*dWdI3*G_contra(i,j);
}
}
}
}
return;
}
void IncompressiblePlaneStressHyperelasticity<StrainEnergy>::compute_stress_and_elasticity_imp( unsigned int /*dim*/,
const libMesh::TensorValue<libMesh::Real>& a_contra,
const libMesh::TensorValue<libMesh::Real>& a_cov,
const libMesh::TensorValue<libMesh::Real>& A_contra,
const libMesh::TensorValue<libMesh::Real>& A_cov,
libMesh::TensorValue<libMesh::Real>& stress,
ElasticityTensor& C)
{
// We're treating a_* and A_* as 2x2, but we're cheating to pick up lambda^2
libMesh::Real lambda_sq = A_cov(2,2);
libMesh::Real A_over_a = 1.0/lambda_sq; // We're incompressible
libMesh::Real I1, I2;
this->compute_I1_I2(a_contra,a_cov,A_contra,A_cov,lambda_sq,A_over_a,I1,I2);
libMesh::Real a_term, A_term;
this->compute_stress_terms( lambda_sq, A_over_a, I1, I2, a_term, A_term );
libMesh::TensorValue<libMesh::Real> daterm_dstrain, dAterm_dstrain;
this->compute_stress_deriv_terms( lambda_sq, A_over_a, I1, I2, a_contra, A_contra, daterm_dstrain, dAterm_dstrain );
ElasticityTensor dAcontra_dstrain;
this->compute_Acontra_deriv( A_contra, dAcontra_dstrain );
// Now compute stress
stress.zero();
for( unsigned int alpha = 0; alpha < 2; alpha++ )
{
for( unsigned int beta = 0; beta < 2; beta++ )
{
stress(alpha,beta) = a_contra(alpha,beta)*a_term + A_contra(alpha,beta)*A_term;
for( unsigned int lambda = 0; lambda < 2; lambda++ )
{
for( unsigned int mu = 0; mu < 2; mu++ )
{
C(alpha,beta,lambda,mu) = a_contra(alpha,beta)*daterm_dstrain(lambda,mu)
+ dAcontra_dstrain(alpha,beta,lambda,mu)*A_term
+ A_contra(alpha,beta)*dAterm_dstrain(lambda,mu);
}
}
}
}
return;
}