RankFourTensor
TensorMechanicsPlasticOrthotropic::dflowPotential_dstress(const RankTwoTensor & stress,
Real /*intnl*/) const
{
if (_associative)
{
const RankTwoTensor j2prime = _l1 * stress;
const RankTwoTensor j3prime = _l2 * stress;
const RankTwoTensor dj2 = dj2_dSkl(j2prime);
const RankTwoTensor dj3 = j3prime.ddet();
const Real j2 = -j2prime.generalSecondInvariant();
const Real j3 = j3prime.det();
const RankFourTensor dr =
dfj2_dj2(j2, j3) *
_l1.innerProductTranspose(dj2).outerProduct(_l1.innerProductTranspose(dj2)) +
dfj2_dj3(j2, j3) *
_l1.innerProductTranspose(dj2).outerProduct(_l2.innerProductTranspose(dj3)) +
dfj3_dj2(j2, j3) *
_l2.innerProductTranspose(dj3).outerProduct(_l1.innerProductTranspose(dj2)) +
dfj3_dj3(j2, j3) *
_l2.innerProductTranspose(dj3).outerProduct(_l2.innerProductTranspose(dj3));
const RankTwoTensor r = _b * dI_sigma() +
dphi_dj2(j2, j3) * _l1.innerProductTranspose(dj2_dSkl(j2prime)) +
dphi_dj3(j2, j3) * _l2.innerProductTranspose(j3prime.ddet());
const Real norm = r.L2norm();
return dr / norm - (r / Utility::pow<3>(norm)).outerProduct(dr.innerProductTranspose(r));
}
else
return TensorMechanicsPlasticJ2::dflowPotential_dstress(stress, 0);
}
void
StressDivergenceTensors::computeFiniteDeformJacobian()
{
const RankTwoTensor I(RankTwoTensor::initIdentity);
const RankFourTensor II_ijkl = I.mixedProductIkJl(I);
// Bring back to unrotated config
const RankTwoTensor unrotated_stress =
(*_rotation_increment)[_qp].transpose() * _stress[_qp] * (*_rotation_increment)[_qp];
// Incremental deformation gradient Fhat
const RankTwoTensor Fhat =
(*_deformation_gradient)[_qp] * (*_deformation_gradient_old)[_qp].inverse();
const RankTwoTensor Fhatinv = Fhat.inverse();
const RankTwoTensor rot_times_stress = (*_rotation_increment)[_qp] * unrotated_stress;
const RankFourTensor dstress_drot =
I.mixedProductIkJl(rot_times_stress) + I.mixedProductJkIl(rot_times_stress);
const RankFourTensor rot_rank_four =
(*_rotation_increment)[_qp].mixedProductIkJl((*_rotation_increment)[_qp]);
const RankFourTensor drot_dUhatinv = Fhat.mixedProductIkJl(I);
const RankTwoTensor A = I - Fhatinv;
// Ctilde = Chat^-1 - I
const RankTwoTensor Ctilde = A * A.transpose() - A - A.transpose();
const RankFourTensor dCtilde_dFhatinv =
-I.mixedProductIkJl(A) - I.mixedProductJkIl(A) + II_ijkl + I.mixedProductJkIl(I);
// Second order approximation of Uhat - consistent with strain increment definition
// const RankTwoTensor Uhat = I - 0.5 * Ctilde - 3.0/8.0 * Ctilde * Ctilde;
RankFourTensor dUhatinv_dCtilde =
0.5 * II_ijkl - 1.0 / 8.0 * (I.mixedProductIkJl(Ctilde) + Ctilde.mixedProductIkJl(I));
RankFourTensor drot_dFhatinv = drot_dUhatinv * dUhatinv_dCtilde * dCtilde_dFhatinv;
drot_dFhatinv -= Fhat.mixedProductIkJl((*_rotation_increment)[_qp].transpose());
_finite_deform_Jacobian_mult[_qp] = dstress_drot * drot_dFhatinv;
const RankFourTensor dstrain_increment_dCtilde =
-0.5 * II_ijkl + 0.25 * (I.mixedProductIkJl(Ctilde) + Ctilde.mixedProductIkJl(I));
_finite_deform_Jacobian_mult[_qp] +=
rot_rank_four * _Jacobian_mult[_qp] * dstrain_increment_dCtilde * dCtilde_dFhatinv;
_finite_deform_Jacobian_mult[_qp] += Fhat.mixedProductJkIl(_stress[_qp]);
const RankFourTensor dFhat_dFhatinv = -Fhat.mixedProductIkJl(Fhat.transpose());
const RankTwoTensor dJ_dFhatinv = dFhat_dFhatinv.innerProductTranspose(Fhat.ddet());
// Component from Jacobian derivative
_finite_deform_Jacobian_mult[_qp] += _stress[_qp].outerProduct(dJ_dFhatinv);
// Derivative of Fhatinv w.r.t. undisplaced coordinates
const RankTwoTensor Finv = (*_deformation_gradient)[_qp].inverse();
const RankFourTensor dFhatinv_dGradu = -Fhatinv.mixedProductIkJl(Finv.transpose());
_finite_deform_Jacobian_mult[_qp] = _finite_deform_Jacobian_mult[_qp] * dFhatinv_dGradu;
}
RankTwoTensor
TensorMechanicsPlasticOrthotropic::dyieldFunction_dstress(const RankTwoTensor & stress,
Real /*intnl*/) const
{
const RankTwoTensor j2prime = _l1 * stress;
const RankTwoTensor j3prime = _l2 * stress;
const Real j2 = -j2prime.generalSecondInvariant();
const Real j3 = j3prime.det();
return _b * dI_sigma() + dphi_dj2(j2, j3) * _l1.innerProductTranspose(dj2_dSkl(j2prime)) +
dphi_dj3(j2, j3) * _l2.innerProductTranspose(j3prime.ddet());
}