void
ComputeSmearedCrackingStress::computeCrackStrainAndOrientation(
RealVectorValue & strain_in_crack_dir)
{
// The rotation tensor is ordered such that directions for pre-existing cracks appear first
// in the list of columns. For example, if there is one existing crack, its direction is in the
// first column in the rotation tensor.
const unsigned int num_known_dirs = getNumKnownCrackDirs();
if (num_known_dirs == 0)
{
std::vector<Real> eigval(3, 0.0);
RankTwoTensor eigvec;
_elastic_strain[_qp].symmetricEigenvaluesEigenvectors(eigval, eigvec);
// If the elastic strain is beyond the cracking strain, save the eigen vectors as
// the rotation tensor. Reverse their order so that the third principal strain
// (most tensile) will correspond to the first crack.
_crack_rotation[_qp].fillColumn(0, eigvec.column(2));
_crack_rotation[_qp].fillColumn(1, eigvec.column(1));
_crack_rotation[_qp].fillColumn(2, eigvec.column(0));
strain_in_crack_dir(0) = eigval[2];
strain_in_crack_dir(1) = eigval[1];
strain_in_crack_dir(2) = eigval[0];
}
else if (num_known_dirs == 1)
{
// This is easily the most complicated case.
// 1. Rotate the elastic strain to the orientation associated with the known
// crack.
// 2. Extract the lower 2x2 block into a separate tensor.
// 3. Run the eigen solver on the result.
// 4. Update the rotation tensor to reflect the effect of the 2 eigenvectors.
// 1.
const RankTwoTensor & R = _crack_rotation[_qp];
RankTwoTensor ePrime(_elastic_strain[_qp]);
ePrime.rotate(R.transpose()); // elastic strain in crack coordinates
// 2.
ColumnMajorMatrix e2x2(2, 2);
e2x2(0, 0) = ePrime(1, 1);
e2x2(1, 0) = ePrime(2, 1);
e2x2(0, 1) = ePrime(1, 2);
e2x2(1, 1) = ePrime(2, 2);
// 3.
ColumnMajorMatrix e_val2x1(2, 1);
ColumnMajorMatrix e_vec2x2(2, 2);
e2x2.eigen(e_val2x1, e_vec2x2);
// 4.
RankTwoTensor eigvec(
1.0, 0.0, 0.0, 0.0, e_vec2x2(0, 1), e_vec2x2(1, 1), 0.0, e_vec2x2(0, 0), e_vec2x2(1, 0));
_crack_rotation[_qp] = _crack_rotation_old[_qp] * eigvec; // Roe implementation
strain_in_crack_dir(0) = ePrime(0, 0);
strain_in_crack_dir(1) = e_val2x1(1, 0);
strain_in_crack_dir(2) = e_val2x1(0, 0);
}
else if (num_known_dirs == 2 || num_known_dirs == 3)
{
// Rotate to cracked orientation and pick off the strains in the rotated
// coordinate directions.
const RankTwoTensor & R = _crack_rotation[_qp];
RankTwoTensor ePrime(_elastic_strain[_qp]);
ePrime.rotate(R.transpose()); // elastic strain in crack coordinates
strain_in_crack_dir(0) = ePrime(0, 0);
strain_in_crack_dir(1) = ePrime(1, 1);
strain_in_crack_dir(2) = ePrime(2, 2);
}
else
mooseError("Invalid number of known crack directions");
}
void
HyperElasticPhaseFieldIsoDamage::computeDamageStress()
{
Real lambda = _elasticity_tensor[_qp](0, 0, 1, 1);
Real mu = _elasticity_tensor[_qp](0, 1, 0, 1);
Real c = _c[_qp];
Real xfac = Utility::pow<2>(1.0 - c) + _kdamage;
std::vector<Real> eigval;
RankTwoTensor evec;
_ee.symmetricEigenvaluesEigenvectors(eigval, evec);
for (unsigned int i = 0; i < LIBMESH_DIM; ++i)
_etens[i].vectorOuterProduct(evec.column(i), evec.column(i));
Real etr = 0.0;
for (unsigned int i = 0; i < LIBMESH_DIM; ++i)
etr += eigval[i];
Real etrpos = (std::abs(etr) + etr) / 2.0;
Real etrneg = (std::abs(etr) - etr) / 2.0;
RankTwoTensor pk2pos, pk2neg;
for (unsigned int i = 0; i < LIBMESH_DIM; ++i)
{
pk2pos += _etens[i] * (lambda * etrpos + 2.0 * mu * (std::abs(eigval[i]) + eigval[i]) / 2.0);
pk2neg += _etens[i] * (lambda * etrneg + 2.0 * mu * (std::abs(eigval[i]) - eigval[i]) / 2.0);
}
_pk2_tmp = pk2pos * xfac - pk2neg;
if (_save_state)
{
std::vector<Real> epos(LIBMESH_DIM), eneg(LIBMESH_DIM);
for (unsigned int i = 0; i < LIBMESH_DIM; ++i)
{
epos[i] = (std::abs(eigval[i]) + eigval[i]) / 2.0;
eneg[i] = (std::abs(eigval[i]) - eigval[i]) / 2.0;
}
// sum squares of epos and eneg
Real pval(0.0), nval(0.0);
for (unsigned int i = 0; i < LIBMESH_DIM; ++i)
{
pval += epos[i] * epos[i];
nval += eneg[i] * eneg[i];
}
// Energy with positive principal strains
const Real G0_pos = lambda * etrpos * etrpos / 2.0 + mu * pval;
const Real G0_neg = lambda * etrneg * etrneg / 2.0 + mu * nval;
// Assign history variable and derivative
if (G0_pos > _hist_old[_qp])
_hist[_qp] = G0_pos;
else
_hist[_qp] = _hist_old[_qp];
Real hist_variable = _hist_old[_qp];
if (_use_current_hist)
hist_variable = _hist[_qp];
// Elastic free energy density
_F[_qp] = hist_variable * xfac - G0_neg + _gc[_qp] / (2 * _l[_qp]) * c * c;
// derivative of elastic free energy density wrt c
_dFdc[_qp] = -hist_variable * 2.0 * (1.0 - c) * (1 - _kdamage) + _gc[_qp] / _l[_qp] * c;
// 2nd derivative of elastic free energy density wrt c
_d2Fdc2[_qp] = hist_variable * 2.0 * (1 - _kdamage) + _gc[_qp] / _l[_qp];
_dG0_dee = pk2pos;
_dpk2_dc = -pk2pos * 2.0 * (1.0 - c);
}
}