//! Compute cosine of the angle between two vectors.
double computeCosineOfAngleBetweenVectors( const Eigen::VectorXd& vector0,
const Eigen::VectorXd& vector1 )
{
assert( vector0.size( ) == vector1.size( ) );
// Get the cosine of the angle by dotting the normalized vectors.
double dotProductOfNormalizedVectors = vector0.normalized( ).dot( vector1.normalized( ) );
// Explicitly define the extreme cases, which can give problems with the acos function.
if ( dotProductOfNormalizedVectors >= 1.0 )
{
return 1.0;
}
else if ( dotProductOfNormalizedVectors <= -1.0 )
{
return -1.0;
}
// Determine the actual angle.
else
{
return dotProductOfNormalizedVectors;
}
}
void MohrCoulomb<DisplacementDim>::computeConstitutiveRelation(
double const t,
ProcessLib::SpatialPosition const& x,
double const aperture0,
Eigen::Ref<Eigen::VectorXd const>
sigma0,
Eigen::Ref<Eigen::VectorXd const>
w_prev,
Eigen::Ref<Eigen::VectorXd const>
w,
Eigen::Ref<Eigen::VectorXd const>
sigma_prev,
Eigen::Ref<Eigen::VectorXd>
sigma,
Eigen::Ref<Eigen::MatrixXd>
Kep,
typename FractureModelBase<DisplacementDim>::MaterialStateVariables&
material_state_variables)
{
material_state_variables.reset();
MaterialPropertyValues const mat(_mp, t, x);
Eigen::VectorXd const dw = w - w_prev;
const int index_ns = DisplacementDim - 1;
double const aperture = w[index_ns] + aperture0;
double const aperture_prev = w_prev[index_ns] + aperture0;
Eigen::MatrixXd Ke;
{ // Elastic tangent stiffness
Ke = Eigen::MatrixXd::Zero(DisplacementDim, DisplacementDim);
for (int i = 0; i < index_ns; i++)
Ke(i, i) = mat.Ks;
Ke(index_ns, index_ns) =
mat.Kn *
logPenaltyDerivative(aperture0, aperture, _penalty_aperture_cutoff);
}
Eigen::MatrixXd Ke_prev;
{ // Elastic tangent stiffness at w_prev
Ke_prev = Eigen::MatrixXd::Zero(DisplacementDim, DisplacementDim);
for (int i = 0; i < index_ns; i++)
Ke_prev(i, i) = mat.Ks;
Ke_prev(index_ns, index_ns) =
mat.Kn * logPenaltyDerivative(
aperture0, aperture_prev, _penalty_aperture_cutoff);
}
// Total plastic aperture compression
// NOTE: Initial condition sigma0 seems to be associated with an initial
// condition of the w0 = 0. Therefore the initial state is not associated
// with a plastic aperture change.
Eigen::VectorXd const w_p_prev =
w_prev - Ke_prev.fullPivLu().solve(sigma_prev - sigma0);
{ // Exact elastic predictor
sigma.noalias() = Ke * (w - w_p_prev);
sigma.coeffRef(index_ns) =
mat.Kn * w[index_ns] *
logPenalty(aperture0, aperture, _penalty_aperture_cutoff);
}
sigma.noalias() += sigma0;
double const sigma_n = sigma[index_ns];
// correction for an opening fracture
if (_tension_cutoff && sigma_n > 0)
{
Kep.setZero();
sigma.setZero();
material_state_variables.setTensileStress(true);
return;
}
// check shear yield function (Fs)
Eigen::VectorXd const sigma_s = sigma.head(DisplacementDim-1);
double const mag_tau = sigma_s.norm(); // magnitude
double const Fs = mag_tau + sigma_n * std::tan(mat.phi) - mat.c;
material_state_variables.setShearYieldFunctionValue(Fs);
if (Fs < .0)
{
Kep = Ke;
return;
}
Eigen::VectorXd dFs_dS(DisplacementDim);
dFs_dS.head(DisplacementDim-1).noalias() = sigma_s.normalized();
dFs_dS[index_ns] = std::tan(mat.phi);
// plastic potential function: Qs = |tau| + Sn * tan da
Eigen::VectorXd dQs_dS = dFs_dS;
dQs_dS[index_ns] = std::tan(mat.psi);
// plastic multiplier
Eigen::RowVectorXd const A = dFs_dS.transpose() * Ke / (dFs_dS.transpose() * Ke * dQs_dS);
double const d_eta = A * dw;
// plastic part of the dispalcement
Eigen::VectorXd const dwp = dQs_dS * d_eta;
// correct stress
sigma.noalias() = sigma_prev + Ke * (dw - dwp);
// Kep
Kep = Ke - Ke * dQs_dS * A;
}