RankTwoTensor
TensorMechanicsPlasticTensile::dyieldFunction_dstress(const RankTwoTensor & stress,
Real /*intnl*/) const
{
Real mean_stress = stress.trace() / 3.0;
RankTwoTensor dmean_stress = stress.dtrace() / 3.0;
Real sin3Lode = stress.sin3Lode(_lode_cutoff, 0);
if (sin3Lode <= _sin3tt)
{
// the non-edge-smoothed version
std::vector<Real> eigvals;
std::vector<RankTwoTensor> deigvals;
stress.dsymmetricEigenvalues(eigvals, deigvals);
Real denom = std::sqrt(smooth(stress) + Utility::pow<2>(eigvals[2] - mean_stress));
return dmean_stress + (0.5 * dsmooth(stress) * dmean_stress +
(eigvals[2] - mean_stress) * (deigvals[2] - dmean_stress)) /
denom;
}
else
{
// the edge-smoothed version
Real kk = _aaa + _bbb * sin3Lode + _ccc * Utility::pow<2>(sin3Lode);
RankTwoTensor dkk = (_bbb + 2.0 * _ccc * sin3Lode) * stress.dsin3Lode(_lode_cutoff);
Real sibar2 = stress.secondInvariant();
RankTwoTensor dsibar2 = stress.dsecondInvariant();
Real denom = std::sqrt(smooth(stress) + sibar2 * Utility::pow<2>(kk));
return dmean_stress + (0.5 * dsmooth(stress) * dmean_stress +
0.5 * dsibar2 * Utility::pow<2>(kk) + sibar2 * kk * dkk) /
denom;
}
}
/**
* Second derivative of Eginvalues are compared with finite difference method
* This method checks all the elements in RankFourTensor
**/
void
RankTwoEigenRoutinesTest::d2symmetricEigenvaluesTest2()
{
Real ep = 1E-5; // small finite-difference parameter
std::vector<Real> eigvals, eigvalsep, eigvalsep_minus; // eigenvalues in ascending order provided by RankTwoTensor
std::vector<RankTwoTensor> deriv,derivep, derivep_minus; // derivatives of these eigenvalues provided by RankTwoTensor
std::vector<RankFourTensor> second_deriv;
RankTwoTensor mep; // the RankTwoTensor with successive entries shifted by ep
_m2.d2symmetricEigenvalues(second_deriv);
_m2.dsymmetricEigenvalues(eigvals, deriv);
mep = _m2;
for (unsigned int m = 0; m < 3; m++)
for (unsigned i = 0; i < 3; i++)
for (unsigned j = 0; j < 3; j++)
{
for (unsigned int k=0; k<3; k++)
for (unsigned int l=0; l<3; l++)
{
mep(k, l) += ep;
mep.dsymmetricEigenvalues(eigvalsep, derivep);
CPPUNIT_ASSERT_DOUBLES_EQUAL((derivep[m](i, j) - deriv[m](i, j))/ep, second_deriv[m](i, j, k, l), ep);
mep(k, l) -= ep;
}
}
_m3.d2symmetricEigenvalues(second_deriv);
_m3.dsymmetricEigenvalues(eigvals, deriv);
mep = _m3;
for (unsigned int m = 0; m < 3; m++)
for (unsigned i = 0; i < 3; i++)
for (unsigned j = 0; j < 3; j++)
{
for (unsigned int k=0; k<3; k++)
for (unsigned int l=0; l<3; l++)
{
mep(k, l) += ep;
mep.dsymmetricEigenvalues(eigvalsep, derivep);
CPPUNIT_ASSERT_DOUBLES_EQUAL((derivep[m](i, j) - deriv[m](i, j))/ep, second_deriv[m](i, j, k, l), ep);
mep(k, l) -= ep;
}
}
}
RankFourTensor
TensorMechanicsPlasticTensile::dflowPotential_dstress(const RankTwoTensor & stress, const Real & intnl) const
{
Real mean_stress = stress.trace()/3.0;
RankTwoTensor dmean_stress = stress.dtrace()/3.0;
Real sin3Lode = stress.sin3Lode(_lode_cutoff, 0);
if (sin3Lode <= _sin3tt)
{
// the non-edge-smoothed version
std::vector<Real> eigvals;
std::vector<RankTwoTensor> deigvals;
std::vector<RankFourTensor> d2eigvals;
stress.dsymmetricEigenvalues(eigvals, deigvals);
stress.d2symmetricEigenvalues(d2eigvals);
Real denom = std::sqrt(_small_smoother2 + std::pow(eigvals[2] - mean_stress, 2));
RankFourTensor dr_dstress = (eigvals[2] - mean_stress)*d2eigvals[2]/denom;
for (unsigned i = 0 ; i < 3 ; ++i)
for (unsigned j = 0 ; j < 3 ; ++j)
for (unsigned k = 0 ; k < 3 ; ++k)
for (unsigned l = 0 ; l < 3 ; ++l)
dr_dstress(i, j, k, l) += (1 - std::pow((eigvals[2] - mean_stress)/denom, 2))*(deigvals[2](i, j) - dmean_stress(i, j))*(deigvals[2](k, l) - dmean_stress(k, l))/denom;
return dr_dstress;
}
else
{
// the edge-smoothed version
RankTwoTensor dsin3Lode = stress.dsin3Lode(_lode_cutoff);
Real kk = _aaa + _bbb*sin3Lode + _ccc*std::pow(sin3Lode, 2);
RankTwoTensor dkk = (_bbb + 2*_ccc*sin3Lode)*dsin3Lode;
RankFourTensor d2kk = (_bbb + 2*_ccc*sin3Lode)*stress.d2sin3Lode(_lode_cutoff);
for (unsigned i = 0 ; i < 3 ; ++i)
for (unsigned j = 0 ; j < 3 ; ++j)
for (unsigned k = 0 ; k < 3 ; ++k)
for (unsigned l = 0 ; l < 3 ; ++l)
d2kk(i, j, k, l) += 2*_ccc*dsin3Lode(i, j)*dsin3Lode(k, l);
Real sibar2 = stress.secondInvariant();
RankTwoTensor dsibar2 = stress.dsecondInvariant();
RankFourTensor d2sibar2 = stress.d2secondInvariant();
Real denom = std::sqrt(_small_smoother2 + sibar2*std::pow(kk, 2));
RankFourTensor dr_dstress = (0.5*d2sibar2*std::pow(kk, 2) + sibar2*kk*d2kk)/denom;
for (unsigned i = 0 ; i < 3 ; ++i)
for (unsigned j = 0 ; j < 3 ; ++j)
for (unsigned k = 0 ; k < 3 ; ++k)
for (unsigned l = 0 ; l < 3 ; ++l)
{
dr_dstress(i, j, k, l) += (dsibar2(i, j)*dkk(k, l)*kk + dkk(i, j)*dsibar2(k, l)*kk + sibar2*dkk(i, j)*dkk(k, l))/denom;
dr_dstress(i, j, k, l) -= (0.5*dsibar2(i, j)*std::pow(kk, 2) + sibar2*kk*dkk(i, j))*(0.5*dsibar2(k, l)*std::pow(kk, 2) + sibar2*kk*dkk(k, l))/std::pow(denom, 3);
}
return dr_dstress;
}
}
void
TensorMechanicsPlasticMohrCoulombMulti::dflowPotential_dintnlV(const RankTwoTensor & stress, Real intnl, std::vector<RankTwoTensor> & dr_dintnl) const
{
const Real cos_angle = std::cos(psi(intnl));
const Real dsin_angle = cos_angle*dpsi(intnl);
std::vector<Real> eigvals;
std::vector<RankTwoTensor> deigvals;
stress.dsymmetricEigenvalues(eigvals, deigvals);
if (eigvals[0] > eigvals[1] - 0.1*_shift || eigvals[1] > eigvals[2] - 0.1*_shift)
perturbStress(stress, eigvals, deigvals);
dr_dintnl.resize(6);
dr_dintnl[0] = dr_dintnl[1] = 0.5*(deigvals[0] + deigvals[1])*dsin_angle;
dr_dintnl[2] = dr_dintnl[3] = 0.5*(deigvals[0] + deigvals[2])*dsin_angle;
dr_dintnl[4] = dr_dintnl[5] = 0.5*(deigvals[1] + deigvals[2])*dsin_angle;
}
void
TensorMechanicsPlasticMohrCoulombMulti::df_dsig(const RankTwoTensor & stress, Real sin_angle, std::vector<RankTwoTensor> & df) const
{
std::vector<Real> eigvals;
std::vector<RankTwoTensor> deigvals;
stress.dsymmetricEigenvalues(eigvals, deigvals);
if (eigvals[0] > eigvals[1] - 0.1*_shift || eigvals[1] > eigvals[2] - 0.1*_shift)
perturbStress(stress, eigvals, deigvals);
df.resize(6);
df[0] = 0.5*(deigvals[0] - deigvals[1]) + 0.5*(deigvals[0] + deigvals[1])*sin_angle;
df[1] = 0.5*(deigvals[1] - deigvals[0]) + 0.5*(deigvals[0] + deigvals[1])*sin_angle;
df[2] = 0.5*(deigvals[0] - deigvals[2]) + 0.5*(deigvals[0] + deigvals[2])*sin_angle;
df[3] = 0.5*(deigvals[2] - deigvals[0]) + 0.5*(deigvals[0] + deigvals[2])*sin_angle;
df[4] = 0.5*(deigvals[1] - deigvals[2]) + 0.5*(deigvals[1] + deigvals[2])*sin_angle;
df[5] = 0.5*(deigvals[2] - deigvals[1]) + 0.5*(deigvals[1] + deigvals[2])*sin_angle;
}
void
TensorMechanicsPlasticTensileMulti::dyieldFunction_dstressV(const RankTwoTensor & stress, const Real & /*intnl*/, std::vector<RankTwoTensor> & df_dstress) const
{
std::vector<Real> eigvals;
stress.dsymmetricEigenvalues(eigvals, df_dstress);
if (eigvals[0] > eigvals[1] - 0.1*_shift || eigvals[1] > eigvals[2] - 0.1*_shift)
{
Real small_perturbation;
RankTwoTensor shifted_stress = stress;
while (eigvals[0] > eigvals[1] - 0.1*_shift || eigvals[1] > eigvals[2] - 0.1*_shift)
{
for (unsigned i = 0; i < 3; ++i)
for (unsigned j = 0; j <= i; ++j)
{
small_perturbation = 0.1*_shift*2*(MooseRandom::rand() - 0.5);
shifted_stress(i, j) += small_perturbation;
shifted_stress(j, i) += small_perturbation;
}
shifted_stress.dsymmetricEigenvalues(eigvals, df_dstress);
}
}
}
RankFourTensor
TensorMechanicsPlasticTensile::dflowPotential_dstress(const RankTwoTensor & stress,
Real /*intnl*/) const
{
Real mean_stress = stress.trace() / 3.0;
RankTwoTensor dmean_stress = stress.dtrace() / 3.0;
Real sin3Lode = stress.sin3Lode(_lode_cutoff, 0);
if (sin3Lode <= _sin3tt)
{
// the non-edge-smoothed version
std::vector<Real> eigvals;
std::vector<RankTwoTensor> deigvals;
std::vector<RankFourTensor> d2eigvals;
stress.dsymmetricEigenvalues(eigvals, deigvals);
stress.d2symmetricEigenvalues(d2eigvals);
Real denom = std::sqrt(smooth(stress) + Utility::pow<2>(eigvals[2] - mean_stress));
Real denom3 = Utility::pow<3>(denom);
RankTwoTensor numer_part = deigvals[2] - dmean_stress;
RankTwoTensor numer_full =
0.5 * dsmooth(stress) * dmean_stress + (eigvals[2] - mean_stress) * numer_part;
Real d2smooth_over_denom = d2smooth(stress) / denom;
RankFourTensor dr_dstress = (eigvals[2] - mean_stress) * d2eigvals[2] / denom;
for (unsigned i = 0; i < 3; ++i)
for (unsigned j = 0; j < 3; ++j)
for (unsigned k = 0; k < 3; ++k)
for (unsigned l = 0; l < 3; ++l)
{
dr_dstress(i, j, k, l) +=
0.5 * d2smooth_over_denom * dmean_stress(i, j) * dmean_stress(k, l);
dr_dstress(i, j, k, l) += numer_part(i, j) * numer_part(k, l) / denom;
dr_dstress(i, j, k, l) -= numer_full(i, j) * numer_full(k, l) / denom3;
}
return dr_dstress;
}
else
{
// the edge-smoothed version
RankTwoTensor dsin3Lode = stress.dsin3Lode(_lode_cutoff);
Real kk = _aaa + _bbb * sin3Lode + _ccc * Utility::pow<2>(sin3Lode);
RankTwoTensor dkk = (_bbb + 2.0 * _ccc * sin3Lode) * dsin3Lode;
RankFourTensor d2kk = (_bbb + 2.0 * _ccc * sin3Lode) * stress.d2sin3Lode(_lode_cutoff);
for (unsigned i = 0; i < 3; ++i)
for (unsigned j = 0; j < 3; ++j)
for (unsigned k = 0; k < 3; ++k)
for (unsigned l = 0; l < 3; ++l)
d2kk(i, j, k, l) += 2.0 * _ccc * dsin3Lode(i, j) * dsin3Lode(k, l);
Real sibar2 = stress.secondInvariant();
RankTwoTensor dsibar2 = stress.dsecondInvariant();
RankFourTensor d2sibar2 = stress.d2secondInvariant();
Real denom = std::sqrt(smooth(stress) + sibar2 * Utility::pow<2>(kk));
Real denom3 = Utility::pow<3>(denom);
Real d2smooth_over_denom = d2smooth(stress) / denom;
RankTwoTensor numer_full =
0.5 * dsmooth(stress) * dmean_stress + 0.5 * dsibar2 * kk * kk + sibar2 * kk * dkk;
RankFourTensor dr_dstress = (0.5 * d2sibar2 * Utility::pow<2>(kk) + sibar2 * kk * d2kk) / denom;
for (unsigned i = 0; i < 3; ++i)
for (unsigned j = 0; j < 3; ++j)
for (unsigned k = 0; k < 3; ++k)
for (unsigned l = 0; l < 3; ++l)
{
dr_dstress(i, j, k, l) +=
0.5 * d2smooth_over_denom * dmean_stress(i, j) * dmean_stress(k, l);
dr_dstress(i, j, k, l) +=
(dsibar2(i, j) * dkk(k, l) * kk + dkk(i, j) * dsibar2(k, l) * kk +
sibar2 * dkk(i, j) * dkk(k, l)) /
denom;
dr_dstress(i, j, k, l) -= numer_full(i, j) * numer_full(k, l) / denom3;
}
return dr_dstress;
}
}
