RankTwoTensor
TensorMechanicsPlasticMeanCapTC::df_dsig(const RankTwoTensor & stress, Real intnl) const
{
const Real tr = stress.trace();
const Real t_str = tensile_strength(intnl);
if (tr >= t_str)
return stress.dtrace();
const Real c_str = compressive_strength(intnl);
if (tr <= c_str)
return -stress.dtrace();
return - std::cos(M_PI * (tr - c_str) / (t_str - c_str)) * stress.dtrace();
}
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;
}
}
RankTwoTensor
TensorMechanicsPlasticDruckerPrager::dflowPotential_dintnl(const RankTwoTensor & stress,
Real intnl) const
{
Real dbbb;
donlyB(intnl, dilation, dbbb);
return stress.dtrace() * dbbb;
}
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;
}
}
RankFourTensor
TensorMechanicsPlasticMeanCapTC::dflowPotential_dstress(const RankTwoTensor & stress, Real intnl) const
{
const Real tr = stress.trace();
const Real t_str = tensile_strength(intnl);
if (tr >= t_str)
return RankFourTensor();
const Real c_str = compressive_strength(intnl);
if (tr <= c_str)
return RankFourTensor();
return M_PI / (t_str - c_str) * std::sin(M_PI * (tr - c_str) / (t_str - c_str)) * stress.dtrace().outerProduct(stress.dtrace());
}
RankTwoTensor
TensorMechanicsPlasticMeanCapTC::dflowPotential_dintnl(const RankTwoTensor & stress, Real intnl) const
{
const Real tr = stress.trace();
const Real t_str = tensile_strength(intnl);
if (tr >= t_str)
return RankTwoTensor();
const Real c_str = compressive_strength(intnl);
if (tr <= c_str)
return RankTwoTensor();
const Real dt = dtensile_strength(intnl);
const Real dc = dcompressive_strength(intnl);
return std::sin(M_PI * (tr - c_str) / (t_str - c_str)) * stress.dtrace() * M_PI / std::pow(t_str - c_str, 2) * ((tr - t_str) * dc - (tr - c_str) * dt);
}
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;
}
}
RankTwoTensor
TensorMechanicsPlasticDruckerPrager::df_dsig(const RankTwoTensor & stress, Real bbb) const
{
return 0.5 * stress.dsecondInvariant() / std::sqrt(stress.secondInvariant()) +
stress.dtrace() * bbb;
}
bool
TensorMechanicsPlasticMeanCapTC::returnMap(const RankTwoTensor & trial_stress, Real intnl_old, const RankFourTensor & E_ijkl,
Real ep_plastic_tolerance, RankTwoTensor & returned_stress, Real & returned_intnl,
std::vector<Real> & dpm, RankTwoTensor & delta_dp, std::vector<Real> & yf,
bool & trial_stress_inadmissible) const
{
if (!(_use_custom_returnMap))
return TensorMechanicsPlasticModel::returnMap(trial_stress, intnl_old, E_ijkl, ep_plastic_tolerance, returned_stress, returned_intnl, dpm, delta_dp, yf, trial_stress_inadmissible);
yf.resize(1);
Real yf_orig = yieldFunction(trial_stress, intnl_old);
yf[0] = yf_orig;
if (yf_orig < _f_tol)
{
// the trial_stress is admissible
trial_stress_inadmissible = false;
return true;
}
trial_stress_inadmissible = true;
// In the following we want to solve
// trial_stress - stress = E_ijkl * dpm * r ...... (1)
// and either
// stress.trace() = tensile_strength(intnl) ...... (2a)
// intnl = intnl_old + dpm ...... (3a)
// or
// stress.trace() = compressive_strength(intnl) ... (2b)
// intnl = intnl_old - dpm ...... (3b)
// The former (2a and 3a) are chosen if
// trial_stress.trace() > tensile_strength(intnl_old)
// while the latter (2b and 3b) are chosen if
// trial_stress.trace() < compressive_strength(intnl_old)
// The variables we want to solve for are stress, dpm
// and intnl. We do this using a Newton approach, starting
// with stress=trial_stress and intnl=intnl_old and dpm=0
const bool tensile_failure = (trial_stress.trace() >= tensile_strength(intnl_old));
const Real dirn = (tensile_failure ? 1.0 : -1.0);
RankTwoTensor n; // flow direction, which is E_ijkl * r
for (unsigned i = 0; i < 3; ++i)
for (unsigned j = 0; j < 3; ++j)
for (unsigned k = 0; k < 3; ++k)
n(i, j) += dirn * E_ijkl(i, j, k, k);
const Real n_trace = n.trace();
// Perform a Newton-Raphson to find dpm when
// residual = trial_stress.trace() - tensile_strength(intnl) - dpm * n.trace() [for tensile_failure=true]
// or
// residual = trial_stress.trace() - compressive_strength(intnl) - dpm * n.trace() [for tensile_failure=false]
Real trial_trace = trial_stress.trace();
Real residual;
Real jac;
dpm[0] = 0;
unsigned int iter = 0;
do {
if (tensile_failure)
{
residual = trial_trace - tensile_strength(intnl_old + dpm[0]) - dpm[0] * n_trace;
jac = -dtensile_strength(intnl_old + dpm[0]) - n_trace;
}
else
{
residual = trial_trace - compressive_strength(intnl_old - dpm[0]) - dpm[0] * n_trace;
jac = -dcompressive_strength(intnl_old - dpm[0]) - n_trace;
}
dpm[0] += -residual/jac;
if (iter > _max_iters) // not converging
return false;
iter++;
} while (residual*residual > _f_tol*_f_tol);
// set the returned values
yf[0] = 0;
returned_intnl = intnl_old + dirn * dpm[0];
returned_stress = trial_stress - dpm[0] * n;
delta_dp = dpm[0] * dirn * returned_stress.dtrace();
return true;
}