void
RankTwoTensorTest::ddetTest()
{
// this derivative is less trivial than dtrace and dsecondInvariant,
// so let's check with a finite-difference approximation
Real ep = 1E-5; // small finite-difference parameter
Real det; // determinant provided by RankTwoTensor
RankTwoTensor deriv; // derivative of det provided by RankTwoTensor
RankTwoTensor mep; // the RankTwoTensor with successive entries shifted by ep
det = _m3.det();
deriv = _m3.ddet();
mep = _m3;
for (unsigned i = 0 ; i < 3 ; ++i)
for (unsigned j = 0 ; j < 3 ; ++j)
{
mep(i, j) += ep;
CPPUNIT_ASSERT_DOUBLES_EQUAL((mep.det() - det)/ep, deriv(i, j), ep);
mep(i, j) -= ep;
}
det = _unsymmetric1.det();
deriv = _unsymmetric1.ddet();
mep = _unsymmetric1;
for (unsigned i = 0 ; i < 3 ; ++i)
for (unsigned j = 0 ; j < 3 ; ++j)
{
mep(i, j) += ep;
CPPUNIT_ASSERT_DOUBLES_EQUAL((mep.det() - det)/ep, deriv(i, j), ep);
mep(i, j) -= ep;
}
}
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);
}
Real
TensorMechanicsPlasticOrthotropic::yieldFunction(const RankTwoTensor & stress, Real intnl) const
{
const RankTwoTensor j2prime = _l1 * stress;
const RankTwoTensor j3prime = _l2 * stress;
return _b * stress.trace() +
std::pow(std::pow(-j2prime.generalSecondInvariant(), 3.0 / 2.0) - j3prime.det(),
1.0 / 3.0) -
yieldStrength(intnl);
}
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());
}