void
RedbackMechMaterialCC::getJac(const RankTwoTensor & sig,
const RankFourTensor & E_ijkl,
Real flow_incr,
Real sig_eqv,
Real pressure,
Real p_yield_stress,
Real q_yield_stress,
Real pc,
RankFourTensor & dresid_dsig)
{
unsigned i, j, k, l;
RankTwoTensor sig_dev, flow_dirn_vol, flow_dirn_dev, fij, flow_dirn, flow_tensor;
RankTwoTensor dfi_dft;
RankFourTensor dfd_dsig, dfi_dsig;
Real f1, f2;
Real dfi_dseqv_dev, dfi_dseqv_vol, dfi_dseqv;
pc *= -1;
sig_dev = sig.deviatoric();
dfi_dseqv = getDerivativeFlowIncrement(sig, pressure, sig_eqv, pc, q_yield_stress, p_yield_stress);
getFlowTensor(sig, sig_eqv, pressure, pc, flow_dirn);
/* The following calculates the tensorial derivative (Jacobian) of the residual with respect to stress, dr_dsig
* It consists of two terms: The first is
* dr_dsig = (dfi_dseqv_dev*flow_dirn_dev(k,l)) * flow_dirn_dev(i,j)
* which is the tensorial product of the flow increment tensor times the flow direction tensor
*
* The second is the product of the flow increment tensor times the derivative of the flow direction tensor
* with respect to the stress tensor. See also REDBACK's documentation
* */
// This loop calculates the first term
for (i = 0; i < 3; ++i)
for (j = 0; j < 3; ++j)
for (k = 0; k < 3; ++k)
for (l = 0; l < 3; ++l)
dfi_dsig(i, j, k, l) = flow_dirn(i, j) * flow_dirn(k, l) * dfi_dseqv;
// Real flow_tensor_norm = flow_dirn.L2norm();
// This loop calculates the second term. Read REDBACK's documentation
// (same as J2 plasticity case)
f1 = 0.0;
f2 = 0.0;
if (sig_eqv > 1e-8)
{
f1 = 3.0 / (_slope_yield_surface * _slope_yield_surface);
f2 = 2.0 / 9.0 - 1.0 / (_slope_yield_surface * _slope_yield_surface);
}
for (i = 0; i < 3; ++i)
for (j = 0; j < 3; ++j)
for (k = 0; k < 3; ++k)
for (l = 0; l < 3; ++l)
dfd_dsig(i, j, k, l) =
f1 * deltaFunc(i, k) * deltaFunc(j, l) -
f2 * deltaFunc(i, j) * deltaFunc(k, l); // d_flow_dirn/d_sig - 2nd part (J2 plasticity)
// dfd_dsig = dft_dsig1/flow_tensor_norm - 3.0 * dft_dsig2 /
// (2*sig_eqv*flow_tensor_norm*flow_tensor_norm*flow_tensor_norm); //d_flow_dirn/d_sig
// TODO: check if the previous two lines (i.e normalizing the flow vector) should be activated or not. Currently we
// are using the non-unitary flow vector
dresid_dsig = E_ijkl.invSymm() + dfd_dsig * flow_incr + dfi_dsig; // Jacobian
}
void
RedbackMechMaterialCC::getJac(const RankTwoTensor & sig,
const RankFourTensor & E_ijkl,
Real flow_incr,
Real sig_eqv,
Real pressure,
Real p_yield_stress,
Real q_yield_stress,
Real yield_stress,
RankFourTensor & dresid_dsig)
{
unsigned i, j, k, l;
RankTwoTensor sig_dev, flow_dirn_vol, flow_dirn_dev, fij, flow_dirn, flow_tensor;
RankTwoTensor dfi_dft;
RankFourTensor dfd_dsig, dfi_dsig;
Real f1, f2, f3;
Real dfi_dp, dfi_dseqv;
Real pc = -yield_stress;
sig_dev = sig.deviatoric();
getDerivativeFlowIncrement(dfi_dp, dfi_dseqv, sig, pressure, sig_eqv, pc, q_yield_stress, p_yield_stress);
getFlowTensor(sig, sig_eqv, pressure, q_yield_stress, p_yield_stress, yield_stress, flow_dirn);
/* The following calculates the tensorial derivative (Jacobian) of the
*residual with respect to stress, dr_dsig
* It consists of two terms: The first is
* dr_dsig = (dfi_dseqv_dev*flow_dirn_dev(k,l)) * flow_dirn_dev(i,j)
* which is the tensorial product of the flow increment tensor times the flow
*direction tensor
*
* The second is the product of the flow increment tensor times the derivative
*of the flow direction tensor
* with respect to the stress tensor. See also REDBACK's documentation
* */
f1 = 0.0;
f2 = 0.0;
f3 = 0.0;
if (sig_eqv > 1e-10)
{
f1 = 3.0 / (_slope_yield_surface * _slope_yield_surface);
f2 = 2.0 / 9.0 - 1.0 / (_slope_yield_surface * _slope_yield_surface);
f3 = 3.0 / (2.0 * sig_eqv);
}
// This loop calculates the first term
for (i = 0; i < 3; ++i)
for (j = 0; j < 3; ++j)
for (k = 0; k < 3; ++k)
for (l = 0; l < 3; ++l)
dfi_dsig(i, j, k, l) = flow_dirn(i, j) * (f3 * sig_dev(k, l) * dfi_dseqv + dfi_dp * deltaFunc(k, l) / 3.0);
// This loop calculates the second term.
for (i = 0; i < 3; ++i)
for (j = 0; j < 3; ++j)
for (k = 0; k < 3; ++k)
for (l = 0; l < 3; ++l)
dfd_dsig(i, j, k, l) = f1 * deltaFunc(i, k) * deltaFunc(j, l) + f2 * deltaFunc(i, j) * deltaFunc(k, l);
dresid_dsig = E_ijkl.invSymm() + dfd_dsig * flow_incr + dfi_dsig; // Jacobian
}
