Real
InteractionIntegral::computeQpIntegral()
{
Real scalar_q = 0.0;
RealVectorValue grad_q(0.0, 0.0, 0.0);
const std::vector<std::vector<Real>> & phi_curr_elem = *_phi_curr_elem;
const std::vector<std::vector<RealGradient>> & dphi_curr_elem = *_dphi_curr_elem;
for (unsigned int i = 0; i < _current_elem->n_nodes(); ++i)
{
scalar_q += phi_curr_elem[i][_qp] * _q_curr_elem[i];
for (unsigned int j = 0; j < _current_elem->dim(); ++j)
grad_q(j) += dphi_curr_elem[i][_qp](j) * _q_curr_elem[i];
}
// In the crack front coordinate system, the crack direction is (1,0,0)
RealVectorValue crack_direction(0.0);
crack_direction(0) = 1.0;
// Calculate (r,theta) position of qp relative to crack front
Point p(_q_point[_qp]);
_crack_front_definition->calculateRThetaToCrackFront(p, _crack_front_point_index, _r, _theta);
RankTwoTensor aux_stress;
RankTwoTensor aux_du;
if (_sif_mode == SifMethod::KI || _sif_mode == SifMethod::KII || _sif_mode == SifMethod::KIII)
computeAuxFields(aux_stress, aux_du);
else if (_sif_mode == SifMethod::T)
computeTFields(aux_stress, aux_du);
RankTwoTensor grad_disp((*_grad_disp[0])[_qp], (*_grad_disp[1])[_qp], (*_grad_disp[2])[_qp]);
// Rotate stress, strain, displacement and temperature to crack front coordinate system
RealVectorValue grad_q_cf =
_crack_front_definition->rotateToCrackFrontCoords(grad_q, _crack_front_point_index);
RankTwoTensor grad_disp_cf =
_crack_front_definition->rotateToCrackFrontCoords(grad_disp, _crack_front_point_index);
RankTwoTensor stress_cf =
_crack_front_definition->rotateToCrackFrontCoords((*_stress)[_qp], _crack_front_point_index);
RankTwoTensor strain_cf =
_crack_front_definition->rotateToCrackFrontCoords((*_strain)[_qp], _crack_front_point_index);
RealVectorValue grad_temp_cf =
_crack_front_definition->rotateToCrackFrontCoords(_grad_temp[_qp], _crack_front_point_index);
RankTwoTensor dq;
dq(0, 0) = crack_direction(0) * grad_q_cf(0);
dq(0, 1) = crack_direction(0) * grad_q_cf(1);
dq(0, 2) = crack_direction(0) * grad_q_cf(2);
// Calculate interaction integral terms
// Term1 = stress * x1-derivative of aux disp * dq
RankTwoTensor tmp1 = dq * stress_cf;
Real term1 = aux_du.doubleContraction(tmp1);
// Term2 = aux stress * x1-derivative of disp * dq
RankTwoTensor tmp2 = dq * aux_stress;
Real term2 = grad_disp_cf(0, 0) * tmp2(0, 0) + grad_disp_cf(1, 0) * tmp2(0, 1) +
grad_disp_cf(2, 0) * tmp2(0, 2);
// Term3 = aux stress * strain * dq_x (= stress * aux strain * dq_x)
Real term3 = dq(0, 0) * aux_stress.doubleContraction(strain_cf);
// Term4 (thermal strain term) = q * aux_stress * alpha * dtheta_x
// - the term including the derivative of alpha is not implemented
Real term4 = 0.0;
if (_has_temp)
{
Real sigma_alpha = aux_stress.doubleContraction((*_total_deigenstrain_dT)[_qp]);
term4 = scalar_q * sigma_alpha * grad_temp_cf(0);
}
Real q_avg_seg = 1.0;
if (!_crack_front_definition->treatAs2D())
{
q_avg_seg =
(_crack_front_definition->getCrackFrontForwardSegmentLength(_crack_front_point_index) +
_crack_front_definition->getCrackFrontBackwardSegmentLength(_crack_front_point_index)) /
2.0;
}
Real eq = term1 + term2 - term3 + term4;
if (_has_symmetry_plane)
eq *= 2.0;
return eq / q_avg_seg;
}
Real
InteractionIntegralSM::computeQpIntegral()
{
Real scalar_q = 0.0;
RealVectorValue grad_q(0.0, 0.0, 0.0);
const std::vector<std::vector<Real>> & phi_curr_elem = *_phi_curr_elem;
const std::vector<std::vector<RealGradient>> & dphi_curr_elem = *_dphi_curr_elem;
for (unsigned int i = 0; i < _current_elem->n_nodes(); ++i)
{
scalar_q += phi_curr_elem[i][_qp] * _q_curr_elem[i];
for (unsigned int j = 0; j < _current_elem->dim(); ++j)
grad_q(j) += dphi_curr_elem[i][_qp](j) * _q_curr_elem[i];
}
// In the crack front coordinate system, the crack direction is (1,0,0)
RealVectorValue crack_direction(0.0);
crack_direction(0) = 1.0;
RankTwoTensor stress;
stress(0, 0) = _stress[_qp].xx();
stress(0, 1) = _stress[_qp].xy();
stress(0, 2) = _stress[_qp].xz();
stress(1, 0) = _stress[_qp].xy();
stress(1, 1) = _stress[_qp].yy();
stress(1, 2) = _stress[_qp].yz();
stress(2, 0) = _stress[_qp].xz();
stress(2, 1) = _stress[_qp].yz();
stress(2, 2) = _stress[_qp].zz();
RankTwoTensor strain;
strain(0, 0) = _strain[_qp].xx();
strain(0, 1) = _strain[_qp].xy();
strain(0, 2) = _strain[_qp].xz();
strain(1, 0) = _strain[_qp].xy();
strain(1, 1) = _strain[_qp].yy();
strain(1, 2) = _strain[_qp].yz();
strain(2, 0) = _strain[_qp].xz();
strain(2, 1) = _strain[_qp].yz();
strain(2, 2) = _strain[_qp].zz();
RankTwoTensor grad_disp;
grad_disp(0, 0) = _grad_disp_x[_qp](0);
grad_disp(0, 1) = _grad_disp_x[_qp](1);
grad_disp(0, 2) = _grad_disp_x[_qp](2);
grad_disp(1, 0) = _grad_disp_y[_qp](0);
grad_disp(1, 1) = _grad_disp_y[_qp](1);
grad_disp(1, 2) = _grad_disp_y[_qp](2);
grad_disp(2, 0) = _grad_disp_z[_qp](0);
grad_disp(2, 1) = _grad_disp_z[_qp](1);
grad_disp(2, 2) = _grad_disp_z[_qp](2);
// Rotate stress, strain, displacement and temperature to crack front coordinate system
RealVectorValue grad_q_cf =
_crack_front_definition->rotateToCrackFrontCoords(grad_q, _crack_front_point_index);
RankTwoTensor grad_disp_cf =
_crack_front_definition->rotateToCrackFrontCoords(grad_disp, _crack_front_point_index);
RankTwoTensor stress_cf =
_crack_front_definition->rotateToCrackFrontCoords(stress, _crack_front_point_index);
RankTwoTensor strain_cf =
_crack_front_definition->rotateToCrackFrontCoords(strain, _crack_front_point_index);
RealVectorValue grad_temp_cf =
_crack_front_definition->rotateToCrackFrontCoords(_grad_temp[_qp], _crack_front_point_index);
RankTwoTensor dq;
dq(0, 0) = crack_direction(0) * grad_q_cf(0);
dq(0, 1) = crack_direction(0) * grad_q_cf(1);
dq(0, 2) = crack_direction(0) * grad_q_cf(2);
// Calculate interaction integral terms
// Term1 = stress * x1-derivative of aux disp * dq
RankTwoTensor tmp1 = dq * stress_cf;
Real term1 = _aux_grad_disp[_qp].doubleContraction(tmp1);
// Term2 = aux stress * x1-derivative of disp * dq
RankTwoTensor tmp2 = dq * _aux_stress[_qp];
Real term2 = grad_disp_cf(0, 0) * tmp2(0, 0) + grad_disp_cf(1, 0) * tmp2(0, 1) +
grad_disp_cf(2, 0) * tmp2(0, 2);
// Term3 = aux stress * strain * dq_x (= stress * aux strain * dq_x)
Real term3 = dq(0, 0) * _aux_stress[_qp].doubleContraction(strain_cf);
// Term4 (thermal strain term) = q * aux_stress * alpha * dtheta_x
// - the term including the derivative of alpha is not implemented
Real term4 = 0.0;
if (_has_temp)
{
Real aux_stress_trace =
_aux_stress[_qp](0, 0) + _aux_stress[_qp](1, 1) + _aux_stress[_qp](2, 2);
term4 = scalar_q * aux_stress_trace * (*_current_instantaneous_thermal_expansion_coef)[_qp] *
grad_temp_cf(0);
}
Real q_avg_seg = 1.0;
if (!_crack_front_definition->treatAs2D())
{
q_avg_seg =
(_crack_front_definition->getCrackFrontForwardSegmentLength(_crack_front_point_index) +
_crack_front_definition->getCrackFrontBackwardSegmentLength(_crack_front_point_index)) /
2.0;
}
Real eq = term1 + term2 - term3 + term4;
if (_has_symmetry_plane)
eq *= 2.0;
return eq / q_avg_seg;
}
Real
InteractionIntegral::computeQpIntegral()
{
RealVectorValue grad_q = _grad_of_scalar_q[_qp];
//In the crack front coordinate system, the crack direction is (1,0,0)
RealVectorValue crack_direction(0.0);
crack_direction(0) = 1.0;
ColumnMajorMatrix aux_du;
aux_du(0,0) = _aux_grad_disp[_qp](0,0);
aux_du(0,1) = _aux_grad_disp[_qp](0,1);
aux_du(0,2) = _aux_grad_disp[_qp](0,2);
ColumnMajorMatrix stress;
stress(0,0) = _stress[_qp].xx();
stress(0,1) = _stress[_qp].xy();
stress(0,2) = _stress[_qp].xz();
stress(1,0) = _stress[_qp].xy();
stress(1,1) = _stress[_qp].yy();
stress(1,2) = _stress[_qp].yz();
stress(2,0) = _stress[_qp].xz();
stress(2,1) = _stress[_qp].yz();
stress(2,2) = _stress[_qp].zz();
ColumnMajorMatrix strain;
strain(0,0) = _strain[_qp].xx();
strain(0,1) = _strain[_qp].xy();
strain(0,2) = _strain[_qp].xz();
strain(1,0) = _strain[_qp].xy();
strain(1,1) = _strain[_qp].yy();
strain(1,2) = _strain[_qp].yz();
strain(2,0) = _strain[_qp].xz();
strain(2,1) = _strain[_qp].yz();
strain(2,2) = _strain[_qp].zz();
ColumnMajorMatrix grad_disp;
grad_disp(0,0) = _grad_disp_x[_qp](0);
grad_disp(0,1) = _grad_disp_x[_qp](1);
grad_disp(0,2) = _grad_disp_x[_qp](2);
grad_disp(1,0) = _grad_disp_y[_qp](0);
grad_disp(1,1) = _grad_disp_y[_qp](1);
grad_disp(1,2) = _grad_disp_y[_qp](2);
grad_disp(2,0) = _grad_disp_z[_qp](0);
grad_disp(2,1) = _grad_disp_z[_qp](1);
grad_disp(2,2) = _grad_disp_z[_qp](2);
//Rotate stress, strain, and displacement to crack front coordinate system
RealVectorValue grad_q_cf = _crack_front_definition->rotateToCrackFrontCoords(grad_q,_crack_front_point_index);
ColumnMajorMatrix grad_disp_cf = _crack_front_definition->rotateToCrackFrontCoords(grad_disp,_crack_front_point_index);
ColumnMajorMatrix stress_cf = _crack_front_definition->rotateToCrackFrontCoords(stress,_crack_front_point_index);
ColumnMajorMatrix strain_cf = _crack_front_definition->rotateToCrackFrontCoords(strain,_crack_front_point_index);
ColumnMajorMatrix dq;
dq(0,0) = crack_direction(0)*grad_q_cf(0);
dq(0,1) = crack_direction(0)*grad_q_cf(1);
dq(0,2) = crack_direction(0)*grad_q_cf(2);
//Calculate interaction integral terms
// Term1 = stress * x1-derivative of aux disp * dq
ColumnMajorMatrix tmp1 = dq * stress_cf;
Real term1 = aux_du.doubleContraction(tmp1);
// Term2 = aux stress * x1-derivative of disp * dq
ColumnMajorMatrix tmp2 = dq * _aux_stress[_qp];
Real term2 = grad_disp_cf(0,0)*tmp2(0,0)+grad_disp_cf(1,0)*tmp2(0,1)+grad_disp_cf(2,0)*tmp2(0,2);
// Term3 = aux stress * strain * dq_x (= stress * aux strain * dq_x)
Real term3 = dq(0,0) * _aux_stress[_qp].doubleContraction(strain_cf);
Real q_avg_seg = 1.0;
if (!_crack_front_definition->treatAs2D())
{
q_avg_seg = (_crack_front_definition->getCrackFrontForwardSegmentLength(_crack_front_point_index) +
_crack_front_definition->getCrackFrontBackwardSegmentLength(_crack_front_point_index)) / 2.0;
}
Real eq = term1 + term2 - term3;
if (_has_symmetry_plane)
eq *= 2.0;
return eq/q_avg_seg;
}