InteractionIntegral.C

Go to the documentation of this file.

//* This file is part of the MOOSE framework
//* https://mooseframework.inl.gov
//*
//* All rights reserved, see COPYRIGHT for full restrictions
//* https://github.com/idaholab/moose/blob/master/COPYRIGHT
//*
//* Licensed under LGPL 2.1, please see LICENSE for details
//* https://www.gnu.org/licenses/lgpl-2.1.html
 
#include "InteractionIntegral.h"
#include "MooseMesh.h"
#include "RankTwoTensor.h"
#include "Conversion.h"
#include "libmesh/string_to_enum.h"
#include "libmesh/quadrature.h"
#include "DerivativeMaterialInterface.h"
#include "libmesh/utility.h"
#include "CrackFrontDefinition.h"
 
registerMooseObject("SolidMechanicsApp", InteractionIntegral);
registerMooseObject("SolidMechanicsApp", ADInteractionIntegral);
 
template <bool is_ad>
MooseEnum
InteractionIntegralTempl<is_ad>::qFunctionType()
{
  return MooseEnum("Geometry Topology", "Geometry");
}
 
template <bool is_ad>
MooseEnum
InteractionIntegralTempl<is_ad>::sifModeType()
{
  return MooseEnum("KI KII KIII T", "KI");
}
 
template <bool is_ad>
InputParameters
InteractionIntegralTempl<is_ad>::validParams()
{
  InputParameters params = ElementVectorPostprocessor::validParams();
  params.addRequiredCoupledVar(
      "displacements",
      "The displacements appropriate for the simulation geometry and coordinate system");
  params.addCoupledVar("temperature",
                       "The temperature (optional). Must be provided to correctly compute "
                       "stress intensity factors in models with thermal strain gradients.");
  params.addParam<MaterialPropertyName>(
      "functionally_graded_youngs_modulus",
      "Spatially varying elasticity modulus variable. This input is required when "
      "using the functionally graded material capability.");
  params.addParam<MaterialPropertyName>(
      "functionally_graded_youngs_modulus_crack_dir_gradient",
      "Gradient of the spatially varying Young's modulus provided in "
      "'functionally_graded_youngs_modulus' in the direction of crack extension.");
  params.addRequiredParam<UserObjectName>("crack_front_definition",
                                          "The CrackFrontDefinition user object name");
  MooseEnum position_type("Angle Distance", "Distance");
  params.addParam<MooseEnum>(
      "position_type",
      position_type,
      "The method used to calculate position along crack front.  Options are: " +
          position_type.getRawNames());
  params.addRequiredParam<Real>(
      "K_factor", "Conversion factor between interaction integral and stress intensity factor K");
  params.addParam<unsigned int>("symmetry_plane",
                                "Account for a symmetry plane passing through "
                                "the plane of the crack, normal to the specified "
                                "axis (0=x, 1=y, 2=z)");
  params.addRequiredParam<Real>("poissons_ratio", "Poisson's ratio for the material.");
  params.addRequiredParam<Real>("youngs_modulus", "Young's modulus of the material.");
  params.set<bool>("use_displaced_mesh") = false;
  params.addRequiredParam<unsigned int>("ring_index", "Ring ID");
  params.addParam<MooseEnum>("q_function_type",
                             InteractionIntegralTempl<is_ad>::qFunctionType(),
                             "The method used to define the integration domain. Options are: " +
                                 InteractionIntegralTempl<is_ad>::qFunctionType().getRawNames());
  params.addRequiredParam<MooseEnum>(
      "sif_mode",
      InteractionIntegralTempl<is_ad>::sifModeType(),
      "Stress intensity factor to calculate. Choices are: " +
          InteractionIntegralTempl<is_ad>::sifModeType().getRawNames());
  params.addParam<MaterialPropertyName>("eigenstrain_gradient",
                                        "Material defining gradient of eigenstrain tensor");
  params.addParam<MaterialPropertyName>("body_force", "Material defining body force");
  params.addCoupledVar("additional_eigenstrain_00",
                       "Optional additional eigenstrain variable that will be accounted for in the "
                       "interaction integral (component 00 or XX).");
  params.addCoupledVar("additional_eigenstrain_01",
                       "Optional additional eigenstrain variable that will be accounted for in the "
                       "interaction integral (component 01 or XY).");
  params.addCoupledVar("additional_eigenstrain_11",
                       "Optional additional eigenstrain variable that will be accounted for in the "
                       "interaction integral (component 11 or YY).");
  params.addCoupledVar("additional_eigenstrain_22",
                       "Optional additional eigenstrain variable that will be accounted for in the "
                       "interaction integral (component 22 or ZZ).");
  params.addCoupledVar("additional_eigenstrain_02",
                       "Optional additional eigenstrain variable that will be accounted for in the "
                       "interaction integral (component 02 or XZ).");
  params.addCoupledVar("additional_eigenstrain_12",
                       "Optional additional eigenstrain variable that will be accounted for in the "
                       "interaction integral (component 12 or XZ).");
  params.addParamNamesToGroup(
      "additional_eigenstrain_00 additional_eigenstrain_01 additional_eigenstrain_11 "
      "additional_eigenstrain_22 additional_eigenstrain_02 additional_eigenstrain_12",
      "Generic eigenstrains for the computation of the interaction integral.");
  params.addClassDescription(
      "Computes the interaction integral, which is used to compute various "
      "fracture mechanics parameters at a crack tip, including KI, KII, KIII, "
      "and the T stress.");
  return params;
}
 
template <bool is_ad>
InteractionIntegralTempl<is_ad>::InteractionIntegralTempl(const InputParameters & parameters)
  : ElementVectorPostprocessor(parameters),
    _ndisp(coupledComponents("displacements")),
    _crack_front_definition(&getUserObject<CrackFrontDefinition>("crack_front_definition")),
    _treat_as_2d(false),
    _stress(getGenericMaterialPropertyByName<RankTwoTensor, is_ad>("stress")),
    _strain(getGenericMaterialPropertyByName<RankTwoTensor, is_ad>("elastic_strain")),
    _fe_vars(getCoupledMooseVars()),
    _fe_type(_fe_vars[0]->feType()),
    _disp(coupledValues("displacements")),
    _grad_disp(3),
    _has_temp(isCoupled("temperature")),
    _grad_temp(_has_temp ? coupledGradient("temperature") : _grad_zero),
    _functionally_graded_youngs_modulus_crack_dir_gradient(
        isParamSetByUser("functionally_graded_youngs_modulus_crack_dir_gradient")
            ? &getMaterialProperty<Real>("functionally_graded_youngs_modulus_crack_dir_gradient")
            : nullptr),
    _functionally_graded_youngs_modulus(
        isParamSetByUser("functionally_graded_youngs_modulus")
            ? &getMaterialProperty<Real>("functionally_graded_youngs_modulus")
            : nullptr),
    _K_factor(getParam<Real>("K_factor")),
    _has_symmetry_plane(isParamValid("symmetry_plane")),
    _poissons_ratio(getParam<Real>("poissons_ratio")),
    _youngs_modulus(getParam<Real>("youngs_modulus")),
    _fgm_crack(isParamSetByUser("functionally_graded_youngs_modulus_crack_dir_gradient") &&
               isParamSetByUser("functionally_graded_youngs_modulus")),
    _ring_index(getParam<unsigned int>("ring_index")),
    _total_deigenstrain_dT(
        hasMaterialProperty<RankTwoTensor>("total_deigenstrain_dT")
            ? &getGenericMaterialProperty<RankTwoTensor, is_ad>("total_deigenstrain_dT")
            : nullptr),
    _has_additional_eigenstrain(false),
    _additional_eigenstrain_gradient_00(isCoupled("additional_eigenstrain_00")
                                            ? &coupledGradient("additional_eigenstrain_00")
                                            : nullptr),
    _additional_eigenstrain_gradient_01(isCoupled("additional_eigenstrain_01")
                                            ? &coupledGradient("additional_eigenstrain_01")
                                            : nullptr),
    _additional_eigenstrain_gradient_11(isCoupled("additional_eigenstrain_11")
                                            ? &coupledGradient("additional_eigenstrain_11")
                                            : nullptr),
    _additional_eigenstrain_gradient_22(isCoupled("additional_eigenstrain_22")
                                            ? &coupledGradient("additional_eigenstrain_22")
                                            : nullptr),
    _additional_eigenstrain_gradient_02(isCoupled("additional_eigenstrain_02")
                                            ? &coupledGradient("additional_eigenstrain_02")
                                            : nullptr),
    _additional_eigenstrain_gradient_12(isCoupled("additional_eigenstrain_12")
                                            ? &coupledGradient("additional_eigenstrain_12")
                                            : nullptr),
    _q_function_type(getParam<MooseEnum>("q_function_type").template getEnum<QMethod>()),
    _position_type(getParam<MooseEnum>("position_type").template getEnum<PositionType>()),
    _sif_mode(getParam<MooseEnum>("sif_mode").template getEnum<SifMethod>()),
    _x(declareVector("x")),
    _y(declareVector("y")),
    _z(declareVector("z")),
    _position(declareVector("id")),
    _interaction_integral(declareVector("II_" + Moose::stringify(getParam<MooseEnum>("sif_mode")) +
                                        "_" + Moose::stringify(_ring_index))),
    _eigenstrain_gradient(nullptr),
    _body_force(nullptr)
{
  if (_has_temp && !_total_deigenstrain_dT)
    mooseError("InteractionIntegral Error: To include thermal strain term in interaction integral, "
               "must both couple temperature in DomainIntegral block and compute "
               "total_deigenstrain_dT using ThermalFractureIntegral material model.");
 
  if ((!_functionally_graded_youngs_modulus &&
       _functionally_graded_youngs_modulus_crack_dir_gradient) ||
      (_functionally_graded_youngs_modulus &&
       !_functionally_graded_youngs_modulus_crack_dir_gradient))
    paramError("functionally_graded_youngs_modulus_crack_dir_gradient",
               "You have selected to compute the interaction integral for a crack in FGM. That "
               "selection requires the user to provide a spatially varying elasticity modulus "
               "that "
               "defines the transition of material properties (i.e. "
               "'functionally_graded_youngs_modulus') and its "
               "spatial derivative in the crack direction (i.e. "
               "'functionally_graded_youngs_modulus_crack_dir_gradient').");
 
  // plane strain
  _kappa = 3.0 - 4.0 * _poissons_ratio;
  _shear_modulus = _youngs_modulus / (2.0 * (1.0 + _poissons_ratio));
 
  // Checking for consistency between mesh size and length of the provided displacements vector
  if (_ndisp != _mesh.dimension())
    paramError("displacements",
               "InteractionIntegral Error: number of variables supplied in 'displacements' must "
               "match the mesh dimension.");
 
  // fetch gradients of coupled variables
  for (std::size_t i = 0; i < _ndisp; ++i)
    _grad_disp[i] = &coupledGradient("displacements", i);
 
  // set unused dimensions to zero
  for (std::size_t i = _ndisp; i < 3; ++i)
    _grad_disp[i] = &_grad_zero;
 
  if (isParamValid("eigenstrain_gradient"))
  {
    _eigenstrain_gradient =
        &getGenericMaterialProperty<RankThreeTensor, is_ad>("eigenstrain_gradient");
    if (_total_deigenstrain_dT)
      paramError("eigenstrain_gradient",
                 "eigenstrain_gradient cannot be specified for materials that provide the "
                 "total_deigenstrain_dT material property");
  }
  if (isParamValid("body_force"))
    _body_force = &getMaterialProperty<RealVectorValue>("body_force");
 
  _has_additional_eigenstrain = _additional_eigenstrain_gradient_00 &&
                                _additional_eigenstrain_gradient_01 &&
                                _additional_eigenstrain_gradient_11;
 
  if (_mesh.dimension() == 3 && _has_additional_eigenstrain)
    if (!(_additional_eigenstrain_gradient_22 && _additional_eigenstrain_gradient_12 &&
          _additional_eigenstrain_gradient_02))
      paramError("additional_eigenstrain_gradient_12",
                 "If additional eigenstrains are provided for the computation of the interaction "
                 "integral in three dimensions, make sure the six components are provided.");
 
  if (_has_additional_eigenstrain)
    mooseInfo("A generic eigenstrain provided by the user will be considered in the interaction "
              "integral (via domain integral action).");
}
 
template <bool is_ad>
void
InteractionIntegralTempl<is_ad>::initialSetup()
{
  _treat_as_2d = _crack_front_definition->treatAs2D();
  _using_mesh_cutter = _crack_front_definition->usingMeshCutter();
}
 
template <bool is_ad>
void
InteractionIntegralTempl<is_ad>::initialize()
{
  std::size_t num_pts;
  if (_treat_as_2d && _using_mesh_cutter == false)
    num_pts = 1;
  else
    num_pts = _crack_front_definition->getNumCrackFrontPoints();
 
  _x.assign(num_pts, 0.0);
  _y.assign(num_pts, 0.0);
  _z.assign(num_pts, 0.0);
  _position.assign(num_pts, 0.0);
  _interaction_integral.assign(num_pts, 0.0);
 
  for (const auto * fe_var : _fe_vars)
  {
    if (fe_var->feType() != _fe_type)
      mooseError("All coupled displacements must have the same type");
  }
}
 
template <bool is_ad>
Real
InteractionIntegralTempl<is_ad>::computeQpIntegral(const std::size_t crack_front_point_index,
                                                   const Real scalar_q,
                                                   const RealVectorValue & grad_of_scalar_q)
{
  // If q is zero, then dq is also zero, so all terms in the interaction integral would
  // return zero. As such, let us avoid unnecessary, frequent computations
  if (scalar_q < TOLERANCE * TOLERANCE * TOLERANCE)
    return 0.0;
 
  // In the crack front coordinate system, the crack direction is (1,0,0)
  RealVectorValue crack_direction(1.0, 0.0, 0.0);
 
  // Calculate (r,theta) position of qp relative to crack front
  _crack_front_definition->calculateRThetaToCrackFront(
      _q_point[_qp], crack_front_point_index, _r, _theta);
 
  RankTwoTensor aux_stress;
  RankTwoTensor aux_du;
  RankTwoTensor aux_strain;
  RankTwoTensor aux_disp;
 
  if (_sif_mode == SifMethod::KI || _sif_mode == SifMethod::KII || _sif_mode == SifMethod::KIII)
    computeAuxFields(aux_stress, aux_du, aux_strain, aux_disp);
  else if (_sif_mode == SifMethod::T)
    computeTFields(aux_stress, aux_du);
 
  auto grad_disp = RankTwoTensor::initializeFromRows(
      (*_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_of_scalar_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(
      MetaPhysicL::raw_value((_stress)[_qp]), crack_front_point_index);
  RankTwoTensor strain_cf = _crack_front_definition->rotateToCrackFrontCoords(
      MetaPhysicL::raw_value((_strain)[_qp]), crack_front_point_index);
  RealVectorValue grad_temp_cf =
      _crack_front_definition->rotateToCrackFrontCoords(_grad_temp[_qp], crack_front_point_index);
 
  if (getBlockCoordSystem() == Moose::COORD_RZ)
    strain_cf(2, 2) = 0.0;
 
  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(MetaPhysicL::raw_value((*_total_deigenstrain_dT)[_qp]));
    term4 = scalar_q * sigma_alpha * grad_temp_cf(0);
  }
 
  Real term4a = 0.0; // Gradient of arbitrary eigenstrain material contribution
  if (_eigenstrain_gradient)
  {
    // Thermal strain gradient term in Nakamura and Parks, IJSS, 1992:
    // alpha * dT/dx_k*aux_stress*scalar_q
    // Generalization to the gradient of an arbitrary eigenstrain:
    // d_eigenstrain/dx_k*aux_stress*scalar_q
    const RealVectorValue & crack_dir =
        _crack_front_definition->getCrackDirection(crack_front_point_index);
    RankTwoTensor eigenstrain_grad_in_crack_dir =
        crack_dir * MetaPhysicL::raw_value((*_eigenstrain_gradient)[_qp]);
    term4a = scalar_q * aux_stress.doubleContraction(eigenstrain_grad_in_crack_dir);
  }
 
  // Gradient of arbitrary eigenstrain variable contribution.
  // Note that, unlike approaches 4 and 4a, approach 4b can integrate any type of eigenstrain, i.e.
  // it does not have to be thermal or volumetric.
  Real term4b = 0.0;
  // Term4b (eigen strain term) = q * aux_stress * d{eigen_strain}_x
  if (_has_additional_eigenstrain)
  {
    RankTwoTensor I_x(RankTwoTensor::initNone);
    I_x(0, 0) = 1.0;
    RankTwoTensor I_xy(RankTwoTensor::initNone);
    I_xy(0, 1) = 1.0;
    I_xy(1, 0) = 1.0;
    RankTwoTensor I_y(RankTwoTensor::initNone);
    I_y(1, 1) = 1.0;
    RankTwoTensor I_z(RankTwoTensor::initNone);
    I_z(2, 2) = 1.0;
 
    const RealVectorValue & crack_dir =
        _crack_front_definition->getCrackDirection(crack_front_point_index);
 
    const auto eigenstrain_gradient_x =
        I_x.mixedProductJkI((*_additional_eigenstrain_gradient_00)[_qp]);
    const auto eigenstrain_gradient_xy =
        I_xy.mixedProductJkI((*_additional_eigenstrain_gradient_01)[_qp]);
    const auto eigenstrain_gradient_y =
        I_y.mixedProductJkI((*_additional_eigenstrain_gradient_11)[_qp]);
 
    GenericRankThreeTensor<is_ad> eigenstrain_gradient_z;
    if (_additional_eigenstrain_gradient_22)
      eigenstrain_gradient_z = I_z.mixedProductJkI((*_additional_eigenstrain_gradient_22)[_qp]);
 
    RankTwoTensor eigenstrain_grad_in_crack_dir =
        crack_dir * (MetaPhysicL::raw_value(eigenstrain_gradient_x) +
                     MetaPhysicL::raw_value(eigenstrain_gradient_xy) +
                     MetaPhysicL::raw_value(eigenstrain_gradient_y) +
                     MetaPhysicL::raw_value(eigenstrain_gradient_z));
 
    GenericRankThreeTensor<is_ad> eigenstrain_gradient_xz;
    GenericRankThreeTensor<is_ad> eigenstrain_gradient_yz;
    // Add terms for three dimensions
    if (_mesh.dimension() == 3)
    {
      // Add influence of xz and yz.
      RankTwoTensor I_xz(RankTwoTensor::initNone);
      I_xz(0, 2) = I_xz(2, 0) = 1.0;
      eigenstrain_gradient_xz = I_xz.mixedProductJkI((*_additional_eigenstrain_gradient_02)[_qp]);
      RankTwoTensor I_yz(RankTwoTensor::initNone);
      I_yz(1, 2) = I_yz(2, 1) = 1.0;
      eigenstrain_gradient_yz = I_yz.mixedProductJkI((*_additional_eigenstrain_gradient_12)[_qp]);
 
      eigenstrain_grad_in_crack_dir = crack_dir * (MetaPhysicL::raw_value(eigenstrain_gradient_xz) +
                                                   MetaPhysicL::raw_value(eigenstrain_gradient_yz) +
                                                   MetaPhysicL::raw_value(eigenstrain_gradient_x) +
                                                   MetaPhysicL::raw_value(eigenstrain_gradient_xy) +
                                                   MetaPhysicL::raw_value(eigenstrain_gradient_y) +
                                                   MetaPhysicL::raw_value(eigenstrain_gradient_z));
    }
 
    term4b = scalar_q * aux_stress.doubleContraction(eigenstrain_grad_in_crack_dir);
  }
 
  Real term5 = 0.0; // Body force
  if (_body_force)
  {
    // Body force term in Nakamura and Parks, IJSS, 1992:
    // b_i*aux_du*crack_dir*scalar_q
    const RealVectorValue & crack_dir =
        _crack_front_definition->getCrackDirection(crack_front_point_index);
    term5 = -scalar_q * MetaPhysicL::raw_value((*_body_force)[_qp]) * aux_du * crack_dir;
  }
 
  Real term6 = 0.0;
  if (_fgm_crack && scalar_q != 0)
  {
    // See Equation 49 of J.-H. Kim and G. H. Paulino. Consistent formulations of the interaction
    // integral method for fracture of functionally graded materials. Journal of Applied Mechanics,
    // 72(3) 351-364 2004.
 
    // Term 6_1 = Cijkl,j(x) epsilon_{kl}^{aux} disp_{i,1}
    RankTwoTensor cijklj_epsilonkl_aux; // Temporary second order tensor.
    RankFourTensor cijklj;
    cijklj.fillSymmetricIsotropicEandNu(1.0, _poissons_ratio);
    cijklj *= (*_functionally_graded_youngs_modulus_crack_dir_gradient)[_qp];
    cijklj_epsilonkl_aux = cijklj * aux_strain;
 
    const Real term6_a = grad_disp_cf(0, 0) * cijklj_epsilonkl_aux(0, 0) +
                         grad_disp_cf(1, 0) * cijklj_epsilonkl_aux(0, 1) +
                         grad_disp_cf(2, 0) * cijklj_epsilonkl_aux(0, 2);
 
    // Term 6_2 = -Cijkl,1(x) epsilon_{kl} epsilon_{ij}^{aux}
    RankTwoTensor cijkl1_epsilonkl;
    cijkl1_epsilonkl = -cijklj * strain_cf;
    const Real term6_b = cijkl1_epsilonkl.doubleContraction(aux_strain);
 
    term6 = (term6_a + term6_b) * scalar_q;
  }
 
  // Add terms for possibly RZ (axisymmetric problem).
  // See Nahta and Moran, 1993, Domain integrals for axisymmetric interface crack problems, 30(15)
  // Note: Sign problem in Eq. (26). Also Kolosov constant is wrong in Eq. (13)
  Real term7 = 0.0;
  if (getBlockCoordSystem() == Moose::COORD_RZ)
  {
    const Real u_r = (*_disp[0])[_qp];
    const Real radius_qp = _q_point[_qp](0);
 
    const Real term7a =
        (u_r / radius_qp * aux_stress(2, 2) + aux_disp(0, 0) / radius_qp * stress_cf(2, 2) -
         aux_stress.doubleContraction(strain_cf)) /
        radius_qp;
 
    // term7b1: sigma_aux_ralpha * u_alpha,r
    const Real term7b1 = aux_stress(0, 0) * grad_disp_cf(0, 0) +
                         aux_stress(0, 1) * grad_disp_cf(1, 0) +
                         aux_stress(0, 2) * grad_disp_cf(2, 0);
 
    const Real term7b = 1.0 / radius_qp *
                        (term7b1 - aux_stress(2, 2) * grad_disp_cf(0, 0) +
                         stress_cf(2, 2) * (aux_du(0, 0) - aux_disp(0, 0) / radius_qp));
 
    term7 = (term7a + term7b) * scalar_q;
  }
 
  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 + term4a + term4b + term5 + term6 + term7;
 
  if (getBlockCoordSystem() == Moose::COORD_RZ)
  {
    const Real radius_qp = _q_point[_qp](0);
    eq *= radius_qp;
 
    std::size_t num_crack_front_points = _crack_front_definition->getNumCrackFrontPoints();
    if (num_crack_front_points != 1)
      mooseError("Crack front has more than one point, but this is a 2D-RZ problem. Please revise "
                 "your input file.");
 
    const Point * crack_front_point = _crack_front_definition->getCrackFrontPoint(0);
    eq = eq / (*crack_front_point)(0);
  }
 
  return eq / q_avg_seg;
}
 
template <bool is_ad>
void
InteractionIntegralTempl<is_ad>::execute()
{
  // calculate phi and dphi for this element
  const std::size_t dim = _current_elem->dim();
  std::unique_ptr<FEBase> fe(FEBase::build(dim, _fe_type));
  fe->attach_quadrature_rule(const_cast<QBase *>(_qrule));
  _phi_curr_elem = &fe->get_phi();
  _dphi_curr_elem = &fe->get_dphi();
  fe->reinit(_current_elem);
 
  // calculate q for all nodes in this element
  std::size_t ring_base = (_q_function_type == QMethod::Topology) ? 0 : 1;
 
  for (auto icfp = beginIndex(_interaction_integral); icfp < _interaction_integral.size(); icfp++)
  {
    _q_curr_elem.clear();
    for (std::size_t i = 0; i < _current_elem->n_nodes(); ++i)
    {
      const Node * this_node = _current_elem->node_ptr(i);
      Real q_this_node;
 
      if (_q_function_type == QMethod::Geometry)
        q_this_node = _crack_front_definition->DomainIntegralQFunction(
            icfp, _ring_index - ring_base, this_node);
      else // QMethod::Topology
        q_this_node = _crack_front_definition->DomainIntegralTopologicalQFunction(
            icfp, _ring_index - ring_base, this_node);
 
      _q_curr_elem.push_back(q_this_node);
    }
 
    for (_qp = 0; _qp < _qrule->n_points(); _qp++)
    {
      Real scalar_q = 0.0;
      RealVectorValue grad_of_scalar_q(0.0, 0.0, 0.0);
 
      for (std::size_t i = 0; i < _current_elem->n_nodes(); ++i)
      {
        scalar_q += (*_phi_curr_elem)[i][_qp] * _q_curr_elem[i];
 
        for (std::size_t j = 0; j < _current_elem->dim(); ++j)
          grad_of_scalar_q(j) += (*_dphi_curr_elem)[i][_qp](j) * _q_curr_elem[i];
      }
 
      if (getBlockCoordSystem() != Moose::COORD_RZ)
        _interaction_integral[icfp] +=
            _JxW[_qp] * _coord[_qp] * computeQpIntegral(icfp, scalar_q, grad_of_scalar_q);
      else
        _interaction_integral[icfp] +=
            _JxW[_qp] * computeQpIntegral(icfp, scalar_q, grad_of_scalar_q);
    }
  }
}
 
template <bool is_ad>
void
InteractionIntegralTempl<is_ad>::finalize()
{
  gatherSum(_interaction_integral);
 
  for (auto i = beginIndex(_interaction_integral); i < _interaction_integral.size(); ++i)
  {
    if (_has_symmetry_plane)
      _interaction_integral[i] *= 2.0;
 
    const auto cfp = _crack_front_definition->getCrackFrontPoint(i);
    _x[i] = (*cfp)(0);
    _y[i] = (*cfp)(1);
    _z[i] = (*cfp)(2);
 
    if (_position_type == PositionType::Angle)
      _position[i] = _crack_front_definition->getAngleAlongFront(i);
    else
      _position[i] = _crack_front_definition->getDistanceAlongFront(i);
 
    if (_sif_mode == SifMethod::T && !_treat_as_2d)
      _interaction_integral[i] +=
          _poissons_ratio * _crack_front_definition->getCrackFrontTangentialStrain(i);
 
    _interaction_integral[i] *= _K_factor;
  }
}
 
template <bool is_ad>
void
InteractionIntegralTempl<is_ad>::threadJoin(const UserObject & y)
{
  const auto & uo = static_cast<const InteractionIntegralTempl<is_ad> &>(y);
 
  for (auto i = beginIndex(_interaction_integral); i < _interaction_integral.size(); ++i)
    _interaction_integral[i] += uo._interaction_integral[i];
}
 
template <bool is_ad>
void
InteractionIntegralTempl<is_ad>::computeAuxFields(RankTwoTensor & aux_stress,
                                                  RankTwoTensor & grad_disp,
                                                  RankTwoTensor & aux_strain,
                                                  RankTwoTensor & aux_disp)
{
  RealVectorValue k(0.0);
  if (_sif_mode == SifMethod::KI)
    k(0) = 1.0;
  else if (_sif_mode == SifMethod::KII)
    k(1) = 1.0;
  else if (_sif_mode == SifMethod::KIII)
    k(2) = 1.0;
 
  Real t = _theta;
  Real t2 = _theta / 2.0;
  Real tt2 = 3.0 * _theta / 2.0;
  Real st = std::sin(t);
  Real ct = std::cos(t);
  Real st2 = std::sin(t2);
  Real ct2 = std::cos(t2);
  Real stt2 = std::sin(tt2);
  Real ctt2 = std::cos(tt2);
  Real ct2sq = Utility::pow<2>(ct2);
  Real st2sq = Utility::pow<2>(st2);
  Real ct2cu = Utility::pow<3>(ct2);
  Real sqrt2PiR = std::sqrt(2.0 * libMesh::pi * _r);
 
  // Calculate auxiliary stress tensor
  aux_stress.zero();
 
  aux_stress(0, 0) =
      1.0 / sqrt2PiR * (k(0) * ct2 * (1.0 - st2 * stt2) - k(1) * st2 * (2.0 + ct2 * ctt2));
  aux_stress(1, 1) = 1.0 / sqrt2PiR * (k(0) * ct2 * (1.0 + st2 * stt2) + k(1) * st2 * ct2 * ctt2);
  aux_stress(0, 1) = 1.0 / sqrt2PiR * (k(0) * ct2 * st2 * ctt2 + k(1) * ct2 * (1.0 - st2 * stt2));
  aux_stress(0, 2) = -1.0 / sqrt2PiR * k(2) * st2;
  aux_stress(1, 2) = 1.0 / sqrt2PiR * k(2) * ct2;
  // plane stress
  // Real s33 = 0;
  // plane strain
  aux_stress(2, 2) = _poissons_ratio * (aux_stress(0, 0) + aux_stress(1, 1));
 
  aux_stress(1, 0) = aux_stress(0, 1);
  aux_stress(2, 0) = aux_stress(0, 2);
  aux_stress(2, 1) = aux_stress(1, 2);
 
  // Calculate auxiliary strain tensor (only needed to FGM)
  if (_fgm_crack)
  {
    aux_strain.zero();
    RankFourTensor spatial_elasticity_tensor;
    spatial_elasticity_tensor.fillSymmetricIsotropicEandNu(
        (*_functionally_graded_youngs_modulus)[_qp], _poissons_ratio);
    aux_strain = spatial_elasticity_tensor.invSymm() * aux_stress;
  }
 
  // Calculate x1 derivative of auxiliary displacements
  grad_disp.zero();
  grad_disp(0, 0) = k(0) / (4.0 * _shear_modulus * sqrt2PiR) *
                        (ct * ct2 * _kappa + ct * ct2 - 2.0 * ct * ct2cu + st * st2 * _kappa +
                         st * st2 - 6.0 * st * st2 * ct2sq) +
                    k(1) / (4.0 * _shear_modulus * sqrt2PiR) *
                        (ct * st2 * _kappa + ct * st2 + 2.0 * ct * st2 * ct2sq - st * ct2 * _kappa +
                         3.0 * st * ct2 - 6.0 * st * ct2cu);
 
  grad_disp(0, 1) = k(0) / (4.0 * _shear_modulus * sqrt2PiR) *
                        (ct * st2 * _kappa + ct * st2 - 2.0 * ct * st2 * ct2sq - st * ct2 * _kappa -
                         5.0 * st * ct2 + 6.0 * st * ct2cu) +
                    k(1) / (4.0 * _shear_modulus * sqrt2PiR) *
                        (-ct * ct2 * _kappa + 3.0 * ct * ct2 - 2.0 * ct * ct2cu -
                         st * st2 * _kappa + 3.0 * st * st2 - 6.0 * st * st2 * ct2sq);
 
  grad_disp(0, 2) = k(2) / (_shear_modulus * sqrt2PiR) * (st2 * ct - ct2 * st);
 
  // Calculate aux displacement field (used in RZ calculations) - Warp3D manual
  aux_disp.zero();
  if (getBlockCoordSystem() == Moose::COORD_RZ)
  {
    const Real prefactor_rz = std::sqrt(_r / 2.0 / libMesh::pi) / 2.0 / _shear_modulus;
    aux_disp(0, 0) = prefactor_rz * ((k(0) * ct2 * (_kappa - 1.0 + 2.0 * st2sq)) +
                                     (k(1) * st2 * (_kappa + 1.0 + 2.0 * ct2sq)));
    aux_disp(0, 1) = prefactor_rz * ((k(0) * st2 * (_kappa + 1.0 - 2.0 * ct2sq)) -
                                     (k(1) * ct2 * (_kappa - 1.0 - 2.0 * st2sq)));
    aux_disp(0, 2) = prefactor_rz * k(2) * 4.0 * st2;
  }
}
 
template <bool is_ad>
void
InteractionIntegralTempl<is_ad>::computeTFields(RankTwoTensor & aux_stress,
                                                RankTwoTensor & grad_disp)
{
 
  Real t = _theta;
  Real st = std::sin(t);
  Real ct = std::cos(t);
  Real stsq = Utility::pow<2>(st);
  Real ctsq = Utility::pow<2>(ct);
  Real ctcu = Utility::pow<3>(ct);
  Real oneOverPiR = 1.0 / (libMesh::pi * _r);
 
  aux_stress.zero();
  aux_stress(0, 0) = -oneOverPiR * ctcu;
  aux_stress(0, 1) = -oneOverPiR * st * ctsq;
  aux_stress(1, 0) = -oneOverPiR * st * ctsq;
  aux_stress(1, 1) = -oneOverPiR * ct * stsq;
  aux_stress(2, 2) = -oneOverPiR * _poissons_ratio * (ctcu + ct * stsq);
 
  grad_disp.zero();
  grad_disp(0, 0) = oneOverPiR / (4.0 * _youngs_modulus) *
                    (ct * (4.0 * Utility::pow<2>(_poissons_ratio) - 3.0 + _poissons_ratio) -
                     std::cos(3.0 * t) * (1.0 + _poissons_ratio));
  grad_disp(0, 1) = -oneOverPiR / (4.0 * _youngs_modulus) *
                    (st * (4.0 * Utility::pow<2>(_poissons_ratio) - 3.0 + _poissons_ratio) +
                     std::sin(3.0 * t) * (1.0 + _poissons_ratio));
}
 
template class InteractionIntegralTempl<false>;
template class InteractionIntegralTempl<true>;