SolidMechanicsPlasticTensileMulti.C

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//* 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 "SolidMechanicsPlasticTensileMulti.h"
#include "RankFourTensor.h"
 
// Following is for perturbing eigvenvalues.  This looks really bodgy, but works quite well!
#include "MooseRandom.h"
 
registerMooseObject("SolidMechanicsApp", SolidMechanicsPlasticTensileMulti);
registerMooseObjectRenamed("SolidMechanicsApp",
                           TensorMechanicsPlasticTensileMulti,
                           "01/01/2025 00:00",
                           SolidMechanicsPlasticTensileMulti);
 
InputParameters
SolidMechanicsPlasticTensileMulti::validParams()
{
  InputParameters params = SolidMechanicsPlasticModel::validParams();
  params.addClassDescription("Associative tensile plasticity with hardening/softening");
  params.addRequiredParam<UserObjectName>(
      "tensile_strength",
      "A SolidMechanicsHardening UserObject that defines hardening of the tensile strength");
  params.addParam<Real>("shift",
                        "Yield surface is shifted by this amount to avoid problems with "
                        "defining derivatives when eigenvalues are equal.  If this is "
                        "larger than f_tol, a warning will be issued.  Default = f_tol.");
  params.addParam<unsigned int>("max_iterations",
                                10,
                                "Maximum number of Newton-Raphson iterations "
                                "allowed in the custom return-map algorithm. "
                                " For highly nonlinear hardening this may "
                                "need to be higher than 10.");
  params.addParam<bool>("use_custom_returnMap",
                        true,
                        "Whether to use the custom returnMap "
                        "algorithm.  Set to true if you are using "
                        "isotropic elasticity.");
  params.addParam<bool>("use_custom_cto",
                        true,
                        "Whether to use the custom consistent tangent "
                        "operator computations.  Set to true if you are "
                        "using isotropic elasticity.");
  return params;
}
 
SolidMechanicsPlasticTensileMulti::SolidMechanicsPlasticTensileMulti(
    const InputParameters & parameters)
  : SolidMechanicsPlasticModel(parameters),
    _strength(getUserObject<SolidMechanicsHardeningModel>("tensile_strength")),
    _max_iters(getParam<unsigned int>("max_iterations")),
    _shift(parameters.isParamValid("shift") ? getParam<Real>("shift") : _f_tol),
    _use_custom_returnMap(getParam<bool>("use_custom_returnMap")),
    _use_custom_cto(getParam<bool>("use_custom_cto"))
{
  if (_shift < 0)
    mooseError("Value of 'shift' in SolidMechanicsPlasticTensileMulti must not be negative\n");
  if (_shift > _f_tol)
    _console << "WARNING: value of 'shift' in SolidMechanicsPlasticTensileMulti is probably set "
                "too high"
             << std::endl;
  if (LIBMESH_DIM != 3)
    mooseError("SolidMechanicsPlasticTensileMulti is only defined for LIBMESH_DIM=3");
  MooseRandom::seed(0);
}
 
unsigned int
SolidMechanicsPlasticTensileMulti::numberSurfaces() const
{
  return 3;
}
 
void
SolidMechanicsPlasticTensileMulti::yieldFunctionV(const RankTwoTensor & stress,
                                                  Real intnl,
                                                  std::vector<Real> & f) const
{
  std::vector<Real> eigvals;
  stress.symmetricEigenvalues(eigvals);
  const Real str = tensile_strength(intnl);
 
  f.resize(3);
  f[0] = eigvals[0] + _shift - str;
  f[1] = eigvals[1] - str;
  f[2] = eigvals[2] - _shift - str;
}
 
void
SolidMechanicsPlasticTensileMulti::dyieldFunction_dstressV(
    const RankTwoTensor & stress, Real /*intnl*/, std::vector<RankTwoTensor> & df_dstress) const
{
  std::vector<Real> eigvals;
  stress.dsymmetricEigenvalues(eigvals, df_dstress);
 
  if (eigvals[0] > eigvals[1] - 0.1 * _shift || eigvals[1] > eigvals[2] - 0.1 * _shift)
  {
    Real small_perturbation;
    RankTwoTensor shifted_stress = stress;
    while (eigvals[0] > eigvals[1] - 0.1 * _shift || eigvals[1] > eigvals[2] - 0.1 * _shift)
    {
      for (unsigned i = 0; i < 3; ++i)
        for (unsigned j = 0; j <= i; ++j)
        {
          small_perturbation = 0.1 * _shift * 2 * (MooseRandom::rand() - 0.5);
          shifted_stress(i, j) += small_perturbation;
          shifted_stress(j, i) += small_perturbation;
        }
      shifted_stress.dsymmetricEigenvalues(eigvals, df_dstress);
    }
  }
}
 
void
SolidMechanicsPlasticTensileMulti::dyieldFunction_dintnlV(const RankTwoTensor & /*stress*/,
                                                          Real intnl,
                                                          std::vector<Real> & df_dintnl) const
{
  df_dintnl.assign(3, -dtensile_strength(intnl));
}
 
void
SolidMechanicsPlasticTensileMulti::flowPotentialV(const RankTwoTensor & stress,
                                                  Real intnl,
                                                  std::vector<RankTwoTensor> & r) const
{
  // This plasticity is associative so
  dyieldFunction_dstressV(stress, intnl, r);
}
 
void
SolidMechanicsPlasticTensileMulti::dflowPotential_dstressV(
    const RankTwoTensor & stress, Real /*intnl*/, std::vector<RankFourTensor> & dr_dstress) const
{
  stress.d2symmetricEigenvalues(dr_dstress);
}
 
void
SolidMechanicsPlasticTensileMulti::dflowPotential_dintnlV(
    const RankTwoTensor & /*stress*/, Real /*intnl*/, std::vector<RankTwoTensor> & dr_dintnl) const
{
  dr_dintnl.assign(3, RankTwoTensor());
}
 
Real
SolidMechanicsPlasticTensileMulti::tensile_strength(const Real internal_param) const
{
  return _strength.value(internal_param);
}
 
Real
SolidMechanicsPlasticTensileMulti::dtensile_strength(const Real internal_param) const
{
  return _strength.derivative(internal_param);
}
 
void
SolidMechanicsPlasticTensileMulti::activeConstraints(const std::vector<Real> & f,
                                                     const RankTwoTensor & stress,
                                                     Real intnl,
                                                     const RankFourTensor & Eijkl,
                                                     std::vector<bool> & act,
                                                     RankTwoTensor & returned_stress) const
{
  act.assign(3, false);
 
  if (f[0] <= _f_tol && f[1] <= _f_tol && f[2] <= _f_tol)
  {
    returned_stress = stress;
    return;
  }
 
  Real returned_intnl;
  std::vector<Real> dpm(3);
  RankTwoTensor delta_dp;
  std::vector<Real> yf(3);
  bool trial_stress_inadmissible;
  doReturnMap(stress,
              intnl,
              Eijkl,
              0.0,
              returned_stress,
              returned_intnl,
              dpm,
              delta_dp,
              yf,
              trial_stress_inadmissible);
 
  for (unsigned i = 0; i < 3; ++i)
    act[i] = (dpm[i] > 0);
}
 
Real
SolidMechanicsPlasticTensileMulti::dot(const std::vector<Real> & a,
                                       const std::vector<Real> & b) const
{
  return a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
}
 
Real
SolidMechanicsPlasticTensileMulti::triple(const std::vector<Real> & a,
                                          const std::vector<Real> & b,
                                          const std::vector<Real> & c) const
{
  return a[0] * (b[1] * c[2] - b[2] * c[1]) - a[1] * (b[0] * c[2] - b[2] * c[0]) +
         a[2] * (b[0] * c[1] - b[1] * c[0]);
}
 
std::string
SolidMechanicsPlasticTensileMulti::modelName() const
{
  return "TensileMulti";
}
 
bool
SolidMechanicsPlasticTensileMulti::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 SolidMechanicsPlasticModel::returnMap(trial_stress,
                                                 intnl_old,
                                                 E_ijkl,
                                                 ep_plastic_tolerance,
                                                 returned_stress,
                                                 returned_intnl,
                                                 dpm,
                                                 delta_dp,
                                                 yf,
                                                 trial_stress_inadmissible);
 
  return doReturnMap(trial_stress,
                     intnl_old,
                     E_ijkl,
                     ep_plastic_tolerance,
                     returned_stress,
                     returned_intnl,
                     dpm,
                     delta_dp,
                     yf,
                     trial_stress_inadmissible);
}
 
bool
SolidMechanicsPlasticTensileMulti::doReturnMap(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
{
  mooseAssert(dpm.size() == 3,
              "SolidMechanicsPlasticTensileMulti size of dpm should be 3 but it is " << dpm.size());
 
  std::vector<Real> eigvals;
  RankTwoTensor eigvecs;
  trial_stress.symmetricEigenvaluesEigenvectors(eigvals, eigvecs);
  eigvals[0] += _shift;
  eigvals[2] -= _shift;
 
  Real str = tensile_strength(intnl_old);
 
  yf.resize(3);
  yf[0] = eigvals[0] - str;
  yf[1] = eigvals[1] - str;
  yf[2] = eigvals[2] - str;
 
  if (yf[0] <= _f_tol && yf[1] <= _f_tol && yf[2] <= _f_tol)
  {
    // purely elastic (trial_stress, intnl_old)
    trial_stress_inadmissible = false;
    return true;
  }
 
  trial_stress_inadmissible = true;
  delta_dp.zero();
  returned_stress.zero();
 
  // In the following i often assume that E_ijkl is
  // for an isotropic situation.  This reduces FLOPS
  // substantially which is important since the returnMap
  // is potentially the most compute-intensive function
  // of a simulation.
  // In many comments i write the general expression, and
  // i hope that might guide future coders if they are
  // generalising to a non-istropic E_ijkl
 
  // n[alpha] = E_ijkl*r[alpha]_kl expressed in principal stress space
  // (alpha = 0, 1, 2, corresponding to the three surfaces)
  // Note that in principal stress space, the flow
  // directions are, expressed in 'vector' form,
  // r[0] = (1,0,0), r[1] = (0,1,0), r[2] = (0,0,1).
  // Similar for _n:
  // so _n[0] = E_ij00*r[0], _n[1] = E_ij11*r[1], _n[2] = E_ij22*r[2]
  // In the following I assume that the E_ijkl is
  // for an isotropic situation.
  // In the anisotropic situation, we couldn't express
  // the flow directions as vectors in the same principal
  // stress space as the stress: they'd be full rank-2 tensors
  std::vector<RealVectorValue> n(3);
  n[0](0) = E_ijkl(0, 0, 0, 0);
  n[0](1) = E_ijkl(1, 1, 0, 0);
  n[0](2) = E_ijkl(2, 2, 0, 0);
  n[1](0) = E_ijkl(0, 0, 1, 1);
  n[1](1) = E_ijkl(1, 1, 1, 1);
  n[1](2) = E_ijkl(2, 2, 1, 1);
  n[2](0) = E_ijkl(0, 0, 2, 2);
  n[2](1) = E_ijkl(1, 1, 2, 2);
  n[2](2) = E_ijkl(2, 2, 2, 2);
 
  // With non-zero Poisson's ratio and hardening
  // it is not computationally cheap to know whether
  // the trial stress will return to the tip, edge,
  // or plane.  The following is correct for zero
  // Poisson's ratio and no hardening, and at least
  // gives a not-completely-stupid guess in the
  // more general case.
  // trial_order[0] = type of return to try first
  // trial_order[1] = type of return to try second
  // trial_order[2] = type of return to try third
  const unsigned int number_of_return_paths = 3;
  std::vector<int> trial_order(number_of_return_paths);
  if (yf[0] > _f_tol) // all the yield functions are positive, since eigvals are ordered eigvals[0]
                      // <= eigvals[1] <= eigvals[2]
  {
    trial_order[0] = tip;
    trial_order[1] = edge;
    trial_order[2] = plane;
  }
  else if (yf[1] > _f_tol) // two yield functions are positive
  {
    trial_order[0] = edge;
    trial_order[1] = tip;
    trial_order[2] = plane;
  }
  else
  {
    trial_order[0] = plane;
    trial_order[1] = edge;
    trial_order[2] = tip;
  }
 
  unsigned trial;
  bool nr_converged = false;
  for (trial = 0; trial < number_of_return_paths; ++trial)
  {
    switch (trial_order[trial])
    {
      case tip:
        nr_converged = returnTip(eigvals, n, dpm, returned_stress, intnl_old, 0);
        break;
      case edge:
        nr_converged = returnEdge(eigvals, n, dpm, returned_stress, intnl_old, 0);
        break;
      case plane:
        nr_converged = returnPlane(eigvals, n, dpm, returned_stress, intnl_old, 0);
        break;
    }
 
    str = tensile_strength(intnl_old + dpm[0] + dpm[1] + dpm[2]);
 
    if (nr_converged && KuhnTuckerOK(returned_stress, dpm, str, ep_plastic_tolerance))
      break;
  }
 
  if (trial == number_of_return_paths)
  {
    Moose::err << "Trial stress = \n";
    trial_stress.print(Moose::err);
    Moose::err << "Internal parameter = " << intnl_old << std::endl;
    mooseError("SolidMechanicsPlasticTensileMulti: FAILURE!  You probably need to implement a "
               "line search\n");
    // failure - must place yield function values at trial stress into yf
    str = tensile_strength(intnl_old);
    yf[0] = eigvals[0] - str;
    yf[1] = eigvals[1] - str;
    yf[2] = eigvals[2] - str;
    return false;
  }
 
  // success
 
  returned_intnl = intnl_old;
  for (unsigned i = 0; i < 3; ++i)
  {
    yf[i] = returned_stress(i, i) - str;
    delta_dp(i, i) = dpm[i];
    returned_intnl += dpm[i];
  }
  returned_stress = eigvecs * returned_stress * (eigvecs.transpose());
  delta_dp = eigvecs * delta_dp * (eigvecs.transpose());
  return true;
}
 
bool
SolidMechanicsPlasticTensileMulti::returnTip(const std::vector<Real> & eigvals,
                                             const std::vector<RealVectorValue> & n,
                                             std::vector<Real> & dpm,
                                             RankTwoTensor & returned_stress,
                                             Real intnl_old,
                                             Real initial_guess) const
{
  // The returned point is defined by f0=f1=f2=0.
  // that is, returned_stress = diag(str, str, str), where
  // str = tensile_strength(intnl),
  // where intnl = intnl_old + dpm[0] + dpm[1] + dpm[2]
  // The 3 plastic multipliers, dpm, are defiend by the normality condition
  //   eigvals - str = dpm[0]*n[0] + dpm[1]*n[1] + dpm[2]*n[2]
  // (Kuhn-Tucker demands that all dpm are non-negative, but we leave
  // that checking for later.)
  // This is a vector equation with solution (A):
  //   dpm[0] = triple(eigvals - str, n[1], n[2])/trip;
  //   dpm[1] = triple(eigvals - str, n[2], n[0])/trip;
  //   dpm[2] = triple(eigvals - str, n[0], n[1])/trip;
  // where trip = triple(n[0], n[1], n[2]).
  // By adding the three components of that solution together
  // we can get an equation for x = dpm[0] + dpm[1] + dpm[2],
  // and then our Newton-Raphson only involves one variable (x).
  // In the following, i specialise to the isotropic situation.
 
  Real x = initial_guess;
  const Real denom = (n[0](0) - n[0](1)) * (n[0](0) + 2 * n[0](1));
  Real str = tensile_strength(intnl_old + x);
 
  if (_strength.modelName().compare("Constant") != 0)
  {
    // Finding x is expensive.  Therefore
    // although x!=0 for Constant Hardening, solution (A)
    // demonstrates that we don't
    // actually need to know x to find the dpm for
    // Constant Hardening.
    //
    // However, for nontrivial Hardening, the following
    // is necessary
    const Real eig = eigvals[0] + eigvals[1] + eigvals[2];
    const Real bul = (n[0](0) + 2 * n[0](1));
 
    // and finally, the equation we want to solve is:
    // bul*x - eig + 3*str = 0
    // where str=tensile_strength(intnl_old + x)
    // and x = dpm[0] + dpm[1] + dpm[2]
    // (Note this has units of stress, so using _f_tol as a convergence check is reasonable.)
    // Use Netwon-Raphson with initial guess x = 0
    Real residual = bul * x - eig + 3 * str;
    Real jacobian;
    unsigned int iter = 0;
    do
    {
      jacobian = bul + 3 * dtensile_strength(intnl_old + x);
      x += -residual / jacobian;
      if (iter > _max_iters) // not converging
        return false;
      str = tensile_strength(intnl_old + x);
      residual = bul * x - eig + 3 * str;
      iter++;
    } while (residual * residual > _f_tol * _f_tol);
  }
 
  // The following is the solution (A) written above
  // (dpm[0] = triple(eigvals - str, n[1], n[2])/trip, etc)
  // in the isotropic situation
  dpm[0] = (n[0](0) * (eigvals[0] - str) + n[0](1) * (eigvals[0] - eigvals[1] - eigvals[2] + str)) /
           denom;
  dpm[1] = (n[0](0) * (eigvals[1] - str) + n[0](1) * (eigvals[1] - eigvals[2] - eigvals[0] + str)) /
           denom;
  dpm[2] = (n[0](0) * (eigvals[2] - str) + n[0](1) * (eigvals[2] - eigvals[0] - eigvals[1] + str)) /
           denom;
  returned_stress(0, 0) = returned_stress(1, 1) = returned_stress(2, 2) = str;
  return true;
}
 
bool
SolidMechanicsPlasticTensileMulti::returnEdge(const std::vector<Real> & eigvals,
                                              const std::vector<RealVectorValue> & n,
                                              std::vector<Real> & dpm,
                                              RankTwoTensor & returned_stress,
                                              Real intnl_old,
                                              Real initial_guess) const
{
  // work out the point to which we would return, "a".  It is defined by
  // f1 = 0 = f2, and the normality condition:
  //   (eigvals - a).(n1 x n2) = 0,
  // where eigvals is the starting position
  // (it is a vector in principal stress space).
  // To get f1=0=f2, we need a = (a0, str, str), and a0 is found
  // by expanding the normality condition to yield:
  //   a0 = (-str*n1xn2[1] - str*n1xn2[2] + edotn1xn2)/n1xn2[0];
  // where edotn1xn2 = eigvals.(n1 x n2)
  //
  // We need to find the plastic multipliers, dpm, defined by
  //   eigvals - a = dpm[1]*n1 + dpm[2]*n2
  // For the isotropic case, and defining eminusa = eigvals - a,
  // the solution is easy:
  //   dpm[0] = 0;
  //   dpm[1] = (eminusa[1] - eminusa[0])/(n[1][1] - n[1][0]);
  //   dpm[2] = (eminusa[2] - eminusa[0])/(n[2][2] - n[2][0]);
  //
  // Now specialise to the isotropic case.  Define
  //   x = dpm[1] + dpm[2] = (eigvals[1] + eigvals[2] - 2*str)/(n[0][0] + n[0][1])
  // Notice that the RHS is a function of x, so we solve using
  // Newton-Raphson starting with x=initial_guess
  Real x = initial_guess;
  const Real denom = n[0](0) + n[0](1);
  Real str = tensile_strength(intnl_old + x);
 
  if (_strength.modelName().compare("Constant") != 0)
  {
    // Finding x is expensive.  Therefore
    // although x!=0 for Constant Hardening, solution
    // for dpm above demonstrates that we don't
    // actually need to know x to find the dpm for
    // Constant Hardening.
    //
    // However, for nontrivial Hardening, the following
    // is necessary
    const Real eig = eigvals[1] + eigvals[2];
    Real residual = denom * x - eig + 2 * str;
    Real jacobian;
    unsigned int iter = 0;
    do
    {
      jacobian = denom + 2 * dtensile_strength(intnl_old + x);
      x += -residual / jacobian;
      if (iter > _max_iters) // not converging
        return false;
      str = tensile_strength(intnl_old + x);
      residual = denom * x - eig + 2 * str;
      iter++;
    } while (residual * residual > _f_tol * _f_tol);
  }
 
  dpm[0] = 0;
  dpm[1] = ((eigvals[1] * n[0](0) - eigvals[2] * n[0](1)) / (n[0](0) - n[0](1)) - str) / denom;
  dpm[2] = ((eigvals[2] * n[0](0) - eigvals[1] * n[0](1)) / (n[0](0) - n[0](1)) - str) / denom;
 
  returned_stress(0, 0) = eigvals[0] - n[0](1) * (dpm[1] + dpm[2]);
  returned_stress(1, 1) = returned_stress(2, 2) = str;
  return true;
}
 
bool
SolidMechanicsPlasticTensileMulti::returnPlane(const std::vector<Real> & eigvals,
                                               const std::vector<RealVectorValue> & n,
                                               std::vector<Real> & dpm,
                                               RankTwoTensor & returned_stress,
                                               Real intnl_old,
                                               Real initial_guess) const
{
  // the returned point, "a", is defined by f2=0 and
  // a = p - dpm[2]*n2.
  // This is a vector equation in
  // principal stress space, and dpm[2] is the third
  // plasticity multiplier (dpm[0]=0=dpm[1] for return
  // to the plane) and "p" is the starting
  // position (p=eigvals).
  // (Kuhn-Tucker demands that dpm[2]>=0, but we leave checking
  // that condition for later.)
  // Since f2=0, we must have a[2]=tensile_strength,
  // so we can just look at the [2] component of the
  // equation, which yields
  // n[2][2]*dpm[2] - eigvals[2] + str = 0
  // For hardening, str=tensile_strength(intnl_old+dpm[2]),
  // and we want to solve for dpm[2].
  // Use Newton-Raphson with initial guess dpm[2] = initial_guess
  dpm[2] = initial_guess;
  Real residual = n[2](2) * dpm[2] - eigvals[2] + tensile_strength(intnl_old + dpm[2]);
  Real jacobian;
  unsigned int iter = 0;
  do
  {
    jacobian = n[2](2) + dtensile_strength(intnl_old + dpm[2]);
    dpm[2] += -residual / jacobian;
    if (iter > _max_iters) // not converging
      return false;
    residual = n[2](2) * dpm[2] - eigvals[2] + tensile_strength(intnl_old + dpm[2]);
    iter++;
  } while (residual * residual > _f_tol * _f_tol);
 
  dpm[0] = 0;
  dpm[1] = 0;
  returned_stress(0, 0) = eigvals[0] - dpm[2] * n[2](0);
  returned_stress(1, 1) = eigvals[1] - dpm[2] * n[2](1);
  returned_stress(2, 2) = eigvals[2] - dpm[2] * n[2](2);
  return true;
}
 
bool
SolidMechanicsPlasticTensileMulti::KuhnTuckerOK(const RankTwoTensor & returned_diagonal_stress,
                                                const std::vector<Real> & dpm,
                                                Real str,
                                                Real ep_plastic_tolerance) const
{
  for (unsigned i = 0; i < 3; ++i)
    if (!SolidMechanicsPlasticModel::KuhnTuckerSingleSurface(
            returned_diagonal_stress(i, i) - str, dpm[i], ep_plastic_tolerance))
      return false;
  return true;
}
 
RankFourTensor
SolidMechanicsPlasticTensileMulti::consistentTangentOperator(
    const RankTwoTensor & trial_stress,
    Real intnl_old,
    const RankTwoTensor & stress,
    Real intnl,
    const RankFourTensor & E_ijkl,
    const std::vector<Real> & cumulative_pm) const
{
  if (!_use_custom_cto)
    return SolidMechanicsPlasticModel::consistentTangentOperator(
        trial_stress, intnl_old, stress, intnl, E_ijkl, cumulative_pm);
 
  mooseAssert(cumulative_pm.size() == 3,
              "SolidMechanicsPlasticTensileMulti size of cumulative_pm should be 3 but it is "
                  << cumulative_pm.size());
 
  if (cumulative_pm[2] <= 0) // All cumulative_pm are non-positive, so this is admissible
    return E_ijkl;
 
  // Need the eigenvalues at the returned configuration
  std::vector<Real> eigvals;
  stress.symmetricEigenvalues(eigvals);
 
  // need to rotate to and from principal stress space
  // using the eigenvectors of the trial configuration
  // (not the returned configuration).
  std::vector<Real> trial_eigvals;
  RankTwoTensor trial_eigvecs;
  trial_stress.symmetricEigenvaluesEigenvectors(trial_eigvals, trial_eigvecs);
 
  // The returnMap will have returned to the Tip, Edge or
  // Plane.  The consistentTangentOperator describes the
  // change in stress for an arbitrary change in applied
  // strain.  I assume that the change in strain will not
  // change the type of return (Tip remains Tip, Edge remains
  // Edge, Plane remains Plane).
  // I assume isotropic elasticity.
  //
  // The consistent tangent operator is a little different
  // than cases where no rotation to principal stress space
  // is made during the returnMap.  Let S_ij be the stress
  // in original coordinates, and s_ij be the stress in the
  // principal stress coordinates, so that
  // s_ij = diag(eigvals[0], eigvals[1], eigvals[2])
  // We want dS_ij under an arbitrary change in strain (ep->ep+dep)
  // dS = S(ep+dep) - S(ep)
  //    = R(ep+dep) s(ep+dep) R(ep+dep)^T - R(ep) s(ep) R(ep)^T
  // Here R = the rotation to principal-stress space, ie
  // R_ij = eigvecs[i][j] = i^th component of j^th eigenvector
  // Expanding to first order in dep,
  // dS = R(ep) (s(ep+dep) - s(ep)) R(ep)^T
  //      + dR/dep s(ep) R^T + R(ep) s(ep) dR^T/dep
  // The first line is all that is usually calculated in the
  // consistent tangent operator calculation, and is called
  // cto below.
  // The second line involves changes in the eigenvectors, and
  // is called sec below.
 
  RankFourTensor cto;
  const Real hard = dtensile_strength(intnl);
  const Real la = E_ijkl(0, 0, 1, 1);
  const Real mu = 0.5 * (E_ijkl(0, 0, 0, 0) - la);
 
  if (cumulative_pm[1] <= 0)
  {
    // only cumulative_pm[2] is positive, so this is return to the Plane
    const Real denom = hard + la + 2 * mu;
    const Real al = la * la / denom;
    const Real be = la * (la + 2 * mu) / denom;
    const Real ga = hard * (la + 2 * mu) / denom;
    std::vector<Real> comps(9);
    comps[0] = comps[4] = la + 2 * mu - al;
    comps[1] = comps[3] = la - al;
    comps[2] = comps[5] = comps[6] = comps[7] = la - be;
    comps[8] = ga;
    cto.fillFromInputVector(comps, RankFourTensor::principal);
  }
  else if (cumulative_pm[0] <= 0)
  {
    // both cumulative_pm[2] and cumulative_pm[1] are positive, so Edge
    const Real denom = 2 * hard + 2 * la + 2 * mu;
    const Real al = hard * 2 * la / denom;
    const Real be = hard * (2 * la + 2 * mu) / denom;
    std::vector<Real> comps(9);
    comps[0] = la + 2 * mu - 2 * la * la / denom;
    comps[1] = comps[2] = al;
    comps[3] = comps[6] = al;
    comps[4] = comps[5] = comps[7] = comps[8] = be;
    cto.fillFromInputVector(comps, RankFourTensor::principal);
  }
  else
  {
    // all cumulative_pm are positive, so Tip
    const Real denom = 3 * hard + 3 * la + 2 * mu;
    std::vector<Real> comps(2);
    comps[0] = hard * (3 * la + 2 * mu) / denom;
    comps[1] = 0;
    cto.fillFromInputVector(comps, RankFourTensor::symmetric_isotropic);
  }
 
  cto.rotate(trial_eigvecs);
 
  // drdsig = change in eigenvectors under a small stress change
  // drdsig(i,j,m,n) = dR(i,j)/dS_mn
  // The formula below is fairly easily derived:
  // S R = R s, so taking the variation
  // dS R + S dR = dR s + R ds, and multiplying by R^T
  // R^T dS R + R^T S dR = R^T dR s + ds .... (eqn 1)
  // I demand that RR^T = 1 = R^T R, and also that
  // (R+dR)(R+dR)^T = 1 = (R+dT)^T (R+dR), which means
  // that dR = R*c, for some antisymmetric c, so Eqn1 reads
  // R^T dS R + s c = c s + ds
  // Grabbing the components of this gives ds/dS (already
  // in RankTwoTensor), and c, which is:
  //   dR_ik/dS_mn = drdsig(i, k, m, n) = trial_eigvecs(m, b)*trial_eigvecs(n, k)*trial_eigvecs(i,
  //   b)/(trial_eigvals[k] - trial_eigvals[b]);
  // (sum over b!=k).
 
  RankFourTensor drdsig;
  for (unsigned k = 0; k < 3; ++k)
    for (unsigned b = 0; b < 3; ++b)
    {
      if (b == k)
        continue;
      for (unsigned m = 0; m < 3; ++m)
        for (unsigned n = 0; n < 3; ++n)
          for (unsigned i = 0; i < 3; ++i)
            drdsig(i, k, m, n) += trial_eigvecs(m, b) * trial_eigvecs(n, k) * trial_eigvecs(i, b) /
                                  (trial_eigvals[k] - trial_eigvals[b]);
    }
 
  // With diagla = diag(eigvals[0], eigvals[1], digvals[2])
  // The following implements
  // ans(i, j, a, b) += (drdsig(i, k, m, n)*trial_eigvecs(j, l)*diagla(k, l) + trial_eigvecs(i,
  // k)*drdsig(j, l, m, n)*diagla(k, l))*E_ijkl(m, n, a, b);
  // (sum over k, l, m and n)
 
  RankFourTensor ans;
  for (unsigned i = 0; i < 3; ++i)
    for (unsigned j = 0; j < 3; ++j)
      for (unsigned a = 0; a < 3; ++a)
        for (unsigned k = 0; k < 3; ++k)
          for (unsigned m = 0; m < 3; ++m)
            ans(i, j, a, a) += (drdsig(i, k, m, m) * trial_eigvecs(j, k) +
                                trial_eigvecs(i, k) * drdsig(j, k, m, m)) *
                               eigvals[k] * la; // E_ijkl(m, n, a, b) = la*(m==n)*(a==b);
 
  for (unsigned i = 0; i < 3; ++i)
    for (unsigned j = 0; j < 3; ++j)
      for (unsigned a = 0; a < 3; ++a)
        for (unsigned b = 0; b < 3; ++b)
          for (unsigned k = 0; k < 3; ++k)
          {
            ans(i, j, a, b) += (drdsig(i, k, a, b) * trial_eigvecs(j, k) +
                                trial_eigvecs(i, k) * drdsig(j, k, a, b)) *
                               eigvals[k] * mu; // E_ijkl(m, n, a, b) = mu*(m==a)*(n==b)
            ans(i, j, a, b) += (drdsig(i, k, b, a) * trial_eigvecs(j, k) +
                                trial_eigvecs(i, k) * drdsig(j, k, b, a)) *
                               eigvals[k] * mu; // E_ijkl(m, n, a, b) = mu*(m==b)*(n==a)
          }
 
  return cto + ans;
}
 
bool
SolidMechanicsPlasticTensileMulti::useCustomReturnMap() const
{
  return _use_custom_returnMap;
}
 
bool
SolidMechanicsPlasticTensileMulti::useCustomCTO() const
{
  return _use_custom_cto;
}