Ehlers.cpp

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#include "Ehlers.h"
 
#include <Eigen/LU>
#include <boost/math/special_functions/pow.hpp>
 
#include "BaseLib/cpp23.h"
#include "LinearElasticIsotropic.h"
#include "MaterialLib/MPL/Utils/GetSymmetricTensor.h"
#include "MathLib/LinAlg/Eigen/EigenMapTools.h"
 
namespace MPL = MaterialPropertyLib;
 
namespace MaterialLib
{
namespace Solids
{
namespace Ehlers
{
template <int DisplacementDim>
double StateVariables<DisplacementDim>::getEquivalentPlasticStrain() const
{
    return std::sqrt(2.0 / 3.0 * Invariants::FrobeniusNorm(eps_p.D.eval()));
}
 
template <int DisplacementDim>
MathLib::KelvinVector::KelvinMatrixType<DisplacementDim> sOdotS(
    MathLib::KelvinVector::KelvinVectorType<DisplacementDim> const& v);
 
template <int DisplacementDim>
struct PhysicalStressWithInvariants final
{
    static int const KelvinVectorSize =
        MathLib::KelvinVector::kelvin_vector_dimensions(DisplacementDim);
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
    using KelvinVector =
        MathLib::KelvinVector::KelvinVectorType<DisplacementDim>;
 
    explicit PhysicalStressWithInvariants(KelvinVector const& stress)
        : value{stress},
          D{Invariants::deviatoric_projection * stress},
          I_1{Invariants::trace(stress)},
          J_2{Invariants::J2(D)},
          J_3{Invariants::J3(D)}
    {
    }
 
    KelvinVector value;
    KelvinVector D;
    double I_1;
    double J_2;
    double J_3;
 
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW;
};
 
struct OnePlusGamma_pTheta final
{
    OnePlusGamma_pTheta(double const gamma_p, double const theta,
                        double const m_p)
        : value{1 + gamma_p * theta},
          pow_m_p{std::pow(value, m_p)},
          pow_m_p1{pow_m_p / value}
    {
    }
 
    double const value;
    double const pow_m_p;
    double const pow_m_p1;
};
 
template <int DisplacementDim>
double plasticFlowVolumetricPart(
    PhysicalStressWithInvariants<DisplacementDim> const& s,
    double const sqrtPhi, double const alpha_p, double const beta_p,
    double const delta_p, double const epsilon_p)
{
    return 3 *
               (alpha_p * s.I_1 +
                4 * boost::math::pow<2>(delta_p) * boost::math::pow<3>(s.I_1)) /
               (2 * sqrtPhi) +
           3 * beta_p + 6 * epsilon_p * s.I_1;
}
 
template <int DisplacementDim>
typename SolidEhlers<DisplacementDim>::KelvinVector plasticFlowDeviatoricPart(
    PhysicalStressWithInvariants<DisplacementDim> const& s,
    OnePlusGamma_pTheta const& one_gt, double const sqrtPhi,
    typename SolidEhlers<DisplacementDim>::KelvinVector const& dtheta_dsigma,
    double const gamma_p, double const m_p)
{
    return (one_gt.pow_m_p *
            (s.D + s.J_2 * m_p * gamma_p * dtheta_dsigma / one_gt.value)) /
           (2 * sqrtPhi);
}
 
template <int DisplacementDim>
double yieldFunction(MaterialProperties const& mp,
                     PhysicalStressWithInvariants<DisplacementDim> const& s,
                     double const k)
{
    double const I_1_squared = boost::math::pow<2>(s.I_1);
    assert(s.J_2 != 0);
 
    return std::sqrt(s.J_2 * std::pow(1 + mp.gamma * s.J_3 /
                                              (s.J_2 * std::sqrt(s.J_2)),
                                      mp.m) +
                     mp.alpha / 2. * I_1_squared +
                     boost::math::pow<2>(mp.delta) *
                         boost::math::pow<2>(I_1_squared)) +
           mp.beta * s.I_1 + mp.epsilon * I_1_squared - k;
}
 
template <int DisplacementDim>
std::pair<MathLib::KelvinVector::KelvinVectorType<DisplacementDim>,
          MathLib::KelvinVector::KelvinMatrixType<DisplacementDim>>
thetaSigmaDerivatives(double theta,
                      PhysicalStressWithInvariants<DisplacementDim> const& s)
{
    constexpr int KelvinVectorSize =
        MathLib::KelvinVector::kelvin_vector_dimensions(DisplacementDim);
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
    using KelvinVector =
        MathLib::KelvinVector::KelvinVectorType<DisplacementDim>;
    using KelvinMatrix =
        MathLib::KelvinVector::KelvinMatrixType<DisplacementDim>;
 
    if (theta == 0)
    {
        return {KelvinVector::Zero(), KelvinMatrix::Zero()};
    }
 
    auto const& P_dev = Invariants::deviatoric_projection;
 
    // inverse of deviatoric stress tensor
    if (Invariants::determinant(s.D) == 0)
    {
        OGS_FATAL("Determinant is zero. Matrix is non-invertable.");
    }
    // inverse of sigma_D
    KelvinVector const sigma_D_inverse = MathLib::KelvinVector::inverse(s.D);
    KelvinVector const sigma_D_inverse_D = P_dev * sigma_D_inverse;
 
    KelvinVector const dtheta_dsigma =
        theta * sigma_D_inverse_D - 3. / 2. * theta / s.J_2 * s.D;
    KelvinMatrix const d2theta_dsigma2 =
        theta * P_dev * sOdotS<DisplacementDim>(sigma_D_inverse) * P_dev +
        sigma_D_inverse_D * dtheta_dsigma.transpose() -
        3. / 2. * theta / s.J_2 * P_dev -
        3. / 2. * dtheta_dsigma / s.J_2 * s.D.transpose() +
        3. / 2. * theta / boost::math::pow<2>(s.J_2) * s.D * s.D.transpose();
 
    return {dtheta_dsigma, d2theta_dsigma2};
}
 
template <int DisplacementDim>
typename SolidEhlers<DisplacementDim>::ResidualVectorType
calculatePlasticResidual(
    MathLib::KelvinVector::KelvinVectorType<DisplacementDim> const& eps_D,
    double const eps_V,
    PhysicalStressWithInvariants<DisplacementDim> const& s,
    MathLib::KelvinVector::KelvinVectorType<DisplacementDim> const& eps_p_D,
    MathLib::KelvinVector::KelvinVectorType<DisplacementDim> const& eps_p_D_dot,
    double const eps_p_V,
    double const eps_p_V_dot,
    double const eps_p_eff_dot,
    double const lambda,
    double const k,
    MaterialProperties const& mp)
{
    static int const KelvinVectorSize =
        MathLib::KelvinVector::kelvin_vector_dimensions(DisplacementDim);
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
    using KelvinVector =
        MathLib::KelvinVector::KelvinVectorType<DisplacementDim>;
 
    auto const& identity2 = Invariants::identity2;
 
    double const theta = s.J_3 / (s.J_2 * std::sqrt(s.J_2));
 
    typename SolidEhlers<DisplacementDim>::ResidualVectorType residual;
    // calculate stress residual
    residual.template segment<KelvinVectorSize>(0).noalias() =
        s.value / mp.G - 2 * (eps_D - eps_p_D) -
        mp.K / mp.G * (eps_V - eps_p_V) * identity2;
 
    auto const [dtheta_dsigma, d2theta_dsigma2] =
        thetaSigmaDerivatives(theta, s);
 
    OnePlusGamma_pTheta const one_gt{mp.gamma_p, theta, mp.m_p};
    double const sqrtPhi = std::sqrt(
        s.J_2 * one_gt.pow_m_p + mp.alpha_p / 2. * boost::math::pow<2>(s.I_1) +
        boost::math::pow<2>(mp.delta_p) * boost::math::pow<4>(s.I_1));
    KelvinVector const flow_D = plasticFlowDeviatoricPart(
        s, one_gt, sqrtPhi, dtheta_dsigma, mp.gamma_p, mp.m_p);
    KelvinVector const lambda_flow_D = lambda * flow_D;
 
    residual.template segment<KelvinVectorSize>(KelvinVectorSize).noalias() =
        eps_p_D_dot - lambda_flow_D;
 
    // plastic volume strain
    {
        double const flow_V = plasticFlowVolumetricPart<DisplacementDim>(
            s, sqrtPhi, mp.alpha_p, mp.beta_p, mp.delta_p, mp.epsilon_p);
        residual(2 * KelvinVectorSize, 0) = eps_p_V_dot - lambda * flow_V;
    }
 
    // evolution of plastic equivalent strain
    residual(2 * KelvinVectorSize + 1) =
        eps_p_eff_dot -
        std::sqrt(2. / 3. * lambda_flow_D.transpose() * lambda_flow_D);
 
    // yield function (for plastic multiplier)
    residual(2 * KelvinVectorSize + 2) = yieldFunction(mp, s, k) / mp.G;
    return residual;
}
 
template <int DisplacementDim>
typename SolidEhlers<DisplacementDim>::JacobianMatrix calculatePlasticJacobian(
    double const dt,
    PhysicalStressWithInvariants<DisplacementDim> const& s,
    double const lambda,
    MaterialProperties const& mp)
{
    static int const KelvinVectorSize =
        MathLib::KelvinVector::kelvin_vector_dimensions(DisplacementDim);
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
    using KelvinVector =
        MathLib::KelvinVector::KelvinVectorType<DisplacementDim>;
    using KelvinMatrix =
        MathLib::KelvinVector::KelvinMatrixType<DisplacementDim>;
 
    auto const& P_dev = Invariants::deviatoric_projection;
    auto const& identity2 = Invariants::identity2;
 
    double const theta = s.J_3 / (s.J_2 * std::sqrt(s.J_2));
    OnePlusGamma_pTheta const one_gt{mp.gamma_p, theta, mp.m_p};
 
    auto const [dtheta_dsigma, d2theta_dsigma2] =
        thetaSigmaDerivatives(theta, s);
 
    // deviatoric flow
    double const sqrtPhi = std::sqrt(
        s.J_2 * one_gt.pow_m_p + mp.alpha_p / 2. * boost::math::pow<2>(s.I_1) +
        boost::math::pow<2>(mp.delta_p) * boost::math::pow<4>(s.I_1));
    KelvinVector const flow_D = plasticFlowDeviatoricPart(
        s, one_gt, sqrtPhi, dtheta_dsigma, mp.gamma_p, mp.m_p);
    KelvinVector const lambda_flow_D = lambda * flow_D;
 
    typename SolidEhlers<DisplacementDim>::JacobianMatrix jacobian =
        SolidEhlers<DisplacementDim>::JacobianMatrix::Zero();
 
    // G_11
    jacobian.template block<KelvinVectorSize, KelvinVectorSize>(0, 0)
        .noalias() = KelvinMatrix::Identity();
 
    // G_12
    jacobian
        .template block<KelvinVectorSize, KelvinVectorSize>(0, KelvinVectorSize)
        .noalias() = 2 * KelvinMatrix::Identity();
 
    // G_13
    jacobian.template block<KelvinVectorSize, 1>(0, 2 * KelvinVectorSize)
        .noalias() = mp.K / mp.G * identity2;
 
    // G_14 and G_15 are zero
 
    // G_21 -- derivative of deviatoric flow
 
    double const gm_p = mp.gamma_p * mp.m_p;
    // intermediate variable for derivative of deviatoric flow
    KelvinVector const M0 = s.J_2 / one_gt.value * dtheta_dsigma;
    // derivative of Phi w.r.t. sigma
    KelvinVector const dPhi_dsigma =
        one_gt.pow_m_p * (s.D + gm_p * M0) +
        (mp.alpha_p * s.I_1 +
         4 * boost::math::pow<2>(mp.delta_p) * boost::math::pow<3>(s.I_1)) *
            identity2;
 
    // intermediate variable for derivative of deviatoric flow
    KelvinMatrix const M1 =
        one_gt.pow_m_p *
        (s.D * dPhi_dsigma.transpose() + gm_p * M0 * dPhi_dsigma.transpose());
    // intermediate variable for derivative of deviatoric flow
    KelvinMatrix const M2 =
        one_gt.pow_m_p * (P_dev + s.D * gm_p * M0.transpose());
 
    // intermediate variable for derivative of deviatoric flow
    KelvinMatrix const M3 =
        gm_p * one_gt.pow_m_p1 *
        ((s.D + (gm_p - mp.gamma_p) * M0) * dtheta_dsigma.transpose() +
         s.J_2 * d2theta_dsigma2);
 
    // derivative of flow_D w.r.t. sigma
    KelvinMatrix const dflow_D_dsigma =
        (-M1 / (4 * boost::math::pow<3>(sqrtPhi)) + (M2 + M3) / (2 * sqrtPhi)) *
        mp.G;
    jacobian
        .template block<KelvinVectorSize, KelvinVectorSize>(KelvinVectorSize, 0)
        .noalias() = -lambda * dflow_D_dsigma;
 
    // G_22
    jacobian
        .template block<KelvinVectorSize, KelvinVectorSize>(KelvinVectorSize,
                                                            KelvinVectorSize)
        .noalias() = KelvinMatrix::Identity() / dt;
 
    // G_23 and G_24 are zero
 
    // G_25
    jacobian
        .template block<KelvinVectorSize, 1>(KelvinVectorSize,
                                             2 * KelvinVectorSize + 2)
        .noalias() = -flow_D;
 
    // G_31
    {
        // derivative of flow_V w.r.t. sigma
        KelvinVector const dflow_V_dsigma =
            3 * mp.G *
            (-(mp.alpha_p * s.I_1 + 4 * boost::math::pow<2>(mp.delta_p) *
                                        boost::math::pow<3>(s.I_1)) /
                 (4 * boost::math::pow<3>(sqrtPhi)) * dPhi_dsigma +
             (mp.alpha_p * identity2 +
              12 * boost::math::pow<2>(mp.delta_p * s.I_1) * identity2) /
                 (2 * sqrtPhi) +
             2 * mp.epsilon_p * identity2);
 
        jacobian.template block<1, KelvinVectorSize>(2 * KelvinVectorSize, 0)
            .noalias() = -lambda * dflow_V_dsigma.transpose();
    }
 
    // G_32 is zero
 
    // G_33
    jacobian(2 * KelvinVectorSize, 2 * KelvinVectorSize) = 1. / dt;
 
    // G_34 is zero
 
    // G_35
    {
        double const flow_V = plasticFlowVolumetricPart<DisplacementDim>(
            s, sqrtPhi, mp.alpha_p, mp.beta_p, mp.delta_p, mp.epsilon_p);
        jacobian(2 * KelvinVectorSize, 2 * KelvinVectorSize + 2) = -flow_V;
    }
 
    // increment of effectiv plastic strain
    double const eff_flow =
        std::sqrt(2. / 3. * lambda_flow_D.transpose() * lambda_flow_D);
 
    if (eff_flow > 0)
    {
        // intermediate variable for derivative of plastic jacobian
        KelvinVector const eff_flow23_lambda_flow_D =
            -2 / 3. / eff_flow * lambda_flow_D;
        // G_41
        jacobian
            .template block<1, KelvinVectorSize>(2 * KelvinVectorSize + 1, 0)
            .noalias() = lambda * dflow_D_dsigma * eff_flow23_lambda_flow_D;
        // G_45
        jacobian(2 * KelvinVectorSize + 1, 2 * KelvinVectorSize + 2) =
            eff_flow23_lambda_flow_D.transpose() * flow_D;
    }
 
    // G_42 and G_43 are zero
 
    // G_44
    jacobian(2 * KelvinVectorSize + 1, 2 * KelvinVectorSize + 1) = 1. / dt;
 
    // G_51
    {
        double const one_gt_pow_m = std::pow(one_gt.value, mp.m);
        double const gm = mp.gamma * mp.m;
        // derivative of yield function w.r.t. sigma
        KelvinVector const dF_dsigma =
            mp.G *
                (one_gt_pow_m * (s.D + gm * M0) +
                 (mp.alpha * s.I_1 + 4 * boost::math::pow<2>(mp.delta) *
                                         boost::math::pow<3>(s.I_1)) *
                     identity2) /
                (2. * sqrtPhi) +
            mp.G * (mp.beta + 2 * mp.epsilon_p * s.I_1) * identity2;
 
        jacobian
            .template block<1, KelvinVectorSize>(2 * KelvinVectorSize + 2, 0)
            .noalias() = dF_dsigma.transpose() / mp.G;
    }
 
    // G_54
    jacobian(2 * KelvinVectorSize + 2, 2 * KelvinVectorSize + 1) =
        -mp.kappa * mp.hardening_coefficient / mp.G;
 
    // G_52, G_53, G_55 are zero
    return jacobian;
}
 
template <int DisplacementDim>
MathLib::KelvinVector::KelvinMatrixType<DisplacementDim> calculateDResidualDEps(
    double const K, double const G)
{
    static int const KelvinVectorSize =
        MathLib::KelvinVector::kelvin_vector_dimensions(DisplacementDim);
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
 
    auto const& P_dev = Invariants::deviatoric_projection;
    auto const& P_sph = Invariants::spherical_projection;
    auto const& I =
        MathLib::KelvinVector::KelvinMatrixType<DisplacementDim>::Identity();
 
    return -2. * I * P_dev - 3. * K / G * I * P_sph;
}
 
inline double calculateIsotropicHardening(double const kappa,
                                          double const hardening_coefficient,
                                          double const eps_p_eff)
{
    return kappa * (1. + eps_p_eff * hardening_coefficient);
}
 
template <int DisplacementDim>
typename SolidEhlers<DisplacementDim>::KelvinVector predict_sigma(
    double const G, double const K,
    typename SolidEhlers<DisplacementDim>::KelvinVector const& sigma_prev,
    typename SolidEhlers<DisplacementDim>::KelvinVector const& eps,
    typename SolidEhlers<DisplacementDim>::KelvinVector const& eps_prev,
    double const eps_V)
{
    static int const KelvinVectorSize =
        MathLib::KelvinVector::kelvin_vector_dimensions(DisplacementDim);
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
    auto const& P_dev = Invariants::deviatoric_projection;
 
    // dimensionless initial hydrostatic pressure
    double const pressure_prev = Invariants::trace(sigma_prev) / (-3. * G);
    // initial strain invariant
    double const e_prev = Invariants::trace(eps_prev);
    // dimensioness hydrostatic stress increment
    double const pressure = pressure_prev - K / G * (eps_V - e_prev);
    // dimensionless deviatoric initial stress
    typename SolidEhlers<DisplacementDim>::KelvinVector const sigma_D_prev =
        P_dev * sigma_prev / G;
    // dimensionless deviatoric stress
    typename SolidEhlers<DisplacementDim>::KelvinVector const sigma_D =
        sigma_D_prev + 2 * P_dev * (eps - eps_prev);
    return sigma_D - pressure * Invariants::identity2;
}
 
template <typename ResidualVector, typename KelvinVector>
std::tuple<KelvinVector, PlasticStrain<KelvinVector>, double>
splitSolutionVector(ResidualVector const& solution)
{
    static auto const size = KelvinVector::SizeAtCompileTime;
    return std::forward_as_tuple(
        solution.template segment<size>(size * 0),
        PlasticStrain<KelvinVector>{solution.template segment<size>(size * 1),
                                    solution[size * 2], solution[size * 2 + 1]},
        solution[size * 2 + 2]);
}
 
template <int DisplacementDim>
SolidEhlers<DisplacementDim>::SolidEhlers(
    NumLib::NewtonRaphsonSolverParameters nonlinear_solver_parameters,
    MaterialPropertiesParameters material_properties,
    std::unique_ptr<DamagePropertiesParameters>&& damage_properties,
    TangentType tangent_type)
    : _nonlinear_solver_parameters(std::move(nonlinear_solver_parameters)),
      _mp(std::move(material_properties)),
      _damage_properties(std::move(damage_properties)),
      _tangent_type(tangent_type)
{
}
 
template <int DisplacementDim>
double SolidEhlers<DisplacementDim>::computeFreeEnergyDensity(
    double const /*t*/,
    ParameterLib::SpatialPosition const& /*x*/,
    double const /*dt*/,
    KelvinVector const& eps,
    KelvinVector const& sigma,
    typename MechanicsBase<DisplacementDim>::MaterialStateVariables const&
        material_state_variables) const
{
    assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
               &material_state_variables) != nullptr);
 
    auto const& eps_p = static_cast<StateVariables<DisplacementDim> const&>(
                            material_state_variables)
                            .eps_p;
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
    auto const& identity2 = Invariants::identity2;
    return (eps - eps_p.D - eps_p.V / 3 * identity2).dot(sigma) / 2;
}
 
template <int DisplacementDim>
std::optional<std::tuple<typename SolidEhlers<DisplacementDim>::KelvinVector,
                         std::unique_ptr<typename MechanicsBase<
                             DisplacementDim>::MaterialStateVariables>,
                         typename SolidEhlers<DisplacementDim>::KelvinMatrix>>
SolidEhlers<DisplacementDim>::integrateStress(
    MaterialPropertyLib::VariableArray const& variable_array_prev,
    MaterialPropertyLib::VariableArray const& variable_array, double const t,
    ParameterLib::SpatialPosition const& x, double const dt,
    typename MechanicsBase<DisplacementDim>::MaterialStateVariables const&
        material_state_variables) const
{
    auto const& eps_m = std::get<MPL::SymmetricTensor<DisplacementDim>>(
        variable_array.mechanical_strain);
    auto const& eps_m_prev = std::get<MPL::SymmetricTensor<DisplacementDim>>(
        variable_array_prev.mechanical_strain);
    auto const& sigma_prev = std::get<MPL::SymmetricTensor<DisplacementDim>>(
        variable_array_prev.stress);
 
    assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
               &material_state_variables) != nullptr);
 
    StateVariables<DisplacementDim> state =
        static_cast<StateVariables<DisplacementDim> const&>(
            material_state_variables);
    state.setInitialConditions();
 
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
 
    // volumetric strain
    double const eps_V = Invariants::trace(eps_m);
 
    auto const& P_dev = Invariants::deviatoric_projection;
    // deviatoric strain
    KelvinVector const eps_D = P_dev * eps_m;
 
    // do the evaluation once per function call.
    MaterialProperties const mp(t, x, _mp);
 
    KelvinVector sigma = predict_sigma<DisplacementDim>(
        mp.G, mp.K, sigma_prev, eps_m, eps_m_prev, eps_V);
 
    KelvinMatrix tangentStiffness;
 
    PhysicalStressWithInvariants<DisplacementDim> s{mp.G * sigma};
    // Quit early if sigma is zero (nothing to do), or dt is zero, or if we are
    // still in elastic zone.
    // dt can be zero if we are in the initialization phase and the tangent
    // stiffness will no be necessary. Anyway the newton loop below would not
    // work because of division by zero.
    if ((sigma.squaredNorm() == 0. || dt == 0. ||
         yieldFunction(
             mp, s,
             calculateIsotropicHardening(mp.kappa, mp.hardening_coefficient,
                                         state.eps_p.eff)) < 0.))
    {
        tangentStiffness = elasticTangentStiffness<DisplacementDim>(
            mp.K - 2. / 3 * mp.G, mp.G);
    }
    else
    {
        // Linear solver for the newton loop is required after the loop with the
        // same matrix. This saves one decomposition.
        Eigen::FullPivLU<Eigen::Matrix<double, JacobianResidualSize,
                                       JacobianResidualSize, Eigen::RowMajor>>
            linear_solver;
 
        {
            using KelvinVector =
                MathLib::KelvinVector::KelvinVectorType<DisplacementDim>;
            using ResidualVectorType =
                Eigen::Matrix<double, JacobianResidualSize, 1>;
            using JacobianMatrix =
                Eigen::Matrix<double, JacobianResidualSize,
                              JacobianResidualSize, Eigen::RowMajor>;
 
            JacobianMatrix jacobian;
 
            // Agglomerated solution vector construction.  It is later split
            // into individual parts by splitSolutionVector().
            ResidualVectorType solution;
            solution << sigma, state.eps_p.D, state.eps_p.V, state.eps_p.eff, 0;
 
            auto const update_residual = [&](ResidualVectorType& residual)
            {
                auto const& eps_p_D =
                    solution.template segment<KelvinVectorSize>(
                        KelvinVectorSize);
                KelvinVector const eps_p_D_dot =
                    (eps_p_D - state.eps_p_prev.D) / dt;
 
                double const& eps_p_V = solution[KelvinVectorSize * 2];
                double const eps_p_V_dot = (eps_p_V - state.eps_p_prev.V) / dt;
 
                double const& eps_p_eff = solution[KelvinVectorSize * 2 + 1];
                double const eps_p_eff_dot =
                    (eps_p_eff - state.eps_p_prev.eff) / dt;
 
                double const k_hardening = calculateIsotropicHardening(
                    mp.kappa, mp.hardening_coefficient,
                    solution[KelvinVectorSize * 2 + 1]);
                residual = calculatePlasticResidual<DisplacementDim>(
                    eps_D, eps_V, s,
                    solution.template segment<KelvinVectorSize>(
                        KelvinVectorSize),
                    eps_p_D_dot, solution[KelvinVectorSize * 2], eps_p_V_dot,
                    eps_p_eff_dot, solution[KelvinVectorSize * 2 + 2],
                    k_hardening, mp);
            };
 
            auto const update_jacobian = [&](JacobianMatrix& jacobian)
            {
                jacobian = calculatePlasticJacobian<DisplacementDim>(
                    dt, s, solution[KelvinVectorSize * 2 + 2], mp);
            };
 
            auto const update_solution =
                [&](ResidualVectorType const& increment)
            {
                solution += increment;
                s = PhysicalStressWithInvariants<DisplacementDim>{
                    mp.G * solution.template segment<KelvinVectorSize>(0)};
            };
 
            auto newton_solver = NumLib::NewtonRaphson(
                linear_solver, update_jacobian, update_residual,
                update_solution, _nonlinear_solver_parameters);
 
            auto const success_iterations = newton_solver.solve(jacobian);
 
            if (!success_iterations)
            {
                return {};
            }
 
            // If the Newton loop didn't run, the linear solver will not be
            // initialized.
            // This happens usually for the first iteration of the first
            // timestep.
            if (*success_iterations == 0)
            {
                linear_solver.compute(jacobian);
            }
 
            std::tie(sigma, state.eps_p, std::ignore) =
                splitSolutionVector<ResidualVectorType, KelvinVector>(solution);
        }
 
        // Calculate residual derivative w.r.t. strain
        Eigen::Matrix<double, JacobianResidualSize, KelvinVectorSize,
                      Eigen::RowMajor>
            dresidual_deps =
                Eigen::Matrix<double, JacobianResidualSize, KelvinVectorSize,
                              Eigen::RowMajor>::Zero();
        dresidual_deps.template block<KelvinVectorSize, KelvinVectorSize>(0, 0)
            .noalias() = calculateDResidualDEps<DisplacementDim>(mp.K, mp.G);
 
        if (_tangent_type == TangentType::Elastic)
        {
            tangentStiffness =
                elasticTangentStiffness<DisplacementDim>(mp.K, mp.G);
        }
        else if (_tangent_type == TangentType::Plastic ||
                 _tangent_type == TangentType::PlasticDamageSecant)
        {
            tangentStiffness =
                mp.G *
                linear_solver.solve(-dresidual_deps)
                    .template block<KelvinVectorSize, KelvinVectorSize>(0, 0);
            if (_tangent_type == TangentType::PlasticDamageSecant)
            {
                tangentStiffness *= 1 - state.damage.value();
            }
        }
        else
        {
            OGS_FATAL(
                "Unimplemented tangent type behaviour for the tangent type "
                "'{}'.",
                BaseLib::to_underlying(_tangent_type));
        }
    }
 
    KelvinVector sigma_final = mp.G * sigma;
 
    return {std::make_tuple(
        sigma_final,
        std::unique_ptr<
            typename MechanicsBase<DisplacementDim>::MaterialStateVariables>{
            new StateVariables<DisplacementDim>{
                static_cast<StateVariables<DisplacementDim> const&>(state)}},
        tangentStiffness)};
}
 
template <int DisplacementDim>
std::vector<typename MechanicsBase<DisplacementDim>::InternalVariable>
SolidEhlers<DisplacementDim>::getInternalVariables() const
{
    return {{"damage.kappa_d", 1,
             [](typename MechanicsBase<
                    DisplacementDim>::MaterialStateVariables const& state,
                std::vector<double>& cache) -> std::vector<double> const&
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto const& ehlers_state =
                     static_cast<StateVariables<DisplacementDim> const&>(state);
 
                 cache.resize(1);
                 cache.front() = ehlers_state.damage.kappa_d();
                 return cache;
             },
             [](typename MechanicsBase<DisplacementDim>::MaterialStateVariables&
                    state) -> std::span<double>
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto& ehlers_state =
                     static_cast<StateVariables<DisplacementDim>&>(state);
 
                 return {&ehlers_state.damage.kappa_d(), 1};
             }},
            {"damage.value", 1,
             [](typename MechanicsBase<
                    DisplacementDim>::MaterialStateVariables const& state,
                std::vector<double>& cache) -> std::vector<double> const&
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto const& ehlers_state =
                     static_cast<StateVariables<DisplacementDim> const&>(state);
 
                 cache.resize(1);
                 cache.front() = ehlers_state.damage.value();
                 return cache;
             },
             [](typename MechanicsBase<DisplacementDim>::MaterialStateVariables&
                    state) -> std::span<double>
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto& ehlers_state =
                     static_cast<StateVariables<DisplacementDim>&>(state);
 
                 return {&ehlers_state.damage.value(), 1};
             }},
            {"eps_p.D", KelvinVector::RowsAtCompileTime,
             [](typename MechanicsBase<
                    DisplacementDim>::MaterialStateVariables const& state,
                std::vector<double>& cache) -> std::vector<double> const&
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto const& ehlers_state =
                     static_cast<StateVariables<DisplacementDim> const&>(state);
 
                 cache.resize(KelvinVector::RowsAtCompileTime);
                 MathLib::toVector<KelvinVector>(
                     cache, KelvinVector::RowsAtCompileTime) =
                     MathLib::KelvinVector::kelvinVectorToSymmetricTensor(
                         ehlers_state.eps_p.D);
 
                 return cache;
             },
             [](typename MechanicsBase<DisplacementDim>::MaterialStateVariables&
                    state) -> std::span<double>
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto& ehlers_state =
                     static_cast<StateVariables<DisplacementDim>&>(state);
 
                 return {
                     ehlers_state.eps_p.D.data(),
                     static_cast<std::size_t>(KelvinVector::RowsAtCompileTime)};
             }},
            {"eps_p.V", 1,
             [](typename MechanicsBase<
                    DisplacementDim>::MaterialStateVariables const& state,
                std::vector<double>& cache) -> std::vector<double> const&
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto const& ehlers_state =
                     static_cast<StateVariables<DisplacementDim> const&>(state);
 
                 cache.resize(1);
                 cache.front() = ehlers_state.eps_p.V;
                 return cache;
             },
             [](typename MechanicsBase<DisplacementDim>::MaterialStateVariables&
                    state) -> std::span<double>
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto& ehlers_state =
                     static_cast<StateVariables<DisplacementDim>&>(state);
 
                 return {&ehlers_state.eps_p.V, 1};
             }},
            {"eps_p.eff", 1,
             [](typename MechanicsBase<
                    DisplacementDim>::MaterialStateVariables const& state,
                std::vector<double>& cache) -> std::vector<double> const&
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto const& ehlers_state =
                     static_cast<StateVariables<DisplacementDim> const&>(state);
 
                 cache.resize(1);
                 cache.front() = ehlers_state.eps_p.eff;
                 return cache;
             },
             [](typename MechanicsBase<DisplacementDim>::MaterialStateVariables&
                    state) -> std::span<double>
             {
                 assert(dynamic_cast<StateVariables<DisplacementDim> const*>(
                            &state) != nullptr);
                 auto& ehlers_state =
                     static_cast<StateVariables<DisplacementDim>&>(state);
 
                 return {&ehlers_state.eps_p.eff, 1};
             }}};
}
 
template class SolidEhlers<2>;
template class SolidEhlers<3>;
 
template struct StateVariables<2>;
template struct StateVariables<3>;
 
template <>
MathLib::KelvinVector::KelvinMatrixType<3> sOdotS<3>(
    MathLib::KelvinVector::KelvinVectorType<3> const& v)
{
    MathLib::KelvinVector::KelvinMatrixType<3> result;
 
    result(0, 0) = v(0) * v(0);
    result(0, 1) = result(1, 0) = v(3) * v(3) / 2.;
    result(0, 2) = result(2, 0) = v(5) * v(5) / 2.;
    result(0, 3) = result(3, 0) = v(0) * v(3);
    result(0, 4) = result(4, 0) = v(3) * v(5) / std::sqrt(2.);
    result(0, 5) = result(5, 0) = v(0) * v(5);
 
    result(1, 1) = v(1) * v(1);
    result(1, 2) = result(2, 1) = v(4) * v(4) / 2.;
    result(1, 3) = result(3, 1) = v(3) * v(1);
    result(1, 4) = result(4, 1) = v(1) * v(4);
    result(1, 5) = result(5, 1) = v(3) * v(4) / std::sqrt(2.);
 
    result(2, 2) = v(2) * v(2);
    result(2, 3) = result(3, 2) = v(5) * v(4) / std::sqrt(2.);
    result(2, 4) = result(4, 2) = v(4) * v(2);
    result(2, 5) = result(5, 2) = v(5) * v(2);
 
    result(3, 3) = v(0) * v(1) + v(3) * v(3) / 2.;
    result(3, 4) = result(4, 3) =
        v(3) * v(4) / 2. + v(5) * v(1) / std::sqrt(2.);
    result(3, 5) = result(5, 3) =
        v(0) * v(4) / std::sqrt(2.) + v(3) * v(5) / 2.;
 
    result(4, 4) = v(1) * v(2) + v(4) * v(4) / 2.;
    result(4, 5) = result(5, 4) =
        v(3) * v(2) / std::sqrt(2.) + v(5) * v(4) / 2.;
 
    result(5, 5) = v(0) * v(2) + v(5) * v(5) / 2.;
    return result;
}
 
template <>
MathLib::KelvinVector::KelvinMatrixType<2> sOdotS<2>(
    MathLib::KelvinVector::KelvinVectorType<2> const& v)
{
    MathLib::KelvinVector::KelvinMatrixType<2> result;
 
    result(0, 0) = v(0) * v(0);
    result(0, 1) = result(1, 0) = v(3) * v(3) / 2.;
    result(0, 2) = result(2, 0) = 0;
    result(0, 3) = result(3, 0) = v(0) * v(3);
 
    result(1, 1) = v(1) * v(1);
    result(1, 2) = result(2, 1) = 0;
    result(1, 3) = result(3, 1) = v(3) * v(1);
 
    result(2, 2) = v(2) * v(2);
    result(2, 3) = result(3, 2) = 0;
 
    result(3, 3) = v(0) * v(1) + v(3) * v(3) / 2.;
 
    return result;
}
 
}  // namespace Ehlers
}  // namespace Solids
}  // namespace MaterialLib