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material_neohookean_inline_impl.hh
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material_neohookean_inline_impl.hh
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/**
* @file material_neohookean_inline_impl.hh
*
* @author Daniel Pino Muñoz <daniel.pinomunoz@epfl.ch>
*
* @date creation: Mon Apr 08 2013
* @date last modification: Thu Feb 20 2020
*
* @brief Implementation of the inline functions of the material elastic
*
*
* @section LICENSE
*
* Copyright (©) 2014-2021 EPFL (Ecole Polytechnique Fédérale de Lausanne)
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* Akantu is free software: you can redistribute it and/or modify it under the
* terms of the GNU Lesser General Public License as published by the Free
* Software Foundation, either version 3 of the License, or (at your option) any
* later version.
*
* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
* details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
*
*/
/* -------------------------------------------------------------------------- */
#include "material_neohookean.hh"
/* -------------------------------------------------------------------------- */
#include <cmath>
#include <iostream>
#include <utility>
/* -------------------------------------------------------------------------- */
namespace akantu {
/* -------------------------------------------------------------------------- */
template <Int dim>
inline void MaterialNeohookean<dim>::computeDeltaStressOnQuad(
__attribute__((unused)) const Matrix<Real> & grad_u,
__attribute__((unused)) const Matrix<Real> & grad_delta_u,
__attribute__((unused)) Matrix<Real> & delta_S) {}
//! computes the second piola kirchhoff stress, called S
template <Int dim>
template <class Args>
inline void MaterialNeohookean<dim>::computeStressOnQuad(Args && args) {
// Neo hookean book
auto && F = Material::gradUToF<dim>(args["grad_u"_n]);
auto && C = Material::rightCauchy<dim>(F);
// the term sqrt(C33) corresponds to the off plane strain (2D plane stress)
auto J = F.determinant() * std::sqrt(args["C33"_n]);
args["sigma"_n] = Matrix<Real, dim, dim>::Identity() * mu +
(lambda * log(J) - mu) * C.inverse();
}
/* -------------------------------------------------------------------------- */
class C33_NR : public Math::NewtonRaphsonFunctor<Real> {
public:
C33_NR(std::string name, const Real & lambda, const Real & mu,
const Matrix<Real> & C)
: NewtonRaphsonFunctor(std::move(name)), lambda(lambda), mu(mu), C(C) {}
inline Real f(const Real & x) const override {
return (this->lambda / 2. *
(std::log(x) + std::log(this->C(0, 0) * this->C(1, 1) -
Math::pow<2>(this->C(0, 1)))) +
this->mu * (x - 1.));
}
inline Real f_prime(const Real & x) const override {
AKANTU_DEBUG_ASSERT(std::abs(x) > Math::getTolerance(),
"x is zero (x should be the off plane right Cauchy"
<< " measure in this function so you made a mistake"
<< " somewhere else that lead to a zero here!!!");
return (this->lambda / (2. * x) + this->mu);
}
private:
const Real & lambda;
const Real & mu;
const Matrix<Real> & C;
};
/* -------------------------------------------------------------------------- */
template <Int dim>
template <class Args>
inline void
MaterialNeohookean<dim>::computeThirdAxisDeformationOnQuad(Args && args) {
// Neo hookean book
auto F = Material::gradUToF<dim>(args["grad_u"_n]);
auto C = Material::rightCauchy<dim>(F);
Math::NewtonRaphson<Real> nr(1e-5, 100);
auto & C33 = args["C33"_n];
C33 = nr.solve(C33_NR("Neohookean_plan_stress", this->lambda, this->mu, C),
C33);
}
/* -------------------------------------------------------------------------- */
template <Int dim>
inline void
MaterialNeohookean<dim>::computePiolaKirchhoffOnQuad(const Matrix<Real> & E,
Matrix<Real> & S) {
/// \f$ \sigma_{ij} = \lambda * (\nabla u)_{kk} * \delta_{ij} + \mu * (\nabla
/// u_{ij} + \nabla u_{ji}) \f$
S = Matrix<Real, dim, dim>::Identity() * lambda * E.trace() + 2.0 * mu * E;
}
/* -------------------------------------------------------------------------- */
template <Int dim>
inline void MaterialNeohookean<dim>::computeFirstPiolaKirchhoffOnQuad(
const Matrix<Real> & grad_u, const Matrix<Real> & S, Matrix<Real> & P) {
auto F = Material::gradUToF<dim>(grad_u);
// first Piola-Kirchhoff is computed as the product of the deformation
// gracient
// tensor and the second Piola-Kirchhoff stress tensor
P = F * S;
}
/**************************************************************************************/
/* Computation of the potential energy for a this neo hookean material */
template <Int dim>
inline void MaterialNeohookean<dim>::computePotentialEnergyOnQuad(
const Matrix<Real> & grad_u, Real & epot) {
auto F = Material::gradUToF<dim>(grad_u);
auto C = Material::rightCauchy<dim>(F);
auto J = F.determinant();
epot =
0.5 * lambda * pow(log(J), 2.) + mu * (-log(J) + 0.5 * (C.trace() - dim));
}
/* -------------------------------------------------------------------------- */
template <Int dim>
template <class Args>
inline void MaterialNeohookean<dim>::computeTangentModuliOnQuad(Args && args) {
auto & tangent = args["tangent_moduli"_n];
// Neo hookean book
auto cols = tangent.cols();
auto rows = tangent.rows();
auto F = Material::gradUToF<dim>(args["grad_u"_n]);
auto C = Material::rightCauchy<dim>(F);
auto J = F.determinant() * sqrt(C33);
auto && Cminus = C.inverse();
for (Int m = 0; m < rows; m++) {
UInt i = VoigtHelper<dim>::vec[m][0];
UInt j = VoigtHelper<dim>::vec[m][1];
for (Int n = 0; n < cols; n++) {
UInt k = VoigtHelper<dim>::vec[n][0];
UInt l = VoigtHelper<dim>::vec[n][1];
// book belytchko
tangent(m, n) = lambda * Cminus(i, j) * Cminus(k, l) +
(mu - lambda * log(J)) * (Cminus(i, k) * Cminus(j, l) +
Cminus(i, l) * Cminus(k, j));
}
}
}
/* -------------------------------------------------------------------------- */
} // namespace akantu

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