RankTwoTensor
RankTwoTensor::dsin3Lode(const Real r0) const
{
Real bar = secondInvariant();
if (bar <= r0)
return RankTwoTensor();
else
return -1.5 * std::sqrt(3.0) * (dthirdInvariant() / std::pow(bar, 1.5) - 1.5 * dsecondInvariant() * thirdInvariant() / std::pow(bar, 2.5));
}
Real
RankTwoTensor::sin3Lode(const Real r0, const Real r0_value) const
{
Real bar = secondInvariant();
if (bar <= r0)
// in this case the Lode angle is not defined
return r0_value;
else
// the min and max here gaurd against precision-loss when bar is tiny but nonzero.
return std::max(std::min(-1.5 * std::sqrt(3.0) * thirdInvariant() / std::pow(bar, 1.5), 1.0), -1.0);
}
/** The elasticity tensor in material description.
* Using expressions (6.193/4) from Holzapfel (2000), identifying the
* invariant \f$ I_2 \f$ from here with his \f$ I_3 \f$, and discarding all
* terms in that book with derivatives w.r.t \f$ I_2 \f$ gives finally
* \f[
* C_{ABCD} =
* \delta_1 \delta_{AB} \delta_{CD} +
* \delta_3 (\delta_{AB} C^{-1}_{CD} + C^{-1}_{AB} \delta_{CD})
* \delta_6 C^{-1}_{AB} C^{-1}_{CD} +
* \frac{\delta_7}{2} (C^{-1}_{AC} C^{-1}_{BD} + C^{-1}_{AD} C^{-1}_{BC} )
* \f]
* where the factors are
* \f[
* \delta_1 = 4 \frac{\partial^2 W}{\partial I_1^2} \qquad
* \delta_3 = 4 \det{C} \frac{\partial^2 W}{\partial I_1 \partial I_2}
* \qquad
* \delta_6 = 4 \left( \det(C) \frac{\partial W}{\partial I_2} +
* \det(C)^2 \frac{\partial^2 W}{\partial I_2^2} \right)
* \qquad
* \delta_7 = -4 \det{C} \frac{\partial W}{\partial I_2}
* \f]
*
* Note: The storage is based on the minor symmetries.
* \param[in] F Deformation gradient
* \param[out] C Elasticity tensor
*/
void materialElasticityTensor( const Tensor& F, ElastTensor& CE ) const
{
// compute Cauchy-Green stretch tensor
Tensor C;
mat::rightCauchyGreen( F, C );
Tensor Cinv;
const double detC = mat::inverseAndDet( C, Cinv );
// invariants
const double I1 = firstInvariant( C );
const double I2 = secondInvariant( C );
// energy derivatives
const double dWdI2 = energy_.dWdI2( I1, I2 );
const double d2WdI1dI1 = energy_.d2WdI1dI1( I1, I2 );
const double d2WdI2dI2 = energy_.d2WdI2dI2( I1, I2 );
const double d2WdI1dI2 = energy_.d2WdI1dI2( I1, I2 );
// constant factors (non-zero ones)
const double d1 = 4. * d2WdI1dI1;
const double d3 = 4. * detC * d2WdI1dI2;
const double d6 = 4. * ( detC * dWdI2 + detC*detC * d2WdI2dI2 );
const double d7 = -4. * detC * dWdI2;
// clear to zero
CE = ElastTensor::Constant( 0. );
// go through all 81 indices
for ( unsigned I = 0; I < 3; I++ ) {
for ( unsigned J = 0; J <=I ; J++ ) {
const unsigned v1 = mat::Voigt::apply( I, J );
for ( unsigned K = 0; K < 3; K++ ) {
for ( unsigned L = 0; L <=K ; L++ ) {
const unsigned v2 = mat::Voigt::apply( K, L );
CE( v1, v2 ) +=
d1 * (I==J and K==L ? 1. : 0.) +
d3 * (I==J ? Cinv(K,L) : 0.) +
d3 * (K==L ? Cinv(I,J) : 0.) +
d6 * Cinv(I,J) * Cinv(K,L) +
d7/2. * (Cinv(I,K) * Cinv(J,L) + Cinv(I,L) * Cinv(J,K));
}
}
}
}
return;
}
RankFourTensor
RankTwoTensor::d2sin3Lode(const Real r0) const
{
Real bar = secondInvariant();
if (bar <= r0)
return RankFourTensor();
Real J3 = thirdInvariant();
RankTwoTensor dII = dsecondInvariant();
RankTwoTensor dIII = dthirdInvariant();
RankFourTensor deriv = d2thirdInvariant()/std::pow(bar, 1.5) - 1.5*d2secondInvariant()*J3/std::pow(bar, 2.5);
for (unsigned i = 0; i < N; ++i)
for (unsigned j = 0; j < N; ++j)
for (unsigned k = 0; k < N; ++k)
for (unsigned l = 0; l < N; ++l)
deriv(i, j, k, l) += (-1.5*dII(i, j)*dIII(k, l) -1.5 * dIII(i, j) * dII(k, l)) / std::pow(bar, 2.5) + 1.5*2.5*dII(i, j) * dII(k, l) * J3 / std::pow(bar, 3.5);
deriv *= -1.5 * std::sqrt(3.0);
return deriv;
}
RankTwoTensor
RankTwoTensor::dthirdInvariant() const
{
RankTwoTensor s = 0.5 * deviatoric();
s += s.transpose();
RankTwoTensor d;
Real sec_over_three = secondInvariant() / 3.0;
d(0, 0) = s(1, 1) * s(2, 2) - s(2, 1) * s(1, 2) + sec_over_three;
d(0, 1) = s(2, 0) * s(1, 2) - s(1, 0) * s(2, 2);
d(0, 2) = s(1, 0) * s(2, 1) - s(2, 0) * s(1, 1);
d(1, 0) = s(2, 1) * s(0, 2) - s(0, 1) * s(2, 2);
d(1, 1) = s(0, 0) * s(2, 2) - s(2, 0) * s(0, 2) + sec_over_three;
d(1, 2) = s(2, 0) * s(0, 1) - s(0, 0) * s(2, 1);
d(2, 0) = s(0, 1) * s(1, 2) - s(1, 1) * s(0, 2);
d(2, 1) = s(1, 0) * s(0, 2) - s(0, 0) * s(1, 2);
d(2, 2) = s(0, 0) * s(1, 1) - s(1, 0) * s(0, 1) + sec_over_three;
return d;
}
/** The second Piola-Kirchhoff stress tensor S.
* The definition \f$ S = \partial \psi / \partial E \f$ yields
* \f[
* S = 2 \left( \frac{\partial W}{\partial I_1} I +
* \frac{\partial W}{\partial I_2} \det{C} C^{-1} \right)
* \f]
* \param[in] F Deformation gradient
* \param[out] S 2nd Piola-Kirchhoff stress tensor
*/
void secondPiolaKirchhoff( const Tensor& F, Tensor& S ) const
{
// compute Cauchy-Green stretch tensor
Tensor C;
mat::rightCauchyGreen( F, C );
Tensor Cinv;
const double detC = mat::inverseAndDet( C, Cinv );
// invariants
const double I1 = firstInvariant( C );
const double I2 = secondInvariant( C );
// energy derivatives
const double dWdI1 = energy_.dWdI1( I1, I2 );
const double dWdI2 = energy_.dWdI2( I1, I2 );
// evaluate the second Piola-Kirchhoff stress tensor
S.noalias() =
2. * dWdI1 * Tensor::Identity() + 2. * dWdI2 * detC * Cinv;
}
