void
ComputeFiniteStrain::computeQpStrain()
{
// inverse of _Fhat
RankTwoTensor invFhat(_Fhat[_qp].inverse());
// A = I - _Fhat^-1
RankTwoTensor A(RankTwoTensor::initIdentity);
A -= invFhat;
// Cinv - I = A A^T - A - A^T;
RankTwoTensor Cinv_I = A * A.transpose() - A - A.transpose();
// strain rate D from Taylor expansion, Chat = (-1/2(Chat^-1 - I) + 1/4*(Chat^-1 - I)^2 + ...
RankTwoTensor total_strain_increment = -Cinv_I * 0.5 + Cinv_I * Cinv_I * 0.25;
_strain_increment[_qp] = total_strain_increment;
if (_no_thermal_eigenstrains) //Deprecated; use ComputeThermalExpansionEigenStrains instead
{
if (_t_step >= 2)
_step_one = false;
if (_step_one)
_strain_increment[_qp].addIa(-_thermal_expansion_coeff * (_T[_qp] - _T0));
else
_strain_increment[_qp].addIa(-_thermal_expansion_coeff * (_T[_qp] - _T_old[_qp]));
}
// Remove the Eigen strain increment
_strain_increment[_qp] -= _stress_free_strain_increment[_qp];
RankTwoTensor D = _strain_increment[_qp] / _dt;
_strain_rate[_qp] = D;
const Real a[3] = {
invFhat(1,2) - invFhat(2,1),
invFhat(2,0) - invFhat(0,2),
invFhat(0,1) - invFhat(1,0)
};
Real q = (a[0]*a[0] + a[1]*a[1] + a[2]*a[2]) / 4.0;
Real trFhatinv_1 = invFhat.trace() - 1.0;
const Real p = trFhatinv_1 * trFhatinv_1 / 4.0;
// cos theta_a
const Real C1 = std::sqrt(p + 3.0*p*p*(1.0 - (p + q))/((p+q)*(p+q)) - 2.0*p*p*p*(1-(p+q))/((p+q)*(p+q)*(p+q)));
Real C2;
if (q > 0.01)
// (1-cos theta_a)/4q
C2 = (1.0 - C1) / (4.0 * q);
else
//alternate form for small q
C2 = 0.125 + q * 0.03125 * (p*p - 12 * (p-1)) / (p*p)
+ q*q * (p - 2.0) * (p*p - 10.0 * p + 32.0) / (p*p*p)
+ q*q*q * (1104.0 - 992.0 * p + 376.0 * p*p - 72 * p*p*p + 5.0 * p*p*p*p) / (512.0*p*p*p*p);
const Real C3 = 0.5 * std::sqrt((p * q * (3.0 - q) + p*p*p + q*q) / ((p + q) * (p + q) * (p + q))); //sin theta_a/(2 sqrt(q))
// Calculate incremental rotation. Note that this value is the transpose of that from Rashid, 93, so we transpose it before storing
RankTwoTensor R_incr;
R_incr.addIa(C1);
for (unsigned int i = 0; i < 3; ++i)
for (unsigned int j = 0; j < 3; ++j)
R_incr(i,j) += C2 * a[i] * a[j];
R_incr(0,1) += C3 * a[2];
R_incr(0,2) -= C3 * a[1];
R_incr(1,0) -= C3 * a[2];
R_incr(1,2) += C3 * a[0];
R_incr(2,0) += C3 * a[1];
R_incr(2,1) -= C3 * a[0];
_rotation_increment[_qp] = R_incr.transpose();
//Update strain in intermediate configuration
_mechanical_strain[_qp] = _mechanical_strain_old[_qp] + _strain_increment[_qp];
_total_strain[_qp] = _total_strain_old[_qp] + total_strain_increment;
//Rotate strain to current configuration
_mechanical_strain[_qp] = _rotation_increment[_qp] * _mechanical_strain[_qp] * _rotation_increment[_qp].transpose();
_total_strain[_qp] = _rotation_increment[_qp] * _total_strain[_qp] * _rotation_increment[_qp].transpose();
}
void
FiniteStrainMaterial::computeQpStrain(const RankTwoTensor & Fhat)
{
//Cinv - I = A A^T - A - A^T;
RankTwoTensor A; //A = I - Fhatinv
A.addIa(1.0);
A -= Fhat.inverse();
RankTwoTensor Cinv_I = A*A.transpose() - A - A.transpose();
//strain rate D from Taylor expansion, Chat = (-1/2(Chat^-1 - I) + 1/4*(Chat^-1 - I)^2 + ...
_strain_increment[_qp] = -Cinv_I*0.5 + Cinv_I*Cinv_I*0.25;
/*RankTwoTensor Chat = Fhat.transpose()*Fhat;
RankTwoTensor A = Chat;
A.addIa(-1.0);
RankTwoTensor B = Chat*0.25;
B.addIa(-0.75);
_strain_increment[_qp] = -B*A;*/
RankTwoTensor D = _strain_increment[_qp]/_t_step;
_strain_rate[_qp] = D;
//Calculate rotation R_incr
RankTwoTensor invFhat(Fhat.inverse());
std::vector<Real> a(3);
a[0] = invFhat(1,2) - invFhat(2,1);
a[1] = invFhat(2,0) - invFhat(0,2);
a[2] = invFhat(0,1) - invFhat(1,0);
Real q = (a[0]*a[0] + a[1]*a[1] + a[2]*a[2])/4.0;
Real trFhatinv_1 = invFhat.trace() - 1.0;
Real p = trFhatinv_1*trFhatinv_1/4.0;
// Real y = 1.0/((q + p)*(q + p)*(q + p));
/*Real C1 = std::sqrt(p * (1 + (p*(q+q+(q+p))) * (1-(q+p)) * y));
Real C2 = 0.125 + q * 0.03125 * (p*p - 12*(p-1)) / (p*p);
Real C3 = 0.5 * std::sqrt( (p*q*(3-q) + p*p*p + q*q)*y );
*/
Real C1 = std::sqrt(p + 3.0*p*p*(1.0 - (p + q))/((p+q)*(p+q)) - 2.0*p*p*p*(1-(p+q))/((p+q)*(p+q)*(p+q))); //cos theta_a
Real C2 = 0.0;
if (q > 0.01)
C2 = (1.0 - C1)/(4.0*q); // (1-cos theta_a)/4q
else //alternate form for small q
C2 = 0.125 + q*0.03125*(p*p - 12*(p-1))/(p*p) + q*q*(p - 2.0)*(p*p - 10.0*p + 32.0)/(p*p*p) + q*q*q*(1104.0 - 992.0*p + 376.0*p*p - 72*p*p*p + 5.0*p*p*p*p)/(512.0*p*p*p*p);
Real C3 = 0.5*std::sqrt((p*q*(3.0 - q) + p*p*p + q*q)/((p + q)*(p + q)*(p + q))); //sin theta_a/(2 sqrt(q))
//Calculate incremental rotation. Note that this value is the transpose of that from Rashid, 93, so we transpose it before storing
RankTwoTensor R_incr;
R_incr.addIa(C1);
for (unsigned int i=0; i<3; ++i)
for (unsigned int j = 0; j < 3; ++j)
R_incr(i,j) += C2*a[i]*a[j];
R_incr(0,1) += C3*a[2];
R_incr(0,2) -= C3*a[1];
R_incr(1,0) -= C3*a[2];
R_incr(1,2) += C3*a[0];
R_incr(2,0) += C3*a[1];
R_incr(2,1) -= C3*a[0];
_rotation_increment[_qp] = R_incr.transpose();
}
void
ComputeFiniteStrain::computeQpIncrements(RankTwoTensor & total_strain_increment, RankTwoTensor & rotation_increment)
{
switch (_decomposition_method)
{
case DecompMethod::TaylorExpansion:
{
// inverse of _Fhat
RankTwoTensor invFhat(_Fhat[_qp].inverse());
// A = I - _Fhat^-1
RankTwoTensor A(RankTwoTensor::initIdentity);
A -= invFhat;
// Cinv - I = A A^T - A - A^T;
RankTwoTensor Cinv_I = A * A.transpose() - A - A.transpose();
// strain rate D from Taylor expansion, Chat = (-1/2(Chat^-1 - I) + 1/4*(Chat^-1 - I)^2 + ...
total_strain_increment = -Cinv_I * 0.5 + Cinv_I * Cinv_I * 0.25;
const Real a[3] = {
invFhat(1, 2) - invFhat(2, 1),
invFhat(2, 0) - invFhat(0, 2),
invFhat(0, 1) - invFhat(1, 0)
};
Real q = (a[0] * a[0] + a[1] * a[1] + a[2] * a[2]) / 4.0;
Real trFhatinv_1 = invFhat.trace() - 1.0;
const Real p = trFhatinv_1 * trFhatinv_1 / 4.0;
// cos theta_a
const Real C1 = std::sqrt(p + 3.0 * std::pow(p, 2.0) * (1.0 - (p + q)) / std::pow(p + q, 2.0) - 2.0 * std::pow(p, 3.0) * (1.0 - (p + q)) / std::pow(p + q, 3.0));
Real C2;
if (q > 0.01)
// (1-cos theta_a)/4q
C2 = (1.0 - C1) / (4.0 * q);
else
//alternate form for small q
C2 = 0.125 + q * 0.03125 * (std::pow(p, 2.0) - 12.0 * (p - 1.0)) / std::pow(p, 2.0)
+ std::pow(q, 2.0) * (p - 2.0) * (std::pow(p, 2.0) - 10.0 * p + 32.0) / std::pow(p, 3.0)
+ std::pow(q, 3.0) * (1104.0 - 992.0 * p + 376.0 * std::pow(p, 2.0) - 72.0 * std::pow(p, 3.0) + 5.0 * std::pow(p, 4.0)) / (512.0 * std::pow(p, 4.0));
const Real C3 = 0.5 * std::sqrt((p * q * (3.0 - q) + std::pow(p, 3.0) + std::pow(q, 2.0)) / std::pow(p + q, 3.0)); //sin theta_a/(2 sqrt(q))
// Calculate incremental rotation. Note that this value is the transpose of that from Rashid, 93, so we transpose it before storing
RankTwoTensor R_incr;
R_incr.addIa(C1);
for (unsigned int i = 0; i < 3; ++i)
for (unsigned int j = 0; j < 3; ++j)
R_incr(i,j) += C2 * a[i] * a[j];
R_incr(0,1) += C3 * a[2];
R_incr(0,2) -= C3 * a[1];
R_incr(1,0) -= C3 * a[2];
R_incr(1,2) += C3 * a[0];
R_incr(2,0) += C3 * a[1];
R_incr(2,1) -= C3 * a[0];
rotation_increment = R_incr.transpose();
break;
}
case DecompMethod::EigenSolution:
{
std::vector<Real> e_value(3);
RankTwoTensor e_vector, N1, N2, N3;
RankTwoTensor Chat = _Fhat[_qp].transpose() * _Fhat[_qp];
Chat.symmetricEigenvaluesEigenvectors(e_value, e_vector);
const Real lambda1 = std::sqrt(e_value[0]);
const Real lambda2 = std::sqrt(e_value[1]);
const Real lambda3 = std::sqrt(e_value[2]);
N1.vectorOuterProduct(e_vector.column(0), e_vector.column(0));
N2.vectorOuterProduct(e_vector.column(1), e_vector.column(1));
N3.vectorOuterProduct(e_vector.column(2), e_vector.column(2));
RankTwoTensor Uhat = N1 * lambda1 + N2 * lambda2 + N3 * lambda3;
RankTwoTensor invUhat(Uhat.inverse());
rotation_increment = _Fhat[_qp] * invUhat;
total_strain_increment = N1 * std::log(lambda1) + N2 * std::log(lambda2) + N3 * std::log(lambda3);
break;
}
default:
mooseError("ComputeFiniteStrain Error: Pass valid decomposition type: TaylorExpansion or EigenSolution.");
}
}
