void Element::polarDecompositionEigen( const ColumnMajorMatrix & Fhat, ColumnMajorMatrix & Rhat, SymmTensor & strain_increment ) { const int ND = 3; ColumnMajorMatrix eigen_value(ND,1), eigen_vector(ND,ND); ColumnMajorMatrix invUhat(ND,ND), logVhat(ND,ND); ColumnMajorMatrix n1(ND,1), n2(ND,1), n3(ND,1), N1(ND,1), N2(ND,1), N3(ND,1); ColumnMajorMatrix Chat = Fhat.transpose() * Fhat; Chat.eigen(eigen_value,eigen_vector); for(int i = 0; i < ND; i++) { N1(i) = eigen_vector(i,0); N2(i) = eigen_vector(i,1); N3(i) = eigen_vector(i,2); } const Real lamda1 = std::sqrt(eigen_value(0)); const Real lamda2 = std::sqrt(eigen_value(1)); const Real lamda3 = std::sqrt(eigen_value(2)); const Real log1 = std::log(lamda1); const Real log2 = std::log(lamda2); const Real log3 = std::log(lamda3); ColumnMajorMatrix Uhat = N1 * N1.transpose() * lamda1 + N2 * N2.transpose() * lamda2 + N3 * N3.transpose() * lamda3; invertMatrix(Uhat,invUhat); Rhat = Fhat * invUhat; strain_increment = N1 * N1.transpose() * log1 + N2 * N2.transpose() * log2 + N3 * N3.transpose() * log3; }
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."); } }