void
AxisymmetricRZ::computeStrain( const unsigned qp,
const SymmTensor & total_strain_old,
SymmTensor & total_strain_new,
SymmTensor & strain_increment )
{
strain_increment.xx() = _grad_disp_r[qp](0);
strain_increment.yy() = _grad_disp_z[qp](1);
strain_increment.zz() = (_solid_model.q_point(qp)(0) != 0.0 ? _disp_r[qp]/_solid_model.q_point(qp)(0) : 0.0);
strain_increment.xy() = 0.5*(_grad_disp_r[qp](1) + _grad_disp_z[qp](0));
if (_large_strain)
{
strain_increment.xx() += 0.5*(_grad_disp_r[qp](0)*_grad_disp_r[qp](0) +
_grad_disp_z[qp](0)*_grad_disp_z[qp](0));
strain_increment.yy() += 0.5*(_grad_disp_r[qp](1)*_grad_disp_r[qp](1) +
_grad_disp_z[qp](1)*_grad_disp_z[qp](1));
strain_increment.zz() += 0.5*(strain_increment.zz()*strain_increment.zz());
strain_increment.xy() += 0.5*(_grad_disp_r[qp](0)*_grad_disp_r[qp](1) +
_grad_disp_z[qp](0)*_grad_disp_z[qp](1));
}
total_strain_new = strain_increment;
strain_increment -= total_strain_old;
}
bool
ConstitutiveModel::applyThermalStrain(unsigned qp,
SymmTensor & strain_increment,
SymmTensor & d_strain_dT)
{
if (_has_temp && _t_step != 0)
{
Real inc_thermal_strain;
Real d_thermal_strain_d_temp;
Real old_temp;
if (_t_step == 1 && _has_stress_free_temp)
old_temp = _stress_free_temp;
else
old_temp = _temperature_old[qp];
Real current_temp = _temperature[qp];
Real delta_t = current_temp - old_temp;
Real alpha = _alpha;
if (_alpha_function)
{
Point p;
Real alpha_current_temp = _alpha_function->value(current_temp,p);
Real alpha_old_temp = _alpha_function->value(old_temp,p);
if (_mean_alpha_function)
{
Real small(1e-6);
Real numerator = alpha_current_temp * (current_temp - _ref_temp) - alpha_old_temp * (old_temp - _ref_temp);
Real denominator = 1.0 + alpha_old_temp * (old_temp - _ref_temp);
if (denominator < small)
mooseError("Denominator too small in thermal strain calculation");
inc_thermal_strain = numerator / denominator;
d_thermal_strain_d_temp = alpha_current_temp * (current_temp - _ref_temp);
}
else
{
inc_thermal_strain = delta_t * 0.5 * (alpha_current_temp + alpha_old_temp);
d_thermal_strain_d_temp = alpha_current_temp;
}
}
else
{
inc_thermal_strain = delta_t * alpha;
d_thermal_strain_d_temp = alpha;
}
strain_increment.addDiag(-inc_thermal_strain);
d_strain_dT.addDiag(-d_thermal_strain_d_temp);
}
bool modified = true;
return modified;
}
Real
volumetricStrain(const SymmTensor & symm_strain)
{
Real value = symm_strain.trace();
value += symm_strain.xx() * symm_strain.yy() + symm_strain.yy() * symm_strain.zz() +
symm_strain.zz() * symm_strain.xx() +
symm_strain.xx() * symm_strain.yy() * symm_strain.zz();
return value;
}
Real
StressDivergence::computeQpJacobian()
{
Real sum_C3x3 = _Jacobian_mult[_qp].sum_3x3();
RealGradient sum_C3x1 = _Jacobian_mult[_qp].sum_3x1();
Real jacobian = 0.0;
// B^T_i * C * B_j
jacobian += _Jacobian_mult[_qp].stiffness(
_component, _component, _grad_test[_i][_qp], _grad_phi[_j][_qp]); // B^T_i * C *B_j
if (_volumetric_locking_correction)
{
// jacobian = Bbar^T_i * C * Bbar_j where Bbar = B + Bvol
// jacobian = B^T_i * C * B_j + Bvol^T_i * C * Bvol_j + Bvol^T_i * C * B_j + B^T_i * C * Bvol_j
// Bvol^T_i * C * Bvol_j
jacobian += sum_C3x3 * (_avg_grad_test[_i][_component] - _grad_test[_i][_qp](_component)) *
(_avg_grad_phi[_j][_component] - _grad_phi[_j][_qp](_component)) / 9.0;
// B^T_i * C * Bvol_j
jacobian += sum_C3x1(_component) * _grad_test[_i][_qp](_component) *
(_avg_grad_phi[_j][_component] - _grad_phi[_j][_qp](_component)) / 3.0;
// Bvol^T_i * C * B_j
SymmTensor phi;
if (_component == 0)
{
phi.xx() = _grad_phi[_j][_qp](0);
phi.xy() = _grad_phi[_j][_qp](1);
phi.xz() = _grad_phi[_j][_qp](2);
}
else if (_component == 1)
{
phi.yy() = _grad_phi[_j][_qp](1);
phi.xy() = _grad_phi[_j][_qp](0);
phi.yz() = _grad_phi[_j][_qp](2);
}
else if (_component == 2)
{
phi.zz() = _grad_phi[_j][_qp](2);
phi.xz() = _grad_phi[_j][_qp](0);
phi.yz() = _grad_phi[_j][_qp](1);
}
SymmTensor tmp(_Jacobian_mult[_qp] * phi);
jacobian += (tmp.xx() + tmp.yy() + tmp.zz()) *
(_avg_grad_test[_i][_component] - _grad_test[_i][_qp](_component)) / 3.0;
}
if (_dt > 0)
return jacobian * (1 + _alpha + _zeta / _dt);
else
return jacobian;
}
void
PlaneStrain::computeStrain( const unsigned qp,
const SymmTensor & total_strain_old,
SymmTensor & total_strain_new,
SymmTensor & strain_increment )
{
strain_increment.xx() = _grad_disp_x[qp](0);
strain_increment.yy() = _grad_disp_y[qp](1);
strain_increment.zz() = 0;
strain_increment.xy() = 0.5*(_grad_disp_x[qp](1) + _grad_disp_y[qp](0));
strain_increment.yz() = 0;
strain_increment.zx() = 0;
if (_large_strain)
{
strain_increment.xx() += 0.5*(_grad_disp_x[qp](0)*_grad_disp_x[qp](0) +
_grad_disp_y[qp](0)*_grad_disp_y[qp](0));
strain_increment.yy() += 0.5*(_grad_disp_x[qp](1)*_grad_disp_x[qp](1) +
_grad_disp_y[qp](1)*_grad_disp_y[qp](1));
strain_increment.xy() += 0.5*(_grad_disp_x[qp](0)*_grad_disp_x[qp](1) +
_grad_disp_y[qp](0)*_grad_disp_y[qp](1));
}
total_strain_new = strain_increment;
strain_increment -= total_strain_old;
}
Real
vonMisesStress(const SymmTensor & symm_stress)
{
Real value = std::pow(symm_stress.xx() - symm_stress.yy(), 2) +
std::pow(symm_stress.yy() - symm_stress.zz(), 2) +
std::pow(symm_stress.zz() - symm_stress.xx(), 2) +
6 * (std::pow(symm_stress.xy(), 2) + std::pow(symm_stress.yz(), 2) +
std::pow(symm_stress.zx(), 2));
return std::sqrt(value / 2.0);
}
Real
StressDivergenceRSpherical::computeQpOffDiagJacobian(unsigned int jvar)
{
if ( _temp_coupled && jvar == _temp_var )
{
SymmTensor test;
test.xx() = _grad_test[_i][_qp](0);
test.yy() = _test[_i][_qp] / _q_point[_qp](0);
test.zz() = test.yy();
return _d_stress_dT[_qp].doubleContraction(test) * _phi[_j][_qp];
}
return 0;
}
bool PoroElasticity::formPermeabilityTensor (SymmTensor& K,
const Vectors&,
const FiniteElement&,
const Vec3& X) const
{
const PoroMaterial* pmat = dynamic_cast<const PoroMaterial*>(material);
if (!pmat) return false;
Vec3 permeability = pmat->getPermeability(X);
K.zero();
for (size_t i = 1; i <= K.dim(); i++)
K(i,i) = permeability(i);
return true;
}
Real
MaterialTensorAux::principalValue( const SymmTensor & tensor, unsigned int index )
{
ColumnMajorMatrix eval(3,1);
ColumnMajorMatrix evec(3,3);
tensor.columnMajorMatrix().eigen(eval, evec);
// Eigen computes low to high. We want high first.
int i = -index + 2;
return eval(i);
}
Real
calcPrincipleValues(const SymmTensor & symm_tensor, unsigned int index, RealVectorValue & direction)
{
ColumnMajorMatrix eval(3,1);
ColumnMajorMatrix evec(3,3);
symm_tensor.columnMajorMatrix().eigen(eval, evec);
// Eigen computes low to high. We want high first.
direction(0) = evec(0,index);
direction(1) = evec(1,index);
direction(2) = evec(2,index);
return eval(index);
}
Real
StressDivergenceRZ::computeQpOffDiagJacobian(unsigned int jvar)
{
if ( _rdisp_coupled && jvar == _rdisp_var )
{
return calculateJacobian( _component, 0 );
}
else if ( _zdisp_coupled && jvar == _zdisp_var )
{
return calculateJacobian( _component, 1 );
}
else if ( _temp_coupled && jvar == _temp_var )
{
SymmTensor test;
if (_component == 0)
{
test.xx() = _grad_test[_i][_qp](0);
test.xy() = 0.5*_grad_test[_i][_qp](1);
test.zz() = _test[_i][_qp] / _q_point[_qp](0);
}
else
{
test.xy() = 0.5*_grad_test[_i][_qp](0);
test.yy() = _grad_test[_i][_qp](1);
}
return _d_stress_dT[_qp].doubleContraction(test) * _phi[_j][_qp];
}
return 0;
}
void
Nonlinear::computeStrainIncrement( const ColumnMajorMatrix & Fhat,
SymmTensor & strain_increment )
{
//
// A calculation of the strain at the mid-interval is probably more
// accurate (second vs. first order). This would require the
// incremental deformation gradient at the mid-step, which we
// currently don't have. We would then have to calculate the
// rotation for the whole step.
//
//
// We are looking for:
// log( Uhat )
// = log( sqrt( Fhat^T*Fhat ) )
// = log( sqrt( Chat ) )
// A Taylor series expansion gives:
// ( Chat - 0.25 * Chat^T*Chat - 0.75 * I )
// = ( - 0.25 * Chat^T*Chat + Chat - 0.75 * I )
// = ( (0.25*Chat - 0.75*I) * (Chat - I) )
// = ( B * A )
// B
// = 0.25*Chat - 0.75*I
// = 0.25*(Chat - I) - 0.5*I
// = 0.25*A - 0.5*I
//
const Real Uxx = Fhat(0,0);
const Real Uxy = Fhat(0,1);
const Real Uxz = Fhat(0,2);
const Real Uyx = Fhat(1,0);
const Real Uyy = Fhat(1,1);
const Real Uyz = Fhat(1,2);
const Real Uzx = Fhat(2,0);
const Real Uzy = Fhat(2,1);
const Real Uzz = Fhat(2,2);
const Real Axx = Uxx*Uxx + Uyx*Uyx + Uzx*Uzx - 1.0;
const Real Axy = Uxx*Uxy + Uyx*Uyy + Uzx*Uzy;
const Real Axz = Uxx*Uxz + Uyx*Uyz + Uzx*Uzz;
const Real Ayy = Uxy*Uxy + Uyy*Uyy + Uzy*Uzy - 1.0;
const Real Ayz = Uxy*Uxz + Uyy*Uyz + Uzy*Uzz;
const Real Azz = Uxz*Uxz + Uyz*Uyz + Uzz*Uzz - 1.0;
const Real Bxx = 0.25 * Axx - 0.5;
const Real Bxy = 0.25 * Axy;
const Real Bxz = 0.25 * Axz;
const Real Byy = 0.25 * Ayy - 0.5;
const Real Byz = 0.25 * Ayz;
const Real Bzz = 0.25 * Azz - 0.5;
strain_increment.xx( -(Bxx*Axx + Bxy*Axy + Bxz*Axz) );
strain_increment.xy( -(Bxx*Axy + Bxy*Ayy + Bxz*Ayz) );
strain_increment.zx( -(Bxx*Axz + Bxy*Ayz + Bxz*Azz) );
strain_increment.yy( -(Bxy*Axy + Byy*Ayy + Byz*Ayz) );
strain_increment.yz( -(Bxy*Axz + Byy*Ayz + Byz*Azz) );
strain_increment.zz( -(Bxz*Axz + Byz*Ayz + Bzz*Azz) );
}
ColumnMajorMatrix
CrackFrontDefinition::rotateToCrackFrontCoords(const SymmTensor tensor, const unsigned int node_index) const
{
ColumnMajorMatrix tensor_CMM;
tensor_CMM(0,0) = tensor.xx();
tensor_CMM(0,1) = tensor.xy();
tensor_CMM(0,2) = tensor.xz();
tensor_CMM(1,0) = tensor.xy();
tensor_CMM(1,1) = tensor.yy();
tensor_CMM(1,2) = tensor.yz();
tensor_CMM(2,0) = tensor.xz();
tensor_CMM(2,1) = tensor.yz();
tensor_CMM(2,2) = tensor.zz();
ColumnMajorMatrix tmp = _rot_matrix[node_index] * tensor_CMM;
ColumnMajorMatrix rotT = _rot_matrix[node_index].transpose();
ColumnMajorMatrix rotated_tensor = tmp * rotT;
return rotated_tensor;
}
void
CardiacHolzapfel2009Material::computeQpStressProperties(const SymmTensor &C, const SymmTensor & /*E*/)
{
const SymmTensor CC(square(C));
// invariants (We will not need all of them. However, defining them avoids to forget any.)
const Real I1(C.trace());
/*const Real I2(0.5*(I1*I1-CC.trace()));*/
/*const Real I3(_J[_qp]); */
const Real I4f(_Ef[_qp]*(C*_Ef[_qp]));
const Real I4s(_Es[_qp]*(C*_Es[_qp]));
/*const Real I4n(_En[_qp]*(C*_En[_qp])); */
/*const Real I5f(_Ef[_qp]*(CC*_Ef[_qp]));*/
/*const Real I5s(_Es[_qp]*(CC*_Es[_qp]));*/
/*const Real I5n(_En[_qp]*(CC*_En[_qp]));*/
const Real I8fs(_Ef[_qp]*(C*_Es[_qp]));
/*const Real I8fn(_Ef[_qp]*(C*_En[_qp]));*/
/*const Real I8sn(_Es[_qp]*(C*_En[_qp]));*/
// the following will be needed in the stress as well as in the energy and stress_derivative
const Real i_term( _p[A1 ]*std::exp(_p[B1 ]*(I1 -3) ) );
const Real f_term( 2*_p[Af ]*std::exp(_p[Bf ]*(I4f-1)*(I4f-1)) );
const Real s_term( 2*_p[As ]*std::exp(_p[Bs ]*(I4s-1)*(I4s-1)) );
const Real fs_term( _p[Afs]*std::exp(_p[Bfs]* I8fs * I8fs ) );
// elastic energy contribution
_W[_qp] = i_term /(2*_p[B1 ])
+ ( f_term - 2*_p[Af ])/(2*_p[Bf ])
+ ( s_term - 2*_p[As ])/(2*_p[Bs ])
+ (fs_term - 2*_p[Afs])/(2*_p[Bfs]);
// tensors for constructing stress and stress_derivative
const SymmTensor EfEf(kron(_Ef[_qp]));
const SymmTensor EsEs(kron(_Es[_qp]));
const SymmTensor EfEs(kronSym(_Ef[_qp],_Es[_qp]));
// stress
_stress[_qp] = scaledID(i_term) + EfEf*(I4f-1)*f_term + EsEs*(I4s-1)*s_term + EfEs*I8fs*fs_term;
// stress derivative /* fancy lambda function syntax makes things much easier here */
_stress_derivative[_qp].fill_from_minor_iter( [&](const unsigned int M,
const unsigned int N,
const unsigned int P,
const unsigned int Q) -> Real { return _p[B1 ] * i_term * _id(M,N) * _id(P,Q)
+ (1 + (I4f-1)*(I4f-1)*2*_p[Bf ]) * f_term * EfEf(M,N) * EfEf(P,Q)
+ (1 + (I4f-1)*(I4f-1)*2*_p[Bf ]) * s_term * EsEs(M,N) * EsEs(P,Q)
+ (1 + I8fs *2*_p[Bfs]) * fs_term * EfEs(M,N) * EfEs(P,Q); } );
}
bool Elasticity::kinematics (const Vector& eV,
const Vector& N, const Matrix& dNdX, double r,
Matrix& B, Tensor&, SymmTensor& eps) const
{
// Evaluate the strain-displacement matrix, B
if (axiSymmetry)
{
if (!this->formBmatrix(B,N,dNdX,r))
return false;
}
else
{
if (!this->formBmatrix(B,dNdX))
return false;
}
if (eV.empty() || eps.dim() == 0)
return true;
// Evaluate the strains
return B.multiply(eV,eps); // eps = B*eV
}
Real
component(const SymmTensor & symm_tensor, unsigned int index, RealVectorValue & direction)
{
direction.zero();
if (index < 3)
direction(index) = 1.0;
else if (index == 3)//xy
{
direction(0) = std::sqrt(0.5);
direction(1) = direction(0);
}
else if (index == 4)//yz
{
direction(1) = std::sqrt(0.5);
direction(2) = direction(1);
}
else if (index == 5)//zx
{
direction(0) = std::sqrt(0.5);
direction(2) = direction(0);
}
return symm_tensor.component(index);
}
void
ReturnMappingModel::computeStress(const Elem & /*current_elem*/,
const SymmElasticityTensor & elasticityTensor,
const SymmTensor & stress_old,
SymmTensor & strain_increment,
SymmTensor & stress_new,
SymmTensor & inelastic_strain_increment)
{
// compute deviatoric trial stress
SymmTensor dev_trial_stress(stress_new);
dev_trial_stress.addDiag(-dev_trial_stress.trace() / 3.0);
// compute effective trial stress
Real dts_squared = dev_trial_stress.doubleContraction(dev_trial_stress);
Real effective_trial_stress = std::sqrt(1.5 * dts_squared);
// compute effective strain increment
SymmTensor dev_strain_increment(strain_increment);
dev_strain_increment.addDiag(-strain_increment.trace() / 3.0);
_effective_strain_increment = dev_strain_increment.doubleContraction(dev_strain_increment);
_effective_strain_increment = std::sqrt(2.0 / 3.0 * _effective_strain_increment);
const SymmIsotropicElasticityTensor * iso_e_t =
dynamic_cast<const SymmIsotropicElasticityTensor *>(&elasticityTensor);
if (!iso_e_t)
mooseError("Models derived from ReturnMappingModel require a SymmIsotropicElasticityTensor");
_three_shear_modulus = 3.0 * iso_e_t->shearModulus();
computeStressInitialize(effective_trial_stress, elasticityTensor);
Real scalar;
returnMappingSolve(effective_trial_stress, scalar, _console);
// compute inelastic and elastic strain increments
if (_legacy_return_mapping)
{
if (effective_trial_stress < 0.01)
effective_trial_stress = 0.01;
inelastic_strain_increment = dev_trial_stress;
inelastic_strain_increment *= (1.5 * scalar / effective_trial_stress);
}
else
{
if (scalar != 0.0)
inelastic_strain_increment = dev_trial_stress * (1.5 * scalar / effective_trial_stress);
else
inelastic_strain_increment = 0.0;
}
strain_increment -= inelastic_strain_increment;
_effective_inelastic_strain[_qp] = _effective_inelastic_strain_old[_qp] + scalar;
// compute stress increment
stress_new = elasticityTensor * strain_increment;
// update stress
stress_new += stress_old;
computeStressFinalize(inelastic_strain_increment);
}
void
SymmIsotropicElasticityTensor::multiply( const SymmTensor & x, SymmTensor & b ) const
{
const Real xx = x.xx();
const Real yy = x.yy();
const Real zz = x.zz();
const Real xy = x.xy();
const Real yz = x.yz();
const Real zx = x.zx();
b.xx() = _val[ 0]*xx + _val[ 1]*yy + _val[ 2]*zz;
b.yy() = _val[ 1]*xx + _val[ 6]*yy + _val[ 7]*zz;
b.zz() = _val[ 2]*xx + _val[ 7]*yy + _val[11]*zz;
b.xy() = 2*_val[15]*xy;
b.yz() = 2*_val[18]*yz;
b.zx() = 2*_val[20]*zx;
}
bool FractureElasticityVoigt::evalStress (double lambda, double mu, double Gc,
const SymmTensor& epsil, double* Phi,
SymmTensor* sigma, Matrix* dSdE,
bool postProc, bool printElm) const
{
PROFILE3("FractureEl::evalStress");
unsigned short int a = 0, b = 0;
// Define a Lambda-function to set up the isotropic constitutive matrix
auto&& setIsotropic = [this,a,b](Matrix& C, double lambda, double mu) mutable
{
for (a = 1; a <= C.rows(); a++)
if (a > nsd)
C(a,a) = mu;
else
{
C(a,a) = 2.0*mu;
for (b = 1; b <= nsd; b++)
C(a,b) += lambda;
}
};
// Define some material constants
double trEps = epsil.trace();
double C0 = trEps >= -epsZ ? Gc*lambda : lambda;
double Cp = Gc*mu;
if (trEps >= -epsZ && trEps <= epsZ)
{
// No strains, stress free configuration
Phi[0] = 0.0;
if (postProc)
Phi[1] = Phi[2] = Phi[3] = 0.0;
if (sigma)
*sigma = 0.0;
if (dSdE)
setIsotropic(*dSdE,C0,Cp);
return true;
}
// Calculate principal strains and the associated directions
Vec3 eps;
std::vector<SymmTensor> M(nsd,SymmTensor(nsd));
{
PROFILE4("Tensor::principal");
if (!epsil.principal(eps,M.data()))
return false;
}
// Split the strain tensor into positive and negative parts
SymmTensor ePos(nsd), eNeg(nsd);
for (a = 0; a < nsd; a++)
if (eps[a] > 0.0)
ePos += eps[a]*M[a];
else if (eps[a] < 0.0)
eNeg += eps[a]*M[a];
if (sigma)
{
// Evaluate the stress tensor
*sigma = C0*trEps;
*sigma += 2.0*mu*(Gc*ePos + eNeg);
}
// Evaluate the tensile energy
Phi[0] = mu*(ePos*ePos).trace();
if (trEps > 0.0) Phi[0] += 0.5*lambda*trEps*trEps;
if (postProc)
{
// Evaluate the compressive energy
Phi[1] = mu*(eNeg*eNeg).trace();
if (trEps < 0.0) Phi[1] += 0.5*lambda*trEps*trEps;
// Evaluate the total strain energy
Phi[2] = Gc*Phi[0] + Phi[1];
// Evaluate the bulk energy
Phi[3] = Gc*(Phi[0] + Phi[1]);
}
else if (sigmaC > 0.0) // Evaluate the crack driving function
Phi[0] = this->MieheCrit56(eps,lambda,mu);
#if INT_DEBUG > 4
std::cout <<"eps_p = "<< eps <<"\n";
for (a = 0; a < nsd; a++)
std::cout <<"M("<< 1+a <<") =\n"<< M[a];
std::cout <<"ePos =\n"<< ePos <<"eNeg =\n"<< eNeg;
if (sigma) std::cout <<"sigma =\n"<< *sigma;
std::cout <<"Phi = "<< Phi[0];
if (postProc) std::cout <<" "<< Phi[1] <<" "<< Phi[2] <<" "<< Phi[3];
std::cout << std::endl;
#else
if (printElm)
{
std::cout <<"g(c) = "<< Gc
<<"\nepsilon =\n"<< epsil <<"eps_p = "<< eps
<<"\nePos =\n"<< ePos <<"eNeg =\n"<< eNeg;
if (sigma) std::cout <<"sigma =\n"<< *sigma;
std::cout <<"Phi = "<< Phi[0];
if (postProc) std::cout <<" "<< Phi[1] <<" "<< Phi[2] <<" "<< Phi[3];
std::cout << std::endl;
}
#endif
if (!dSdE)
return true;
else if (eps[0] == eps[nsd-1])
{
// Hydrostatic pressure
setIsotropic(*dSdE, C0, eps.x > 0.0 ? Cp : mu);
return true;
}
typedef unsigned short int s_ind; // Convenience type definition
// Define a Lambda-function to calculate (lower triangle of) the matrix Qa
auto&& getQ = [this](Matrix& Q, const SymmTensor& Ma, double C)
{
if (C == 0.0) return;
auto&& Mult = [Ma](s_ind i, s_ind j, s_ind k, s_ind l)
{
return Ma(i,j)*Ma(k,l);
};
s_ind i, j, k, l, is, js;
// Normal stresses and strains
for (i = 1; i <= nsd; i++)
for (j = 1; j <= i; j++)
Q(i,j) += C*Mult(i,i,j,j);
is = nsd+1;
for (i = 1; i < nsd; i++)
for (j = i+1; j <= nsd; j++, is++)
{
// Shear stress coupled to normal strain
for (k = 1; k <= nsd; k++)
Q(is,k) += C*Mult(i,j,k,k);
// Shear stress coupled to shear strain
js = nsd+1;
for (k = 1; k < nsd; k++)
for (l = k+1; js <= is; l++, js++)
Q(is,js) += C*Mult(i,j,k,l);
}
};
// Define a Lambda-function to calculate (lower triangle of) the matrix Gab
auto&& getG = [this](Matrix& G, const SymmTensor& Ma,
const SymmTensor& Mb, double C)
{
if (C == 0.0) return;
auto&& Mult = [Ma,Mb](s_ind i, s_ind j, s_ind k, s_ind l)
{
return Ma(i,k)*Mb(j,l) + Ma(i,l)*Mb(j,k) +
Mb(i,k)*Ma(j,l) + Mb(i,l)*Ma(j,k);
};
s_ind i, j, k, l, is, js;
// Normal stresses and strains
for (i = 1; i <= nsd; i++)
for (j = 1; j <= i; j++)
G(i,j) += C*Mult(i,i,j,j);
is = nsd+1;
for (i = 1; i < nsd; i++)
for (j = i+1; j <= nsd; j++, is++)
{
// Shear stress coupled to normal strain
for (k = 1; k <= nsd; k++)
G(is,k) += C*Mult(i,j,k,k);
// Shear stress coupled to shear strain
js = nsd+1;
for (k = 1; k < nsd; k++)
for (l = k+1; js <= is; l++, js++)
G(is,js) += C*Mult(i,j,k,l);
}
};
// Evaluate the stress tangent assuming Voigt notation and symmetry
for (a = 1; a <= nsd; a++)
for (b = 1; b <= a; b++)
(*dSdE)(a,b) = C0;
for (a = 0; a < nsd; a++)
{
double C1 = eps[a] >= 0.0 ? Cp : mu;
getQ(*dSdE, M[a], 2.0*C1);
if (eps[a] != 0.0)
for (b = 0; b < nsd; b++)
if (a != b && eps[a] != eps[b])
getG(*dSdE,M[a],M[b],C1/(1.0-eps[b]/eps[a]));
}
// Account for symmetry
for (b = 2; b <= dSdE->rows(); b++)
for (a = 1; a < b; a++)
(*dSdE)(a,b) = (*dSdE)(b,a);
return true;
}
Real
StressDivergence::computeQpOffDiagJacobian(unsigned int jvar)
{
unsigned int coupled_component = 0;
bool active = false;
if (_xdisp_coupled && jvar == _xdisp_var)
{
coupled_component = 0;
active = true;
}
else if (_ydisp_coupled && jvar == _ydisp_var)
{
coupled_component = 1;
active = true;
}
else if (_zdisp_coupled && jvar == _zdisp_var)
{
coupled_component = 2;
active = true;
}
if (active)
{
Real sum_C3x3 = _Jacobian_mult[_qp].sum_3x3();
RealGradient sum_C3x1 = _Jacobian_mult[_qp].sum_3x1();
Real jacobian = 0.0;
jacobian += _Jacobian_mult[_qp].stiffness(
_component, coupled_component, _grad_test[_i][_qp], _grad_phi[_j][_qp]); // B^T_i * C *B_j
if (_volumetric_locking_correction)
{
// jacobian = Bbar^T_i * C * Bbar_j where Bbar = B + Bvol
// jacobian = B^T_i * C * B_j + Bvol^T_i * C * Bvol_j + Bvol^T_i * C * B_j + B^T_i * C *
// Bvol_j
// Bvol^T_i * C * Bvol_j
jacobian += sum_C3x3 * (_avg_grad_test[_i][_component] - _grad_test[_i][_qp](_component)) *
(_avg_grad_phi[_j][coupled_component] - _grad_phi[_j][_qp](coupled_component)) /
9.0;
// B^T_i * C * Bvol_j
jacobian += sum_C3x1(_component) * _grad_test[_i][_qp](_component) *
(_avg_grad_phi[_j][coupled_component] - _grad_phi[_j][_qp](coupled_component)) /
3.0;
// Bvol^T_i * C * B_i
SymmTensor phi;
if (coupled_component == 0)
{
phi.xx() = _grad_phi[_j][_qp](0);
phi.xy() = _grad_phi[_j][_qp](1);
phi.xz() = _grad_phi[_j][_qp](2);
}
else if (coupled_component == 1)
{
phi.yy() = _grad_phi[_j][_qp](1);
phi.xy() = _grad_phi[_j][_qp](0);
phi.yz() = _grad_phi[_j][_qp](2);
}
else if (coupled_component == 2)
{
phi.zz() = _grad_phi[_j][_qp](2);
phi.xz() = _grad_phi[_j][_qp](0);
phi.yz() = _grad_phi[_j][_qp](1);
}
SymmTensor tmp(_Jacobian_mult[_qp] * phi);
jacobian += (tmp.xx() + tmp.yy() + tmp.zz()) *
(_avg_grad_test[_i][_component] - _grad_test[_i][_qp](_component)) / 3.0;
}
if (_dt > 0)
return jacobian * (1 + _alpha + _zeta / _dt);
else
return jacobian;
}
if (_temp_coupled && jvar == _temp_var)
return _d_stress_dT[_qp].rowDot(_component, _grad_test[_i][_qp]) * _phi[_j][_qp];
return 0;
}
Real
equivalentPlasticStrain(const SymmTensor & symm_strain)
{
return std::sqrt(2.0 / 3.0 * symm_strain.doubleContraction(symm_strain));
}
void
PLC_LSH::computeLSH( const SymmTensor & strain_increment,
SymmTensor & plastic_strain_increment,
SymmTensor & stress_new )
{
// compute deviatoric trial stress
SymmTensor dev_trial_stress(stress_new);
dev_trial_stress.addDiag( -stress_new.trace()/3 );
// effective trial stress
Real dts_squared = dev_trial_stress.doubleContraction(dev_trial_stress);
Real effective_trial_stress = std::sqrt(1.5 * dts_squared);
// determine if yield condition is satisfied
Real yield_condition = effective_trial_stress - _hardening_variable_old[_qp] - _yield_stress;
_hardening_variable[_qp] = _hardening_variable_old[_qp];
_plastic_strain[_qp] = _plastic_strain_old[_qp];
if (yield_condition > 0) // then use newton iteration to determine effective plastic strain increment
{
unsigned int it = 0;
Real plastic_residual = 0;
Real norm_plas_residual = 10;
Real first_norm_plas_residual = 10;
Real scalar_plastic_strain_increment = 0;
while (it < _max_its &&
norm_plas_residual > _absolute_tolerance &&
(norm_plas_residual/first_norm_plas_residual) > _relative_tolerance)
{
plastic_residual = effective_trial_stress - (3. * _shear_modulus * scalar_plastic_strain_increment) -
_hardening_variable[_qp] - _yield_stress;
norm_plas_residual = std::abs(plastic_residual);
if (it == 0)
{
first_norm_plas_residual = norm_plas_residual;
}
scalar_plastic_strain_increment += plastic_residual / (3. * _shear_modulus + _hardening_constant);
_hardening_variable[_qp] = _hardening_variable_old[_qp] + (_hardening_constant * scalar_plastic_strain_increment);
if (_output_iteration_info == true)
{
_console
<< "pls_it=" << it
<< " trl_strs=" << effective_trial_stress
<< " del_p=" << scalar_plastic_strain_increment
<< " harden=" << _hardening_variable[_qp]
<< " rel_res=" << norm_plas_residual/first_norm_plas_residual
<< " rel_tol=" << _relative_tolerance
<< " abs_res=" << norm_plas_residual
<< " abs_tol=" << _absolute_tolerance
<< std::endl;
}
++it;
}
if (it == _max_its &&
norm_plas_residual > _absolute_tolerance &&
(norm_plas_residual/first_norm_plas_residual) > _relative_tolerance)
{
mooseError("Max sub-newton iteration hit during plasticity increment solve!");
}
if (effective_trial_stress < 0.01)
{
effective_trial_stress = 0.01;
}
plastic_strain_increment = dev_trial_stress;
plastic_strain_increment *= (1.5*scalar_plastic_strain_increment/effective_trial_stress);
SymmTensor elastic_strain_increment(strain_increment);
elastic_strain_increment -= plastic_strain_increment;
// compute stress increment
stress_new = *elasticityTensor() * elastic_strain_increment;
// update stress and plastic strain
stress_new += _stress_old;
_plastic_strain[_qp] += plastic_strain_increment;
} // end of if statement
}
Real
hydrostatic(const SymmTensor & symm_tensor)
{
return symm_tensor.trace() / 3.0;
}
Real
firstInvariant(const SymmTensor & symm_tensor)
{
return symm_tensor.trace();
}
Real
thirdInvariant(const SymmTensor & symm_tensor)
{
Real val = 0.0;
val = symm_tensor.xx() * symm_tensor.yy() * symm_tensor.zz() -
symm_tensor.xx() * symm_tensor.yz() * symm_tensor.yz() +
symm_tensor.xy() * symm_tensor.yz() * symm_tensor.zx() -
symm_tensor.xy() * symm_tensor.xy() * symm_tensor.zz() +
symm_tensor.zx() * symm_tensor.xy() * symm_tensor.yz() -
symm_tensor.zx() * symm_tensor.yy() * symm_tensor.zx();
return val;
}
Real
secondInvariant(const SymmTensor & symm_tensor)
{
Real value = symm_tensor.xx() * symm_tensor.yy() +
symm_tensor.yy() * symm_tensor.zz() +
symm_tensor.zz() * symm_tensor.xx() -
symm_tensor.xy() * symm_tensor.xy() -
symm_tensor.yz() * symm_tensor.yz() -
symm_tensor.zx() * symm_tensor.zx();
return value;
}
void
Element::unrotateSymmetricTensor( const ColumnMajorMatrix & R,
const SymmTensor & T,
SymmTensor & result )
{
// Rt T R
// 00 10 20 00 01 02 00 01 02
// 01 11 21 * 10 11 12 * 10 11 12
// 02 12 22 20 21 22 20 21 22
//
const Real T00 = R(0,0)*T.xx() + R(1,0)*T.xy() + R(2,0)*T.zx();
const Real T01 = R(0,0)*T.xy() + R(1,0)*T.yy() + R(2,0)*T.yz();
const Real T02 = R(0,0)*T.zx() + R(1,0)*T.yz() + R(2,0)*T.zz();
const Real T10 = R(0,1)*T.xx() + R(1,1)*T.xy() + R(2,1)*T.zx();
const Real T11 = R(0,1)*T.xy() + R(1,1)*T.yy() + R(2,1)*T.yz();
const Real T12 = R(0,1)*T.zx() + R(1,1)*T.yz() + R(2,1)*T.zz();
const Real T20 = R(0,2)*T.xx() + R(1,2)*T.xy() + R(2,2)*T.zx();
const Real T21 = R(0,2)*T.xy() + R(1,2)*T.yy() + R(2,2)*T.yz();
const Real T22 = R(0,2)*T.zx() + R(1,2)*T.yz() + R(2,2)*T.zz();
result.xx( T00 * R(0,0) + T01 * R(1,0) + T02 * R(2,0) );
result.yy( T10 * R(0,1) + T11 * R(1,1) + T12 * R(2,1) );
result.zz( T20 * R(0,2) + T21 * R(1,2) + T22 * R(2,2) );
result.xy( T00 * R(0,1) + T01 * R(1,1) + T02 * R(2,1) );
result.yz( T10 * R(0,2) + T11 * R(1,2) + T12 * R(2,2) );
result.zx( T00 * R(0,2) + T01 * R(1,2) + T02 * R(2,2) );
}
void
SymmElasticityTensor::multiply(const SymmTensor & x, SymmTensor & b) const
{
const Real xx = x.xx();
const Real yy = x.yy();
const Real zz = x.zz();
const Real xy = x.xy();
const Real yz = x.yz();
const Real zx = x.zx();
b.xx() =
_val[0] * xx + _val[1] * yy + _val[2] * zz + 2 * (_val[3] * xy + _val[4] * yz + _val[5] * zx);
b.yy() = _val[1] * xx + _val[6] * yy + _val[7] * zz +
2 * (_val[8] * xy + _val[9] * yz + _val[10] * zx);
b.zz() = _val[2] * xx + _val[7] * yy + _val[11] * zz +
2 * (_val[12] * xy + _val[13] * yz + _val[14] * zx);
b.xy() = _val[3] * xx + _val[8] * yy + _val[12] * zz +
2 * (_val[15] * xy + _val[16] * yz + _val[17] * zx);
b.yz() = _val[4] * xx + _val[9] * yy + _val[13] * zz +
2 * (_val[16] * xy + _val[18] * yz + _val[19] * zx);
b.zx() = _val[5] * xx + _val[10] * yy + _val[14] * zz +
2 * (_val[17] * xy + _val[19] * yz + _val[20] * zx);
}
void
Linear::computeStrain( const unsigned qp,
const SymmTensor & total_strain_old,
SymmTensor & total_strain_new,
SymmTensor & strain_increment )
{
strain_increment.xx( _grad_disp_x[qp](0) );
strain_increment.yy( _grad_disp_y[qp](1) );
strain_increment.zz( _grad_disp_z[qp](2) );
strain_increment.xy( 0.5*(_grad_disp_x[qp](1)+_grad_disp_y[qp](0)) );
strain_increment.yz( 0.5*(_grad_disp_y[qp](2)+_grad_disp_z[qp](1)) );
strain_increment.zx( 0.5*(_grad_disp_z[qp](0)+_grad_disp_x[qp](2)) );
if (_large_strain)
{
strain_increment.xx() += 0.5*(_grad_disp_x[qp](0)*_grad_disp_x[qp](0) +
_grad_disp_y[qp](0)*_grad_disp_y[qp](0) +
_grad_disp_z[qp](0)*_grad_disp_z[qp](0));
strain_increment.yy() += 0.5*(_grad_disp_x[qp](1)*_grad_disp_x[qp](1) +
_grad_disp_y[qp](1)*_grad_disp_y[qp](1) +
_grad_disp_z[qp](1)*_grad_disp_z[qp](1));
strain_increment.zz() += 0.5*(_grad_disp_x[qp](2)*_grad_disp_x[qp](2) +
_grad_disp_y[qp](2)*_grad_disp_y[qp](2) +
_grad_disp_z[qp](2)*_grad_disp_z[qp](2));
strain_increment.xy() += 0.5*(_grad_disp_x[qp](0)*_grad_disp_x[qp](1) +
_grad_disp_y[qp](0)*_grad_disp_y[qp](1) +
_grad_disp_z[qp](0)*_grad_disp_z[qp](1));
strain_increment.yz() += 0.5*(_grad_disp_x[qp](1)*_grad_disp_x[qp](2) +
_grad_disp_y[qp](1)*_grad_disp_y[qp](2) +
_grad_disp_z[qp](1)*_grad_disp_z[qp](2));
strain_increment.zx() += 0.5*(_grad_disp_x[qp](2)*_grad_disp_x[qp](0) +
_grad_disp_y[qp](2)*_grad_disp_y[qp](0) +
_grad_disp_z[qp](2)*_grad_disp_z[qp](0));
}
if (_volumetric_locking_correction)
{
// volumetric locking correction - averaging the volumertic strain over the element
Real volumetric_strain = 0.0;
Real volume = 0.0;
for (unsigned int qp_loop = 0; qp_loop < _solid_model.qrule()->n_points(); ++qp_loop)
{
volumetric_strain += (_grad_disp_x[qp_loop](0) + _grad_disp_y[qp_loop](1) + _grad_disp_z[qp_loop](2)) / 3.0 * _solid_model.JxW(qp_loop);
volume += _solid_model.JxW(qp_loop);
if (_large_strain)
{
volumetric_strain += 0.5 * (_grad_disp_x[qp](0) * _grad_disp_x[qp](0) +
_grad_disp_y[qp](0) * _grad_disp_y[qp](0) +
_grad_disp_z[qp](0) * _grad_disp_z[qp](0)) / 3.0 * _solid_model.JxW(qp_loop);
volumetric_strain += 0.5 * (_grad_disp_x[qp](1) * _grad_disp_x[qp](1) +
_grad_disp_y[qp](1) * _grad_disp_y[qp](1) +
_grad_disp_z[qp](1) * _grad_disp_z[qp](1)) / 3.0 * _solid_model.JxW(qp_loop);
volumetric_strain += 0.5 * (_grad_disp_x[qp](2) * _grad_disp_x[qp](2) +
_grad_disp_y[qp](2) * _grad_disp_y[qp](2) +
_grad_disp_z[qp](2) * _grad_disp_z[qp](2)) / 3.0 * _solid_model.JxW(qp_loop);
}
}
volumetric_strain /= volume; // average volumetric strain
// strain increment at _qp
Real trace = strain_increment.trace();
strain_increment.xx() += volumetric_strain - trace / 3.0;
strain_increment.yy() += volumetric_strain - trace / 3.0;
strain_increment.zz() += volumetric_strain - trace / 3.0;
}
total_strain_new = strain_increment;
strain_increment -= total_strain_old;
}
Real
component(const SymmTensor & symm_tensor, unsigned int index)
{
return symm_tensor.component(index);
}