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;
}
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;
}
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));
strain_increment.yz() = 0;
strain_increment.zx() = 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;
}
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) );
}
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;
}
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;
}
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) );
}
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);
}
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;
}
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);
if (_have_strain_zz)
strain_increment.zz() = _strain_zz[qp];
else if (_have_scalar_strain_zz && _scalar_strain_zz.size()>0)
strain_increment.zz() = _scalar_strain_zz[0];
else
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));
}
if (_volumetric_locking_correction)
{
// volumetric locking correction
Real volumetric_strain = 0.0;
Real volume = 0.0;
Real dim = 3.0;
for (unsigned int qp_loop = 0; qp_loop < _solid_model.qrule()->n_points(); ++qp_loop)
{
if (_have_strain_zz)
volumetric_strain += (_grad_disp_x[qp_loop](0) + _grad_disp_y[qp_loop](1) + _strain_zz[qp_loop]) / dim * _solid_model.JxW(qp_loop);
else if (_have_scalar_strain_zz && _scalar_strain_zz.size() > 0)
volumetric_strain += (_grad_disp_x[qp_loop](0) + _grad_disp_y[qp_loop](1) + _scalar_strain_zz[0]) / dim * _solid_model.JxW(qp_loop);
else
volumetric_strain += (_grad_disp_x[qp_loop](0) + _grad_disp_y[qp_loop](1)) / dim * _solid_model.JxW(qp_loop);
volume += _solid_model.JxW(qp_loop);
if (_large_strain)
{
volumetric_strain += 0.5 * (_grad_disp_x[qp_loop](0) * _grad_disp_x[qp_loop](0) +
_grad_disp_y[qp_loop](0) * _grad_disp_y[qp_loop](0)) / dim * _solid_model.JxW(qp_loop);
volumetric_strain += 0.5 * (_grad_disp_x[qp_loop](1) * _grad_disp_x[qp_loop](1) +
_grad_disp_y[qp_loop](1) * _grad_disp_y[qp_loop](1)) / dim * _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 / dim;
strain_increment.yy() += volumetric_strain - trace / dim;
strain_increment.zz() += volumetric_strain - trace / dim;
}
total_strain_new = strain_increment;
strain_increment -= total_strain_old;
}
Real
MaterialTensorAux::getTensorQuantity(const SymmTensor & tensor,
const MTA_ENUM quantity,
const MooseEnum & quantity_moose_enum,
const int index,
const Point * curr_point,
const Point * p1,
const Point * p2)
{
Real value(0);
if (quantity == MTA_COMPONENT)
{
value = tensor.component(index);
}
else if ( quantity == MTA_VONMISES )
{
value = std::sqrt(0.5*(
std::pow(tensor.xx() - tensor.yy(), 2) +
std::pow(tensor.yy() - tensor.zz(), 2) +
std::pow(tensor.zz() - tensor.xx(), 2) + 6 * (
std::pow(tensor.xy(), 2) +
std::pow(tensor.yz(), 2) +
std::pow(tensor.zx(), 2))));
}
else if ( quantity == MTA_PLASTICSTRAINMAG )
{
value = std::sqrt(2.0/3.0 * tensor.doubleContraction(tensor));
}
else if ( quantity == MTA_HYDROSTATIC )
{
value = tensor.trace()/3.0;
}
else if ( quantity == MTA_HOOP )
{
// This is the location of the stress. A vector from the cylindrical axis to this point will define the x' axis.
Point p0( *curr_point );
// The vector p1 + t*(p2-p1) defines the cylindrical axis. The point along this
// axis closest to p0 is found by the following for t:
const Point p2p1( *p2 - *p1 );
const Point p2p0( *p2 - p0 );
const Point p1p0( *p1 - p0 );
const Real t( -(p1p0*p2p1)/p2p1.size_sq() );
// The nearest point on the cylindrical axis to p0 is p.
const Point p( *p1 + t * p2p1 );
Point xp( p0 - p );
xp /= xp.size();
Point yp( p2p1/p2p1.size() );
Point zp( xp.cross( yp ));
//
// The following works but does more than we need
//
// // Rotation matrix R
// ColumnMajorMatrix R(3,3);
// // Fill with direction cosines
// R(0,0) = xp(0);
// R(1,0) = xp(1);
// R(2,0) = xp(2);
// R(0,1) = yp(0);
// R(1,1) = yp(1);
// R(2,1) = yp(2);
// R(0,2) = zp(0);
// R(1,2) = zp(1);
// R(2,2) = zp(2);
// // Rotate
// ColumnMajorMatrix tensor( _tensor[_qp].columnMajorMatrix() );
// ColumnMajorMatrix tensorp( R.transpose() * ( tensor * R ));
// // Hoop stress is the zz stress
// value = tensorp(2,2);
//
// Instead, tensorp(2,2) = R^T(2,:)*tensor*R(:,2)
//
const Real zp0( zp(0) );
const Real zp1( zp(1) );
const Real zp2( zp(2) );
value = (zp0*tensor(0,0)+zp1*tensor(1,0)+zp2*tensor(2,0))*zp0 +
(zp0*tensor(0,1)+zp1*tensor(1,1)+zp2*tensor(2,1))*zp1 +
(zp0*tensor(0,2)+zp1*tensor(1,2)+zp2*tensor(2,2))*zp2;
}
else if ( quantity == MTA_RADIAL )
{
// This is the location of the stress. A vector from the cylindrical axis to this point will define the x' axis
// which is the radial direction in which we want the stress.
Point p0( *curr_point );
// The vector p1 + t*(p2-p1) defines the cylindrical axis. The point along this
// axis closest to p0 is found by the following for t:
const Point p2p1( *p2 - *p1 );
const Point p2p0( *p2 - p0 );
const Point p1p0( *p1 - p0 );
const Real t( -(p1p0*p2p1)/p2p1.size_sq() );
// The nearest point on the cylindrical axis to p0 is p.
const Point p( *p1 + t * p2p1 );
Point xp( p0 - p );
xp /= xp.size();
const Real xp0( xp(0) );
const Real xp1( xp(1) );
const Real xp2( xp(2) );
value = (xp0*tensor(0,0)+xp1*tensor(1,0)+xp2*tensor(2,0))*xp0 +
(xp0*tensor(0,1)+xp1*tensor(1,1)+xp2*tensor(2,1))*xp1 +
(xp0*tensor(0,2)+xp1*tensor(1,2)+xp2*tensor(2,2))*xp2;
}
else if ( quantity == MTA_AXIAL )
{
// The vector p2p1=(p2-p1) defines the axis, which is the direction in which we want the stress.
Point p2p1( *p2 - *p1 );
p2p1 /= p2p1.size();
const Real axis0( p2p1(0) );
const Real axis1( p2p1(1) );
const Real axis2( p2p1(2) );
value = (axis0*tensor(0,0)+axis1*tensor(1,0)+axis2*tensor(2,0))*axis0 +
(axis0*tensor(0,1)+axis1*tensor(1,1)+axis2*tensor(2,1))*axis1 +
(axis0*tensor(0,2)+axis1*tensor(1,2)+axis2*tensor(2,2))*axis2;
}
else if ( quantity == MTA_MAXPRINCIPAL )
{
value = principalValue( tensor, 0 );
}
else if ( quantity == MTA_MEDPRINCIPAL )
{
value = principalValue( tensor, 1 );
}
else if ( quantity == MTA_MINPRINCIPAL )
{
value = principalValue( tensor, 2 );
}
else if ( quantity == MTA_FIRSTINVARIANT )
{
value = tensor.trace();
}
else if ( quantity == MTA_SECONDINVARIANT )
{
value =
tensor.xx()*tensor.yy() +
tensor.yy()*tensor.zz() +
tensor.zz()*tensor.xx() -
tensor.xy()*tensor.xy() -
tensor.yz()*tensor.yz() -
tensor.zx()*tensor.zx();
}
else if ( quantity == MTA_THIRDINVARIANT )
{
value =
tensor.xx()*tensor.yy()*tensor.zz() -
tensor.xx()*tensor.yz()*tensor.yz() +
tensor.xy()*tensor.zx()*tensor.yz() -
tensor.xy()*tensor.xy()*tensor.zz() +
tensor.zx()*tensor.xy()*tensor.yz() -
tensor.zx()*tensor.zx()*tensor.yy();
}
else if ( quantity == MTA_TRIAXIALITY )
{
Real hydrostatic = tensor.trace()/3.0;
Real von_mises = std::sqrt(0.5*(
std::pow(tensor.xx() - tensor.yy(), 2) +
std::pow(tensor.yy() - tensor.zz(), 2) +
std::pow(tensor.zz() - tensor.xx(), 2) + 6 * (
std::pow(tensor.xy(), 2) +
std::pow(tensor.yz(), 2) +
std::pow(tensor.zx(), 2))));
value = std::abs(hydrostatic / von_mises);
}
else if ( quantity == MTA_VOLUMETRICSTRAIN )
{
value =
tensor.trace() +
tensor.xx()*tensor.yy() +
tensor.yy()*tensor.zz() +
tensor.zz()*tensor.xx() +
tensor.xx()*tensor.yy()*tensor.zz();
}
else
{
mooseError("Unknown quantity in MaterialTensorAux: " + quantity_moose_enum.operator std::string());
}
return value;
}