void
MisesMat :: give3dLSMaterialStiffnessMatrix(FloatMatrix &answer, MatResponseMode mode, GaussPoint *gp, TimeStep *tStep)
{
MisesMatStatus *status = static_cast< MisesMatStatus * >( this->giveStatus(gp) );
// start from the elastic stiffness
FloatMatrix I(6, 6);
I.at(1, 1) = I.at(2, 2) = I.at(3, 3) = 1;
I.at(4, 4) = I.at(5, 5) = I.at(6, 6) = 0.5;
FloatArray delta(6);
delta.at(1) = delta.at(2) = delta.at(3) = 1;
FloatMatrix F;
F.beMatrixForm( status->giveTempFVector() );
double J = F.giveDeterminant();
const FloatArray &trialStressDev = status->giveTrialStressDev();
double trialStressVol = status->giveTrialStressVol();
double trialS = computeStressNorm(trialStressDev);
FloatArray n = trialStressDev;
if ( trialS == 0 ) {
n.resize(6);
} else {
n.times(1 / trialS);
}
FloatMatrix C;
FloatMatrix help;
help.beDyadicProductOf(delta, delta);
C = help;
help.times(-1. / 3.);
FloatMatrix C1 = I;
C1.add(help);
C1.times(2 * trialStressVol);
FloatMatrix n1, n2;
n1.beDyadicProductOf(n, delta);
n2.beDyadicProductOf(delta, n);
help = n1;
help.add(n2);
C1.add(-2. / 3. * trialS, help);
C.times(K * J * J);
C.add(-K * ( J * J - 1 ), I);
C.add(C1);
//////////////////////////////////////////////////////////////////////////////////////////////////////////
FloatMatrix invF;
FloatMatrix T(6, 6);
invF.beInverseOf(F);
//////////////////////////////////////////////////
//first row of pull back transformation matrix
T.at(1, 1) = invF.at(1, 1) * invF.at(1, 1);
T.at(1, 2) = invF.at(1, 2) * invF.at(1, 2);
T.at(1, 3) = invF.at(1, 3) * invF.at(1, 3);
T.at(1, 4) = 2. * invF.at(1, 2) * invF.at(1, 3);
T.at(1, 5) = 2. * invF.at(1, 1) * invF.at(1, 3);
T.at(1, 6) = 2. * invF.at(1, 1) * invF.at(1, 2);
//second row of pull back transformation matrix
T.at(2, 1) = invF.at(2, 1) * invF.at(2, 1);
T.at(2, 2) = invF.at(2, 2) * invF.at(2, 2);
T.at(2, 3) = invF.at(2, 3) * invF.at(2, 3);
T.at(2, 4) = 2. * invF.at(2, 2) * invF.at(2, 3);
T.at(2, 5) = 2. * invF.at(2, 1) * invF.at(2, 3);
T.at(2, 6) = 2. * invF.at(2, 1) * invF.at(2, 2);
//third row of pull back transformation matrix
T.at(3, 1) = invF.at(3, 1) * invF.at(3, 1);
T.at(3, 2) = invF.at(3, 2) * invF.at(3, 2);
T.at(3, 3) = invF.at(3, 3) * invF.at(3, 3);
T.at(3, 4) = 2. * invF.at(3, 2) * invF.at(3, 3);
T.at(3, 5) = 2. * invF.at(3, 1) * invF.at(3, 3);
T.at(3, 6) = 2. * invF.at(3, 1) * invF.at(3, 2);
//fourth row of pull back transformation matrix
T.at(4, 1) = invF.at(2, 1) * invF.at(3, 1);
T.at(4, 2) = invF.at(2, 2) * invF.at(3, 2);
T.at(4, 3) = invF.at(2, 3) * invF.at(3, 3);
T.at(4, 4) = ( invF.at(2, 2) * invF.at(3, 3) + invF.at(2, 3) * invF.at(3, 2) );
T.at(4, 5) = ( invF.at(2, 1) * invF.at(3, 3) + invF.at(2, 3) * invF.at(3, 1) );
T.at(4, 6) = ( invF.at(2, 1) * invF.at(3, 2) + invF.at(2, 2) * invF.at(3, 1) );
//fifth row of pull back transformation matrix
T.at(5, 1) = invF.at(1, 1) * invF.at(3, 1);
T.at(5, 2) = invF.at(1, 2) * invF.at(3, 2);
T.at(5, 3) = invF.at(1, 3) * invF.at(3, 3);
T.at(5, 4) = ( invF.at(1, 2) * invF.at(3, 3) + invF.at(1, 3) * invF.at(3, 2) );
T.at(5, 5) = ( invF.at(1, 1) * invF.at(3, 3) + invF.at(1, 3) * invF.at(3, 1) );
T.at(5, 6) = ( invF.at(1, 1) * invF.at(3, 2) + invF.at(1, 2) * invF.at(3, 1) );
//sixth row of pull back transformation matrix
T.at(6, 1) = invF.at(1, 1) * invF.at(2, 1);
T.at(6, 2) = invF.at(1, 2) * invF.at(2, 2);
T.at(6, 3) = invF.at(1, 3) * invF.at(2, 3);
T.at(6, 4) = ( invF.at(1, 2) * invF.at(2, 3) + invF.at(1, 3) * invF.at(2, 2) );
T.at(6, 5) = ( invF.at(1, 1) * invF.at(2, 3) + invF.at(1, 3) * invF.at(2, 1) );
T.at(6, 6) = ( invF.at(1, 1) * invF.at(2, 2) + invF.at(1, 2) * invF.at(2, 1) );
///////////////////////////////////////////
if ( mode != TangentStiffness ) {
help.beProductTOf(C, T);
answer.beProductOf(T, help);
return;
}
double kappa = status->giveCumulativePlasticStrain();
// increment of cumulative plastic strain as an indicator of plastic loading
double dKappa = sqrt(3. / 2.) * ( status->giveTempCumulativePlasticStrain() - kappa );
//double dKappa = ( status->giveTempCumulativePlasticStrain() - kappa);
if ( dKappa <= 0.0 ) { // elastic loading - elastic stiffness plays the role of tangent stiffness
help.beProductTOf(C, T);
answer.beProductOf(T, help);
return;
}
// === plastic loading ===
//dKappa = dKappa*sqrt(3./2.);
// trial deviatoric stress and its norm
double beta0, beta1, beta2, beta3, beta4;
if ( trialS == 0 ) {
beta1 = 0;
} else {
beta1 = 2 * trialStressVol * dKappa / trialS;
}
if ( trialStressVol == 0 ) {
beta0 = 0;
beta2 = 0;
beta3 = beta1;
beta4 = 0;
} else {
beta0 = 1 + H / 3 / trialStressVol;
beta2 = ( 1 - 1 / beta0 ) * 2. / 3. * trialS * dKappa / trialStressVol;
beta3 = 1 / beta0 - beta1 + beta2;
beta4 = ( 1 / beta0 - beta1 ) * trialS / trialStressVol;
}
FloatMatrix N;
N.beDyadicProductOf(n, n);
N.times(-2 * trialStressVol * beta3);
C1.times(-beta1);
FloatMatrix mN(3, 3);
mN.at(1, 1) = n.at(1);
mN.at(1, 2) = n.at(6);
mN.at(1, 3) = n.at(5);
mN.at(2, 1) = n.at(6);
mN.at(2, 2) = n.at(2);
mN.at(2, 3) = n.at(4);
mN.at(3, 1) = n.at(5);
mN.at(3, 2) = n.at(4);
mN.at(3, 3) = n.at(3);
FloatMatrix mN2;
mN2.beProductOf(mN, mN);
double volN2 = 1. / 3. * ( mN2.at(1, 1) + mN2.at(2, 2) + mN2.at(3, 3) );
FloatArray devN2(6);
devN2.at(1) = mN2.at(1, 1) - volN2;
devN2.at(2) = mN2.at(2, 2) - volN2;
devN2.at(3) = mN2.at(3, 3) - volN2;
devN2.at(4) = mN2.at(2, 3);
devN2.at(5) = mN2.at(1, 3);
devN2.at(6) = mN2.at(1, 2);
FloatMatrix nonSymPart;
nonSymPart.beDyadicProductOf(n, devN2);
//symP.beTranspositionOf(nonSymPart);
//symP.add(nonSymPart);
//symP.times(1./2.);
nonSymPart.times(-2 * trialStressVol * beta4);
C.add(C1);
C.add(N);
C.add(nonSymPart);
help.beProductTOf(C, T);
answer.beProductOf(T, help);
}
