//send back the tangent
const XC::Matrix& XC::DruckerPragerPlaneStrain::getInitialTangent()
{
tangent(0,0) = mCep(0,0);
tangent(0,1) = mCep(0,1);
tangent(0,2) = mCep(0,3);
tangent(1,0) = mCep(1,0);
tangent(1,1) = mCep(1,1);
tangent(1,2) = mCep(1,3);
tangent(2,0) = mCep(3,0);
tangent(2,1) = mCep(3,1);
tangent(2,2) = mCep(3,3);
return tangent;
}
//--------------------Plasticity-------------------------------------
//plasticity integration routine
void DruckerPrager:: plastic_integrator( )
{
bool okay; // boolean variable to ensure satisfaction of multisurface kuhn tucker conditions
double f1;
double f2;
double norm_eta;
double Invariant_1;
double Invariant_ep;
double norm_ep;
double norm_dev_ep;
Vector epsilon_e(6);
Vector s(6);
Vector eta(6);
Vector dev_ep(6);
Vector Jact(2);
double fTOL;
double gTOL;
fTOL = 0.0;
gTOL = -1.0e-10;
double NormCep;
double alpha1; // hardening parameter for DP surface
double alpha2; // hardening parameter for tension cut-off
Vector n(6); // normal to the yield surface in strain space
Vector R(2); // residual vector
Vector gamma(2); // vector of consistency parameters
Vector dgamma(2); // incremental vector of consistency parameters
Matrix g(2,2); // jacobian of the corner region (return map)
Matrix g_contra(2,2); // inverse of jacobian of the corner region
// set trial state:
// epsilon_n1_p_trial = ..._n1_p = ..._n_p
mEpsilon_n1_p = mEpsilon_n_p;
// alpha1_n+1_trial
mAlpha1_n1 = mAlpha1_n;
// alpha2_n+1_trial
mAlpha2_n1 = mAlpha2_n;
// beta_n+1_trial
mBeta_n1 = mBeta_n;
// epsilon_elastic = epsilon_n+1 - epsilon_n_p
epsilon_e = mEpsilon - mEpsilon_n1_p;
// trial stress
mSigma = mCe*epsilon_e;
// deviator stress tensor: s = 2G * IIdev * epsilon_e
//I1_trial
Invariant_1 = ( mSigma(0) + mSigma(1) + mSigma(2) );
// s_n+1_trial
s = mSigma - (Invariant_1/3.0)*mI1;
//eta_trial = s_n+1_trial - beta_n;
eta = s - mBeta_n;
// compute yield function value (contravariant norm)
norm_eta = sqrt(eta(0)*eta(0) + eta(1)*eta(1) + eta(2)*eta(2) + 2*(eta(3)*eta(3) + eta(4)*eta(4) + eta(5)*eta(5)));
// f1_n+1_trial
f1 = norm_eta + mrho*Invariant_1 - root23*Kiso(mAlpha1_n1);
// f2_n+1_trial
f2 = Invariant_1 - T(mAlpha2_n1);
// update elastic bulk and shear moduli
this->updateElasticParam();
// check trial state
int count = 1;
if ((f1<=fTOL) && (f2<=fTOL) || mElastFlag < 2) {
okay = true;
// trial state = elastic state - don't need to do any updates.
mCep = mCe;
count = 0;
// set state variables for recorders
Invariant_ep = mEpsilon_n1_p(0)+mEpsilon_n1_p(1)+mEpsilon_n1_p(2);
norm_ep = sqrt(mEpsilon_n1_p(0)*mEpsilon_n1_p(0) + mEpsilon_n1_p(1)*mEpsilon_n1_p(1) + mEpsilon_n1_p(2)*mEpsilon_n1_p(2)
+ 0.5*(mEpsilon_n1_p(3)*mEpsilon_n1_p(3) + mEpsilon_n1_p(4)*mEpsilon_n1_p(4) + mEpsilon_n1_p(5)*mEpsilon_n1_p(5)));
dev_ep = mEpsilon_n1_p - one3*Invariant_ep*mI1;
norm_dev_ep = sqrt(dev_ep(0)*dev_ep(0) + dev_ep(1)*dev_ep(1) + dev_ep(2)*dev_ep(2)
+ 0.5*(dev_ep(3)*dev_ep(3) + dev_ep(4)*dev_ep(4) + dev_ep(5)*dev_ep(5)));
mState(0) = Invariant_1;
mState(1) = norm_eta;
mState(2) = Invariant_ep;
mState(3) = norm_dev_ep;
mState(4) = norm_ep;
return;
}
else {
// plastic correction required
okay = false;
// determine number of active surfaces. size & fill Jact
if ( (f1 > fTOL ) && (f2 <= fTOL) ) {
// f1 surface only
Jact(0) = 1;
Jact(1) = 0;
}
else if ( (f1 <= fTOL ) && (f2 > fTOL) ) {
// f2 surface only
Jact(0) = 0;
Jact(1) = 1;
}
else if ( (f1 > fTOL ) && (f2 > fTOL) ) {
// both surfaces active
Jact(0) = 1;
Jact(1) = 1;
}
}
//-----------------MultiSurface Placity Return Map--------------------------------------
while (!okay) {
alpha1 = mAlpha1_n;
alpha2 = mAlpha2_n;
// n = eta / norm_eta; (contravaraint)
if (norm_eta < 1.0e-13) {
n.Zero();
} else {
n = eta/norm_eta;
}
// initialize R, gamma1, gamma2, dgamma1, dgamma2 = 0
R.Zero();
gamma.Zero();
dgamma.Zero();
// initialize g such that det(g) = 1
g(0,0) = 1;
g(1,1) = 1;
g(1,0) = 0;
g(0,1) = 0;
// Newton procedure to compute nonlinear gamma1 and gamma2
//initialize terms
for (int i = 0; i < 2; i++) {
if (Jact(i) == 1) {
R(0) = norm_eta - (2*mG + two3*mHprime)*gamma(0) + mrho*Invariant_1
- 9*mK*mrho*mrho_bar*gamma(0) - 9*mK*mrho*gamma(1) - root23*Kiso(alpha1);
g(0,0) = -2*mG - two3*(mHprime + Kisoprime(alpha1)) - 9*mK*mrho*mrho_bar;
} else if (Jact(i) == 2) {
R(1) = Invariant_1 - 9*mK*mrho_bar*gamma(0) - 9*mK*gamma(1) - T(alpha2);
g(1,1) = -9*mK + mdelta2*T(alpha2);
}
}
if (Jact(0) == 1 && Jact(1) == 1) {
g(0,1) = -9*mK*mrho;
g(1,0) = mrho_bar*(-9*mK + mdelta2*T(alpha2));
}
g.Invert(g_contra);
// iteration counter
int m = 0;
//iterate
while ((fabs(R.Norm()) > 1e-10) && (m < 10)) {
dgamma = -1*g_contra * R;
gamma += dgamma;
//update alpha1 and alpha2
alpha1 = mAlpha1_n + root23*gamma(0);
alpha2 = mAlpha2_n + mrho_bar*gamma(0) + gamma(1);
// reset g & R matrices
g(0,0) = 1;
g(1,1) = 1;
g(1,0) = 0;
g(0,1) = 0;
R.Zero();
for (int i = 0; i < 2; i++) {
if (Jact(i) == 1) {
R(0) = norm_eta - (2*mG + two3*mHprime)*gamma(0) + mrho*Invariant_1
- 9*mK*mrho*mrho_bar*gamma(0) - 9*mK*mrho*gamma(1) - root23*Kiso(alpha1);
g(0,0) = -2*mG - two3*(mHprime + Kisoprime(alpha1)) - 9*mK*mrho*mrho_bar;
} else if (Jact(i) == 2) {
R(1) = Invariant_1 - 9*mK*mrho_bar*gamma(0) - 9*mK*gamma(1) - T(alpha2);
g(1,1) = -9*mK + mdelta2*T(alpha2);
}
}
if (Jact(0) == 1 && Jact(1) == 1) {
g(0,1) = -9*mK*mrho;
g(1,0) = mrho_bar*(-9*mK + mdelta2*T(alpha2));
}
g.Invert(g_contra);
m++;
}
// check maintain Kuhn-Tucker conditions
f1 = norm_eta - (2*mG + two3*mHprime)*gamma(0) + mrho*Invariant_1
-9*mK*mrho*mrho_bar*gamma(0) - 9*mK*mrho*gamma(1) - root23*Kiso(alpha1);
f2 = Invariant_1 - 9*mK*mrho_bar*gamma(0) - 9*mK*gamma(1) - T(alpha2);
if ( count > 100 ) {
okay = true;
break;
}
// check active surfaces
if ((Jact(0) == 1) && (Jact(1) == 0)) {
// f2 may be > or < f2_tr because of softening of f2 related to alpha1
if (f2 >= fTOL) {
// okay = false;
Jact(0) = 1;
Jact(1) = 1;
count += 1;
} else {
okay = true;
}
} else if ((Jact(0) == 0) && (Jact(1) == 1)) {
// f1 will always be less than f1_tr
okay = true;
} else if ((Jact(0) == 1) && (Jact(1) == 1)) {
if ((gamma(0) <= gTOL) && (gamma(1) > gTOL)){
// okay = false;
Jact(0) = 0;
Jact(1) = 1;
count += 1;
} else if ((gamma(0) > gTOL) && (gamma(1) <= gTOL)){
// okay = false;
Jact(0) = 1;
Jact(1) = 0;
count += 1;
} else if ((gamma(0) > gTOL) && (gamma(1) > gTOL)) {
okay = true;
}
}
if ( (count > 3) && (!okay) ) {
Jact(0) = 1;
Jact(1) = 1;
count += 100;
}
if ( count > 3 ) {
opserr << "Jact = " << Jact;
opserr << "count = " << count << endln;
}
} // end of while(!okay) loop
//update everything and exit!
Vector b1(6);
Vector b2(6);
Vector n_covar(6);
Vector temp1(6);
Vector temp2(6);
// update alpha1 and alpha2
mAlpha1_n1 = alpha1;
mAlpha2_n1 = alpha2;
//update epsilon_n1_p
//first calculate n_covar
// n_a = G_ab * n^b = covariant
n_covar(0) = n(0);
n_covar(1) = n(1);
n_covar(2) = n(2);
n_covar(3) = 2*n(3);
n_covar(4) = 2*n(4);
n_covar(5) = 2*n(5);
mEpsilon_n1_p = mEpsilon_n_p + (mrho_bar*gamma(0) + gamma(1))*mI1 + gamma(0)*n_covar;
Invariant_ep = mEpsilon_n1_p(0)+mEpsilon_n1_p(1)+mEpsilon_n1_p(2);
norm_ep = sqrt(mEpsilon_n1_p(0)*mEpsilon_n1_p(0) + mEpsilon_n1_p(1)*mEpsilon_n1_p(1) + mEpsilon_n1_p(2)*mEpsilon_n1_p(2)
+ 0.5*(mEpsilon_n1_p(3)*mEpsilon_n1_p(3) + mEpsilon_n1_p(4)*mEpsilon_n1_p(4) + mEpsilon_n1_p(5)*mEpsilon_n1_p(5)));
dev_ep = mEpsilon_n1_p - one3*Invariant_ep*mI1;
norm_dev_ep = sqrt(dev_ep(0)*dev_ep(0) + dev_ep(1)*dev_ep(1) + dev_ep(2)*dev_ep(2)
+ 0.5*(dev_ep(3)*dev_ep(3) + dev_ep(4)*dev_ep(4) + dev_ep(5)*dev_ep(5)));
// update sigma
mSigma -= (3*mK*mrho_bar*gamma(0) + 3*mK*gamma(1))*mI1 + 2*mG*gamma(0)*n;
s -= 2*mG*gamma(0) * n;
Invariant_1 -= 9*mK*mrho_bar*gamma(0) + 9*mK*gamma(1);
//mSigma = s + Invariant_1/3.0 * mI1;
//update beta_n1
mBeta_n1 = mBeta_n - (two3*mHprime*gamma(0))*n;
//eta_n+1 = s_n+1 - beta_n+1;
eta = s - mBeta_n1;
norm_eta = sqrt(eta(0)*eta(0) + eta(1)*eta(1) + eta(2)*eta(2) + 2*(eta(3)*eta(3) + eta(4)*eta(4) + eta(5)*eta(5)));
// update Cep
// note: Cep is contravariant
if ((Jact(0) == 1) && (Jact(1) == 0)) {
b1 = 2*mG*n + 3*mK*mrho*mI1;
b2.Zero();
} else if ((Jact(0) == 0) && (Jact(1) == 1)){
b1.Zero();
b2 = 3*mK*mI1;
} else if ((Jact(0) == 1) && (Jact(1) == 1)){
b1 = 2*mG*n + 3*mK*mrho*mI1;
b2 = 3*mK*mI1;
}
temp1 = g_contra(0,0)*b1 + g_contra(0,1)*b2;
temp2 = mrho_bar*temp1 + g_contra(1,0)*b1 + g_contra(1,1)*b2;
NormCep = 0.0;
for (int i = 0; i < 6; i++){
for (int j = 0; j < 6; j++) {
mCep(i,j) = mCe(i,j)
+ 3*mK * mI1(i)*temp2(j)
+ 2*mG * n(i)*temp1(j)
- 4*mG*mG/norm_eta*gamma(0) * (mIIdev(i,j) - n(i)*n(j));
NormCep += mCep(i,j)*mCep(i,j);
}
}
if ( NormCep < 1e-10){
mCep = 1.0e-3 * mCe;
opserr << "NormCep = " << NormCep << endln;
}
mState(0) = Invariant_1;
mState(1) = norm_eta;
mState(2) = Invariant_ep;
mState(3) = norm_dev_ep;
mState(4) = norm_ep;
return;
}
