// Get V-I which is Biot strain rotated into current particle orientation
// It is based on total strain
Matrix3 MPMBase::GetBiotStrain(void) const
{
Matrix3 F = GetDeformationGradientMatrix();
Matrix3 V = F.LeftDecompose(NULL,NULL);
V(0,0) -= 1.;
V(1,1) -= 1.;
V(2,2) -= 1.;
return V;
}
// Entry point for large rotation
void IsoPlasticity::LRConstitutiveLaw(MPMBase *mptr,Matrix3 du,double delTime,int np,void *properties,ResidualStrains *res) const
{
// current previous deformation gradient and stretch
Matrix3 pFnm1 = mptr->GetDeformationGradientMatrix();
// get incremental deformation gradient and decompose it
const Matrix3 dF = du.Exponential(incrementalDefGradTerms);
Matrix3 dR;
Matrix3 dV = dF.LeftDecompose(&dR,NULL);
// decompose to get previous stretch
Matrix3 Vnm1 = pFnm1.LeftDecompose(NULL,NULL);
// get strain increments de = (dV-I) dR Vnm1 dRT
dV(0,0) -= 1.;
dV(1,1) -= 1.;
dV(2,2) -= 1.;
Matrix3 de = dV*Vnm1.RMRT(dR);
// Update total deformation gradient
Matrix3 pF = dF*pFnm1;
mptr->SetDeformationGradientMatrix(pF);
// Effective strain by deducting thermal strain (no shear thermal strain because isotropic)
// (note: using unreduced terms in CTE3 and CME3)
double eres=CTE3*res->dT;
if(DiffusionTask::active)
eres+=CME3*res->dC;
// Trial update assuming elastic response
double delV;
// 3D or 2D
PlasticProperties *p = (PlasticProperties *)properties;
if(np==THREED_MPM)
{ delV = de.trace() - 3.*eres;
LRPlasticityConstLaw(mptr,de(0,0),de(1,1),de(2,2),2*de(0,1),2.*de(0,2),2.*de(1,2),
delTime,np,delV,eres,p,res,&dR);
return;
}
else if(np==PLANE_STRESS_MPM)
delV = p->psRed*(de(0,0)+de(1,1)-2.*eres);
else
delV = de.trace() - 3.*eres;
LRPlasticityConstLaw(mptr,de(0,0),de(1,1),2.*de(0,1),de(2,2),delTime,np,delV,eres,p,res,&dR);
}
/* Take increments in strain and calculate new Particle: strains, rotation strain,
stresses, strain energy,
du are (gradient rates X time increment) to give deformation gradient change
For Axisymmetry: x->R, y->Z, z->theta, np==AXISYMMETRIC_MPM, otherwise dvzz=0
This material tracks pressure and stores deviatoric stress only in particle stress tensor
*/
void Viscoelastic::MPMConstitutiveLaw(MPMBase *mptr,Matrix3 du,double delTime,int np,void *properties,ResidualStrains *res,int historyOffset) const
{
// current previous deformation gradient and stretch
Matrix3 pFnm1 = mptr->GetDeformationGradientMatrix();
// get incremental deformation gradient and decompose it
const Matrix3 dF = du.Exponential(incrementalDefGradTerms);
Matrix3 dR;
Matrix3 dVstretch = dF.LeftDecompose(&dR,NULL);
// decompose to get previous stretch
Matrix3 Vnm1 = pFnm1.LeftDecompose(NULL,NULL);
// get strain increments de = (dV-I) dR Vnma dRT
dVstretch(0,0) -= 1.;
dVstretch(1,1) -= 1.;
dVstretch(2,2) -= 1.;
Matrix3 Vrot = Vnm1.RMRT(dR);
Matrix3 detot = dVstretch*Vrot;
// Update total deformation gradient
Matrix3 pF = dF*pFnm1;
mptr->SetDeformationGradientMatrix(pF);
// Effective strain by deducting thermal strain (no shear thermal strain because isotropic)
double eres=CTE*res->dT;
if(DiffusionTask::active)
eres+=CME*res->dC;
// update pressure
double dTq0 = 0.,dispEnergy = 0.;
double traceDe = detot.trace();
double delV = traceDe - 3.*eres;
#ifdef USE_KIRCHOFF_STRESS
// tracking J
double detdF = dF.determinant();
double **h =(double **)(mptr->GetHistoryPtr(0));
double J = detdF*h[mptrHistory][MGJ_HISTORY];
h[mptrHistory][MGJ_HISTORY] = J;
UpdatePressure(mptr,delV,res,eres,detdF,J,delTime,dTq0,dispEnergy);
#else
UpdatePressure(mptr,delV,res,eres,&dF,delTime,dTq0,dispEnergy);
#endif
// deviatoric strains increment in de
// Actually de is finding 2*(dev e) to avoid many multiplies by two
Tensor de;
double dV = traceDe/3.;
de.xx = 2.*(detot(0,0) - dV);
de.yy = 2.*(detot(1,1) - dV);
de.zz = 2.*(detot(2,2) - dV);
de.xy = 2.*detot(0,1);
if(np==THREED_MPM)
{ de.xz = 2.*detot(0,2);
de.yz = 2.*detot(1,2);
}
// Find initial 2*e(t) (deviatoric strain) in ed
Tensor ed;
double thirdV = Vrot.trace()/3.;
ed.xx = 2.*(Vrot(0,0)-thirdV);
ed.yy = 2.*(Vrot(1,1)-thirdV);
ed.zz = 2.*(Vrot(2,2)-thirdV);
ed.xy = 2.*Vrot(0,1);
if(np==THREED_MPM)
{ ed.xz = 2.*Vrot(0,2);
ed.yz = 2.*Vrot(1,2);
}
// increment particle deviatoric stresses - elastic part
double dsig[6];
dsig[XX] = Gered*de.xx;
dsig[YY] = Gered*de.yy;
dsig[ZZ] = Gered*de.zz;
dsig[XY] = Gered*de.xy;
if(np==THREED_MPM)
{ dsig[XZ] = Gered*de.xz;
dsig[YZ] = Gered*de.yz;
}
// get internal variable increments, update them, add to incremental stress, and get dissipated energy6
Tensor dak;
double **ak =(double **)(mptr->GetHistoryPtr(0));
int k;
for(k=0;k<ntaus;k++)
{ double tmp = exp(-delTime/tauk[k]);
double tmpm1 = tmp-1.;
double tmpp1 = tmp+1.;
double arg = 0.25*delTime/tauk[k]; // 0.25 because e's have factor of 2
dak.xx = tmpm1*ak[XX_HISTORY][k] + arg*(tmpp1*ed.xx + de.xx);
dak.yy = tmpm1*ak[YY_HISTORY][k] + arg*(tmpp1*ed.yy + de.yy);
dak.xy = tmpm1*ak[XY_HISTORY][k] + arg*(tmpp1*ed.xy + de.xy);
dak.zz = tmpm1*ak[ZZ_HISTORY][k] + arg*(tmpp1*ed.zz + de.zz);
ak[XX_HISTORY][k] += dak.xx;
ak[YY_HISTORY][k] += dak.yy;
ak[ZZ_HISTORY][k] += dak.zz;
ak[XY_HISTORY][k] += dak.xy;
// add to stress increments
dsig[XX] -= TwoGkred[k]*dak.xx;
dsig[YY] -= TwoGkred[k]*dak.yy;
dsig[ZZ] -= TwoGkred[k]*dak.zz;
dsig[XY] -= TwoGkred[k]*dak.xy;
// dissipation
dispEnergy += TwoGkred[k]*(dak.xx*(0.5*(ed.xx+0.5*de.xx)-ak[XX_HISTORY][k]+0.5*dak.xx)
+ dak.yy*(0.5*(ed.yy+0.5*de.yy)-ak[YY_HISTORY][k]+0.5*dak.yy)
+ dak.zz*(0.5*(ed.zz+0.5*de.zz)-ak[ZZ_HISTORY][k]+0.5*dak.zz)
+ dak.xy*(0.5*(ed.xy+0.5*de.xy)-ak[XY_HISTORY][k]+0.5*dak.xy));
// extra terms for 3D
if(np==THREED_MPM)
{ // internal variables
dak.xz = tmpm1*ak[XZ_HISTORY][k] + arg*(tmpp1*ed.xz + de.xz);
dak.yz = tmpm1*ak[YZ_HISTORY][k] + arg*(tmpp1*ed.yz + de.yz);
ak[XZ_HISTORY][k] += dak.xz;
ak[YZ_HISTORY][k] += dak.yz;
// stresses
dsig[XZ] -= TwoGkred[k]*dak.xz;
dsig[YZ] -= TwoGkred[k]*dak.yz;
// dissipation
dispEnergy += TwoGkred[k]*(dak.xz*(0.5*(ed.xz+0.5*de.xz)-ak[XZ_HISTORY][k]+0.5*dak.xz)
+ dak.yz*(0.5*(ed.yz+0.5*de.yz)-ak[YZ_HISTORY][k]+0.5*dak.yz));
}
}
// Update particle deviatoric stresses
Tensor *sp=mptr->GetStressTensor();
//Tensor st0 = *sp;
if(np==THREED_MPM)
{ // incremental rotate of prior stress
Matrix3 stn(sp->xx,sp->xy,sp->xz,sp->xy,sp->yy,sp->yz,sp->xz,sp->yz,sp->zz);
Matrix3 str = stn.RMRT(dR);
#ifdef USE_KIRCHOFF_STRESS
// convert sigma(n)/rho(n) to sigma(n)/rho(n+1) and add dsigma/rho(n+1)
sp->xx = detdF*str(0,0)+J*dsig[XX];
sp->yy = detdF*str(1,1)+J*dsig[YY];
sp->xy = detdF*str(0,1)+J*dsig[XY];
sp->zz = detdF*str(2,2)+J*dsig[ZZ];
sp->yz = detdF*str(1,2)+J*dsig[YZ];
sp->xz = detdF*str(0,2)+J*dsig[XZ];
#else
sp->xx = str(0,0)+dsig[XX];
sp->yy = str(1,1)+dsig[YY];
sp->xy = str(0,1)+dsig[XY];
sp->zz = str(2,2)+dsig[ZZ];
sp->yz = str(1,2)+dsig[YZ];
sp->xz = str(0,2)+dsig[XZ];
#endif
}
else
{ // incremental rotate of prior stress
Matrix3 stn(sp->xx,sp->xy,sp->xy,sp->yy,sp->zz);
Matrix3 str = stn.RMRT(dR);
#ifdef USE_KIRCHOFF_STRESS
// convert sigma(n)/rho(n) to sigma(n)/rho(n+1) and add dsigma/rho(n+1)
sp->xx = detdF*str(0,0)+J*dsig[XX];
sp->yy = detdF*str(1,1)+J*dsig[YY];
sp->xy = detdF*str(0,1)+J*dsig[XY];
sp->zz = detdF*sp->zz+J*dsig[ZZ];
#else
sp->xx = str(0,0)+dsig[XX];
sp->yy = str(1,1)+dsig[YY];
sp->xy = str(0,1)+dsig[XY];
sp->zz += dsig[ZZ];
#endif
}
// incremental work energy = shear energy (dilation and residual energy done in update pressure)
double shearEnergy = sp->xx*detot(0,0) + sp->yy*detot(1,1) + sp->zz*detot(2,2) + sp->xy*de.xy;
if(np==THREED_MPM)
{ shearEnergy += sp->xz*de.xz + sp->yz*de.yz;
}
mptr->AddWorkEnergyAndResidualEnergy(shearEnergy,0.);
// disispated energy per unit mass (dPhi/(rho0 V0)) (nJ/g)
mptr->AddPlastEnergy(dispEnergy);
// heat energy is Cv dT - dPhi
// The dPhi is subtracted here because it will show up in next
// time step within Cv dT (if adibatic heating occurs)
// The Cv dT was done in update pressure
IncrementHeatEnergy(mptr,res->dT,dTq0,dispEnergy);
}