/* Take increments in strain and calculate new Particle: strains, rotation strain, plastic strain,
stresses, strain energy, plastic energy, dissipated energy, angle
dvij 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) const
{
// 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 delV = du.trace() - 3.*eres;
UpdatePressure(mptr,delV,res,eres);
// deviatoric strains increment
// Actually find 2*de to avoid many multiples by two
Tensor de;
double dV = du.trace()/3.;
de.xx = 2.*(du(0,0) - dV);
de.yy = 2.*(du(1,1) - dV);
de.zz = 2.*(du(2,2) - dV);
de.xy = du(0,1)+du(1,0);
if(np==THREED_MPM)
{ de.xz = du(0,2)+du(2,0);
de.yz = du(1,2)+du(2,1);
}
// Find initial 2*e(t) in ed
Tensor *ep=mptr->GetStrainTensor();
Tensor ed = *ep;
double thirdV = (ed.xx+ed.yy+ed.zz)/3.;
ed.xx = 2.*(ed.xx-thirdV);
ed.yy = 2.*(ed.yy-thirdV);
ed.zz = 2.*(ed.zz-thirdV);
// Increment total strains on the particle
ep->xx += du(0,0);
ep->yy += du(1,1);
ep->zz += du(2,2);
ep->xy += de.xy;
if(np==THREED_MPM)
{ ep->xz += de.xz;
ep->yz += de.yz;
}
// get internal variable increments and update them too
Tensor dak;
double **ak =(double **)(mptr->GetHistoryPtr());
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;
if(np==THREED_MPM)
{ 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;
}
}
// increment particle deviatoric stresses
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;
}
for(k=0;k<ntaus;k++)
{ dsig[XX] -= TwoGkred[k]*dak.xx;
dsig[YY] -= TwoGkred[k]*dak.yy;
dsig[ZZ] -= TwoGkred[k]*dak.zz;
dsig[XY] -= TwoGkred[k]*dak.xy;
if(np==THREED_MPM)
{ dsig[XZ] -= TwoGkred[k]*dak.xz;
dsig[YZ] -= TwoGkred[k]*dak.yz;
}
}
// Hypoelastic increment of particle deviatoric stresses
Tensor *sp=mptr->GetStressTensor();
Tensor st0 = *sp;
double dwrotxy = du(1,0)-du(0,1);
if(np==THREED_MPM)
{ double dwrotxz = du(2,0)-du(0,2);
double dwrotyz = du(2,1)-du(1,2);
Hypo3DCalculations(mptr,dwrotxy,dwrotxz,dwrotyz,dsig);
}
else
{ Hypo2DCalculations(mptr,-dwrotxy,dsig[XX],dsig[YY],dsig[XY]);
sp->zz += dsig[ZZ];
}
// incremental work energy = shear energy (dilation and residual energy done in update pressure)
double shearEnergy = 0.5*((sp->xx+st0.xx)*du(0,0) + (sp->yy+st0.yy)*du(1,1) + (sp->zz+st0.zz)*du(2,2)+
(sp->xy+st0.xy)*de.xy);
if(np==THREED_MPM)
{ shearEnergy += 0.5*((sp->xz+st0.xz)*de.xz + (sp->yz+st0.yz)*de.yz);
}
mptr->AddWorkEnergyAndResidualEnergy(shearEnergy,0.);
// disispated energy per unit mass (dPhi/(rho0 V0)) (uJ/g)
double dispEnergy = 0.;
for(k=0;k<ntaus;k++)
{ 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));
if(np==THREED_MPM)
{ 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));
}
}
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,0.,0.,dispEnergy);
}
// To allow some subclasses to support large deformations, the initial calculation for incremental
// deformation gradient (the dvij), volume change (delV) and relative volume (Jnew) can be
// handled first by the subclass. This mtehod then finishes the constitutive law
void IsoPlasticity::PlasticityConstLaw(MPMBase *mptr,double dvxx,double dvyy,double dvzz,double dvxy,double dvyx,
double dvxz,double dvzx,double dvyz,double dvzy,double delTime,int np,double delV,double Jnew,double eres,
PlasticProperties *p,ResidualStrains *res) const
{
// here dvij is total strain increment, dexxr is relative strain by subtracting off eres
double dexxr=dvxx-eres;
double deyyr=dvyy-eres;
double dezzr=dvzz-eres; // use in plane strain only
double dgxy=dvxy+dvyx;
double dgxz=dvxz+dvzx;
double dgyz=dvyz+dvzy;
// rotational strain increments (particle updated by Hypo3D)
double dwrotxy=dvyx-dvxy;
double dwrotxz=dvzx-dvxz;
double dwrotyz=dvzy-dvyz;
// allow arbitrary equation of state for pressure
UpdatePressure(mptr,delV,Jnew,np,p,res,eres);
// Elastic deviatoric stress increment
Tensor *ep=mptr->GetStrainTensor();
Tensor *sp=mptr->GetStressTensor();
Tensor stk,st0=*sp;
double dsig[6];
double thirdDelV = delV/3.;
dsig[XX] = 2.*p->Gred*(dexxr-thirdDelV);
dsig[YY] = 2.*p->Gred*(deyyr-thirdDelV);
dsig[ZZ] = 2.*p->Gred*(dezzr-thirdDelV);
dsig[YZ] = p->Gred*dgyz;
dsig[XZ] = p->Gred*dgxz;
dsig[XY] = p->Gred*dgxy;
stk.xx = st0.xx + dsig[XX];
stk.yy = st0.yy + dsig[YY];
stk.zz = st0.zz + dsig[ZZ];
stk.yz = st0.yz + dsig[YZ];
stk.xz = st0.xz + dsig[XZ];
stk.xy = st0.xy + dsig[XY];
// Calculate plastic potential f = ||s|| - sqrt(2/3)*sy(alpha,rate,...)
HardeningAlpha alpha;
plasticLaw->UpdateTrialAlpha(mptr,np,&alpha); // initialize to last value
double strial = GetMagnitudeSFromDev(&stk,np);
double ftrial = strial - SQRT_TWOTHIRDS*plasticLaw->GetYield(mptr,np,delTime,&alpha,p->hardProps);
if(ftrial<0.)
{ // elastic, update stress and strain energy as usual
// Add to total strain
ep->xx+=dvxx;
ep->yy+=dvyy;
ep->zz+=dvzz;
ep->xy+=dgxy;
ep->xz+=dgxz;
ep->yz+=dgyz;
// update stress (need to make hypoelastic)
Hypo3DCalculations(mptr,dwrotxy,dwrotxz,dwrotyz,dsig);
// work energy increment per unit mass (dU/(rho0 V0)) (uJ/g)
mptr->AddWorkEnergy(0.5*((st0.xx+sp->xx)*dvxx
+ (st0.yy+sp->yy)*dvyy
+ (st0.zz+sp->zz)*dvzz
+ (st0.yz+sp->yz)*dgyz
+ (st0.xz+sp->xz)*dgxz
+ (st0.xy+sp->xy)*dgxy));
// heat energy is Cv(dT-dTq0) - dPhi, but dPhi is zero here
// and Cv(dT-dTq0) was done in Update Pressure
// give material chance to update history variables that change in elastic updates
ElasticUpdateFinished(mptr,np,delTime);
return;
}
// Find direction of plastic strain and lambda for this plastic state
// Base class finds it numerically, subclass can override if solvable by more efficient meethods
Tensor dfds;
GetDfDsigma(strial,&stk,np,&dfds);
double lambdak = plasticLaw->SolveForLambdaBracketed(mptr,np,strial,&stk,p->Gred,1.,1.,delTime,&alpha,p->hardProps);
// Now have lambda, finish update on this particle
// Plastic strain increments on particle
double dexxp = lambdak*dfds.xx;
double deyyp = lambdak*dfds.yy;
double dezzp = lambdak*dfds.zz;
double dgxyp = 2.*lambdak*dfds.xy; // 2 for engineering plastic shear strain
double dgxzp = 2.*lambdak*dfds.xz; // 2 for engineering plastic shear strain
double dgyzp = 2.*lambdak*dfds.yz; // 2 for engineering plastic shear strain
Tensor *eplast=mptr->GetPlasticStrainTensor();
eplast->xx += dexxp;
eplast->yy += deyyp;
eplast->zz += dezzp;
eplast->xy += dgxyp;
eplast->xz += dgxzp;
eplast->yz += dgyzp;
// Elastic strain increments on particle
ep->xx += (dvxx-dexxp);
ep->yy += (dvyy-deyyp);
ep->zz += (dvzz-dezzp);
dgxy -= dgxyp;
ep->xy += dgxy;
dgxz -= dgxzp;
ep->xz += dgxz;
dgyz -= dgyzp;
ep->yz += dgyz;
// Elastic strain increment minus the residual terms by now subtracting plastic parts
dexxr -= dexxp;
deyyr -= deyyp;
dezzr -= dezzp; // plain strain only
//dgxy, dgxz, dgyz done above
// increment particle deviatoric stresses
dsig[XX] -= 2.*p->Gred*dexxp;
dsig[YY] -= 2.*p->Gred*deyyp;
dsig[ZZ] -= 2.*p->Gred*dezzp;
dsig[YZ] -= p->Gred*dgyzp;
dsig[XZ] -= p->Gred*dgxzp;
dsig[XY] -= p->Gred*dgxyp;
Hypo3DCalculations(mptr,dwrotxy,dwrotxz,dwrotyz,dsig);
// work energy increment per unit mass (dU/(rho0 V0)) (uJ/g)
double workEnergy = 0.5*((st0.xx+sp->xx)*dvxx + (st0.yy+sp->yy)*dvzz
+ (st0.zz+sp->zz)*dvzz + (st0.yz+sp->yz)*dgyz
+ (st0.xz+sp->xz)*dgxz + (st0.xy+sp->xy)*dgxy);
// plastic strain work
double plastEnergy = lambdak*(sp->xx*dfds.xx + sp->yy*dfds.yy + sp->zz*dfds.zz
+ 2.*sp->xy*dfds.xy + 2.*sp->xz*dfds.xz + 2.*sp->yz*dfds.yz);
// total work
mptr->AddWorkEnergy(plastEnergy + workEnergy);
// disispated energy per unit mass (dPhi/(rho0 V0)) (uJ/g)
double qdalphaTerm = lambdak*SQRT_TWOTHIRDS*plasticLaw->GetYieldIncrement(mptr,np,delTime,&alpha,p->hardProps);
double dispEnergy = plastEnergy - qdalphaTerm;
// heat energy is Cv(dT-dTq0) - dPhi
// The dPhi is subtracted here because it will show up in next
// time step within Cv dT (if adiabatic heating occurs)
// The Cv(dT-dTq0) was done already in update pressure
IncrementHeatEnergy(mptr,0.,0.,dispEnergy);
// The cumulative dissipated energy is tracked in plastic energy
// Setting the disp energy allows heating if mechanical energy is on
mptr->AddPlastEnergy(dispEnergy);
// update internal variables
plasticLaw->UpdatePlasticInternal(mptr,np,&alpha);
}
/* For 3D MPM analysis, take increments in strain and calculate new
Particle: strains, rotation strain, stresses, strain energy, angle
dvij are (gradient rates X time increment) to give deformation gradient change
Assumes linear elastic, uses hypoelastic correction
*/
void Elastic::MPMConstLaw(MPMBase *mptr,double dvxx,double dvyy,double dvzz,double dvxy,double dvyx,
double dvxz,double dvzx,double dvyz,double dvzy,double delTime,int np,void *properties,ResidualStrains *res) const
{
// cast pointer to material-specific data
ElasticProperties *p = (ElasticProperties *)properties;
// Add to total strain
Tensor *ep = mptr->GetStrainTensor();
ep->xx += dvxx;
ep->yy += dvyy;
ep->zz += dvzz;
double dgamxy = dvxy+dvyx;
ep->xy += dgamxy;
double dgamxz = dvxz+dvzx;
ep->xz += dgamxz;
double dgamyz = dvyz+dvzy;
ep->yz += dgamyz;
// rotational strain increments (particle updated by Hypo3D)
double dwrotxy = dvyx-dvxy;
double dwrotxz = dvzx-dvxz;
double dwrotyz = dvzy-dvyz;
// residual strains (thermal and moisture) (isotropic only)
double exxr = p->alpha[0]*res->dT;
double eyyr = p->alpha[1]*res->dT;
double ezzr = p->alpha[2]*res->dT;
double eyzr = p->alpha[3]*res->dT;
double exzr = p->alpha[4]*res->dT;
double exyr = p->alpha[5]*res->dT;
if(DiffusionTask::active)
{ exxr += p->beta[0]*res->dC;
eyyr += p->beta[1]*res->dC;
ezzr += p->beta[2]*res->dC;
eyzr += p->beta[3]*res->dC;
exzr += p->beta[4]*res->dC;
exyr += p->beta[5]*res->dC;
}
// effective strains
double dvxxeff = dvxx-exxr;
double dvyyeff = dvyy-eyyr;
double dvzzeff = dvzz-ezzr;
double dgamyzeff = dgamyz-eyzr;
double dgamxzeff = dgamxz-exzr;
double dgamxyeff = dgamxy-exyr;
// save initial stresses
Tensor *sp=mptr->GetStressTensor();
Tensor st0=*sp;
// stress increments
double delsp[6];
int i;
for(i=0;i<6;i++)
{ delsp[i] = p->C[i][0]*dvxxeff + p->C[i][1]*dvyyeff + p->C[i][2]*dvzzeff
+ p->C[i][3]*dgamyzeff + p->C[i][4]*dgamxzeff + p->C[i][5]*dgamxyeff;
}
// update stress (need to make hypoelastic)
Hypo3DCalculations(mptr,dwrotxy,dwrotxz,dwrotyz,delsp);
// work energy increment per unit mass (dU/(rho0 V0))
mptr->AddWorkEnergyAndResidualEnergy(
0.5*((st0.xx+sp->xx)*dvxx + (st0.yy+sp->yy)*dvyy
+ (st0.zz+sp->zz)*dvzz + (st0.yz+sp->yz)*dgamyz
+ (st0.xz+sp->xz)*dgamxz + (st0.xy+sp->xy)*dgamxy),
0.5*((st0.xx+sp->xx)*exxr + (st0.yy+sp->yy)*eyyr
+ (st0.zz+sp->zz)*ezzr + (st0.yz+sp->yz)*eyzr
+ (st0.xz+sp->xz)*exzr + (st0.xy+sp->xy)*exyr));
// track heat energy
IncrementHeatEnergy(mptr,res->dT,0.,0.);
}
