/* 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 dvxy,double dvyx,
double dvzz,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; // trial dexx=dvxx
double deyyr = dvyy-eres; // trial deyy=dvyy
double dezzr = dvzz-eres; // In plane strain trial dezz=0, dvzz=0 except in axisymmetric
double dgxy = dvxy+dvyx; // no need to substract residual strain
double dwrotxy = dvyx-dvxy;
// allow arbitrary equation of state for pressure
double P0 = mptr->GetPressure();
UpdatePressure(mptr,delV,Jnew,np,p,res,eres);
double Pfinal = mptr->GetPressure();
// Deviatoric stress increment
Tensor *ep=mptr->GetStrainTensor();
Tensor *sp=mptr->GetStressTensor();
Tensor dels,stk,st0=*sp;
double thirdDelV = delV/3.;
dels.xx = 2.*p->Gred*(dexxr-thirdDelV);
dels.yy = 2.*p->Gred*(deyyr-thirdDelV);
if(np==PLANE_STRESS_MPM)
dels.zz = Pfinal-P0;
else
dels.zz = 2.*p->Gred*(dezzr-thirdDelV);
dels.xy = p->Gred*dgxy;
// trial deviatoric stress
stk.xx = st0.xx + dels.xx;
stk.yy = st0.yy + dels.yy;
stk.zz = st0.zz + dels.zz;
stk.xy = st0.xy + dels.xy;
// Calculate plastic potential f = ||s|| - sqrt(2/3)*sy(alpha,rate,...)
HardeningAlpha alpha;
plasticLaw->UpdateTrialAlpha(mptr,np,&alpha); // initialize to last value and zero plastic strain rate
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 input strain increment to elastic strain on particle
ep->xx += dvxx;
ep->yy += dvyy;
ep->xy += dgxy;
// increment deviatoric stress (Units N/m^2 cm^3/g) (pressure not needed here)
Hypo2DCalculations(mptr,-dwrotxy,dels.xx,dels.yy,dels.xy);
// work energy increment per unit mass (dU/(rho0 V0)) (by midpoint rule) (uJ/g)
// energy units are also Pa cm^3/g, i.e., same as stress units
sp->zz = stk.zz;
if(np==AXISYMMETRIC_MPM)
{ ep->zz += dvzz;
mptr->AddWorkEnergy(0.5*((st0.xx+sp->xx)*dvxx + (st0.yy+sp->yy)*dvyy
+ (st0.xy+sp->xy)*dgxy + (st0.zz+sp->zz)*dvzz));
}
else
{ mptr->AddWorkEnergy(0.5*((st0.xx+sp->xx)*dvxx + (st0.yy+sp->yy)*dvyy
+ (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 subclass material chance to update history variables that change in elastic updates
ElasticUpdateFinished(mptr,np,delTime);
return;
}
// Find lambda for this plastic state
// Base class finds it numerically, subclass can override if solvable by more efficient methods
double lambdak = plasticLaw->SolveForLambdaBracketed(mptr,np,strial,&stk,p->Gred,p->psKred,Pfinal,delTime,&alpha,p->hardProps);
// Now have lambda, finish update on this particle
Tensor dfds;
if(np==PLANE_STRESS_MPM)
{ // get final stress state
double d1 = (1 + p->psKred*lambdak);
double d2 = (1.+2.*p->Gred*lambdak);
double n1 = (stk.xx+stk.yy-2.*Pfinal)/d1;
double n2 = (-stk.xx+stk.yy)/d2;
double sxx = (n1-n2)/2.;
double syy = (n1+n2)/2.;
double txy = stk.xy/d2;
// find increment in deviatoric stress
dels.xx = sxx+Pfinal-st0.xx;
dels.yy = syy+Pfinal-st0.yy;
dels.xy = txy-st0.xy;
// get final direction
dfds.xx = (2.*sxx-syy)/3.;
dfds.yy = (2.*syy-sxx)/3.;
dfds.zz = -(dfds.xx+dfds.yy);
dfds.xy = txy; // tensorial shear strain
// If fully plastic, the increment in deviatoric stress should be zero
// sxx = sigmaxx+Pfinal = st0.xx, syy = sigmayy+Pfinal = st0.yy, szz = Pfinal
// But first two must be wrong because trace is no longer zero?
//
// This is probably wrong
// i.e.: n1 + 2*Pfinal = st0.xx+st0.yy
// (stk.xx+stk.yy)/d1 - 2*Pfinal*(1/d1-1) = st0.xx+st0.yy
// But (stk.xx+stk.yy) = (st0.xx+st0.yy) + 2.*Gred*(dexxr+deyyr-2*thirdDelV)
// (st0.xx+st0.yy)(1/d1-1) - 2*Pfinal*(1/d1-1) = -2.*Gred*(dexxr+deyyr-2*thirdDelV)/d1
// (st0.xx+st0.yy-2*Pfinal)*(1-d1) = -2.*Gred*(dexxr+deyyr-2*thirdDelV)
// B*Kred*lam = 2.*Gred*A or lam = 2.*Gred*A/(B*Kred)
// where B = st0.xx+st0.yy-2*Pfinal, and A = (dexxr+deyyr-2*thirdDelV)
// Then d1 = 1+2*Gred*A/B, n1 = ((st0.xx+st0.yy) + 2*Gred*A - 2*Pfinal)/d1 = (B+2*Gred*A)*B/(B+2*Gred*A) = B
//
// Also expect syy-sxx = sigmayy-sigmaxx = st0.yy-st0.xx = n2
// (stk.yy-stk.xx)/d2 = st0.yy-st0.xx
// (st0.yy-st0.xx)/d2 + 2*Gred*(deyyr-dexxr)/d2 = (st0.yy-st0.xx)
// (st0.yy-st0.xx)(d2-1) = - 2*Gred*(deyyr-dexxr)
// lam = (deyyr-dexxr)/(st0.yy-st0.xx)
// Then n2 = (st0.yy-st0.xx + 2*Gred*(deyyr-dexxr))/(1+2*Gred*(deyyr-dexxr)/(st0.yy-st0.xx)) = st0.yy-st0.xx
//
// But these to lam's seem to differ?
// We can write
}
else
{ // get final direction
GetDfDsigma(strial,&stk,np,&dfds);
}
// 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
Tensor *eplast=mptr->GetPlasticStrainTensor();
eplast->xx += dexxp;
eplast->yy += deyyp;
eplast->xy += dgxyp;
eplast->zz += dezzp;
// Elastic strain increments on particle
ep->xx += (dvxx-dexxp);
ep->yy += (dvyy-deyyp);
dgxy -= dgxyp;
ep->xy += dgxy;
if(np==PLANE_STRESS_MPM)
ep->zz += eres - p->psLr2G*(dexxr+deyyr+dezzp);
else
{ // plain strain and axisymmetric when plastic increments is dezzp
// In plain strain, dvzz=0 elastic increment is -dezzp to zz strain total is zero
// Axisymmetry can have non-zero hoop strain
ep->zz += (dvzz-dezzp);
dezzr -= dezzp;
}
// Elastic strain increment minus the residual terms by now subtracting plastic parts
dexxr -= dexxp;
deyyr -= deyyp;
//dgxy -= dgxyp; // done above
//dezzr -= dezzp; // plane strain and axisymmetry done above, plain stress not needed
// increment particle deviatoric stresses (plane stress increments found above)
if(np!=PLANE_STRESS_MPM)
{ dels.xx -= 2.*p->Gred*dexxp;
dels.yy -= 2.*p->Gred*deyyp;
dels.xy -= p->Gred*dgxyp;
dels.zz -= 2.*p->Gred*dezzp;
sp->zz += dels.zz;
}
else
sp->zz = Pfinal; // now equal to Pfinal
Hypo2DCalculations(mptr,-dwrotxy,dels.xx,dels.yy,dels.xy);
// Elastic work increment per unit mass (dU/(rho0 V0)) (uJ/g)
double workEnergy = 0.5*((st0.xx+sp->xx)*dvxx
+ (st0.yy+sp->yy)*dvyy
+ (st0.xy+sp->xy)*dgxy);
if(np==AXISYMMETRIC_MPM)
{ workEnergy += 0.5*(st0.zz+sp->zz)*dvzz;
}
// plastic strain work
double plastEnergy = lambdak*(sp->xx*dfds.xx + sp->yy*dfds.yy + sp->zz*dfds.zz + 2.*sp->xy*dfds.xy);
// 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 adibatic heating occurs)
// The Cv(dT-dTq0) was done in update pressure
IncrementHeatEnergy(mptr,0.,0.,dispEnergy);
// The cumulative dissipated energy is tracked in plastic energy
mptr->AddPlastEnergy(dispEnergy);
// update internal variables
plasticLaw->UpdatePlasticInternal(mptr,np,&alpha);
}
/* For 2D 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
For Axisymmetric MPM, x->R, y->Z, x->theta, and dvzz is change in hoop strain
(i.e., du/r on particle and dvzz will be zero if not axisymmetric)
*/
void Elastic::MPMConstLaw(MPMBase *mptr,double dvxx,double dvyy,double dvxy,double dvyx,double dvzz,
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;
double dgam=dvxy+dvyx;
ep->xy+=dgam;
double dwrotxy=dvyx-dvxy;
// residual strains (thermal and moisture)
double erxx = p->alpha[1]*res->dT;
double eryy = p->alpha[2]*res->dT;
double erxy = p->alpha[3]*res->dT;
double erzz = CTE3*res->dT;
if(DiffusionTask::active)
{ erxx += p->beta[1]*res->dC;
eryy += p->beta[2]*res->dC;
erxy += p->beta[3]*res->dC;
erzz += CME3*res->dC;
}
// moisture and thermal strain and temperature change
// (when diffusion, conduction, OR thermal ramp active)
double dvxxeff = dvxx - erxx;
double dvyyeff = dvyy - eryy;
double dgameff = dgam - erxy;
// save initial stresses
Tensor *sp=mptr->GetStressTensor();
Tensor st0=*sp;
// find stress (Units N/m^2 cm^3/g)
// this does xx, yy, ans xy only. zz do later if needed
double c1,c2,c3;
if(np==AXISYMMETRIC_MPM)
{ // axisymmetric strain
ep->zz += dvzz;
// hoop stress affect on RR, ZZ, and RZ stresses
double dvzzeff = dvzz - erzz;
c1=p->C[1][1]*dvxxeff + p->C[1][2]*dvyyeff + p->C[4][1]*dvzzeff + p->C[1][3]*dgameff;
c2=p->C[1][2]*dvxxeff + p->C[2][2]*dvyyeff + p->C[4][2]*dvzzeff + p->C[2][3]*dgameff;
c3=p->C[1][3]*dvxxeff + p->C[2][3]*dvyyeff + p->C[4][3]*dvzzeff + p->C[3][3]*dgameff;
}
else
{ c1=p->C[1][1]*dvxxeff + p->C[1][2]*dvyyeff + p->C[1][3]*dgameff;
c2=p->C[1][2]*dvxxeff + p->C[2][2]*dvyyeff + p->C[2][3]*dgameff;
c3=p->C[1][3]*dvxxeff + p->C[2][3]*dvyyeff + p->C[3][3]*dgameff;
}
Hypo2DCalculations(mptr,-dwrotxy,c1,c2,c3);
// work and resdiaul strain energu increments
double workEnergy = 0.5*((st0.xx+sp->xx)*dvxx + (st0.yy+sp->yy)*dvyy + (st0.xy+sp->xy)*dgam);
double resEnergy = 0.5*((st0.xx+sp->xx)*erxx + (st0.yy+sp->yy)*eryy + (st0.xy+sp->xy)*erxy);
if(np==PLANE_STRAIN_MPM)
{ // need to add back terms to get from reduced cte to actual cte
sp->zz += p->C[4][1]*(dvxx+p->alpha[5]*erzz)+p->C[4][2]*(dvyy+p->alpha[6]*erzz)
+p->C[4][3]*(dgam+p->alpha[7]*erzz)-p->C[4][4]*erzz;
// extra residual energy increment per unit mass (dU/(rho0 V0)) (by midpoint rule) (uJ/g)
resEnergy += 0.5*(st0.zz+sp->zz)*erzz;
}
else if(np==PLANE_STRESS_MPM)
{ ep->zz += p->C[4][1]*dvxxeff+p->C[4][2]*dvyyeff+p->C[4][3]*dgameff+erzz;
}
else
{ // axisymmetric hoop stress
sp->zz += p->C[4][1]*dvxxeff + p->C[4][2]*dvyyeff + p->C[4][4]*(dvzz - erzz) + p->C[4][3]*dgameff;
// extra work and residual energy increment per unit mass (dU/(rho0 V0)) (by midpoint rule) (uJ/g)
workEnergy += 0.5*(st0.zz+sp->zz)*dvzz;
resEnergy += 0.5*(st0.zz+sp->zz)*erzz;
}
mptr->AddWorkEnergyAndResidualEnergy(workEnergy, resEnergy);
// track heat energy
IncrementHeatEnergy(mptr,res->dT,0.,0.);
}
