// decompose current F to get rotation
// apply initial rotation to account for aniostropy
Matrix3 MPMBase::GetRtot(void)
{
// get deformation gradient
Matrix3 pF = GetDeformationGradientMatrix();
// Decompose for R
Matrix3 R;
Matrix3 U = pF.RightDecompose(&R,NULL);
// apply initial rotation total rotation from initial material axes
Matrix3 R0 = GetInitialRotation();
Matrix3 Rtot = R*R0;
// return it
return Rtot;
}
// Apply Constitutive law, check np to know what type
void TaitLiquid::MPMConstitutiveLaw(MPMBase *mptr,Matrix3 du,double delTime,int np,void *properties,ResidualStrains *res) const
{
#ifdef NO_SHEAR_MODEL
// get incremental deformation gradient
const Matrix3 dF = du.Exponential(incrementalDefGradTerms);
// decompose dF into dR and dU
Matrix3 dR;
Matrix3 dU = dF.RightDecompose(&dR,NULL);
// current deformation gradient
double detdF = dF.determinant();
Matrix3 pF = mptr->GetDeformationGradientMatrix();
Matrix3 F = dR*pF;
if(np==THREED_MPM)
F.Scale(pow(detdF,1./3.));
else
F.Scale2D(sqrt(detdF));
// new deformation matrix with volume change onle
mptr->SetDeformationGradientMatrix(F);
#else
// Update total deformation gradient, and calculate trial B
double detdF = IncrementDeformation(mptr,du,NULL,np);
#endif
// Get new J and save result on the particle
double J = detdF * mptr->GetHistoryDble(J_history);
mptr->SetHistoryDble(J_history,J);
// account for residual stresses
double dresStretch,resStretch = GetResidualStretch(mptr,dresStretch,res);
double Jres = resStretch*resStretch*resStretch;
double Jeff = J/Jres;
// new Kirchhoff pressure (over rho0) from Tait equation
double p0=mptr->GetPressure();
double pressure = J*TAIT_C*Ksp*(exp((1.-Jeff)/TAIT_C)-1.);
mptr->SetPressure(pressure);
// incremental energy per unit mass - dilational part
double avgP = 0.5*(p0+pressure);
double delV = 1. - 1./detdF;
double workEnergy = -avgP*delV;
// incremental residual energy per unit mass
double delVres = 1. - 1./(dresStretch*dresStretch*dresStretch);
double resEnergy = -avgP*delVres;
// viscosity term = 2 eta (0.5(grad v) + 0.5*(grad V)^T - (1/3) tr(grad v) I)
// (i.e., divatoric part of the symmetric strain tensor, 2 is for conversion to engineering shear strain)
Matrix3 shear;
double c[3][3];
c[0][0] = (2.*du(0,0)-du(1,1)-du(2,2))/3.;
c[1][1] = (2.*du(1,1)-du(0,0)-du(2,2))/3.;
c[2][2] = (2.*du(2,2)-du(0,0)-du(1,1))/3.;
c[0][1] = 0.5*(du(0,1)+du(1,0));
c[1][0] = c[0][1];
if(np==THREED_MPM)
{ c[0][2] = 0.5*(du(0,2)+du(2,0));
c[2][0] = c[0][2];
c[1][2] = 0.5*(du(1,2)+du(2,1));
c[2][1] = c[1][2];
shear.set(c);
}
else
shear.set(c[0][0],c[0][1],c[1][0],c[1][1],c[2][2]);
// Get Kirchoff shear stress (over rho0)
shear.Scale(J*TwoEtasp/delTime);
// update deviatoric stress
Tensor *sp=mptr->GetStressTensor();
sp->xx = shear(0,0);
sp->yy = shear(1,1);
sp->zz = shear(2,2);
sp->xy = shear(0,1);
if(np==THREED_MPM)
{ sp->xz = shear(0,2);
sp->yz = shear(1,2);
}
// shear work per unit mass = tau.du = tau.tau*delTime/TwoEtasp
double shearWork = sp->xx*sp->xx + sp->yy*sp->yy + sp->zz*sp->zz + 2.*sp->xy*sp->xy;
if(np==THREED_MPM) shearWork += 2.*(sp->xz*sp->xz + sp->yz*sp->yz);
shearWork *= delTime/TwoEtasp;
mptr->AddWorkEnergyAndResidualEnergy(workEnergy+shearWork,resEnergy);
// particle isentropic temperature increment dT/T = - J (K/K0) gamma0 Delta(V)/V
// Delta(V)/V = 1. - 1/detdF (total volume)
double Kratio = Jeff*(1.+pressure/(TAIT_C*Ksp));
double dTq0 = -J*Kratio*gamma0*mptr->pPreviousTemperature*delV;
// heat energy is Cv (dT - dTq0) -dPhi
// Here do Cv (dT - dTq0)
// dPhi = shearWork is lost due to shear term
IncrementHeatEnergy(mptr,res->dT,dTq0,shearWork);
}
// Apply Constitutive law, check np to know what type
void TaitLiquid::MPMConstitutiveLaw(MPMBase *mptr,Matrix3 du,double delTime,int np,void *properties,
ResidualStrains *res,int historyOffset) const
{
#ifdef NO_SHEAR_MODEL
// get incremental deformation gradient
const Matrix3 dF = du.Exponential(incrementalDefGradTerms);
// decompose dF into dR and dU
Matrix3 dR;
Matrix3 dU = dF.RightDecompose(&dR,NULL);
// current deformation gradient
double detdF = dF.determinant();
Matrix3 pF = mptr->GetDeformationGradientMatrix();
Matrix3 F = dR*pF;
if(np==THREED_MPM)
F.Scale(pow(detdF,1./3.));
else
F.Scale2D(sqrt(detdF));
// new deformation matrix with volume change onle
mptr->SetDeformationGradientMatrix(F);
#else
#ifdef ELASTIC_B_MODEL
// get incremental deformation gradient
const Matrix3 dF = du.Exponential(incrementalDefGradTerms);
double detdF = dF.determinant();
// current deformation gradient
Matrix3 pF = mptr->GetDeformationGradientMatrix();
// new deformation matrix
const Matrix3 F = dF*pF;
mptr->SetDeformationGradientMatrix(F);
#else
// Update total deformation gradient, and calculate trial B
double detdF = IncrementDeformation(mptr,du,NULL,np);
#endif
#endif
// Get new J and save result on the particle
double J = detdF * mptr->GetHistoryDble(J_History,historyOffset);
mptr->SetHistoryDble(J_History,J,historyOffset);
#ifdef ELASTIC_B_MODEL
// store pressure strain as elastic B
Tensor *pB = mptr->GetAltStrainTensor() ;
if(np==THREED_MPM || np==AXISYMMETRIC_MPM)
{ double J23 = pow(J,2./3.);
pB->xx = J23;
pB->yy = J23;
pB->zz = J23;
}
else
{ pB->xx = J;
pB->yy = J;
}
#endif
// account for residual stresses
double dJres = GetIncrementalResJ(mptr,res);
double Jres = dJres * mptr->GetHistoryDble(J_History+1,historyOffset);
mptr->SetHistoryDble(J_History+1,Jres,historyOffset);
double Jeff = J/Jres;
// new Kirchhoff pressure (over rho0) from Tait equation
double p0=mptr->GetPressure();
double pressure = J*TAIT_C*Ksp*(exp((1.-Jeff)/TAIT_C)-1.);
mptr->SetPressure(pressure);
// incremental energy per unit mass - dilational part
double avgP = 0.5*(p0+pressure);
double delV = 1. - 1./detdF;
double workEnergy = -avgP*delV;
// incremental residual energy per unit mass
double delVres = 1. - 1./dJres;
double resEnergy = -avgP*delVres;
// viscosity term = 2 eta (0.5(grad v) + 0.5*(grad V)^T - (1/3) tr(grad v) I) = 2 eta dev(grad v)
// (i.e., deviatoric part of the symmetric strain tensor, 2 is for conversion to engineering shear strain)
// simple shear rate = |2 dev(grad v)|
Matrix3 shear;
double c[3][3];
double shearRate;
c[0][0] = (2.*du(0,0)-du(1,1)-du(2,2))/3.;
c[1][1] = (2.*du(1,1)-du(0,0)-du(2,2))/3.;
c[2][2] = (2.*du(2,2)-du(0,0)-du(1,1))/3.;
c[0][1] = 0.5*(du(0,1)+du(1,0));
c[1][0] = c[0][1];
shearRate = c[0][0]*c[0][0] + c[1][1]*c[1][1] + c[2][2]*c[2][2]
+ 2.*c[0][1]*c[0][1];
if(np==THREED_MPM)
{ c[0][2] = 0.5*(du(0,2)+du(2,0));
c[2][0] = c[0][2];
c[1][2] = 0.5*(du(1,2)+du(2,1));
c[2][1] = c[1][2];
shearRate += 2.*(c[0][2]*c[0][2] + c[1][2]*c[1][2]);
shear.set(c);
}
else
shear.set(c[0][0],c[0][1],c[1][0],c[1][1],c[2][2]);
shearRate = 2.*sqrt(shearRate)/delTime;
// Store shear rate
mptr->SetHistoryDble(J_History+2,shearRate,historyOffset);
// Get effective visocisy
double twoetaspRate = 0.;
if(numViscosity==1)
{ twoetaspRate = TwoEtasp[0];
}
else
{ shearRate = log10(shearRate);
if(shearRate < logShearRate[0])
twoetaspRate = TwoEtasp[0];
else if(shearRate > logShearRate[numViscosity-1])
twoetaspRate = TwoEtasp[numViscosity-1];
else
{ // interpolate
for(int i=1;i<numViscosity;i++)
{ if(shearRate <= logShearRate[i])
{ // between i-1 and i
double fract = (logShearRate[i]-shearRate)/(logShearRate[i]-logShearRate[i-1]);
twoetaspRate = fract*TwoEtasp[i-1] + (1.-fract)*TwoEtasp[i];
break;
}
}
}
}
// Get Kirchoff shear stress (over rho0)
shear.Scale(J*twoetaspRate/delTime);
// update deviatoric stress
Tensor *sp=mptr->GetStressTensor();
sp->xx = shear(0,0);
sp->yy = shear(1,1);
sp->zz = shear(2,2);
sp->xy = shear(0,1);
if(np==THREED_MPM)
{ sp->xz = shear(0,2);
sp->yz = shear(1,2);
}
// shear work per unit mass = tau.du = tau.tau*delTime/twoetaspRate
double shearWork = sp->xx*sp->xx + sp->yy*sp->yy + sp->zz*sp->zz + 2.*sp->xy*sp->xy;
if(np==THREED_MPM) shearWork += 2.*(sp->xz*sp->xz + sp->yz*sp->yz);
shearWork *= delTime/twoetaspRate;
mptr->AddWorkEnergyAndResidualEnergy(workEnergy+shearWork,resEnergy);
// particle isentropic temperature increment dT/T = - J (K/K0) gamma0 Delta(V)/V
// Delta(V)/V = 1. - 1/detdF (total volume)
double Kratio = Jeff*(1.+pressure/(TAIT_C*Ksp));
double dTq0 = -J*Kratio*gamma0*mptr->pPreviousTemperature*delV;
// heat energy is Cv (dT - dTq0) -dPhi
// Here do Cv (dT - dTq0)
// dPhi = shearWork is lost due to shear term
IncrementHeatEnergy(mptr,res->dT,dTq0,shearWork);
}
// Entry point for large rotation
void Elastic::LRConstitutiveLaw(MPMBase *mptr,Matrix3 du,double delTime,int np,void *properties,ResidualStrains *res) const
{
// get previous deformation gradient
Matrix3 pFnm1 = mptr->GetDeformationGradientMatrix();
// get incremental deformation gradient and decompose it
const Matrix3 dF = du.Exponential(incrementalDefGradTerms);
Matrix3 dR;
Matrix3 dU = dF.RightDecompose(&dR,NULL);
// get pinitial rotation R0
Matrix3 R0 = mptr->GetInitialRotation();
// get previous rotation and stretch
Matrix3 Rnm1;
Matrix3 Unm1 = pFnm1.RightDecompose(&Rnm1,NULL);
// get strain increments de = R0T.[(Rnm1T.dU.Rnm1 - I).Unm1].R0
Matrix3 dUrot = dU.RTMR(Rnm1);
dUrot(0,0) -= 1.;
dUrot(1,1) -= 1.;
dUrot(2,2) -= 1.;
dUrot *= Unm1;
// apply initial rotation to get strain increment in the material coordinates
Matrix3 de = dUrot.RTMR(R0);
Matrix3 Rtotnm1M3 = Rnm1*R0;
Matrix3 *Rtotnm1 = &Rtotnm1M3;
// get total rotation
Matrix3 Rtot = dR*Rtotnm1M3;
if(np==THREED_MPM) mptr->SetRtot(Rtot);
// Update total deformation gradient
Matrix3 pF = dF*pFnm1;
mptr->SetDeformationGradientMatrix(pF);
// cast pointer to material-specific data
ElasticProperties *p = GetElasticPropertiesPointer(properties);
// residual strains (thermal and moisture) in material axes
double exxr,eyyr,ezzr;
if(np==THREED_MPM)
{ exxr = p->alpha[0]*res->dT;
eyyr = p->alpha[1]*res->dT;
ezzr = p->alpha[2]*res->dT;
if(DiffusionTask::active)
{ exxr += p->beta[0]*res->dC;
eyyr += p->beta[1]*res->dC;
ezzr += p->beta[2]*res->dC;
}
}
else
{ exxr = p->alpha[1]*res->dT;
eyyr = p->alpha[2]*res->dT;
ezzr = p->alpha[4]*res->dT;
if(DiffusionTask::active)
{ exxr += p->beta[1]*res->dC;
eyyr += p->beta[2]*res->dC;
ezzr += p->beta[4]*res->dC;
}
}
Matrix3 er = Matrix3(exxr,0.,0.,eyyr,ezzr);
// finish up
LRElasticConstitutiveLaw(mptr,de,er,Rtot,dR,Rtotnm1,np,properties,res);
}