// Increment small strain material deformation gradient using F(n) = (I + grad u)F(n-1)
void Elastic::HypoIncrementDeformation(MPMBase *mptr,Matrix3 du) const
{
// get incremental deformation gradient
const Matrix3 dF = du.Exponential(1);
// current deformation gradient
Matrix3 pF = mptr->GetDeformationGradientMatrix();
// new deformation matrix
const Matrix3 F = dF*pF;
mptr->SetDeformationGradientMatrix(F);
}
// 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);
}
/* Given matrix of incremental deformation dF = exp(dt*grad v), increment particle strain,
rotation, and LeftCauchy Green strain (latter is assumed to be stored in the particle's
plastic strain tensor (which is accessed also with GetAltStrainTensor().
New new F is dF.F, which is used to find new strain
New B = dF.(Old B).dF^T
Returns |dF|
*/
double HyperElastic::IncrementDeformation(MPMBase *mptr,Matrix3 du,Tensor *Btrial,int np) const
{
// get incremental deformation gradient
const Matrix3 dF = du.Exponential(incrementalDefGradTerms);
// current deformation gradient
Matrix3 pF = mptr->GetDeformationGradientMatrix();
// new deformation matrix
const Matrix3 F = dF*pF;
mptr->SetDeformationGradientMatrix(F);
// increment Left Cauchy tensor B = F.F^T = dF.old B.dF^T
// plain stress will need to update B.zz when known
Matrix3 pBold = mptr->GetElasticLeftCauchyMatrix();
// elements of dF.B
Matrix3 dFoldB = dF*pBold;
// return trial B (if provided) or store new B on the particle
Tensor *pB = Btrial!=NULL ? Btrial : mptr->GetAltStrainTensor() ;
pB->xx = dFoldB(0,0)*dF(0,0) + dFoldB(0,1)*dF(0,1) + dFoldB(0,2)*dF(0,2);
pB->xy = dFoldB(0,0)*dF(1,0) + dFoldB(0,1)*dF(1,1) + dFoldB(0,2)*dF(1,2);
pB->yy = dFoldB(1,0)*dF(1,0) + dFoldB(1,1)*dF(1,1) + dFoldB(1,2)*dF(1,2);
pB->zz = dFoldB(2,0)*dF(2,0) + dFoldB(2,1)*dF(2,1) + dFoldB(2,2)*dF(2,2);
if(np == THREED_MPM)
{ pB->xz = dFoldB(0,0)*dF(2,0) + dFoldB(0,1)*dF(2,1) + dFoldB(0,2)*dF(2,2);
pB->yz = dFoldB(1,0)*dF(2,0) + dFoldB(1,1)*dF(2,1) + dFoldB(1,2)*dF(2,2);
}
// return |dF|
return dF.determinant();
}
// 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);
}
/* 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);
}
// 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);
}
// 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);
}