// If needed, a material can initialize particle state
// For example, ideal gas initializes to base line pressure
// Such a class must pass on the super class after its own initializations
void MaterialBase::SetInitialParticleState(MPMBase *mptr,int np) const
{
if(isolatedSystemAndParticles)
{ // need to add initial heat energy, because special cases in this mode
// will ignore the Cv dT term
double Cv = GetHeatCapacity(mptr); // in nJ/(g-K)
mptr->AddHeatEnergy(Cv*(mptr->pTemperature-thermal.reference));
// initial entropy kept to zero, could add something here if ever needed
}
}
// Increment heat energy using Cv(dT-dTq0) - dPhi, where Cv*dTq0 + dPhi = Cv dTad is total
// dispated energy (it can by provided as either a temperature rise or an energy)
// dTq0 is temperature rise due to material mechanisms if the process was adiabatic
// dPhi is dissipated energy that is converted to temperature rise
void MaterialBase::IncrementHeatEnergy(MPMBase *mptr,double dT,double dTq0,double dPhi) const
{
double Cv = GetHeatCapacity(mptr); // in nJ/(g-K)
double dispEnergy = Cv*dTq0 + dPhi; // = Cv dTad
// Isolated means no conduction and no thermal ramp (and in future if have other ways
// to change particle temperature, those are not active either)
// In this mode, adiabatic has dq=0 and isothermal releases all as heat
if(isolatedSystemAndParticles)
{ // Here dText = 0
// If adiabatic, dq = 0 (nothing to add)
// If isothermal dq = -Cv dTad
if(!ConductionTask::adiabatic)
{ mptr->AddHeatEnergy(-dispEnergy);
mptr->AddEntropy(-dispEnergy/mptr->pPreviousTemperature);
}
}
else
{ // For non isolated particle use dq = Cv(dT-dTad)
// If adiabatic, the Cv dT term in next step will include Cv dTad from
// this step to give particle dq = 0. If isothermal, it will not and
// dq will be disspated heat
double totalHeat = Cv*dT - dispEnergy;
mptr->AddHeatEnergy(totalHeat);
if(ConductionTask::adiabatic)
{ double dTad = dispEnergy/Cv;
mptr->AddEntropy(Cv*dT/mptr->pPreviousTemperature - dispEnergy/(mptr->pPreviousTemperature+dTad));
}
else
{ mptr->AddEntropy(totalHeat/mptr->pPreviousTemperature);
}
}
// The dispated energy is added here, but only if adiabatic, in which case it will be
// converted to particle temperature rise later in the calculations. This temperature
// change works with conduction on or off
if(ConductionTask::adiabatic)
mptr->AddDispEnergy(dispEnergy);
}
// For Cp heat capacity in nJ/(g-K)
double MaterialBase::GetCpHeatCapacity(MPMBase *mptr) const { return GetHeatCapacity(mptr)+GetCpMinusCv(mptr); }
/* Take increments in strain and calculate new Particle: strains, rotation strain,
stresses, strain energy,
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 Mooney::MPMConstitutiveLaw(MPMBase *mptr,Matrix3 du,double delTime,int np,void *properties,
ResidualStrains *res,int historyOffset) const
{
// incremental energy, store initial stress
Tensor *sporig=mptr->GetStressTensor();
Tensor st0 = *sporig;
// Update strains and rotations and Left Cauchy strain
double detDf = IncrementDeformation(mptr,du,NULL,np);
// get pointer to new left Cauchy strain
Tensor *B = mptr->GetAltStrainTensor();
// 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);
// Deformation gradients and Cauchy tensor differ in plane stress
if(np==PLANE_STRESS_MPM)
{ // Find B->zz required to have zero stress in z direction
// fixed arguments
double arg = B->xx*B->yy - B->xy*B->xy;
double arg12 = sqrt(arg);
double arg16 = pow(arg,1./6.);
double arg2 = B->xx+B->yy;
// Newton-Rapheson starting at B.zz = 1
// In tests finds answer in 3 or less steps
Tensor *ep=mptr->GetStrainTensor();
double xn16,xn12,xnp1,xn = (1.+ep->zz)*(1.+ep->zz);
double fx,fxp,J13,J0,J2,Jeff;
int iter=1;
double mJ2P,mdJ2PdJ;
// solution for B.zz in xn and J = sqrt(xn*arg) with dJ/dxn = arg/(2 sqrt(xn*arg))
// Solving f=0 where f = 3J^2 Kterm + Jres*G1(2*xn-arg2)J^(1/3) + Jres*G2(xn*arg2 - 2*arg)/J^(1/3)
// where J^(1/3) = (xn*arg)^(1/6)
// df/dxn = d(3J^2 Kterm)/dJ dJ/dxn
// + Jres*G1((14*xn-arg2)/(6*xn))J^(1/3)
// + Jres*G2((5*xn*arg2+2*arg)/(6*xn))/J^(1/3)
while(iter<20)
{ xn16 = pow(xn,1./6.);
xn12 = sqrt(xn);
J13 = xn16*arg16;
J0 = xn12*arg12;
J2 = J0*J0;
Jeff = J0/Jres;
// get f and df/dxn
GetNewtonPressureTerms(Jeff, Ksp, mJ2P, mdJ2PdJ);
fx = 3.*Jres*mJ2P + G1sp*(2.*xn-arg2)*J13 + G2sp*(xn*arg2-2.*arg)/J13;
fxp = (1.5*J0/xn)*mdJ2PdJ + G1sp*J13*(14.*xn-arg2)/(6.*xn) + G2sp*(2.*arg+5.*xn*arg2)/(6.*J13*xn);
// new prediction for solution
xnp1 = xn - fx/fxp;
//cout << iter << ": " << xn << "," << xnp1 << "," << fabs(xn-xnp1) << endl;
if(fabs(xn-xnp1)<1e-10) break;
xn = xnp1;
iter+=1;
}
if(iter>=20) cout << "# Not enough iterations in plane stress Mooney-Rivlin material" << endl;
// Done and xn = new B->zz = Fzz^2 = dFzz*(old Bzz)*dFzz = dFzz^2*(old Bzz),
// and Fzz = dFzz*(old Fzz) = 1 + ep->zz
double dFzz = sqrt(xn/B->zz);
B->zz = xn;
// particle strain ezz now known
ep->zz = dFzz*(1.+ep->zz) - 1.;
// incremental J changes
detDf *= dFzz;
}
// Increment J and save it in history data
double J = detDf*mptr->GetHistoryDble(J_History,historyOffset);
mptr->SetHistoryDble(J_History,J,historyOffset);
// account for residual stresses
double Jeff = J/Jres;
// update pressure
double p0=mptr->GetPressure();
double Kterm = J*GetVolumetricTerms(Jeff,Ksp); // times J to get Kirchoff stress
// artifical viscosity
double delV = 1. - 1./detDf; // total volume change
double QAVred = 0.,AVEnergy=0.;
if(delV<0. && artificialViscosity)
{ QAVred = GetArtificalViscosity(delV/delTime,sqrt(Ksp*J),mptr);
if(ConductionTask::AVHeating) AVEnergy = fabs(QAVred*delV);
}
double Pfinal = -Kterm + QAVred;
// set the pressure
mptr->SetPressure(Pfinal);
// incremental energy - dilational part
double avgP = 0.5*(p0+Pfinal);
double dilEnergy = -avgP*delV;
// incremental residual energy
double delVres = 1. - 1./dJres;
double resEnergy = -avgP*delVres;
// Account for density change in specific stress
// i.e.. Get (Cauchy Stress)/rho = J*(Cauchy Stress)/rho0 = (Kirchoff Stress)/rho0
double J23 = pow(J, 2./3.);
double J43 = J23*J23;
double JforG1 = J23/Jres; // J^(5/3)/(Jres J) = J^(2/3)/Jres to get Kirchoff stress
double JforG2 = J43/Jres; // J^(7/3)/(Jres J) = J^(4/3)/Jres to get Kirchoff stress
//JforG1 *= Jres; // this uses Jeff to get Kirchoff stress
//JforG2 *= Jres; // this uses Jeff to get Kirchoff stress
// find deviatoric (Cauchy stress)J/rho0 = deviatoric (Kirchoff stress)/rho0
Tensor *sp=mptr->GetStressTensor();
sp->xx = (2*B->xx-B->yy-B->zz)*G1sp/(3.*JforG1)
+ (B->xx*(B->yy+B->zz)-2*B->yy*B->zz-B->xy*B->xy)*G2sp/(3.*JforG2);
sp->yy = (2*B->yy-B->xx-B->zz)*G1sp/(3.*JforG1)
+ (B->yy*(B->xx+B->zz)-2*B->xx*B->zz-B->xy*B->xy)*G2sp/(3.*JforG2);
sp->zz = (2*B->zz-B->xx-B->yy)*G1sp/(3.*JforG1)
+ (B->zz*(B->xx+B->yy)-2*B->xx*B->yy+2.*B->xy*B->xy)*G2sp/(3.*JforG2);
sp->xy = B->xy*G1sp/JforG1 + (B->zz*B->xy)*G2sp/JforG2;
if(np==THREED_MPM)
{ sp->xx += (2.*B->yz*B->yz-B->xz*B->xz)*G2sp/(3.*JforG2);
sp->yy += (2.*B->xz*B->xz-B->yz*B->yz)*G2sp/(3.*JforG2);
sp->zz -= (B->xz*B->xz+B->yz*B->yz)*G2sp/(3.*JforG2);
sp->xy -= B->xz*B->yz*G2sp/JforG2;
sp->xz = B->xz*G1sp/JforG1 + (B->yy*B->xz-B->xy*B->yz)*G2sp/JforG2;
sp->yz = B->yz*G1sp/JforG1 + (B->xx*B->yz-B->xy*B->xz)*G2sp/JforG2;
}
// incremental work energy = shear energy
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)*(du(0,1)+du(1,0)));
if(np==THREED_MPM)
{ shearEnergy += 0.5*((sp->xz+st0.xz)*(du(0,2)+du(2,0)) + (sp->yz+st0.yz)*(du(1,2)+du(2,1)));
}
// strain energy
double dU = dilEnergy + shearEnergy;
mptr->AddWorkEnergyAndResidualEnergy(dU,resEnergy);
// thermodynamics depends on whether or not this is a rubber
double dTq0;
if(rubber)
{ // convert internal energy to temperature change
dTq0 = dU/GetHeatCapacity(mptr);
}
else
{ // elastic particle isentropic temperature increment
double Kratio; // = rho_0 K/(rho K_0)
switch(UofJOption)
{ case J_MINUS_1_SQUARED:
Kratio = Jeff;
break;
case LN_J_SQUARED:
Kratio = (1-log(Jeff))/(Jeff*Jeff);
break;
case HALF_J_SQUARED_MINUS_1_MINUS_LN_J:
default:
Kratio = 0.5*(Jeff + 1./Jeff);
break;
}
dTq0 = -J*Kratio*gamma0*mptr->pPreviousTemperature*delV;
}
IncrementHeatEnergy(mptr,res->dT,dTq0,QAVred);
}
double delTime,double &dTq0,double &AVEnergy) const
#else
void Viscoelastic::UpdatePressure(MPMBase *mptr,double delV,ResidualStrains *res,double eres,const Matrix3 *dF,
double delTime,double &dTq0,double &AVEnergy) const
#endif
{
if(pressureLaw==LINEAR_PRESSURE)
{ // pressure change
#ifdef USE_KIRCHOFF_STRESS
double dP = (detdF-1.)*mptr->GetPressure()-J*Kered*delV;
#else
double dP = -Kered*delV;
#endif
// artifical viscosity
// delV is total incremental volumetric strain = total Delta(V)/V
double QAVred = 0.;
if(delV<0. && artificialViscosity)
{ // Wants K/rho
#ifdef USE_KIRCHOFF_STRESS
QAVred = GetArtificalViscosity(delV/delTime,sqrt(Kered*J));
#else
QAVred = GetArtificalViscosity(delV/delTime,sqrt(Kered));
#endif
if(ConductionTask::AVHeating) AVEnergy += fabs(QAVred*delV);
}
// increment pressure
mptr->IncrementPressure(dP+QAVred);
// work energy is dU = -P dV + s.de(total)
// Here do hydrostatic term
// Work energy increment per unit mass (dU/(rho0 V0)) (nJ/g)
double avgP = mptr->GetPressure()-0.5*dP;
mptr->AddWorkEnergyAndResidualEnergy(-avgP*delV,-3.*avgP*eres);
// Isoentropic temperature rise = -(K alpha T)/(rho Cv) (dV/V)
dTq0 += -Kered*CTE*mptr->pPreviousTemperature*(delV+3*eres)/GetHeatCapacity(mptr);;
}
else
{ // J is total volume change - may need to reference to free-swelling volume if that works
// Note that swelling looks like a problem because the sums of strains needs to be adjusted
// to stress-free state
// history pointer
double **h =(double **)(mptr->GetHistoryPtr(0));
#ifndef USE_KIRCHOFF_STRESS
// tracking J
double detdF = dF->determinant();
double J = detdF*h[mptrHistory][MGJ_HISTORY];
h[mptrHistory][MGJ_HISTORY] = J;
#endif
// account for residual strains (needed in tension, but must always track)
double dJres = exp(3.*CTE*res->dT);
double Jres = dJres*h[mptrHistory][MGJRES_HISTORY];
h[mptrHistory][MGJRES_HISTORY] = Jres;
// previous pressure
double P,P0 = mptr->GetPressure();
double delVMG = 1. - 1./detdF; // (Vnp1-Vn)/Vnp1
double Kred;
// M-G EOS
// Want specific pressure or pressure over current density (using J = rho0/rho)
if(J<1.)
{ // new compression J(k+1) = 1-x(k+1)
double x = 1.-J;
// compression law
// denominator = 1 - S1*x - S2*x^2 - S3*x^3
double denom = 1./(1. - x*(S1 + x*(S2 + x*S3)));
// law not valid if denominator passes zero
if(denom<0)
{
#pragma omp critical (output)
{ cout << "# Excessive x = " << x << endl;
mptr->Describe();
}
}
// current effective and reduced (by rho0) bulk modulus
Kred = C0squared*(1.-0.5*gamma0*x)*denom*denom;
// Pressure from bulk modulus and an energy term
double e = mptr->GetInternalEnergy();
#ifdef USE_KIRCHOFF_STRESS
P = J*(Kred*x + gamma0*e);
#else
P = Kred*x + gamma0*e;
#endif
// particle isentropic temperature increment
dTq0 += -J*gamma0*mptr->pPreviousTemperature*delVMG;
}
else
{ // In tension hyperelastic law P = - K0(J-1)
double Jeff = J/Jres;
Kred = C0squared*Jeff;
#ifdef USE_KIRCHOFF_STRESS
P = -J*C0squared*(Jeff-1.);
#else
P = -C0squared*(Jeff-1.);
#endif
// particle isentropic temperature increment
double Kratio = Jeff;
dTq0 += -J*Kratio*gamma0*mptr->pPreviousTemperature*delVMG;
}
// artifical viscosity
// delVMG is total incremental volumetric strain = total Delta(V)/V
double QAVred = 0.;
if(delVMG<0. && artificialViscosity)
{ QAVred = GetArtificalViscosity(delVMG/delTime,sqrt(Kred*J));
if(ConductionTask::AVHeating) AVEnergy += fabs(QAVred*delVMG);
}
// set final pressure
mptr->SetPressure(P+QAVred);
// work energy is dU = -P dV + s.de(total)
// Here do hydrostatic terms, deviatoric later
double avgP = 0.5*(P0+P);
double delVres = 1. - 1./dJres;
mptr->AddWorkEnergyAndResidualEnergy(-avgP*delVMG,-avgP*delVres);
}
}
