in kinetics: non schmid stresses only influence peierls mechanism, but not solid solution hardening; as a result the derivative of the velocity with respect to the resolved stress has to be split into a Schmid and a non-Schmid part
This commit is contained in:
Christoph Kords
parent
ec377a6e8e
commit
4f8664baa3
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@ -1545,7 +1545,8 @@ endsubroutine
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!*********************************************************************
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!* calculates kinetics *
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!*********************************************************************
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subroutine constitutive_nonlocal_kinetics(v, dv_dtau, tau, tauThreshold, c, Temperature, g, ip, el)
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subroutine constitutive_nonlocal_kinetics(v, dv_dtau, dv_dtauNS, tau, tauNS, &
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tauThreshold, c, Temperature, g, ip, el)
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use debug, only: debug_level, &
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debug_constitutive, &
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@ -1567,23 +1568,23 @@ integer(pInt), intent(in) :: g, & !< curre
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c !< dislocation character (1:edge, 2:screw)
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real(pReal), intent(in) :: Temperature !< temperature
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real(pReal), dimension(totalNslip(phase_plasticityInstance(material_phase(g,ip,el)))), &
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intent(in) :: tau, & !< resolved external shear stress (for bcc this already contains non Schmid effects)
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intent(in) :: tau, & !< resolved external shear stress (without non Schmid effects)
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tauNS, & !< resolved external shear stress (including non Schmid effects)
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tauThreshold !< threshold shear stress
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!*** output variables
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real(pReal), dimension(totalNslip(phase_plasticityInstance(material_phase(g,ip,el)))), &
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intent(out) :: v !< velocity
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real(pReal), dimension(totalNslip(phase_plasticityInstance(material_phase(g,ip,el)))), &
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intent(out) :: dv_dtau !< velocity derivative with respect to resolved shear stress
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intent(out) :: v, & !< velocity
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dv_dtau, & !< velocity derivative with respect to resolved shear stress (without non Schmid contributions)
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dv_dtauNS !< velocity derivative with respect to resolved shear stress (including non Schmid contributions)
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!*** local variables
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integer(pInt) :: instance, & !< current instance of this plasticity
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ns, & !< short notation for the total number of active slip systems
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s !< index of my current slip system
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real(pReal), dimension(totalNslip(phase_plasticityInstance(material_phase(g,ip,el)))) :: &
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tauEff !< effective shear stress
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real(pReal) tauRel_P, &
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tauRel_S, &
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tauEff, & !< effective shear stress
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tPeierls, & !< waiting time in front of a peierls barriers
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tSolidSolution, & !< waiting time in front of a solid solution obstacle
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vViscous, & !< viscous glide velocity
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@ -1607,30 +1608,31 @@ real(pReal) tauRel_P, &
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instance = phase_plasticityInstance(material_phase(g,ip,el))
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ns = totalNslip(instance)
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tauEff = abs(tau) - tauThreshold
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v = 0.0_pReal
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dv_dtau = 0.0_pReal
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dv_dtauNS = 0.0_pReal
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if (Temperature > 0.0_pReal) then
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do s = 1_pInt,ns
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if (tauEff(s) > 0.0_pReal) then
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if (abs(tau(s)) > tauThreshold(s)) then
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!* Peierls contribution
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!* Effective stress includes non Schmid constributions
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!* The derivative only gives absolute values; the correct sign is taken care of in the formula for the derivative of the velocity
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tauEff = max(0.0_pReal, abs(tauNS(s)) - tauThreshold(s)) ! ensure that the effective stress is positive
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meanfreepath_P = burgers(s,instance)
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jumpWidth_P = burgers(s,instance)
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activationLength_P = doublekinkwidth(instance) * burgers(s,instance)
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activationVolume_P = activationLength_P * jumpWidth_P * burgers(s,instance)
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criticalStress_P = peierlsStress(s,c,instance)
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activationEnergy_P = criticalStress_P * activationVolume_P
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tauRel_P = min(1.0_pReal, tauEff(s) / criticalStress_P) ! ensure that the activation probability cannot become greater than one
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tauRel_P = min(1.0_pReal, tauEff / criticalStress_P) ! ensure that the activation probability cannot become greater than one
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tPeierls = 1.0_pReal / fattack(instance) &
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* exp(activationEnergy_P / (KB * Temperature) &
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* (1.0_pReal - tauRel_P**pParam(instance))**qParam(instance))
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if (tauEff(s) < criticalStress_P) then
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if (tauEff < criticalStress_P) then
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dtPeierls_dtau = tPeierls * pParam(instance) * qParam(instance) * activationVolume_P / (KB * Temperature) &
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* (1.0_pReal - tauRel_P**pParam(instance))**(qParam(instance)-1.0_pReal) &
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* tauRel_P**(pParam(instance)-1.0_pReal)
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@ -1642,17 +1644,18 @@ if (Temperature > 0.0_pReal) then
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!* Contribution from solid solution strengthening
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!* The derivative only gives absolute values; the correct sign is taken care of in the formula for the derivative of the velocity
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tauEff = abs(tau(s)) - tauThreshold(s)
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meanfreepath_S = burgers(s,instance) / sqrt(solidSolutionConcentration(instance))
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jumpWidth_S = solidSolutionSize(instance) * burgers(s,instance)
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activationLength_S = burgers(s,instance) / sqrt(solidSolutionConcentration(instance))
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activationVolume_S = activationLength_S * jumpWidth_S * burgers(s,instance)
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activationEnergy_S = solidSolutionEnergy(instance)
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criticalStress_S = activationEnergy_S / activationVolume_S
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tauRel_S = min(1.0_pReal, tauEff(s) / criticalStress_S) ! ensure that the activation probability cannot become greater than one
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tauRel_S = min(1.0_pReal, tauEff / criticalStress_S) ! ensure that the activation probability cannot become greater than one
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tSolidSolution = 1.0_pReal / fattack(instance) &
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* exp(activationEnergy_S / (KB * Temperature) &
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* (1.0_pReal - tauRel_S**pParam(instance))**qParam(instance))
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if (tauEff(s) < criticalStress_S) then
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if (tauEff < criticalStress_S) then
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dtSolidSolution_dtau = tSolidSolution * pParam(instance) * qParam(instance) &
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* activationVolume_S / (KB * Temperature) &
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* (1.0_pReal - tauRel_S**pParam(instance))**(qParam(instance)-1.0_pReal) &
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@ -1664,8 +1667,9 @@ if (Temperature > 0.0_pReal) then
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!* viscous glide velocity
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tauEff = abs(tau(s)) - tauThreshold(s)
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mobility = burgers(s,instance) / viscosity(instance)
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vViscous = mobility * tauEff(s)
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vViscous = mobility * tauEff
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!* Mean velocity results from waiting time at peierls barriers and solid solution obstacles with respective meanfreepath of
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@ -1674,11 +1678,9 @@ if (Temperature > 0.0_pReal) then
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v(s) = sign(1.0_pReal,tau(s)) &
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/ (tPeierls / meanfreepath_P + tSolidSolution / meanfreepath_S + 1.0_pReal / vViscous)
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dv_dtau(s) = v(s) * v(s) &
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* (dtPeierls_dtau / meanfreepath_P &
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+ dtSolidSolution_dtau / meanfreepath_S &
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+ 1.0_pReal / (mobility * tauEff(s)*tauEff(s)))
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dv_dtau(s) = v(s) * v(s) * (dtSolidSolution_dtau / meanfreepath_S &
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+ mobility / (vViscous * vViscous))
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dv_dtauNS(s) = v(s) * v(s) * dtPeierls_dtau / meanfreepath_P
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endif
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enddo
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endif
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@ -1691,10 +1693,12 @@ endif
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write(6,*)
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write(6,'(a,i8,1x,i2,1x,i1)') '<< CONST >> nonlocal_kinetics at el ip g',el,ip,g
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write(6,*)
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write(6,'(a,/,12x,12(f12.5,1x))') '<< CONST >> tauThreshold / MPa', tauThreshold / 1e6_pReal
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write(6,'(a,/,12x,12(f12.5,1x))') '<< CONST >> tau / MPa', tau / 1e6_pReal
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write(6,'(a,/,12x,12(f12.5,1x))') '<< CONST >> tauEff / MPa', tauEff / 1e6_pReal
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write(6,'(a,/,12x,12(f12.5,1x))') '<< CONST >> tauNS / MPa', tauNS / 1e6_pReal
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write(6,'(a,/,12x,12(f12.5,1x))') '<< CONST >> v / 1e-3m/s', v * 1e3
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write(6,'(a,/,12x,12(e12.5,1x))') '<< CONST >> dv_dtau', dv_dtau
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write(6,'(a,/,12x,12(e12.5,1x))') '<< CONST >> dv_dtauNS', dv_dtauNS
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endif
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#endif
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@ -1760,10 +1764,11 @@ real(pReal), dimension(totalNslip(phase_plasticityInstance(material_phase(g,ip,e
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rhoSgl !< single dislocation densities (including blocked)
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real(pReal), dimension(totalNslip(phase_plasticityInstance(material_phase(g,ip,el))),4) :: &
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v, & !< velocity
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tau, & !< resolved shear stress including non Schmid and backstress terms
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dgdot_dtau, & !< derivative of the shear rate with respect to the shear stress
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dv_dtau !< velocity derivative with respect to the shear stress
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tauNS, & !< resolved shear stress including non Schmid and backstress terms
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dv_dtau, & !< velocity derivative with respect to the shear stress
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dv_dtauNS !< velocity derivative with respect to the shear stress
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real(pReal), dimension(totalNslip(phase_plasticityInstance(material_phase(g,ip,el)))) :: &
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tau, & !< resolved shear stress including backstress terms
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gdotTotal, & !< shear rate
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tauBack, & !< back stress from dislocation gradients on same slip system
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tauThreshold !< threshold shear stress
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@ -1799,26 +1804,31 @@ tauThreshold = state%p(iTauF(1:ns,myInstance))
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do s = 1_pInt,ns
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sLattice = slipSystemLattice(s,myInstance)
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tau(s,1:2) = math_mul6x6(Tstar_v, lattice_Sslip_v(1:6,1,sLattice,myStructure))
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if (tau(s,1) > 0.0_pReal) then
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tau(s,3) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,1,s,myInstance))
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tau(s,4) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,3,s,myInstance))
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tau(s) = math_mul6x6(Tstar_v, lattice_Sslip_v(1:6,1,sLattice,myStructure))
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tauNS(s,1) = tau(s)
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tauNS(s,2) = tau(s)
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if (tau(s) > 0.0_pReal) then
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tauNS(s,3) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,1,s,myInstance))
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tauNS(s,4) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,3,s,myInstance))
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else
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tau(s,3) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,2,s,myInstance))
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tau(s,4) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,4,s,myInstance))
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tauNS(s,3) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,2,s,myInstance))
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tauNS(s,4) = math_mul33xx33(math_Mandel6to33(Tstar_v), nonSchmidProjection(1:3,1:3,4,s,myInstance))
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endif
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forall (t = 1_pInt:4_pInt) &
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tau(s,t) = tau(s,t) + tauBack(s) ! add backstress
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enddo
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forall (t = 1_pInt:4_pInt) &
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tauNS(1:ns,t) = tauNS(1:ns,t) + tauBack ! add backstress
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tau = tau + tauBack ! add backstress
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!*** get dislocation velocity and its tangent and store the velocity in the state array
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! edges
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call constitutive_nonlocal_kinetics(v(1:ns,1), dv_dtau(1:ns,1), tau(1:ns,1), tauThreshold(1:ns), &
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call constitutive_nonlocal_kinetics(v(1:ns,1), dv_dtau(1:ns,1), dv_dtauNS(1:ns,1), &
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tau(1:ns), tauNS(1:ns,1), tauThreshold(1:ns), &
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1_pInt, Temperature, g, ip, el)
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v(1:ns,2) = v(1:ns,1)
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dv_dtau(1:ns,2) = dv_dtau(1:ns,1)
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dv_dtauNS(1:ns,2) = dv_dtauNS(1:ns,1)
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state%p(iV(1:ns,2,myInstance)) = v(1:ns,1)
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!screws
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@ -1826,11 +1836,13 @@ if (lattice_NnonSchmid(myStructure) == 0_pInt) then
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forall(t = 3_pInt:4_pInt)
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v(1:ns,t) = v(1:ns,1)
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dv_dtau(1:ns,t) = dv_dtau(1:ns,1)
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dv_dtauNS(1:ns,t) = dv_dtauNS(1:ns,1)
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state%p(iV(1:ns,t,myInstance)) = v(1:ns,1)
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endforall
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else ! take non-Schmid contributions into account
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do t = 3_pInt,4_pInt
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call constitutive_nonlocal_kinetics(v(1:ns,t), dv_dtau(1:ns,t), tau(1:ns,t), tauThreshold(1:ns), &
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call constitutive_nonlocal_kinetics(v(1:ns,t), dv_dtau(1:ns,t), dv_dtauNS(1:ns,t), &
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tau(1:ns), tauNS(1:ns,t), tauThreshold(1:ns), &
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2_pInt , Temperature, g, ip, el)
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state%p(iV(1:ns,t,myInstance)) = v(1:ns,t)
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enddo
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@ -1843,32 +1855,36 @@ forall (s = 1_pInt:ns, t = 5_pInt:8_pInt, rhoSgl(s,t) * v(s,t-4_pInt) < 0.0_pRea
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rhoSgl(s,t-4_pInt) = rhoSgl(s,t-4_pInt) + abs(rhoSgl(s,t))
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!*** Calculation of gdot and its tangent
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gdotTotal = sum(rhoSgl(1:ns,1:4) * v, 2) * burgers(1:ns,myInstance)
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forall (t = 1_pInt:4_pInt) &
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dgdot_dtau(1:ns,t) = rhoSgl(1:ns,t) * dv_dtau(1:ns,t) * burgers(1:ns,myInstance)
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!*** Calculation of Lp and its tangent
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gdotTotal = sum(rhoSgl(1:ns,1:4) * v, 2) * burgers(1:ns,myInstance)
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do s = 1_pInt,ns
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sLattice = slipSystemLattice(s,myInstance)
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Lp = Lp + gdotTotal(s) * lattice_Sslip(1:3,1:3,1,sLattice,myStructure)
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if (tau(s,1) > 0.0_pReal) then
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! Schmid contributions to tangent
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forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt,k=1_pInt:3_pInt,l=1_pInt:3_pInt) &
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dLp_dTstar3333(i,j,k,l) = dLp_dTstar3333(i,j,k,l) &
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+ lattice_Sslip(i,j,1,sLattice,myStructure) * lattice_Sslip(k,l,1,sLattice,myStructure) &
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* sum(rhoSgl(s,1:4) * dv_dtau(s,1:4)) * burgers(s,myInstance)
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! non Schmid contributions to tangent
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if (tau(s) > 0.0_pReal) then
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forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt,k=1_pInt:3_pInt,l=1_pInt:3_pInt) &
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dLp_dTstar3333(i,j,k,l) = dLp_dTstar3333(i,j,k,l) &
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+ dgdot_dtau(s,1) * lattice_Sslip(i,j,1,sLattice,myStructure) * lattice_Sslip(k,l,1,sLattice,myStructure) &
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+ dgdot_dtau(s,2) * lattice_Sslip(i,j,1,sLattice,myStructure) * lattice_Sslip(k,l,1,sLattice,myStructure) &
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+ dgdot_dtau(s,3) * lattice_Sslip(i,j,1,sLattice,myStructure) * nonSchmidProjection(k,l,1,s,myInstance) &
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+ dgdot_dtau(s,4) * lattice_Sslip(i,j,1,sLattice,myStructure) * nonSchmidProjection(k,l,3,s,myInstance)
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+ lattice_Sslip(i,j,1,sLattice,myStructure) &
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* ( nonSchmidProjection(k,l,1,s,myInstance) * rhoSgl(s,3) * dv_dtauNS(s,3) &
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+ nonSchmidProjection(k,l,3,s,myInstance) * rhoSgl(s,4) * dv_dtauNS(s,4) ) &
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* burgers(s,myInstance)
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else
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forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt,k=1_pInt:3_pInt,l=1_pInt:3_pInt) &
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dLp_dTstar3333(i,j,k,l) = dLp_dTstar3333(i,j,k,l) &
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+ dgdot_dtau(s,1) * lattice_Sslip(i,j,1,sLattice,myStructure) * lattice_Sslip(k,l,1,sLattice,myStructure) &
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+ dgdot_dtau(s,2) * lattice_Sslip(i,j,1,sLattice,myStructure) * lattice_Sslip(k,l,1,sLattice,myStructure) &
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+ dgdot_dtau(s,3) * lattice_Sslip(i,j,1,sLattice,myStructure) * nonSchmidProjection(k,l,2,s,myInstance) &
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+ dgdot_dtau(s,4) * lattice_Sslip(i,j,1,sLattice,myStructure) * nonSchmidProjection(k,l,4,s,myInstance)
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+ lattice_Sslip(i,j,1,sLattice,myStructure) &
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* ( nonSchmidProjection(k,l,2,s,myInstance) * rhoSgl(s,3) * dv_dtauNS(s,3) &
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+ nonSchmidProjection(k,l,4,s,myInstance) * rhoSgl(s,4) * dv_dtauNS(s,4) ) &
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* burgers(s,myInstance)
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endif
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enddo
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dLp_dTstar99 = math_Plain3333to99(dLp_dTstar3333)
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@ -2331,11 +2347,11 @@ rhoDotMultiplication = 0.0_pReal
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if (myStructure == 2_pInt) then ! BCC
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forall (s = 1:ns, sum(abs(v(s,1:4))) > 0.0_pReal)
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rhoDotMultiplication(s,1:2) = sum(abs(gdot(s,3:4))) / burgers(s,myInstance) & ! assuming double-cross-slip of screws to be decisive for multiplication
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* sqrt(rhoForest(s)) / lambda0(s,myInstance) !& ! mean free path
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!* sum(abs(v(s,3:4))) / sum(abs(v(s,1:4))) ! ratio of screw to overall velocity determines edge generation
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* sqrt(rhoForest(s)) / lambda0(s,myInstance) ! & ! mean free path
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! * 2.0_pReal * sum(abs(v(s,3:4))) / sum(abs(v(s,1:4))) ! ratio of screw to overall velocity determines edge generation
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rhoDotMultiplication(s,3:4) = sum(abs(gdot(s,3:4))) / burgers(s,myInstance) & ! assuming double-cross-slip of screws to be decisive for multiplication
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* sqrt(rhoForest(s)) / lambda0(s,myInstance) !& ! mean free path
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!* sum(abs(v(s,1:2))) / sum(abs(v(s,1:4))) ! ratio of edge to overall velocity determines screw generation
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* sqrt(rhoForest(s)) / lambda0(s,myInstance) ! & ! mean free path
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! * 2.0_pReal * sum(abs(v(s,1:2))) / sum(abs(v(s,1:4))) ! ratio of edge to overall velocity determines screw generation
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endforall
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else ! ALL OTHER STRUCTURES
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