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NickelSuperalloy.f
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NickelSuperalloy.f
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C Christos Skamniotis
C University of Oxford
C December 2021
C
C Simplified Double exponent slip rule and empirical creep law for tertiary creep:
subroutine NickelSuperalloy(xNorm,xDir,tau,signtau,tauc,
+ burgerv,dtime,nSys,iphase,CurrentTemperature,Lp,
+ tmat,gammaDot, cubicslip,creep, usvars, nsvars)
implicit none
! number of slip system
integer, intent(in):: nSys
! activation flag for cubic slip (additional 6 systems activated when loading is along 111)
INTEGER,intent(in) :: cubicslip
! activation flag for tertiary creep
INTEGER,intent(in) :: creep
! phase
integer, intent(in):: iphase
! number of Abaqus state variables
INTEGER,intent(in) :: nsvars
! slip directions and normals
real*8, intent(in) :: xNorm(nSys,3),xDir(nSys,3)
! resolved shear stress and critical resolved shear stress
! and sign of the resolved shear stress
! tauc is positive by definition
real*8, intent(in) :: tau(nSys), tauc(nSys), signtau(nSys)
! Burgers vectors
real*8, intent(in) :: burgerv(nSys)
! time step
real*8, intent(in) :: dtime
! Temperature in Kelvin
real*8, intent(in) :: CurrentTemperature
! Abaqus state variables
REAL*8,intent(in) :: usvars(nsvars)
! plastic velocity gradient
real*8, intent(out) :: Lp(3,3)
! and its derivative with respect to the stress
real*8, intent(out) :: tmat(6,6)
! plastic strain rate on each slip system
real*8, intent(out) :: gammaDot(nSys)
! Boltzmann constant (J/K)
real*8, parameter :: kB = 1.38e-23
! Gas constant (J*mol/K)
real*8, parameter :: R = 8.314462
******************************************
** The following parameters must be set **
c
*** RATE DEPENDENT PLASTICITY (thermally activated glide)
! Activation energy for octahedral slip (J)
real*8, parameter :: Foctahedral = 9.39e-19
! Activation energy for cubic slip (J)
real*8, parameter :: Fcubic = 1.17e-18
! reference strain rate (1/s)
real*8, parameter :: gammadot0 = 1.0e7 ! real 1.0e-7
! rate sensitivity exponents
real*8, parameter :: p = 0.78 ! real 0.78
real*8, parameter :: q = 1.15 ! real 1.15
c
*** TERTIARY CREEP (dislocation climb & damage)
!!!! Initial creep rate constants
! Activation energy for creep (J/mol)
real*8, parameter :: Qo = 460000.0
! reference rate (1/s)
real*8, parameter :: ao = 4.0e8
! stress multiplier (1/MPa)
real*8, parameter :: bo = 3.2e-2
!!!! Climb/damage constants
! Activation energy for damage (J/mol)
real*8, parameter :: QD = 340000.0
! reference rate (1/s)
real*8, parameter :: aD = 6000000.0
! stress multiplier (1/MPa)
real*8, parameter :: bD = -5.0e-08
!!!! Rafting
C real*8, parameter :: SS = 100
C real*8, parameter :: TT = 1000
C real*8, parameter :: QQ = 20000
C real*8, parameter :: m = -3
** End of parameters to set **
******************************************
! slip system index
integer :: i
! Schmid tensor and its transpose
real*8 :: SNij(3,3), NSij(3,3)
! Schmid tensor and its transpose in Voigt notation
real*8 :: sni(6), nsi(6)
! higher order Schmid tensor in Voigt notation
real*8 :: SNNS(6,6)
! temporary slip normal and slip direction
real*8 :: tempNorm(3), tempDir(3)
! temporary variable to calculate the Jacobian
real*8 :: result1
! Jacobian
real*8 :: result4(6,6)
! activation energy
real*8 :: dF
! RSS/CRSS ratio
real*8 :: tau_ratio
C
C
C *** CALCULATE LP AND THE DERIVATIVE OF PLASTIC STRAIN INCREMENT WITH
C RESPECT TO THE STRESS DEFINED AS tmat***
C
C
tmat = 0.0
Lp = 0.0
result4 = 0.0
! contribution to Lp of all slip systems
do i=1,nSys
tau_ratio=tau(i)/tauc(i)
if (tau_ratio >= 0.0) then
if (tau_ratio >= 1.0) then ! avoid negative values before elevating to power q
gammaDot(i) = signtau(i)*gammadot0 !*exp(-dF/(kB*CurrentTemperature))
else ! standard case
dF=Foctahedral
if (i .gt. 12) then
dF=Fcubic
end if
! strain rate due to thermally activated glide (rate dependent plasticity)
gammaDot(i) = signtau(i)*gammadot0*
+ exp(-(dF/(kB*CurrentTemperature))*(1- tau_ratio**p)**q)
end if
! add tertiary creep strain rate
if (creep == 1) then
gammaDot(i) = gammaDot(i) +
+ signtau(i)*ao*exp(bo*tau(i) -
+ Qo/(R*CurrentTemperature)) +
+ signtau(i)*abs(usvars(89+i))*aD*exp(bD*tau(i) -
+ QD/(R*CurrentTemperature))
end if
tempNorm = xNorm(i,:)
tempDir = xDir(i,:)
SNij = spread(tempDir,2,3)*spread(tempNorm,1,3)
NSij = spread(tempNorm,2,3)*spread(tempDir,1,3)
call KGMATVEC6(SNij,sni)
call KGMATVEC6(NSij,nsi)
SNNS = spread(sni,2,6)*spread(nsi,1,6)
! calculate derivative d ( gammaDot(i) ) / d ( tau(i) )
result1 = abs(gammaDot(i))
result1 = result1 * dF/(kB*CurrentTemperature)
result1 = result1 * q
result1 = result1 * (1- tau_ratio**p)**(q-1.0)
result1 = result1 * p
result1 = result1 / tauc(i)
result1 = result1 * tau_ratio**(p-1.0)
if (creep == 1) then
result1 = result1 + ao*bo*
+ exp(bo*tau(i)-Qo/(R*CurrentTemperature)) +
+ abs(usvars(89+i))*aD*bD*
+ exp(bD*tau(i)-QD/(R*CurrentTemperature))
end if
! contribution to Jacobian
result4 = result4 + dtime*result1*SNNS
! plastic velocity gradient contribution
Lp = Lp + gammaDot(i)*SNij
else
gammaDot(i) = 0.0
end if
end do
tmat = 0.5*(result4+transpose(result4))
return
end