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second order expansion of temperature-dependent variables

This commit is contained in:
Martin Diehl 2022-01-26 07:24:29 +01:00
parent e7a999ffb1
commit 5267794ff2
1 changed files with 37 additions and 22 deletions
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59
src/phase_mechanical_plastic_dislotwin.f90
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@ -25,14 +25,14 @@ submodule(phase:plastic) dislotwin
xi_sb = 1.0_pReal, & !< value for shearband resistance xi_sb = 1.0_pReal, & !< value for shearband resistance
v_sb = 1.0_pReal, & !< value for shearband velocity_0 v_sb = 1.0_pReal, & !< value for shearband velocity_0
E_sb = 1.0_pReal, & !< activation energy for shear bands E_sb = 1.0_pReal, & !< activation energy for shear bands
delta_G = 1.0_pReal, & !< free energy difference between austensite and martensite
i_tr = 1.0_pReal, & !< adjustment parameter to calculate MFP for transformation i_tr = 1.0_pReal, & !< adjustment parameter to calculate MFP for transformation
h = 1.0_pReal, & !< stack height of hex nucleus h = 1.0_pReal, & !< stack height of hex nucleus
T_ref = T_ROOM, & T_ref = T_ROOM, &
a_cI = 1.0_pReal, & a_cI = 1.0_pReal, &
a_cF = 1.0_pReal a_cF = 1.0_pReal
real(pReal), dimension(2) :: & real(pReal), dimension(3) :: &
Gamma_sf = 0.0_pReal Gamma_sf = 0.0_pReal, & !< stacking fault energy
Delta_G = 0.0_pReal !< free energy difference between austensite and martensite
real(pReal), allocatable, dimension(:) :: & real(pReal), allocatable, dimension(:) :: &
b_sl, & !< absolute length of Burgers vector [m] for each slip system b_sl, & !< absolute length of Burgers vector [m] for each slip system
b_tw, & !< absolute length of Burgers vector [m] for each twin system b_tw, & !< absolute length of Burgers vector [m] for each twin system
@ -303,7 +303,9 @@ module function plastic_dislotwin_init() result(myPlasticity)
prm%h = pl%get_asFloat('h', defaultVal=0.0_pReal) ! ToDo: This is not optional! prm%h = pl%get_asFloat('h', defaultVal=0.0_pReal) ! ToDo: This is not optional!
prm%i_tr = pl%get_asFloat('i_tr', defaultVal=0.0_pReal) ! ToDo: This is not optional! prm%i_tr = pl%get_asFloat('i_tr', defaultVal=0.0_pReal) ! ToDo: This is not optional!
prm%delta_G = pl%get_asFloat('delta_G') prm%Delta_G(1) = pl%get_asFloat('Delta_G')
prm%Delta_G(2) = pl%get_asFloat('Delta_G,T', defaultVal=0.0_pReal)
prm%Delta_G(3) = pl%get_asFloat('Delta_G,T^2',defaultVal=0.0_pReal)
prm%x_c_tr = pl%get_asFloat('x_c_tr', defaultVal=0.0_pReal) ! ToDo: This is not optional! prm%x_c_tr = pl%get_asFloat('x_c_tr', defaultVal=0.0_pReal) ! ToDo: This is not optional!
prm%L_tr = pl%get_asFloat('L_tr') prm%L_tr = pl%get_asFloat('L_tr')
prm%a_cI = pl%get_asFloat('a_cI', defaultVal=0.0_pReal) prm%a_cI = pl%get_asFloat('a_cI', defaultVal=0.0_pReal)
@ -363,6 +365,7 @@ module function plastic_dislotwin_init() result(myPlasticity)
prm%T_ref = pl%get_asFloat('T_ref') prm%T_ref = pl%get_asFloat('T_ref')
prm%Gamma_sf(1) = pl%get_asFloat('Gamma_sf') prm%Gamma_sf(1) = pl%get_asFloat('Gamma_sf')
prm%Gamma_sf(2) = pl%get_asFloat('Gamma_sf,T',defaultVal=0.0_pReal) prm%Gamma_sf(2) = pl%get_asFloat('Gamma_sf,T',defaultVal=0.0_pReal)
prm%Gamma_sf(3) = pl%get_asFloat('Gamma_sf,T^2',defaultVal=0.0_pReal)
end if end if
slipAndTwinActive: if (prm%sum_N_sl * prm%sum_N_tw > 0) then slipAndTwinActive: if (prm%sum_N_sl * prm%sum_N_tw > 0) then
@ -559,7 +562,7 @@ module subroutine dislotwin_LpAndItsTangent(Lp,dLp_dMp,Mp,ph,en)
+ ddot_gamma_dtau_sl(i) * prm%P_sl(k,l,i) * prm%P_sl(m,n,i) + ddot_gamma_dtau_sl(i) * prm%P_sl(k,l,i) * prm%P_sl(m,n,i)
end do slipContribution end do slipContribution
call kinetics_tw(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tw,ddot_gamma_dtau_tw) if (prm%sum_N_tw > 0) call kinetics_tw(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tw,ddot_gamma_dtau_tw)
twinContibution: do i = 1, prm%sum_N_tw twinContibution: do i = 1, prm%sum_N_tw
Lp = Lp + dot_gamma_tw(i)*prm%P_tw(1:3,1:3,i) Lp = Lp + dot_gamma_tw(i)*prm%P_tw(1:3,1:3,i)
forall (k=1:3,l=1:3,m=1:3,n=1:3) & forall (k=1:3,l=1:3,m=1:3,n=1:3) &
@ -567,7 +570,7 @@ module subroutine dislotwin_LpAndItsTangent(Lp,dLp_dMp,Mp,ph,en)
+ ddot_gamma_dtau_tw(i)* prm%P_tw(k,l,i)*prm%P_tw(m,n,i) + ddot_gamma_dtau_tw(i)* prm%P_tw(k,l,i)*prm%P_tw(m,n,i)
end do twinContibution end do twinContibution
call kinetics_tr(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tr,ddot_gamma_dtau_tr) if (prm%sum_N_tr > 0) call kinetics_tr(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tr,ddot_gamma_dtau_tr)
transContibution: do i = 1, prm%sum_N_tr transContibution: do i = 1, prm%sum_N_tr
Lp = Lp + dot_gamma_tr(i)*prm%P_tr(1:3,1:3,i) Lp = Lp + dot_gamma_tr(i)*prm%P_tr(1:3,1:3,i)
forall (k=1:3,l=1:3,m=1:3,n=1:3) & forall (k=1:3,l=1:3,m=1:3,n=1:3) &
@ -641,13 +644,12 @@ module subroutine dislotwin_dotState(Mp,T,ph,en)
real(pReal) :: & real(pReal) :: &
mu, & mu, &
nu, & nu, &
Gamma Gamma_sf
associate(prm => param(ph), stt => state(ph), dot => dotState(ph), dst => dependentState(ph)) associate(prm => param(ph), stt => state(ph), dot => dotState(ph), dst => dependentState(ph))
mu = elastic_mu(ph,en) mu = elastic_mu(ph,en)
nu = elastic_nu(ph,en) nu = elastic_nu(ph,en)
Gamma = prm%Gamma_sf(1) + prm%Gamma_sf(2) * (T-prm%T_ref)
f_matrix = 1.0_pReal & f_matrix = 1.0_pReal &
- sum(stt%f_tw(1:prm%sum_N_tw,en)) & - sum(stt%f_tw(1:prm%sum_N_tw,en)) &
@ -674,8 +676,11 @@ module subroutine dislotwin_dotState(Mp,T,ph,en)
dot_rho_dip_climb(i) = 0.0_pReal dot_rho_dip_climb(i) = 0.0_pReal
else else
! Argon & Moffat, Acta Metallurgica, Vol. 29, pg 293 to 299, 1981 ! Argon & Moffat, Acta Metallurgica, Vol. 29, pg 293 to 299, 1981
Gamma_sf = prm%Gamma_sf(1) &
+ prm%Gamma_sf(2) * (T-prm%T_ref) &
+ prm%Gamma_sf(3) * (T-prm%T_ref)**2
sigma_cl = dot_product(prm%n0_sl(1:3,i),matmul(Mp,prm%n0_sl(1:3,i))) sigma_cl = dot_product(prm%n0_sl(1:3,i),matmul(Mp,prm%n0_sl(1:3,i)))
b_d = merge(24.0_pReal*PI*(1.0_pReal - nu)/(2.0_pReal + nu) * Gamma / (mu*prm%b_sl(i)), & b_d = merge(24.0_pReal*PI*(1.0_pReal - nu)/(2.0_pReal + nu) * Gamma_sf / (mu*prm%b_sl(i)), &
1.0_pReal, & 1.0_pReal, &
prm%ExtendedDislocations) prm%ExtendedDislocations)
v_cl = 2.0_pReal*prm%omega*b_d**2*exp(-prm%Q_cl/(K_B*T)) & v_cl = 2.0_pReal*prm%omega*b_d**2*exp(-prm%Q_cl/(K_B*T)) &
@ -695,10 +700,10 @@ module subroutine dislotwin_dotState(Mp,T,ph,en)
- 2.0_pReal*prm%d_caron/prm%b_sl * stt%rho_dip(:,en)*abs(dot_gamma_sl) & - 2.0_pReal*prm%d_caron/prm%b_sl * stt%rho_dip(:,en)*abs(dot_gamma_sl) &
- dot_rho_dip_climb - dot_rho_dip_climb
call kinetics_tw(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tw) if (prm%sum_N_tw > 0) call kinetics_tw(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tw)
dot%f_tw(:,en) = f_matrix*dot_gamma_tw/prm%gamma_char dot%f_tw(:,en) = f_matrix*dot_gamma_tw/prm%gamma_char
call kinetics_tr(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tr) if (prm%sum_N_tr > 0) call kinetics_tr(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tr)
dot%f_tr(:,en) = f_matrix*dot_gamma_tr dot%f_tr(:,en) = f_matrix*dot_gamma_tr
end associate end associate
@ -718,7 +723,8 @@ module subroutine dislotwin_dependentState(T,ph,en)
T T
real(pReal) :: & real(pReal) :: &
sumf_tw,Gamma,sumf_tr Gamma_sf, Delta_G, &
sumf_tw, sumf_tr
real(pReal), dimension(param(ph)%sum_N_sl) :: & real(pReal), dimension(param(ph)%sum_N_sl) :: &
inv_lambda_sl inv_lambda_sl
real(pReal), dimension(param(ph)%sum_N_tw) :: & real(pReal), dimension(param(ph)%sum_N_tw) :: &
@ -739,7 +745,12 @@ module subroutine dislotwin_dependentState(T,ph,en)
sumf_tw = sum(stt%f_tw(1:prm%sum_N_tw,en)) sumf_tw = sum(stt%f_tw(1:prm%sum_N_tw,en))
sumf_tr = sum(stt%f_tr(1:prm%sum_N_tr,en)) sumf_tr = sum(stt%f_tr(1:prm%sum_N_tr,en))
Gamma = prm%Gamma_sf(1) + prm%Gamma_sf(2) * (T-prm%T_ref) Gamma_sf = prm%Gamma_sf(1) &
+ prm%Gamma_sf(2) * (T-prm%T_ref) &
+ prm%Gamma_sf(3) * (T-prm%T_ref)**2
Delta_G = prm%Delta_G(1) &
+ prm%Delta_G(2) * (T-prm%T_ref) &
+ prm%Delta_G(3) * (T-prm%T_ref)**2
!* rescaled volume fraction for topology !* rescaled volume fraction for topology
f_over_t_tw = stt%f_tw(1:prm%sum_N_tw,en)/prm%t_tw ! this is per system ... f_over_t_tw = stt%f_tw(1:prm%sum_N_tw,en)/prm%t_tw ! this is per system ...
@ -763,11 +774,11 @@ module subroutine dislotwin_dependentState(T,ph,en)
dst%tau_pass(:,en) = mu*prm%b_sl* sqrt(matmul(prm%h_sl_sl,stt%rho_mob(:,en)+stt%rho_dip(:,en))) dst%tau_pass(:,en) = mu*prm%b_sl* sqrt(matmul(prm%h_sl_sl,stt%rho_mob(:,en)+stt%rho_dip(:,en)))
!* threshold stress for growing twin/martensite !* threshold stress for growing twin/martensite
dst%tau_hat_tw(:,en) = Gamma/(3.0_pReal*prm%b_tw) & dst%tau_hat_tw(:,en) = Gamma_sf/(3.0_pReal*prm%b_tw) &
+ 3.0_pReal*prm%b_tw*mu/(prm%L_tw*prm%b_tw) + 3.0_pReal*prm%b_tw*mu/(prm%L_tw*prm%b_tw)
dst%tau_hat_tr(:,en) = Gamma/(3.0_pReal*prm%b_tr) & dst%tau_hat_tr(:,en) = Gamma_sf/(3.0_pReal*prm%b_tr) &
+ 3.0_pReal*prm%b_tr*mu/(prm%L_tr*prm%b_tr) & + 3.0_pReal*prm%b_tr*mu/(prm%L_tr*prm%b_tr) &
+ prm%h*prm%delta_G/(3.0_pReal*prm%b_tr) + prm%h*Delta_G/(3.0_pReal*prm%b_tr)
end associate end associate
@ -936,7 +947,7 @@ pure subroutine kinetics_tw(Mp,T,dot_gamma_sl,ph,en,&
tau, tau_r, & tau, tau_r, &
dot_N_0, & dot_N_0, &
x0, V, & x0, V, &
Gamma, & Gamma_sf, &
mu, nu, & mu, nu, &
P_ncs, dP_ncs_dtau, & P_ncs, dP_ncs_dtau, &
P, dP_dtau P, dP_dtau
@ -950,11 +961,13 @@ pure subroutine kinetics_tw(Mp,T,dot_gamma_sl,ph,en,&
isFCC: if (prm%fccTwinTransNucleation) then isFCC: if (prm%fccTwinTransNucleation) then
mu = elastic_mu(ph,en) mu = elastic_mu(ph,en)
nu = elastic_nu(ph,en) nu = elastic_nu(ph,en)
Gamma = prm%Gamma_sf(1) + prm%Gamma_sf(2) * (T-prm%T_ref) Gamma_sf = prm%Gamma_sf(1) &
+ prm%Gamma_sf(2) * (T-prm%T_ref) &
+ prm%Gamma_sf(3) * (T-prm%T_ref)**2
do i = 1, prm%sum_N_tw do i = 1, prm%sum_N_tw
tau = math_tensordot(Mp,prm%P_tw(1:3,1:3,i)) tau = math_tensordot(Mp,prm%P_tw(1:3,1:3,i))
x0 = mu*prm%b_tw(i)**2*(2.0_pReal+nu)/(Gamma*8.0_pReal*PI*(1.0_pReal-nu)) ! ToDo: In the paper, the Burgers vector for slip is used x0 = mu*prm%b_tw(i)**2*(2.0_pReal+nu)/(Gamma_sf*8.0_pReal*PI*(1.0_pReal-nu)) ! ToDo: In the paper, the Burgers vector for slip is used
tau_r = mu*prm%b_tw(i)/(2.0_pReal*PI)*(1.0_pReal/(x0+prm%x_c_tw)+cos(PI/3.0_pReal)/x0) ! ToDo: In the paper, the Burgers vector for slip is used tau_r = mu*prm%b_tw(i)/(2.0_pReal*PI)*(1.0_pReal/(x0+prm%x_c_tw)+cos(PI/3.0_pReal)/x0) ! ToDo: In the paper, the Burgers vector for slip is used
if (tau > tol_math_check .and. tau < tau_r) then if (tau > tol_math_check .and. tau < tau_r) then
@ -1026,7 +1039,7 @@ pure subroutine kinetics_tr(Mp,T,dot_gamma_sl,ph,en,&
tau, tau_r, & tau, tau_r, &
dot_N_0, & dot_N_0, &
x0, V, & x0, V, &
Gamma, & Gamma_sf, &
mu, nu, & mu, nu, &
P_ncs, dP_ncs_dtau, & P_ncs, dP_ncs_dtau, &
P, dP_dtau P, dP_dtau
@ -1039,11 +1052,13 @@ pure subroutine kinetics_tr(Mp,T,dot_gamma_sl,ph,en,&
mu = elastic_mu(ph,en) mu = elastic_mu(ph,en)
nu = elastic_nu(ph,en) nu = elastic_nu(ph,en)
Gamma = prm%Gamma_sf(1) + prm%Gamma_sf(2) * (T-prm%T_ref) Gamma_sf = prm%Gamma_sf(1) &
+ prm%Gamma_sf(2) * (T-prm%T_ref) &
+ prm%Gamma_sf(3) * (T-prm%T_ref)**2
do i = 1, prm%sum_N_tr do i = 1, prm%sum_N_tr
tau = math_tensordot(Mp,prm%P_tr(1:3,1:3,i)) tau = math_tensordot(Mp,prm%P_tr(1:3,1:3,i))
x0 = mu*prm%b_tr(i)**2*(2.0_pReal+nu)/(Gamma*8.0_pReal*PI*(1.0_pReal-nu)) ! ToDo: In the paper, the Burgers vector for slip is used x0 = mu*prm%b_tr(i)**2*(2.0_pReal+nu)/(Gamma_sf*8.0_pReal*PI*(1.0_pReal-nu)) ! ToDo: In the paper, the Burgers vector for slip is used
tau_r = mu*prm%b_tr(i)/(2.0_pReal*PI)*(1.0_pReal/(x0+prm%x_c_tr)+cos(PI/3.0_pReal)/x0) ! ToDo: In the paper, the Burgers vector for slip is used tau_r = mu*prm%b_tr(i)/(2.0_pReal*PI)*(1.0_pReal/(x0+prm%x_c_tr)+cos(PI/3.0_pReal)/x0) ! ToDo: In the paper, the Burgers vector for slip is used
if (tau > tol_math_check .and. tau < tau_r) then if (tau > tol_math_check .and. tau < tau_r) then