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Merge branch 'dislotwin-fix-TWIP-TRIP' into 'development' 

dislotwin following paper

See merge request damask/DAMASK!527
This commit is contained in:
Franz Roters 2022-02-17 14:08:58 +00:00
parent 138b0e03f2 eca57d3b5f
commit b92ad3e31b
4 changed files with 231 additions and 234 deletions
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73
examples/config/Phase_Dislotwin_TWIP-Steel-FeMnC.yaml
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@ -1,31 +1,62 @@
type: dislotwin
output: [rho_mob, rho_dip, gamma_sl, Lambda_sl, tau_pass, f_tw, Lambda_tw, tau_hat_tw, f_tr]
D: 2.0e-5
output: [rho_mob, rho_dip, gamma_sl, Lambda_sl, tau_pass, f_tw, Lambda_tw, f_tr]
# Slip
N_sl: [12]
b_sl: [2.56e-10]
Q_sl: [3.5e-19]
p_sl: [0.325]
q_sl: [1.55]
B: [0.001]
i_sl: [30.0]
rho_mob_0: [1.0e+12]
rho_dip_0: [1.0]
v_0: [1.0e+4]
Q_sl: [3.7e-19]
p_sl: [1.0]
q_sl: [1.0]
tau_0: [1.5e+8]
i_sl: [10.0] # Adj. parameter controlling dislocation mean free path
D_0: 4.0e-5 # Vacancy diffusion prefactor / m^2/s
tau_0: [3.5e+8]
D_a: 1.0 # minimum dipole distance / b
Q_cl: 4.5e-19 # Activation energy for climb / J
h_sl-sl: [0.122, 0.122, 0.625, 0.07, 0.137, 0.137, 0.122] # Interaction coefficients (Kubin et al. 2008)
# twinning parameters
h_tw-sl: [0.0] # ToDo: values not known
# Twin
N_tw: [12]
b_tw: [1.47e-10] # Burgers vector length of twin system / b
t_tw: [5.0e-8] # Twin stack mean thickness / m
L_tw: 442.0 # Length of twin nuclei / b
x_c_tw: 1.0e-9 # critical distance for formation of twin nucleus / m
V_cs: 1.67e-29 # cross slip volume / m^3
p_tw: [10.0] # r-exponent in twin formation probability
i_tw: 1.0 # Adj. parameter controlling twin mean free path
h_sl-tw: [0.0, 1.0, 1.0] # dislocation-twin interaction coefficients
h_tw-tw: [0.0, 1.0] # twin-twin interaction coefficients
T_ref: 0.0
Gamma_sf: -0.0396 # stacking fault energy / J/m^2 at zero K; TWIP steel: -0.0526; Cu: -0.0396
Gamma_sf,T: 0.0002 # temperature dependence / J/(m^2 K) of stacking fault energy
b_tw: [1.40e-10]
L_tw: 1.8e-7
i_tw: 10.0
t_tw: [5.0e-8]
p_tw: [3.0] # A
h_tw-tw: [0.0, 1.0] # ToDo: values not known
h_sl-tw: [0.0, 1.0, 1.0] # ToDo: values not known
# Transformation
N_tr: [12]
b_tr: [1.40e-10]
L_tr: 1.8e-7
i_tr: 3.0
t_tr: [1.0e-7]
p_tr: [3.0] # B
V_mol: 7.09e-6
Delta_G: 1.33343932e+02
Delta_G,T: 2.56640412
Delta_G,T^2: 1.49524179e-03
h_tr-tr: [1.0, 1.0] # guessing
h_sl-tr: [0.0, 1.0, 1.0] # guessing
# Twin & Transformation
T_ref: 298.15
Gamma_sf: 2.89394017e-02
Gamma_sf,T: 1.22823814e-04
Gamma_sf,T^2: 1.47322968e-07
x_c: 1.0e-9
V_cs: 1.67e-29
a_cF: 3.62e-10
c/a_hP: 1.633
# Slip & Twin & Transformation
D: 2.0e-5

35
src/lattice.f90
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@ -528,22 +528,22 @@ end function lattice_C66_twin
!> @brief Rotated elasticity matrices for transformation in 6x6-matrix notation
!--------------------------------------------------------------------------------------------------
function lattice_C66_trans(Ntrans,C_parent66,lattice_target, &
cOverA_trans,a_bcc,a_fcc)
cOverA_trans,a_fcc,a_bcc)
integer, dimension(:), intent(in) :: Ntrans !< number of active twin systems per family
character(len=2), intent(in) :: lattice_target !< Bravais lattice (Pearson symbol)
real(pReal), dimension(6,6), intent(in) :: C_parent66
real(pReal), optional, intent(in) :: cOverA_trans, a_fcc, a_bcc
real(pReal), dimension(6,6,sum(Ntrans)) :: lattice_C66_trans
real(pReal), dimension(6,6) :: C_bar66, C_target_unrotated66
real(pReal), dimension(3,3,sum(Ntrans)) :: Q,S
type(tRotation) :: R
real(pReal) :: a_bcc, a_fcc, cOverA_trans
integer :: i
!--------------------------------------------------------------------------------------------------
! elasticity matrix of the target phase in cube orientation
if (lattice_target == 'hP') then
if (lattice_target == 'hP' .and. present(cOverA_trans)) then
! https://doi.org/10.1063/1.1663858 eq. (16), eq. (18), eq. (19)
! https://doi.org/10.1016/j.actamat.2016.07.032 eq. (47), eq. (48)
if (cOverA_trans < 1.0_pReal .or. cOverA_trans > 2.0_pReal) &
@ -562,7 +562,7 @@ function lattice_C66_trans(Ntrans,C_parent66,lattice_target, &
C_target_unrotated66(3,3) = C_bar66(3,3)
C_target_unrotated66(4,4) = C_bar66(4,4) - C_bar66(1,4)**2/(0.5_pReal*(C_bar66(1,1) - C_bar66(1,2)))
C_target_unrotated66 = lattice_symmetrize_C66(C_target_unrotated66,'hP')
elseif (lattice_target == 'cI') then
elseif (lattice_target == 'cI' .and. present(a_fcc) .and. present(a_bcc)) then
if (a_bcc <= 0.0_pReal .or. a_fcc <= 0.0_pReal) &
call IO_error(134,ext_msg='lattice_C66_trans: '//trim(lattice_target))
C_target_unrotated66 = C_parent66
@ -1484,26 +1484,27 @@ end function lattice_SchmidMatrix_twin
!> @brief Schmid matrix for transformation
!> details only active twin systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_SchmidMatrix_trans(Ntrans,lattice_target,cOverA,a_bcc,a_fcc) result(SchmidMatrix)
function lattice_SchmidMatrix_trans(Ntrans,lattice_target,cOverA,a_fcc,a_bcc) result(SchmidMatrix)
integer, dimension(:), intent(in) :: Ntrans !< number of active twin systems per family
character(len=2), intent(in) :: lattice_target !< Bravais lattice (Pearson symbol)
real(pReal), intent(in) :: cOverA !< c/a ratio
real(pReal), optional, intent(in) :: cOverA, a_bcc, a_fcc
real(pReal), dimension(3,3,sum(Ntrans)) :: SchmidMatrix
real(pReal), dimension(3,3,sum(Ntrans)) :: devNull
real(pReal) :: a_bcc, a_fcc
if (lattice_target /= 'cI' .and. lattice_target /= 'hP') &
call IO_error(137,ext_msg='lattice_SchmidMatrix_trans: '//trim(lattice_target))
if (lattice_target == 'hP' .and. (cOverA < 1.0_pReal .or. cOverA > 2.0_pReal)) &
if (lattice_target == 'hP' .and. present(cOverA)) then
if (cOverA < 1.0_pReal .or. cOverA > 2.0_pReal) &
call IO_error(131,ext_msg='lattice_SchmidMatrix_trans: '//trim(lattice_target))
if (lattice_target == 'cI' .and. (a_bcc <= 0.0_pReal .or. a_fcc <= 0.0_pReal)) &
call buildTransformationSystem(devNull,SchmidMatrix,Ntrans,cOverA=cOverA)
else if (lattice_target == 'cI' .and. present(a_fcc) .and. present(a_bcc)) then
if (a_bcc <= 0.0_pReal .or. a_fcc <= 0.0_pReal) &
call IO_error(134,ext_msg='lattice_SchmidMatrix_trans: '//trim(lattice_target))
call buildTransformationSystem(devNull,SchmidMatrix,Ntrans,cOverA,a_fcc,a_bcc)
call buildTransformationSystem(devNull,SchmidMatrix,Ntrans,a_fcc=a_fcc,a_bcc=a_bcc)
else
call IO_error(131,ext_msg='lattice_SchmidMatrix_trans: '//trim(lattice_target))
end if
end function lattice_SchmidMatrix_trans
@ -1971,7 +1972,7 @@ subroutine buildTransformationSystem(Q,S,Ntrans,cOverA,a_fcc,a_bcc)
real(pReal), dimension(3,3,sum(Ntrans)), intent(out) :: &
Q, & !< Total rotation: Q = R*B
S !< Eigendeformation tensor for phase transformation
real(pReal), intent(in) :: &
real(pReal), optional, intent(in) :: &
cOverA, & !< c/a for target hex lattice
a_bcc, & !< lattice parameter a for bcc target lattice
a_fcc !< lattice parameter a for fcc parent lattice
@ -2050,7 +2051,7 @@ subroutine buildTransformationSystem(Q,S,Ntrans,cOverA,a_fcc,a_bcc)
0.0, 0.0, 1.0, 45.0 &
],shape(FCCTOBCC_BAINROT))
if (a_bcc > 0.0_pReal .and. a_fcc > 0.0_pReal .and. dEq0(cOverA)) then ! fcc -> bcc
if (present(a_bcc) .and. present(a_fcc)) then
do i = 1,sum(Ntrans)
call R%fromAxisAngle(FCCTOBCC_SYSTEMTRANS(:,i),degrees=.true.,P=1)
call B%fromAxisAngle(FCCTOBCC_BAINROT(:,i), degrees=.true.,P=1)
@ -2064,7 +2065,7 @@ subroutine buildTransformationSystem(Q,S,Ntrans,cOverA,a_fcc,a_bcc)
Q(1:3,1:3,i) = matmul(R%asMatrix(),B%asMatrix())
S(1:3,1:3,i) = matmul(R%asMatrix(),U) - MATH_I3
enddo
elseif (cOverA > 0.0_pReal .and. dEq0(a_bcc)) then ! fcc -> hex
else if (present(cOverA)) then
ss = MATH_I3
sd = MATH_I3
ss(1,3) = sqrt(2.0_pReal)/4.0_pReal

17
src/phase_mechanical_plastic.f90
View File

@ -140,15 +140,15 @@ submodule(phase:mechanical) plastic
dotState
end function plastic_kinehardening_dotState
module subroutine dislotwin_dotState(Mp,T,ph,en)
module function dislotwin_dotState(Mp,ph,en) result(dotState)
real(pReal), dimension(3,3), intent(in) :: &
Mp !< Mandel stress
real(pReal), intent(in) :: &
T
integer, intent(in) :: &
ph, &
en
end subroutine dislotwin_dotState
real(pReal), dimension(plasticState(ph)%sizeDotState) :: &
dotState
end function dislotwin_dotState
module function dislotungsten_dotState(Mp,ph,en) result(dotState)
real(pReal), dimension(3,3), intent(in) :: &
@ -170,12 +170,10 @@ submodule(phase:mechanical) plastic
en
end subroutine nonlocal_dotState
module subroutine dislotwin_dependentState(T,ph,en)
module subroutine dislotwin_dependentState(ph,en)
integer, intent(in) :: &
ph, &
en
real(pReal), intent(in) :: &
T
end subroutine dislotwin_dependentState
module subroutine dislotungsten_dependentState(ph,en)
@ -326,8 +324,7 @@ module function plastic_dotState(subdt,ph,en) result(dotState)
dotState = plastic_kinehardening_dotState(Mp,ph,en)
case (PLASTIC_DISLOTWIN_ID) plasticType
call dislotwin_dotState(Mp,thermal_T(ph,en),ph,en)
dotState = plasticState(ph)%dotState(:,en)
dotState = dislotwin_dotState(Mp,ph,en)
case (PLASTIC_DISLOTUNGSTEN_ID) plasticType
dotState = dislotungsten_dotState(Mp,ph,en)
@ -355,7 +352,7 @@ module subroutine plastic_dependentState(ph,en)
plasticType: select case (phase_plasticity(ph))
case (PLASTIC_DISLOTWIN_ID) plasticType
call dislotwin_dependentState(thermal_T(ph,en),ph,en)
call dislotwin_dependentState(ph,en)
case (PLASTIC_DISLOTUNGSTEN_ID) plasticType
call dislotungsten_dependentState(ph,en)

288
src/phase_mechanical_plastic_dislotwin.f90
View File

@ -17,22 +17,23 @@ submodule(phase:plastic) dislotwin
p_sb = 1.0_pReal, & !< p-exponent in shear band velocity
q_sb = 1.0_pReal, & !< q-exponent in shear band velocity
i_tw = 1.0_pReal, & !< adjustment parameter to calculate MFP for twinning
L_tw = 1.0_pReal, & !< length of twin nuclei in Burgers vectors: TODO unit should be meters
L_tr = 1.0_pReal, & !< length of trans nuclei in Burgers vectors: TODO unit should be meters
x_c_tw = 1.0_pReal, & !< critical distance for formation of twin nucleus
x_c_tr = 1.0_pReal, & !< critical distance for formation of trans nucleus
i_tr = 1.0_pReal, & !< adjustment parameter to calculate MFP for transformation
L_tw = 1.0_pReal, & !< length of twin nuclei
L_tr = 1.0_pReal, & !< length of trans nuclei
x_c = 1.0_pReal, & !< critical distance for formation of twin/trans nucleus
V_cs = 1.0_pReal, & !< cross slip volume
xi_sb = 1.0_pReal, & !< value for shearband resistance
v_sb = 1.0_pReal, & !< value for shearband velocity_0
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
h = 1.0_pReal, & !< stack height of hex nucleus
T_ref = T_ROOM, &
a_cI = 1.0_pReal, &
a_cF = 1.0_pReal
real(pReal), dimension(2) :: &
Gamma_sf = 0.0_pReal
gamma_char_tr = sqrt(0.125_pReal), & !< Characteristic shear for transformation
a_cF = 1.0_pReal, &
cOverA_hP = 1.0_pReal, &
V_mol = 1.0_pReal, &
rho = 1.0_pReal
type(tPolynomial) :: &
Gamma_sf, & !< stacking fault energy
Delta_G !< free energy difference between austensite and martensite
real(pReal), allocatable, dimension(:) :: &
b_sl, & !< absolute length of Burgers vector [m] for each slip system
b_tw, & !< absolute length of Burgers vector [m] for each twin system
@ -48,7 +49,7 @@ submodule(phase:plastic) dislotwin
r, & !< exponent in twin nucleation rate
s, & !< exponent in trans nucleation rate
tau_0, & !< strength due to elements in solid solution
gamma_char, & !< characteristic shear for twins
gamma_char_tw, & !< characteristic shear for twins
B, & !< drag coefficient
d_caron !< distance of spontaneous annhihilation
real(pReal), allocatable, dimension(:,:) :: &
@ -77,13 +78,22 @@ submodule(phase:plastic) dislotwin
character(len=pStringLen), allocatable, dimension(:) :: &
output
logical :: &
ExtendedDislocations, & !< consider split into partials for climb calculation
extendedDislocations, & !< consider split into partials for climb calculation
fccTwinTransNucleation, & !< twinning and transformation models are for fcc
omitDipoles !< flag controlling consideration of dipole formation
character(len=:), allocatable, dimension(:) :: &
systems_sl, &
systems_tw
end type !< container type for internal constitutive parameters
end type tParameters !< container type for internal constitutive parameters
type :: tIndexDotState
integer, dimension(2) :: &
rho_mob, &
rho_dip, &
gamma_sl, &
f_tw, &
f_tr
end type tIndexDotState
type :: tDislotwinState
real(pReal), dimension(:,:), pointer :: &
@ -99,17 +109,14 @@ submodule(phase:plastic) dislotwin
Lambda_sl, & !< mean free path between 2 obstacles seen by a moving dislocation
Lambda_tw, & !< mean free path between 2 obstacles seen by a growing twin
Lambda_tr, & !< mean free path between 2 obstacles seen by a growing martensite
tau_pass, & !< threshold stress for slip
tau_hat_tw, & !< threshold stress for twinning
tau_hat_tr !< threshold stress for transformation
tau_pass !< threshold stress for slip
end type tDislotwinDependentState
!--------------------------------------------------------------------------------------------------
! containers for parameters and state
type(tParameters), allocatable, dimension(:) :: param
type(tDislotwinState), allocatable, dimension(:) :: &
dotState, &
state
type(tIndexDotState), allocatable, dimension(:) :: indexDotState
type(tDislotwinState), allocatable, dimension(:) :: state
type(tDislotwinDependentState), allocatable, dimension(:) :: dependentState
contains
@ -129,6 +136,7 @@ module function plastic_dislotwin_init() result(myPlasticity)
startIndex, endIndex
integer, dimension(:), allocatable :: &
N_sl
real(pReal) :: a_cF
real(pReal), allocatable, dimension(:) :: &
rho_mob_0, & !< initial unipolar dislocation density per slip system
rho_dip_0 !< initial dipole dislocation density per slip system
@ -159,15 +167,15 @@ module function plastic_dislotwin_init() result(myPlasticity)
phases => config_material%get('phase')
allocate(param(phases%length))
allocate(indexDotState(phases%length))
allocate(state(phases%length))
allocate(dotState(phases%length))
allocate(dependentState(phases%length))
do ph = 1, phases%length
if (.not. myPlasticity(ph)) cycle
associate(prm => param(ph), dot => dotState(ph), stt => state(ph), dst => dependentState(ph))
associate(prm => param(ph), stt => state(ph), dst => dependentState(ph), &
idx_dot => indexDotState(ph))
phase => phases%get(ph)
mech => phase%get('mechanical')
@ -179,7 +187,6 @@ module function plastic_dislotwin_init() result(myPlasticity)
prm%output = pl%get_as1dString('output',defaultVal=emptyStringArray)
#endif
!--------------------------------------------------------------------------------------------------
! slip related parameters
N_sl = pl%get_as1dInt('N_sl',defaultVal=emptyIntArray)
@ -209,7 +216,7 @@ module function plastic_dislotwin_init() result(myPlasticity)
prm%Q_cl = pl%get_asFloat('Q_cl')
prm%ExtendedDislocations = pl%get_asBool('extend_dislocations',defaultVal = .false.)
prm%extendedDislocations = pl%get_asBool('extend_dislocations',defaultVal = .false.)
prm%omitDipoles = pl%get_asBool('omit_dipoles',defaultVal = .false.)
! multiplication factor according to crystal structure (nearest neighbors bcc vs fcc/hex)
@ -262,16 +269,10 @@ module function plastic_dislotwin_init() result(myPlasticity)
prm%t_tw = pl%get_as1dFloat('t_tw', requiredSize=size(prm%N_tw))
prm%r = pl%get_as1dFloat('p_tw', requiredSize=size(prm%N_tw))
prm%x_c_tw = pl%get_asFloat('x_c_tw')
prm%L_tw = pl%get_asFloat('L_tw')
prm%i_tw = pl%get_asFloat('i_tw')
prm%gamma_char= lattice_characteristicShear_Twin(prm%N_tw,phase_lattice(ph),phase_cOverA(ph))
if (.not. prm%fccTwinTransNucleation) then
prm%dot_N_0_tw = pl%get_as1dFloat('dot_N_0_tw')
prm%dot_N_0_tw = math_expand(prm%dot_N_0_tw,prm%N_tw)
endif
prm%gamma_char_tw = lattice_characteristicShear_Twin(prm%N_tw,phase_lattice(ph),phase_cOverA(ph))
! expand: family => system
prm%b_tw = math_expand(prm%b_tw,prm%N_tw)
@ -279,17 +280,14 @@ module function plastic_dislotwin_init() result(myPlasticity)
prm%r = math_expand(prm%r,prm%N_tw)
! sanity checks
if ( prm%x_c_tw < 0.0_pReal) extmsg = trim(extmsg)//' x_c_tw'
if (.not. prm%fccTwinTransNucleation) extmsg = trim(extmsg)//' TWIP for non-fcc'
if ( prm%L_tw < 0.0_pReal) extmsg = trim(extmsg)//' L_tw'
if ( prm%i_tw < 0.0_pReal) extmsg = trim(extmsg)//' i_tw'
if (any(prm%b_tw < 0.0_pReal)) extmsg = trim(extmsg)//' b_tw'
if (any(prm%t_tw < 0.0_pReal)) extmsg = trim(extmsg)//' t_tw'
if (any(prm%r < 0.0_pReal)) extmsg = trim(extmsg)//' p_tw'
if (.not. prm%fccTwinTransNucleation) then
if (any(prm%dot_N_0_tw < 0.0_pReal)) extmsg = trim(extmsg)//' dot_N_0_tw'
end if
else twinActive
allocate(prm%gamma_char,prm%b_tw,prm%dot_N_0_tw,prm%t_tw,prm%r,source=emptyRealArray)
allocate(prm%gamma_char_tw,prm%b_tw,prm%t_tw,prm%r,source=emptyRealArray)
allocate(prm%h_tw_tw(0,0))
end if twinActive
@ -301,32 +299,28 @@ module function plastic_dislotwin_init() result(myPlasticity)
prm%b_tr = pl%get_as1dFloat('b_tr')
prm%b_tr = math_expand(prm%b_tr,prm%N_tr)
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%delta_G = pl%get_asFloat('delta_G')
prm%x_c_tr = pl%get_asFloat('x_c_tr', defaultVal=0.0_pReal) ! ToDo: This is not optional!
prm%i_tr = pl%get_asFloat('i_tr')
prm%Delta_G = polynomial(pl%asDict(),'Delta_G','T')
prm%L_tr = pl%get_asFloat('L_tr')
prm%a_cI = pl%get_asFloat('a_cI', defaultVal=0.0_pReal)
prm%a_cF = pl%get_asFloat('a_cF', defaultVal=0.0_pReal)
prm%lattice_tr = pl%get_asString('lattice_tr')
a_cF = pl%get_asFloat('a_cF')
prm%h = 5.0_pReal * a_cF/sqrt(3.0_pReal)
prm%cOverA_hP = pl%get_asFloat('c/a_hP')
prm%rho = 4.0_pReal/(sqrt(3.0_pReal)*a_cF**2)/N_A
prm%V_mol = pl%get_asFloat('V_mol')
prm%h_tr_tr = lattice_interaction_TransByTrans(prm%N_tr,pl%get_as1dFloat('h_tr-tr'),&
phase_lattice(ph))
prm%P_tr = lattice_SchmidMatrix_trans(prm%N_tr,prm%lattice_tr, &
0.0_pReal, &
prm%a_cI, &
prm%a_cF)
prm%P_tr = lattice_SchmidMatrix_trans(prm%N_tr,'hP',prm%cOverA_hP)
prm%t_tr = pl%get_as1dFloat('t_tr')
prm%t_tr = math_expand(prm%t_tr,prm%N_tr)
prm%s = pl%get_as1dFloat('p_tr',defaultVal=[0.0_pReal])
prm%s = pl%get_as1dFloat('p_tr')
prm%s = math_expand(prm%s,prm%N_tr)
! sanity checks
if ( prm%x_c_tr < 0.0_pReal) extmsg = trim(extmsg)//' x_c_tr'
if (.not. prm%fccTwinTransNucleation) extmsg = trim(extmsg)//' TRIP for non-fcc'
if ( prm%L_tr < 0.0_pReal) extmsg = trim(extmsg)//' L_tr'
if ( prm%V_mol < 0.0_pReal) extmsg = trim(extmsg)//' V_mol'
if ( prm%i_tr < 0.0_pReal) extmsg = trim(extmsg)//' i_tr'
if (any(prm%t_tr < 0.0_pReal)) extmsg = trim(extmsg)//' t_tr'
if (any(prm%s < 0.0_pReal)) extmsg = trim(extmsg)//' p_tr'
@ -356,15 +350,16 @@ module function plastic_dislotwin_init() result(myPlasticity)
if (prm%sum_N_sl + prm%sum_N_tw + prm%sum_N_tw > 0) &
prm%D = pl%get_asFloat('D')
if (prm%sum_N_tw + prm%sum_N_tr > 0) &
if (prm%sum_N_tw + prm%sum_N_tr > 0) then
prm%x_c = pl%get_asFloat('x_c')
prm%V_cs = pl%get_asFloat('V_cs')
if (prm%sum_N_tw + prm%sum_N_tr > 0 .or. prm%ExtendedDislocations) then
prm%T_ref = pl%get_asFloat('T_ref')
prm%Gamma_sf(1) = pl%get_asFloat('Gamma_sf')
prm%Gamma_sf(2) = pl%get_asFloat('Gamma_sf,T',defaultVal=0.0_pReal)
if (prm%x_c < 0.0_pReal) extmsg = trim(extmsg)//' x_c'
if (prm%V_cs < 0.0_pReal) extmsg = trim(extmsg)//' V_cs'
end if
if (prm%sum_N_tw + prm%sum_N_tr > 0 .or. prm%extendedDislocations) &
prm%Gamma_sf = polynomial(pl%asDict(),'Gamma_sf','T')
slipAndTwinActive: if (prm%sum_N_sl * prm%sum_N_tw > 0) then
prm%h_sl_tw = lattice_interaction_SlipByTwin(N_sl,prm%N_tw,pl%get_as1dFloat('h_sl-tw'), &
phase_lattice(ph))
@ -385,55 +380,51 @@ module function plastic_dislotwin_init() result(myPlasticity)
+ size(['f_tr']) * prm%sum_N_tr
sizeState = sizeDotState
call phase_allocateState(plasticState(ph),Nmembers,sizeState,sizeDotState,0)
deallocate(plasticState(ph)%dotState) ! ToDo: remove dotState completely
!--------------------------------------------------------------------------------------------------
! locally defined state aliases and initialization of state0 and atol
! state aliases and initialization
startIndex = 1
endIndex = prm%sum_N_sl
idx_dot%rho_mob = [startIndex,endIndex]
stt%rho_mob=>plasticState(ph)%state(startIndex:endIndex,:)
stt%rho_mob= spread(rho_mob_0,2,Nmembers)
dot%rho_mob=>plasticState(ph)%dotState(startIndex:endIndex,:)
plasticState(ph)%atol(startIndex:endIndex) = pl%get_asFloat('atol_rho',defaultVal=1.0_pReal)
if (any(plasticState(ph)%atol(startIndex:endIndex) < 0.0_pReal)) extmsg = trim(extmsg)//' atol_rho'
startIndex = endIndex + 1
endIndex = endIndex + prm%sum_N_sl
idx_dot%rho_dip = [startIndex,endIndex]
stt%rho_dip=>plasticState(ph)%state(startIndex:endIndex,:)
stt%rho_dip= spread(rho_dip_0,2,Nmembers)
dot%rho_dip=>plasticState(ph)%dotState(startIndex:endIndex,:)
plasticState(ph)%atol(startIndex:endIndex) = pl%get_asFloat('atol_rho',defaultVal=1.0_pReal)
startIndex = endIndex + 1
endIndex = endIndex + prm%sum_N_sl
idx_dot%gamma_sl = [startIndex,endIndex]
stt%gamma_sl=>plasticState(ph)%state(startIndex:endIndex,:)
dot%gamma_sl=>plasticState(ph)%dotState(startIndex:endIndex,:)
plasticState(ph)%atol(startIndex:endIndex) = pl%get_asFloat('atol_gamma',defaultVal=1.0e-6_pReal)
if (any(plasticState(ph)%atol(startIndex:endIndex) < 0.0_pReal)) extmsg = trim(extmsg)//' atol_gamma'
startIndex = endIndex + 1
endIndex = endIndex + prm%sum_N_tw
idx_dot%f_tw = [startIndex,endIndex]
stt%f_tw=>plasticState(ph)%state(startIndex:endIndex,:)
dot%f_tw=>plasticState(ph)%dotState(startIndex:endIndex,:)
plasticState(ph)%atol(startIndex:endIndex) = pl%get_asFloat('atol_f_tw',defaultVal=1.0e-6_pReal)
if (any(plasticState(ph)%atol(startIndex:endIndex) < 0.0_pReal)) extmsg = trim(extmsg)//' atol_f_tw'
startIndex = endIndex + 1
endIndex = endIndex + prm%sum_N_tr
idx_dot%f_tr = [startIndex,endIndex]
stt%f_tr=>plasticState(ph)%state(startIndex:endIndex,:)
dot%f_tr=>plasticState(ph)%dotState(startIndex:endIndex,:)
plasticState(ph)%atol(startIndex:endIndex) = pl%get_asFloat('atol_f_tr',defaultVal=1.0e-6_pReal)
if (any(plasticState(ph)%atol(startIndex:endIndex) < 0.0_pReal)) extmsg = trim(extmsg)//' atol_f_tr'
allocate(dst%Lambda_sl (prm%sum_N_sl,Nmembers),source=0.0_pReal)
allocate(dst%tau_pass (prm%sum_N_sl,Nmembers),source=0.0_pReal)
allocate(dst%Lambda_sl(prm%sum_N_sl,Nmembers),source=0.0_pReal)
allocate(dst%Lambda_tw(prm%sum_N_tw,Nmembers),source=0.0_pReal)
allocate(dst%tau_hat_tw (prm%sum_N_tw,Nmembers),source=0.0_pReal)
allocate(dst%Lambda_tr(prm%sum_N_tr,Nmembers),source=0.0_pReal)
allocate(dst%tau_hat_tr (prm%sum_N_tr,Nmembers),source=0.0_pReal)
end associate
@ -482,7 +473,7 @@ module function plastic_dislotwin_homogenizedC(ph,en) result(homogenizedC)
end if twinActive
transActive: if (prm%sum_N_tr > 0) then
C66_tr = lattice_C66_trans(prm%N_tr,C,prm%lattice_tr,0.0_pReal,prm%a_cI,prm%a_cF)
C66_tr = lattice_C66_trans(prm%N_tr,C,'hP',prm%cOverA_hP)
do i=1,prm%sum_N_tr
homogenizedC = homogenizedC &
+ stt%f_tr(i,en)*C66_tr(1:6,1:6,i)
@ -559,7 +550,7 @@ 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)
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
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) &
@ -567,7 +558,7 @@ 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)
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
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) &
@ -612,15 +603,15 @@ end subroutine dislotwin_LpAndItsTangent
!--------------------------------------------------------------------------------------------------
!> @brief Calculate the rate of change of microstructure.
!--------------------------------------------------------------------------------------------------
module subroutine dislotwin_dotState(Mp,T,ph,en)
module function dislotwin_dotState(Mp,ph,en) result(dotState)
real(pReal), dimension(3,3), intent(in):: &
Mp !< Mandel stress
real(pReal), intent(in) :: &
T !< temperature at integration point
integer, intent(in) :: &
ph, &
en
real(pReal), dimension(plasticState(ph)%sizeDotState) :: &
dotState
integer :: i
real(pReal) :: &
@ -639,22 +630,27 @@ module subroutine dislotwin_dotState(Mp,T,ph,en)
real(pReal), dimension(param(ph)%sum_N_tr) :: &
dot_gamma_tr
real(pReal) :: &
mu, &
nu, &
Gamma
mu, nu, &
T
associate(prm => param(ph), stt => state(ph), dot => dotState(ph), dst => dependentState(ph))
associate(prm => param(ph), stt => state(ph), dst => dependentState(ph), &
dot_rho_mob => dotState(indexDotState(ph)%rho_mob(1):indexDotState(ph)%rho_mob(2)), &
dot_rho_dip => dotState(indexDotState(ph)%rho_dip(1):indexDotState(ph)%rho_dip(2)), &
abs_dot_gamma_sl => dotState(indexDotState(ph)%gamma_sl(1):indexDotState(ph)%gamma_sl(2)), &
dot_f_tw => dotState(indexDotState(ph)%f_tw(1):indexDotState(ph)%f_tw(2)), &
dot_f_tr => dotState(indexDotState(ph)%f_tr(1):indexDotState(ph)%f_tr(2)))
mu = elastic_mu(ph,en)
nu = elastic_nu(ph,en)
Gamma = prm%Gamma_sf(1) + prm%Gamma_sf(2) * (T-prm%T_ref)
T = thermal_T(ph,en)
f_matrix = 1.0_pReal &
- sum(stt%f_tw(1:prm%sum_N_tw,en)) &
- sum(stt%f_tr(1:prm%sum_N_tr,en))
call kinetics_sl(Mp,T,ph,en,dot_gamma_sl)
dot%gamma_sl(:,en) = abs(dot_gamma_sl)
abs_dot_gamma_sl = abs(dot_gamma_sl)
slipState: do i = 1, prm%sum_N_sl
tau = math_tensordot(Mp,prm%P_sl(1:3,1:3,i))
@ -668,16 +664,18 @@ module subroutine dislotwin_dotState(Mp,T,ph,en)
d_hat = math_clip(d_hat, left = prm%d_caron(i))
dot_rho_dip_formation(i) = 2.0_pReal*(d_hat-prm%d_caron(i))/prm%b_sl(i) &
* stt%rho_mob(i,en)*abs(dot_gamma_sl(i))
* stt%rho_mob(i,en)*abs_dot_gamma_sl(i)
if (dEq(d_hat,prm%d_caron(i))) then
dot_rho_dip_climb(i) = 0.0_pReal
else
! Argon & Moffat, Acta Metallurgica, Vol. 29, pg 293 to 299, 1981
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)), &
1.0_pReal, &
prm%ExtendedDislocations)
if (prm%extendedDislocations) then
b_d = 24.0_pReal*PI*(1.0_pReal - nu)/(2.0_pReal + nu) * prm%Gamma_sf%at(T) / (mu*prm%b_sl(i))
else
b_d = 1.0_pReal
end if
v_cl = 2.0_pReal*prm%omega*b_d**2*exp(-prm%Q_cl/(K_B*T)) &
* (exp(abs(sigma_cl)*prm%b_sl(i)**3/(K_B*T)) - 1.0_pReal)
@ -687,38 +685,36 @@ module subroutine dislotwin_dotState(Mp,T,ph,en)
end if significantSlipStress
end do slipState
dot%rho_mob(:,en) = abs(dot_gamma_sl)/(prm%b_sl*dst%Lambda_sl(:,en)) &
dot_rho_mob = abs_dot_gamma_sl/(prm%b_sl*dst%Lambda_sl(:,en)) &
- dot_rho_dip_formation &
- 2.0_pReal*prm%d_caron/prm%b_sl * stt%rho_mob(:,en)*abs(dot_gamma_sl)
- 2.0_pReal*prm%d_caron/prm%b_sl * stt%rho_mob(:,en)*abs_dot_gamma_sl
dot%rho_dip(:,en) = dot_rho_dip_formation &
- 2.0_pReal*prm%d_caron/prm%b_sl * stt%rho_dip(:,en)*abs(dot_gamma_sl) &
dot_rho_dip = dot_rho_dip_formation &
- 2.0_pReal*prm%d_caron/prm%b_sl * stt%rho_dip(:,en)*abs_dot_gamma_sl &
- dot_rho_dip_climb
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
if (prm%sum_N_tw > 0) call kinetics_tw(Mp,T,abs_dot_gamma_sl,ph,en,dot_gamma_tw)
dot_f_tw = f_matrix*dot_gamma_tw/prm%gamma_char_tw
call kinetics_tr(Mp,T,dot_gamma_sl,ph,en,dot_gamma_tr)
dot%f_tr(:,en) = f_matrix*dot_gamma_tr
if (prm%sum_N_tr > 0) call kinetics_tr(Mp,T,abs_dot_gamma_sl,ph,en,dot_gamma_tr)
dot_f_tr = f_matrix*dot_gamma_tr/prm%gamma_char_tr
end associate
end subroutine dislotwin_dotState
end function dislotwin_dotState
!--------------------------------------------------------------------------------------------------
!> @brief Calculate derived quantities from state.
!--------------------------------------------------------------------------------------------------
module subroutine dislotwin_dependentState(T,ph,en)
module subroutine dislotwin_dependentState(ph,en)
integer, intent(in) :: &
ph, &
en
real(pReal), intent(in) :: &
T
real(pReal) :: &
sumf_tw,Gamma,sumf_tr
sumf_tw, sumf_tr
real(pReal), dimension(param(ph)%sum_N_sl) :: &
inv_lambda_sl
real(pReal), dimension(param(ph)%sum_N_tw) :: &
@ -728,19 +724,15 @@ module subroutine dislotwin_dependentState(T,ph,en)
inv_lambda_tr_tr, & !< 1/mean free distance between 2 martensite stacks from different systems seen by a growing martensite
f_over_t_tr
real(pReal) :: &
mu, &
nu
mu
associate(prm => param(ph), stt => state(ph), dst => dependentState(ph))
mu = elastic_mu(ph,en)
nu = elastic_nu(ph,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))
Gamma = prm%Gamma_sf(1) + prm%Gamma_sf(2) * (T-prm%T_ref)
!* 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_tr = sumf_tr/prm%t_tr ! but this not
@ -762,13 +754,6 @@ module subroutine dislotwin_dependentState(T,ph,en)
!* threshold stress for dislocation motion
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
dst%tau_hat_tw(:,en) = Gamma/(3.0_pReal*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) &
+ 3.0_pReal*prm%b_tr*mu/(prm%L_tr*prm%b_tr) &
+ prm%h*prm%delta_G/(3.0_pReal*prm%b_tr)
end associate
end subroutine dislotwin_dependentState
@ -813,9 +798,6 @@ module subroutine plastic_dislotwin_results(ph,group)
case('Lambda_tw')
call results_writeDataset(dst%Lambda_tw,group,trim(prm%output(ou)), &
'mean free path for twinning','m',prm%systems_tw)
case('tau_hat_tw')
call results_writeDataset(dst%tau_hat_tw,group,trim(prm%output(ou)), &
'threshold stress for twinning','Pa',prm%systems_tw)
case('f_tr')
if (prm%sum_N_tr>0) call results_writeDataset(stt%f_tr,group,trim(prm%output(ou)), &
@ -847,15 +829,14 @@ pure subroutine kinetics_sl(Mp,T,ph,en, &
integer, intent(in) :: &
ph, &
en
real(pReal), dimension(param(ph)%sum_N_sl), intent(out) :: &
dot_gamma_sl
real(pReal), dimension(param(ph)%sum_N_sl), optional, intent(out) :: &
ddot_gamma_dtau_sl, &
tau_sl
real(pReal), dimension(param(ph)%sum_N_sl) :: &
ddot_gamma_dtau
real(pReal), dimension(param(ph)%sum_N_sl) :: &
tau, &
stressRatio, &
@ -914,7 +895,7 @@ end subroutine kinetics_sl
! NOTE: Against the common convention, the result (i.e. intent(out)) variables are the last to
! have the optional arguments at the end.
!--------------------------------------------------------------------------------------------------
pure subroutine kinetics_tw(Mp,T,dot_gamma_sl,ph,en,&
pure subroutine kinetics_tw(Mp,T,abs_dot_gamma_sl,ph,en,&
dot_gamma_tw,ddot_gamma_dtau_tw)
real(pReal), dimension(3,3), intent(in) :: &
@ -925,18 +906,17 @@ pure subroutine kinetics_tw(Mp,T,dot_gamma_sl,ph,en,&
ph, &
en
real(pReal), dimension(param(ph)%sum_N_sl), intent(in) :: &
dot_gamma_sl
abs_dot_gamma_sl
real(pReal), dimension(param(ph)%sum_N_tw), intent(out) :: &
dot_gamma_tw
real(pReal), dimension(param(ph)%sum_N_tw), optional, intent(out) :: &
ddot_gamma_dtau_tw
real :: &
tau, tau_r, &
tau, tau_r, tau_hat, &
dot_N_0, &
x0, V, &
Gamma, &
Gamma_sf, &
mu, nu, &
P_ncs, dP_ncs_dtau, &
P, dP_dtau
@ -947,51 +927,38 @@ pure subroutine kinetics_tw(Mp,T,dot_gamma_sl,ph,en,&
associate(prm => param(ph), stt => state(ph), dst => dependentState(ph))
isFCC: if (prm%fccTwinTransNucleation) then
mu = elastic_mu(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%at(T)
tau_hat = 3.0_pReal*prm%b_tw(1)*mu/prm%L_tw &
+ Gamma_sf/(3.0_pReal*prm%b_tw(1))
x0 = mu*prm%b_sl(1)**2*(2.0_pReal+nu)/(Gamma_sf*8.0_pReal*PI*(1.0_pReal-nu))
tau_r = mu*prm%b_sl(1)/(2.0_pReal*PI)*(1.0_pReal/(x0+prm%x_c)+cos(PI/3.0_pReal)/x0)
do i = 1, prm%sum_N_tw
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
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
P = exp(-(dst%tau_hat_tw(i,en)/tau)**prm%r(i))
dP_dTau = prm%r(i) * (dst%tau_hat_tw(i,en)/tau)**prm%r(i)/tau * P
P = exp(-(tau_hat/tau)**prm%r(i))
dP_dTau = prm%r(i) * (tau_hat/tau)**prm%r(i)/tau * P
s = prm%fcc_twinNucleationSlipPair(1:2,i)
dot_N_0 = sum(abs(dot_gamma_sl(s(2:1:-1)))*(stt%rho_mob(s,en)+stt%rho_dip(s,en))) &
/ (prm%L_tw*prm%b_sl(i))
dot_N_0 = sum(abs_dot_gamma_sl(s(2:1:-1))*(stt%rho_mob(s,en)+stt%rho_dip(s,en)))/(prm%L_tw*3.0_pReal)
P_ncs = 1.0_pReal-exp(-prm%V_cs/(K_B*T)*(tau_r-tau))
dP_ncs_dtau = prm%V_cs / (K_B * T) * (P_ncs - 1.0_pReal)
V = PI/4.0_pReal*dst%Lambda_tw(i,en)**2*prm%t_tw(i)
dot_gamma_tw(i) = V*dot_N_0*P_ncs*P
dot_gamma_tw(i) = V*dot_N_0*P_ncs*P*prm%gamma_char_tw(i)
if (present(ddot_gamma_dtau_tw)) &
ddot_gamma_dtau_tw(i) = V*dot_N_0*(P*dP_ncs_dtau + P_ncs*dP_dtau)
ddot_gamma_dtau_tw(i) = V*dot_N_0*(P*dP_ncs_dtau + P_ncs*dP_dtau)*prm%gamma_char_tw(i)
else
dot_gamma_tw(i) = 0.0_pReal
if (present(ddot_gamma_dtau_tw)) ddot_gamma_dtau_tw(i) = 0.0_pReal
end if
end do
else isFCC
do i = 1, prm%sum_N_tw
error stop 'not implemented'
tau = math_tensordot(Mp,prm%P_tw(1:3,1:3,i))
if (tau > tol_math_check) then
dot_gamma_tw(i) = 0.0_pReal
if (present(ddot_gamma_dtau_tw)) ddot_gamma_dtau_tw(i) = 0.0_pReal
else
dot_gamma_tw(i) = 0.0_pReal
if (present(ddot_gamma_dtau_tw)) ddot_gamma_dtau_tw(i) = 0.0_pReal
end if
end do
end if isFCC
end associate
end subroutine kinetics_tw
@ -1004,7 +971,7 @@ end subroutine kinetics_tw
! NOTE: Against the common convention, the result (i.e. intent(out)) variables are the last to
! have the optional arguments at the end.
!--------------------------------------------------------------------------------------------------
pure subroutine kinetics_tr(Mp,T,dot_gamma_sl,ph,en,&
pure subroutine kinetics_tr(Mp,T,abs_dot_gamma_sl,ph,en,&
dot_gamma_tr,ddot_gamma_dtau_tr)
real(pReal), dimension(3,3), intent(in) :: &
@ -1015,18 +982,17 @@ pure subroutine kinetics_tr(Mp,T,dot_gamma_sl,ph,en,&
ph, &
en
real(pReal), dimension(param(ph)%sum_N_sl), intent(in) :: &
dot_gamma_sl
abs_dot_gamma_sl
real(pReal), dimension(param(ph)%sum_N_tr), intent(out) :: &
dot_gamma_tr
real(pReal), dimension(param(ph)%sum_N_tr), optional, intent(out) :: &
ddot_gamma_dtau_tr
real :: &
tau, tau_r, &
tau, tau_r, tau_hat, &
dot_N_0, &
x0, V, &
Gamma, &
Gamma_sf, &
mu, nu, &
P_ncs, dP_ncs_dtau, &
P, dP_dtau
@ -1039,28 +1005,30 @@ pure subroutine kinetics_tr(Mp,T,dot_gamma_sl,ph,en,&
mu = elastic_mu(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%at(T)
tau_hat = 3.0_pReal*prm%b_tr(1)*mu/prm%L_tr &
+ (Gamma_sf + (prm%h/prm%V_mol - 2.0_pReal*prm%rho)*prm%Delta_G%at(T))/(3.0_pReal*prm%b_tr(1))
x0 = mu*prm%b_sl(1)**2*(2.0_pReal+nu)/(Gamma_sf*8.0_pReal*PI*(1.0_pReal-nu))
tau_r = mu*prm%b_sl(1)/(2.0_pReal*PI)*(1.0_pReal/(x0+prm%x_c)+cos(PI/3.0_pReal)/x0)
do i = 1, prm%sum_N_tr
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
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
P = exp(-(dst%tau_hat_tr(i,en)/tau)**prm%s(i))
dP_dTau = prm%s(i) * (dst%tau_hat_tr(i,en)/tau)**prm%s(i)/tau * P
P = exp(-(tau_hat/tau)**prm%s(i))
dP_dTau = prm%s(i) * (tau_hat/tau)**prm%s(i)/tau * P
s = prm%fcc_twinNucleationSlipPair(1:2,i)
dot_N_0 = sum(abs(dot_gamma_sl(s(2:1:-1)))*(stt%rho_mob(s,en)+stt%rho_dip(s,en))) &
/ (prm%L_tr*prm%b_sl(i))
dot_N_0 = sum(abs_dot_gamma_sl(s(2:1:-1))*(stt%rho_mob(s,en)+stt%rho_dip(s,en)))/(prm%L_tr*3.0_pReal)
P_ncs = 1.0_pReal-exp(-prm%V_cs/(K_B*T)*(tau_r-tau))
dP_ncs_dtau = prm%V_cs / (K_B * T) * (P_ncs - 1.0_pReal)
V = PI/4.0_pReal*dst%Lambda_tr(i,en)**2*prm%t_tr(i)
dot_gamma_tr(i) = V*dot_N_0*P_ncs*P
dot_gamma_tr(i) = V*dot_N_0*P_ncs*P*prm%gamma_char_tr
if (present(ddot_gamma_dtau_tr)) &
ddot_gamma_dtau_tr(i) = V*dot_N_0*(P*dP_ncs_dtau + P_ncs*dP_dtau)
ddot_gamma_dtau_tr(i) = V*dot_N_0*(P*dP_ncs_dtau + P_ncs*dP_dtau)*prm%gamma_char_tr
else
dot_gamma_tr(i) = 0.0_pReal
if (present(ddot_gamma_dtau_tr)) ddot_gamma_dtau_tr(i) = 0.0_pReal