!-------------------------------------------------------------------------------------------------- !> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH !> @author Su Leen Wong, Max-Planck-Institut für Eisenforschung GmbH !> @author Nan Jia, Max-Planck-Institut für Eisenforschung GmbH !> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH !> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH !> @brief material subroutine incoprorating dislocation and twinning physics !> @details to be done !-------------------------------------------------------------------------------------------------- submodule(phase:plastic) dislotwin real(pREAL), parameter :: gamma_char_tr = sqrt(0.125_pREAL) !< Characteristic shear for transformation type :: tParameters real(pREAL) :: & Q_cl = 1.0_pREAL, & !< activation energy for dislocation climb omega = 1.0_pREAL, & !< frequency factor for dislocation climb D = 1.0_pREAL, & !< grain size 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 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 tau_sb = 1.0_pREAL, & !< value for shearband resistance gamma_0_sb = 1.0_pREAL, & !< value for shearband velocity_0 E_sb = 1.0_pREAL, & !< activation energy for shear bands h = 1.0_pREAL, & !< stack height of hex nucleus 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, & !< magnitude of Burgers vector (m) for each slip system b_tw, & !< magnitude of Burgers vector (m) for each twin system b_tr, & !< magnitude of Burgers vector (m) for each transformation system Q_sl,& !< activation energy for glide (J) for each slip system v_0, & !< dislocation velocity prefactor (m/s) for each slip system dot_N_0_tw, & !< twin nucleation rate (1/m³s) for each twin system t_tw, & !< twin thickness (m) for each twin system i_sl, & !< Adj. parameter for distance between 2 forest dislocations for each slip system t_tr, & !< martensite lamellar thickness (m) for each trans system p, & !< p-exponent in glide velocity q, & !< q-exponent in glide velocity r, & !< exponent in twin nucleation rate s, & !< exponent in trans nucleation rate tau_0, & !< strength due to elements in solid solution gamma_char_tw, & !< characteristic shear for twins B, & !< drag coefficient d_caron !< distance of spontaneous annhihilation real(pREAL), allocatable, dimension(:,:) :: & h_sl_sl, & !< components of slip-slip interaction matrix h_sl_tw, & !< components of slip-twin interaction matrix h_sl_tr, & !< components of slip-trans interaction matrix h_tw_tw, & !< components of twin-twin interaction matrix h_tr_tr, & !< components of trans-trans interaction matrix n0_sl, & !< slip system normal forestProjection real(pREAL), allocatable, dimension(:,:,:) :: & P_sl, & P_tw, & P_tr integer :: & sum_N_sl, & !< total number of active slip systems sum_N_tw, & !< total number of active twin systems sum_N_tr !< total number of active transformation systems integer, allocatable, dimension(:) :: & N_tw, & N_tr integer, allocatable, dimension(:,:) :: & fcc_twinNucleationSlipPair ! ToDo: Better name? Is also used for trans character(len=:), allocatable :: & crystal_tr, & isotropic_bound character(len=pSTRLEN), allocatable, dimension(:) :: & output logical :: & 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 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 :: & rho_mob, & rho_dip, & gamma_sl, & f_tw, & f_tr end type tDislotwinState type :: tDislotwinDependentState real(pREAL), dimension(:,:), allocatable :: & 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 end type tDislotwinDependentState !-------------------------------------------------------------------------------------------------- ! containers for parameters and state type(tParameters), allocatable, dimension(:) :: param type(tIndexDotState), allocatable, dimension(:) :: indexDotState type(tDislotwinState), allocatable, dimension(:) :: state type(tDislotwinDependentState), allocatable, dimension(:) :: dependentState contains !-------------------------------------------------------------------------------------------------- !> @brief Perform module initialization. !> @details reads in material parameters, allocates arrays, and does sanity checks !-------------------------------------------------------------------------------------------------- module function plastic_dislotwin_init() result(myPlasticity) logical, dimension(:), allocatable :: myPlasticity integer :: & ph, i, & Nmembers, & sizeState, sizeDotState, & startIndex, endIndex integer, dimension(:), allocatable :: & N_sl real(pREAL) :: a_cF real(pREAL), allocatable, dimension(:) :: & f_edge, & !< edge character fraction of total dislocation density rho_mob_0, & !< initial unipolar dislocation density per slip system rho_dip_0 !< initial dipole dislocation density per slip system character(len=:), allocatable :: & refs, & extmsg type(tDict), pointer :: & phases, & phase, & mech, & pl myPlasticity = plastic_active('dislotwin') if (count(myPlasticity) == 0) return print'(/,1x,a)', '<<<+- phase:mechanical:plastic:dislotwin init -+>>>' print'(/,1x,a)', 'A. Ma and F. Roters, Acta Materialia 52(12):3603–3612, 2004' print'( 1x,a)', 'https://doi.org/10.1016/j.actamat.2004.04.012' print'(/,1x,a)', 'F. Roters et al., Computational Materials Science 39:91–95, 2007' print'( 1x,a)', 'https://doi.org/10.1016/j.commatsci.2006.04.014' print'(/,1x,a)', 'S.L. Wong et al., Acta Materialia 118:140–151, 2016' print'( 1x,a)', 'https://doi.org/10.1016/j.actamat.2016.07.032' print'(/,1x,a,1x,i0)', '# phases:',count(myPlasticity); flush(IO_STDOUT) phases => config_material%get_dict('phase') allocate(param(phases%length)) allocate(indexDotState(phases%length)) allocate(state(phases%length)) allocate(dependentState(phases%length)) extmsg = '' do ph = 1, phases%length if (.not. myPlasticity(ph)) cycle associate(prm => param(ph), & stt => state(ph), dst => dependentState(ph), & idx_dot => indexDotState(ph)) phase => phases%get_dict(ph) mech => phase%get_dict('mechanical') pl => mech%get_dict('plastic') print'(/,1x,a,1x,i0,a)', 'phase',ph,': '//phases%key(ph) refs = config_listReferences(pl,indent=3) if (len(refs) > 0) print'(/,1x,a)', refs #if defined (__GFORTRAN__) prm%output = output_as1dStr(pl) #else prm%output = pl%get_as1dStr('output',defaultVal=emptyStrArray) #endif prm%isotropic_bound = pl%get_asStr('isotropic_bound',defaultVal='isostrain') !-------------------------------------------------------------------------------------------------- ! slip related parameters N_sl = pl%get_as1dInt('N_sl',defaultVal=emptyIntArray) prm%sum_N_sl = sum(abs(N_sl)) slipActive: if (prm%sum_N_sl > 0) then prm%systems_sl = crystal_labels_slip(N_sl,phase_lattice(ph)) prm%P_sl = crystal_SchmidMatrix_slip(N_sl,phase_lattice(ph),phase_cOverA(ph)) prm%n0_sl = crystal_slip_normal(N_sl,phase_lattice(ph),phase_cOverA(ph)) prm%extendedDislocations = pl%get_asBool('extend_dislocations',defaultVal=.false.) prm%omitDipoles = pl%get_asBool('omit_dipoles', defaultVal=.false.) prm%Q_cl = pl%get_asReal('Q_cl') f_edge = math_expand(pl%get_as1dReal('f_edge', requiredSize=size(N_sl), & defaultVal=[(0.5_pREAL,i=1,size(N_sl))]),N_sl) rho_mob_0 = math_expand(pl%get_as1dReal('rho_mob_0', requiredSize=size(N_sl)),N_sl) rho_dip_0 = math_expand(pl%get_as1dReal('rho_dip_0', requiredSize=size(N_sl)),N_sl) prm%v_0 = math_expand(pl%get_as1dReal('v_0', requiredSize=size(N_sl)),N_sl) prm%b_sl = math_expand(pl%get_as1dReal('b_sl', requiredSize=size(N_sl)),N_sl) prm%Q_sl = math_expand(pl%get_as1dReal('Q_sl', requiredSize=size(N_sl)),N_sl) prm%i_sl = math_expand(pl%get_as1dReal('i_sl', requiredSize=size(N_sl)),N_sl) prm%p = math_expand(pl%get_as1dReal('p_sl', requiredSize=size(N_sl)),N_sl) prm%q = math_expand(pl%get_as1dReal('q_sl', requiredSize=size(N_sl)),N_sl) prm%tau_0 = math_expand(pl%get_as1dReal('tau_0', requiredSize=size(N_sl)),N_sl) prm%B = math_expand(pl%get_as1dReal('B', requiredSize=size(N_sl), & defaultVal=[(0.0_pREAL,i=1,size(N_sl))]),N_sl) prm%d_caron = prm%b_sl * pl%get_asReal('D_a') prm%h_sl_sl = crystal_interaction_SlipBySlip(N_sl,pl%get_as1dReal('h_sl-sl'),phase_lattice(ph)) prm%forestProjection = spread( f_edge,1,prm%sum_N_sl) & * crystal_forestProjection_edge (N_sl,phase_lattice(ph),phase_cOverA(ph)) & + spread(1.0_pREAL-f_edge,1,prm%sum_N_sl) & * crystal_forestProjection_screw(N_sl,phase_lattice(ph),phase_cOverA(ph)) prm%fccTwinTransNucleation = phase_lattice(ph) == 'cF' .and. N_sl(1) == 12 if (prm%fccTwinTransNucleation) prm%fcc_twinNucleationSlipPair = crystal_CF_TWINNUCLEATIONSLIPPAIR ! multiplication factor according to crystal structure (nearest neighbors bcc vs fcc/hex) ! details: Argon & Moffat, Acta Metallurgica, Vol. 29, pg 293 to 299, 1981 prm%omega = pl%get_asReal('omega', defaultVal=1000.0_pREAL) & * merge(12.0_pREAL,8.0_pREAL,any(phase_lattice(ph) == ['cF','hP'])) ! sanity checks if ( prm%Q_cl <= 0.0_pREAL) extmsg = trim(extmsg)//' Q_cl' if (any(rho_mob_0 < 0.0_pREAL)) extmsg = trim(extmsg)//' rho_mob_0' if (any(rho_dip_0 < 0.0_pREAL)) extmsg = trim(extmsg)//' rho_dip_0' if (any(prm%v_0 < 0.0_pREAL)) extmsg = trim(extmsg)//' v_0' if (any(prm%b_sl <= 0.0_pREAL)) extmsg = trim(extmsg)//' b_sl' if (any(prm%Q_sl <= 0.0_pREAL)) extmsg = trim(extmsg)//' Q_sl' if (any(prm%i_sl <= 0.0_pREAL)) extmsg = trim(extmsg)//' i_sl' if (any(prm%B < 0.0_pREAL)) extmsg = trim(extmsg)//' B' if (any(prm%d_caron < 0.0_pREAL)) extmsg = trim(extmsg)//' d_caron(D_a,b_sl)' if (any(prm%p<=0.0_pREAL .or. prm%p>1.0_pREAL)) extmsg = trim(extmsg)//' p_sl' if (any(prm%q< 1.0_pREAL .or. prm%q>2.0_pREAL)) extmsg = trim(extmsg)//' q_sl' else slipActive rho_mob_0 = emptyRealArray rho_dip_0 = emptyRealArray allocate(prm%v_0, & prm%b_sl, & prm%Q_sl, & prm%i_sl, & prm%p, & prm%q, & prm%tau_0, & prm%B, & source=emptyRealArray) allocate(prm%forestProjection(0,0), & prm%h_sl_sl(0,0)) ! PE: What about P_sl and systems_sl? end if slipActive !-------------------------------------------------------------------------------------------------- ! twin related parameters prm%N_tw = pl%get_as1dInt('N_tw', defaultVal=emptyIntArray) prm%sum_N_tw = sum(abs(prm%N_tw)) twinActive: if (prm%sum_N_tw > 0) then prm%systems_tw = crystal_labels_twin(prm%N_tw,phase_lattice(ph)) prm%P_tw = crystal_SchmidMatrix_twin(prm%N_tw,phase_lattice(ph),phase_cOverA(ph)) prm%gamma_char_tw = crystal_characteristicShear_Twin(prm%N_tw,phase_lattice(ph),phase_cOverA(ph)) prm%L_tw = pl%get_asReal('L_tw') prm%i_tw = pl%get_asReal('i_tw') prm%b_tw = math_expand(pl%get_as1dReal('b_tw', requiredSize=size(prm%N_tw)),prm%N_tw) prm%t_tw = math_expand(pl%get_as1dReal('t_tw', requiredSize=size(prm%N_tw)),prm%N_tw) prm%r = math_expand(pl%get_as1dReal('p_tw', requiredSize=size(prm%N_tw)),prm%N_tw) prm%h_tw_tw = crystal_interaction_TwinByTwin(prm%N_tw,pl%get_as1dReal('h_tw-tw'), & phase_lattice(ph)) ! sanity checks 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' else twinActive 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 !-------------------------------------------------------------------------------------------------- ! transformation related parameters prm%N_tr = pl%get_as1dInt('N_tr', defaultVal=emptyIntArray) prm%sum_N_tr = sum(abs(prm%N_tr)) transActive: if (prm%sum_N_tr > 0) then prm%P_tr = crystal_SchmidMatrix_trans(prm%N_tr,'hP',prm%cOverA_hP) prm%Delta_G = polynomial(pl,'Delta_G','T') prm%i_tr = pl%get_asReal('i_tr') prm%L_tr = pl%get_asReal('L_tr') prm%cOverA_hP = pl%get_asReal('c/a_hP') prm%V_mol = pl%get_asReal('V_mol') prm%b_tr = math_expand(pl%get_as1dReal('b_tr'),prm%N_tr) prm%t_tr = math_expand(pl%get_as1dReal('t_tr'),prm%N_tr) prm%s = math_expand(pl%get_as1dReal('p_tr'),prm%N_tr) a_cF = prm%b_tr(1)*sqrt(6.0_pREAL) ! b_tr is Shockley partial prm%h = 5.0_pREAL * a_cF/sqrt(3.0_pREAL) prm%rho = 4.0_pREAL/(sqrt(3.0_pREAL)*a_cF**2)/N_A prm%h_tr_tr = crystal_interaction_TransByTrans(prm%N_tr,pl%get_as1dReal('h_tr-tr'),& phase_lattice(ph)) ! sanity checks 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' else transActive allocate(prm%s,prm%b_tr,prm%t_tr,source=emptyRealArray) allocate(prm%h_tr_tr(0,0)) end if transActive !-------------------------------------------------------------------------------------------------- ! shearband related parameters prm%gamma_0_sb = pl%get_asReal('gamma_0_sb',defaultVal=0.0_pREAL) if (prm%gamma_0_sb > 0.0_pREAL) then prm%tau_sb = pl%get_asReal('tau_sb') prm%E_sb = pl%get_asReal('Q_sb') prm%p_sb = pl%get_asReal('p_sb') prm%q_sb = pl%get_asReal('q_sb') ! sanity checks if (prm%tau_sb < 0.0_pREAL) extmsg = trim(extmsg)//' tau_sb' if (prm%E_sb < 0.0_pREAL) extmsg = trim(extmsg)//' Q_sb' if (prm%p_sb <= 0.0_pREAL) extmsg = trim(extmsg)//' p_sb' if (prm%q_sb <= 0.0_pREAL) extmsg = trim(extmsg)//' q_sb' end if !-------------------------------------------------------------------------------------------------- ! parameters required for several mechanisms and their interactions if (prm%sum_N_sl + prm%sum_N_tw + prm%sum_N_tr > 0) & prm%D = pl%get_asReal('D') if (prm%sum_N_tw + prm%sum_N_tr > 0) then prm%x_c = pl%get_asReal('x_c') prm%V_cs = pl%get_asReal('V_cs') 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,'Gamma_sf','T') slipAndTwinActive: if (prm%sum_N_sl * prm%sum_N_tw > 0) then prm%h_sl_tw = crystal_interaction_SlipByTwin(N_sl,prm%N_tw,pl%get_as1dReal('h_sl-tw'), & phase_lattice(ph)) if (prm%fccTwinTransNucleation .and. size(prm%N_tw) /= 1) extmsg = trim(extmsg)//' N_tw: nucleation' end if slipAndTwinActive slipAndTransActive: if (prm%sum_N_sl * prm%sum_N_tr > 0) then prm%h_sl_tr = crystal_interaction_SlipByTrans(N_sl,prm%N_tr,pl%get_as1dReal('h_sl-tr'), & phase_lattice(ph)) if (prm%fccTwinTransNucleation .and. size(prm%N_tr) /= 1) extmsg = trim(extmsg)//' N_tr: nucleation' end if slipAndTransActive twinAndTransActive: if (prm%sum_N_tw * prm%sum_N_tr > 0) then if (dNeq(prm%b_tw(1),prm%b_tr(1))) extmsg = trim(extmsg)//' b_tw != b_tr' end if twinAndTransActive !-------------------------------------------------------------------------------------------------- ! allocate state arrays Nmembers = count(material_ID_phase == ph) sizeDotState = size(['rho_mob ','rho_dip ','gamma_sl']) * prm%sum_N_sl & + size(['f_tw']) * prm%sum_N_tw & + 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 !-------------------------------------------------------------------------------------------------- ! 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) plasticState(ph)%atol(startIndex:endIndex) = pl%get_asReal('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) plasticState(ph)%atol(startIndex:endIndex) = pl%get_asReal('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,:) plasticState(ph)%atol(startIndex:endIndex) = pl%get_asReal('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,:) plasticState(ph)%atol(startIndex:endIndex) = pl%get_asReal('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,:) plasticState(ph)%atol(startIndex:endIndex) = pl%get_asReal('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%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%Lambda_tr(prm%sum_N_tr,Nmembers),source=0.0_pREAL) end associate !-------------------------------------------------------------------------------------------------- ! exit if any parameter is out of range if (extmsg /= '') call IO_error(211,ext_msg=trim(extmsg)) end do end function plastic_dislotwin_init !-------------------------------------------------------------------------------------------------- !> @brief Return the homogenized elasticity matrix. !-------------------------------------------------------------------------------------------------- module function plastic_dislotwin_homogenizedC(ph,en) result(homogenizedC) integer, intent(in) :: & ph, en real(pREAL), dimension(6,6) :: & homogenizedC, & C real(pREAL), dimension(:,:,:), allocatable :: & C66_tw, & C66_tr integer :: i real(pREAL) :: f_matrix C = elastic_C66(ph,en) associate(prm => param(ph), stt => state(ph)) 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)) homogenizedC = f_matrix * C twinActive: if (prm%sum_N_tw > 0) then C66_tw = crystal_C66_twin(prm%N_tw,C,phase_lattice(ph),phase_cOverA(ph)) do i = 1, prm%sum_N_tw homogenizedC = homogenizedC & + stt%f_tw(i,en)*C66_tw(1:6,1:6,i) end do end if twinActive transActive: if (prm%sum_N_tr > 0) then C66_tr = crystal_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) end do end if transActive end associate end function plastic_dislotwin_homogenizedC !-------------------------------------------------------------------------------------------------- !> @brief Calculate plastic velocity gradient and its tangent. !-------------------------------------------------------------------------------------------------- module subroutine dislotwin_LpAndItsTangent(Lp,dLp_dMp,Mp,ph,en) real(pREAL), dimension(3,3), intent(out) :: Lp real(pREAL), dimension(3,3,3,3), intent(out) :: dLp_dMp real(pREAL), dimension(3,3), intent(in) :: Mp integer, intent(in) :: ph,en integer :: i,k,l,m,n real(pREAL) :: & f_matrix,StressRatio_p,& E_kB_T, & ddot_gamma_dtau, & tau, & T real(pREAL), dimension(param(ph)%sum_N_sl) :: & dot_gamma_sl,ddot_gamma_dtau_sl real(pREAL), dimension(param(ph)%sum_N_tw) :: & dot_gamma_tw,ddot_gamma_dtau_tw real(pREAL), dimension(param(ph)%sum_N_tr) :: & dot_gamma_tr,ddot_gamma_dtau_tr real(pREAL):: dot_gamma_sb real(pREAL), dimension(3,3) :: eigVectors, P_sb real(pREAL), dimension(3) :: eigValues real(pREAL), dimension(3,6), parameter :: & sb_sComposition = & reshape(real([& 1, 0, 1, & 1, 0,-1, & 1, 1, 0, & 1,-1, 0, & 0, 1, 1, & 0, 1,-1 & ],pREAL),[ 3,6]), & sb_mComposition = & reshape(real([& 1, 0,-1, & 1, 0,+1, & 1,-1, 0, & 1, 1, 0, & 0, 1,-1, & 0, 1, 1 & ],pREAL),[ 3,6]) T = thermal_T(ph,en) Lp = 0.0_pREAL dLp_dMp = 0.0_pREAL associate(prm => param(ph), stt => state(ph)) 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,ddot_gamma_dtau_sl) slipContribution: do i = 1, prm%sum_N_sl Lp = Lp + dot_gamma_sl(i)*prm%P_sl(1:3,1:3,i) forall (k=1:3,l=1:3,m=1:3,n=1:3) & dLp_dMp(k,l,m,n) = dLp_dMp(k,l,m,n) & + ddot_gamma_dtau_sl(i) * prm%P_sl(k,l,i) * prm%P_sl(m,n,i) end do slipContribution 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) & dLp_dMp(k,l,m,n) = dLp_dMp(k,l,m,n) & + ddot_gamma_dtau_tw(i)* prm%P_tw(k,l,i)*prm%P_tw(m,n,i) end do twinContibution 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) & dLp_dMp(k,l,m,n) = dLp_dMp(k,l,m,n) & + ddot_gamma_dtau_tr(i)* prm%P_tr(k,l,i)*prm%P_tr(m,n,i) end do transContibution Lp = Lp * f_matrix dLp_dMp = dLp_dMp * f_matrix shearBandingContribution: if (dNeq0(prm%gamma_0_sb)) then E_kB_T = prm%E_sb/(K_B*T) call math_eigh33(eigValues,eigVectors,Mp) ! is Mp symmetric by design? do i = 1,6 P_sb = 0.5_pREAL * math_outer(matmul(eigVectors,sb_sComposition(1:3,i)),& matmul(eigVectors,sb_mComposition(1:3,i))) tau = math_tensordot(Mp,P_sb) significantShearBandStress: if (abs(tau) > tol_math_check) then StressRatio_p = (abs(tau)/prm%tau_sb)**prm%p_sb dot_gamma_sb = sign(prm%gamma_0_sb*exp(-E_kB_T*(1-StressRatio_p)**prm%q_sb), tau) ddot_gamma_dtau = abs(dot_gamma_sb)*E_kB_T*prm%p_sb*prm%q_sb/prm%tau_sb & * (abs(tau)/prm%tau_sb)**(prm%p_sb-1.0_pREAL) & * (1.0_pREAL-StressRatio_p)**(prm%q_sb-1.0_pREAL) Lp = Lp + dot_gamma_sb * P_sb forall (k=1:3,l=1:3,m=1:3,n=1:3) & dLp_dMp(k,l,m,n) = dLp_dMp(k,l,m,n) & + ddot_gamma_dtau * P_sb(k,l) * P_sb(m,n) end if significantShearBandStress end do end if shearBandingContribution end associate end subroutine dislotwin_LpAndItsTangent !-------------------------------------------------------------------------------------------------- !> @brief Calculate the rate of change of microstructure. !-------------------------------------------------------------------------------------------------- module function dislotwin_dotState(Mp,ph,en) result(dotState) real(pREAL), dimension(3,3), intent(in):: & Mp !< Mandel stress integer, intent(in) :: & ph, & en real(pREAL), dimension(plasticState(ph)%sizeDotState) :: & dotState integer :: i real(pREAL) :: & f_matrix, & d_hat, & v_cl, & !< climb velocity tau, & sigma_cl, & !< climb stress b_d !< ratio of Burgers vector to stacking fault width real(pREAL), dimension(param(ph)%sum_N_sl) :: & dot_rho_dip_formation, & dot_rho_dip_climb, & dot_gamma_sl real(pREAL), dimension(param(ph)%sum_N_tw) :: & dot_gamma_tw real(pREAL), dimension(param(ph)%sum_N_tr) :: & dot_gamma_tr real(pREAL) :: & mu, nu, & T 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,prm%isotropic_bound) nu = elastic_nu(ph,en,prm%isotropic_bound) 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) 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)) significantSlipStress: if (dEq0(tau) .or. prm%omitDipoles) then dot_rho_dip_formation(i) = 0.0_pREAL dot_rho_dip_climb(i) = 0.0_pREAL else significantSlipStress d_hat = 3.0_pREAL*mu*prm%b_sl(i)/(16.0_pREAL*PI*abs(tau)) d_hat = math_clip(d_hat, right = dst%Lambda_sl(i,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) 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))) 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) dot_rho_dip_climb(i) = 4.0_pREAL*v_cl*stt%rho_dip(i,en) & / (d_hat-prm%d_caron(i)) end if end if significantSlipStress end do slipState 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 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 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 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/gamma_char_tr end associate end function dislotwin_dotState !-------------------------------------------------------------------------------------------------- !> @brief Calculate derived quantities from state. !-------------------------------------------------------------------------------------------------- module subroutine dislotwin_dependentState(ph,en) integer, intent(in) :: & ph, & en real(pREAL) :: & sumf_tw, sumf_tr real(pREAL), dimension(param(ph)%sum_N_sl) :: & inv_lambda_sl real(pREAL), dimension(param(ph)%sum_N_tw) :: & inv_lambda_tw_tw, & !< 1/mean free distance between 2 twin stacks from different systems seen by a growing twin f_over_t_tw real(pREAL), dimension(param(ph)%sum_N_tr) :: & 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 associate(prm => param(ph), stt => state(ph), dst => dependentState(ph)) mu = elastic_mu(ph,en,prm%isotropic_bound) sumf_tw = sum(stt%f_tw(1:prm%sum_N_tw,en)) sumf_tr = sum(stt%f_tr(1:prm%sum_N_tr,en)) !* 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 ! ToDo ...Physically correct, but naming could be adjusted inv_lambda_sl = sqrt(matmul(prm%forestProjection,stt%rho_mob(:,en)+stt%rho_dip(:,en)))/prm%i_sl if (prm%sum_N_tw > 0 .and. prm%sum_N_sl > 0) & inv_lambda_sl = inv_lambda_sl + matmul(prm%h_sl_tw,f_over_t_tw)/(1.0_pREAL-sumf_tw) if (prm%sum_N_tr > 0 .and. prm%sum_N_sl > 0) & inv_lambda_sl = inv_lambda_sl + matmul(prm%h_sl_tr,f_over_t_tr)/(1.0_pREAL-sumf_tr) dst%Lambda_sl(:,en) = prm%D / (1.0_pREAL+prm%D*inv_lambda_sl) inv_lambda_tw_tw = matmul(prm%h_tw_tw,f_over_t_tw)/(1.0_pREAL-sumf_tw) dst%Lambda_tw(:,en) = prm%i_tw*prm%D/(1.0_pREAL+prm%D*inv_lambda_tw_tw) inv_lambda_tr_tr = matmul(prm%h_tr_tr,f_over_t_tr)/(1.0_pREAL-sumf_tr) dst%Lambda_tr(:,en) = prm%i_tr*prm%D/(1.0_pREAL+prm%D*inv_lambda_tr_tr) !* 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))) end associate end subroutine dislotwin_dependentState !-------------------------------------------------------------------------------------------------- !> @brief Write results to HDF5 output file. !-------------------------------------------------------------------------------------------------- module subroutine plastic_dislotwin_result(ph,group) integer, intent(in) :: ph character(len=*), intent(in) :: group integer :: ou associate(prm => param(ph), stt => state(ph), dst => dependentState(ph)) do ou = 1,size(prm%output) select case(trim(prm%output(ou))) case('rho_mob') call result_writeDataset(stt%rho_mob,group,trim(prm%output(ou)), & 'mobile dislocation density','1/m²',prm%systems_sl) case('rho_dip') call result_writeDataset(stt%rho_dip,group,trim(prm%output(ou)), & 'dislocation dipole density','1/m²',prm%systems_sl) case('gamma_sl') call result_writeDataset(stt%gamma_sl,group,trim(prm%output(ou)), & 'plastic shear','1',prm%systems_sl) case('Lambda_sl') call result_writeDataset(dst%Lambda_sl,group,trim(prm%output(ou)), & 'mean free path for slip','m',prm%systems_sl) case('tau_pass') call result_writeDataset(dst%tau_pass,group,trim(prm%output(ou)), & 'passing stress for slip','Pa',prm%systems_sl) case('f_tw') call result_writeDataset(stt%f_tw,group,trim(prm%output(ou)), & 'twinned volume fraction','m³/m³',prm%systems_tw) case('Lambda_tw') call result_writeDataset(dst%Lambda_tw,group,trim(prm%output(ou)), & 'mean free path for twinning','m',prm%systems_tw) case('f_tr') if (prm%sum_N_tr>0) call result_writeDataset(stt%f_tr,group,trim(prm%output(ou)), & 'martensite volume fraction','m³/m³') end select end do end associate end subroutine plastic_dislotwin_result !-------------------------------------------------------------------------------------------------- !> @brief Calculate shear rates on slip systems, their derivatives with respect to resolved ! stress, and the resolved stress. !> @details Derivatives and resolved stress are calculated only optionally. ! NOTE: Contrary to common convention, here the result (i.e. intent(out)) variables have to be put ! at the end since some of them are optional. !-------------------------------------------------------------------------------------------------- pure subroutine kinetics_sl(Mp,T,ph,en, & dot_gamma_sl,ddot_gamma_dtau_sl,tau_sl) real(pREAL), dimension(3,3), intent(in) :: & Mp !< Mandel stress real(pREAL), intent(in) :: & T !< temperature 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, & StressRatio_p, & Q_kB_T, & v_wait_inverse, & !< inverse of the effective velocity of a dislocation waiting at obstacles (unsigned) v_run_inverse, & !< inverse of the velocity of a free moving dislocation (unsigned) dV_wait_inverse_dTau, & dV_run_inverse_dTau, & dV_dTau, & tau_eff !< effective resolved stress integer :: i associate(prm => param(ph), stt => state(ph), dst => dependentState(ph)) tau = [(math_tensordot(Mp,prm%P_sl(1:3,1:3,i)),i = 1, prm%sum_N_sl)] tau_eff = abs(tau)-dst%tau_pass(:,en) significantStress: where(tau_eff > tol_math_check) stressRatio = tau_eff/prm%tau_0 StressRatio_p = stressRatio** prm%p Q_kB_T = prm%Q_sl/(K_B*T) v_wait_inverse = exp(Q_kB_T*(1.0_pREAL-StressRatio_p)** prm%q) & / prm%v_0 v_run_inverse = prm%B/(tau_eff*prm%b_sl) dot_gamma_sl = sign(stt%rho_mob(:,en)*prm%b_sl/(v_wait_inverse+v_run_inverse),tau) dV_wait_inverse_dTau = -1.0_pREAL * v_wait_inverse * prm%p * prm%q * Q_kB_T & * (stressRatio**(prm%p-1.0_pREAL)) & * (1.0_pREAL-StressRatio_p)**(prm%q-1.0_pREAL) & / prm%tau_0 dV_run_inverse_dTau = -1.0_pREAL * v_run_inverse/tau_eff dV_dTau = -1.0_pREAL * (dV_wait_inverse_dTau+dV_run_inverse_dTau) & / (v_wait_inverse+v_run_inverse)**2 ddot_gamma_dtau = dV_dTau*stt%rho_mob(:,en)*prm%b_sl else where significantStress dot_gamma_sl = 0.0_pREAL ddot_gamma_dtau = 0.0_pREAL end where significantStress end associate if (present(ddot_gamma_dtau_sl)) ddot_gamma_dtau_sl = ddot_gamma_dtau if (present(tau_sl)) tau_sl = tau end subroutine kinetics_sl !-------------------------------------------------------------------------------------------------- !> @brief Calculate shear rates on twin systems and their derivatives with respect to resolved ! stress. !> @details Derivatives are calculated only optionally. ! NOTE: Contrary to common convention, here the result (i.e. intent(out)) variables have to be put ! at the end since some of them are optional. !-------------------------------------------------------------------------------------------------- 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) :: & Mp !< Mandel stress real(pREAL), intent(in) :: & T !< temperature integer, intent(in) :: & ph, & en real(pREAL), dimension(param(ph)%sum_N_sl), intent(in) :: & 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(pREAL) :: & tau, tau_r, tau_hat, & dot_N_0, & x0, V, & Gamma_sf, & mu, nu, & P_ncs, dP_ncs_dtau, & P, dP_dtau integer, dimension(2) :: & s integer :: i associate(prm => param(ph), stt => state(ph), dst => dependentState(ph)) mu = elastic_mu(ph,en,prm%isotropic_bound) nu = elastic_nu(ph,en,prm%isotropic_bound) 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)) if (tau > tol_math_check .and. tau < tau_r) then 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*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*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)*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 end associate end subroutine kinetics_tw !-------------------------------------------------------------------------------------------------- !> @brief Calculate shear rates on transformation systems and their derivatives with respect to ! resolved stress. !> @details Derivatives are calculated only optionally. ! NOTE: Contrary to common convention, here the result (i.e. intent(out)) variables have to be put ! at the end since some of them are optional. !-------------------------------------------------------------------------------------------------- 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) :: & Mp !< Mandel stress real(pREAL), intent(in) :: & T !< temperature integer, intent(in) :: & ph, & en real(pREAL), dimension(param(ph)%sum_N_sl), intent(in) :: & 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(pREAL) :: & tau, tau_r, tau_hat, & dot_N_0, & x0, V, & Gamma_sf, & mu, nu, & P_ncs, dP_ncs_dtau, & P, dP_dtau integer, dimension(2) :: & s integer :: i associate(prm => param(ph), stt => state(ph), dst => dependentState(ph)) mu = elastic_mu(ph,en,prm%isotropic_bound) nu = elastic_nu(ph,en,prm%isotropic_bound) 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)) if (tau > tol_math_check .and. tau < tau_r) then 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*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*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)*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 end if end do end associate end subroutine kinetics_tr end submodule dislotwin