!-------------------------------------------------------------------------------------------------- !> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH !> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH !> @author David Cereceda, Lawrence Livermore National Laboratory !> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH !> @brief crystal plasticity model for bcc metals, especially Tungsten !-------------------------------------------------------------------------------------------------- submodule(phase:plastic) dislotungsten type :: tParameters real(pREAL) :: & D = 1.0_pREAL, & !< grain size D_0 = 1.0_pREAL, & !< prefactor for self-diffusion coefficient Q_cl = 1.0_pREAL !< activation energy for dislocation climb real(pREAL), allocatable, dimension(:) :: & b_sl, & !< magnitude of Burgers vector [m] d_caron, & !< distance of spontaneous annhihilation i_sl, & !< Adj. parameter for distance between 2 forest dislocations f_at, & !< factor to calculate atomic volume tau_Peierls, & !< Peierls stress !* mobility law parameters Q_s, & !< activation energy for glide [J] p, & !< p-exponent in glide velocity q, & !< q-exponent in glide velocity B, & !< friction coefficient h, & !< height of the kink pair w, & !< width of the kink pair omega !< attempt frequency for kink pair nucleation real(pREAL), allocatable, dimension(:,:) :: & h_sl_sl, & !< slip resistance from slip activity forestProjection real(pREAL), allocatable, dimension(:,:,:) :: & P_sl, & P_nS_pos, & P_nS_neg integer :: & sum_N_sl !< total number of active slip system character(len=:), allocatable :: & isotropic_bound character(len=pSTRLEN), allocatable, dimension(:) :: & output logical :: & dipoleFormation !< flag indicating consideration of dipole formation character(len=:), allocatable, dimension(:) :: & systems_sl end type tParameters !< container type for internal constitutive parameters type :: tIndexDotState integer, dimension(2) :: & rho_mob, & rho_dip, & gamma_sl end type tIndexDotState type :: tDislotungstenState real(pREAL), dimension(:,:), pointer :: & rho_mob, & rho_dip, & gamma_sl end type tDislotungstenState type :: tDislotungstenDependentState real(pREAL), dimension(:,:), allocatable :: & Lambda_sl, & tau_pass end type tDislotungstenDependentState !-------------------------------------------------------------------------------------------------- ! containers for parameters and state type(tParameters), allocatable, dimension(:) :: param type(tIndexDotState), allocatable, dimension(:) :: indexDotState type(tDisloTungstenState), allocatable, dimension(:) :: state type(tDisloTungstenDependentState), allocatable, dimension(:) :: dependentState contains !-------------------------------------------------------------------------------------------------- !> @brief Perform module initialization. !> @details reads in material parameters, allocates arrays, and does sanity checks !-------------------------------------------------------------------------------------------------- module function plastic_dislotungsten_init() result(myPlasticity) logical, dimension(:), allocatable :: myPlasticity integer :: & ph, i, & Nmembers, & sizeState, sizeDotState, & startIndex, endIndex integer, dimension(:), allocatable :: & N_sl real(pREAL),dimension(:), allocatable :: & f_edge, & !< edge character fraction of total dislocation density rho_mob_0, & !< initial dislocation density rho_dip_0 !< initial dipole density real(pREAL), dimension(:,:), allocatable :: & a_nS !< non-Schmid coefficients character(len=:), allocatable :: & refs, & extmsg type(tDict), pointer :: & phases, & phase, & mech, & pl myPlasticity = plastic_active('dislotungsten') if (count(myPlasticity) == 0) return print'(/,1x,a)', '<<<+- phase:mechanical:plastic:dislotungsten init -+>>>' print'(/,1x,a)', 'D. Cereceda et al., International Journal of Plasticity 78:242–256, 2016' print'( 1x,a)', 'https://doi.org/10.1016/j.ijplas.2015.09.002' 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)) if (phase_lattice(ph) == 'cI') then allocate(a_nS(3,size(pl%get_as1dReal('a_nonSchmid_110',defaultVal=emptyRealArray)))) a_nS(1,:) = pl%get_as1dReal('a_nonSchmid_110',defaultVal=emptyRealArray) prm%P_nS_pos = crystal_SchmidMatrix_slip(N_sl,phase_lattice(ph),phase_cOverA(ph),nonSchmidCoefficients=a_nS,sense=+1) prm%P_nS_neg = crystal_SchmidMatrix_slip(N_sl,phase_lattice(ph),phase_cOverA(ph),nonSchmidCoefficients=a_nS,sense=-1) deallocate(a_nS) else prm%P_nS_pos = +prm%P_sl prm%P_nS_neg = -prm%P_sl end if prm%dipoleformation = .not. pl%get_asBool('no_dipole_formation', defaultVal=.false.) prm%D = pl%get_asReal('D') prm%D_0 = pl%get_asReal('D_0') 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%b_sl = math_expand(pl%get_as1dReal('b_sl', requiredSize=size(N_sl)),N_sl) prm%Q_s = math_expand(pl%get_as1dReal('Q_s', requiredSize=size(N_sl)),N_sl) prm%i_sl = math_expand(pl%get_as1dReal('i_sl', requiredSize=size(N_sl)),N_sl) prm%tau_Peierls = math_expand(pl%get_as1dReal('tau_Peierls', 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%h = math_expand(pl%get_as1dReal('h', requiredSize=size(N_sl)),N_sl) prm%w = math_expand(pl%get_as1dReal('w', requiredSize=size(N_sl)),N_sl) prm%omega = math_expand(pl%get_as1dReal('omega', requiredSize=size(N_sl)),N_sl) prm%B = math_expand(pl%get_as1dReal('B', requiredSize=size(N_sl)),N_sl) prm%d_caron = prm%b_sl * pl%get_asReal('D_a') prm%f_at = prm%b_sl**3*pl%get_asReal('f_at') 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)) ! sanity checks if ( prm%D_0 < 0.0_pREAL) extmsg = trim(extmsg)//' D_0' 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%b_sl <= 0.0_pREAL)) extmsg = trim(extmsg)//' b_sl' if (any(prm%Q_s <= 0.0_pREAL)) extmsg = trim(extmsg)//' Q_s' if (any(prm%tau_Peierls < 0.0_pREAL)) extmsg = trim(extmsg)//' tau_Peierls' 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%f_at <= 0.0_pREAL)) extmsg = trim(extmsg)//' f_at or b_sl' else slipActive rho_mob_0 = emptyRealArray rho_dip_0 = emptyRealArray allocate(prm%b_sl, & prm%d_caron, & prm%i_sl, & prm%f_at, & prm%tau_Peierls, & prm%Q_s, & prm%p, & prm%q, & prm%B, & prm%h, & prm%w, & prm%omega, & source = emptyRealArray) allocate(prm%forestProjection(0,0)) allocate(prm%h_sl_sl (0,0)) end if slipActive !-------------------------------------------------------------------------------------------------- ! allocate state arrays Nmembers = count(material_ID_phase == ph) sizeDotState = size(['rho_mob ','rho_dip ','gamma_sl']) * prm%sum_N_sl 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' 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) 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_dislotungsten_init !-------------------------------------------------------------------------------------------------- !> @brief Calculate plastic velocity gradient and its tangent. !-------------------------------------------------------------------------------------------------- pure module subroutine dislotungsten_LpAndItsTangent(Lp,dLp_dMp, & Mp,ph,en) real(pREAL), dimension(3,3), intent(out) :: & Lp !< plastic velocity gradient real(pREAL), dimension(3,3,3,3), intent(out) :: & dLp_dMp !< derivative of Lp with respect to the Mandel stress real(pREAL), dimension(3,3), intent(in) :: & Mp !< Mandel stress integer, intent(in) :: & ph, & en integer :: & i,k,l,m,n real(pREAL) :: & T !< temperature real(pREAL), dimension(param(ph)%sum_N_sl) :: & dot_gamma, ddot_gamma_dtau real(pREAL), dimension(3,3,param(ph)%sum_N_sl) :: & P_nS T = thermal_T(ph,en) Lp = 0.0_pREAL dLp_dMp = 0.0_pREAL associate(prm => param(ph)) call kinetics(Mp,T,ph,en, dot_gamma,ddot_gamma_dtau) P_nS = merge(prm%P_nS_pos,prm%P_nS_neg, spread(spread(dot_gamma,1,3),2,3)>0.0_pREAL) ! faster than 'merge' in loop do i = 1, prm%sum_N_sl Lp = Lp + dot_gamma(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(i) * prm%P_sl(k,l,i) * P_nS(m,n,i) end do end associate end subroutine dislotungsten_LpAndItsTangent !-------------------------------------------------------------------------------------------------- !> @brief Calculate the rate of change of microstructure. !-------------------------------------------------------------------------------------------------- module function dislotungsten_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 real(pREAL), dimension(param(ph)%sum_N_sl) :: & tau_eff, & v_cl, & dot_rho_dip_formation, & dot_rho_dip_climb, & d_hat 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)), & dot_gamma => dotState(indexDotState(ph)%gamma_sl(1):indexDotState(ph)%gamma_sl(2))) mu = elastic_mu(ph,en,prm%isotropic_bound) nu = elastic_nu(ph,en,prm%isotropic_bound) T = thermal_T(ph,en) call kinetics(Mp,T,ph,en,& dot_gamma, tau = tau_eff) dot_gamma = abs(dot_gamma) where(dEq0(dot_gamma)) dot_rho_dip_formation = 0.0_pREAL dot_rho_dip_climb = 0.0_pREAL else where d_hat = math_clip(mu*prm%b_sl/(8.0_pREAL*PI*(1.0_pREAL-nu)*tau_eff), & left = prm%d_caron, & ! lower limit right = dst%Lambda_sl(:,en)) ! upper limit dot_rho_dip_formation = merge(dot_gamma * 2.0_pREAL*(d_hat-prm%d_caron)/prm%b_sl * stt%rho_mob(:,en), & 0.0_pREAL, & prm%dipoleformation) v_cl = (3.0_pREAL*mu*prm%D_0*exp(-prm%Q_cl/(K_B*T))*prm%f_at/(2.0_pREAL*PI*K_B*T)) & * (1.0_pREAL/(d_hat+prm%d_caron)) dot_rho_dip_climb = (4.0_pREAL*v_cl*stt%rho_dip(:,en))/(d_hat-prm%d_caron) ! ToDo: Discuss with Franz: Stress dependency? end where dot_rho_mob = dot_gamma / (prm%b_sl*dst%Lambda_sl(:,en)) & ! multiplication - dot_rho_dip_formation & - dot_gamma * 2.0_pREAL*prm%d_caron/prm%b_sl * stt%rho_mob(:,en) ! spontaneous annihilation of 2 edges dot_rho_dip = dot_rho_dip_formation & - dot_rho_dip_climb & - dot_gamma * 2.0_pREAL*prm%d_caron/prm%b_sl * stt%rho_dip(:,en) ! spontaneous annihilation of an edge with a dipole end associate end function dislotungsten_dotState !-------------------------------------------------------------------------------------------------- !> @brief Calculate derived quantities from state. !-------------------------------------------------------------------------------------------------- module subroutine dislotungsten_dependentState(ph,en) integer, intent(in) :: & ph, & en real(pREAL), dimension(param(ph)%sum_N_sl) :: & Lambda_sl_inv associate(prm => param(ph), stt => state(ph), dst => dependentState(ph)) dst%tau_pass(:,en) = elastic_mu(ph,en,prm%isotropic_bound)*prm%b_sl & * sqrt(matmul(prm%h_sl_sl,stt%rho_mob(:,en)+stt%rho_dip(:,en))) Lambda_sl_inv = 1.0_pREAL/prm%D & + sqrt(matmul(prm%forestProjection,stt%rho_mob(:,en)+stt%rho_dip(:,en)))/prm%i_sl dst%Lambda_sl(:,en) = Lambda_sl_inv**(-1.0_pREAL) end associate end subroutine dislotungsten_dependentState !-------------------------------------------------------------------------------------------------- !> @brief Write results to HDF5 output file. !-------------------------------------------------------------------------------------------------- module subroutine plastic_dislotungsten_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)), & 'threshold stress for slip','Pa',prm%systems_sl) end select end do end associate end subroutine plastic_dislotungsten_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(Mp,T,ph,en, & dot_gamma,ddot_gamma_dtau,tau) 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 real(pREAL), dimension(param(ph)%sum_N_sl), optional, intent(out) :: & ddot_gamma_dtau, & tau real(pREAL), dimension(param(ph)%sum_N_sl) :: & StressRatio, & StressRatio_p,StressRatio_pminus1, & tau_pos, tau_neg, tau_eff, & t_n,t_k, dtk,dtn integer :: i associate(prm => param(ph), stt => state(ph), dst => dependentState(ph)) tau_pos = [(math_tensordot(Mp,prm%P_nS_pos(1:3,1:3,i)),i=1,prm%sum_N_sl)] tau_neg = [(math_tensordot(Mp,prm%P_nS_neg(1:3,1:3,i)),i=1,prm%sum_N_sl)] tau_eff = math_clip(max(tau_pos,tau_neg) - dst%tau_pass(:,en),left = 0.0_pREAL) if (present(tau)) tau = tau_eff associate(BoltzmannRatio => prm%Q_s/(K_B*T), & b_rho => stt%rho_mob(:,en) * prm%b_sl, & effectiveLength => dst%Lambda_sl(:,en) - prm%w) where(tau_eff > tol_math_check) StressRatio = tau_eff/prm%tau_Peierls StressRatio_p = StressRatio** prm%p StressRatio_pminus1 = StressRatio**(prm%p-1.0_pREAL) t_n = prm%b_sl*exp(BoltzmannRatio*(1.0_pREAL-StressRatio_p) ** prm%q) & / (prm%omega*effectiveLength) t_k = effectiveLength * prm%B /(2.0_pREAL*prm%b_sl*tau_eff) ! corrected eq. (14) dot_gamma = b_rho * prm%h/(t_n + t_k) * merge(+1.0_pREAL,-1.0_pREAL, tau_pos>tau_neg) else where dot_gamma = 0.0_pREAL end where if (present(ddot_gamma_dtau)) then where(tau_eff > tol_math_check) dtn = -1.0_pREAL * t_n * BoltzmannRatio * prm%p * prm%q * (1.0_pREAL-StressRatio_p)**(prm%q - 1.0_pREAL) & * StressRatio_pminus1 / prm%tau_Peierls dtk = -1.0_pREAL * t_k / tau_eff ddot_gamma_dtau = -1.0_pREAL * dot_gamma * (dtn + dtk) / (t_n + t_k) else where ddot_gamma_dtau = 0.0_pREAL end where end if end associate end associate end subroutine kinetics end submodule dislotungsten