!-------------------------------------------------------------------------------------------------- !> @author Christoph Kords, 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 for plasticity including dislocation flux !-------------------------------------------------------------------------------------------------- submodule(phase:plastic) nonlocal use geometry_plastic_nonlocal, only: & nIPneighbors => geometry_plastic_nonlocal_nIPneighbors, & IPneighborhood => geometry_plastic_nonlocal_IPneighborhood, & IPvolume => geometry_plastic_nonlocal_IPvolume0, & IParea => geometry_plastic_nonlocal_IParea0, & IPareaNormal => geometry_plastic_nonlocal_IPareaNormal0, & geometry_plastic_nonlocal_disable type :: tGeometry real(pReal), dimension(:), allocatable :: V_0 end type tGeometry type(tGeometry), dimension(:), allocatable :: geom real(pReal), parameter :: & kB = 1.38e-23_pReal !< Boltzmann constant in J/Kelvin ! storage order of dislocation types integer, dimension(*), parameter :: & sgl = [1,2,3,4,5,6,7,8] !< signed (single) integer, dimension(*), parameter :: & edg = [1,2,5,6,9], & !< edge scr = [3,4,7,8,10] !< screw integer, dimension(*), parameter :: & mob = [1,2,3,4], & !< mobile imm = [5,6,7,8] !< immobile (blocked) integer, dimension(*), parameter :: & dip = [9,10], & !< dipole imm_edg = imm(1:2), & !< immobile edge imm_scr = imm(3:4) !< immobile screw integer, parameter :: & mob_edg_pos = 1, & !< mobile edge positive mob_edg_neg = 2, & !< mobile edge negative mob_scr_pos = 3, & !< mobile screw positive mob_scr_neg = 4 !< mobile screw positive ! BEGIN DEPRECATED integer, dimension(:,:,:), allocatable :: & iRhoU, & !< state indices for unblocked density iV, & !< state indices for dislocation velocities iD !< state indices for stable dipole height !END DEPRECATED real(pReal), dimension(:,:,:,:,:,:), allocatable :: & compatibility !< slip system compatibility between me and my neighbors type :: tInitialParameters !< container type for internal constitutive parameters real(pReal) :: & sigma_rho_u, & !< standard deviation of scatter in initial dislocation density random_rho_u, & random_rho_u_binning real(pReal), dimension(:), allocatable :: & rho_u_ed_pos_0, & !< initial edge_pos dislocation density rho_u_ed_neg_0, & !< initial edge_neg dislocation density rho_u_sc_pos_0, & !< initial screw_pos dislocation density rho_u_sc_neg_0, & !< initial screw_neg dislocation density rho_d_ed_0, & !< initial edge dipole dislocation density rho_d_sc_0 !< initial screw dipole dislocation density integer, dimension(:), allocatable :: & N_sl end type tInitialParameters type :: tParameters !< container type for internal constitutive parameters real(pReal) :: & V_at, & !< atomic volume D_0, & !< prefactor for self-diffusion coefficient Q_cl, & !< activation enthalpy for diffusion atol_rho, & !< absolute tolerance for dislocation density in state integration rho_significant, & !< density considered significant rho_min, & !< number of dislocations considered significant w, & !< width of a doubkle kink in multiples of the Burgers vector length b Q_sol, & !< activation energy for solid solution in J f_sol, & !< solid solution obstacle size in multiples of the Burgers vector length c_sol, & !< concentration of solid solution in atomic parts p, & !< parameter for kinetic law (Kocks,Argon,Ashby) q, & !< parameter for kinetic law (Kocks,Argon,Ashby) eta, & !< viscosity for dislocation glide in Pa s nu_a, & !< attack frequency in Hz chi_surface, & !< transmissivity at free surface chi_GB, & !< transmissivity at grain boundary (identified by different texture) f_c, & !< safety factor for CFL flux condition f_ed_mult, & !< factor that determines how much edge dislocations contribute to multiplication (0...1) f_F, & f_ed, & mu, & nu real(pReal), dimension(:), allocatable :: & d_ed, & !< minimum stable edge dipole height d_sc, & !< minimum stable screw dipole height tau_Peierls_ed, & tau_Peierls_sc, & i_sl, & !< mean free path prefactor for each b_sl !< absolute length of Burgers vector [m] real(pReal), dimension(:,:), allocatable :: & slip_normal, & slip_direction, & slip_transverse, & minDipoleHeight, & ! edge and screw peierlsstress, & ! edge and screw h_sl_sl ,& !< coefficients for slip-slip interaction forestProjection_Edge, & !< matrix of forest projections of edge dislocations forestProjection_Screw !< matrix of forest projections of screw dislocations real(pReal), dimension(:,:,:), allocatable :: & Schmid, & !< Schmid contribution nonSchmid_pos, & nonSchmid_neg !< combined projection of Schmid and non-Schmid contributions to the resolved shear stress (only for screws) integer :: & sum_N_sl integer, dimension(:), allocatable :: & colinearSystem !< colinear system to the active slip system (only valid for fcc!) character(len=pStringLen), dimension(:), allocatable :: & output logical :: & shortRangeStressCorrection, & !< use of short range stress correction by excess density gradient term nonSchmidActive = .false. end type tParameters type :: tNonlocalMicrostructure real(pReal), allocatable, dimension(:,:) :: & tau_pass, & tau_Back end type tNonlocalMicrostructure type :: tNonlocalState real(pReal), pointer, dimension(:,:) :: & rho, & ! < all dislocations rhoSgl, & rhoSglMobile, & ! iRhoU rho_sgl_mob_edg_pos, & rho_sgl_mob_edg_neg, & rho_sgl_mob_scr_pos, & rho_sgl_mob_scr_neg, & rhoSglImmobile, & rho_sgl_imm_edg_pos, & rho_sgl_imm_edg_neg, & rho_sgl_imm_scr_pos, & rho_sgl_imm_scr_neg, & rhoDip, & rho_dip_edg, & rho_dip_scr, & rho_forest, & gamma, & v, & v_edg_pos, & v_edg_neg, & v_scr_pos, & v_scr_neg end type tNonlocalState type(tNonlocalState), allocatable, dimension(:) :: & deltaState, & dotState, & state, & state0 type(tParameters), dimension(:), allocatable :: param !< containers of constitutive parameters type(tNonlocalMicrostructure), dimension(:), allocatable :: microstructure contains !-------------------------------------------------------------------------------------------------- !> @brief Perform module initialization. !> @details reads in material parameters, allocates arrays, and does sanity checks !-------------------------------------------------------------------------------------------------- module function plastic_nonlocal_init() result(myPlasticity) logical, dimension(:), allocatable :: myPlasticity integer :: & Ninstances, & ph, & Nmembers, & sizeState, sizeDotState, sizeDependentState, sizeDeltaState, & s1, s2, & s, t, l real(pReal), dimension(:), allocatable :: & a character(len=pStringLen) :: & extmsg = '' type(tInitialParameters) :: & ini class(tNode), pointer :: & phases, & phase, & mech, & pl myPlasticity = plastic_active('nonlocal') Ninstances = count(myPlasticity) if(Ninstances == 0) then call geometry_plastic_nonlocal_disable return endif print'(/,a)', ' <<<+- phase:mechanical:plastic:nonlocal init -+>>>' print'(a,i0)', ' # phases: ',Ninstances; flush(IO_STDOUT) print*, 'C. Reuber et al., Acta Materialia 71:333–348, 2014' print*, 'https://doi.org/10.1016/j.actamat.2014.03.012'//IO_EOL print*, 'C. Kords, Dissertation RWTH Aachen, 2014' print*, 'http://publications.rwth-aachen.de/record/229993' phases => config_material%get('phase') allocate(geom(phases%length)) allocate(param(phases%length)) allocate(state(phases%length)) allocate(state0(phases%length)) allocate(dotState(phases%length)) allocate(deltaState(phases%length)) allocate(microstructure(phases%length)) do ph = 1, phases%length if(.not. myPlasticity(ph)) cycle associate(prm => param(ph), dot => dotState(ph), stt => state(ph), & st0 => state0(ph), del => deltaState(ph), dst => microstructure(ph)) phase => phases%get(ph) mech => phase%get('mechanical') pl => mech%get('plastic') phase_localPlasticity(ph) = .not. pl%contains('nonlocal') #if defined (__GFORTRAN__) prm%output = output_as1dString(pl) #else prm%output = pl%get_as1dString('output',defaultVal=emptyStringArray) #endif prm%atol_rho = pl%get_asFloat('atol_rho',defaultVal=1.0e4_pReal) ! This data is read in already in lattice prm%mu = lattice_mu(ph) prm%nu = lattice_nu(ph) ini%N_sl = pl%get_as1dInt('N_sl',defaultVal=emptyIntArray) prm%sum_N_sl = sum(abs(ini%N_sl)) slipActive: if (prm%sum_N_sl > 0) then prm%Schmid = lattice_SchmidMatrix_slip(ini%N_sl,phase%get_asString('lattice'),& phase%get_asFloat('c/a',defaultVal=0.0_pReal)) if(trim(phase%get_asString('lattice')) == 'cI') then a = pl%get_as1dFloat('a_nonSchmid',defaultVal = emptyRealArray) if(size(a) > 0) prm%nonSchmidActive = .true. prm%nonSchmid_pos = lattice_nonSchmidMatrix(ini%N_sl,a,+1) prm%nonSchmid_neg = lattice_nonSchmidMatrix(ini%N_sl,a,-1) else prm%nonSchmid_pos = prm%Schmid prm%nonSchmid_neg = prm%Schmid endif prm%h_sl_sl = lattice_interaction_SlipBySlip(ini%N_sl, & pl%get_as1dFloat('h_sl_sl'), & phase%get_asString('lattice')) prm%forestProjection_edge = lattice_forestProjection_edge (ini%N_sl,phase%get_asString('lattice'),& phase%get_asFloat('c/a',defaultVal=0.0_pReal)) prm%forestProjection_screw = lattice_forestProjection_screw(ini%N_sl,phase%get_asString('lattice'),& phase%get_asFloat('c/a',defaultVal=0.0_pReal)) prm%slip_direction = lattice_slip_direction (ini%N_sl,phase%get_asString('lattice'),& phase%get_asFloat('c/a',defaultVal=0.0_pReal)) prm%slip_transverse = lattice_slip_transverse(ini%N_sl,phase%get_asString('lattice'),& phase%get_asFloat('c/a',defaultVal=0.0_pReal)) prm%slip_normal = lattice_slip_normal (ini%N_sl,phase%get_asString('lattice'),& phase%get_asFloat('c/a',defaultVal=0.0_pReal)) ! collinear systems (only for octahedral slip systems in fcc) allocate(prm%colinearSystem(prm%sum_N_sl), source = -1) do s1 = 1, prm%sum_N_sl do s2 = 1, prm%sum_N_sl if (all(dEq0 (math_cross(prm%slip_direction(1:3,s1),prm%slip_direction(1:3,s2)))) .and. & any(dNeq0(math_cross(prm%slip_normal (1:3,s1),prm%slip_normal (1:3,s2))))) & prm%colinearSystem(s1) = s2 enddo enddo ini%rho_u_ed_pos_0 = pl%get_as1dFloat('rho_u_ed_pos_0', requiredSize=size(ini%N_sl)) ini%rho_u_ed_neg_0 = pl%get_as1dFloat('rho_u_ed_neg_0', requiredSize=size(ini%N_sl)) ini%rho_u_sc_pos_0 = pl%get_as1dFloat('rho_u_sc_pos_0', requiredSize=size(ini%N_sl)) ini%rho_u_sc_neg_0 = pl%get_as1dFloat('rho_u_sc_neg_0', requiredSize=size(ini%N_sl)) ini%rho_d_ed_0 = pl%get_as1dFloat('rho_d_ed_0', requiredSize=size(ini%N_sl)) ini%rho_d_sc_0 = pl%get_as1dFloat('rho_d_sc_0', requiredSize=size(ini%N_sl)) prm%i_sl = pl%get_as1dFloat('i_sl', requiredSize=size(ini%N_sl)) prm%b_sl = pl%get_as1dFloat('b_sl', requiredSize=size(ini%N_sl)) prm%i_sl = math_expand(prm%i_sl,ini%N_sl) prm%b_sl = math_expand(prm%b_sl,ini%N_sl) prm%d_ed = pl%get_as1dFloat('d_ed', requiredSize=size(ini%N_sl)) prm%d_sc = pl%get_as1dFloat('d_sc', requiredSize=size(ini%N_sl)) prm%d_ed = math_expand(prm%d_ed,ini%N_sl) prm%d_sc = math_expand(prm%d_sc,ini%N_sl) allocate(prm%minDipoleHeight(prm%sum_N_sl,2)) prm%minDipoleHeight(:,1) = prm%d_ed prm%minDipoleHeight(:,2) = prm%d_sc prm%tau_Peierls_ed = pl%get_as1dFloat('tau_Peierls_ed', requiredSize=size(ini%N_sl)) prm%tau_Peierls_sc = pl%get_as1dFloat('tau_Peierls_sc', requiredSize=size(ini%N_sl)) prm%tau_Peierls_ed = math_expand(prm%tau_Peierls_ed,ini%N_sl) prm%tau_Peierls_sc = math_expand(prm%tau_Peierls_sc,ini%N_sl) allocate(prm%peierlsstress(prm%sum_N_sl,2)) prm%peierlsstress(:,1) = prm%tau_Peierls_ed prm%peierlsstress(:,2) = prm%tau_Peierls_sc prm%rho_significant = pl%get_asFloat('rho_significant') prm%rho_min = pl%get_asFloat('rho_min', 0.0_pReal) prm%f_c = pl%get_asFloat('f_c',defaultVal=2.0_pReal) prm%V_at = pl%get_asFloat('V_at') prm%D_0 = pl%get_asFloat('D_0') prm%Q_cl = pl%get_asFloat('Q_cl') prm%f_F = pl%get_asFloat('f_F') prm%f_ed = pl%get_asFloat('f_ed') !,'edgejogs' prm%w = pl%get_asFloat('w') prm%Q_sol = pl%get_asFloat('Q_sol') prm%f_sol = pl%get_asFloat('f_sol') prm%c_sol = pl%get_asFloat('c_sol') prm%p = pl%get_asFloat('p_sl') prm%q = pl%get_asFloat('q_sl') prm%eta = pl%get_asFloat('eta') prm%nu_a = pl%get_asFloat('nu_a') ! ToDo: discuss logic ini%sigma_rho_u = pl%get_asFloat('sigma_rho_u') ini%random_rho_u = pl%get_asFloat('random_rho_u',defaultVal= 0.0_pReal) if (pl%contains('random_rho_u')) & ini%random_rho_u_binning = pl%get_asFloat('random_rho_u_binning',defaultVal=0.0_pReal) !ToDo: useful default? ! if (rhoSglRandom(instance) < 0.0_pReal) & ! if (rhoSglRandomBinning(instance) <= 0.0_pReal) & prm%chi_surface = pl%get_asFloat('chi_surface',defaultVal=1.0_pReal) prm%chi_GB = pl%get_asFloat('chi_GB', defaultVal=-1.0_pReal) prm%f_ed_mult = pl%get_asFloat('f_ed_mult') prm%shortRangeStressCorrection = pl%get_asBool('short_range_stress_correction', defaultVal = .false.) !-------------------------------------------------------------------------------------------------- ! sanity checks if (any(prm%b_sl < 0.0_pReal)) extmsg = trim(extmsg)//' b_sl' if (any(prm%i_sl <= 0.0_pReal)) extmsg = trim(extmsg)//' i_sl' if (any(ini%rho_u_ed_pos_0 < 0.0_pReal)) extmsg = trim(extmsg)//' rho_u_ed_pos_0' if (any(ini%rho_u_ed_neg_0 < 0.0_pReal)) extmsg = trim(extmsg)//' rho_u_ed_neg_0' if (any(ini%rho_u_sc_pos_0 < 0.0_pReal)) extmsg = trim(extmsg)//' rho_u_sc_pos_0' if (any(ini%rho_u_sc_neg_0 < 0.0_pReal)) extmsg = trim(extmsg)//' rho_u_sc_neg_0' if (any(ini%rho_d_ed_0 < 0.0_pReal)) extmsg = trim(extmsg)//' rho_d_ed_0' if (any(ini%rho_d_sc_0 < 0.0_pReal)) extmsg = trim(extmsg)//' rho_d_sc_0' if (any(prm%peierlsstress < 0.0_pReal)) extmsg = trim(extmsg)//' tau_peierls' if (any(prm%minDipoleHeight < 0.0_pReal)) extmsg = trim(extmsg)//' d_ed or d_sc' if (prm%eta <= 0.0_pReal) extmsg = trim(extmsg)//' eta' if (prm%Q_cl <= 0.0_pReal) extmsg = trim(extmsg)//' Q_cl' if (prm%nu_a <= 0.0_pReal) extmsg = trim(extmsg)//' nu_a' if (prm%w <= 0.0_pReal) extmsg = trim(extmsg)//' w' if (prm%D_0 < 0.0_pReal) extmsg = trim(extmsg)//' D_0' if (prm%V_at <= 0.0_pReal) extmsg = trim(extmsg)//' V_at' ! ToDo: in dislotungsten, the atomic volume is given as a factor if (prm%rho_min < 0.0_pReal) extmsg = trim(extmsg)//' rho_min' if (prm%rho_significant < 0.0_pReal) extmsg = trim(extmsg)//' rho_significant' if (prm%atol_rho < 0.0_pReal) extmsg = trim(extmsg)//' atol_rho' if (prm%f_c < 0.0_pReal) extmsg = trim(extmsg)//' f_c' if (prm%p <= 0.0_pReal .or. prm%p > 1.0_pReal) extmsg = trim(extmsg)//' p_sl' if (prm%q < 1.0_pReal .or. prm%q > 2.0_pReal) extmsg = trim(extmsg)//' q_sl' if (prm%f_F < 0.0_pReal .or. prm%f_F > 1.0_pReal) & extmsg = trim(extmsg)//' f_F' if (prm%f_ed < 0.0_pReal .or. prm%f_ed > 1.0_pReal) & extmsg = trim(extmsg)//' f_ed' if (prm%Q_sol <= 0.0_pReal) extmsg = trim(extmsg)//' Q_sol' if (prm%f_sol <= 0.0_pReal) extmsg = trim(extmsg)//' f_sol' if (prm%c_sol <= 0.0_pReal) extmsg = trim(extmsg)//' c_sol' if (prm%chi_GB > 1.0_pReal) extmsg = trim(extmsg)//' chi_GB' if (prm%chi_surface < 0.0_pReal .or. prm%chi_surface > 1.0_pReal) & extmsg = trim(extmsg)//' chi_surface' if (prm%f_ed_mult < 0.0_pReal .or. prm%f_ed_mult > 1.0_pReal) & extmsg = trim(extmsg)//' f_ed_mult' endif slipActive !-------------------------------------------------------------------------------------------------- ! allocate state arrays Nmembers = count(material_phaseAt2 == ph) sizeDotState = size([ 'rhoSglEdgePosMobile ','rhoSglEdgeNegMobile ', & 'rhoSglScrewPosMobile ','rhoSglScrewNegMobile ', & 'rhoSglEdgePosImmobile ','rhoSglEdgeNegImmobile ', & 'rhoSglScrewPosImmobile','rhoSglScrewNegImmobile', & 'rhoDipEdge ','rhoDipScrew ', & 'gamma ' ]) * prm%sum_N_sl !< "basic" microstructural state variables that are independent from other state variables sizeDependentState = size([ 'rhoForest ']) * prm%sum_N_sl !< microstructural state variables that depend on other state variables sizeState = sizeDotState + sizeDependentState & + size([ 'velocityEdgePos ','velocityEdgeNeg ', & 'velocityScrewPos ','velocityScrewNeg ', & 'maxDipoleHeightEdge ','maxDipoleHeightScrew' ]) * prm%sum_N_sl !< other dependent state variables that are not updated by microstructure sizeDeltaState = sizeDotState call phase_allocateState(plasticState(ph),Nmembers,sizeState,sizeDotState,sizeDeltaState) allocate(geom(ph)%V_0(Nmembers)) call storeGeometry(ph) plasticState(ph)%nonlocal = pl%get_asBool('nonlocal') if(plasticState(ph)%nonlocal .and. .not. allocated(IPneighborhood)) & call IO_error(212,ext_msg='IPneighborhood does not exist') plasticState(ph)%offsetDeltaState = 0 ! ToDo: state structure does not follow convention st0%rho => plasticState(ph)%state0 (0*prm%sum_N_sl+1:10*prm%sum_N_sl,:) stt%rho => plasticState(ph)%state (0*prm%sum_N_sl+1:10*prm%sum_N_sl,:) dot%rho => plasticState(ph)%dotState (0*prm%sum_N_sl+1:10*prm%sum_N_sl,:) del%rho => plasticState(ph)%deltaState (0*prm%sum_N_sl+1:10*prm%sum_N_sl,:) plasticState(ph)%atol(1:10*prm%sum_N_sl) = prm%atol_rho stt%rhoSgl => plasticState(ph)%state (0*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) dot%rhoSgl => plasticState(ph)%dotState (0*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) del%rhoSgl => plasticState(ph)%deltaState (0*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) stt%rhoSglMobile => plasticState(ph)%state (0*prm%sum_N_sl+1: 4*prm%sum_N_sl,:) dot%rhoSglMobile => plasticState(ph)%dotState (0*prm%sum_N_sl+1: 4*prm%sum_N_sl,:) del%rhoSglMobile => plasticState(ph)%deltaState (0*prm%sum_N_sl+1: 4*prm%sum_N_sl,:) stt%rho_sgl_mob_edg_pos => plasticState(ph)%state (0*prm%sum_N_sl+1: 1*prm%sum_N_sl,:) dot%rho_sgl_mob_edg_pos => plasticState(ph)%dotState (0*prm%sum_N_sl+1: 1*prm%sum_N_sl,:) del%rho_sgl_mob_edg_pos => plasticState(ph)%deltaState (0*prm%sum_N_sl+1: 1*prm%sum_N_sl,:) stt%rho_sgl_mob_edg_neg => plasticState(ph)%state (1*prm%sum_N_sl+1: 2*prm%sum_N_sl,:) dot%rho_sgl_mob_edg_neg => plasticState(ph)%dotState (1*prm%sum_N_sl+1: 2*prm%sum_N_sl,:) del%rho_sgl_mob_edg_neg => plasticState(ph)%deltaState (1*prm%sum_N_sl+1: 2*prm%sum_N_sl,:) stt%rho_sgl_mob_scr_pos => plasticState(ph)%state (2*prm%sum_N_sl+1: 3*prm%sum_N_sl,:) dot%rho_sgl_mob_scr_pos => plasticState(ph)%dotState (2*prm%sum_N_sl+1: 3*prm%sum_N_sl,:) del%rho_sgl_mob_scr_pos => plasticState(ph)%deltaState (2*prm%sum_N_sl+1: 3*prm%sum_N_sl,:) stt%rho_sgl_mob_scr_neg => plasticState(ph)%state (3*prm%sum_N_sl+1: 4*prm%sum_N_sl,:) dot%rho_sgl_mob_scr_neg => plasticState(ph)%dotState (3*prm%sum_N_sl+1: 4*prm%sum_N_sl,:) del%rho_sgl_mob_scr_neg => plasticState(ph)%deltaState (3*prm%sum_N_sl+1: 4*prm%sum_N_sl,:) stt%rhoSglImmobile => plasticState(ph)%state (4*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) dot%rhoSglImmobile => plasticState(ph)%dotState (4*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) del%rhoSglImmobile => plasticState(ph)%deltaState (4*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) stt%rho_sgl_imm_edg_pos => plasticState(ph)%state (4*prm%sum_N_sl+1: 5*prm%sum_N_sl,:) dot%rho_sgl_imm_edg_pos => plasticState(ph)%dotState (4*prm%sum_N_sl+1: 5*prm%sum_N_sl,:) del%rho_sgl_imm_edg_pos => plasticState(ph)%deltaState (4*prm%sum_N_sl+1: 5*prm%sum_N_sl,:) stt%rho_sgl_imm_edg_neg => plasticState(ph)%state (5*prm%sum_N_sl+1: 6*prm%sum_N_sl,:) dot%rho_sgl_imm_edg_neg => plasticState(ph)%dotState (5*prm%sum_N_sl+1: 6*prm%sum_N_sl,:) del%rho_sgl_imm_edg_neg => plasticState(ph)%deltaState (5*prm%sum_N_sl+1: 6*prm%sum_N_sl,:) stt%rho_sgl_imm_scr_pos => plasticState(ph)%state (6*prm%sum_N_sl+1: 7*prm%sum_N_sl,:) dot%rho_sgl_imm_scr_pos => plasticState(ph)%dotState (6*prm%sum_N_sl+1: 7*prm%sum_N_sl,:) del%rho_sgl_imm_scr_pos => plasticState(ph)%deltaState (6*prm%sum_N_sl+1: 7*prm%sum_N_sl,:) stt%rho_sgl_imm_scr_neg => plasticState(ph)%state (7*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) dot%rho_sgl_imm_scr_neg => plasticState(ph)%dotState (7*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) del%rho_sgl_imm_scr_neg => plasticState(ph)%deltaState (7*prm%sum_N_sl+1: 8*prm%sum_N_sl,:) stt%rhoDip => plasticState(ph)%state (8*prm%sum_N_sl+1:10*prm%sum_N_sl,:) dot%rhoDip => plasticState(ph)%dotState (8*prm%sum_N_sl+1:10*prm%sum_N_sl,:) del%rhoDip => plasticState(ph)%deltaState (8*prm%sum_N_sl+1:10*prm%sum_N_sl,:) stt%rho_dip_edg => plasticState(ph)%state (8*prm%sum_N_sl+1: 9*prm%sum_N_sl,:) dot%rho_dip_edg => plasticState(ph)%dotState (8*prm%sum_N_sl+1: 9*prm%sum_N_sl,:) del%rho_dip_edg => plasticState(ph)%deltaState (8*prm%sum_N_sl+1: 9*prm%sum_N_sl,:) stt%rho_dip_scr => plasticState(ph)%state (9*prm%sum_N_sl+1:10*prm%sum_N_sl,:) dot%rho_dip_scr => plasticState(ph)%dotState (9*prm%sum_N_sl+1:10*prm%sum_N_sl,:) del%rho_dip_scr => plasticState(ph)%deltaState (9*prm%sum_N_sl+1:10*prm%sum_N_sl,:) stt%gamma => plasticState(ph)%state (10*prm%sum_N_sl + 1:11*prm%sum_N_sl,1:Nmembers) dot%gamma => plasticState(ph)%dotState (10*prm%sum_N_sl + 1:11*prm%sum_N_sl,1:Nmembers) del%gamma => plasticState(ph)%deltaState (10*prm%sum_N_sl + 1:11*prm%sum_N_sl,1:Nmembers) plasticState(ph)%atol(10*prm%sum_N_sl+1:11*prm%sum_N_sl ) = pl%get_asFloat('atol_gamma', defaultVal = 1.0e-2_pReal) if(any(plasticState(ph)%atol(10*prm%sum_N_sl+1:11*prm%sum_N_sl) < 0.0_pReal)) & extmsg = trim(extmsg)//' atol_gamma' plasticState(ph)%slipRate => plasticState(ph)%dotState (10*prm%sum_N_sl + 1:11*prm%sum_N_sl,1:Nmembers) stt%rho_forest => plasticState(ph)%state (11*prm%sum_N_sl + 1:12*prm%sum_N_sl,1:Nmembers) stt%v => plasticState(ph)%state (12*prm%sum_N_sl + 1:16*prm%sum_N_sl,1:Nmembers) stt%v_edg_pos => plasticState(ph)%state (12*prm%sum_N_sl + 1:13*prm%sum_N_sl,1:Nmembers) stt%v_edg_neg => plasticState(ph)%state (13*prm%sum_N_sl + 1:14*prm%sum_N_sl,1:Nmembers) stt%v_scr_pos => plasticState(ph)%state (14*prm%sum_N_sl + 1:15*prm%sum_N_sl,1:Nmembers) stt%v_scr_neg => plasticState(ph)%state (15*prm%sum_N_sl + 1:16*prm%sum_N_sl,1:Nmembers) allocate(dst%tau_pass(prm%sum_N_sl,Nmembers),source=0.0_pReal) allocate(dst%tau_back(prm%sum_N_sl,Nmembers),source=0.0_pReal) end associate if (Nmembers > 0) call stateInit(ini,ph,Nmembers) plasticState(ph)%state0 = plasticState(ph)%state !-------------------------------------------------------------------------------------------------- ! exit if any parameter is out of range if (extmsg /= '') call IO_error(211,ext_msg=trim(extmsg)//'(nonlocal)') enddo allocate(compatibility(2,maxval(param%sum_N_sl),maxval(param%sum_N_sl),nIPneighbors,& discretization_nIPs,discretization_Nelems), source=0.0_pReal) ! BEGIN DEPRECATED---------------------------------------------------------------------------------- allocate(iRhoU(maxval(param%sum_N_sl),4,phases%length), source=0) allocate(iV(maxval(param%sum_N_sl),4,phases%length), source=0) allocate(iD(maxval(param%sum_N_sl),2,phases%length), source=0) do ph = 1, phases%length if(.not. myPlasticity(ph)) cycle phase => phases%get(ph) Nmembers = count(material_phaseAt2 == ph) l = 0 do t = 1,4 do s = 1,param(ph)%sum_N_sl l = l + 1 iRhoU(s,t,ph) = l enddo enddo l = l + (4+2+1+1)*param(ph)%sum_N_sl ! immobile(4), dipole(2), shear, forest do t = 1,4 do s = 1,param(ph)%sum_N_sl l = l + 1 iV(s,t,ph) = l enddo enddo do t = 1,2 do s = 1,param(ph)%sum_N_sl l = l + 1 iD(s,t,ph) = l enddo enddo if (iD(param(ph)%sum_N_sl,2,ph) /= plasticState(ph)%sizeState) & error stop 'state indices not properly set (nonlocal)' enddo end function plastic_nonlocal_init !-------------------------------------------------------------------------------------------------- !> @brief calculates quantities characterizing the microstructure !-------------------------------------------------------------------------------------------------- module subroutine nonlocal_dependentState(ph, me, ip, el) integer, intent(in) :: & ph, & me, & ip, & el integer :: & no, & !< neighbor offset neighbor_el, & ! element number of neighboring material point neighbor_ip, & ! integration point of neighboring material point c, & ! index of dilsocation character (edge, screw) s, & ! slip system index dir, & n real(pReal) :: & FVsize, & nRealNeighbors ! number of really existing neighbors integer, dimension(2) :: & neighbors real(pReal), dimension(2) :: & rhoExcessGradient, & rhoExcessGradient_over_rho, & rhoTotal real(pReal), dimension(3) :: & rhoExcessDifferences, & normal_latticeConf real(pReal), dimension(3,3) :: & invFe, & !< inverse of elastic deformation gradient invFp, & !< inverse of plastic deformation gradient connections, & invConnections real(pReal), dimension(3,nIPneighbors) :: & connection_latticeConf real(pReal), dimension(2,param(ph)%sum_N_sl) :: & rhoExcess real(pReal), dimension(param(ph)%sum_N_sl) :: & rho_edg_delta, & rho_scr_delta real(pReal), dimension(param(ph)%sum_N_sl,10) :: & rho, & rho0, & rho_neighbor0 real(pReal), dimension(param(ph)%sum_N_sl,param(ph)%sum_N_sl) :: & myInteractionMatrix ! corrected slip interaction matrix real(pReal), dimension(param(ph)%sum_N_sl,nIPneighbors) :: & rho_edg_delta_neighbor, & rho_scr_delta_neighbor real(pReal), dimension(2,maxval(param%sum_N_sl),nIPneighbors) :: & neighbor_rhoExcess, & ! excess density at neighboring material point neighbor_rhoTotal ! total density at neighboring material point real(pReal), dimension(3,param(ph)%sum_N_sl,2) :: & m ! direction of dislocation motion associate(prm => param(ph),dst => microstructure(ph), stt => state(ph)) rho = getRho(ph,me) stt%rho_forest(:,me) = matmul(prm%forestProjection_Edge, sum(abs(rho(:,edg)),2)) & + matmul(prm%forestProjection_Screw,sum(abs(rho(:,scr)),2)) ! coefficients are corrected for the line tension effect ! (see Kubin,Devincre,Hoc; 2008; Modeling dislocation storage rates and mean free paths in face-centered cubic crystals) if (any(lattice_structure(material_phaseAt(1,el)) == [LATTICE_bcc_ID,LATTICE_fcc_ID])) then myInteractionMatrix = prm%h_sl_sl & * spread(( 1.0_pReal - prm%f_F & + prm%f_F & * log(0.35_pReal * prm%b_sl * sqrt(max(stt%rho_forest(:,me),prm%rho_significant))) & / log(0.35_pReal * prm%b_sl * 1e6_pReal))** 2.0_pReal,2,prm%sum_N_sl) else myInteractionMatrix = prm%h_sl_sl endif dst%tau_pass(:,me) = prm%mu * prm%b_sl & * sqrt(matmul(myInteractionMatrix,sum(abs(rho),2))) !*** calculate the dislocation stress of the neighboring excess dislocation densities !*** zero for material points of local plasticity !################################################################################################# ! ToDo: MD: this is most likely only correct for F_i = I !################################################################################################# rho0 = getRho0(ph,me) if (.not. phase_localPlasticity(material_phaseAt(1,el)) .and. prm%shortRangeStressCorrection) then invFp = math_inv33(phase_mechanical_Fp(ph)%data(1:3,1:3,me)) invFe = math_inv33(phase_mechanical_Fe(ph)%data(1:3,1:3,me)) rho_edg_delta = rho0(:,mob_edg_pos) - rho0(:,mob_edg_neg) rho_scr_delta = rho0(:,mob_scr_pos) - rho0(:,mob_scr_neg) rhoExcess(1,:) = rho_edg_delta rhoExcess(2,:) = rho_scr_delta FVsize = geom(ph)%V_0(me) ** (1.0_pReal/3.0_pReal) !* loop through my neighborhood and get the connection vectors (in lattice frame) and the excess densities nRealNeighbors = 0.0_pReal neighbor_rhoTotal = 0.0_pReal do n = 1,nIPneighbors neighbor_el = IPneighborhood(1,n,ip,el) neighbor_ip = IPneighborhood(2,n,ip,el) no = material_phasememberAt(1,neighbor_ip,neighbor_el) if (neighbor_el > 0 .and. neighbor_ip > 0) then if (material_phaseAt(1,neighbor_el) == ph) then nRealNeighbors = nRealNeighbors + 1.0_pReal rho_neighbor0 = getRho0(ph,no) rho_edg_delta_neighbor(:,n) = rho_neighbor0(:,mob_edg_pos) - rho_neighbor0(:,mob_edg_neg) rho_scr_delta_neighbor(:,n) = rho_neighbor0(:,mob_scr_pos) - rho_neighbor0(:,mob_scr_neg) neighbor_rhoTotal(1,:,n) = sum(abs(rho_neighbor0(:,edg)),2) neighbor_rhoTotal(2,:,n) = sum(abs(rho_neighbor0(:,scr)),2) connection_latticeConf(1:3,n) = matmul(invFe, discretization_IPcoords(1:3,neighbor_el+neighbor_ip-1) & - discretization_IPcoords(1:3,el+neighbor_ip-1)) normal_latticeConf = matmul(transpose(invFp), IPareaNormal(1:3,n,ip,el)) if (math_inner(normal_latticeConf,connection_latticeConf(1:3,n)) < 0.0_pReal) & ! neighboring connection points in opposite direction to face normal: must be periodic image connection_latticeConf(1:3,n) = normal_latticeConf * IPvolume(ip,el)/IParea(n,ip,el) ! instead take the surface normal scaled with the diameter of the cell else ! local neighbor or different lattice structure or different constitution instance -> use central values instead connection_latticeConf(1:3,n) = 0.0_pReal rho_edg_delta_neighbor(:,n) = rho_edg_delta rho_scr_delta_neighbor(:,n) = rho_scr_delta endif else ! free surface -> use central values instead connection_latticeConf(1:3,n) = 0.0_pReal rho_edg_delta_neighbor(:,n) = rho_edg_delta rho_scr_delta_neighbor(:,n) = rho_scr_delta endif enddo neighbor_rhoExcess(1,:,:) = rho_edg_delta_neighbor neighbor_rhoExcess(2,:,:) = rho_scr_delta_neighbor !* loop through the slip systems and calculate the dislocation gradient by !* 1. interpolation of the excess density in the neighorhood !* 2. interpolation of the dead dislocation density in the central volume m(1:3,:,1) = prm%slip_direction m(1:3,:,2) = -prm%slip_transverse do s = 1,prm%sum_N_sl ! gradient from interpolation of neighboring excess density ... do c = 1,2 do dir = 1,3 neighbors(1) = 2 * dir - 1 neighbors(2) = 2 * dir connections(dir,1:3) = connection_latticeConf(1:3,neighbors(1)) & - connection_latticeConf(1:3,neighbors(2)) rhoExcessDifferences(dir) = neighbor_rhoExcess(c,s,neighbors(1)) & - neighbor_rhoExcess(c,s,neighbors(2)) enddo invConnections = math_inv33(connections) if (all(dEq0(invConnections))) call IO_error(-1,ext_msg='back stress calculation: inversion error') rhoExcessGradient(c) = math_inner(m(1:3,s,c), matmul(invConnections,rhoExcessDifferences)) enddo ! ... plus gradient from deads ... rhoExcessGradient(1) = rhoExcessGradient(1) + sum(rho(s,imm_edg)) / FVsize rhoExcessGradient(2) = rhoExcessGradient(2) + sum(rho(s,imm_scr)) / FVsize ! ... normalized with the total density ... rhoTotal(1) = (sum(abs(rho(s,edg))) + sum(neighbor_rhoTotal(1,s,:))) / (1.0_pReal + nRealNeighbors) rhoTotal(2) = (sum(abs(rho(s,scr))) + sum(neighbor_rhoTotal(2,s,:))) / (1.0_pReal + nRealNeighbors) rhoExcessGradient_over_rho = 0.0_pReal where(rhoTotal > 0.0_pReal) rhoExcessGradient_over_rho = rhoExcessGradient / rhoTotal ! ... gives the local stress correction when multiplied with a factor dst%tau_back(s,me) = - prm%mu * prm%b_sl(s) / (2.0_pReal * PI) & * ( rhoExcessGradient_over_rho(1) / (1.0_pReal - prm%nu) & + rhoExcessGradient_over_rho(2)) enddo endif #ifdef DEBUG if (debugConstitutive%extensive & .and. ((debugConstitutive%element == el .and. debugConstitutive%ip == ip)& .or. .not. debugConstitutive%selective)) then print'(/,a,i8,1x,i2,1x,i1,/)', '<< CONST >> nonlocal_microstructure at el ip ',el,ip print'(a,/,12x,12(e10.3,1x))', '<< CONST >> rhoForest', stt%rho_forest(:,me) print'(a,/,12x,12(f10.5,1x))', '<< CONST >> tauThreshold / MPa', dst%tau_pass(:,me)*1e-6 print'(a,/,12x,12(f10.5,1x),/)', '<< CONST >> tauBack / MPa', dst%tau_back(:,me)*1e-6 endif #endif end associate end subroutine nonlocal_dependentState !-------------------------------------------------------------------------------------------------- !> @brief calculates plastic velocity gradient and its tangent !-------------------------------------------------------------------------------------------------- module subroutine nonlocal_LpAndItsTangent(Lp,dLp_dMp, & Mp,Temperature,ph,me) real(pReal), dimension(3,3), intent(out) :: & Lp !< plastic velocity gradient real(pReal), dimension(3,3,3,3), intent(out) :: & dLp_dMp integer, intent(in) :: & ph, & me real(pReal), intent(in) :: & Temperature !< temperature real(pReal), dimension(3,3), intent(in) :: & Mp !< derivative of Lp with respect to Mp integer :: & ns, & !< short notation for the total number of active slip systems i, & j, & k, & l, & t, & !< dislocation type s !< index of my current slip system real(pReal), dimension(param(ph)%sum_N_sl,8) :: & rhoSgl !< single dislocation densities (including blocked) real(pReal), dimension(param(ph)%sum_N_sl,10) :: & rho real(pReal), dimension(param(ph)%sum_N_sl,4) :: & v, & !< velocity tauNS, & !< resolved shear stress including non Schmid and backstress terms dv_dtau, & !< velocity derivative with respect to the shear stress dv_dtauNS !< velocity derivative with respect to the shear stress real(pReal), dimension(param(ph)%sum_N_sl) :: & tau, & !< resolved shear stress including backstress terms gdotTotal !< shear rate associate(prm => param(ph),dst=>microstructure(ph),& stt=>state(ph)) ns = prm%sum_N_sl !*** shortcut to state variables rho = getRho(ph,me) rhoSgl = rho(:,sgl) do s = 1,ns tau(s) = math_tensordot(Mp, prm%Schmid(1:3,1:3,s)) tauNS(s,1) = tau(s) tauNS(s,2) = tau(s) if (tau(s) > 0.0_pReal) then tauNS(s,3) = math_tensordot(Mp, +prm%nonSchmid_pos(1:3,1:3,s)) tauNS(s,4) = math_tensordot(Mp, -prm%nonSchmid_neg(1:3,1:3,s)) else tauNS(s,3) = math_tensordot(Mp, +prm%nonSchmid_neg(1:3,1:3,s)) tauNS(s,4) = math_tensordot(Mp, -prm%nonSchmid_pos(1:3,1:3,s)) endif enddo tauNS = tauNS + spread(dst%tau_back(:,me),2,4) tau = tau + dst%tau_back(:,me) ! edges call kinetics(v(:,1), dv_dtau(:,1), dv_dtauNS(:,1), & tau, tauNS(:,1), dst%tau_pass(:,me),1,Temperature, ph) v(:,2) = v(:,1) dv_dtau(:,2) = dv_dtau(:,1) dv_dtauNS(:,2) = dv_dtauNS(:,1) !screws if (prm%nonSchmidActive) then do t = 3,4 call kinetics(v(:,t), dv_dtau(:,t), dv_dtauNS(:,t), & tau, tauNS(:,t), dst%tau_pass(:,me),2,Temperature, ph) enddo else v(:,3:4) = spread(v(:,1),2,2) dv_dtau(:,3:4) = spread(dv_dtau(:,1),2,2) dv_dtauNS(:,3:4) = spread(dv_dtauNS(:,1),2,2) endif stt%v(:,me) = pack(v,.true.) !*** Bauschinger effect forall (s = 1:ns, t = 5:8, rhoSgl(s,t) * v(s,t-4) < 0.0_pReal) & rhoSgl(s,t-4) = rhoSgl(s,t-4) + abs(rhoSgl(s,t)) gdotTotal = sum(rhoSgl(:,1:4) * v, 2) * prm%b_sl Lp = 0.0_pReal dLp_dMp = 0.0_pReal do s = 1,ns Lp = Lp + gdotTotal(s) * prm%Schmid(1:3,1:3,s) forall (i=1:3,j=1:3,k=1:3,l=1:3) & dLp_dMp(i,j,k,l) = dLp_dMp(i,j,k,l) & + prm%Schmid(i,j,s) * prm%Schmid(k,l,s) & * sum(rhoSgl(s,1:4) * dv_dtau(s,1:4)) * prm%b_sl(s) & + prm%Schmid(i,j,s) & * (+ prm%nonSchmid_pos(k,l,s) * rhoSgl(s,3) * dv_dtauNS(s,3) & - prm%nonSchmid_neg(k,l,s) * rhoSgl(s,4) * dv_dtauNS(s,4)) * prm%b_sl(s) enddo end associate end subroutine nonlocal_LpAndItsTangent !-------------------------------------------------------------------------------------------------- !> @brief (instantaneous) incremental change of microstructure !-------------------------------------------------------------------------------------------------- module subroutine plastic_nonlocal_deltaState(Mp,ph,me) real(pReal), dimension(3,3), intent(in) :: & Mp !< MandelStress integer, intent(in) :: & ph, & me integer :: & ns, & ! short notation for the total number of active slip systems c, & ! character of dislocation t, & ! type of dislocation s ! index of my current slip system real(pReal), dimension(param(ph)%sum_N_sl,10) :: & deltaRhoRemobilization, & ! density increment by remobilization deltaRhoDipole2SingleStress ! density increment by dipole dissociation (by stress change) real(pReal), dimension(param(ph)%sum_N_sl,10) :: & rho ! current dislocation densities real(pReal), dimension(param(ph)%sum_N_sl,4) :: & v ! dislocation glide velocity real(pReal), dimension(param(ph)%sum_N_sl) :: & tau ! current resolved shear stress real(pReal), dimension(param(ph)%sum_N_sl,2) :: & rhoDip, & ! current dipole dislocation densities (screw and edge dipoles) dUpper, & ! current maximum stable dipole distance for edges and screws dUpperOld, & ! old maximum stable dipole distance for edges and screws deltaDUpper ! change in maximum stable dipole distance for edges and screws associate(prm => param(ph),dst => microstructure(ph),del => deltaState(ph)) ns = prm%sum_N_sl !*** shortcut to state variables forall (s = 1:ns, t = 1:4) v(s,t) = plasticState(ph)%state(iV(s,t,ph),me) forall (s = 1:ns, c = 1:2) dUpperOld(s,c) = plasticState(ph)%state(iD(s,c,ph),me) rho = getRho(ph,me) rhoDip = rho(:,dip) !**************************************************************************** !*** dislocation remobilization (bauschinger effect) where(rho(:,imm) * v < 0.0_pReal) deltaRhoRemobilization(:,mob) = abs(rho(:,imm)) deltaRhoRemobilization(:,imm) = - rho(:,imm) rho(:,mob) = rho(:,mob) + abs(rho(:,imm)) rho(:,imm) = 0.0_pReal elsewhere deltaRhoRemobilization(:,mob) = 0.0_pReal deltaRhoRemobilization(:,imm) = 0.0_pReal endwhere deltaRhoRemobilization(:,dip) = 0.0_pReal !**************************************************************************** !*** calculate dipole formation and dissociation by stress change !*** calculate limits for stable dipole height do s = 1,prm%sum_N_sl tau(s) = math_tensordot(Mp, prm%Schmid(1:3,1:3,s)) +dst%tau_back(s,me) if (abs(tau(s)) < 1.0e-15_pReal) tau(s) = 1.0e-15_pReal enddo dUpper(:,1) = prm%mu * prm%b_sl/(8.0_pReal * PI * (1.0_pReal - prm%nu) * abs(tau)) dUpper(:,2) = prm%mu * prm%b_sl/(4.0_pReal * PI * abs(tau)) where(dNeq0(sqrt(sum(abs(rho(:,edg)),2)))) & dUpper(:,1) = min(1.0_pReal/sqrt(sum(abs(rho(:,edg)),2)),dUpper(:,1)) where(dNeq0(sqrt(sum(abs(rho(:,scr)),2)))) & dUpper(:,2) = min(1.0_pReal/sqrt(sum(abs(rho(:,scr)),2)),dUpper(:,2)) dUpper = max(dUpper,prm%minDipoleHeight) deltaDUpper = dUpper - dUpperOld !*** dissociation by stress increase deltaRhoDipole2SingleStress = 0.0_pReal forall (c=1:2, s=1:ns, deltaDUpper(s,c) < 0.0_pReal .and. & dNeq0(dUpperOld(s,c) - prm%minDipoleHeight(s,c))) & deltaRhoDipole2SingleStress(s,8+c) = rhoDip(s,c) * deltaDUpper(s,c) & / (dUpperOld(s,c) - prm%minDipoleHeight(s,c)) forall (t=1:4) deltaRhoDipole2SingleStress(:,t) = -0.5_pReal * deltaRhoDipole2SingleStress(:,(t-1)/2+9) forall (s = 1:ns, c = 1:2) plasticState(ph)%state(iD(s,c,ph),me) = dUpper(s,c) plasticState(ph)%deltaState(:,me) = 0.0_pReal del%rho(:,me) = reshape(deltaRhoRemobilization + deltaRhoDipole2SingleStress, [10*ns]) end associate end subroutine plastic_nonlocal_deltaState !--------------------------------------------------------------------------------------------------- !> @brief calculates the rate of change of microstructure !--------------------------------------------------------------------------------------------------- module subroutine nonlocal_dotState(Mp, Temperature,timestep, & ph,me,ip,el) real(pReal), dimension(3,3), intent(in) :: & Mp !< MandelStress real(pReal), intent(in) :: & Temperature, & !< temperature timestep !< substepped crystallite time increment integer, intent(in) :: & ph, & me, & ip, & !< current integration point el !< current element number integer :: & ns, & !< short notation for the total number of active slip systems c, & !< character of dislocation t, & !< type of dislocation s !< index of my current slip system real(pReal), dimension(param(ph)%sum_N_sl,10) :: & rho, & rho0, & !< dislocation density at beginning of time step rhoDot, & !< density evolution rhoDotMultiplication, & !< density evolution by multiplication rhoDotSingle2DipoleGlide, & !< density evolution by dipole formation (by glide) rhoDotAthermalAnnihilation, & !< density evolution by athermal annihilation rhoDotThermalAnnihilation !< density evolution by thermal annihilation real(pReal), dimension(param(ph)%sum_N_sl,8) :: & rhoSgl, & !< current single dislocation densities (positive/negative screw and edge without dipoles) my_rhoSgl0 !< single dislocation densities of central ip (positive/negative screw and edge without dipoles) real(pReal), dimension(param(ph)%sum_N_sl,4) :: & v, & !< current dislocation glide velocity v0, & gdot !< shear rates real(pReal), dimension(param(ph)%sum_N_sl) :: & tau, & !< current resolved shear stress vClimb !< climb velocity of edge dipoles real(pReal), dimension(param(ph)%sum_N_sl,2) :: & rhoDip, & !< current dipole dislocation densities (screw and edge dipoles) dLower, & !< minimum stable dipole distance for edges and screws dUpper !< current maximum stable dipole distance for edges and screws real(pReal) :: & selfDiffusion !< self diffusion if (timestep <= 0.0_pReal) then plasticState(ph)%dotState = 0.0_pReal return endif associate(prm => param(ph), & dst => microstructure(ph), & dot => dotState(ph), & stt => state(ph)) ns = prm%sum_N_sl tau = 0.0_pReal gdot = 0.0_pReal rho = getRho(ph,me) rhoSgl = rho(:,sgl) rhoDip = rho(:,dip) rho0 = getRho0(ph,me) my_rhoSgl0 = rho0(:,sgl) forall (s = 1:ns, t = 1:4) v(s,t) = plasticState(ph)%state(iV(s,t,ph),me) gdot = rhoSgl(:,1:4) * v * spread(prm%b_sl,2,4) #ifdef DEBUG if (debugConstitutive%basic & .and. ((debugConstitutive%element == el .and. debugConstitutive%ip == ip) & .or. .not. debugConstitutive%selective)) then print'(a,/,10(12x,12(e12.5,1x),/))', '<< CONST >> rho / 1/m^2', rhoSgl, rhoDip print'(a,/,4(12x,12(e12.5,1x),/))', '<< CONST >> gdot / 1/s',gdot endif #endif !**************************************************************************** !*** limits for stable dipole height do s = 1,ns tau(s) = math_tensordot(Mp, prm%Schmid(1:3,1:3,s)) + dst%tau_back(s,me) if (abs(tau(s)) < 1.0e-15_pReal) tau(s) = 1.0e-15_pReal enddo dLower = prm%minDipoleHeight dUpper(:,1) = prm%mu * prm%b_sl/(8.0_pReal * PI * (1.0_pReal - prm%nu) * abs(tau)) dUpper(:,2) = prm%mu * prm%b_sl/(4.0_pReal * PI * abs(tau)) where(dNeq0(sqrt(sum(abs(rho(:,edg)),2)))) & dUpper(:,1) = min(1.0_pReal/sqrt(sum(abs(rho(:,edg)),2)),dUpper(:,1)) where(dNeq0(sqrt(sum(abs(rho(:,scr)),2)))) & dUpper(:,2) = min(1.0_pReal/sqrt(sum(abs(rho(:,scr)),2)),dUpper(:,2)) dUpper = max(dUpper,dLower) !**************************************************************************** !*** dislocation multiplication rhoDotMultiplication = 0.0_pReal isBCC: if (lattice_structure(ph) == LATTICE_bcc_ID) then forall (s = 1:ns, sum(abs(v(s,1:4))) > 0.0_pReal) rhoDotMultiplication(s,1:2) = sum(abs(gdot(s,3:4))) / prm%b_sl(s) & ! assuming double-cross-slip of screws to be decisive for multiplication * sqrt(stt%rho_forest(s,me)) / prm%i_sl(s) ! & ! mean free path ! * 2.0_pReal * sum(abs(v(s,3:4))) / sum(abs(v(s,1:4))) ! ratio of screw to overall velocity determines edge generation rhoDotMultiplication(s,3:4) = sum(abs(gdot(s,3:4))) /prm%b_sl(s) & ! assuming double-cross-slip of screws to be decisive for multiplication * sqrt(stt%rho_forest(s,me)) / prm%i_sl(s) ! & ! mean free path ! * 2.0_pReal * sum(abs(v(s,1:2))) / sum(abs(v(s,1:4))) ! ratio of edge to overall velocity determines screw generation endforall else isBCC rhoDotMultiplication(:,1:4) = spread( & (sum(abs(gdot(:,1:2)),2) * prm%f_ed_mult + sum(abs(gdot(:,3:4)),2)) & * sqrt(stt%rho_forest(:,me)) / prm%i_sl / prm%b_sl, 2, 4) endif isBCC forall (s = 1:ns, t = 1:4) v0(s,t) = plasticState(ph)%state0(iV(s,t,ph),me) !**************************************************************************** !*** calculate dipole formation and annihilation !*** formation by glide do c = 1,2 rhoDotSingle2DipoleGlide(:,2*c-1) = -2.0_pReal * dUpper(:,c) / prm%b_sl & * ( rhoSgl(:,2*c-1) * abs(gdot(:,2*c)) & ! negative mobile --> positive mobile + rhoSgl(:,2*c) * abs(gdot(:,2*c-1)) & ! positive mobile --> negative mobile + abs(rhoSgl(:,2*c+4)) * abs(gdot(:,2*c-1))) ! positive mobile --> negative immobile rhoDotSingle2DipoleGlide(:,2*c) = -2.0_pReal * dUpper(:,c) / prm%b_sl & * ( rhoSgl(:,2*c-1) * abs(gdot(:,2*c)) & ! negative mobile --> positive mobile + rhoSgl(:,2*c) * abs(gdot(:,2*c-1)) & ! positive mobile --> negative mobile + abs(rhoSgl(:,2*c+3)) * abs(gdot(:,2*c))) ! negative mobile --> positive immobile rhoDotSingle2DipoleGlide(:,2*c+3) = -2.0_pReal * dUpper(:,c) / prm%b_sl & * rhoSgl(:,2*c+3) * abs(gdot(:,2*c)) ! negative mobile --> positive immobile rhoDotSingle2DipoleGlide(:,2*c+4) = -2.0_pReal * dUpper(:,c) / prm%b_sl & * rhoSgl(:,2*c+4) * abs(gdot(:,2*c-1)) ! positive mobile --> negative immobile rhoDotSingle2DipoleGlide(:,c+8) = abs(rhoDotSingle2DipoleGlide(:,2*c+3)) & + abs(rhoDotSingle2DipoleGlide(:,2*c+4)) & - rhoDotSingle2DipoleGlide(:,2*c-1) & - rhoDotSingle2DipoleGlide(:,2*c) enddo !*** athermal annihilation rhoDotAthermalAnnihilation = 0.0_pReal forall (c=1:2) & rhoDotAthermalAnnihilation(:,c+8) = -2.0_pReal * dLower(:,c) / prm%b_sl & * ( 2.0_pReal * (rhoSgl(:,2*c-1) * abs(gdot(:,2*c)) + rhoSgl(:,2*c) * abs(gdot(:,2*c-1))) & ! was single hitting single + 2.0_pReal * (abs(rhoSgl(:,2*c+3)) * abs(gdot(:,2*c)) + abs(rhoSgl(:,2*c+4)) * abs(gdot(:,2*c-1))) & ! was single hitting immobile single or was immobile single hit by single + rhoDip(:,c) * (abs(gdot(:,2*c-1)) + abs(gdot(:,2*c)))) ! single knocks dipole constituent ! annihilated screw dipoles leave edge jogs behind on the colinear system if (lattice_structure(ph) == LATTICE_fcc_ID) & forall (s = 1:ns, prm%colinearSystem(s) > 0) & rhoDotAthermalAnnihilation(prm%colinearSystem(s),1:2) = - rhoDotAthermalAnnihilation(s,10) & * 0.25_pReal * sqrt(stt%rho_forest(s,me)) * (dUpper(s,2) + dLower(s,2)) * prm%f_ed !*** thermally activated annihilation me edge dipoles by climb rhoDotThermalAnnihilation = 0.0_pReal selfDiffusion = prm%D_0 * exp(-prm%Q_cl / (kB * Temperature)) vClimb = prm%V_at * selfDiffusion * prm%mu & / ( kB * Temperature * PI * (1.0_pReal-prm%nu) * (dUpper(:,1) + dLower(:,1))) forall (s = 1:ns, dUpper(s,1) > dLower(s,1)) & rhoDotThermalAnnihilation(s,9) = max(- 4.0_pReal * rhoDip(s,1) * vClimb(s) / (dUpper(s,1) - dLower(s,1)), & - rhoDip(s,1) / timestep - rhoDotAthermalAnnihilation(s,9) & - rhoDotSingle2DipoleGlide(s,9)) ! make sure that we do not annihilate more dipoles than we have rhoDot = rhoDotFlux(timestep, ph,me,ip,el) & + rhoDotMultiplication & + rhoDotSingle2DipoleGlide & + rhoDotAthermalAnnihilation & + rhoDotThermalAnnihilation if ( any(rho(:,mob) + rhoDot(:,1:4) * timestep < -prm%atol_rho) & .or. any(rho(:,dip) + rhoDot(:,9:10) * timestep < -prm%atol_rho)) then #ifdef DEBUG if (debugConstitutive%extensive) then print'(a,i5,a,i2)', '<< CONST >> evolution rate leads to negative density at el ',el,' ip ',ip print'(a)', '<< CONST >> enforcing cutback !!!' endif #endif plasticState(ph)%dotState = IEEE_value(1.0_pReal,IEEE_quiet_NaN) else dot%rho(:,me) = pack(rhoDot,.true.) dot%gamma(:,me) = sum(gdot,2) endif end associate end subroutine nonlocal_dotState !--------------------------------------------------------------------------------------------------- !> @brief calculates the rate of change of microstructure !--------------------------------------------------------------------------------------------------- function rhoDotFlux(timestep,ph,me,ip,el) real(pReal), intent(in) :: & timestep !< substepped crystallite time increment integer, intent(in) :: & ph, & me, & ip, & !< current integration point el !< current element number integer :: & neighbor_ph, & !< phase of my neighbor's plasticity ns, & !< short notation for the total number of active slip systems c, & !< character of dislocation n, & !< index of my current neighbor neighbor_el, & !< element number of my neighbor neighbor_ip, & !< integration point of my neighbor neighbor_n, & !< neighbor index pointing to me when looking from my neighbor opposite_neighbor, & !< index of my opposite neighbor opposite_ip, & !< ip of my opposite neighbor opposite_el, & !< element index of my opposite neighbor opposite_n, & !< neighbor index pointing to me when looking from my opposite neighbor t, & !< type of dislocation no,& !< neighbor offset shortcut np,& !< neighbor phase shortcut topp, & !< type of dislocation with opposite sign to t s !< index of my current slip system real(pReal), dimension(param(ph)%sum_N_sl,10) :: & rho, & rho0, & !< dislocation density at beginning of time step rhoDotFlux !< density evolution by flux real(pReal), dimension(param(ph)%sum_N_sl,8) :: & rhoSgl, & !< current single dislocation densities (positive/negative screw and edge without dipoles) neighbor_rhoSgl0, & !< current single dislocation densities of neighboring ip (positive/negative screw and edge without dipoles) my_rhoSgl0 !< single dislocation densities of central ip (positive/negative screw and edge without dipoles) real(pReal), dimension(param(ph)%sum_N_sl,4) :: & v, & !< current dislocation glide velocity v0, & neighbor_v0, & !< dislocation glide velocity of enighboring ip gdot !< shear rates real(pReal), dimension(3,param(ph)%sum_N_sl,4) :: & m !< direction of dislocation motion real(pReal), dimension(3,3) :: & my_F, & !< my total deformation gradient neighbor_F, & !< total deformation gradient of my neighbor my_Fe, & !< my elastic deformation gradient neighbor_Fe, & !< elastic deformation gradient of my neighbor Favg !< average total deformation gradient of me and my neighbor real(pReal), dimension(3) :: & normal_neighbor2me, & !< interface normal pointing from my neighbor to me in neighbor's lattice configuration normal_neighbor2me_defConf, & !< interface normal pointing from my neighbor to me in shared deformed configuration normal_me2neighbor, & !< interface normal pointing from me to my neighbor in my lattice configuration normal_me2neighbor_defConf !< interface normal pointing from me to my neighbor in shared deformed configuration real(pReal) :: & area, & !< area of the current interface transmissivity, & !< overall transmissivity of dislocation flux to neighboring material point lineLength !< dislocation line length leaving the current interface associate(prm => param(ph), & dst => microstructure(ph), & dot => dotState(ph), & stt => state(ph)) ns = prm%sum_N_sl gdot = 0.0_pReal rho = getRho(ph,me) rhoSgl = rho(:,sgl) rho0 = getRho0(ph,me) my_rhoSgl0 = rho0(:,sgl) forall (s = 1:ns, t = 1:4) v(s,t) = plasticState(ph)%state(iV(s,t,ph),me) !ToDo: MD: I think we should use state0 here gdot = rhoSgl(:,1:4) * v * spread(prm%b_sl,2,4) forall (s = 1:ns, t = 1:4) v0(s,t) = plasticState(ph)%state0(iV(s,t,ph),me) !**************************************************************************** !*** calculate dislocation fluxes (only for nonlocal plasticity) rhoDotFlux = 0.0_pReal if (.not. phase_localPlasticity(material_phaseAt(1,el))) then !*** check CFL (Courant-Friedrichs-Lewy) condition for flux if (any( abs(gdot) > 0.0_pReal & ! any active slip system ... .and. prm%f_c * abs(v0) * timestep & > IPvolume(ip,el) / maxval(IParea(:,ip,el)))) then ! ...with velocity above critical value (we use the reference volume and area for simplicity here) #ifdef DEBUG if (debugConstitutive%extensive) then print'(a,i5,a,i2)', '<< CONST >> CFL condition not fullfilled at el ',el,' ip ',ip print'(a,e10.3,a,e10.3)', '<< CONST >> velocity is at ', & maxval(abs(v0), abs(gdot) > 0.0_pReal & .and. prm%f_c * abs(v0) * timestep & > IPvolume(ip,el) / maxval(IParea(:,ip,el))), & ' at a timestep of ',timestep print*, '<< CONST >> enforcing cutback !!!' endif #endif rhoDotFlux = IEEE_value(1.0_pReal,IEEE_quiet_NaN) ! enforce cutback return endif !*** be aware of the definition of slip_transverse = slip_direction x slip_normal !!! !*** opposite sign to our t vector in the (s,t,n) triplet !!! m(1:3,:,1) = prm%slip_direction m(1:3,:,2) = -prm%slip_direction m(1:3,:,3) = -prm%slip_transverse m(1:3,:,4) = prm%slip_transverse my_F = phase_mechanical_F(ph)%data(1:3,1:3,me) my_Fe = matmul(my_F, math_inv33(phase_mechanical_Fp(ph)%data(1:3,1:3,me))) neighbors: do n = 1,nIPneighbors neighbor_el = IPneighborhood(1,n,ip,el) neighbor_ip = IPneighborhood(2,n,ip,el) neighbor_n = IPneighborhood(3,n,ip,el) np = material_phaseAt(1,neighbor_el) no = material_phasememberAt(1,neighbor_ip,neighbor_el) opposite_neighbor = n + mod(n,2) - mod(n+1,2) opposite_el = IPneighborhood(1,opposite_neighbor,ip,el) opposite_ip = IPneighborhood(2,opposite_neighbor,ip,el) opposite_n = IPneighborhood(3,opposite_neighbor,ip,el) if (neighbor_n > 0) then ! if neighbor exists, average deformation gradient neighbor_ph = material_phaseAt(1,neighbor_el) neighbor_F = phase_mechanical_F(np)%data(1:3,1:3,no) neighbor_Fe = matmul(neighbor_F, math_inv33(phase_mechanical_Fp(np)%data(1:3,1:3,no))) Favg = 0.5_pReal * (my_F + neighbor_F) else ! if no neighbor, take my value as average Favg = my_F endif neighbor_v0 = 0.0_pReal ! needed for check of sign change in flux density below !* FLUX FROM MY NEIGHBOR TO ME !* This is only considered, if I have a neighbor of nonlocal plasticity !* (also nonlocal constitutive law with local properties) that is at least a little bit !* compatible. !* If it's not at all compatible, no flux is arriving, because everything is dammed in front of !* my neighbor's interface. !* The entering flux from my neighbor will be distributed on my slip systems according to the !* compatibility if (neighbor_n > 0) then if (phase_plasticity(material_phaseAt(1,neighbor_el)) == PLASTICITY_NONLOCAL_ID .and. & any(compatibility(:,:,:,n,ip,el) > 0.0_pReal)) then forall (s = 1:ns, t = 1:4) neighbor_v0(s,t) = plasticState(np)%state0(iV (s,t,neighbor_ph),no) neighbor_rhoSgl0(s,t) = max(plasticState(np)%state0(iRhoU(s,t,neighbor_ph),no),0.0_pReal) endforall where (neighbor_rhoSgl0 * IPvolume(neighbor_ip,neighbor_el) ** 0.667_pReal < prm%rho_min & .or. neighbor_rhoSgl0 < prm%rho_significant) & neighbor_rhoSgl0 = 0.0_pReal normal_neighbor2me_defConf = math_det33(Favg) * matmul(math_inv33(transpose(Favg)), & IPareaNormal(1:3,neighbor_n,neighbor_ip,neighbor_el)) ! normal of the interface in (average) deformed configuration (pointing neighbor => me) normal_neighbor2me = matmul(transpose(neighbor_Fe), normal_neighbor2me_defConf) & / math_det33(neighbor_Fe) ! interface normal in the lattice configuration of my neighbor area = IParea(neighbor_n,neighbor_ip,neighbor_el) * norm2(normal_neighbor2me) normal_neighbor2me = normal_neighbor2me / norm2(normal_neighbor2me) ! normalize the surface normal to unit length do s = 1,ns do t = 1,4 c = (t + 1) / 2 topp = t + mod(t,2) - mod(t+1,2) if (neighbor_v0(s,t) * math_inner(m(1:3,s,t), normal_neighbor2me) > 0.0_pReal & ! flux from my neighbor to me == entering flux for me .and. v0(s,t) * neighbor_v0(s,t) >= 0.0_pReal ) then ! ... only if no sign change in flux density lineLength = neighbor_rhoSgl0(s,t) * neighbor_v0(s,t) & * math_inner(m(1:3,s,t), normal_neighbor2me) * area ! positive line length that wants to enter through this interface where (compatibility(c,:,s,n,ip,el) > 0.0_pReal) & rhoDotFlux(:,t) = rhoDotFlux(1:ns,t) & + lineLength/IPvolume(ip,el)*compatibility(c,:,s,n,ip,el)**2.0_pReal ! transferring to equally signed mobile dislocation type where (compatibility(c,:,s,n,ip,el) < 0.0_pReal) & rhoDotFlux(:,topp) = rhoDotFlux(:,topp) & + lineLength/IPvolume(ip,el)*compatibility(c,:,s,n,ip,el)**2.0_pReal ! transferring to opposite signed mobile dislocation type endif enddo enddo endif; endif !* FLUX FROM ME TO MY NEIGHBOR !* This is not considered, if my opposite neighbor has a different constitutive law than nonlocal (still considered for nonlocal law with local properties). !* Then, we assume, that the opposite(!) neighbor sends an equal amount of dislocations to me. !* So the net flux in the direction of my neighbor is equal to zero: !* leaving flux to neighbor == entering flux from opposite neighbor !* In case of reduced transmissivity, part of the leaving flux is stored as dead dislocation density. !* That means for an interface of zero transmissivity the leaving flux is fully converted to dead dislocations. if (opposite_n > 0) then if (phase_plasticity(material_phaseAt(1,opposite_el)) == PLASTICITY_NONLOCAL_ID) then normal_me2neighbor_defConf = math_det33(Favg) & * matmul(math_inv33(transpose(Favg)),IPareaNormal(1:3,n,ip,el)) ! normal of the interface in (average) deformed configuration (pointing me => neighbor) normal_me2neighbor = matmul(transpose(my_Fe), normal_me2neighbor_defConf) & / math_det33(my_Fe) ! interface normal in my lattice configuration area = IParea(n,ip,el) * norm2(normal_me2neighbor) normal_me2neighbor = normal_me2neighbor / norm2(normal_me2neighbor) ! normalize the surface normal to unit length do s = 1,ns do t = 1,4 c = (t + 1) / 2 if (v0(s,t) * math_inner(m(1:3,s,t), normal_me2neighbor) > 0.0_pReal ) then ! flux from me to my neighbor == leaving flux for me (might also be a pure flux from my mobile density to dead density if interface not at all transmissive) if (v0(s,t) * neighbor_v0(s,t) >= 0.0_pReal) then ! no sign change in flux density transmissivity = sum(compatibility(c,:,s,n,ip,el)**2.0_pReal) ! overall transmissivity from this slip system to my neighbor else ! sign change in flux density means sign change in stress which does not allow for dislocations to arive at the neighbor transmissivity = 0.0_pReal endif lineLength = my_rhoSgl0(s,t) * v0(s,t) & * math_inner(m(1:3,s,t), normal_me2neighbor) * area ! positive line length of mobiles that wants to leave through this interface rhoDotFlux(s,t) = rhoDotFlux(s,t) - lineLength / IPvolume(ip,el) ! subtract dislocation flux from current type rhoDotFlux(s,t+4) = rhoDotFlux(s,t+4) & + lineLength / IPvolume(ip,el) * (1.0_pReal - transmissivity) & * sign(1.0_pReal, v0(s,t)) ! dislocation flux that is not able to leave through interface (because of low transmissivity) will remain as immobile single density at the material point endif enddo enddo endif; endif enddo neighbors endif end associate end function rhoDotFlux !-------------------------------------------------------------------------------------------------- !> @brief Compatibility update !> @detail Compatibility is defined as normalized product of signed cosine of the angle between the slip ! plane normals and signed cosine of the angle between the slip directions. Only the largest values ! that sum up to a total of 1 are considered, all others are set to zero. !-------------------------------------------------------------------------------------------------- module subroutine plastic_nonlocal_updateCompatibility(orientation,ph,i,e) type(rotation), dimension(1,discretization_nIPs,discretization_Nelems), intent(in) :: & orientation ! crystal orientation integer, intent(in) :: & ph, & i, & e integer :: & n, & ! neighbor index me, & neighbor_e, & ! element index of my neighbor neighbor_i, & ! integration point index of my neighbor neighbor_me, & neighbor_phase, & ns, & ! number of active slip systems s1, & ! slip system index (me) s2 ! slip system index (my neighbor) real(pReal), dimension(2,param(ph)%sum_N_sl,param(ph)%sum_N_sl,nIPneighbors) :: & my_compatibility ! my_compatibility for current element and ip real(pReal) :: & my_compatibilitySum, & thresholdValue, & nThresholdValues logical, dimension(param(ph)%sum_N_sl) :: & belowThreshold type(rotation) :: mis associate(prm => param(ph)) ns = prm%sum_N_sl me = material_phaseMemberAt(1,i,e) !*** start out fully compatible my_compatibility = 0.0_pReal forall(s1 = 1:ns) my_compatibility(:,s1,s1,:) = 1.0_pReal neighbors: do n = 1,nIPneighbors neighbor_e = IPneighborhood(1,n,i,e) neighbor_i = IPneighborhood(2,n,i,e) neighbor_me = material_phaseMemberAt(1,neighbor_i,neighbor_e) neighbor_phase = material_phaseAt(1,neighbor_e) if (neighbor_e <= 0 .or. neighbor_i <= 0) then !* FREE SURFACE !* Set surface transmissivity to the value specified in the material.config forall(s1 = 1:ns) my_compatibility(:,s1,s1,n) = sqrt(prm%chi_surface) elseif (neighbor_phase /= ph) then !* PHASE BOUNDARY !* If we encounter a different nonlocal phase at the neighbor, !* we consider this to be a real "physical" phase boundary, so completely incompatible. !* If one of the two phases has a local plasticity law, !* we do not consider this to be a phase boundary, so completely compatible. if (.not. phase_localPlasticity(neighbor_phase) .and. .not. phase_localPlasticity(ph)) & forall(s1 = 1:ns) my_compatibility(:,s1,s1,n) = 0.0_pReal elseif (prm%chi_GB >= 0.0_pReal) then !* GRAIN BOUNDARY ! !* fixed transmissivity for adjacent ips with different texture (only if explicitly given in material.config) if (any(dNeq(material_orientation0(1,ph,me)%asQuaternion(), & material_orientation0(1,neighbor_phase,neighbor_me)%asQuaternion())) .and. & (.not. phase_localPlasticity(neighbor_phase))) & forall(s1 = 1:ns) my_compatibility(:,s1,s1,n) = sqrt(prm%chi_GB) else !* GRAIN BOUNDARY ? !* Compatibility defined by relative orientation of slip systems: !* The my_compatibility value is defined as the product of the slip normal projection and the slip direction projection. !* Its sign is always positive for screws, for edges it has the same sign as the slip normal projection. !* Since the sum for each slip system can easily exceed one (which would result in a transmissivity larger than one), !* only values above or equal to a certain threshold value are considered. This threshold value is chosen, such that !* the number of compatible slip systems is minimized with the sum of the original compatibility values exceeding one. !* Finally the smallest compatibility value is decreased until the sum is exactly equal to one. !* All values below the threshold are set to zero. mis = orientation(1,i,e)%misorientation(orientation(1,neighbor_i,neighbor_e)) mySlipSystems: do s1 = 1,ns neighborSlipSystems: do s2 = 1,ns my_compatibility(1,s2,s1,n) = math_inner(prm%slip_normal(1:3,s1), & mis%rotate(prm%slip_normal(1:3,s2))) & * abs(math_inner(prm%slip_direction(1:3,s1), & mis%rotate(prm%slip_direction(1:3,s2)))) my_compatibility(2,s2,s1,n) = abs(math_inner(prm%slip_normal(1:3,s1), & mis%rotate(prm%slip_normal(1:3,s2)))) & * abs(math_inner(prm%slip_direction(1:3,s1), & mis%rotate(prm%slip_direction(1:3,s2)))) enddo neighborSlipSystems my_compatibilitySum = 0.0_pReal belowThreshold = .true. do while (my_compatibilitySum < 1.0_pReal .and. any(belowThreshold)) thresholdValue = maxval(my_compatibility(2,:,s1,n), belowThreshold) ! screws always positive nThresholdValues = real(count(my_compatibility(2,:,s1,n) >= thresholdValue),pReal) where (my_compatibility(2,:,s1,n) >= thresholdValue) belowThreshold = .false. if (my_compatibilitySum + thresholdValue * nThresholdValues > 1.0_pReal) & where (abs(my_compatibility(:,:,s1,n)) >= thresholdValue) & my_compatibility(:,:,s1,n) = sign((1.0_pReal - my_compatibilitySum)/nThresholdValues,& my_compatibility(:,:,s1,n)) my_compatibilitySum = my_compatibilitySum + nThresholdValues * thresholdValue enddo where(belowThreshold) my_compatibility(1,:,s1,n) = 0.0_pReal where(belowThreshold) my_compatibility(2,:,s1,n) = 0.0_pReal enddo mySlipSystems endif enddo neighbors compatibility(:,:,:,:,i,e) = my_compatibility end associate end subroutine plastic_nonlocal_updateCompatibility !-------------------------------------------------------------------------------------------------- !> @brief writes results to HDF5 output file !-------------------------------------------------------------------------------------------------- module subroutine plastic_nonlocal_results(ph,group) integer, intent(in) :: ph character(len=*),intent(in) :: group integer :: o associate(prm => param(ph),dst => microstructure(ph),stt=>state(ph)) outputsLoop: do o = 1,size(prm%output) select case(trim(prm%output(o))) case('rho_u_ed_pos') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_mob_edg_pos, trim(prm%output(o)), & 'positive mobile edge density','1/m²') case('rho_b_ed_pos') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_imm_edg_pos, trim(prm%output(o)), & 'positive immobile edge density','1/m²') case('rho_u_ed_neg') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_mob_edg_neg, trim(prm%output(o)), & 'negative mobile edge density','1/m²') case('rho_b_ed_neg') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_imm_edg_neg, trim(prm%output(o)), & 'negative immobile edge density','1/m²') case('rho_d_ed') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_dip_edg, trim(prm%output(o)), & 'edge dipole density','1/m²') case('rho_u_sc_pos') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_mob_scr_pos, trim(prm%output(o)), & 'positive mobile screw density','1/m²') case('rho_b_sc_pos') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_imm_scr_pos, trim(prm%output(o)), & 'positive immobile screw density','1/m²') case('rho_u_sc_neg') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_mob_scr_neg, trim(prm%output(o)), & 'negative mobile screw density','1/m²') case('rho_b_sc_neg') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_sgl_imm_scr_neg, trim(prm%output(o)), & 'negative immobile screw density','1/m²') case('rho_d_sc') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_dip_scr, trim(prm%output(o)), & 'screw dipole density','1/m²') case('rho_f') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%rho_forest, trim(prm%output(o)), & 'forest density','1/m²') case('v_ed_pos') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%v_edg_pos, trim(prm%output(o)), & 'positive edge velocity','m/s') case('v_ed_neg') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%v_edg_neg, trim(prm%output(o)), & 'negative edge velocity','m/s') case('v_sc_pos') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%v_scr_pos, trim(prm%output(o)), & 'positive srew velocity','m/s') case('v_sc_neg') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%v_scr_neg, trim(prm%output(o)), & 'negative screw velocity','m/s') case('gamma') if(prm%sum_N_sl>0) call results_writeDataset(group,stt%gamma, trim(prm%output(o)), & 'plastic shear','1') case('tau_pass') if(prm%sum_N_sl>0) call results_writeDataset(group,dst%tau_pass, trim(prm%output(o)), & 'passing stress for slip','Pa') end select enddo outputsLoop end associate end subroutine plastic_nonlocal_results !-------------------------------------------------------------------------------------------------- !> @brief populates the initial dislocation density !-------------------------------------------------------------------------------------------------- subroutine stateInit(ini,phase,Nmembers) type(tInitialParameters) :: & ini integer,intent(in) :: & phase, & Nmembers integer :: & i, & e, & f, & from, & upto, & s, & phasemember real(pReal), dimension(2) :: & noise, & rnd real(pReal) :: & meanDensity, & totalVolume, & densityBinning, & minimumIpVolume real(pReal), dimension(Nmembers) :: & volume associate(stt => state(phase)) if (ini%random_rho_u > 0.0_pReal) then ! randomly distribute dislocation segments on random slip system and of random type in the volume do e = 1,discretization_Nelems do i = 1,discretization_nIPs if (material_phaseAt(1,e) == phase) volume(material_phasememberAt(1,i,e)) = IPvolume(i,e) enddo enddo totalVolume = sum(volume) minimumIPVolume = minval(volume) densityBinning = ini%random_rho_u_binning / minimumIpVolume ** (2.0_pReal / 3.0_pReal) ! fill random material points with dislocation segments until the desired overall density is reached meanDensity = 0.0_pReal do while(meanDensity < ini%random_rho_u) call random_number(rnd) phasemember = nint(rnd(1)*real(Nmembers,pReal) + 0.5_pReal) s = nint(rnd(2)*real(sum(ini%N_sl),pReal)*4.0_pReal + 0.5_pReal) meanDensity = meanDensity + densityBinning * volume(phasemember) / totalVolume stt%rhoSglMobile(s,phasemember) = densityBinning enddo else ! homogeneous distribution with noise do e = 1, Nmembers do f = 1,size(ini%N_sl,1) from = 1 + sum(ini%N_sl(1:f-1)) upto = sum(ini%N_sl(1:f)) do s = from,upto noise = [math_sampleGaussVar(0.0_pReal, ini%sigma_rho_u), & math_sampleGaussVar(0.0_pReal, ini%sigma_rho_u)] stt%rho_sgl_mob_edg_pos(s,e) = ini%rho_u_ed_pos_0(f) + noise(1) stt%rho_sgl_mob_edg_neg(s,e) = ini%rho_u_ed_neg_0(f) + noise(1) stt%rho_sgl_mob_scr_pos(s,e) = ini%rho_u_sc_pos_0(f) + noise(2) stt%rho_sgl_mob_scr_neg(s,e) = ini%rho_u_sc_neg_0(f) + noise(2) enddo stt%rho_dip_edg(from:upto,e) = ini%rho_d_ed_0(f) stt%rho_dip_scr(from:upto,e) = ini%rho_d_sc_0(f) enddo enddo endif end associate end subroutine stateInit !-------------------------------------------------------------------------------------------------- !> @brief calculates kinetics !-------------------------------------------------------------------------------------------------- pure subroutine kinetics(v, dv_dtau, dv_dtauNS, tau, tauNS, tauThreshold, c, Temperature, ph) integer, intent(in) :: & c, & !< dislocation character (1:edge, 2:screw) ph real(pReal), intent(in) :: & Temperature !< temperature real(pReal), dimension(param(ph)%sum_N_sl), intent(in) :: & tau, & !< resolved external shear stress (without non Schmid effects) tauNS, & !< resolved external shear stress (including non Schmid effects) tauThreshold !< threshold shear stress real(pReal), dimension(param(ph)%sum_N_sl), intent(out) :: & v, & !< velocity dv_dtau, & !< velocity derivative with respect to resolved shear stress (without non Schmid contributions) dv_dtauNS !< velocity derivative with respect to resolved shear stress (including non Schmid contributions) integer :: & s !< index of my current slip system real(pReal) :: & tauRel_P, & tauRel_S, & tauEff, & !< effective shear stress tPeierls, & !< waiting time in front of a peierls barriers tSolidSolution, & !< waiting time in front of a solid solution obstacle vViscous, & !< viscous glide velocity dtPeierls_dtau, & !< derivative with respect to resolved shear stress dtSolidSolution_dtau, & !< derivative with respect to resolved shear stress meanfreepath_S, & !< mean free travel distance for dislocations between two solid solution obstacles meanfreepath_P, & !< mean free travel distance for dislocations between two Peierls barriers jumpWidth_P, & !< depth of activated area jumpWidth_S, & !< depth of activated area activationLength_P, & !< length of activated dislocation line activationLength_S, & !< length of activated dislocation line activationVolume_P, & !< volume that needs to be activated to overcome barrier activationVolume_S, & !< volume that needs to be activated to overcome barrier activationEnergy_P, & !< energy that is needed to overcome barrier activationEnergy_S, & !< energy that is needed to overcome barrier criticalStress_P, & !< maximum obstacle strength criticalStress_S, & !< maximum obstacle strength mobility !< dislocation mobility associate(prm => param(ph)) v = 0.0_pReal dv_dtau = 0.0_pReal dv_dtauNS = 0.0_pReal do s = 1,prm%sum_N_sl if (abs(tau(s)) > tauThreshold(s)) then !* Peierls contribution !* Effective stress includes non Schmid constributions !* The derivative only gives absolute values; the correct sign is taken care of in the formula for the derivative of the velocity tauEff = max(0.0_pReal, abs(tauNS(s)) - tauThreshold(s)) ! ensure that the effective stress is positive meanfreepath_P = prm%b_sl(s) jumpWidth_P = prm%b_sl(s) activationLength_P = prm%w *prm%b_sl(s) activationVolume_P = activationLength_P * jumpWidth_P * prm%b_sl(s) criticalStress_P = prm%peierlsStress(s,c) activationEnergy_P = criticalStress_P * activationVolume_P tauRel_P = min(1.0_pReal, tauEff / criticalStress_P) ! ensure that the activation probability cannot become greater than one tPeierls = 1.0_pReal / prm%nu_a & * exp(activationEnergy_P / (kB * Temperature) & * (1.0_pReal - tauRel_P**prm%p)**prm%q) if (tauEff < criticalStress_P) then dtPeierls_dtau = tPeierls * prm%p * prm%q * activationVolume_P / (kB * Temperature) & * (1.0_pReal - tauRel_P**prm%p)**(prm%q-1.0_pReal) * tauRel_P**(prm%p-1.0_pReal) else dtPeierls_dtau = 0.0_pReal endif !* Contribution from solid solution strengthening !* The derivative only gives absolute values; the correct sign is taken care of in the formula for the derivative of the velocity tauEff = abs(tau(s)) - tauThreshold(s) meanfreepath_S = prm%b_sl(s) / sqrt(prm%c_sol) jumpWidth_S = prm%f_sol * prm%b_sl(s) activationLength_S = prm%b_sl(s) / sqrt(prm%c_sol) activationVolume_S = activationLength_S * jumpWidth_S * prm%b_sl(s) activationEnergy_S = prm%Q_sol criticalStress_S = activationEnergy_S / activationVolume_S tauRel_S = min(1.0_pReal, tauEff / criticalStress_S) ! ensure that the activation probability cannot become greater than one tSolidSolution = 1.0_pReal / prm%nu_a & * exp(activationEnergy_S / (kB * Temperature)* (1.0_pReal - tauRel_S**prm%p)**prm%q) if (tauEff < criticalStress_S) then dtSolidSolution_dtau = tSolidSolution * prm%p * prm%q * activationVolume_S / (kB * Temperature) & * (1.0_pReal - tauRel_S**prm%p)**(prm%q-1.0_pReal)* tauRel_S**(prm%p-1.0_pReal) else dtSolidSolution_dtau = 0.0_pReal endif !* viscous glide velocity tauEff = abs(tau(s)) - tauThreshold(s) mobility = prm%b_sl(s) / prm%eta vViscous = mobility * tauEff !* Mean velocity results from waiting time at peierls barriers and solid solution obstacles with respective meanfreepath of !* free flight at glide velocity in between. !* adopt sign from resolved stress v(s) = sign(1.0_pReal,tau(s)) & / (tPeierls / meanfreepath_P + tSolidSolution / meanfreepath_S + 1.0_pReal / vViscous) dv_dtau(s) = v(s)**2.0_pReal * (dtSolidSolution_dtau / meanfreepath_S + mobility /vViscous**2.0_pReal) dv_dtauNS(s) = v(s)**2.0_pReal * dtPeierls_dtau / meanfreepath_P endif enddo end associate end subroutine kinetics !-------------------------------------------------------------------------------------------------- !> @brief returns copy of current dislocation densities from state !> @details raw values is rectified !-------------------------------------------------------------------------------------------------- pure function getRho(ph,me) integer, intent(in) :: ph, me real(pReal), dimension(param(ph)%sum_N_sl,10) :: getRho associate(prm => param(ph)) getRho = reshape(state(ph)%rho(:,me),[prm%sum_N_sl,10]) ! ensure positive densities (not for imm, they have a sign) getRho(:,mob) = max(getRho(:,mob),0.0_pReal) getRho(:,dip) = max(getRho(:,dip),0.0_pReal) where(abs(getRho) < max(prm%rho_min/geom(ph)%V_0(me)**(2.0_pReal/3.0_pReal),prm%rho_significant)) & getRho = 0.0_pReal end associate end function getRho !-------------------------------------------------------------------------------------------------- !> @brief returns copy of current dislocation densities from state !> @details raw values is rectified !-------------------------------------------------------------------------------------------------- pure function getRho0(ph,me) integer, intent(in) :: ph, me real(pReal), dimension(param(ph)%sum_N_sl,10) :: getRho0 associate(prm => param(ph)) getRho0 = reshape(state0(ph)%rho(:,me),[prm%sum_N_sl,10]) ! ensure positive densities (not for imm, they have a sign) getRho0(:,mob) = max(getRho0(:,mob),0.0_pReal) getRho0(:,dip) = max(getRho0(:,dip),0.0_pReal) where (abs(getRho0) < max(prm%rho_min/geom(ph)%V_0(me)**(2.0_pReal/3.0_pReal),prm%rho_significant)) & getRho0 = 0.0_pReal end associate end function getRho0 subroutine storeGeometry(ph) integer, intent(in) :: ph integer :: ce, co real(pReal), dimension(:), allocatable :: V V = reshape(IPvolume,[product(shape(IPvolume))]) do ce = 1, size(material_homogenizationMemberAt2,1) do co = 1, homogenization_maxNconstituents if (material_phaseAt2(co,ce) == ph) geom(ph)%V_0(material_phaseMemberAt2(co,ce)) = V(ce) enddo enddo end subroutine end submodule nonlocal