!-------------------------------------------------------------------------------------------------- !> @author Philip Eisenlohr, Michigan State University !> @author Zhuowen Zhao, Michigan State University !> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH !> @brief Phenomenological crystal plasticity using a power-law formulation for the shear rates !! and a Voce-type kinematic hardening rule. !-------------------------------------------------------------------------------------------------- submodule(phase:plastic) kinehardening type :: tParameters real(pREAL) :: & n = 1.0_pREAL, & !< stress exponent for slip dot_gamma_0 = 1.0_pREAL !< reference shear strain rate for slip real(pREAL), allocatable, dimension(:) :: & h_0_xi, & !< initial hardening rate of forest stress per slip family !! θ_0,for h_0_chi, & !< initial hardening rate of back stress per slip family !! θ_0,bs h_inf_xi, & !< asymptotic hardening rate of forest stress per slip family !! θ_1,for h_inf_chi, & !< asymptotic hardening rate of back stress per slip family !! θ_1,bs xi_inf, & !< back-extrapolated forest stress from terminal linear hardening chi_inf !< back-extrapolated back stress from terminal linear hardening real(pREAL), allocatable, dimension(:,:) :: & h_sl_sl !< slip resistance change per slip activity real(pREAL), allocatable, dimension(:,:,:) :: & P, & P_nS_pos, & P_nS_neg integer :: & sum_N_sl logical :: & nonSchmidActive = .false. character(len=pSTRLEN), allocatable, dimension(:) :: & output character(len=:), allocatable, dimension(:) :: & systems_sl end type tParameters type :: tIndexDotState integer, dimension(2) :: & xi, & chi, & gamma end type tIndexDotState type :: tKinehardeningState real(pREAL), pointer, dimension(:,:) :: & xi, & !< forest stress !! τ_for chi, & !< back stress !! τ_bs chi_flip, & !< back stress at last reversal of stress sense !! χ_0 gamma, & !< accumulated (absolute) shear gamma_flip, & !< accumulated shear at last reversal of stress sense !! γ_0 sgn_gamma !< sense of acting shear stress (-1 or +1) end type tKinehardeningState !-------------------------------------------------------------------------------------------------- ! containers for parameters and state type(tParameters), allocatable, dimension(:) :: param type(tIndexDotState), allocatable, dimension(:) :: indexDotState type(tKinehardeningState), allocatable, dimension(:) :: state, deltaState contains !-------------------------------------------------------------------------------------------------- !> @brief Perform module initialization. !> @details reads in material parameters, allocates arrays, and does sanity checks !-------------------------------------------------------------------------------------------------- module function plastic_kinehardening_init() result(myPlasticity) logical, dimension(:), allocatable :: myPlasticity integer :: & ph, o, & Nmembers, & sizeState, sizeDeltaState, sizeDotState, & startIndex, endIndex integer, dimension(:), allocatable :: & N_sl real(pREAL), dimension(:), allocatable :: & xi_0, & !< initial forest stress !! τ_for,0 a !< non-Schmid coefficients character(len=:), allocatable :: & refs, & extmsg type(tDict), pointer :: & phases, & phase, & mech, & pl myPlasticity = plastic_active('kinehardening') if (count(myPlasticity) == 0) return print'(/,1x,a)', '<<<+- phase:mechanical:plastic:kinehardening init -+>>>' print'(/,a,i0)', ' # phases: ',count(myPlasticity); flush(IO_STDOUT) print'(/,1x,a)', 'J.A. Wollmershauser et al., International Journal of Fatigue 36:181–193, 2012' print'( 1x,a)', 'https://doi.org/10.1016/j.ijfatigue.2011.07.008' phases => config_material%get_dict('phase') allocate(param(phases%length)) allocate(indexDotState(phases%length)) allocate(state(phases%length)) allocate(deltaState(phases%length)) extmsg = '' do ph = 1, phases%length if (.not. myPlasticity(ph)) cycle associate(prm => param(ph), & stt => state(ph), dlt => deltaState(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 !-------------------------------------------------------------------------------------------------- ! 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 = crystal_SchmidMatrix_slip(N_sl,phase_lattice(ph),phase_cOverA(ph)) if (phase_lattice(ph) == 'cI') then a = pl%get_as1dReal('a_nonSchmid',defaultVal=emptyRealArray) prm%nonSchmidActive = size(a) > 0 prm%P_nS_pos = crystal_nonSchmidMatrix(N_sl,a,+1) prm%P_nS_neg = crystal_nonSchmidMatrix(N_sl,a,-1) else prm%P_nS_pos = prm%P prm%P_nS_neg = prm%P end if prm%dot_gamma_0 = pl%get_asReal('dot_gamma_0') prm%n = pl%get_asReal('n') prm%h_sl_sl = crystal_interaction_SlipBySlip(N_sl,pl%get_as1dReal('h_sl-sl'), & phase_lattice(ph)) xi_0 = math_expand(pl%get_as1dReal('xi_0', requiredSize=size(N_sl)),N_sl) prm%xi_inf = math_expand(pl%get_as1dReal('xi_inf', requiredSize=size(N_sl)),N_sl) prm%chi_inf = math_expand(pl%get_as1dReal('chi_inf', requiredSize=size(N_sl)),N_sl) prm%h_0_xi = math_expand(pl%get_as1dReal('h_0_xi', requiredSize=size(N_sl)),N_sl) prm%h_0_chi = math_expand(pl%get_as1dReal('h_0_chi', requiredSize=size(N_sl)),N_sl) prm%h_inf_xi = math_expand(pl%get_as1dReal('h_inf_xi', requiredSize=size(N_sl)),N_sl) prm%h_inf_chi = math_expand(pl%get_as1dReal('h_inf_chi', requiredSize=size(N_sl)),N_sl) !-------------------------------------------------------------------------------------------------- ! sanity checks if ( prm%dot_gamma_0 <= 0.0_pREAL) extmsg = trim(extmsg)//' dot_gamma_0' if ( prm%n <= 0.0_pREAL) extmsg = trim(extmsg)//' n' if (any(xi_0 <= 0.0_pREAL)) extmsg = trim(extmsg)//' xi_0' if (any(prm%xi_inf <= 0.0_pREAL)) extmsg = trim(extmsg)//' xi_inf' if (any(prm%chi_inf <= 0.0_pREAL)) extmsg = trim(extmsg)//' chi_inf' else slipActive xi_0 = emptyRealArray allocate(prm%xi_inf, & prm%chi_inf, & prm%h_0_xi, & prm%h_0_chi, & prm%h_inf_xi, & prm%h_inf_chi, & source=emptyRealArray) allocate(prm%h_sl_sl(0,0)) end if slipActive !-------------------------------------------------------------------------------------------------- ! allocate state arrays Nmembers = count(material_ID_phase == ph) sizeDotState = prm%sum_N_sl * size(['xi ',& 'chi ',& 'gamma']) sizeDeltaState = prm%sum_N_sl * size(['sgn_gamma ',& 'chi_flip ',& 'gamma_flip']) sizeState = sizeDotState + sizeDeltaState call phase_allocateState(plasticState(ph),Nmembers,sizeState,sizeDotState,sizeDeltaState) deallocate(plasticState(ph)%dotState) ! ToDo: remove dotState completely !-------------------------------------------------------------------------------------------------- ! state aliases and initialization startIndex = 1 endIndex = prm%sum_N_sl idx_dot%xi = [startIndex,endIndex] stt%xi => plasticState(ph)%state(startIndex:endIndex,:) stt%xi = spread(xi_0, 2, Nmembers) plasticState(ph)%atol(startIndex:endIndex) = pl%get_asReal('atol_xi',defaultVal=1.0_pREAL) if (any(plasticState(ph)%atol(startIndex:endIndex) < 0.0_pREAL)) extmsg = trim(extmsg)//' atol_xi' startIndex = endIndex + 1 endIndex = endIndex + prm%sum_N_sl idx_dot%chi = [startIndex,endIndex] stt%chi => plasticState(ph)%state(startIndex:endIndex,:) plasticState(ph)%atol(startIndex:endIndex) = pl%get_asReal('atol_xi',defaultVal=1.0_pREAL) startIndex = endIndex + 1 endIndex = endIndex + prm%sum_N_sl idx_dot%gamma = [startIndex,endIndex] stt%gamma => 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' o = plasticState(ph)%offsetDeltaState startIndex = endIndex + 1 endIndex = endIndex + prm%sum_N_sl stt%sgn_gamma => plasticState(ph)%state (startIndex :endIndex ,:) dlt%sgn_gamma => plasticState(ph)%deltaState(startIndex-o:endIndex-o,:) startIndex = endIndex + 1 endIndex = endIndex + prm%sum_N_sl stt%chi_flip => plasticState(ph)%state (startIndex :endIndex ,:) dlt%chi_flip => plasticState(ph)%deltaState(startIndex-o:endIndex-o,:) startIndex = endIndex + 1 endIndex = endIndex + prm%sum_N_sl stt%gamma_flip => plasticState(ph)%state (startIndex :endIndex ,:) dlt%gamma_flip => plasticState(ph)%deltaState(startIndex-o:endIndex-o,:) 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_kinehardening_init !-------------------------------------------------------------------------------------------------- !> @brief Calculate plastic velocity gradient and its tangent. !-------------------------------------------------------------------------------------------------- pure module subroutine kinehardening_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), dimension(param(ph)%sum_N_sl) :: & dot_gamma_pos,dot_gamma_neg, & ddot_gamma_dtau_pos,ddot_gamma_dtau_neg Lp = 0.0_pREAL dLp_dMp = 0.0_pREAL associate(prm => param(ph)) call kinetics(Mp,ph,en, dot_gamma_pos,dot_gamma_neg,ddot_gamma_dtau_pos,ddot_gamma_dtau_neg) do i = 1, prm%sum_N_sl Lp = Lp + (dot_gamma_pos(i)+dot_gamma_neg(i))*prm%P(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_pos(i) * prm%P(k,l,i) * prm%P_nS_pos(m,n,i) & + ddot_gamma_dtau_neg(i) * prm%P(k,l,i) * prm%P_nS_neg(m,n,i) end do end associate end subroutine kinehardening_LpAndItsTangent !-------------------------------------------------------------------------------------------------- !> @brief Calculate the rate of change of microstructure. !-------------------------------------------------------------------------------------------------- module function plastic_kinehardening_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) :: & sumGamma real(pREAL), dimension(param(ph)%sum_N_sl) :: & dot_gamma_pos,dot_gamma_neg associate(prm => param(ph), stt => state(ph), & dot_xi => dotState(IndexDotState(ph)%xi(1):IndexDotState(ph)%xi(2)),& dot_chi => dotState(IndexDotState(ph)%chi(1):IndexDotState(ph)%chi(2)),& dot_gamma => dotState(IndexDotState(ph)%gamma(1):IndexDotState(ph)%gamma(2))) call kinetics(Mp,ph,en, dot_gamma_pos,dot_gamma_neg) dot_gamma = abs(dot_gamma_pos+dot_gamma_neg) sumGamma = sum(stt%gamma(:,en)) dot_xi = matmul(prm%h_sl_sl,dot_gamma) & * ( prm%h_inf_xi & + ( prm%h_0_xi & - prm%h_inf_xi * (1_pREAL -sumGamma*prm%h_0_xi/prm%xi_inf) ) & * exp(-sumGamma*prm%h_0_xi/prm%xi_inf) & ) dot_chi = stt%sgn_gamma(:,en)*dot_gamma & * ( prm%h_inf_chi & + ( prm%h_0_chi & - prm%h_inf_chi*(1_pREAL -(stt%gamma(:,en)-stt%gamma_flip(:,en))*prm%h_0_chi/(prm%chi_inf+stt%chi_flip(:,en))) ) & * exp(-(stt%gamma(:,en)-stt%gamma_flip(:,en))*prm%h_0_chi/(prm%chi_inf+stt%chi_flip(:,en))) & ) end associate end function plastic_kinehardening_dotState !-------------------------------------------------------------------------------------------------- !> @brief Calculate (instantaneous) incremental change of microstructure. !-------------------------------------------------------------------------------------------------- module subroutine plastic_kinehardening_deltaState(Mp,ph,en) real(pREAL), dimension(3,3), intent(in) :: & Mp !< Mandel stress integer, intent(in) :: & ph, & en real(pREAL), dimension(param(ph)%sum_N_sl) :: & dot_gamma_pos,dot_gamma_neg, & sgn_gamma associate(prm => param(ph), stt => state(ph), dlt => deltaState(ph)) call kinetics(Mp,ph,en, dot_gamma_pos,dot_gamma_neg) sgn_gamma = merge(state(ph)%sgn_gamma(:,en), & sign(1.0_pREAL,dot_gamma_pos+dot_gamma_neg), & dEq0(dot_gamma_pos+dot_gamma_neg,1e-10_pREAL)) where(dNeq(sgn_gamma,stt%sgn_gamma(:,en),0.1_pREAL)) ! ToDo sgn_gamma*stt%sgn_gamma(:,en)<0 dlt%sgn_gamma (:,en) = sgn_gamma - stt%sgn_gamma (:,en) dlt%chi_flip (:,en) = abs(stt%chi (:,en)) - stt%chi_flip (:,en) dlt%gamma_flip(:,en) = stt%gamma(:,en) - stt%gamma_flip(:,en) else where dlt%sgn_gamma (:,en) = 0.0_pREAL dlt%chi_flip (:,en) = 0.0_pREAL dlt%gamma_flip(:,en) = 0.0_pREAL end where end associate end subroutine plastic_kinehardening_deltaState !-------------------------------------------------------------------------------------------------- !> @brief Write results to HDF5 output file. !-------------------------------------------------------------------------------------------------- module subroutine plastic_kinehardening_result(ph,group) integer, intent(in) :: ph character(len=*), intent(in) :: group integer :: ou associate(prm => param(ph), stt => state(ph)) do ou = 1,size(prm%output) select case(trim(prm%output(ou))) case ('xi') call result_writeDataset(stt%xi,group,trim(prm%output(ou)), & 'forest stress','Pa',prm%systems_sl) case ('chi') call result_writeDataset(stt%chi,group,trim(prm%output(ou)), & 'back stress','Pa',prm%systems_sl) case ('sgn(gamma)') call result_writeDataset(int(stt%sgn_gamma),group,trim(prm%output(ou)), & 'sense of shear','1',prm%systems_sl) case ('chi_flip') call result_writeDataset(stt%chi_flip,group,trim(prm%output(ou)), & 'back stress at last reversal of stress sense','Pa',prm%systems_sl) case ('gamma_flip') call result_writeDataset(stt%gamma_flip,group,trim(prm%output(ou)), & 'plastic shear at last reversal of stress sense','1',prm%systems_sl) case ('gamma') call result_writeDataset(stt%gamma,group,trim(prm%output(ou)), & 'plastic shear','1',prm%systems_sl) end select end do end associate end subroutine plastic_kinehardening_result !-------------------------------------------------------------------------------------------------- !> @brief Calculate shear rates on slip systems and their derivatives with respect to resolved ! stress. !> @details: Derivatives are calculated only optionally. ! NOTE: Contrary to common convention, here the result (i.e. intent(out)) variables have to be put ! at the end since some of them are optional. !-------------------------------------------------------------------------------------------------- pure subroutine kinetics(Mp,ph,en, & dot_gamma_pos,dot_gamma_neg,ddot_gamma_dtau_pos,ddot_gamma_dtau_neg) real(pREAL), dimension(3,3), intent(in) :: & Mp !< Mandel stress integer, intent(in) :: & ph, & en real(pREAL), intent(out), dimension(param(ph)%sum_N_sl) :: & dot_gamma_pos, & dot_gamma_neg real(pREAL), intent(out), dimension(param(ph)%sum_N_sl), optional :: & ddot_gamma_dtau_pos, & ddot_gamma_dtau_neg real(pREAL), dimension(param(ph)%sum_N_sl) :: & tau_pos, & tau_neg integer :: i associate(prm => param(ph), stt => state(ph)) do i = 1, prm%sum_N_sl tau_pos(i) = math_tensordot(Mp,prm%P_nS_pos(1:3,1:3,i)) - stt%chi(i,en) tau_neg(i) = merge(math_tensordot(Mp,prm%P_nS_neg(1:3,1:3,i)) - stt%chi(i,en), & 0.0_pREAL, prm%nonSchmidActive) end do where(dNeq0(tau_pos)) dot_gamma_pos = prm%dot_gamma_0 * merge(0.5_pREAL,1.0_pREAL, prm%nonSchmidActive) & ! 1/2 if non-Schmid active * sign(abs(tau_pos/stt%xi(:,en))**prm%n, tau_pos) else where dot_gamma_pos = 0.0_pREAL end where where(dNeq0(tau_neg)) dot_gamma_neg = prm%dot_gamma_0 * 0.5_pREAL & ! only used if non-Schmid active, always 1/2 * sign(abs(tau_neg/stt%xi(:,en))**prm%n, tau_neg) else where dot_gamma_neg = 0.0_pREAL end where if (present(ddot_gamma_dtau_pos)) then where(dNeq0(dot_gamma_pos)) ddot_gamma_dtau_pos = dot_gamma_pos*prm%n/tau_pos else where ddot_gamma_dtau_pos = 0.0_pREAL end where end if if (present(ddot_gamma_dtau_neg)) then where(dNeq0(dot_gamma_neg)) ddot_gamma_dtau_neg = dot_gamma_neg*prm%n/tau_neg else where ddot_gamma_dtau_neg = 0.0_pREAL end where end if end associate end subroutine kinetics end submodule kinehardening