!-------------------------------------------------------------------------------------------------- !> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH !> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH !> @author Pratheek Shanthraj, Max-Planck-Institut für Eisenforschung GmbH !> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH !> @brief contains lattice definitions including Schmid matrices for slip, twin, trans, ! and cleavage as well as interaction among the various systems !-------------------------------------------------------------------------------------------------- module lattice use prec use IO use config use math use rotations implicit none private !-------------------------------------------------------------------------------------------------- ! face centered cubic (cF) integer, dimension(*), parameter :: & FCC_NSLIPSYSTEM = [12, 6] !< # of slip systems per family for fcc integer, dimension(*), parameter :: & FCC_NTWINSYSTEM = [12] !< # of twin systems per family for fcc integer, dimension(*), parameter :: & FCC_NTRANSSYSTEM = [12] !< # of transformation systems per family for fcc integer, dimension(*), parameter :: & FCC_NCLEAVAGESYSTEM = [3] !< # of cleavage systems per family for fcc integer, parameter :: & FCC_NSLIP = sum(FCC_NSLIPSYSTEM), & !< total # of slip systems for fcc FCC_NTWIN = sum(FCC_NTWINSYSTEM), & !< total # of twin systems for fcc FCC_NTRANS = sum(FCC_NTRANSSYSTEM), & !< total # of transformation systems for fcc FCC_NCLEAVAGE = sum(FCC_NCLEAVAGESYSTEM) !< total # of cleavage systems for fcc real(pReal), dimension(3+3,FCC_NSLIP), parameter :: & FCC_SYSTEMSLIP = reshape(real([& ! <110>{111} systems 0, 1,-1, 1, 1, 1, & ! B2 -1, 0, 1, 1, 1, 1, & ! B4 1,-1, 0, 1, 1, 1, & ! B5 0,-1,-1, -1,-1, 1, & ! C1 1, 0, 1, -1,-1, 1, & ! C3 -1, 1, 0, -1,-1, 1, & ! C5 0,-1, 1, 1,-1,-1, & ! A2 -1, 0,-1, 1,-1,-1, & ! A3 1, 1, 0, 1,-1,-1, & ! A6 0, 1, 1, -1, 1,-1, & ! D1 1, 0,-1, -1, 1,-1, & ! D4 -1,-1, 0, -1, 1,-1, & ! D6 ! <110>{110}/non-octahedral systems 1, 1, 0, 1,-1, 0, & 1,-1, 0, 1, 1, 0, & 1, 0, 1, 1, 0,-1, & 1, 0,-1, 1, 0, 1, & 0, 1, 1, 0, 1,-1, & 0, 1,-1, 0, 1, 1 & ],pReal),shape(FCC_SYSTEMSLIP)) !< fcc slip systems real(pReal), dimension(3+3,FCC_NTWIN), parameter :: & FCC_SYSTEMTWIN = reshape(real( [& ! <112>{111} systems -2, 1, 1, 1, 1, 1, & 1,-2, 1, 1, 1, 1, & 1, 1,-2, 1, 1, 1, & 2,-1, 1, -1,-1, 1, & -1, 2, 1, -1,-1, 1, & -1,-1,-2, -1,-1, 1, & -2,-1,-1, 1,-1,-1, & 1, 2,-1, 1,-1,-1, & 1,-1, 2, 1,-1,-1, & 2, 1,-1, -1, 1,-1, & -1,-2,-1, -1, 1,-1, & -1, 1, 2, -1, 1,-1 & ],pReal),shape(FCC_SYSTEMTWIN)) !< fcc twin systems integer, dimension(2,FCC_NTWIN), parameter, public :: & lattice_FCC_TWINNUCLEATIONSLIPPAIR = reshape( [& 2,3, & 1,3, & 1,2, & 5,6, & 4,6, & 4,5, & 8,9, & 7,9, & 7,8, & 11,12, & 10,12, & 10,11 & ],shape(lattice_FCC_TWINNUCLEATIONSLIPPAIR)) real(pReal), dimension(3+3,FCC_NCLEAVAGE), parameter :: & FCC_SYSTEMCLEAVAGE = reshape(real([& ! <001>{001} systems 0, 1, 0, 1, 0, 0, & 0, 0, 1, 0, 1, 0, & 1, 0, 0, 0, 0, 1 & ],pReal),shape(FCC_SYSTEMCLEAVAGE)) !< fcc cleavage systems !-------------------------------------------------------------------------------------------------- ! body centered cubic (cI) integer, dimension(*), parameter :: & BCC_NSLIPSYSTEM = [12, 12, 24] !< # of slip systems per family for bcc integer, dimension(*), parameter :: & BCC_NTWINSYSTEM = [12] !< # of twin systems per family for bcc integer, dimension(*), parameter :: & BCC_NCLEAVAGESYSTEM = [3] !< # of cleavage systems per family for bcc integer, parameter :: & BCC_NSLIP = sum(BCC_NSLIPSYSTEM), & !< total # of slip systems for bcc BCC_NTWIN = sum(BCC_NTWINSYSTEM), & !< total # of twin systems for bcc BCC_NCLEAVAGE = sum(BCC_NCLEAVAGESYSTEM) !< total # of cleavage systems for bcc real(pReal), dimension(3+3,BCC_NSLIP), parameter :: & BCC_SYSTEMSLIP = reshape(real([& ! <111>{110} systems 1,-1, 1, 0, 1, 1, & ! D1 -1,-1, 1, 0, 1, 1, & ! C1 1, 1, 1, 0,-1, 1, & ! B2 -1, 1, 1, 0,-1, 1, & ! A2 -1, 1, 1, 1, 0, 1, & ! A3 -1,-1, 1, 1, 0, 1, & ! C3 1, 1, 1, -1, 0, 1, & ! B4 1,-1, 1, -1, 0, 1, & ! D4 -1, 1, 1, 1, 1, 0, & ! A6 -1, 1,-1, 1, 1, 0, & ! D6 1, 1, 1, -1, 1, 0, & ! B5 1, 1,-1, -1, 1, 0, & ! C5 ! <111>{112} systems -1, 1, 1, 2, 1, 1, & ! A-4 1, 1, 1, -2, 1, 1, & ! B-3 1, 1,-1, 2,-1, 1, & ! C-10 1,-1, 1, 2, 1,-1, & ! D-9 1,-1, 1, 1, 2, 1, & ! D-6 1, 1,-1, -1, 2, 1, & ! C-5 1, 1, 1, 1,-2, 1, & ! B-12 -1, 1, 1, 1, 2,-1, & ! A-11 1, 1,-1, 1, 1, 2, & ! C-2 1,-1, 1, -1, 1, 2, & ! D-1 -1, 1, 1, 1,-1, 2, & ! A-8 1, 1, 1, 1, 1,-2, & ! B-7 ! Slip system <111>{123} 1, 1,-1, 1, 2, 3, & 1,-1, 1, -1, 2, 3, & -1, 1, 1, 1,-2, 3, & 1, 1, 1, 1, 2,-3, & 1,-1, 1, 1, 3, 2, & 1, 1,-1, -1, 3, 2, & 1, 1, 1, 1,-3, 2, & -1, 1, 1, 1, 3,-2, & 1, 1,-1, 2, 1, 3, & 1,-1, 1, -2, 1, 3, & -1, 1, 1, 2,-1, 3, & 1, 1, 1, 2, 1,-3, & 1,-1, 1, 2, 3, 1, & 1, 1,-1, -2, 3, 1, & 1, 1, 1, 2,-3, 1, & -1, 1, 1, 2, 3,-1, & -1, 1, 1, 3, 1, 2, & 1, 1, 1, -3, 1, 2, & 1, 1,-1, 3,-1, 2, & 1,-1, 1, 3, 1,-2, & -1, 1, 1, 3, 2, 1, & 1, 1, 1, -3, 2, 1, & 1, 1,-1, 3,-2, 1, & 1,-1, 1, 3, 2,-1 & ],pReal),shape(BCC_SYSTEMSLIP)) !< bcc slip systems real(pReal), dimension(3+3,BCC_NTWIN), parameter :: & BCC_SYSTEMTWIN = reshape(real([& ! <111>{112} systems -1, 1, 1, 2, 1, 1, & 1, 1, 1, -2, 1, 1, & 1, 1,-1, 2,-1, 1, & 1,-1, 1, 2, 1,-1, & 1,-1, 1, 1, 2, 1, & 1, 1,-1, -1, 2, 1, & 1, 1, 1, 1,-2, 1, & -1, 1, 1, 1, 2,-1, & 1, 1,-1, 1, 1, 2, & 1,-1, 1, -1, 1, 2, & -1, 1, 1, 1,-1, 2, & 1, 1, 1, 1, 1,-2 & ],pReal),shape(BCC_SYSTEMTWIN)) !< bcc twin systems real(pReal), dimension(3+3,BCC_NCLEAVAGE), parameter :: & BCC_SYSTEMCLEAVAGE = reshape(real([& ! <001>{001} systems 0, 1, 0, 1, 0, 0, & 0, 0, 1, 0, 1, 0, & 1, 0, 0, 0, 0, 1 & ],pReal),shape(BCC_SYSTEMCLEAVAGE)) !< bcc cleavage systems !-------------------------------------------------------------------------------------------------- ! hexagonal (hP) integer, dimension(*), parameter :: & HEX_NSLIPSYSTEM = [3, 3, 6, 12, 6] !< # of slip systems per family for hex integer, dimension(*), parameter :: & HEX_NTWINSYSTEM = [6, 6, 6, 6] !< # of slip systems per family for hex integer, parameter :: & HEX_NSLIP = sum(HEX_NSLIPSYSTEM), & !< total # of slip systems for hex HEX_NTWIN = sum(HEX_NTWINSYSTEM) !< total # of twin systems for hex real(pReal), dimension(4+4,HEX_NSLIP), parameter :: & HEX_SYSTEMSLIP = reshape(real([& ! <-1-1.0>{00.1}/basal systems (independent of c/a-ratio) 2, -1, -1, 0, 0, 0, 0, 1, & -1, 2, -1, 0, 0, 0, 0, 1, & -1, -1, 2, 0, 0, 0, 0, 1, & ! <-1-1.0>{1-1.0}/prismatic systems (independent of c/a-ratio) 2, -1, -1, 0, 0, 1, -1, 0, & -1, 2, -1, 0, -1, 0, 1, 0, & -1, -1, 2, 0, 1, -1, 0, 0, & ! <-1-1.0>{-11.1}/1. order pyramidal systems (direction independent of c/a-ratio) -1, 2, -1, 0, 1, 0, -1, 1, & -2, 1, 1, 0, 0, 1, -1, 1, & -1, -1, 2, 0, -1, 1, 0, 1, & 1, -2, 1, 0, -1, 0, 1, 1, & 2, -1, -1, 0, 0, -1, 1, 1, & 1, 1, -2, 0, 1, -1, 0, 1, & ! <11.3>{-10.1}/1. order pyramidal systems (direction independent of c/a-ratio) -2, 1, 1, 3, 1, 0, -1, 1, & -1, -1, 2, 3, 1, 0, -1, 1, & -1, -1, 2, 3, 0, 1, -1, 1, & 1, -2, 1, 3, 0, 1, -1, 1, & 1, -2, 1, 3, -1, 1, 0, 1, & 2, -1, -1, 3, -1, 1, 0, 1, & 2, -1, -1, 3, -1, 0, 1, 1, & 1, 1, -2, 3, -1, 0, 1, 1, & 1, 1, -2, 3, 0, -1, 1, 1, & -1, 2, -1, 3, 0, -1, 1, 1, & -1, 2, -1, 3, 1, -1, 0, 1, & -2, 1, 1, 3, 1, -1, 0, 1, & ! <11.3>{-1-1.2}/2. order pyramidal systems -1, -1, 2, 3, 1, 1, -2, 2, & 1, -2, 1, 3, -1, 2, -1, 2, & 2, -1, -1, 3, -2, 1, 1, 2, & 1, 1, -2, 3, -1, -1, 2, 2, & -1, 2, -1, 3, 1, -2, 1, 2, & -2, 1, 1, 3, 2, -1, -1, 2 & ],pReal),shape(HEX_SYSTEMSLIP)) !< hex slip systems, sorted by P. Eisenlohr CCW around starting next to a_1 axis real(pReal), dimension(4+4,HEX_NTWIN), parameter :: & HEX_SYSTEMTWIN = reshape(real([& ! <-10.1>{10.2} systems, shear = (3-(c/a)^2)/(sqrt(3) c/a) ! tension in Co, Mg, Zr, Ti, and Be; compression in Cd and Zn -1, 0, 1, 1, 1, 0, -1, 2, & ! 0, -1, 1, 1, 0, 1, -1, 2, & 1, -1, 0, 1, -1, 1, 0, 2, & 1, 0, -1, 1, -1, 0, 1, 2, & 0, 1, -1, 1, 0, -1, 1, 2, & -1, 1, 0, 1, 1, -1, 0, 2, & ! <11.6>{-1-1.1} systems, shear = 1/(c/a) ! tension in Co, Re, and Zr -1, -1, 2, 6, 1, 1, -2, 1, & 1, -2, 1, 6, -1, 2, -1, 1, & 2, -1, -1, 6, -2, 1, 1, 1, & 1, 1, -2, 6, -1, -1, 2, 1, & -1, 2, -1, 6, 1, -2, 1, 1, & -2, 1, 1, 6, 2, -1, -1, 1, & ! <10.-2>{10.1} systems, shear = (4(c/a)^2-9)/(4 sqrt(3) c/a) ! compression in Mg 1, 0, -1, -2, 1, 0, -1, 1, & 0, 1, -1, -2, 0, 1, -1, 1, & -1, 1, 0, -2, -1, 1, 0, 1, & -1, 0, 1, -2, -1, 0, 1, 1, & 0, -1, 1, -2, 0, -1, 1, 1, & 1, -1, 0, -2, 1, -1, 0, 1, & ! <11.-3>{11.2} systems, shear = 2((c/a)^2-2)/(3 c/a) ! compression in Ti and Zr 1, 1, -2, -3, 1, 1, -2, 2, & -1, 2, -1, -3, -1, 2, -1, 2, & -2, 1, 1, -3, -2, 1, 1, 2, & -1, -1, 2, -3, -1, -1, 2, 2, & 1, -2, 1, -3, 1, -2, 1, 2, & 2, -1, -1, -3, 2, -1, -1, 2 & ],pReal),shape(HEX_SYSTEMTWIN)) !< hex twin systems, sorted by P. Eisenlohr CCW around starting next to a_1 axis !-------------------------------------------------------------------------------------------------- ! body centered tetragonal (tI) integer, dimension(*), parameter :: & BCT_NSLIPSYSTEM = [2, 2, 2, 4, 2, 4, 2, 2, 4, 8, 4, 8, 8 ] !< # of slip systems per family for bct integer, parameter :: & BCT_NSLIP = sum(BCT_NSLIPSYSTEM) !< total # of slip systems for bct real(pReal), dimension(3+3,BCT_NSLIP), parameter :: & BCT_SYSTEMSLIP = reshape(real([& ! {100)<001] systems 0, 0, 1, 1, 0, 0, & 0, 0, 1, 0, 1, 0, & ! {110)<001] systems 0, 0, 1, 1, 1, 0, & 0, 0, 1, -1, 1, 0, & ! {100)<010] systems 0, 1, 0, 1, 0, 0, & 1, 0, 0, 0, 1, 0, & ! {110)<1-11]/2 systems 1,-1, 1, 1, 1, 0, & 1,-1,-1, 1, 1, 0, & -1,-1,-1, -1, 1, 0, & -1,-1, 1, -1, 1, 0, & ! {110)<1-10] systems 1, -1, 0, 1, 1, 0, & 1, 1, 0, 1,-1, 0, & ! {100)<011] systems 0, 1, 1, 1, 0, 0, & 0,-1, 1, 1, 0, 0, & -1, 0, 1, 0, 1, 0, & 1, 0, 1, 0, 1, 0, & ! {001)<010] systems 0, 1, 0, 0, 0, 1, & 1, 0, 0, 0, 0, 1, & ! {001)<110] systems 1, 1, 0, 0, 0, 1, & -1, 1, 0, 0, 0, 1, & ! {011)<01-1] systems 0, 1,-1, 0, 1, 1, & 0,-1,-1, 0,-1, 1, & -1, 0,-1, -1, 0, 1, & 1, 0,-1, 1, 0, 1, & ! {011)<1-11]/2 systems 1,-1, 1, 0, 1, 1, & 1, 1,-1, 0, 1, 1, & 1, 1, 1, 0, 1,-1, & -1, 1, 1, 0, 1,-1, & 1,-1,-1, 1, 0, 1, & -1,-1, 1, 1, 0, 1, & 1, 1, 1, 1, 0,-1, & 1,-1, 1, 1, 0,-1, & ! {011)<100] systems 1, 0, 0, 0, 1, 1, & 1, 0, 0, 0, 1,-1, & 0, 1, 0, 1, 0, 1, & 0, 1, 0, 1, 0,-1, & ! {211)<01-1] systems 0, 1,-1, 2, 1, 1, & 0,-1,-1, 2,-1, 1, & 1, 0,-1, 1, 2, 1, & -1, 0,-1, -1, 2, 1, & 0, 1,-1, -2, 1, 1, & 0,-1,-1, -2,-1, 1, & -1, 0,-1, -1,-2, 1, & 1, 0,-1, 1,-2, 1, & ! {211)<-111]/2 systems -1, 1, 1, 2, 1, 1, & -1,-1, 1, 2,-1, 1, & 1,-1, 1, 1, 2, 1, & -1,-1, 1, -1, 2, 1, & 1, 1, 1, -2, 1, 1, & 1,-1, 1, -2,-1, 1, & -1, 1, 1, -1,-2, 1, & 1, 1, 1, 1,-2, 1 & ],pReal),shape(BCT_SYSTEMSLIP)) !< bct slip systems for c/a = 0.5456 (Sn), sorted by Bieler 2009 (https://doi.org/10.1007/s11664-009-0909-x) interface lattice_forestProjection_edge module procedure slipProjection_transverse end interface lattice_forestProjection_edge interface lattice_forestProjection_screw module procedure slipProjection_direction end interface lattice_forestProjection_screw public :: & lattice_init, & lattice_equivalent_nu, & lattice_equivalent_mu, & lattice_symmetrize_33, & lattice_symmetrize_C66, & lattice_SchmidMatrix_slip, & lattice_SchmidMatrix_twin, & lattice_SchmidMatrix_trans, & lattice_SchmidMatrix_cleavage, & lattice_nonSchmidMatrix, & lattice_interaction_SlipBySlip, & lattice_interaction_TwinByTwin, & lattice_interaction_TransByTrans, & lattice_interaction_SlipByTwin, & lattice_interaction_SlipByTrans, & lattice_interaction_TwinBySlip, & lattice_characteristicShear_Twin, & lattice_C66_twin, & lattice_C66_trans, & lattice_forestProjection_edge, & lattice_forestProjection_screw, & lattice_slip_normal, & lattice_slip_direction, & lattice_slip_transverse, & lattice_labels_slip, & lattice_labels_twin contains !-------------------------------------------------------------------------------------------------- !> @brief Module initialization !-------------------------------------------------------------------------------------------------- subroutine lattice_init print'(/,1x,a)', '<<<+- lattice init -+>>>'; flush(IO_STDOUT) call selfTest end subroutine lattice_init !-------------------------------------------------------------------------------------------------- !> @brief Characteristic shear for twinning !-------------------------------------------------------------------------------------------------- function lattice_characteristicShear_Twin(Ntwin,lattice,CoverA) result(characteristicShear) integer, dimension(:), intent(in) :: Ntwin !< number of active twin systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(sum(Ntwin)) :: characteristicShear integer :: & a, & !< index of active system p, & !< index in potential system list f, & !< index of my family s !< index of my system in current family integer, dimension(HEX_NTWIN), parameter :: & HEX_SHEARTWIN = reshape( [& 1, & ! <-10.1>{10.2} 1, & 1, & 1, & 1, & 1, & 2, & ! <11.6>{-1-1.1} 2, & 2, & 2, & 2, & 2, & 3, & ! <10.-2>{10.1} 3, & 3, & 3, & 3, & 3, & 4, & ! <11.-3>{11.2} 4, & 4, & 4, & 4, & 4 & ],[HEX_NTWIN]) ! indicator to formulas below a = 0 myFamilies: do f = 1,size(Ntwin,1) mySystems: do s = 1,Ntwin(f) a = a + 1 select case(lattice) case('cF','cI') characteristicShear(a) = 0.5_pReal*sqrt(2.0_pReal) case('hP') if (cOverA < 1.0_pReal .or. cOverA > 2.0_pReal) & call IO_error(131,ext_msg='lattice_characteristicShear_Twin') p = sum(HEX_NTWINSYSTEM(1:f-1))+s select case(HEX_SHEARTWIN(p)) ! from Christian & Mahajan 1995 p.29 case (1) ! <-10.1>{10.2} characteristicShear(a) = (3.0_pReal-cOverA**2)/sqrt(3.0_pReal)/CoverA case (2) ! <11.6>{-1-1.1} characteristicShear(a) = 1.0_pReal/cOverA case (3) ! <10.-2>{10.1} characteristicShear(a) = (4.0_pReal*cOverA**2-9.0_pReal)/sqrt(48.0_pReal)/cOverA case (4) ! <11.-3>{11.2} characteristicShear(a) = 2.0_pReal*(cOverA**2-2.0_pReal)/3.0_pReal/cOverA end select case default call IO_error(137,ext_msg='lattice_characteristicShear_Twin: '//trim(lattice)) end select enddo mySystems enddo myFamilies end function lattice_characteristicShear_Twin !-------------------------------------------------------------------------------------------------- !> @brief Rotated elasticity matrices for twinning in 6x6-matrix notation !-------------------------------------------------------------------------------------------------- function lattice_C66_twin(Ntwin,C66,lattice,CoverA) integer, dimension(:), intent(in) :: Ntwin !< number of active twin systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), dimension(6,6), intent(in) :: C66 !< unrotated parent stiffness matrix real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(6,6,sum(Ntwin)) :: lattice_C66_twin real(pReal), dimension(3,3,sum(Ntwin)):: coordinateSystem type(rotation) :: R integer :: i select case(lattice) case('cF') coordinateSystem = buildCoordinateSystem(Ntwin,FCC_NSLIPSYSTEM,FCC_SYSTEMTWIN,& lattice,0.0_pReal) case('cI') coordinateSystem = buildCoordinateSystem(Ntwin,BCC_NSLIPSYSTEM,BCC_SYSTEMTWIN,& lattice,0.0_pReal) case('hP') coordinateSystem = buildCoordinateSystem(Ntwin,HEX_NSLIPSYSTEM,HEX_SYSTEMTWIN,& lattice,cOverA) case default call IO_error(137,ext_msg='lattice_C66_twin: '//trim(lattice)) end select do i = 1, sum(Ntwin) call R%fromAxisAngle([coordinateSystem(1:3,2,i),PI],P=1) ! ToDo: Why always 180 deg? lattice_C66_twin(1:6,1:6,i) = R%rotStiffness(C66) enddo end function lattice_C66_twin !-------------------------------------------------------------------------------------------------- !> @brief Rotated elasticity matrices for transformation in 6x6-matrix notation !-------------------------------------------------------------------------------------------------- function lattice_C66_trans(Ntrans,C_parent66,lattice_target, & cOverA_trans,a_bcc,a_fcc) integer, dimension(:), intent(in) :: Ntrans !< number of active twin systems per family character(len=2), intent(in) :: lattice_target !< Bravais lattice (Pearson symbol) real(pReal), dimension(6,6), intent(in) :: C_parent66 real(pReal), dimension(6,6,sum(Ntrans)) :: lattice_C66_trans real(pReal), dimension(6,6) :: C_bar66, C_target_unrotated66 real(pReal), dimension(3,3,sum(Ntrans)) :: Q,S type(rotation) :: R real(pReal) :: a_bcc, a_fcc, cOverA_trans integer :: i !-------------------------------------------------------------------------------------------------- ! elasticity matrix of the target phase in cube orientation if (lattice_target == 'hP') then ! https://doi.org/10.1063/1.1663858 eq. (16), eq. (18), eq. (19) ! https://doi.org/10.1016/j.actamat.2016.07.032 eq. (47), eq. (48) if (cOverA_trans < 1.0_pReal .or. cOverA_trans > 2.0_pReal) & call IO_error(131,ext_msg='lattice_C66_trans: '//trim(lattice_target)) C_bar66(1,1) = (C_parent66(1,1) + C_parent66(1,2) + 2.0_pReal*C_parent66(4,4))/2.0_pReal C_bar66(1,2) = (C_parent66(1,1) + 5.0_pReal*C_parent66(1,2) - 2.0_pReal*C_parent66(4,4))/6.0_pReal C_bar66(3,3) = (C_parent66(1,1) + 2.0_pReal*C_parent66(1,2) + 4.0_pReal*C_parent66(4,4))/3.0_pReal C_bar66(1,3) = (C_parent66(1,1) + 2.0_pReal*C_parent66(1,2) - 2.0_pReal*C_parent66(4,4))/3.0_pReal C_bar66(4,4) = (C_parent66(1,1) - C_parent66(1,2) + C_parent66(4,4))/3.0_pReal C_bar66(1,4) = (C_parent66(1,1) - C_parent66(1,2) - 2.0_pReal*C_parent66(4,4)) /(3.0_pReal*sqrt(2.0_pReal)) C_target_unrotated66 = 0.0_pReal C_target_unrotated66(1,1) = C_bar66(1,1) - C_bar66(1,4)**2/C_bar66(4,4) C_target_unrotated66(1,2) = C_bar66(1,2) + C_bar66(1,4)**2/C_bar66(4,4) C_target_unrotated66(1,3) = C_bar66(1,3) C_target_unrotated66(3,3) = C_bar66(3,3) C_target_unrotated66(4,4) = C_bar66(4,4) - C_bar66(1,4)**2/(0.5_pReal*(C_bar66(1,1) - C_bar66(1,2))) C_target_unrotated66 = lattice_symmetrize_C66(C_target_unrotated66,'hP') elseif (lattice_target == 'cI') then if (a_bcc <= 0.0_pReal .or. a_fcc <= 0.0_pReal) & call IO_error(134,ext_msg='lattice_C66_trans: '//trim(lattice_target)) C_target_unrotated66 = C_parent66 else call IO_error(137,ext_msg='lattice_C66_trans : '//trim(lattice_target)) endif do i = 1,6 if (abs(C_target_unrotated66(i,i)) @brief Non-schmid projections for bcc with up to 6 coefficients ! https://doi.org/10.1016/j.actamat.2012.03.053, eq. (17) ! https://doi.org/10.1016/j.actamat.2008.07.037, table 1 !-------------------------------------------------------------------------------------------------- function lattice_nonSchmidMatrix(Nslip,nonSchmidCoefficients,sense) result(nonSchmidMatrix) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family real(pReal), dimension(:), intent(in) :: nonSchmidCoefficients !< non-Schmid coefficients for projections integer, intent(in) :: sense !< sense (-1,+1) real(pReal), dimension(1:3,1:3,sum(Nslip)) :: nonSchmidMatrix real(pReal), dimension(1:3,1:3,sum(Nslip)) :: coordinateSystem !< coordinate system of slip system real(pReal), dimension(3) :: direction, normal, np type(rotation) :: R integer :: i if (abs(sense) /= 1) error stop 'Sense in lattice_nonSchmidMatrix' coordinateSystem = buildCoordinateSystem(Nslip,BCC_NSLIPSYSTEM,BCC_SYSTEMSLIP,'cI',0.0_pReal) coordinateSystem(1:3,1,1:sum(Nslip)) = coordinateSystem(1:3,1,1:sum(Nslip))*real(sense,pReal) ! convert unidirectional coordinate system nonSchmidMatrix = lattice_SchmidMatrix_slip(Nslip,'cI',0.0_pReal) ! Schmid contribution do i = 1,sum(Nslip) direction = coordinateSystem(1:3,1,i) normal = coordinateSystem(1:3,2,i) call R%fromAxisAngle([direction,60.0_pReal],degrees=.true.,P=1) np = R%rotate(normal) if (size(nonSchmidCoefficients)>0) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) & + nonSchmidCoefficients(1) * math_outer(direction, np) if (size(nonSchmidCoefficients)>1) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) & + nonSchmidCoefficients(2) * math_outer(math_cross(normal, direction), normal) if (size(nonSchmidCoefficients)>2) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) & + nonSchmidCoefficients(3) * math_outer(math_cross(np, direction), np) if (size(nonSchmidCoefficients)>3) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) & + nonSchmidCoefficients(4) * math_outer(normal, normal) if (size(nonSchmidCoefficients)>4) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) & + nonSchmidCoefficients(5) * math_outer(math_cross(normal, direction), & math_cross(normal, direction)) if (size(nonSchmidCoefficients)>5) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) & + nonSchmidCoefficients(6) * math_outer(direction, direction) enddo end function lattice_nonSchmidMatrix !-------------------------------------------------------------------------------------------------- !> @brief Slip-slip interaction matrix !> @details only active slip systems are considered !> @details https://doi.org/10.1016/j.actamat.2016.12.040 (fcc: Tab S4-1, bcc: Tab S5-1) !> @details https://doi.org/10.1016/j.ijplas.2014.06.010 (hex: Tab 3b) !-------------------------------------------------------------------------------------------------- function lattice_interaction_SlipBySlip(Nslip,interactionValues,lattice) result(interactionMatrix) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family real(pReal), dimension(:), intent(in) :: interactionValues !< values for slip-slip interaction character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), dimension(sum(Nslip),sum(Nslip)) :: interactionMatrix integer, dimension(:), allocatable :: NslipMax integer, dimension(:,:), allocatable :: interactionTypes integer, dimension(FCC_NSLIP,FCC_NSLIP), parameter :: & FCC_INTERACTIONSLIPSLIP = reshape( [& 1, 2, 2, 4, 7, 5, 3, 5, 5, 4, 6, 7, 10,11,10,11,12,13, & ! -----> acting (forest) 2, 1, 2, 7, 4, 5, 6, 4, 7, 5, 3, 5, 10,11,12,13,10,11, & ! | 2, 2, 1, 5, 5, 3, 6, 7, 4, 7, 6, 4, 12,13,10,11,10,11, & ! | 4, 7, 6, 1, 2, 2, 4, 6, 7, 3, 5, 5, 10,11,11,10,13,12, & ! v 7, 4, 6, 2, 1, 2, 5, 3, 5, 6, 4, 7, 10,11,13,12,11,10, & ! reacting (primary) 5, 5, 3, 2, 2, 1, 7, 6, 4, 6, 7, 4, 12,13,11,10,11,10, & 3, 5, 5, 4, 6, 7, 1, 2, 2, 4, 7, 6, 11,10,11,10,12,13, & 6, 4, 7, 5, 3, 5, 2, 1, 2, 7, 4, 6, 11,10,13,12,10,11, & 6, 7, 4, 7, 6, 4, 2, 2, 1, 5, 5, 3, 13,12,11,10,10,11, & 4, 6, 7, 3, 5, 5, 4, 7, 6, 1, 2, 2, 11,10,10,11,13,12, & 5, 3, 5, 6, 4, 7, 7, 4, 6, 2, 1, 2, 11,10,12,13,11,10, & 7, 6, 4, 6, 7, 4, 5, 5, 3, 2, 2, 1, 13,12,10,11,11,10, & 10,10,12,10,10,12,11,11,13,11,11,13, 1, 8, 9, 9, 9, 9, & 11,11,13,11,11,13,10,10,12,10,10,12, 8, 1, 9, 9, 9, 9, & 10,12,10,11,13,11,11,13,11,10,12,10, 9, 9, 1, 8, 9, 9, & 11,13,11,10,12,10,10,12,10,11,13,11, 9, 9, 8, 1, 9, 9, & 12,10,10,13,11,11,12,10,10,13,11,11, 9, 9, 9, 9, 1, 8, & 13,11,11,12,10,10,13,11,11,12,10,10, 9, 9, 9, 9, 8, 1 & ],shape(FCC_INTERACTIONSLIPSLIP)) !< Slip-slip interaction types for fcc / Madec 2017 (https://doi.org/10.1016/j.actamat.2016.12.040) !< 1: self interaction --> alpha 0 !< 2: coplanar interaction --> alpha copla !< 3: collinear interaction --> alpha coli !< 4: Hirth locks --> alpha 1 !< 5: glissile junctions I --> alpha 2 !< 6: glissile junctions II --> alpha 2* !< 7: Lomer locks --> alpha 3 !< 8: crossing (similar to Hirth locks in <110>{111} for two {110} planes) !< 9: similar to Lomer locks in <110>{111} for two {110} planes !<10: similar to Lomer locks in <110>{111} btw one {110} and one {111} plane !<11: similar to glissile junctions in <110>{111} btw one {110} and one {111} plane !<12: crossing btw one {110} and one {111} plane !<13: collinear btw one {110} and one {111} plane integer, dimension(BCC_NSLIP,BCC_NSLIP), parameter :: & BCC_INTERACTIONSLIPSLIP = reshape( [& 1, 3, 6, 6, 7, 5, 4, 2, 4, 2, 7, 5, 18,18,11, 8, 9,13,17,14,13, 9,17,14, 28,25,28,28,25,28,28,28,28,25,28,28,25,28,28,28,28,28,28,25,28,28,28,25, &! -----> acting (forest) 3, 1, 6, 6, 4, 2, 7, 5, 7, 5, 4, 2, 18,18, 8,11,13, 9,14,17, 9,13,14,17, 25,28,28,28,28,25,28,28,25,28,28,28,28,25,28,28,28,28,25,28,28,28,25,28, &! | 6, 6, 1, 3, 5, 7, 2, 4, 5, 7, 2, 4, 11, 8,18,18,17,14, 9,13,17,14,13, 9, 28,28,28,25,28,28,25,28,28,28,28,25,28,28,25,28,28,25,28,28,28,25,28,28, &! | 6, 6, 3, 1, 2, 4, 5, 7, 2, 4, 5, 7, 8,11,18,18,14,17,13, 9,14,17, 9,13, 28,28,25,28,28,28,28,25,28,28,25,28,28,28,28,25,25,28,28,28,25,28,28,28, &! v 7, 5, 4, 2, 1, 3, 6, 6, 2, 4, 7, 5, 9,17,13,14,18,11,18, 8,13,17, 9,14, 28,28,25,28,28,28,28,25,28,28,25,28,28,28,28,25,25,28,28,28,25,28,28,28, &! reacting (primary) 4, 2, 7, 5, 3, 1, 6, 6, 5, 7, 4, 2, 13,14, 9,17,18, 8,18,11, 9,14,13,17, 25,28,28,28,28,25,28,28,25,28,28,28,28,25,28,28,28,28,25,28,28,28,25,28, & 5, 7, 2, 4, 6, 6, 1, 3, 7, 5, 2, 4, 17, 9,14,13,11,18, 8,18,17,13,14, 9, 28,28,28,25,28,28,25,28,28,28,28,25,28,28,25,28,28,25,28,28,28,25,28,28, & 2, 4, 5, 7, 6, 6, 3, 1, 4, 2, 5, 7, 14,13,17, 9, 8,18,11,18,14, 9,17,13, 28,25,28,28,25,28,28,28,28,25,28,28,25,28,28,28,28,28,28,25,28,28,28,25, & 5, 7, 4, 2, 2, 4, 7, 5, 1, 3, 6, 6, 9,17,14,13,13,17,14, 9,18,11, 8,18, 28,28,25,28,28,28,28,25,28,28,25,28,28,28,28,25,25,28,28,28,25,28,28,28, & 2, 4, 7, 5, 5, 7, 4, 2, 3, 1, 6, 6, 13,14,17, 9, 9,14,17,13,18, 8,11,18, 28,25,28,28,25,28,28,28,28,25,28,28,25,28,28,28,28,28,28,25,28,28,28,25, & 7, 5, 2, 4, 7, 5, 2, 4, 6, 6, 1, 3, 17, 9,13,14,17,13, 9,14,11,18,18, 8, 28,28,28,25,28,28,25,28,28,28,28,25,28,28,25,28,28,25,28,28,28,25,28,28, & 4, 2, 5, 7, 4, 2, 5, 7, 6, 6, 3, 1, 14,13, 9,17,14, 9,13,17, 8,18,18,11, 25,28,28,28,28,25,28,28,25,28,28,28,28,25,28,28,28,28,25,28,28,28,25,28, & 19,19,10, 8, 9,12,16,15, 9,12,16,15, 1,20,24,24,23,22,21, 2,23,22, 2,21, 28,28,26,28,28,28,28,26,28,28,26,28,28,28,28,26,26,28,28,28,26,28,28,28, & 19,19, 8,10,16,15, 9,12,16,15, 9,12, 20, 1,24,24,22,23, 2,21,22,23,21, 2, 28,28,28,26,28,28,26,28,28,28,28,26,28,28,26,28,28,26,28,28,28,26,28,28, & 10, 8,19,19,12, 9,15,16,15,16,12, 9, 24,24, 1,20,21, 2,23,22, 2,21,23,22, 26,28,28,28,28,26,28,28,26,28,28,28,28,26,28,28,28,28,26,28,28,28,26,28, & 8,10,19,19,15,16,12, 9,12, 9,15,16, 24,24,20, 1, 2,21,22,23,21, 2,22,23, 28,26,28,28,26,28,28,28,28,26,28,28,26,28,28,28,28,28,28,26,28,28,28,26, & 9,12,16,15,19,19,10, 8,12, 9,16,15, 23,21,22, 2, 1,24,20,24,23, 2,22,21, 28,26,28,28,26,28,28,28,28,26,28,28,26,28,28,28,28,28,28,26,28,28,28,26, & 12, 9,15,16,10, 8,19,19,16,15,12, 9, 21,23, 2,21,24, 1,24,20, 2,23,21,22, 26,28,28,28,28,26,28,28,26,28,28,28,28,26,28,28,28,28,26,28,28,28,26,28, & 16,15, 9,12,19,19, 8,10,15,16, 9,12, 22, 2,23,22,20,24, 1,24,22,21,23, 2, 28,28,28,26,28,28,26,28,28,28,28,26,28,28,26,28,28,26,28,28,28,26,28,28, & 15,16,12, 9, 8,10,19,19, 9,12,15,16, 2,22,21,23,24,20,24, 1,21,22, 2,23, 28,28,26,28,28,28,28,26,28,28,26,28,28,28,28,26,26,28,28,28,26,28,28,28, & 12, 9,16,15,12, 9,16,15,19,19,10, 8, 23,21, 2,22,23, 2,21,22, 1,24,24,20, 26,28,28,28,28,26,28,28,26,28,28,28,28,26,28,28,28,28,26,28,28,28,26,28, & 9,12,15,16,16,15,12, 9,10, 8,19,19, 21,23,22, 2, 2,23,22,21,24, 1,20,24, 28,26,28,28,26,28,28,28,28,26,28,28,26,28,28,28,28,28,28,26,28,28,28,26, & 16,15,12, 9, 9,12,15,16, 8,10,19,19, 2,22,23,21,21,22,23, 2,24,20, 1,24, 28,28,26,28,28,28,28,26,28,28,26,28,28,28,28,26,26,28,28,28,26,28,28,28, & 15,16, 9,12,15,16, 9,12,19,19, 8,10, 22, 2,21,23,22,21, 2,23,20,24,24, 1, 28,28,28,26,28,28,26,28,28,28,28,26,28,28,26,28,28,26,28,28,28,26,28,28, & 28,25,28,28,28,25,28,28,28,28,28,25, 28,28,26,28,28,26,28,28,26,28,28,28, 1,28,28,28,28,27,28,28,27,28,28,28,28,27,28,28,28,28,27,28,28,28,27,28, & 25,28,28,28,28,28,28,25,28,25,28,28, 28,28,28,26,26,28,28,28,28,26,28,28, 28, 1,28,28,27,28,28,28,28,27,28,28,27,28,28,28,28,28,28,27,28,28,28,27, & 28,28,28,25,25,28,28,28,25,28,28,28, 26,28,28,28,28,28,28,26,28,28,26,28, 28,28, 1,28,28,28,28,27,28,28,27,28,28,28,28,27,27,28,28,28,27,28,28,28, & 28,28,25,28,28,28,25,28,28,28,25,28, 28,26,28,28,28,28,26,28,28,28,28,26, 28,28,28, 1,28,28,27,28,28,28,28,27,28,28,27,28,28,27,28,28,28,27,28,28, & 25,28,28,28,28,28,28,25,28,25,28,28, 28,28,28,26,26,28,28,28,28,26,28,28, 28,27,28,28, 1,28,28,28,28,27,28,28,27,28,28,28,28,28,28,27,28,28,28,27, & 28,25,28,28,28,25,28,28,28,28,28,25, 28,28,26,28,28,26,28,28,26,28,28,28, 27,28,28,28,28, 1,28,28,27,28,28,28,28,27,28,28,28,28,27,28,28,28,27,28, & 28,28,25,28,28,28,25,28,28,28,25,28, 28,26,28,28,28,28,26,28,28,28,28,26, 28,28,28,27,28,28, 1,28,28,28,28,27,28,28,27,28,28,27,28,28,28,27,28,28, & 28,28,28,25,25,28,28,28,25,28,28,28, 26,28,28,28,28,28,28,26,28,28,26,28, 28,28,27,28,28,28,28, 1,28,28,27,28,28,28,28,27,27,28,28,28,27,28,28,28, & 28,25,28,28,28,25,28,28,28,28,28,25, 28,28,26,28,28,26,28,28,26,28,28,28, 27,28,28,28,28,27,28,28, 1,28,28,28,28,27,28,28,28,28,27,28,28,28,27,28, & 25,28,28,28,28,28,28,25,28,25,28,28, 28,28,28,26,26,28,28,28,28,26,28,28, 28,27,28,28,27,28,28,28,28, 1,28,28,27,28,28,28,28,28,28,27,28,28,28,27, & 28,28,28,25,25,28,28,28,25,28,28,28, 26,28,28,28,28,28,28,26,28,28,26,28, 28,28,27,28,28,28,28,27,28,28, 1,28,28,28,28,27,27,28,28,28,27,28,28,28, & 28,28,25,28,28,28,25,28,28,28,25,28, 28,26,28,28,28,28,26,28,28,28,28,26, 28,28,28,27,28,28,27,28,28,28,28, 1,28,28,27,28,28,27,28,28,28,27,28,28, & 25,28,28,28,28,28,28,25,28,25,28,28, 28,28,28,26,26,28,28,28,28,26,28,28, 28,27,28,28,27,28,28,28,28,27,28,28, 1,28,28,28,28,28,28,27,28,28,28,27, & 28,25,28,28,28,25,28,28,28,28,28,25, 28,28,26,28,28,26,28,28,26,28,28,28, 27,28,28,28,28,27,28,28,27,28,28,28,28, 1,28,28,28,28,27,28,28,28,27,28, & 28,28,25,28,28,28,25,28,28,28,25,28, 28,26,28,28,28,28,26,28,28,28,28,26, 28,28,28,27,28,28,27,28,28,28,28,27,28,28, 1,28,28,27,28,28,28,27,28,28, & 28,28,28,25,25,28,28,28,25,28,28,28, 26,28,28,28,28,28,28,26,28,28,26,28, 28,28,27,28,28,28,28,27,28,28,27,28,28,28,28, 1,27,28,28,28,27,28,28,28, & 28,28,28,25,25,28,28,28,25,28,28,28, 26,28,28,28,28,28,28,26,28,28,26,28, 28,28,27,28,28,28,28,27,28,28,27,28,28,28,28,27, 1,28,28,28,27,28,28,28, & 28,28,25,28,28,28,25,28,28,28,25,28, 28,26,28,28,28,28,26,28,28,28,28,26, 28,28,28,27,28,28,27,28,28,28,28,27,28,28,27,28,28, 1,28,28,28,27,28,28, & 28,25,28,28,28,25,28,28,28,28,28,25, 28,28,26,28,28,26,28,28,26,28,28,28, 27,28,28,28,28,27,28,28,27,28,28,28,28,27,28,28,28,28, 1,28,28,28,27,28, & 25,28,28,28,28,28,28,25,28,25,28,28, 28,28,28,26,26,28,28,28,28,26,28,28, 28,27,28,28,27,28,28,28,28,27,28,28,27,28,28,28,28,28,28, 1,28,28,28,27, & 28,28,28,25,25,28,28,28,25,28,28,28, 26,28,28,28,28,28,28,26,28,28,26,28, 28,28,27,28,28,28,28,27,28,28,27,28,28,28,28,27,27,28,28,28, 1,28,28,28, & 28,28,25,28,28,28,25,28,28,28,25,28, 28,26,28,28,28,28,26,28,28,28,28,26, 28,28,28,27,28,28,27,28,28,28,28,27,28,28,27,28,28,27,28,28,28, 1,28,28, & 28,25,28,28,28,25,28,28,28,28,28,25, 28,28,26,28,28,26,28,28,26,28,28,28, 27,28,28,28,28,27,28,28,27,28,28,28,28,27,28,28,28,28,27,28,28,28, 1,28, & 25,28,28,28,28,28,28,25,28,25,28,28, 28,28,28,26,26,28,28,28,28,26,28,28, 28,27,28,28,27,28,28,28,28,27,28,28,27,28,28,28,28,28,28,27,28,28,28, 1 & ],shape(BCC_INTERACTIONSLIPSLIP)) !< Slip-slip interaction types for bcc / Madec 2017 (https://doi.org/10.1016/j.actamat.2016.12.040) !< 1: self interaction --> alpha 0 !< 2: collinear interaction --> alpha 1 !< 3: coplanar interaction --> alpha 2 !< 4-7: other coefficients !< 8: {110}-{112} collinear and perpendicular planes --> alpha 6 !< 9: {110}-{112} collinear --> alpha 7 !< 10-24: other coefficients !< 25: {110}-{123} collinear !< 26: {112}-{123} collinear !< 27: {123}-{123} collinear !< 28: other interaction integer, dimension(HEX_NSLIP,HEX_NSLIP), parameter :: & HEX_INTERACTIONSLIPSLIP = reshape( [& ! basal prism 1. pyr 1. pyr 2. pyr 1, 2, 2, 3, 4, 4, 9,10, 9, 9,10, 9, 20,21,22,22,21,20,20,21,22,22,21,20, 47,47,48,47,47,48, & ! -----> acting (forest) 2, 1, 2, 4, 3, 4, 10, 9, 9,10, 9, 9, 22,22,21,20,20,21,22,22,21,20,20,21, 47,48,47,47,48,47, & ! | basal 2, 2, 1, 4, 4, 3, 9, 9,10, 9, 9,10, 21,20,20,21,22,22,21,20,20,21,22,22, 48,47,47,48,47,47, & ! | ! v 7, 8, 8, 5, 6, 6, 11,12,11,11,12,11, 23,24,25,25,24,23,23,24,25,25,24,23, 49,49,50,49,49,50, & ! reacting (primary) 8, 7, 8, 6, 5, 6, 12,11,11,12,11,11, 25,25,24,23,23,24,25,25,24,23,23,24, 49,50,49,49,50,49, & ! prism 8, 8, 7, 6, 6, 5, 11,11,12,11,11,12, 24,23,23,24,25,25,24,23,23,24,25,25, 50,49,49,50,49,49, & 18,19,18, 16,17,16, 13,14,14,15,14,14, 26,26,27,28,28,27,29,29,27,28,28,27, 51,52,51,51,52,51, & 19,18,18, 17,16,16, 14,13,14,14,15,14, 28,27,26,26,27,28,28,27,29,29,27,28, 51,51,52,51,51,52, & 18,18,19, 16,16,17, 14,14,13,14,14,15, 27,28,28,27,26,26,27,28,28,27,29,29, 52,51,51,52,51,51, & 18,19,18, 16,17,16, 15,14,14,13,14,14, 29,29,27,28,28,27,26,26,27,28,28,27, 51,52,51,51,52,51, & ! 1. pyr 19,18,18, 17,16,16, 14,15,14,14,13,14, 28,27,29,29,27,28,28,27,26,26,27,28, 51,51,52,51,51,52, & 18,18,19, 16,16,17, 14,14,15,14,14,13, 27,28,28,27,29,29,27,28,28,27,26,26, 52,51,51,52,51,51, & 44,45,46, 41,42,43, 37,38,39,40,38,39, 30,31,32,32,32,33,34,35,32,32,32,36, 53,54,55,53,54,56, & 46,45,44, 43,42,41, 37,39,38,40,39,38, 31,30,36,32,32,32,35,34,33,32,32,32, 56,54,53,55,54,53, & 45,46,44, 42,43,41, 39,37,38,39,40,38, 32,36,30,31,32,32,32,33,34,35,32,32, 56,53,54,55,53,54, & 45,44,46, 42,41,43, 38,37,39,38,40,39, 32,32,31,30,36,32,32,32,35,34,33,32, 53,56,54,53,55,54, & 46,44,45, 43,41,42, 38,39,37,38,39,40, 32,32,32,36,30,31,32,32,32,33,34,35, 54,56,53,54,55,53, & 44,46,45, 41,43,42, 39,38,37,39,38,40, 33,32,32,32,31,30,36,32,32,32,35,34, 54,53,56,54,53,55, & 44,45,46, 41,42,43, 40,38,39,37,38,39, 34,35,32,32,32,36,30,31,32,32,32,33, 53,54,56,53,54,55, & ! 1. pyr 46,45,44, 43,42,41, 40,39,38,37,39,38, 35,34,33,32,32,32,31,30,36,32,32,32, 55,54,53,56,54,53, & 45,46,44, 42,43,41, 39,40,38,39,37,38, 32,33,34,35,32,32,32,36,30,31,32,32, 55,53,54,56,53,54, & 45,44,46, 42,41,43, 38,40,39,38,37,39, 32,32,35,34,33,32,32,32,31,30,36,32, 53,55,54,53,56,54, & 46,44,45, 43,41,42, 38,39,40,38,39,37, 32,32,32,33,34,35,32,32,32,36,30,31, 54,55,53,54,56,53, & 44,46,45, 41,43,42, 39,38,40,39,38,37, 36,32,32,32,35,34,33,32,32,32,31,30, 54,53,55,54,53,56, & 68,68,69, 66,66,67, 64,64,65,64,65,65, 60,61,61,60,62,62,60,63,63,60,62,62, 57,58,58,59,58,58, & 68,69,68, 66,67,66, 65,64,64,65,64,64, 62,62,60,61,61,60,62,62,60,63,63,60, 58,57,58,58,59,58, & 69,68,68, 67,66,66, 64,65,64,64,65,64, 63,60,62,62,60,61,61,60,62,62,60,63, 58,58,57,58,58,59, & 68,68,69, 66,66,67, 64,64,65,64,64,65, 60,63,63,60,62,62,60,61,61,60,62,62, 59,58,58,57,58,58, & ! 2. pyr 68,69,68, 66,67,66, 65,64,64,65,64,64, 62,62,60,63,63,60,62,62,60,61,61,60, 58,59,58,58,57,58, & 69,68,68, 67,66,66, 64,65,64,64,65,64, 61,60,62,62,60,63,63,60,62,62,60,61, 58,58,59,58,58,57 & ],shape(HEX_INTERACTIONSLIPSLIP)) !< Slip-slip interaction types for hex (onion peel naming scheme) !< 10.1016/j.ijplas.2014.06.010 table 3 !< 10.1080/14786435.2012.699689 table 2 and 3 !< 1: S1 basal self-interaction !< 2: 1 coplanar basal/basal !< 3: 3 collinear basal/prismatic !< 4: 4 non-collinear basal/prismatic !< 5: S2 prismatic self-interaction !< 6: 2 prismatic/prismatic !< 7: 5 collinear prismatic/basal !< 8: 6 non-collinear prismatic/basal !< 9: non-collinear basal/pyr a !< 10: collinear basal/pyr a !< 11: non-collinear prismatic/pyr a !< 12: collinear prismatic/pyr a !< 16: non-collinear pyr a/prismatic !< 17: collinear pyr a/prismatic !< 18: non-collinear pyr a/basal !< 19: collinear pyr a/basal !< 47: 8 non-collinear basal/pyramidal !< 48: 7 collinear basal/pyramidal !< 49: 10 non-collinear prismatic/pyramidal !< 50: 9 collinear prismatic/pyramidal !< 57: S3 pyramidal c+a self-interaction !< 58: 16 non-collinear pyramidal/pyramidal !< 59: 15 collinear pyramidal/pyramidal !< 66: 14 non-collienar pyramidal/prismatic !< 67: 13 semi-collienar pyramidal/prismatic !< 68: 12 non-collinear pyramidal/basal !< 69: 11 semi-collinear pyramidal/basal integer, dimension(BCT_NSLIP,BCT_NSLIP), parameter :: & BCT_INTERACTIONSLIPSLIP = reshape( [& 1, 2, 3, 3, 7, 7, 13, 13, 13, 13, 21, 21, 31, 31, 31, 31, 43, 43, 57, 57, 73, 73, 73, 73, 91, 91, 91, 91, 91, 91, 91, 91, 111, 111, 111, 111, 133,133,133,133,133,133,133,133, 157,157,157,157,157,157,157,157, & ! -----> acting 2, 1, 3, 3, 7, 7, 13, 13, 13, 13, 21, 21, 31, 31, 31, 31, 43, 43, 57, 57, 73, 73, 73, 73, 91, 91, 91, 91, 91, 91, 91, 91, 111, 111, 111, 111, 133,133,133,133,133,133,133,133, 157,157,157,157,157,157,157,157, & ! | ! | 6, 6, 4, 5, 8, 8, 14, 14, 14, 14, 22, 22, 32, 32, 32, 32, 44, 44, 58, 58, 74, 74, 74, 74, 92, 92, 92, 92, 92, 92, 92, 92, 112, 112, 112, 112, 134,134,134,134,134,134,134,134, 158,158,158,158,158,158,158,158, & ! v 6, 6, 5, 4, 8, 8, 14, 14, 14, 14, 22, 22, 32, 32, 32, 32, 44, 44, 58, 58, 74, 74, 74, 74, 92, 92, 92, 92, 92, 92, 92, 92, 112, 112, 112, 112, 134,134,134,134,134,134,134,134, 158,158,158,158,158,158,158,158, & ! reacting 12, 12, 11, 11, 9, 10, 15, 15, 15, 15, 23, 23, 33, 33, 33, 33, 45, 45, 59, 59, 75, 75, 75, 75, 93, 93, 93, 93, 93, 93, 93, 93, 113, 113, 113, 113, 135,135,135,135,135,135,135,135, 159,159,159,159,159,159,159,159, & 12, 12, 11, 11, 10, 9, 15, 15, 15, 15, 23, 23, 33, 33, 33, 33, 45, 45, 59, 59, 75, 75, 75, 75, 93, 93, 93, 93, 93, 93, 93, 93, 113, 113, 113, 113, 135,135,135,135,135,135,135,135, 159,159,159,159,159,159,159,159, & 20, 20, 19, 19, 18, 18, 16, 17, 17, 17, 24, 24, 34, 34, 34, 34, 46, 46, 60, 60, 76, 76, 76, 76, 94, 94, 94, 94, 94, 94, 94, 94, 114, 114, 114, 114, 136,136,136,136,136,136,136,136, 160,160,160,160,160,160,160,160, & 20, 20, 19, 19, 18, 18, 17, 16, 17, 17, 24, 24, 34, 34, 34, 34, 46, 46, 60, 60, 76, 76, 76, 76, 94, 94, 94, 94, 94, 94, 94, 94, 114, 114, 114, 114, 136,136,136,136,136,136,136,136, 160,160,160,160,160,160,160,160, & 20, 20, 19, 19, 18, 18, 17, 17, 16, 17, 24, 24, 34, 34, 34, 34, 46, 46, 60, 60, 76, 76, 76, 76, 94, 94, 94, 94, 94, 94, 94, 94, 114, 114, 114, 114, 136,136,136,136,136,136,136,136, 160,160,160,160,160,160,160,160, & 20, 20, 19, 19, 18, 18, 17, 17, 17, 16, 24, 24, 34, 34, 34, 34, 46, 46, 60, 60, 76, 76, 76, 76, 94, 94, 94, 94, 94, 94, 94, 94, 114, 114, 114, 114, 136,136,136,136,136,136,136,136, 160,160,160,160,160,160,160,160, & 30, 30, 29, 29, 28, 28, 27, 27, 27, 27, 25, 26, 35, 35, 35, 35, 47, 47, 61, 61, 77, 77, 77, 77, 95, 95, 95, 95, 95, 95, 95, 95, 115, 115, 115, 115, 137,137,137,137,137,137,137,137, 161,161,161,161,161,161,161,161, & 30, 30, 29, 29, 28, 28, 27, 27, 27, 27, 26, 25, 35, 35, 35, 35, 47, 47, 61, 61, 77, 77, 77, 77, 95, 95, 95, 95, 95, 95, 95, 95, 115, 115, 115, 115, 137,137,137,137,137,137,137,137, 161,161,161,161,161,161,161,161, & 42, 42, 41, 41, 40, 40, 39, 39, 39, 39, 38, 38, 36, 37, 37, 37, 48, 48, 62, 62, 78, 78, 78, 78, 96, 96, 96, 96, 96, 96, 96, 96, 116, 116, 116, 116, 138,138,138,138,138,138,138,138, 162,162,162,162,162,162,162,162, & 42, 42, 41, 41, 40, 40, 39, 39, 39, 39, 38, 38, 37, 36, 37, 37, 48, 48, 62, 62, 78, 78, 78, 78, 96, 96, 96, 96, 96, 96, 96, 96, 116, 116, 116, 116, 138,138,138,138,138,138,138,138, 162,162,162,162,162,162,162,162, & 42, 42, 41, 41, 40, 40, 39, 39, 39, 39, 38, 38, 37, 37, 36, 37, 48, 48, 62, 62, 78, 78, 78, 78, 96, 96, 96, 96, 96, 96, 96, 96, 116, 116, 116, 116, 138,138,138,138,138,138,138,138, 162,162,162,162,162,162,162,162, & 42, 42, 41, 41, 40, 40, 39, 39, 39, 39, 38, 38, 37, 37, 37, 36, 48, 48, 62, 62, 78, 78, 78, 78, 96, 96, 96, 96, 96, 96, 96, 96, 116, 116, 116, 116, 138,138,138,138,138,138,138,138, 162,162,162,162,162,162,162,162, & 56, 56, 55, 55, 54, 54, 53, 53, 53, 53, 52, 52, 51, 51, 51, 51, 49, 50, 63, 63, 79, 79, 79, 79, 97, 97, 97, 97, 97, 97, 97, 97, 117, 117, 117, 117, 139,139,139,139,139,139,139,139, 163,163,163,163,163,163,163,163, & 56, 56, 55, 55, 54, 54, 53, 53, 53, 53, 52, 52, 51, 51, 51, 51, 50, 49, 63, 63, 79, 79, 79, 79, 97, 97, 97, 97, 97, 97, 97, 97, 117, 117, 117, 117, 139,139,139,139,139,139,139,139, 163,163,163,163,163,163,163,163, & 72, 72, 71, 71, 70, 70, 69, 69, 69, 69, 68, 68, 67, 67, 67, 67, 66, 66, 64, 65, 80, 80, 80, 80, 98, 98, 98, 98, 98, 98, 98, 98, 118, 118, 118, 118, 140,140,140,140,140,140,140,140, 164,164,164,164,164,164,164,164, & 72, 72, 71, 71, 70, 70, 69, 69, 69, 69, 68, 68, 67, 67, 67, 67, 66, 66, 65, 64, 80, 80, 80, 80, 98, 98, 98, 98, 98, 98, 98, 98, 118, 118, 118, 118, 140,140,140,140,140,140,140,140, 164,164,164,164,164,164,164,164, & 90, 90, 89, 89, 88, 88, 87, 87, 87, 87, 86, 86, 85, 85, 85, 85, 84, 84, 83, 83, 81, 82, 82, 82, 99, 99, 99, 99, 99, 99, 99, 99, 119, 119, 119, 119, 141,141,141,141,141,141,141,141, 165,165,165,165,165,165,165,165, & 90, 90, 89, 89, 88, 88, 87, 87, 87, 87, 86, 86, 85, 85, 85, 85, 84, 84, 83, 83, 82, 81, 82, 82, 99, 99, 99, 99, 99, 99, 99, 99, 119, 119, 119, 119, 141,141,141,141,141,141,141,141, 165,165,165,165,165,165,165,165, & 90, 90, 89, 89, 88, 88, 87, 87, 87, 87, 86, 86, 85, 85, 85, 85, 84, 84, 83, 83, 82, 82, 81, 82, 99, 99, 99, 99, 99, 99, 99, 99, 119, 119, 119, 119, 141,141,141,141,141,141,141,141, 165,165,165,165,165,165,165,165, & 90, 90, 89, 89, 88, 88, 87, 87, 87, 87, 86, 86, 85, 85, 85, 85, 84, 84, 83, 83, 82, 82, 82, 81, 99, 99, 99, 99, 99, 99, 99, 99, 119, 119, 119, 119, 141,141,141,141,141,141,141,141, 165,165,165,165,165,165,165,165, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 100,101,101,101,101,101,101,101, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 101,100,101,101,101,101,101,101, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 101,101,100,101,101,101,101,101, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 101,101,101,100,101,101,101,101, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 101,101,101,101,100,101,101,101, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 101,101,101,101,101,100,101,101, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 101,101,101,101,101,101,100,101, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 110,110, 109,109, 108,108, 107,107,107,107, 106,106, 105,105,105,105, 104,104, 103,103, 102,102,102,102, 101,101,101,101,101,101,101,100, 120, 120, 120, 120, 142,142,142,142,142,142,142,142, 166,166,166,166,166,166,166,166, & 132,132, 131,131, 130,130, 129,129,129,129, 128,128, 127,127,127,127, 126,126, 125,125, 124,124,124,124, 123,123,123,123,123,123,123,123, 121, 122, 122, 122, 143,143,143,143,143,143,143,143, 167,167,167,167,167,167,167,167, & 132,132, 131,131, 130,130, 129,129,129,129, 128,128, 127,127,127,127, 126,126, 125,125, 124,124,124,124, 123,123,123,123,123,123,123,123, 121, 121, 122, 122, 143,143,143,143,143,143,143,143, 167,167,167,167,167,167,167,167, & 132,132, 131,131, 130,130, 129,129,129,129, 128,128, 127,127,127,127, 126,126, 125,125, 124,124,124,124, 123,123,123,123,123,123,123,123, 121, 122, 121, 122, 143,143,143,143,143,143,143,143, 167,167,167,167,167,167,167,167, & 132,132, 131,131, 130,130, 129,129,129,129, 128,128, 127,127,127,127, 126,126, 125,125, 124,124,124,124, 123,123,123,123,123,123,123,123, 121, 122, 122, 121, 143,143,143,143,143,143,143,143, 167,167,167,167,167,167,167,167, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 144,145,145,145,145,145,145,145, 168,168,168,168,168,168,168,168, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 145,144,145,145,145,145,145,145, 168,168,168,168,168,168,168,168, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 145,145,144,145,145,145,145,145, 168,168,168,168,168,168,168,168, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 145,145,145,144,145,145,145,145, 168,168,168,168,168,168,168,168, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 145,145,145,145,144,145,145,145, 168,168,168,168,168,168,168,168, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 145,145,145,145,145,144,145,145, 168,168,168,168,168,168,168,168, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 145,145,145,145,145,145,144,145, 168,168,168,168,168,168,168,168, & 156,156, 155,155, 154,154, 153,153,153,153, 152,152, 151,151,151,151, 150,150, 149,149, 148,148,148,148, 147,147,147,147,147,147,147,147, 146, 146, 146, 146, 145,145,145,145,145,145,145,144, 168,168,168,168,168,168,168,168, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 169,170,170,170,170,170,170,170, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 170,169,170,170,170,170,170,170, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 170,170,169,170,170,170,170,170, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 170,170,170,169,170,170,170,170, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 170,170,170,170,169,170,170,170, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 169,170,170,170,170,169,170,170, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 169,170,170,170,170,170,169,170, & 182,182, 181,181, 180,180, 179,179,179,179, 178,178, 177,177,177,177, 176,176, 175,175, 174,174,174,174, 173,173,173,173,173,173,173,173, 172, 172, 172, 172, 171,171,171,171,171,171,171,171, 169,170,170,170,170,170,170,169 & ],shape(BCT_INTERACTIONSLIPSLIP)) select case(lattice) case('cF') interactionTypes = FCC_INTERACTIONSLIPSLIP NslipMax = FCC_NSLIPSYSTEM case('cI') interactionTypes = BCC_INTERACTIONSLIPSLIP NslipMax = BCC_NSLIPSYSTEM case('hP') interactionTypes = HEX_INTERACTIONSLIPSLIP NslipMax = HEX_NSLIPSYSTEM case('tI') interactionTypes = BCT_INTERACTIONSLIPSLIP NslipMax = BCT_NSLIPSYSTEM case default call IO_error(137,ext_msg='lattice_interaction_SlipBySlip: '//trim(lattice)) end select interactionMatrix = buildInteraction(Nslip,Nslip,NslipMax,NslipMax,interactionValues,interactionTypes) end function lattice_interaction_SlipBySlip !-------------------------------------------------------------------------------------------------- !> @brief Twin-twin interaction matrix !> details only active twin systems are considered !-------------------------------------------------------------------------------------------------- function lattice_interaction_TwinByTwin(Ntwin,interactionValues,lattice) result(interactionMatrix) integer, dimension(:), intent(in) :: Ntwin !< number of active twin systems per family real(pReal), dimension(:), intent(in) :: interactionValues !< values for twin-twin interaction character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), dimension(sum(Ntwin),sum(Ntwin)) :: interactionMatrix integer, dimension(:), allocatable :: NtwinMax integer, dimension(:,:), allocatable :: interactionTypes integer, dimension(FCC_NTWIN,FCC_NTWIN), parameter :: & FCC_INTERACTIONTWINTWIN = reshape( [& 1,1,1,2,2,2,2,2,2,2,2,2, & ! -----> acting 1,1,1,2,2,2,2,2,2,2,2,2, & ! | 1,1,1,2,2,2,2,2,2,2,2,2, & ! | 2,2,2,1,1,1,2,2,2,2,2,2, & ! v 2,2,2,1,1,1,2,2,2,2,2,2, & ! reacting 2,2,2,1,1,1,2,2,2,2,2,2, & 2,2,2,2,2,2,1,1,1,2,2,2, & 2,2,2,2,2,2,1,1,1,2,2,2, & 2,2,2,2,2,2,1,1,1,2,2,2, & 2,2,2,2,2,2,2,2,2,1,1,1, & 2,2,2,2,2,2,2,2,2,1,1,1, & 2,2,2,2,2,2,2,2,2,1,1,1 & ],shape(FCC_INTERACTIONTWINTWIN)) !< Twin-twin interaction types for fcc integer, dimension(BCC_NTWIN,BCC_NTWIN), parameter :: & BCC_INTERACTIONTWINTWIN = reshape( [& 1,3,3,3,3,3,3,2,3,3,2,3, & ! -----> acting 3,1,3,3,3,3,2,3,3,3,3,2, & ! | 3,3,1,3,3,2,3,3,2,3,3,3, & ! | 3,3,3,1,2,3,3,3,3,2,3,3, & ! v 3,3,3,2,1,3,3,3,3,2,3,3, & ! reacting 3,3,2,3,3,1,3,3,2,3,3,3, & 3,2,3,3,3,3,1,3,3,3,3,2, & 2,3,3,3,3,3,3,1,3,3,2,3, & 3,3,2,3,3,2,3,3,1,3,3,3, & 3,3,3,2,2,3,3,3,3,1,3,3, & 2,3,3,3,3,3,3,2,3,3,1,3, & 3,2,3,3,3,3,2,3,3,3,3,1 & ],shape(BCC_INTERACTIONTWINTWIN)) !< Twin-twin interaction types for bcc !< 1: self interaction !< 2: collinear interaction !< 3: other interaction integer, dimension(HEX_NTWIN,HEX_NTWIN), parameter :: & HEX_INTERACTIONTWINTWIN = reshape( [& ! <-10.1>{10.2} <11.6>{-1-1.1} <10.-2>{10.1} <11.-3>{11.2} 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, & ! -----> acting 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, & ! | 2, 2, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, & ! | 2, 2, 2, 1, 2, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, & ! v <-10.1>{10.2} 2, 2, 2, 2, 1, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, & ! reacting 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, & 6, 6, 6, 6, 6, 6, 4, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 14,14,14,14,14,14, & 6, 6, 6, 6, 6, 6, 5, 4, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 14,14,14,14,14,14, & 6, 6, 6, 6, 6, 6, 5, 5, 4, 5, 5, 5, 8, 8, 8, 8, 8, 8, 14,14,14,14,14,14, & 6, 6, 6, 6, 6, 6, 5, 5, 5, 4, 5, 5, 8, 8, 8, 8, 8, 8, 14,14,14,14,14,14, & ! <11.6>{-1-1.1} 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 4, 5, 8, 8, 8, 8, 8, 8, 14,14,14,14,14,14, & 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 4, 8, 8, 8, 8, 8, 8, 14,14,14,14,14,14, & 12,12,12,12,12,12, 11,11,11,11,11,11, 9,10,10,10,10,10, 15,15,15,15,15,15, & 12,12,12,12,12,12, 11,11,11,11,11,11, 10, 9,10,10,10,10, 15,15,15,15,15,15, & 12,12,12,12,12,12, 11,11,11,11,11,11, 10,10, 9,10,10,10, 15,15,15,15,15,15, & 12,12,12,12,12,12, 11,11,11,11,11,11, 10,10,10, 9,10,10, 15,15,15,15,15,15, & ! <10.-2>{10.1} 12,12,12,12,12,12, 11,11,11,11,11,11, 10,10,10,10, 9,10, 15,15,15,15,15,15, & 12,12,12,12,12,12, 11,11,11,11,11,11, 10,10,10,10,10, 9, 15,15,15,15,15,15, & 20,20,20,20,20,20, 19,19,19,19,19,19, 18,18,18,18,18,18, 16,17,17,17,17,17, & 20,20,20,20,20,20, 19,19,19,19,19,19, 18,18,18,18,18,18, 17,16,17,17,17,17, & 20,20,20,20,20,20, 19,19,19,19,19,19, 18,18,18,18,18,18, 17,17,16,17,17,17, & 20,20,20,20,20,20, 19,19,19,19,19,19, 18,18,18,18,18,18, 17,17,17,16,17,17, & ! <11.-3>{11.2} 20,20,20,20,20,20, 19,19,19,19,19,19, 18,18,18,18,18,18, 17,17,17,17,16,17, & 20,20,20,20,20,20, 19,19,19,19,19,19, 18,18,18,18,18,18, 17,17,17,17,17,16 & ],shape(HEX_INTERACTIONTWINTWIN)) !< Twin-twin interaction types for hex select case(lattice) case('cF') interactionTypes = FCC_INTERACTIONTWINTWIN NtwinMax = FCC_NTWINSYSTEM case('cI') interactionTypes = BCC_INTERACTIONTWINTWIN NtwinMax = BCC_NTWINSYSTEM case('hP') interactionTypes = HEX_INTERACTIONTWINTWIN NtwinMax = HEX_NTWINSYSTEM case default call IO_error(137,ext_msg='lattice_interaction_TwinByTwin: '//trim(lattice)) end select interactionMatrix = buildInteraction(Ntwin,Ntwin,NtwinMax,NtwinMax,interactionValues,interactionTypes) end function lattice_interaction_TwinByTwin !-------------------------------------------------------------------------------------------------- !> @brief Trans-trans interaction matrix !> details only active trans systems are considered !-------------------------------------------------------------------------------------------------- function lattice_interaction_TransByTrans(Ntrans,interactionValues,lattice) result(interactionMatrix) integer, dimension(:), intent(in) :: Ntrans !< number of active trans systems per family real(pReal), dimension(:), intent(in) :: interactionValues !< values for trans-trans interaction character(len=2), intent(in) :: lattice ! acting 1,1,1,2,2,2,2,2,2,2,2,2, & ! | 1,1,1,2,2,2,2,2,2,2,2,2, & ! | 2,2,2,1,1,1,2,2,2,2,2,2, & ! v 2,2,2,1,1,1,2,2,2,2,2,2, & ! reacting 2,2,2,1,1,1,2,2,2,2,2,2, & 2,2,2,2,2,2,1,1,1,2,2,2, & 2,2,2,2,2,2,1,1,1,2,2,2, & 2,2,2,2,2,2,1,1,1,2,2,2, & 2,2,2,2,2,2,2,2,2,1,1,1, & 2,2,2,2,2,2,2,2,2,1,1,1, & 2,2,2,2,2,2,2,2,2,1,1,1 & ],shape(FCC_INTERACTIONTRANSTRANS)) !< Trans-trans interaction types for fcc if (lattice == 'cF') then interactionTypes = FCC_INTERACTIONTRANSTRANS NtransMax = FCC_NTRANSSYSTEM else call IO_error(137,ext_msg='lattice_interaction_TransByTrans: '//trim(lattice)) end if interactionMatrix = buildInteraction(Ntrans,Ntrans,NtransMax,NtransMax,interactionValues,interactionTypes) end function lattice_interaction_TransByTrans !-------------------------------------------------------------------------------------------------- !> @brief Slip-twin interaction matrix !> details only active slip and twin systems are considered !-------------------------------------------------------------------------------------------------- function lattice_interaction_SlipByTwin(Nslip,Ntwin,interactionValues,lattice) result(interactionMatrix) integer, dimension(:), intent(in) :: Nslip, & !< number of active slip systems per family Ntwin !< number of active twin systems per family real(pReal), dimension(:), intent(in) :: interactionValues !< values for slip-twin interaction character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), dimension(sum(Nslip),sum(Ntwin)) :: interactionMatrix integer, dimension(:), allocatable :: NslipMax, & NtwinMax integer, dimension(:,:), allocatable :: interactionTypes integer, dimension(FCC_NTWIN,FCC_NSLIP), parameter :: & FCC_INTERACTIONSLIPTWIN = reshape( [& 1,1,1,3,3,3,2,2,2,3,3,3, & ! -----> twin (acting) 1,1,1,3,3,3,3,3,3,2,2,2, & ! | 1,1,1,2,2,2,3,3,3,3,3,3, & ! | 3,3,3,1,1,1,3,3,3,2,2,2, & ! v 3,3,3,1,1,1,2,2,2,3,3,3, & ! slip (reacting) 2,2,2,1,1,1,3,3,3,3,3,3, & 2,2,2,3,3,3,1,1,1,3,3,3, & 3,3,3,2,2,2,1,1,1,3,3,3, & 3,3,3,3,3,3,1,1,1,2,2,2, & 3,3,3,2,2,2,3,3,3,1,1,1, & 2,2,2,3,3,3,3,3,3,1,1,1, & 3,3,3,3,3,3,2,2,2,1,1,1, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4 & ],shape(FCC_INTERACTIONSLIPTWIN)) !< Slip-twin interaction types for fcc !< 1: coplanar interaction !< 2: screw trace between slip system and twin habit plane (easy cross slip) !< 3: other interaction integer, dimension(BCC_NTWIN,BCC_NSLIP), parameter :: & BCC_INTERACTIONSLIPTWIN = reshape( [& 3,3,3,2,2,3,3,3,3,2,3,3, & ! -----> twin (acting) 3,3,2,3,3,2,3,3,2,3,3,3, & ! | 3,2,3,3,3,3,2,3,3,3,3,2, & ! | 2,3,3,3,3,3,3,2,3,3,2,3, & ! v 2,3,3,3,3,3,3,2,3,3,2,3, & ! slip (reacting) 3,3,2,3,3,2,3,3,2,3,3,3, & 3,2,3,3,3,3,2,3,3,3,3,2, & 3,3,3,2,2,3,3,3,3,2,3,3, & 2,3,3,3,3,3,3,2,3,3,2,3, & 3,3,3,2,2,3,3,3,3,2,3,3, & 3,2,3,3,3,3,2,3,3,3,3,2, & 3,3,2,3,3,2,3,3,2,3,3,3, & 1,3,3,3,3,3,3,2,3,3,2,3, & 3,1,3,3,3,3,2,3,3,3,3,2, & 3,3,1,3,3,2,3,3,2,3,3,3, & 3,3,3,1,2,3,3,3,3,2,3,3, & 3,3,3,2,1,3,3,3,3,2,3,3, & 3,3,2,3,3,1,3,3,2,3,3,3, & 3,2,3,3,3,3,1,3,3,3,3,2, & 2,3,3,3,3,3,3,1,3,3,2,3, & 3,3,2,3,3,2,3,3,1,3,3,3, & 3,3,3,2,2,3,3,3,3,1,3,3, & 2,3,3,3,3,3,3,2,3,3,1,3, & 3,2,3,3,3,3,2,3,3,3,3,1, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4 & ],shape(BCC_INTERACTIONSLIPTWIN)) !< Slip-twin interaction types for bcc !< 1: coplanar interaction !< 2: screw trace between slip system and twin habit plane (easy cross slip) !< 3: other interaction !< 4: other interaction with slip family {123} integer, dimension(HEX_NTWIN,HEX_NSLIP), parameter :: & HEX_INTERACTIONSLIPTWIN = reshape( [& ! <-10.1>{10.2} <11.6>{-1-1.1} <10.-2>{10.1} <11.-3>{11.2} 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, & ! ----> twin (acting) 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, & ! | basal 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, & ! | ! v 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, & ! slip (reacting) 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, & ! prism 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, & 9, 9, 9, 9, 9, 9, 10,10,10,10,10,10, 11,11,11,11,11,11, 12,12,12,12,12,12, & 9, 9, 9, 9, 9, 9, 10,10,10,10,10,10, 11,11,11,11,11,11, 12,12,12,12,12,12, & 9, 9, 9, 9, 9, 9, 10,10,10,10,10,10, 11,11,11,11,11,11, 12,12,12,12,12,12, & 9, 9, 9, 9, 9, 9, 10,10,10,10,10,10, 11,11,11,11,11,11, 12,12,12,12,12,12, & ! 1. pyr 9, 9, 9, 9, 9, 9, 10,10,10,10,10,10, 11,11,11,11,11,11, 12,12,12,12,12,12, & 9, 9, 9, 9, 9, 9, 10,10,10,10,10,10, 11,11,11,11,11,11, 12,12,12,12,12,12, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & ! 1. pyr 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 13,13,13,13,13,13, 14,14,14,14,14,14, 15,15,15,15,15,15, 16,16,16,16,16,16, & 17,17,17,17,17,17, 18,18,18,18,18,18, 19,19,19,19,19,19, 20,20,20,20,20,20, & 17,17,17,17,17,17, 18,18,18,18,18,18, 19,19,19,19,19,19, 20,20,20,20,20,20, & 17,17,17,17,17,17, 18,18,18,18,18,18, 19,19,19,19,19,19, 20,20,20,20,20,20, & 17,17,17,17,17,17, 18,18,18,18,18,18, 19,19,19,19,19,19, 20,20,20,20,20,20, & ! 2. pyr 17,17,17,17,17,17, 18,18,18,18,18,18, 19,19,19,19,19,19, 20,20,20,20,20,20, & 17,17,17,17,17,17, 18,18,18,18,18,18, 19,19,19,19,19,19, 20,20,20,20,20,20 & ],shape(HEX_INTERACTIONSLIPTWIN)) !< Slip-twin interaction types for hex select case(lattice) case('cF') interactionTypes = FCC_INTERACTIONSLIPTWIN NslipMax = FCC_NSLIPSYSTEM NtwinMax = FCC_NTWINSYSTEM case('cI') interactionTypes = BCC_INTERACTIONSLIPTWIN NslipMax = BCC_NSLIPSYSTEM NtwinMax = BCC_NTWINSYSTEM case('hP') interactionTypes = HEX_INTERACTIONSLIPTWIN NslipMax = HEX_NSLIPSYSTEM NtwinMax = HEX_NTWINSYSTEM case default call IO_error(137,ext_msg='lattice_interaction_SlipByTwin: '//trim(lattice)) end select interactionMatrix = buildInteraction(Nslip,Ntwin,NslipMax,NtwinMax,interactionValues,interactionTypes) end function lattice_interaction_SlipByTwin !-------------------------------------------------------------------------------------------------- !> @brief Slip-trans interaction matrix !> details only active slip and trans systems are considered !-------------------------------------------------------------------------------------------------- function lattice_interaction_SlipByTrans(Nslip,Ntrans,interactionValues,lattice) result(interactionMatrix) integer, dimension(:), intent(in) :: Nslip, & !< number of active slip systems per family Ntrans !< number of active trans systems per family real(pReal), dimension(:), intent(in) :: interactionValues !< values for slip-trans interaction character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) (parent crystal) real(pReal), dimension(sum(Nslip),sum(Ntrans)) :: interactionMatrix integer, dimension(:), allocatable :: NslipMax, & NtransMax integer, dimension(:,:), allocatable :: interactionTypes integer, dimension(FCC_NTRANS,FCC_NSLIP), parameter :: & FCC_INTERACTIONSLIPTRANS = reshape( [& 1,1,1,3,3,3,2,2,2,3,3,3, & ! -----> trans (acting) 1,1,1,3,3,3,3,3,3,2,2,2, & ! | 1,1,1,2,2,2,3,3,3,3,3,3, & ! | 3,3,3,1,1,1,3,3,3,2,2,2, & ! v 3,3,3,1,1,1,2,2,2,3,3,3, & ! slip (reacting) 2,2,2,1,1,1,3,3,3,3,3,3, & 2,2,2,3,3,3,1,1,1,3,3,3, & 3,3,3,2,2,2,1,1,1,3,3,3, & 3,3,3,3,3,3,1,1,1,2,2,2, & 3,3,3,2,2,2,3,3,3,1,1,1, & 2,2,2,3,3,3,3,3,3,1,1,1, & 3,3,3,3,3,3,2,2,2,1,1,1, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4, & 4,4,4,4,4,4,4,4,4,4,4,4 & ],shape(FCC_INTERACTIONSLIPTRANS)) !< Slip-trans interaction types for fcc select case(lattice) case('cF') interactionTypes = FCC_INTERACTIONSLIPTRANS NslipMax = FCC_NSLIPSYSTEM NtransMax = FCC_NTRANSSYSTEM case default call IO_error(137,ext_msg='lattice_interaction_SlipByTrans: '//trim(lattice)) end select interactionMatrix = buildInteraction(Nslip,Ntrans,NslipMax,NtransMax,interactionValues,interactionTypes) end function lattice_interaction_SlipByTrans !-------------------------------------------------------------------------------------------------- !> @brief Twin-slip interaction matrix !> details only active twin and slip systems are considered !-------------------------------------------------------------------------------------------------- function lattice_interaction_TwinBySlip(Ntwin,Nslip,interactionValues,lattice) result(interactionMatrix) integer, dimension(:), intent(in) :: Ntwin, & !< number of active twin systems per family Nslip !< number of active slip systems per family real(pReal), dimension(:), intent(in) :: interactionValues !< values for twin-twin interaction character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), dimension(sum(Ntwin),sum(Nslip)) :: interactionMatrix integer, dimension(:), allocatable :: NtwinMax, & NslipMax integer, dimension(:,:), allocatable :: interactionTypes integer, dimension(FCC_NSLIP,FCC_NTWIN), parameter :: & FCC_INTERACTIONTWINSLIP = 1 !< Twin-slip interaction types for fcc integer, dimension(BCC_NSLIP,BCC_NTWIN), parameter :: & BCC_INTERACTIONTWINSLIP = 1 !< Twin-slip interaction types for bcc integer, dimension(HEX_NSLIP,HEX_NTWIN), parameter :: & HEX_INTERACTIONTWINSLIP = reshape( [& ! basal prism 1. pyr 1. pyr 2. pyr 1, 1, 1, 5, 5, 5, 9, 9, 9, 9, 9, 9, 13,13,13,13,13,13,13,13,13,13,13,13, 17,17,17,17,17,17, & ! ----> slip (acting) 1, 1, 1, 5, 5, 5, 9, 9, 9, 9, 9, 9, 13,13,13,13,13,13,13,13,13,13,13,13, 17,17,17,17,17,17, & ! | 1, 1, 1, 5, 5, 5, 9, 9, 9, 9, 9, 9, 13,13,13,13,13,13,13,13,13,13,13,13, 17,17,17,17,17,17, & ! | 1, 1, 1, 5, 5, 5, 9, 9, 9, 9, 9, 9, 13,13,13,13,13,13,13,13,13,13,13,13, 17,17,17,17,17,17, & ! v <-10.1>{10.2} 1, 1, 1, 5, 5, 5, 9, 9, 9, 9, 9, 9, 13,13,13,13,13,13,13,13,13,13,13,13, 17,17,17,17,17,17, & ! twin (reacting) 1, 1, 1, 5, 5, 5, 9, 9, 9, 9, 9, 9, 13,13,13,13,13,13,13,13,13,13,13,13, 17,17,17,17,17,17, & 2, 2, 2, 6, 6, 6, 10,10,10,10,10,10, 14,14,14,14,14,14,14,14,14,14,14,14, 18,18,18,18,18,18, & 2, 2, 2, 6, 6, 6, 10,10,10,10,10,10, 14,14,14,14,14,14,14,14,14,14,14,14, 18,18,18,18,18,18, & 2, 2, 2, 6, 6, 6, 10,10,10,10,10,10, 14,14,14,14,14,14,14,14,14,14,14,14, 18,18,18,18,18,18, & 2, 2, 2, 6, 6, 6, 10,10,10,10,10,10, 14,14,14,14,14,14,14,14,14,14,14,14, 18,18,18,18,18,18, & ! <11.6>{-1-1.1} 2, 2, 2, 6, 6, 6, 10,10,10,10,10,10, 14,14,14,14,14,14,14,14,14,14,14,14, 18,18,18,18,18,18, & 2, 2, 2, 6, 6, 6, 10,10,10,10,10,10, 14,14,14,14,14,14,14,14,14,14,14,14, 18,18,18,18,18,18, & 3, 3, 3, 7, 7, 7, 11,11,11,11,11,11, 15,15,15,15,15,15,15,15,15,15,15,15, 19,19,19,19,19,19, & 3, 3, 3, 7, 7, 7, 11,11,11,11,11,11, 15,15,15,15,15,15,15,15,15,15,15,15, 19,19,19,19,19,19, & 3, 3, 3, 7, 7, 7, 11,11,11,11,11,11, 15,15,15,15,15,15,15,15,15,15,15,15, 19,19,19,19,19,19, & 3, 3, 3, 7, 7, 7, 11,11,11,11,11,11, 15,15,15,15,15,15,15,15,15,15,15,15, 19,19,19,19,19,19, & ! <10.-2>{10.1} 3, 3, 3, 7, 7, 7, 11,11,11,11,11,11, 15,15,15,15,15,15,15,15,15,15,15,15, 19,19,19,19,19,19, & 3, 3, 3, 7, 7, 7, 11,11,11,11,11,11, 15,15,15,15,15,15,15,15,15,15,15,15, 19,19,19,19,19,19, & 4, 4, 4, 8, 8, 8, 12,12,12,12,12,12, 16,16,16,16,16,16,16,16,16,16,16,16, 20,20,20,20,20,20, & 4, 4, 4, 8, 8, 8, 12,12,12,12,12,12, 16,16,16,16,16,16,16,16,16,16,16,16, 20,20,20,20,20,20, & 4, 4, 4, 8, 8, 8, 12,12,12,12,12,12, 16,16,16,16,16,16,16,16,16,16,16,16, 20,20,20,20,20,20, & 4, 4, 4, 8, 8, 8, 12,12,12,12,12,12, 16,16,16,16,16,16,16,16,16,16,16,16, 20,20,20,20,20,20, & ! <11.-3>{11.2} 4, 4, 4, 8, 8, 8, 12,12,12,12,12,12, 16,16,16,16,16,16,16,16,16,16,16,16, 20,20,20,20,20,20, & 4, 4, 4, 8, 8, 8, 12,12,12,12,12,12, 16,16,16,16,16,16,16,16,16,16,16,16, 20,20,20,20,20,20 & ],shape(HEX_INTERACTIONTWINSLIP)) !< Twin-slip interaction types for hex select case(lattice) case('cF') interactionTypes = FCC_INTERACTIONTWINSLIP NtwinMax = FCC_NTWINSYSTEM NslipMax = FCC_NSLIPSYSTEM case('cI') interactionTypes = BCC_INTERACTIONTWINSLIP NtwinMax = BCC_NTWINSYSTEM NslipMax = BCC_NSLIPSYSTEM case('hP') interactionTypes = HEX_INTERACTIONTWINSLIP NtwinMax = HEX_NTWINSYSTEM NslipMax = HEX_NSLIPSYSTEM case default call IO_error(137,ext_msg='lattice_interaction_TwinBySlip: '//trim(lattice)) end select interactionMatrix = buildInteraction(Ntwin,Nslip,NtwinMax,NslipMax,interactionValues,interactionTypes) end function lattice_interaction_TwinBySlip !-------------------------------------------------------------------------------------------------- !> @brief Schmid matrix for slip !> details only active slip systems are considered !-------------------------------------------------------------------------------------------------- function lattice_SchmidMatrix_slip(Nslip,lattice,cOverA) result(SchmidMatrix) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA real(pReal), dimension(3,3,sum(Nslip)) :: SchmidMatrix real(pReal), dimension(3,3,sum(Nslip)) :: coordinateSystem real(pReal), dimension(:,:), allocatable :: slipSystems integer, dimension(:), allocatable :: NslipMax integer :: i select case(lattice) case('cF') NslipMax = FCC_NSLIPSYSTEM slipSystems = FCC_SYSTEMSLIP case('cI') NslipMax = BCC_NSLIPSYSTEM slipSystems = BCC_SYSTEMSLIP case('hP') NslipMax = HEX_NSLIPSYSTEM slipSystems = HEX_SYSTEMSLIP case('tI') NslipMax = BCT_NSLIPSYSTEM slipSystems = BCT_SYSTEMSLIP case default allocate(NslipMax(0)) call IO_error(137,ext_msg='lattice_SchmidMatrix_slip: '//trim(lattice)) end select if (any(NslipMax(1:size(Nslip)) - Nslip < 0)) & call IO_error(145,ext_msg='Nslip '//trim(lattice)) if (any(Nslip < 0)) & call IO_error(144,ext_msg='Nslip '//trim(lattice)) coordinateSystem = buildCoordinateSystem(Nslip,NslipMax,slipSystems,lattice,cOverA) do i = 1, sum(Nslip) SchmidMatrix(1:3,1:3,i) = math_outer(coordinateSystem(1:3,1,i),coordinateSystem(1:3,2,i)) if (abs(math_trace33(SchmidMatrix(1:3,1:3,i))) > tol_math_check) & call IO_error(0,i,ext_msg = 'dilatational Schmid matrix for slip') enddo end function lattice_SchmidMatrix_slip !-------------------------------------------------------------------------------------------------- !> @brief Schmid matrix for twinning !> details only active twin systems are considered !-------------------------------------------------------------------------------------------------- function lattice_SchmidMatrix_twin(Ntwin,lattice,cOverA) result(SchmidMatrix) integer, dimension(:), intent(in) :: Ntwin !< number of active twin systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(3,3,sum(Ntwin)) :: SchmidMatrix real(pReal), dimension(3,3,sum(Ntwin)) :: coordinateSystem real(pReal), dimension(:,:), allocatable :: twinSystems integer, dimension(:), allocatable :: NtwinMax integer :: i select case(lattice) case('cF') NtwinMax = FCC_NTWINSYSTEM twinSystems = FCC_SYSTEMTWIN case('cI') NtwinMax = BCC_NTWINSYSTEM twinSystems = BCC_SYSTEMTWIN case('hP') NtwinMax = HEX_NTWINSYSTEM twinSystems = HEX_SYSTEMTWIN case default allocate(NtwinMax(0)) call IO_error(137,ext_msg='lattice_SchmidMatrix_twin: '//trim(lattice)) end select if (any(NtwinMax(1:size(Ntwin)) - Ntwin < 0)) & call IO_error(145,ext_msg='Ntwin '//trim(lattice)) if (any(Ntwin < 0)) & call IO_error(144,ext_msg='Ntwin '//trim(lattice)) coordinateSystem = buildCoordinateSystem(Ntwin,NtwinMax,twinSystems,lattice,cOverA) do i = 1, sum(Ntwin) SchmidMatrix(1:3,1:3,i) = math_outer(coordinateSystem(1:3,1,i),coordinateSystem(1:3,2,i)) if (abs(math_trace33(SchmidMatrix(1:3,1:3,i))) > tol_math_check) & call IO_error(0,i,ext_msg = 'dilatational Schmid matrix for twin') enddo end function lattice_SchmidMatrix_twin !-------------------------------------------------------------------------------------------------- !> @brief Schmid matrix for transformation !> details only active twin systems are considered !-------------------------------------------------------------------------------------------------- function lattice_SchmidMatrix_trans(Ntrans,lattice_target,cOverA,a_bcc,a_fcc) result(SchmidMatrix) integer, dimension(:), intent(in) :: Ntrans !< number of active twin systems per family character(len=2), intent(in) :: lattice_target !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(3,3,sum(Ntrans)) :: SchmidMatrix real(pReal), dimension(3,3,sum(Ntrans)) :: devNull real(pReal) :: a_bcc, a_fcc if (lattice_target /= 'cI' .and. lattice_target /= 'hP') & call IO_error(137,ext_msg='lattice_SchmidMatrix_trans: '//trim(lattice_target)) if (lattice_target == 'hP' .and. (cOverA < 1.0_pReal .or. cOverA > 2.0_pReal)) & call IO_error(131,ext_msg='lattice_SchmidMatrix_trans: '//trim(lattice_target)) if (lattice_target == 'cI' .and. (a_bcc <= 0.0_pReal .or. a_fcc <= 0.0_pReal)) & call IO_error(134,ext_msg='lattice_SchmidMatrix_trans: '//trim(lattice_target)) call buildTransformationSystem(devNull,SchmidMatrix,Ntrans,cOverA,a_fcc,a_bcc) end function lattice_SchmidMatrix_trans !-------------------------------------------------------------------------------------------------- !> @brief Schmid matrix for cleavage !> details only active cleavage systems are considered !-------------------------------------------------------------------------------------------------- function lattice_SchmidMatrix_cleavage(Ncleavage,lattice,cOverA) result(SchmidMatrix) integer, dimension(:), intent(in) :: Ncleavage !< number of active cleavage systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(3,3,3,sum(Ncleavage)) :: SchmidMatrix real(pReal), dimension(3,3,sum(Ncleavage)) :: coordinateSystem real(pReal), dimension(:,:), allocatable :: cleavageSystems integer, dimension(:), allocatable :: NcleavageMax integer :: i select case(lattice) case('cF') NcleavageMax = FCC_NCLEAVAGESYSTEM cleavageSystems = FCC_SYSTEMCLEAVAGE case('cI') NcleavageMax = BCC_NCLEAVAGESYSTEM cleavageSystems = BCC_SYSTEMCLEAVAGE case default allocate(NcleavageMax(0)) call IO_error(137,ext_msg='lattice_SchmidMatrix_cleavage: '//trim(lattice)) end select if (any(NcleavageMax(1:size(Ncleavage)) - Ncleavage < 0)) & call IO_error(145,ext_msg='Ncleavage '//trim(lattice)) if (any(Ncleavage < 0)) & call IO_error(144,ext_msg='Ncleavage '//trim(lattice)) coordinateSystem = buildCoordinateSystem(Ncleavage,NcleavageMax,cleavageSystems,lattice,cOverA) do i = 1, sum(Ncleavage) SchmidMatrix(1:3,1:3,1,i) = math_outer(coordinateSystem(1:3,1,i),coordinateSystem(1:3,2,i)) SchmidMatrix(1:3,1:3,2,i) = math_outer(coordinateSystem(1:3,3,i),coordinateSystem(1:3,2,i)) SchmidMatrix(1:3,1:3,3,i) = math_outer(coordinateSystem(1:3,2,i),coordinateSystem(1:3,2,i)) enddo end function lattice_SchmidMatrix_cleavage !-------------------------------------------------------------------------------------------------- !> @brief Slip direction of slip systems (|| b) !-------------------------------------------------------------------------------------------------- function lattice_slip_direction(Nslip,lattice,cOverA) result(d) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(3,sum(Nslip)) :: d real(pReal), dimension(3,3,sum(Nslip)) :: coordinateSystem coordinateSystem = coordinateSystem_slip(Nslip,lattice,cOverA) d = coordinateSystem(1:3,1,1:sum(Nslip)) end function lattice_slip_direction !-------------------------------------------------------------------------------------------------- !> @brief Normal direction of slip systems (|| n) !-------------------------------------------------------------------------------------------------- function lattice_slip_normal(Nslip,lattice,cOverA) result(n) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(3,sum(Nslip)) :: n real(pReal), dimension(3,3,sum(Nslip)) :: coordinateSystem coordinateSystem = coordinateSystem_slip(Nslip,lattice,cOverA) n = coordinateSystem(1:3,2,1:sum(Nslip)) end function lattice_slip_normal !-------------------------------------------------------------------------------------------------- !> @brief Transverse direction of slip systems (|| t = b x n) !-------------------------------------------------------------------------------------------------- function lattice_slip_transverse(Nslip,lattice,cOverA) result(t) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(3,sum(Nslip)) :: t real(pReal), dimension(3,3,sum(Nslip)) :: coordinateSystem coordinateSystem = coordinateSystem_slip(Nslip,lattice,cOverA) t = coordinateSystem(1:3,3,1:sum(Nslip)) end function lattice_slip_transverse !-------------------------------------------------------------------------------------------------- !> @brief Labels of slip systems !> details only active slip systems are considered !-------------------------------------------------------------------------------------------------- function lattice_labels_slip(Nslip,lattice) result(labels) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) character(len=:), dimension(:), allocatable :: labels real(pReal), dimension(:,:), allocatable :: slipSystems integer, dimension(:), allocatable :: NslipMax select case(lattice) case('cF') NslipMax = FCC_NSLIPSYSTEM slipSystems = FCC_SYSTEMSLIP case('cI') NslipMax = BCC_NSLIPSYSTEM slipSystems = BCC_SYSTEMSLIP case('hP') NslipMax = HEX_NSLIPSYSTEM slipSystems = HEX_SYSTEMSLIP case('tI') NslipMax = BCT_NSLIPSYSTEM slipSystems = BCT_SYSTEMSLIP case default call IO_error(137,ext_msg='lattice_labels_slip: '//trim(lattice)) end select if (any(NslipMax(1:size(Nslip)) - Nslip < 0)) & call IO_error(145,ext_msg='Nslip '//trim(lattice)) if (any(Nslip < 0)) & call IO_error(144,ext_msg='Nslip '//trim(lattice)) labels = getLabels(Nslip,NslipMax,slipSystems) end function lattice_labels_slip !-------------------------------------------------------------------------------------------------- !> @brief Return 3x3 tensor with symmetry according to given Bravais lattice !-------------------------------------------------------------------------------------------------- pure function lattice_symmetrize_33(T,lattice) result(T_sym) real(pReal), dimension(3,3) :: T_sym real(pReal), dimension(3,3), intent(in) :: T character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) T_sym = 0.0_pReal select case(lattice) case('cF','cI') T_sym(1,1) = T(1,1) T_sym(2,2) = T(1,1) T_sym(3,3) = T(1,1) case('hP','tI') T_sym(1,1) = T(1,1) T_sym(2,2) = T(1,1) T_sym(3,3) = T(3,3) end select end function lattice_symmetrize_33 !-------------------------------------------------------------------------------------------------- !> @brief Return stiffness matrix in 6x6 notation with symmetry according to given Bravais lattice !> @details J. A. Rayne and B. S. Chandrasekhar Phys. Rev. 120, 1658 Erratum Phys. Rev. 122, 1962 !-------------------------------------------------------------------------------------------------- pure function lattice_symmetrize_C66(C66,lattice) result(C66_sym) real(pReal), dimension(6,6) :: C66_sym real(pReal), dimension(6,6), intent(in) :: C66 character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) integer :: i,j C66_sym = 0.0_pReal select case(lattice) case ('cF','cI') C66_sym(1,1) = C66(1,1); C66_sym(2,2) = C66(1,1); C66_sym(3,3) = C66(1,1) C66_sym(1,2) = C66(1,2); C66_sym(1,3) = C66(1,2); C66_sym(2,3) = C66(1,2) C66_sym(4,4) = C66(4,4); C66_sym(5,5) = C66(4,4); C66_sym(6,6) = C66(4,4) ! isotropic C_44 = (C_11-C_12)/2 case ('hP') C66_sym(1,1) = C66(1,1); C66_sym(2,2) = C66(1,1) C66_sym(3,3) = C66(3,3) C66_sym(1,2) = C66(1,2) C66_sym(1,3) = C66(1,3); C66_sym(2,3) = C66(1,3) C66_sym(4,4) = C66(4,4); C66_sym(5,5) = C66(4,4) C66_sym(6,6) = 0.5_pReal*(C66(1,1)-C66(1,2)) case ('tI') C66_sym(1,1) = C66(1,1); C66_sym(2,2) = C66(1,1) C66_sym(3,3) = C66(3,3) C66_sym(1,2) = C66(1,2) C66_sym(1,3) = C66(1,3); C66_sym(2,3) = C66(1,3) C66_sym(4,4) = C66(4,4); C66_sym(5,5) = C66(4,4) C66_sym(6,6) = C66(6,6) end select do i = 1, 6 do j = i+1, 6 C66_sym(j,i) = C66_sym(i,j) enddo enddo end function lattice_symmetrize_C66 !-------------------------------------------------------------------------------------------------- !> @brief Labels for twin systems !> details only active twin systems are considered !-------------------------------------------------------------------------------------------------- function lattice_labels_twin(Ntwin,lattice) result(labels) integer, dimension(:), intent(in) :: Ntwin !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) character(len=:), dimension(:), allocatable :: labels real(pReal), dimension(:,:), allocatable :: twinSystems integer, dimension(:), allocatable :: NtwinMax select case(lattice) case('cF') NtwinMax = FCC_NTWINSYSTEM twinSystems = FCC_SYSTEMTWIN case('cI') NtwinMax = BCC_NTWINSYSTEM twinSystems = BCC_SYSTEMTWIN case('hP') NtwinMax = HEX_NTWINSYSTEM twinSystems = HEX_SYSTEMTWIN case default call IO_error(137,ext_msg='lattice_labels_twin: '//trim(lattice)) end select if (any(NtwinMax(1:size(Ntwin)) - Ntwin < 0)) & call IO_error(145,ext_msg='Ntwin '//trim(lattice)) if (any(Ntwin < 0)) & call IO_error(144,ext_msg='Ntwin '//trim(lattice)) labels = getLabels(Ntwin,NtwinMax,twinSystems) end function lattice_labels_twin !-------------------------------------------------------------------------------------------------- !> @brief Projection of the transverse direction onto the slip plane !> @details: This projection is used to calculate forest hardening for edge dislocations !-------------------------------------------------------------------------------------------------- function slipProjection_transverse(Nslip,lattice,cOverA) result(projection) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(sum(Nslip),sum(Nslip)) :: projection real(pReal), dimension(3,sum(Nslip)) :: n, t integer :: i, j n = lattice_slip_normal (Nslip,lattice,cOverA) t = lattice_slip_transverse(Nslip,lattice,cOverA) do i=1, sum(Nslip); do j=1, sum(Nslip) projection(i,j) = abs(math_inner(n(:,i),t(:,j))) enddo; enddo end function slipProjection_transverse !-------------------------------------------------------------------------------------------------- !> @brief Projection of the slip direction onto the slip plane !> @details: This projection is used to calculate forest hardening for screw dislocations !-------------------------------------------------------------------------------------------------- function slipProjection_direction(Nslip,lattice,cOverA) result(projection) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(sum(Nslip),sum(Nslip)) :: projection real(pReal), dimension(3,sum(Nslip)) :: n, d integer :: i, j n = lattice_slip_normal (Nslip,lattice,cOverA) d = lattice_slip_direction(Nslip,lattice,cOverA) do i=1, sum(Nslip); do j=1, sum(Nslip) projection(i,j) = abs(math_inner(n(:,i),d(:,j))) enddo; enddo end function slipProjection_direction !-------------------------------------------------------------------------------------------------- !> @brief build a local coordinate system on slip systems !> @details Order: Direction, plane (normal), and common perpendicular !-------------------------------------------------------------------------------------------------- function coordinateSystem_slip(Nslip,lattice,cOverA) result(coordinateSystem) integer, dimension(:), intent(in) :: Nslip !< number of active slip systems per family character(len=2), intent(in) :: lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: cOverA !< c/a ratio real(pReal), dimension(3,3,sum(Nslip)) :: coordinateSystem real(pReal), dimension(:,:), allocatable :: slipSystems integer, dimension(:), allocatable :: NslipMax select case(lattice) case('cF') NslipMax = FCC_NSLIPSYSTEM slipSystems = FCC_SYSTEMSLIP case('cI') NslipMax = BCC_NSLIPSYSTEM slipSystems = BCC_SYSTEMSLIP case('hP') NslipMax = HEX_NSLIPSYSTEM slipSystems = HEX_SYSTEMSLIP case('tI') NslipMax = BCT_NSLIPSYSTEM slipSystems = BCT_SYSTEMSLIP case default allocate(NslipMax(0)) call IO_error(137,ext_msg='coordinateSystem_slip: '//trim(lattice)) end select if (any(NslipMax(1:size(Nslip)) - Nslip < 0)) & call IO_error(145,ext_msg='Nslip '//trim(lattice)) if (any(Nslip < 0)) & call IO_error(144,ext_msg='Nslip '//trim(lattice)) coordinateSystem = buildCoordinateSystem(Nslip,NslipMax,slipSystems,lattice,cOverA) end function coordinateSystem_slip !-------------------------------------------------------------------------------------------------- !> @brief Populate reduced interaction matrix !-------------------------------------------------------------------------------------------------- function buildInteraction(reacting_used,acting_used,reacting_max,acting_max,values,matrix) integer, dimension(:), intent(in) :: & reacting_used, & !< # of reacting systems per family as specified in material.config acting_used, & !< # of acting systems per family as specified in material.config reacting_max, & !< max # of reacting systems per family for given lattice acting_max !< max # of acting systems per family for given lattice real(pReal), dimension(:), intent(in) :: values !< interaction values integer, dimension(:,:), intent(in) :: matrix !< interaction types real(pReal), dimension(sum(reacting_used),sum(acting_used)) :: buildInteraction integer :: & acting_family_index, acting_family, acting_system, & reacting_family_index, reacting_family, reacting_system, & i,j,k,l do acting_family = 1,size(acting_used,1) acting_family_index = sum(acting_used(1:acting_family-1)) do acting_system = 1,acting_used(acting_family) do reacting_family = 1,size(reacting_used,1) reacting_family_index = sum(reacting_used(1:reacting_family-1)) do reacting_system = 1,reacting_used(reacting_family) i = sum( acting_max(1: acting_family-1)) + acting_system j = sum(reacting_max(1:reacting_family-1)) + reacting_system k = acting_family_index + acting_system l = reacting_family_index + reacting_system if (matrix(i,j) > size(values)) call IO_error(138,ext_msg='buildInteraction') buildInteraction(l,k) = values(matrix(i,j)) enddo; enddo enddo; enddo end function buildInteraction !-------------------------------------------------------------------------------------------------- !> @brief Build a local coordinate system on slip, twin, trans, cleavage systems !> @details Order: Direction, plane (normal), and common perpendicular !-------------------------------------------------------------------------------------------------- function buildCoordinateSystem(active,potential,system,lattice,cOverA) integer, dimension(:), intent(in) :: & active, & !< # of active systems per family potential !< # of potential systems per family real(pReal), dimension(:,:), intent(in) :: & system character(len=2), intent(in) :: & lattice !< Bravais lattice (Pearson symbol) real(pReal), intent(in) :: & cOverA real(pReal), dimension(3,3,sum(active)) :: & buildCoordinateSystem real(pReal), dimension(3) :: & direction, normal integer :: & a, & !< index of active system p, & !< index in potential system matrix f, & !< index of my family s !< index of my system in current family if (lattice == 'tI' .and. cOverA > 2.0_pReal) & call IO_error(131,ext_msg='buildCoordinateSystem:'//trim(lattice)) if (lattice == 'hP' .and. (cOverA < 1.0_pReal .or. cOverA > 2.0_pReal)) & call IO_error(131,ext_msg='buildCoordinateSystem:'//trim(lattice)) a = 0 activeFamilies: do f = 1,size(active,1) activeSystems: do s = 1,active(f) a = a + 1 p = sum(potential(1:f-1))+s select case(lattice) case ('cF','cI','tI') direction = system(1:3,p) normal = system(4:6,p) case ('hP') direction = [ system(1,p)*1.5_pReal, & (system(1,p)+2.0_pReal*system(2,p))*sqrt(0.75_pReal), & system(4,p)*cOverA ] ! direction [uvtw]->[3u/2 (u+2v)*sqrt(3)/2 w*(p/a)]) normal = [ system(5,p), & (system(5,p)+2.0_pReal*system(6,p))/sqrt(3.0_pReal), & system(8,p)/cOverA ] ! plane (hkil)->(h (h+2k)/sqrt(3) l/(p/a)) case default call IO_error(137,ext_msg='buildCoordinateSystem: '//trim(lattice)) end select buildCoordinateSystem(1:3,1,a) = direction/norm2(direction) buildCoordinateSystem(1:3,2,a) = normal /norm2(normal) buildCoordinateSystem(1:3,3,a) = math_cross(direction/norm2(direction),& normal /norm2(normal)) enddo activeSystems enddo activeFamilies end function buildCoordinateSystem !-------------------------------------------------------------------------------------------------- !> @brief Helper function to define transformation systems ! Needed to calculate Schmid matrix and rotated stiffness matrices. ! @details: set c/a = 0.0 for fcc -> bcc transformation ! set a_Xcc = 0.0 for fcc -> hex transformation !-------------------------------------------------------------------------------------------------- subroutine buildTransformationSystem(Q,S,Ntrans,cOverA,a_fcc,a_bcc) integer, dimension(:), intent(in) :: & Ntrans real(pReal), dimension(3,3,sum(Ntrans)), intent(out) :: & Q, & !< Total rotation: Q = R*B S !< Eigendeformation tensor for phase transformation real(pReal), intent(in) :: & cOverA, & !< c/a for target hex lattice a_bcc, & !< lattice parameter a for bcc target lattice a_fcc !< lattice parameter a for fcc parent lattice type(rotation) :: & R, & !< Pitsch rotation B !< Rotation of fcc to Bain coordinate system real(pReal), dimension(3,3) :: & U, & !< Bain deformation ss, sd real(pReal), dimension(3) :: & x, y, z integer :: & i real(pReal), dimension(3+3,FCC_NTRANS), parameter :: & FCCTOHEX_SYSTEMTRANS = reshape(real( [& -2, 1, 1, 1, 1, 1, & 1,-2, 1, 1, 1, 1, & 1, 1,-2, 1, 1, 1, & 2,-1, 1, -1,-1, 1, & -1, 2, 1, -1,-1, 1, & -1,-1,-2, -1,-1, 1, & -2,-1,-1, 1,-1,-1, & 1, 2,-1, 1,-1,-1, & 1,-1, 2, 1,-1,-1, & 2, 1,-1, -1, 1,-1, & -1,-2,-1, -1, 1,-1, & -1, 1, 2, -1, 1,-1 & ],pReal),shape(FCCTOHEX_SYSTEMTRANS)) real(pReal), dimension(4,fcc_Ntrans), parameter :: & FCCTOBCC_SYSTEMTRANS = reshape([& 0.0, 1.0, 0.0, 10.26, & ! Pitsch OR (Ma & Hartmaier 2014, Table 3) 0.0,-1.0, 0.0, 10.26, & 0.0, 0.0, 1.0, 10.26, & 0.0, 0.0,-1.0, 10.26, & 1.0, 0.0, 0.0, 10.26, & -1.0, 0.0, 0.0, 10.26, & 0.0, 0.0, 1.0, 10.26, & 0.0, 0.0,-1.0, 10.26, & 1.0, 0.0, 0.0, 10.26, & -1.0, 0.0, 0.0, 10.26, & 0.0, 1.0, 0.0, 10.26, & 0.0,-1.0, 0.0, 10.26 & ],shape(FCCTOBCC_SYSTEMTRANS)) integer, dimension(9,fcc_Ntrans), parameter :: & FCCTOBCC_BAINVARIANT = reshape( [& 1, 0, 0, 0, 1, 0, 0, 0, 1, & ! Pitsch OR (Ma & Hartmaier 2014, Table 3) 1, 0, 0, 0, 1, 0, 0, 0, 1, & 1, 0, 0, 0, 1, 0, 0, 0, 1, & 1, 0, 0, 0, 1, 0, 0, 0, 1, & 0, 1, 0, 1, 0, 0, 0, 0, 1, & 0, 1, 0, 1, 0, 0, 0, 0, 1, & 0, 1, 0, 1, 0, 0, 0, 0, 1, & 0, 1, 0, 1, 0, 0, 0, 0, 1, & 0, 0, 1, 1, 0, 0, 0, 1, 0, & 0, 0, 1, 1, 0, 0, 0, 1, 0, & 0, 0, 1, 1, 0, 0, 0, 1, 0, & 0, 0, 1, 1, 0, 0, 0, 1, 0 & ],shape(FCCTOBCC_BAINVARIANT)) real(pReal), dimension(4,fcc_Ntrans), parameter :: & FCCTOBCC_BAINROT = reshape([& 1.0, 0.0, 0.0, 45.0, & ! Rotate fcc austensite to bain variant 1.0, 0.0, 0.0, 45.0, & 1.0, 0.0, 0.0, 45.0, & 1.0, 0.0, 0.0, 45.0, & 0.0, 1.0, 0.0, 45.0, & 0.0, 1.0, 0.0, 45.0, & 0.0, 1.0, 0.0, 45.0, & 0.0, 1.0, 0.0, 45.0, & 0.0, 0.0, 1.0, 45.0, & 0.0, 0.0, 1.0, 45.0, & 0.0, 0.0, 1.0, 45.0, & 0.0, 0.0, 1.0, 45.0 & ],shape(FCCTOBCC_BAINROT)) if (a_bcc > 0.0_pReal .and. a_fcc > 0.0_pReal .and. dEq0(cOverA)) then ! fcc -> bcc do i = 1,sum(Ntrans) call R%fromAxisAngle(FCCTOBCC_SYSTEMTRANS(:,i),degrees=.true.,P=1) call B%fromAxisAngle(FCCTOBCC_BAINROT(:,i), degrees=.true.,P=1) x = real(FCCTOBCC_BAINVARIANT(1:3,i),pReal) y = real(FCCTOBCC_BAINVARIANT(4:6,i),pReal) z = real(FCCTOBCC_BAINVARIANT(7:9,i),pReal) U = (a_bcc/a_fcc)*math_outer(x,x) & + (a_bcc/a_fcc)*math_outer(y,y) * sqrt(2.0_pReal) & + (a_bcc/a_fcc)*math_outer(z,z) * sqrt(2.0_pReal) Q(1:3,1:3,i) = matmul(R%asMatrix(),B%asMatrix()) S(1:3,1:3,i) = matmul(R%asMatrix(),U) - MATH_I3 enddo elseif (cOverA > 0.0_pReal .and. dEq0(a_bcc)) then ! fcc -> hex ss = MATH_I3 sd = MATH_I3 ss(1,3) = sqrt(2.0_pReal)/4.0_pReal sd(3,3) = cOverA/sqrt(8.0_pReal/3.0_pReal) do i = 1,sum(Ntrans) x = FCCTOHEX_SYSTEMTRANS(1:3,i)/norm2(FCCTOHEX_SYSTEMTRANS(1:3,i)) z = FCCTOHEX_SYSTEMTRANS(4:6,i)/norm2(FCCTOHEX_SYSTEMTRANS(4:6,i)) y = -math_cross(x,z) Q(1:3,1,i) = x Q(1:3,2,i) = y Q(1:3,3,i) = z S(1:3,1:3,i) = matmul(Q(1:3,1:3,i), matmul(matmul(sd,ss), transpose(Q(1:3,1:3,i)))) - MATH_I3 ! ToDo: This is of interest for the Schmid matrix only enddo else call IO_error(132,ext_msg='buildTransformationSystem') endif end subroutine buildTransformationSystem !-------------------------------------------------------------------------------------------------- !> @brief select active systems as strings !-------------------------------------------------------------------------------------------------- function getlabels(active,potential,system) result(labels) integer, dimension(:), intent(in) :: & active, & !< # of active systems per family potential !< # of potential systems per family real(pReal), dimension(:,:), intent(in) :: & system character(len=:), dimension(:), allocatable :: labels character(len=:), allocatable :: label integer :: i,j integer :: & a, & !< index of active system p, & !< index in potential system matrix f, & !< index of my family s !< index of my system in current family i = 2*size(system,1) + (size(system,1) - 2) + 4 ! 2 letters per index + spaces + brackets allocate(character(len=i) :: labels(sum(active)), label) a = 0 activeFamilies: do f = 1,size(active,1) activeSystems: do s = 1,active(f) a = a + 1 p = sum(potential(1:f-1))+s i = 1 label(i:i) = '[' direction: do j = 1, size(system,1)/2 write(label(i+1:i+2),'(I2.1)') int(system(j,p)) label(i+3:i+3) = ' ' i = i + 3 enddo direction label(i:i) = ']' i = i +1 label(i:i) = '(' normal: do j = size(system,1)/2+1, size(system,1) write(label(i+1:i+2),'(I2.1)') int(system(j,p)) label(i+3:i+3) = ' ' i = i + 3 enddo normal label(i:i) = ')' labels(a) = label enddo activeSystems enddo activeFamilies end function getlabels !-------------------------------------------------------------------------------------------------- !> @brief Equivalent Poisson's ratio (ν) !> @details https://doi.org/10.1143/JPSJ.20.635 !-------------------------------------------------------------------------------------------------- pure function lattice_equivalent_nu(C,assumption) result(nu) real(pReal), dimension(6,6), intent(in) :: C !< Stiffness tensor (Voigt notation) character(len=5), intent(in) :: assumption !< Assumption ('Voigt' = isostrain, 'Reuss' = isostress) real(pReal) :: nu real(pReal) :: K, mu logical :: error real(pReal), dimension(6,6) :: S if (IO_lc(assumption) == 'voigt') then K = (C(1,1)+C(2,2)+C(3,3) +2.0_pReal*(C(1,2)+C(2,3)+C(1,3))) & / 9.0_pReal elseif (IO_lc(assumption) == 'reuss') then call math_invert(S,error,C) if (error) error stop 'matrix inversion failed' K = 1.0_pReal & / (S(1,1)+S(2,2)+S(3,3) +2.0_pReal*(S(1,2)+S(2,3)+S(1,3))) else error stop 'invalid assumption' endif mu = lattice_equivalent_mu(C,assumption) nu = (1.5_pReal*K-mu)/(3.0_pReal*K+mu) end function lattice_equivalent_nu !-------------------------------------------------------------------------------------------------- !> @brief Equivalent shear modulus (μ) !> @details https://doi.org/10.1143/JPSJ.20.635 !-------------------------------------------------------------------------------------------------- pure function lattice_equivalent_mu(C,assumption) result(mu) real(pReal), dimension(6,6), intent(in) :: C !< Stiffness tensor (Voigt notation) character(len=5), intent(in) :: assumption !< Assumption ('Voigt' = isostrain, 'Reuss' = isostress) real(pReal) :: mu logical :: error real(pReal), dimension(6,6) :: S if (IO_lc(assumption) == 'voigt') then mu = (1.0_pReal*(C(1,1)+C(2,2)+C(3,3)) -1.0_pReal*(C(1,2)+C(2,3)+C(1,3)) +3.0_pReal*(C(4,4)+C(5,5)+C(6,6))) & / 15.0_pReal elseif (IO_lc(assumption) == 'reuss') then call math_invert(S,error,C) if (error) error stop 'matrix inversion failed' mu = 15.0_pReal & / (4.0_pReal*(S(1,1)+S(2,2)+S(3,3)) -4.0_pReal*(S(1,2)+S(2,3)+S(1,3)) +3.0_pReal*(S(4,4)+S(5,5)+S(6,6))) else error stop 'invalid assumption' endif end function lattice_equivalent_mu !-------------------------------------------------------------------------------------------------- !> @brief Check correctness of some lattice functions. !-------------------------------------------------------------------------------------------------- subroutine selfTest real(pReal), dimension(:,:,:), allocatable :: CoSy real(pReal), dimension(:,:), allocatable :: system real(pReal), dimension(6,6) :: C, C_cF, C_cI, C_hP, C_tI real(pReal), dimension(3,3) :: T, T_cF, T_cI, T_hP, T_tI real(pReal), dimension(2) :: r real(pReal) :: lambda integer :: i call random_number(r) system = reshape([1.0_pReal+r(1),0.0_pReal,0.0_pReal, 0.0_pReal,1.0_pReal+r(2),0.0_pReal],[6,1]) CoSy = buildCoordinateSystem([1],[1],system,'cF',0.0_pReal) if (any(dNeq(CoSy(1:3,1:3,1),math_I3))) error stop 'buildCoordinateSystem' do i = 1, 10 call random_number(C) C_cF = lattice_symmetrize_C66(C,'cI') C_cI = lattice_symmetrize_C66(C,'cF') C_hP = lattice_symmetrize_C66(C,'hP') C_tI = lattice_symmetrize_C66(C,'tI') if (any(dNeq(C_cI,transpose(C_cF)))) error stop 'SymmetryC66/cI-cF' if (any(dNeq(C_cF,transpose(C_cI)))) error stop 'SymmetryC66/cF-cI' if (any(dNeq(C_hP,transpose(C_hP)))) error stop 'SymmetryC66/hP' if (any(dNeq(C_tI,transpose(C_tI)))) error stop 'SymmetryC66/tI' if (any(dNeq(C(1,1),[C_cF(1,1),C_cF(2,2),C_cF(3,3)]))) error stop 'SymmetryC_11-22-33/c' if (any(dNeq(C(1,2),[C_cF(1,2),C_cF(1,3),C_cF(2,3)]))) error stop 'SymmetryC_12-13-23/c' if (any(dNeq(C(4,4),[C_cF(4,4),C_cF(5,5),C_cF(6,6)]))) error stop 'SymmetryC_44-55-66/c' if (any(dNeq(C(1,1),[C_hP(1,1),C_hP(2,2)]))) error stop 'SymmetryC_11-22/hP' if (any(dNeq(C(1,3),[C_hP(1,3),C_hP(2,3)]))) error stop 'SymmetryC_13-23/hP' if (any(dNeq(C(4,4),[C_hP(4,4),C_hP(5,5)]))) error stop 'SymmetryC_44-55/hP' if (any(dNeq(C(1,1),[C_tI(1,1),C_tI(2,2)]))) error stop 'SymmetryC_11-22/tI' if (any(dNeq(C(1,3),[C_tI(1,3),C_tI(2,3)]))) error stop 'SymmetryC_13-23/tI' if (any(dNeq(C(4,4),[C_tI(4,4),C_tI(5,5)]))) error stop 'SymmetryC_44-55/tI' call random_number(T) T_cF = lattice_symmetrize_33(T,'cI') T_cI = lattice_symmetrize_33(T,'cF') T_hP = lattice_symmetrize_33(T,'hP') T_tI = lattice_symmetrize_33(T,'tI') if (any(dNeq0(T_cF) .and. math_I3<1.0_pReal)) error stop 'Symmetry33/c' if (any(dNeq0(T_hP) .and. math_I3<1.0_pReal)) error stop 'Symmetry33/hP' if (any(dNeq0(T_tI) .and. math_I3<1.0_pReal)) error stop 'Symmetry33/tI' if (any(dNeq(T(1,1),[T_cI(1,1),T_cI(2,2),T_cI(3,3)]))) error stop 'Symmetry33_11-22-33/c' if (any(dNeq(T(1,1),[T_hP(1,1),T_hP(2,2)]))) error stop 'Symmetry33_11-22/hP' if (any(dNeq(T(1,1),[T_tI(1,1),T_tI(2,2)]))) error stop 'Symmetry33_11-22/tI' enddo call random_number(C) C(1,1) = C(1,1) + C(1,2) + 0.1_pReal C(4,4) = 0.5_pReal * (C(1,1) - C(1,2)) C = lattice_symmetrize_C66(C,'cI') if (dNeq(C(4,4),lattice_equivalent_mu(C,'voigt'),1.0e-12_pReal)) error stop 'equivalent_mu/voigt' if (dNeq(C(4,4),lattice_equivalent_mu(C,'reuss'),1.0e-12_pReal)) error stop 'equivalent_mu/reuss' lambda = C(1,2) if (dNeq(lambda*0.5_pReal/(lambda+lattice_equivalent_mu(C,'voigt')), & lattice_equivalent_nu(C,'voigt'),1.0e-12_pReal)) error stop 'equivalent_nu/voigt' if (dNeq(lambda*0.5_pReal/(lambda+lattice_equivalent_mu(C,'reuss')), & lattice_equivalent_nu(C,'reuss'),1.0e-12_pReal)) error stop 'equivalent_nu/reuss' end subroutine selfTest end module lattice