DAMASK_EICMD/src/lattice.f90

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!--------------------------------------------------------------------------------------------------
!> @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
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!> @brief contains lattice structure definitions including Schmid matrices for slip, twin, trans,
! and cleavage as well as interaction among the various systems
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!--------------------------------------------------------------------------------------------------
module lattice
use prec, only: &
pReal, &
pInt
implicit none
private
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! BEGIN DEPRECATED
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integer(pInt), parameter, public :: &
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LATTICE_maxNslipFamily = 13_pInt, & !< max # of slip system families over lattice structures
LATTICE_maxNcleavageFamily = 3_pInt !< max # of transformation system families over lattice structures
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integer(pInt), allocatable, dimension(:,:), protected, public :: &
lattice_NslipSystem, & !< total # of slip systems in each family
lattice_NcleavageSystem !< total # of transformation systems in each family
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integer(pInt), allocatable, dimension(:,:,:), protected, public :: &
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lattice_interactionSlipSlip !< Slip--slip interaction type
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real(pReal), allocatable, dimension(:,:,:,:,:), protected, public :: &
lattice_Sslip, & !< Schmid and non-Schmid matrices
lattice_Scleavage !< Schmid matrices for cleavage systems
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real(pReal), allocatable, dimension(:,:,:,:), protected, public :: &
lattice_Sslip_v, & !< Mandel notation of lattice_Sslip
lattice_Scleavage_v !< Mandel notation of lattice_Scleavege
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real(pReal), allocatable, dimension(:,:,:), protected, public :: &
lattice_sn, & !< normal direction of slip system
lattice_st, & !< sd x sn
lattice_sd !< slip direction of slip system
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integer(pInt), allocatable, dimension(:), protected, public :: &
lattice_NnonSchmid !< total # of non-Schmid contributions for each structure
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! END DEPRECATED
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!--------------------------------------------------------------------------------------------------
! face centered cubic
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integer(pInt), dimension(LATTICE_maxNslipFamily), parameter, public :: &
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LATTICE_FCC_NSLIPSYSTEM = int([12, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],pInt) !< # of slip systems per family for fcc
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integer(pInt), dimension(1), parameter, public :: &
LATTICE_FCC_NTWINSYSTEM = int([12],pInt) !< # of twin systems per family for fcc
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integer(pInt), dimension(1), parameter, public :: &
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LATTICE_FCC_NTRANSSYSTEM = int([12],pInt) !< # of transformation systems per family for fcc
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integer(pInt), dimension(LATTICE_maxNcleavageFamily), parameter, public :: &
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LATTICE_FCC_NCLEAVAGESYSTEM = int([3, 4, 0],pInt) !< # of cleavage systems per family for fcc
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integer(pInt), parameter, private :: &
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LATTICE_FCC_NSLIP = sum(LATTICE_FCC_NSLIPSYSTEM), & !< total # of slip systems for fcc
LATTICE_FCC_NTWIN = sum(LATTICE_FCC_NTWINSYSTEM), & !< total # of twin systems for fcc
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LATTICE_FCC_NTRANS = sum(LATTICE_FCC_NTRANSSYSTEM), & !< total # of transformation systems for fcc
LATTICE_FCC_NCLEAVAGE = sum(LATTICE_FCC_NCLEAVAGESYSTEM) !< total # of cleavage systems for fcc
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real(pReal), dimension(3+3,LATTICE_FCC_NSLIP), parameter, private :: &
LATTICE_FCC_SYSTEMSLIP = reshape(real([&
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! Slip direction Plane normal ! SCHMID-BOAS notation
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
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-1,-1, 0, -1, 1,-1, & ! D6
! Slip system <110>{110}
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(LATTICE_FCC_SYSTEMSLIP)) !< Slip system <110>{111} directions. Sorted according to Eisenlohr & Hantcherli
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character(len=*), dimension(2), parameter, public :: LATTICE_FCC_SLIPFAMILY_NAME = &
['<0 1 -1>{1 1 1}', &
'<0 1 -1>{0 1 1}']
real(pReal), dimension(3+3,LATTICE_FCC_NTWIN), parameter, private :: &
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LATTICE_fcc_systemTwin = 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 &
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],pReal),shape(LATTICE_FCC_SYSTEMTWIN)) !< Twin system <112>{111} directions. Sorted according to Eisenlohr & Hantcherli
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character(len=*), dimension(1), parameter, public :: LATTICE_FCC_TWINFAMILY_NAME = &
['<-2 1 1>{1 1 1}']
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integer(pInt), dimension(2_pInt,LATTICE_FCC_NTWIN), parameter, public :: &
LATTICE_FCC_TWINNUCLEATIONSLIPPAIR = reshape(int( [&
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2,3, &
1,3, &
1,2, &
5,6, &
4,6, &
4,5, &
8,9, &
7,9, &
7,8, &
11,12, &
10,12, &
10,11 &
],pInt),shape(LATTICE_FCC_TWINNUCLEATIONSLIPPAIR))
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! ToDo: should be in the interaction function
integer(pInt), dimension(LATTICE_FCC_NSLIP,LATTICE_FCC_NSLIP), parameter, public :: &
LATTICE_FCC_INTERACTIONSLIPSLIP = reshape(int( [&
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1, 2, 2, 4, 6, 5, 3, 5, 5, 4, 5, 6, 9,10, 9,10,11,12, & ! ---> slip
2, 1, 2, 6, 4, 5, 5, 4, 6, 5, 3, 5, 9,10,11,12, 9,10, & ! |
2, 2, 1, 5, 5, 3, 5, 6, 4, 6, 5, 4, 11,12, 9,10, 9,10, & ! |
4, 6, 5, 1, 2, 2, 4, 5, 6, 3, 5, 5, 9,10,10, 9,12,11, & ! v slip
6, 4, 5, 2, 1, 2, 5, 3, 5, 5, 4, 6, 9,10,12,11,10, 9, &
5, 5, 3, 2, 2, 1, 6, 5, 4, 5, 6, 4, 11,12,10, 9,10, 9, &
3, 5, 5, 4, 5, 6, 1, 2, 2, 4, 6, 5, 10, 9,10, 9,11,12, &
5, 4, 6, 5, 3, 5, 2, 1, 2, 6, 4, 5, 10, 9,12,11, 9,10, &
5, 6, 4, 6, 5, 4, 2, 2, 1, 5, 5, 3, 12,11,10, 9, 9,10, &
4, 5, 6, 3, 5, 5, 4, 6, 5, 1, 2, 2, 10, 9, 9,10,12,11, &
5, 3, 5, 5, 4, 6, 6, 4, 5, 2, 1, 2, 10, 9,11,12,10, 9, &
6, 5, 4, 5, 6, 4, 5, 5, 3, 2, 2, 1, 12,11, 9,10,10, 9, &
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9, 9,11, 9, 9,11,10,10,12,10,10,12, 1, 7, 8, 8, 8, 8, &
10,10,12,10,10,12, 9, 9,11, 9, 9,11, 7, 1, 8, 8, 8, 8, &
9,11, 9,10,12,10,10,12,10, 9,11, 9, 8, 8, 1, 7, 8, 8, &
10,12,10, 9,11, 9, 9,11, 9,10,12,10, 8, 8, 7, 1, 8, 8, &
11, 9, 9,12,10,10,11, 9, 9,12,10,10, 8, 8, 8, 8, 1, 7, &
12,10,10,11, 9, 9,12,10,10,11, 9, 9, 8, 8, 8, 8, 7, 1 &
],pInt),shape(LATTICE_FCC_INTERACTIONSLIPSLIP),order=[2,1]) !< Slip--slip interaction types for fcc
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!< 1: self interaction
!< 2: coplanar interaction
!< 3: collinear interaction
!< 4: Hirth locks
!< 5: glissile junctions
!< 6: Lomer locks
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!< 7: crossing (similar to Hirth locks in <110>{111} for two {110} planes)
!< 8: similar to Lomer locks in <110>{111} for two {110} planes
!< 9: similar to Lomer locks in <110>{111} btw one {110} and one {111} plane
!<10: similar to glissile junctions in <110>{111} btw one {110} and one {111} plane
!<11: crossing btw one {110} and one {111} plane
!<12: collinear btw one {110} and one {111} plane
real(pReal), dimension(3+3,LATTICE_fcc_Ncleavage), parameter, private :: &
LATTICE_fcc_systemCleavage = reshape(real([&
! Cleavage direction Plane normal
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0, 1, 0, 1, 0, 0, &
0, 0, 1, 0, 1, 0, &
1, 0, 0, 0, 0, 1, &
0, 1,-1, 1, 1, 1, &
0,-1,-1, -1,-1, 1, &
-1, 0,-1, 1,-1,-1, &
0, 1, 1, -1, 1,-1 &
],pReal),shape(LATTICE_FCC_SYSTEMCLEAVAGE))
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!--------------------------------------------------------------------------------------------------
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! body centered cubic
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integer(pInt), dimension(LATTICE_maxNslipFamily), parameter, public :: &
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LATTICE_BCC_NSLIPSYSTEM = int([ 12, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], pInt) !< # of slip systems per family for bcc
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integer(pInt), dimension(1), parameter, public :: &
LATTICE_BCC_NTWINSYSTEM = int([12], pInt) !< # of twin systems per family for bcc
integer(pInt), dimension(LATTICE_maxNcleavageFamily), parameter, public :: &
LATTICE_bcc_NcleavageSystem = int([3, 6, 0],pInt) !< # of cleavage systems per family for bcc
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integer(pInt), parameter, private :: &
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LATTICE_BCC_NSLIP = sum(LATTICE_BCC_NSLIPSYSTEM), & !< total # of slip systems for bcc
LATTICE_BCC_NTWIN = sum(LATTICE_BCC_NTWINSYSTEM), & !< total # of twin systems for bcc
LATTICE_bcc_NnonSchmid = 6_pInt, & !< total # of non-Schmid contributions for bcc (A. Koester, A. Ma, A. Hartmaier 2012)
LATTICE_bcc_Ncleavage = sum(lattice_bcc_NcleavageSystem) !< total # of cleavage systems for bcc
real(pReal), dimension(3+3,LATTICE_BCC_NSLIP), parameter, private :: &
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LATTICE_bcc_systemSlip = reshape(real([&
! Slip direction Plane normal
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! Slip system <111>{110}
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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, &
-1, 1, 1, 1, 1, 0, &
-1, 1,-1, 1, 1, 0, &
1, 1, 1, -1, 1, 0, &
1, 1,-1, -1, 1, 0, &
! Slip system <111>{112}
-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 &
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],pReal),shape(LATTICE_BCC_SYSTEMSLIP))
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character(len=*), dimension(2), parameter, public :: LATTICE_BCC_SLIPFAMILY_NAME = &
['<1 -1 1>{0 1 1}', &
'<1 -1 1>{2 1 1}']
real(pReal), dimension(3+3,LATTICE_BCC_NTWIN), parameter, private :: &
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LATTICE_bcc_systemTwin = reshape(real([&
! Twin system <111>{112}
-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 &
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],pReal),shape(LATTICE_BCC_SYSTEMTWIN))
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character(len=*), dimension(1), parameter, public :: LATTICE_BCC_TWINFAMILY_NAME = &
['<1 1 1>{2 1 1}']
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integer(pInt), dimension(LATTICE_BCC_NSLIP,LATTICE_BCC_NSLIP), parameter, public :: &
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LATTICE_bcc_interactionSlipSlip = reshape(int( [&
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1,2,6,6,5,4,4,3,4,3,5,4, 6,6,4,3,3,4,6,6,4,3,6,6, & ! ---> slip
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2,1,6,6,4,3,5,4,5,4,4,3, 6,6,3,4,4,3,6,6,3,4,6,6, & ! |
6,6,1,2,4,5,3,4,4,5,3,4, 4,3,6,6,6,6,3,4,6,6,4,3, & ! |
6,6,2,1,3,4,4,5,3,4,4,5, 3,4,6,6,6,6,4,3,6,6,3,4, & ! v slip
5,4,4,3,1,2,6,6,3,4,5,4, 3,6,4,6,6,4,6,3,4,6,3,6, &
4,3,5,4,2,1,6,6,4,5,4,3, 4,6,3,6,6,3,6,4,3,6,4,6, &
4,5,3,4,6,6,1,2,5,4,3,4, 6,3,6,4,4,6,3,6,6,4,6,3, &
3,4,4,5,6,6,2,1,4,3,4,5, 6,4,6,3,3,6,4,6,6,3,6,4, &
4,5,4,3,3,4,5,4,1,2,6,6, 3,6,6,4,4,6,6,3,6,4,3,6, &
3,4,5,4,4,5,4,3,2,1,6,6, 4,6,6,3,3,6,6,4,6,3,4,6, &
5,4,3,4,5,4,3,4,6,6,1,2, 6,3,4,6,6,4,3,6,4,6,6,3, &
4,3,4,5,4,3,4,5,6,6,2,1, 6,4,3,6,6,3,4,6,3,6,6,4, &
!
6,6,4,3,3,4,6,6,3,4,6,6, 1,5,6,6,5,6,6,3,5,6,3,6, &
6,6,3,4,6,6,3,4,6,6,3,4, 5,1,6,6,6,5,3,6,6,5,6,3, &
4,3,6,6,4,3,6,6,6,6,4,3, 6,6,1,5,6,3,5,6,3,6,5,6, &
3,4,6,6,6,6,4,3,4,3,6,6, 6,6,5,1,3,6,6,5,6,3,6,5, &
3,4,6,6,6,6,4,3,4,3,6,6, 5,6,6,3,1,6,5,6,5,3,6,6, &
4,3,6,6,4,3,6,6,6,6,4,3, 6,5,3,6,6,1,6,5,3,5,6,6, &
6,6,3,4,6,6,3,4,6,6,3,4, 6,3,5,6,5,6,1,6,6,6,5,3, &
6,6,4,3,3,4,6,6,3,4,6,6, 3,6,6,5,6,5,6,1,6,6,3,5, &
4,3,6,6,4,3,6,6,6,6,4,3, 5,6,3,6,5,3,6,6,1,6,6,5, &
3,4,6,6,6,6,4,3,4,3,6,6, 6,5,6,3,3,5,6,6,6,1,5,6, &
6,6,4,3,3,4,6,6,3,4,6,6, 3,6,5,6,6,6,5,3,6,5,1,6, &
6,6,3,4,6,6,3,4,6,6,3,4, 6,3,6,5,6,6,3,5,5,6,6,1 &
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],pInt),shape(LATTICE_BCC_INTERACTIONSLIPSLIP),order=[2,1]) !< Slip--slip interaction types for bcc from Queyreau et al. Int J Plast 25 (2009) 361377
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!< 1: self interaction
!< 2: coplanar interaction
!< 3: collinear interaction
!< 4: mixed-asymmetrical junction
!< 5: mixed-symmetrical junction
!< 6: edge junction
real(pReal), dimension(3+3,LATTICE_bcc_Ncleavage), parameter, private :: &
LATTICE_bcc_systemCleavage = reshape(real([&
! Cleavage direction Plane normal
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0, 1, 0, 1, 0, 0, &
0, 0, 1, 0, 1, 0, &
1, 0, 0, 0, 0, 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, 1, 0, &
1, 1, 1, -1, 1, 0 &
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],pReal),shape(LATTICE_BCC_SYSTEMCLEAVAGE))
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!--------------------------------------------------------------------------------------------------
! hexagonal
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integer(pInt), dimension(LATTICE_maxNslipFamily), parameter, public :: &
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LATTICE_HEX_NSLIPSYSTEM = int([ 3, 3, 3, 6, 12, 6, 0, 0, 0, 0, 0, 0, 0],pInt) !< # of slip systems per family for hex
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integer(pInt), dimension(4), parameter, public :: &
LATTICE_HEX_NTWINSYSTEM = int([ 6, 6, 6, 6],pInt) !< # of slip systems per family for hex
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integer(pInt), dimension(LATTICE_maxNcleavageFamily), parameter, public :: &
LATTICE_hex_NcleavageSystem = int([3, 0, 0],pInt) !< # of cleavage systems per family for hex
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integer(pInt), parameter, private :: &
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LATTICE_HEX_NSLIP = sum(LATTICE_HEX_NSLIPSystem), & !< total # of slip systems for hex
LATTICE_HEX_NTWIN = sum(LATTICE_HEX_NTWINSYSTEM), & !< total # of twin systems for hex
LATTICE_hex_Ncleavage = sum(lattice_hex_NcleavageSystem) !< total # of cleavage systems for hex
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real(pReal), dimension(4+4,LATTICE_HEX_NSLIP), parameter, private :: &
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LATTICE_hex_systemSlip = reshape(real([&
! Slip direction Plane normal
! Basal systems <11.0>{00.1} (independent of c/a-ratio, Bravais notation (4 coordinate base))
2, -1, -1, 0, 0, 0, 0, 1, &
-1, 2, -1, 0, 0, 0, 0, 1, &
-1, -1, 2, 0, 0, 0, 0, 1, &
! 1st type prismatic systems <11.0>{10.0} (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, &
! 2nd type prismatic systems <10.0>{11.0} -- a slip; plane normals independent of c/a-ratio
0, 1, -1, 0, 2, -1, -1, 0, &
-1, 0, 1, 0, -1, 2, -1, 0, &
1, -1, 0, 0, -1, -1, 2, 0, &
! 1st type 1st order pyramidal systems <11.0>{-11.1} -- plane normals depend on the c/a-ratio
2, -1, -1, 0, 0, 1, -1, 1, &
-1, 2, -1, 0, -1, 0, 1, 1, &
-1, -1, 2, 0, 1, -1, 0, 1, &
1, 1, -2, 0, -1, 1, 0, 1, &
-2, 1, 1, 0, 0, -1, 1, 1, &
1, -2, 1, 0, 1, 0, -1, 1, &
! pyramidal system: c+a slip <11.3>{-10.1} -- plane normals depend on the c/a-ratio
2, -1, -1, 3, -1, 1, 0, 1, &
1, -2, 1, 3, -1, 1, 0, 1, &
-1, -1, 2, 3, 1, 0, -1, 1, &
-2, 1, 1, 3, 1, 0, -1, 1, &
-1, 2, -1, 3, 0, -1, 1, 1, &
1, 1, -2, 3, 0, -1, 1, 1, &
-2, 1, 1, 3, 1, -1, 0, 1, &
-1, 2, -1, 3, 1, -1, 0, 1, &
1, 1, -2, 3, -1, 0, 1, 1, &
2, -1, -1, 3, -1, 0, 1, 1, &
1, -2, 1, 3, 0, 1, -1, 1, &
-1, -1, 2, 3, 0, 1, -1, 1, &
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! pyramidal system: c+a slip <11.3>{-1-1.2} -- as for hexagonal ice (Castelnau et al. 1996, similar to twin system found below)
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2, -1, -1, 3, -2, 1, 1, 2, & ! sorted according to similar twin system
-1, 2, -1, 3, 1, -2, 1, 2, & ! <11.3>{-1-1.2} shear = 2((c/a)^2-2)/(3 c/a)
-1, -1, 2, 3, 1, 1, -2, 2, &
-2, 1, 1, 3, 2, -1, -1, 2, &
1, -2, 1, 3, -1, 2, -1, 2, &
1, 1, -2, 3, -1, -1, 2, 2 &
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],pReal),shape(LATTICE_HEX_SYSTEMSLIP)) !< slip systems for hex sorted by A. Alankar & P. Eisenlohr
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character(len=*), dimension(6), parameter, public :: LATTICE_HEX_SLIPFAMILY_NAME = &
['<1 1 . 1>{0 0 . 1} ', &
'<1 1 . 1>{1 0 . 0} ', &
'<1 0 . 0>{1 1 . 0} ', &
'<1 1 . 0>{-1 1 . 1} ', &
'<1 1 . 3>{-1 0 . 1} ', &
'<1 1 . 3>{-1 -1 . 2}']
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real(pReal), dimension(4+4,LATTICE_hex_Ntwin), parameter, private :: &
LATTICE_hex_systemTwin = reshape(real([&
! Compression or Tension =f(twinning shear=f(c/a)) for each metal ! (according to Yoo 1981)
1, -1, 0, 1, -1, 1, 0, 2, & ! <-10.1>{10.2} shear = (3-(c/a)^2)/(sqrt(3) c/a)
-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, &
!
2, -1, -1, 6, -2, 1, 1, 1, & ! <11.6>{-1-1.1} shear = 1/(c/a)
-1, 2, -1, 6, 1, -2, 1, 1, &
-1, -1, 2, 6, 1, 1, -2, 1, &
-2, 1, 1, 6, 2, -1, -1, 1, &
1, -2, 1, 6, -1, 2, -1, 1, &
1, 1, -2, 6, -1, -1, 2, 1, &
!
-1, 1, 0, -2, -1, 1, 0, 1, & !! <10.-2>{10.1} shear = (4(c/a)^2-9)/(4 sqrt(3) c/a)
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, &
!
2, -1, -1, -3, 2, -1, -1, 2, & ! <11.-3>{11.2} shear = 2((c/a)^2-2)/(3 c/a)
-1, 2, -1, -3, -1, 2, -1, 2, &
-1, -1, 2, -3, -1, -1, 2, 2, &
-2, 1, 1, -3, -2, 1, 1, 2, &
1, -2, 1, -3, 1, -2, 1, 2, &
1, 1, -2, -3, 1, 1, -2, 2 &
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],pReal),shape(LATTICE_HEX_SYSTEMTWIN)) !< twin systems for hex, order follows Prof. Tom Bieler's scheme; but numbering in data was restarted from 1
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character(len=*), dimension(4), parameter, public :: LATTICE_HEX_TWINFAMILY_NAME = &
['<-1 0 . 1>{1 0 . 2} ', &
'<1 1 . 6>{-1 -1 . 1}', &
'<1 0 . -2>{1 0 . 1} ', &
'<1 1 . -3>{1 1 . 2} ']
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integer(pInt), dimension(LATTICE_HEX_NSLIP,LATTICE_HEX_NSLIP), parameter, public :: &
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LATTICE_hex_interactionSlipSlip = reshape(int( [&
1, 2, 2, 3, 3, 3, 7, 7, 7, 13,13,13,13,13,13, 21,21,21,21,21,21,21,21,21,21,21,21, 31,31,31,31,31,31, & ! ---> slip
2, 1, 2, 3, 3, 3, 7, 7, 7, 13,13,13,13,13,13, 21,21,21,21,21,21,21,21,21,21,21,21, 31,31,31,31,31,31, & ! |
2, 2, 1, 3, 3, 3, 7, 7, 7, 13,13,13,13,13,13, 21,21,21,21,21,21,21,21,21,21,21,21, 31,31,31,31,31,31, & ! |
! v slip
6, 6, 6, 4, 5, 5, 8, 8, 8, 14,14,14,14,14,14, 22,22,22,22,22,22,22,22,22,22,22,22, 32,32,32,32,32,32, &
6, 6, 6, 5, 4, 5, 8, 8, 8, 14,14,14,14,14,14, 22,22,22,22,22,22,22,22,22,22,22,22, 32,32,32,32,32,32, &
6, 6, 6, 5, 5, 4, 8, 8, 8, 14,14,14,14,14,14, 22,22,22,22,22,22,22,22,22,22,22,22, 32,32,32,32,32,32, &
!
12,12,12, 11,11,11, 9,10,10, 15,15,15,15,15,15, 23,23,23,23,23,23,23,23,23,23,23,23, 33,33,33,33,33,33, &
12,12,12, 11,11,11, 10, 9,10, 15,15,15,15,15,15, 23,23,23,23,23,23,23,23,23,23,23,23, 33,33,33,33,33,33, &
12,12,12, 11,11,11, 10,10, 9, 15,15,15,15,15,15, 23,23,23,23,23,23,23,23,23,23,23,23, 33,33,33,33,33,33, &
!
20,20,20, 19,19,19, 18,18,18, 16,17,17,17,17,17, 24,24,24,24,24,24,24,24,24,24,24,24, 34,34,34,34,34,34, &
20,20,20, 19,19,19, 18,18,18, 17,16,17,17,17,17, 24,24,24,24,24,24,24,24,24,24,24,24, 34,34,34,34,34,34, &
20,20,20, 19,19,19, 18,18,18, 17,17,16,17,17,17, 24,24,24,24,24,24,24,24,24,24,24,24, 34,34,34,34,34,34, &
20,20,20, 19,19,19, 18,18,18, 17,17,17,16,17,17, 24,24,24,24,24,24,24,24,24,24,24,24, 34,34,34,34,34,34, &
20,20,20, 19,19,19, 18,18,18, 17,17,17,17,16,17, 24,24,24,24,24,24,24,24,24,24,24,24, 34,34,34,34,34,34, &
20,20,20, 19,19,19, 18,18,18, 17,17,17,17,17,16, 24,24,24,24,24,24,24,24,24,24,24,24, 34,34,34,34,34,34, &
!
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 25,26,26,26,26,26,26,26,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,25,26,26,26,26,26,26,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,25,26,26,26,26,26,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,25,26,26,26,26,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,25,26,26,26,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,26,25,26,26,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,26,26,25,26,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,26,26,26,25,26,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,26,26,26,26,25,26,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,26,26,26,26,26,25,26,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,26,26,26,26,26,26,25,26, 35,35,35,35,35,35, &
30,30,30, 29,29,29, 28,28,28, 27,27,27,27,27,27, 26,26,26,26,26,26,26,26,26,26,26,25, 35,35,35,35,35,35, &
!
42,42,42, 41,41,41, 40,40,40, 39,39,39,39,39,39, 38,38,38,38,38,38,38,38,38,38,38,38, 36,37,37,37,37,37, &
42,42,42, 41,41,41, 40,40,40, 39,39,39,39,39,39, 38,38,38,38,38,38,38,38,38,38,38,38, 37,36,37,37,37,37, &
42,42,42, 41,41,41, 40,40,40, 39,39,39,39,39,39, 38,38,38,38,38,38,38,38,38,38,38,38, 37,37,36,37,37,37, &
42,42,42, 41,41,41, 40,40,40, 39,39,39,39,39,39, 38,38,38,38,38,38,38,38,38,38,38,38, 37,37,37,36,37,37, &
42,42,42, 41,41,41, 40,40,40, 39,39,39,39,39,39, 38,38,38,38,38,38,38,38,38,38,38,38, 37,37,37,37,36,37, &
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42,42,42, 41,41,41, 40,40,40, 39,39,39,39,39,39, 38,38,38,38,38,38,38,38,38,38,38,38, 37,37,37,37,37,36 &
],pInt),shape(LATTICE_HEX_INTERACTIONSLIPSLIP),order=[2,1]) !< Slip--slip interaction types for hex (onion peel naming scheme)
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real(pReal), dimension(4+4,LATTICE_hex_Ncleavage), parameter, private :: &
LATTICE_hex_systemCleavage = reshape(real([&
! Cleavage direction Plane normal
2,-1,-1, 0, 0, 0, 0, 1, &
0, 0, 0, 1, 2,-1,-1, 0, &
0, 0, 0, 1, 0, 1,-1, 0 &
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],pReal),shape(LATTICE_HEX_SYSTEMCLEAVAGE))
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!--------------------------------------------------------------------------------------------------
! body centered tetragonal
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integer(pInt), dimension(LATTICE_maxNslipFamily), parameter, public :: &
LATTICE_bct_NslipSystem = int([2, 2, 2, 4, 2, 4, 2, 2, 4, 8, 4, 8, 8 ],pInt) !< # of slip systems per family for bct (Sn) Bieler J. Electr Mater 2009
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integer(pInt), parameter, private :: &
LATTICE_bct_Nslip = sum(lattice_bct_NslipSystem) !< total # of slip systems for bct
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real(pReal), dimension(3+3,LATTICE_bct_Nslip), parameter, private :: &
LATTICE_bct_systemSlip = reshape(real([&
! Slip direction Plane normal
! Slip family 1 {100)<001] (Bravais notation {hkl)<uvw] for bct c/a = 0.5456)
0, 0, 1, 1, 0, 0, &
0, 0, 1, 0, 1, 0, &
! Slip family 2 {110)<001]
0, 0, 1, 1, 1, 0, &
0, 0, 1, -1, 1, 0, &
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! slip family 3 {100)<010]
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0, 1, 0, 1, 0, 0, &
1, 0, 0, 0, 1, 0, &
! Slip family 4 {110)<1-11]/2
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1,-1, 1, 1, 1, 0, &
1,-1,-1, 1, 1, 0, &
-1,-1,-1, -1, 1, 0, &
-1,-1, 1, -1, 1, 0, &
! Slip family 5 {110)<1-10]
1, -1, 0, 1, 1, 0, &
1, 1, 0, 1,-1, 0, &
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! Slip family 6 {100)<011]
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0, 1, 1, 1, 0, 0, &
0,-1, 1, 1, 0, 0, &
-1, 0, 1, 0, 1, 0, &
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1, 0, 1, 0, 1, 0, &
! Slip family 7 {001)<010]
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0, 1, 0, 0, 0, 1, &
1, 0, 0, 0, 0, 1, &
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! Slip family 8 {001)<110]
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1, 1, 0, 0, 0, 1, &
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-1, 1, 0, 0, 0, 1, &
! Slip family 9 {011)<01-1]
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0, 1,-1, 0, 1, 1, &
0,-1,-1, 0,-1, 1, &
-1, 0,-1, -1, 0, 1, &
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1, 0,-1, 1, 0, 1, &
! Slip family 10 {011)<1-11]/2
1,-1, 1, 0, 1, 1, &
1, 1,-1, 0, 1, 1, &
1, 1, 1, 0, 1,-1, &
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-1, 1, 1, 0, 1,-1, &
1,-1,-1, 1, 0, 1, &
-1,-1, 1, 1, 0, 1, &
1, 1, 1, 1, 0,-1, &
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1,-1, 1, 1, 0,-1, &
! Slip family 11 {011)<100]
1, 0, 0, 0, 1, 1, &
1, 0, 0, 0, 1,-1, &
0, 1, 0, 1, 0, 1, &
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0, 1, 0, 1, 0,-1, &
! Slip family 12 {211)<01-1]
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0, 1,-1, 2, 1, 1, &
0,-1,-1, 2,-1, 1, &
1, 0,-1, 1, 2, 1, &
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-1, 0,-1, -1, 2, 1, &
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0, 1,-1, -2, 1, 1, &
0,-1,-1, -2,-1, 1, &
-1, 0,-1, -1,-2, 1, &
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1, 0,-1, 1,-2, 1, &
! Slip family 13 {211)<-111]/2
-1, 1, 1, 2, 1, 1, &
-1,-1, 1, 2,-1, 1, &
1,-1, 1, 1, 2, 1, &
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-1,-1, 1, -1, 2, 1, &
1, 1, 1, -2, 1, 1, &
1,-1, 1, -2,-1, 1, &
-1, 1, 1, -1,-2, 1, &
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1, 1, 1, 1,-2, 1 &
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],pReal),[ 3_pInt + 3_pInt,LATTICE_bct_Nslip]) !< slip systems for bct sorted by Bieler
character(len=*), dimension(13), parameter, public :: LATTICE_BCT_SLIPFAMILY_NAME = &
['{1 0 0)<0 0 1] ', &
'{1 1 0)<0 0 1] ', &
'{1 0 0)<0 1 0] ', &
'{1 1 0)<1 -1 1]', &
'{1 1 0)<1 -1 0]', &
'{1 0 0)<0 1 1] ', &
'{0 0 1)<0 1 0] ', &
'{0 0 1)<1 1 0] ', &
'{0 1 1)<0 1 -1]', &
'{0 1 1)<1 -1 1]', &
'{0 1 1)<1 0 0] ', &
'{2 1 1)<0 1 -1]', &
'{2 1 1)<-1 1 1]']
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integer(pInt), dimension(LATTICE_bct_Nslip,LATTICE_bct_Nslip), parameter, public :: &
LATTICE_bct_interactionSlipSlip = reshape(int( [&
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, &
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, &
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!
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, &
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, &
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!
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, &
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!
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, &
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!
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, &
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!
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, &
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!
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, &
2015-07-02 15:15:25 +05:30
!
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, &
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!
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, &
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!
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 &
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],pInt),[lattice_bct_Nslip,lattice_bct_Nslip],order=[2,1])
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!--------------------------------------------------------------------------------------------------
! isotropic
integer(pInt), dimension(LATTICE_maxNcleavageFamily), parameter, public :: &
LATTICE_iso_NcleavageSystem = int([3, 0, 0],pInt) !< # of cleavage systems per family for iso
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integer(pInt), parameter, private :: &
LATTICE_iso_Ncleavage = sum(lattice_iso_NcleavageSystem) !< total # of cleavage systems for iso
real(pReal), dimension(3+3,LATTICE_iso_Ncleavage), parameter, private :: &
LATTICE_iso_systemCleavage = reshape(real([&
! Cleavage direction Plane normal
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0, 1, 0, 1, 0, 0, &
0, 0, 1, 0, 1, 0, &
1, 0, 0, 0, 0, 1 &
],pReal),[ 3_pInt + 3_pInt,LATTICE_iso_Ncleavage])
!--------------------------------------------------------------------------------------------------
! orthorhombic
integer(pInt), dimension(LATTICE_maxNcleavageFamily), parameter, public :: &
LATTICE_ort_NcleavageSystem = int([1, 1, 1],pInt) !< # of cleavage systems per family for ortho
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integer(pInt), parameter, private :: &
LATTICE_ort_Ncleavage = sum(lattice_ort_NcleavageSystem) !< total # of cleavage systems for ortho
real(pReal), dimension(3+3,LATTICE_ort_Ncleavage), parameter, private :: &
LATTICE_ort_systemCleavage = reshape(real([&
! Cleavage direction Plane normal
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0, 1, 0, 1, 0, 0, &
0, 0, 1, 0, 1, 0, &
1, 0, 0, 0, 0, 1 &
],pReal),[ 3_pInt + 3_pInt,LATTICE_ort_Ncleavage])
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! BEGIN DEPRECATED
integer(pInt), parameter, public :: &
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LATTICE_maxNslip = max(LATTICE_FCC_NSLIP,LATTICE_BCC_NSLIP,LATTICE_HEX_NSLIP, &
LATTICE_bct_Nslip), & !< max # of slip systems over lattice structures
LATTICE_maxNnonSchmid = LATTICE_bcc_NnonSchmid, & !< max # of non-Schmid contributions over lattice structures
LATTICE_maxNcleavage = max(LATTICE_fcc_Ncleavage,LATTICE_bcc_Ncleavage, &
LATTICE_hex_Ncleavage, &
LATTICE_iso_Ncleavage,LATTICE_ort_Ncleavage), & !< max # of cleavage systems over lattice structures
LATTICE_maxNinteraction = 182_pInt
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!END DEPRECATED
real(pReal), dimension(:,:,:), allocatable, public, protected :: &
lattice_C66
real(pReal), dimension(:,:,:,:,:), allocatable, public, protected :: &
lattice_C3333
real(pReal), dimension(:), allocatable, public, protected :: &
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lattice_mu, lattice_nu
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! SHOULD NOT BE PART OF LATTICE BEGIN
real(pReal), dimension(:,:,:,:), allocatable, public, protected :: & ! with higher-order parameters (e.g. temperature-dependent)
lattice_thermalExpansion33
real(pReal), dimension(:,:,:), allocatable, public, protected :: &
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lattice_thermalConductivity33, &
lattice_damageDiffusion33
real(pReal), dimension(:), allocatable, public, protected :: &
lattice_damageMobility, &
lattice_massDensity, &
lattice_specificHeat, &
lattice_referenceTemperature
! SHOULD NOT BE PART OF LATTICE END
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enum, bind(c)
enumerator :: LATTICE_undefined_ID, &
LATTICE_iso_ID, &
LATTICE_fcc_ID, &
LATTICE_bcc_ID, &
LATTICE_hex_ID, &
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LATTICE_bct_ID, &
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LATTICE_ort_ID
end enum
integer(kind(LATTICE_undefined_ID)), dimension(:), allocatable, public, protected :: &
lattice_structure, trans_lattice_structure
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public :: &
lattice_init, &
lattice_qDisorientation, &
LATTICE_fcc_ID, &
LATTICE_bcc_ID, &
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LATTICE_bct_ID, &
LATTICE_hex_ID, &
lattice_SchmidMatrix_slip, &
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lattice_SchmidMatrix_twin, &
lattice_SchmidMatrix_trans, &
lattice_SchmidMatrix_cleavage, &
lattice_nonSchmidMatrix, &
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lattice_interaction_SlipSlip, &
lattice_interaction_TwinTwin, &
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lattice_interaction_TransTrans, &
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lattice_interaction_SlipTwin, &
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lattice_interaction_SlipTrans, &
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lattice_interaction_TwinSlip, &
lattice_forestProjection, &
lattice_characteristicShear_Twin, &
lattice_C66_twin, &
lattice_C66_trans
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contains
!--------------------------------------------------------------------------------------------------
!> @brief Module initialization
!--------------------------------------------------------------------------------------------------
subroutine lattice_init
#if defined(__GFORTRAN__) || __INTEL_COMPILER >= 1800
use, intrinsic :: iso_fortran_env, only: &
compiler_version, &
compiler_options
#endif
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use IO, only: &
IO_error, &
IO_timeStamp
use config, only: &
config_phase
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implicit none
integer(pInt) :: Nphases
character(len=65536) :: &
tag = ''
integer(pInt) :: i,p
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real(pReal), dimension(:), allocatable :: &
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temp, &
CoverA !< c/a ratio for low symmetry type lattice
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write(6,'(/,a)') ' <<<+- lattice init -+>>>'
write(6,'(a15,a)') ' Current time: ',IO_timeStamp()
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#include "compilation_info.f90"
Nphases = size(config_phase)
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allocate(lattice_structure(Nphases),source = LATTICE_undefined_ID)
allocate(trans_lattice_structure(Nphases),source = LATTICE_undefined_ID)
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allocate(lattice_C66(6,6,Nphases), source=0.0_pReal)
allocate(lattice_C3333(3,3,3,3,Nphases), source=0.0_pReal)
allocate(lattice_thermalExpansion33 (3,3,3,Nphases), source=0.0_pReal) ! constant, linear, quadratic coefficients
allocate(lattice_thermalConductivity33 (3,3,Nphases), source=0.0_pReal)
allocate(lattice_damageDiffusion33 (3,3,Nphases), source=0.0_pReal)
allocate(lattice_damageMobility ( Nphases), source=0.0_pReal)
allocate(lattice_massDensity ( Nphases), source=0.0_pReal)
allocate(lattice_specificHeat ( Nphases), source=0.0_pReal)
allocate(lattice_referenceTemperature ( Nphases), source=300.0_pReal)
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allocate(lattice_mu(Nphases), source=0.0_pReal)
allocate(lattice_nu(Nphases), source=0.0_pReal)
allocate(lattice_NnonSchmid(Nphases), source=0_pInt)
allocate(lattice_Sslip(3,3,1+2*lattice_maxNnonSchmid,lattice_maxNslip,Nphases),source=0.0_pReal)
allocate(lattice_Sslip_v(6,1+2*lattice_maxNnonSchmid,lattice_maxNslip,Nphases),source=0.0_pReal)
allocate(lattice_NslipSystem(lattice_maxNslipFamily,Nphases),source=0_pInt)
allocate(lattice_interactionSlipSlip(lattice_maxNslip,lattice_maxNslip,Nphases),source=0_pInt) ! other:me
allocate(lattice_Scleavage(3,3,3,lattice_maxNslip,Nphases),source=0.0_pReal)
allocate(lattice_Scleavage_v(6,3,lattice_maxNslip,Nphases),source=0.0_pReal)
allocate(lattice_NcleavageSystem(lattice_maxNcleavageFamily,Nphases),source=0_pInt)
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allocate(CoverA(Nphases),source=0.0_pReal)
allocate(lattice_sd(3,lattice_maxNslip,Nphases),source=0.0_pReal)
allocate(lattice_st(3,lattice_maxNslip,Nphases),source=0.0_pReal)
allocate(lattice_sn(3,lattice_maxNslip,Nphases),source=0.0_pReal)
do p = 1, size(config_phase)
tag = config_phase(p)%getString('lattice_structure')
select case(trim(tag))
case('iso','isotropic')
lattice_structure(p) = LATTICE_iso_ID
case('fcc')
lattice_structure(p) = LATTICE_fcc_ID
case('bcc')
lattice_structure(p) = LATTICE_bcc_ID
case('hex','hexagonal')
lattice_structure(p) = LATTICE_hex_ID
case('bct')
lattice_structure(p) = LATTICE_bct_ID
case('ort','orthorhombic')
lattice_structure(p) = LATTICE_ort_ID
end select
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tag = 'undefined'
tag = config_phase(p)%getString('trans_lattice_structure',defaultVal=tag)
select case(trim(tag))
case('bcc')
trans_lattice_structure(p) = LATTICE_bcc_ID
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case('hex','hexagonal')
trans_lattice_structure(p) = LATTICE_hex_ID
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end select
lattice_C66(1,1,p) = config_phase(p)%getFloat('c11',defaultVal=0.0_pReal)
lattice_C66(1,2,p) = config_phase(p)%getFloat('c12',defaultVal=0.0_pReal)
lattice_C66(1,3,p) = config_phase(p)%getFloat('c13',defaultVal=0.0_pReal)
lattice_C66(2,2,p) = config_phase(p)%getFloat('c22',defaultVal=0.0_pReal)
lattice_C66(2,3,p) = config_phase(p)%getFloat('c23',defaultVal=0.0_pReal)
lattice_C66(3,3,p) = config_phase(p)%getFloat('c33',defaultVal=0.0_pReal)
lattice_C66(4,4,p) = config_phase(p)%getFloat('c44',defaultVal=0.0_pReal)
lattice_C66(5,5,p) = config_phase(p)%getFloat('c55',defaultVal=0.0_pReal)
lattice_C66(6,6,p) = config_phase(p)%getFloat('c66',defaultVal=0.0_pReal)
CoverA(p) = config_phase(p)%getFloat('c/a',defaultVal=0.0_pReal)
lattice_thermalConductivity33(1,1,p) = config_phase(p)%getFloat('thermal_conductivity11',defaultVal=0.0_pReal)
lattice_thermalConductivity33(2,2,p) = config_phase(p)%getFloat('thermal_conductivity22',defaultVal=0.0_pReal)
lattice_thermalConductivity33(3,3,p) = config_phase(p)%getFloat('thermal_conductivity33',defaultVal=0.0_pReal)
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temp = config_phase(p)%getFloats('thermal_expansion11',defaultVal=[0.0_pReal]) ! read up to three parameters (constant, linear, quadratic with T)
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lattice_thermalExpansion33(1,1,1:size(temp),p) = temp
temp = config_phase(p)%getFloats('thermal_expansion22',defaultVal=[0.0_pReal]) ! read up to three parameters (constant, linear, quadratic with T)
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lattice_thermalExpansion33(2,2,1:size(temp),p) = temp
temp = config_phase(p)%getFloats('thermal_expansion33',defaultVal=[0.0_pReal]) ! read up to three parameters (constant, linear, quadratic with T)
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lattice_thermalExpansion33(3,3,1:size(temp),p) = temp
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lattice_specificHeat(p) = config_phase(p)%getFloat( 'specific_heat',defaultVal=0.0_pReal)
lattice_massDensity(p) = config_phase(p)%getFloat( 'mass_density',defaultVal=0.0_pReal)
lattice_referenceTemperature(p) = config_phase(p)%getFloat( 'reference_temperature',defaultVal=0.0_pReal)
lattice_DamageDiffusion33(1,1,p) = config_phase(p)%getFloat( 'damage_diffusion11',defaultVal=0.0_pReal)
lattice_DamageDiffusion33(2,2,p) = config_phase(p)%getFloat( 'damage_diffusion22',defaultVal=0.0_pReal)
lattice_DamageDiffusion33(3,3,p) = config_phase(p)%getFloat( 'damage_diffusion33',defaultVal=0.0_pReal)
lattice_DamageMobility(p) = config_phase(p)%getFloat( 'damage_mobility',defaultVal=0.0_pReal)
enddo
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do i = 1_pInt,Nphases
if ((CoverA(i) < 1.0_pReal .or. CoverA(i) > 2.0_pReal) &
.and. lattice_structure(i) == LATTICE_hex_ID) call IO_error(131_pInt,el=i) ! checking physical significance of c/a
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if ((CoverA(i) > 2.0_pReal) &
.and. lattice_structure(i) == LATTICE_bct_ID) call IO_error(131_pInt,el=i) ! checking physical significance of c/a
call lattice_initializeStructure(i, CoverA(i))
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enddo
end subroutine lattice_init
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!--------------------------------------------------------------------------------------------------
!> @brief !!!!!!!DEPRECTATED!!!!!!
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!--------------------------------------------------------------------------------------------------
subroutine lattice_initializeStructure(myPhase,CoverA)
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use prec, only: &
tol_math_check
use math, only: &
math_crossproduct, &
math_tensorproduct33, &
math_mul33x33, &
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math_mul33x3, &
math_trace33, &
math_symmetric33, &
math_Mandel33to6, &
math_Mandel3333to66, &
math_Voigt66to3333, &
math_axisAngleToR, &
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INRAD, &
MATH_I3
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use IO, only: &
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IO_error, &
IO_warning
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implicit none
integer(pInt), intent(in) :: myPhase
real(pReal), intent(in) :: &
CoverA
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real(pReal), dimension(3) :: &
sdU, snU, &
np, nn
real(pReal), dimension(3,lattice_maxNslip) :: &
sd, sn
real(pReal), dimension(3,3,2,lattice_maxNnonSchmid,lattice_maxNslip) :: &
sns
integer(pInt) :: &
j, i, &
myNslip, myNcleavage
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lattice_C66(1:6,1:6,myPhase) = lattice_symmetrizeC66(lattice_structure(myPhase),&
lattice_C66(1:6,1:6,myPhase))
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lattice_mu(myPhase) = 0.2_pReal *( lattice_C66(1,1,myPhase) &
- lattice_C66(1,2,myPhase) &
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+ 3.0_pReal*lattice_C66(4,4,myPhase)) ! (C11iso-C12iso)/2 with C11iso=(3*C11+2*C12+4*C44)/5 and C12iso=(C11+4*C12-2*C44)/5
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lattice_nu(myPhase) = ( lattice_C66(1,1,myPhase) &
+ 4.0_pReal*lattice_C66(1,2,myPhase) &
- 2.0_pReal*lattice_C66(4,4,myPhase)) &
/( 4.0_pReal*lattice_C66(1,1,myPhase) &
+ 6.0_pReal*lattice_C66(1,2,myPhase) &
+ 2.0_pReal*lattice_C66(4,4,myPhase))! C12iso/(C11iso+C12iso) with C11iso=(3*C11+2*C12+4*C44)/5 and C12iso=(C11+4*C12-2*C44)/5
lattice_C3333(1:3,1:3,1:3,1:3,myPhase) = math_Voigt66to3333(lattice_C66(1:6,1:6,myPhase)) ! Literature data is Voigt
lattice_C66(1:6,1:6,myPhase) = math_Mandel3333to66(lattice_C3333(1:3,1:3,1:3,1:3,myPhase)) ! DAMASK uses Mandel
do i = 1_pInt, 6_pInt
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if (abs(lattice_C66(i,i,myPhase))<tol_math_check) &
call IO_error(135_pInt,el=i,ip=myPhase,ext_msg='matrix diagonal "el"ement of phase "ip"')
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enddo
forall (i = 1_pInt:3_pInt) &
lattice_thermalExpansion33 (1:3,1:3,i,myPhase) = lattice_symmetrize33(lattice_structure(myPhase),&
lattice_thermalExpansion33 (1:3,1:3,i,myPhase))
lattice_thermalConductivity33 (1:3,1:3,myPhase) = lattice_symmetrize33(lattice_structure(myPhase),&
lattice_thermalConductivity33 (1:3,1:3,myPhase))
lattice_DamageDiffusion33 (1:3,1:3,myPhase) = lattice_symmetrize33(lattice_structure(myPhase),&
lattice_DamageDiffusion33 (1:3,1:3,myPhase))
myNslip = 0_pInt
myNcleavage = 0_pInt
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select case(lattice_structure(myPhase))
!--------------------------------------------------------------------------------------------------
! fcc
case (LATTICE_fcc_ID)
myNslip = LATTICE_FCC_NSLIP
myNcleavage = lattice_fcc_Ncleavage
lattice_NslipSystem (1:lattice_maxNslipFamily,myPhase) = lattice_fcc_NslipSystem
lattice_NcleavageSystem(1:lattice_maxNcleavageFamily,myPhase) = lattice_fcc_NcleavageSystem
lattice_interactionSlipSlip(1:myNslip,1:myNslip,myPhase) = lattice_fcc_interactionSlipSlip
lattice_Scleavage(1:3,1:3,1:3,1:myNcleavage,myPhase) = &
lattice_SchmidMatrix_cleavage(lattice_fcc_ncleavageSystem,'fcc',covera)
do i = 1_pInt,myNslip
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sd(1:3,i) = lattice_fcc_systemSlip(1:3,i)
sn(1:3,i) = lattice_fcc_systemSlip(4:6,i)
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enddo
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!--------------------------------------------------------------------------------------------------
! bcc
case (LATTICE_bcc_ID)
myNslip = LATTICE_BCC_NSLIP
myNcleavage = lattice_bcc_Ncleavage
lattice_NslipSystem(1:lattice_maxNslipFamily,myPhase) = lattice_bcc_NslipSystem
lattice_NcleavageSystem(1:lattice_maxNcleavageFamily,myPhase) = lattice_bcc_NcleavageSystem
lattice_NnonSchmid(myPhase) = lattice_bcc_NnonSchmid
lattice_interactionSlipSlip(1:myNslip,1:myNslip,myPhase) = lattice_bcc_interactionSlipSlip
lattice_Scleavage(1:3,1:3,1:3,1:myNcleavage,myPhase) = &
lattice_SchmidMatrix_cleavage(lattice_bcc_ncleavagesystem,'bcc',covera)
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do i = 1_pInt,myNslip ! assign slip system vectors
sd(1:3,i) = lattice_bcc_systemSlip(1:3,i)
sn(1:3,i) = lattice_bcc_systemSlip(4:6,i)
sdU = sd(1:3,i) / norm2(sd(1:3,i))
snU = sn(1:3,i) / norm2(sn(1:3,i))
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! "np" and "nn" according to Gröger_etal2008, Acta Materialia 56 (2008) 54125425, table 1 (corresponds to their "n1" for positive and negative slip direction respectively)
np = math_mul33x3(math_axisAngleToR(sdU,60.0_pReal*INRAD), snU)
nn = math_mul33x3(math_axisAngleToR(-sdU,60.0_pReal*INRAD), snU)
! Schmid matrices with non-Schmid contributions according to Koester_etal2012, Acta Materialia 60 (2012) 38943901, eq. (17) ("n1" is replaced by either "np" or "nn" according to either positive or negative slip direction)
sns(1:3,1:3,1,1,i) = math_tensorproduct33(sdU, np)
sns(1:3,1:3,2,1,i) = math_tensorproduct33(-sdU, nn)
sns(1:3,1:3,1,2,i) = math_tensorproduct33(math_crossproduct(snU, sdU), snU)
sns(1:3,1:3,2,2,i) = math_tensorproduct33(math_crossproduct(snU, -sdU), snU)
sns(1:3,1:3,1,3,i) = math_tensorproduct33(math_crossproduct(np, sdU), np)
sns(1:3,1:3,2,3,i) = math_tensorproduct33(math_crossproduct(nn, -sdU), nn)
sns(1:3,1:3,1,4,i) = math_tensorproduct33(snU, snU)
sns(1:3,1:3,2,4,i) = math_tensorproduct33(snU, snU)
sns(1:3,1:3,1,5,i) = math_tensorproduct33(math_crossproduct(snU, sdU), math_crossproduct(snU, sdU))
sns(1:3,1:3,2,5,i) = math_tensorproduct33(math_crossproduct(snU, -sdU), math_crossproduct(snU, -sdU))
sns(1:3,1:3,1,6,i) = math_tensorproduct33(sdU, sdU)
sns(1:3,1:3,2,6,i) = math_tensorproduct33(-sdU, -sdU)
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enddo
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!--------------------------------------------------------------------------------------------------
! hex (including conversion from miller-bravais (a1=a2=a3=c) to miller (a, b, c) indices)
case (LATTICE_hex_ID)
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myNslip = LATTICE_HEX_NSLIP
myNcleavage = lattice_hex_Ncleavage
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lattice_NslipSystem(1:lattice_maxNslipFamily,myPhase) = LATTICE_HEX_NSLIPSystem
lattice_NcleavageSystem(1:lattice_maxNcleavageFamily,myPhase) = lattice_hex_NcleavageSystem
lattice_interactionSlipSlip(1:myNslip,1:myNslip,myPhase) = lattice_hex_interactionSlipSlip
lattice_Scleavage(1:3,1:3,1:3,1:myNcleavage,myPhase) = &
lattice_SchmidMatrix_cleavage(lattice_hex_ncleavagesystem,'hex',covera)
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do i = 1_pInt,myNslip ! assign slip system vectors
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sd(1,i) = lattice_hex_systemSlip(1,i)*1.5_pReal ! direction [uvtw]->[3u/2 (u+2v)*sqrt(3)/2 w*(c/a)]
sd(2,i) = (lattice_hex_systemSlip(1,i)+2.0_pReal*lattice_hex_systemSlip(2,i))*&
0.5_pReal*sqrt(3.0_pReal)
sd(3,i) = lattice_hex_systemSlip(4,i)*CoverA
sn(1,i) = lattice_hex_systemSlip(5,i) ! plane (hkil)->(h (h+2k)/sqrt(3) l/(c/a))
sn(2,i) = (lattice_hex_systemSlip(5,i)+2.0_pReal*lattice_hex_systemSlip(6,i))/sqrt(3.0_pReal)
sn(3,i) = lattice_hex_systemSlip(8,i)/CoverA
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enddo
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!--------------------------------------------------------------------------------------------------
! bct
case (LATTICE_bct_ID)
myNslip = lattice_bct_Nslip
lattice_NslipSystem(1:lattice_maxNslipFamily,myPhase) = lattice_bct_NslipSystem
lattice_interactionSlipSlip(1:myNslip,1:myNslip,myPhase) = lattice_bct_interactionSlipSlip
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do i = 1_pInt,myNslip ! assign slip system vectors
sd(1:2,i) = lattice_bct_systemSlip(1:2,i)
sd(3,i) = lattice_bct_systemSlip(3,i)*CoverA
sn(1:2,i) = lattice_bct_systemSlip(4:5,i)
sn(3,i) = lattice_bct_systemSlip(6,i)/CoverA
sdU = sd(1:3,i) / norm2(sd(1:3,i))
snU = sn(1:3,i) / norm2(sn(1:3,i))
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enddo
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!--------------------------------------------------------------------------------------------------
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! orthorhombic (no crystal plasticity)
case (LATTICE_ort_ID)
myNcleavage = lattice_ort_Ncleavage
lattice_NcleavageSystem(1:lattice_maxNcleavageFamily,myPhase) = lattice_ort_NcleavageSystem
lattice_Scleavage(1:3,1:3,1:3,1:myNcleavage,myPhase) = &
lattice_SchmidMatrix_cleavage(lattice_ort_NcleavageSystem,'ort',covera)
!--------------------------------------------------------------------------------------------------
! isotropic (no crystal plasticity)
case (LATTICE_iso_ID)
myNcleavage = lattice_iso_Ncleavage
lattice_NcleavageSystem(1:lattice_maxNcleavageFamily,myPhase) = lattice_iso_NcleavageSystem
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lattice_Scleavage(1:3,1:3,1:3,1:myNcleavage,myPhase) = &
lattice_SchmidMatrix_cleavage(lattice_iso_NcleavageSystem,'iso',covera)
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!--------------------------------------------------------------------------------------------------
! something went wrong
case default
call IO_error(130_pInt,ext_msg='lattice_initializeStructure')
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end select
do i = 1_pInt,myNslip ! store slip system vectors and Schmid matrix for my structure
lattice_sd(1:3,i,myPhase) = sd(1:3,i)/norm2(sd(1:3,i)) ! make unit vector
lattice_sn(1:3,i,myPhase) = sn(1:3,i)/norm2(sn(1:3,i)) ! make unit vector
lattice_st(1:3,i,myPhase) = math_crossproduct(lattice_sd(1:3,i,myPhase), &
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lattice_sn(1:3,i,myPhase))
lattice_Sslip(1:3,1:3,1,i,myPhase) = math_tensorproduct33(lattice_sd(1:3,i,myPhase), &
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lattice_sn(1:3,i,myPhase)) ! calculate Schmid matrix d \otimes n
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do j = 1_pInt,lattice_NnonSchmid(myPhase)
lattice_Sslip(1:3,1:3,2*j ,i,myPhase) = sns(1:3,1:3,1,j,i)
lattice_Sslip(1:3,1:3,2*j+1,i,myPhase) = sns(1:3,1:3,2,j,i)
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enddo
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do j = 1_pInt,1_pInt+2_pInt*lattice_NnonSchmid(myPhase)
lattice_Sslip_v(1:6,j,i,myPhase) = &
math_Mandel33to6(math_symmetric33(lattice_Sslip(1:3,1:3,j,i,myPhase)))
enddo
enddo
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do i = 1_pInt,myNcleavage ! store slip system vectors and Schmid matrix for my structure
do j = 1_pInt,3_pInt
lattice_Scleavage_v(1:6,j,i,myPhase) = &
math_Mandel33to6(math_symmetric33(lattice_Scleavage(1:3,1:3,j,i,myPhase)))
enddo
enddo
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end subroutine lattice_initializeStructure
!--------------------------------------------------------------------------------------------------
!> @brief Symmetrizes stiffness matrix according to lattice type
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!> @details J. A. Rayne and B. S. Chandrasekhar Phys. Rev. 120, 1658 Erratum Phys. Rev. 122, 1962
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!--------------------------------------------------------------------------------------------------
pure function lattice_symmetrizeC66(struct,C66)
implicit none
integer(kind(LATTICE_undefined_ID)), intent(in) :: struct
real(pReal), dimension(6,6), intent(in) :: C66
real(pReal), dimension(6,6) :: lattice_symmetrizeC66
integer(pInt) :: j,k
lattice_symmetrizeC66 = 0.0_pReal
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select case(struct)
case (LATTICE_iso_ID)
forall(k=1_pInt:3_pInt)
forall(j=1_pInt:3_pInt) lattice_symmetrizeC66(k,j) = C66(1,2)
lattice_symmetrizeC66(k,k) = C66(1,1)
lattice_symmetrizeC66(k+3,k+3) = 0.5_pReal*(C66(1,1)-C66(1,2))
end forall
case (LATTICE_fcc_ID,LATTICE_bcc_ID)
forall(k=1_pInt:3_pInt)
forall(j=1_pInt:3_pInt) lattice_symmetrizeC66(k,j) = C66(1,2)
lattice_symmetrizeC66(k,k) = C66(1,1)
lattice_symmetrizeC66(k+3_pInt,k+3_pInt) = C66(4,4)
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end forall
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case (LATTICE_hex_ID)
lattice_symmetrizeC66(1,1) = C66(1,1)
lattice_symmetrizeC66(2,2) = C66(1,1)
lattice_symmetrizeC66(3,3) = C66(3,3)
lattice_symmetrizeC66(1,2) = C66(1,2)
lattice_symmetrizeC66(2,1) = C66(1,2)
lattice_symmetrizeC66(1,3) = C66(1,3)
lattice_symmetrizeC66(3,1) = C66(1,3)
lattice_symmetrizeC66(2,3) = C66(1,3)
lattice_symmetrizeC66(3,2) = C66(1,3)
lattice_symmetrizeC66(4,4) = C66(4,4)
lattice_symmetrizeC66(5,5) = C66(4,4)
lattice_symmetrizeC66(6,6) = 0.5_pReal*(C66(1,1)-C66(1,2))
case (LATTICE_ort_ID)
lattice_symmetrizeC66(1,1) = C66(1,1)
lattice_symmetrizeC66(2,2) = C66(2,2)
lattice_symmetrizeC66(3,3) = C66(3,3)
lattice_symmetrizeC66(1,2) = C66(1,2)
lattice_symmetrizeC66(2,1) = C66(1,2)
lattice_symmetrizeC66(1,3) = C66(1,3)
lattice_symmetrizeC66(3,1) = C66(1,3)
lattice_symmetrizeC66(2,3) = C66(2,3)
lattice_symmetrizeC66(3,2) = C66(2,3)
lattice_symmetrizeC66(4,4) = C66(4,4)
lattice_symmetrizeC66(5,5) = C66(5,5)
lattice_symmetrizeC66(6,6) = C66(6,6)
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case (LATTICE_bct_ID)
lattice_symmetrizeC66(1,1) = C66(1,1)
lattice_symmetrizeC66(2,2) = C66(1,1)
lattice_symmetrizeC66(3,3) = C66(3,3)
lattice_symmetrizeC66(1,2) = C66(1,2)
lattice_symmetrizeC66(2,1) = C66(1,2)
lattice_symmetrizeC66(1,3) = C66(1,3)
lattice_symmetrizeC66(3,1) = C66(1,3)
lattice_symmetrizeC66(2,3) = C66(1,3)
lattice_symmetrizeC66(3,2) = C66(1,3)
lattice_symmetrizeC66(4,4) = C66(4,4)
lattice_symmetrizeC66(5,5) = C66(4,4)
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lattice_symmetrizeC66(6,6) = C66(6,6)
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case default
lattice_symmetrizeC66 = C66
end select
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end function lattice_symmetrizeC66
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!--------------------------------------------------------------------------------------------------
!> @brief Symmetrizes 2nd order tensor according to lattice type
!--------------------------------------------------------------------------------------------------
pure function lattice_symmetrize33(struct,T33)
implicit none
integer(kind(LATTICE_undefined_ID)), intent(in) :: struct
real(pReal), dimension(3,3), intent(in) :: T33
real(pReal), dimension(3,3) :: lattice_symmetrize33
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integer(pInt) :: k
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lattice_symmetrize33 = 0.0_pReal
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select case(struct)
case (LATTICE_iso_ID,LATTICE_fcc_ID,LATTICE_bcc_ID)
forall(k=1_pInt:3_pInt) lattice_symmetrize33(k,k) = T33(1,1)
case (LATTICE_hex_ID)
lattice_symmetrize33(1,1) = T33(1,1)
lattice_symmetrize33(2,2) = T33(1,1)
lattice_symmetrize33(3,3) = T33(3,3)
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case (LATTICE_ort_ID,lattice_bct_ID)
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lattice_symmetrize33(1,1) = T33(1,1)
lattice_symmetrize33(2,2) = T33(2,2)
lattice_symmetrize33(3,3) = T33(3,3)
case default
lattice_symmetrize33 = T33
end select
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end function lattice_symmetrize33
!--------------------------------------------------------------------------------------------------
!> @brief figures whether unit quat falls into stereographic standard triangle
!--------------------------------------------------------------------------------------------------
logical pure function lattice_qInSST(Q, struct)
use, intrinsic :: &
IEEE_arithmetic
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use math, only: &
math_qToRodrig
implicit none
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real(pReal), dimension(4), intent(in) :: Q ! orientation
integer(kind(LATTICE_undefined_ID)), intent(in) :: struct ! lattice structure
real(pReal), dimension(3) :: Rodrig ! Rodrigues vector of Q
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Rodrig = math_qToRodrig(Q)
if (any(IEEE_is_NaN(Rodrig))) then
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lattice_qInSST = .false.
else
select case (struct)
case (LATTICE_bcc_ID,LATTICE_fcc_ID)
lattice_qInSST = Rodrig(1) > Rodrig(2) .and. &
Rodrig(2) > Rodrig(3) .and. &
Rodrig(3) > 0.0_pReal
case (LATTICE_hex_ID)
lattice_qInSST = Rodrig(1) > sqrt(3.0_pReal)*Rodrig(2) .and. &
Rodrig(2) > 0.0_pReal .and. &
Rodrig(3) > 0.0_pReal
case default
lattice_qInSST = .true.
end select
endif
end function lattice_qInSST
!--------------------------------------------------------------------------------------------------
!> @brief calculates the disorientation for 2 unit quaternions
!--------------------------------------------------------------------------------------------------
pure function lattice_qDisorientation(Q1, Q2, struct)
use prec, only: &
tol_math_check
use math, only: &
math_qMul, &
math_qConj
implicit none
real(pReal), dimension(4) :: lattice_qDisorientation
real(pReal), dimension(4), intent(in) :: &
Q1, & ! 1st orientation
Q2 ! 2nd orientation
integer(kind(LATTICE_undefined_ID)), optional, intent(in) :: & ! if given, symmetries between the two orientation will be considered
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struct
real(pReal), dimension(4) :: dQ,dQsymA,mis
integer(pInt) :: i,j,k,s,symmetry
integer(kind(LATTICE_undefined_ID)) :: myStruct
integer(pInt), dimension(2), parameter :: &
NsymOperations = [24_pInt,12_pInt]
real(pReal), dimension(4,36), parameter :: &
symOperations = reshape([&
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1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! cubic symmetry operations
0.0_pReal, 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal), & ! 2-fold symmetry
0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), &
0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, &
0.0_pReal, 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), -1.0_pReal/sqrt(2.0_pReal), &
0.0_pReal, -1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), &
0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), -1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, &
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0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, & ! 3-fold symmetry
-0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, &
0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, &
-0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, &
0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, &
-0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, &
0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, &
-0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, &
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1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 0.0_pReal, & ! 4-fold symmetry
0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, &
-1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 0.0_pReal, &
1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, &
0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, &
-1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, &
1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), &
0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal, &
-1.0_pReal/sqrt(2.0_pReal), 0.0_pReal, 0.0_pReal, 1.0_pReal/sqrt(2.0_pReal), &
!
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1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! hexagonal symmetry operations
0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, & ! 2-fold symmetry
0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, &
0.0_pReal, 0.5_pReal, 2.0_pReal/sqrt(3.0_pReal), 0.0_pReal, &
0.0_pReal, -0.5_pReal, 2.0_pReal/sqrt(3.0_pReal), 0.0_pReal, &
0.0_pReal, 2.0_pReal/sqrt(3.0_pReal), 0.5_pReal, 0.0_pReal, &
0.0_pReal, -2.0_pReal/sqrt(3.0_pReal), 0.5_pReal, 0.0_pReal, &
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2.0_pReal/sqrt(3.0_pReal), 0.0_pReal, 0.0_pReal, 0.5_pReal, & ! 6-fold symmetry
-2.0_pReal/sqrt(3.0_pReal), 0.0_pReal, 0.0_pReal, 0.5_pReal, &
0.5_pReal, 0.0_pReal, 0.0_pReal, 2.0_pReal/sqrt(3.0_pReal), &
-0.5_pReal, 0.0_pReal, 0.0_pReal, 2.0_pReal/sqrt(3.0_pReal), &
0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal &
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],[4,36]) !< Symmetry operations as quaternions 24 for cubic, 12 for hexagonal = 36
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!--------------------------------------------------------------------------------------------------
! check if a structure with known symmetries is given
if (present(struct)) then
myStruct = struct
select case (struct)
case(LATTICE_fcc_ID,LATTICE_bcc_ID)
symmetry = 1_pInt
case(LATTICE_hex_ID)
symmetry = 2_pInt
case default
symmetry = 0_pInt
end select
else
symmetry = 0_pInt
myStruct = LATTICE_undefined_ID
endif
!--------------------------------------------------------------------------------------------------
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! calculate misorientation, for cubic and hexagonal structure find symmetries
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dQ = math_qMul(math_qConj(Q1),Q2)
lattice_qDisorientation = dQ
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select case(symmetry)
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case (1_pInt,2_pInt)
s = sum(NsymOperations(1:symmetry-1_pInt))
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do i = 1_pInt,2_pInt
dQ = math_qConj(dQ) ! switch order of "from -- to"
do j = 1_pInt,NsymOperations(symmetry) ! run through first crystal's symmetries
dQsymA = math_qMul(symOperations(1:4,s+j),dQ) ! apply sym
do k = 1_pInt,NsymOperations(symmetry) ! run through 2nd crystal's symmetries
mis = math_qMul(dQsymA,symOperations(1:4,s+k)) ! apply sym
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if (mis(1) < 0.0_pReal) & ! want positive angle
mis = -mis
if (mis(1)-lattice_qDisorientation(1) > -tol_math_check &
.and. lattice_qInSST(mis,LATTICE_undefined_ID)) lattice_qDisorientation = mis ! found better one
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enddo; enddo; enddo
case (0_pInt)
if (lattice_qDisorientation(1) < 0.0_pReal) lattice_qDisorientation = -lattice_qDisorientation ! keep omega within 0 to 180 deg
end select
end function lattice_qDisorientation
!--------------------------------------------------------------------------------------------------
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!> @brief Characteristic shear for twinning
!--------------------------------------------------------------------------------------------------
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function lattice_characteristicShear_Twin(Ntwin,structure,CoverA) result(characteristicShear)
use IO, only: &
IO_error
implicit none
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integer(pInt), dimension(:), intent(in) :: Ntwin !< number of active twin systems per family
character(len=3), intent(in) :: structure !< lattice structure
real(pReal), intent(in) :: cOverA !< c/a ratio
real(pReal), dimension(sum(Ntwin)) :: characteristicShear
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integer(pInt) :: &
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a, & !< index of active system
c, & !< index in complete system list
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mf, & !< index of my family
ms !< index of my system in current family
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integer(pInt), dimension(LATTICE_HEX_NTWIN), parameter :: &
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HEX_SHEARTWIN = reshape(int( [&
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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 &
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],pInt),[LATTICE_HEX_NTWIN]) ! indicator to formulas below
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a = 0_pInt
myFamilies: do mf = 1_pInt,size(Ntwin,1)
mySystems: do ms = 1_pInt,Ntwin(mf)
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a = a + 1_pInt
select case(trim(structure))
case('fcc','bcc')
characteristicShear(a) = 0.5_pReal*sqrt(2.0_pReal)
case('hex')
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if (cOverA < 1.0_pReal .or. cOverA > 2.0_pReal) &
call IO_error(131_pInt,ext_msg='lattice_characteristicShear_Twin')
c = sum(LATTICE_HEX_NTWINSYSTEM(1:mf-1))+ms
select case(HEX_SHEARTWIN(c)) ! from Christian & Mahajan 1995 p.29
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case (1_pInt) ! <-10.1>{10.2}
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characteristicShear(a) = (3.0_pReal-cOverA**2.0_pReal)/sqrt(3.0_pReal)/CoverA
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case (2_pInt) ! <11.6>{-1-1.1}
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characteristicShear(a) = 1.0_pReal/cOverA
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case (3_pInt) ! <10.-2>{10.1}
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characteristicShear(a) = (4.0_pReal*cOverA**2.0_pReal-9.0_pReal)/sqrt(48.0_pReal)/cOverA
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case (4_pInt) ! <11.-3>{11.2}
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characteristicShear(a) = 2.0_pReal*(cOverA**2.0_pReal-2.0_pReal)/3.0_pReal/cOverA
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end select
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case default
call IO_error(137_pInt,ext_msg='lattice_characteristicShear_Twin: '//trim(structure))
end select
enddo mySystems
enddo myFamilies
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end function lattice_characteristicShear_Twin
!--------------------------------------------------------------------------------------------------
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!> @brief Rotated elasticity matrices for twinning in Mandel notation
!--------------------------------------------------------------------------------------------------
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function lattice_C66_twin(Ntwin,C66,structure,CoverA)
use IO, only: &
IO_error
use math, only: &
INRAD, &
math_axisAngleToR, &
math_Mandel3333to66, &
math_Mandel66to3333, &
math_rotate_forward3333
implicit none
integer(pInt), dimension(:), intent(in) :: Ntwin !< number of active twin systems per family
character(len=*), intent(in) :: structure !< lattice structure
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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
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real(pReal), dimension(3,3,sum(Ntwin)) :: coordinateSystem
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real(pReal), dimension(3,3) :: R
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integer(pInt) :: i
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select case(trim(structure))
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case('fcc')
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coordinateSystem = buildCoordinateSystem(Ntwin,LATTICE_FCC_NSLIPSYSTEM,LATTICE_FCC_SYSTEMTWIN,&
trim(structure),0.0_pReal)
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case('bcc')
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coordinateSystem = buildCoordinateSystem(Ntwin,LATTICE_BCC_NSLIPSYSTEM,LATTICE_BCC_SYSTEMTWIN,&
trim(structure),0.0_pReal)
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case('hex','hexagonal') !ToDo: "No alias policy": long or short?
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coordinateSystem = buildCoordinateSystem(Ntwin,LATTICE_HEX_NSLIPSYSTEM,LATTICE_HEX_SYSTEMTWIN,&
'hex',cOverA)
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case default
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call IO_error(137_pInt,ext_msg='lattice_C66_twin: '//trim(structure))
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end select
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do i = 1, sum(Ntwin)
R = math_axisAngleToR(coordinateSystem(1:3,2,i), 180.0_pReal * INRAD) ! ToDo: Why always 180 deg?
lattice_C66_twin(1:6,1:6,i) = math_Mandel3333to66(math_rotate_forward3333(math_Mandel66to3333(C66),R))
enddo
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end function lattice_C66_twin
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!--------------------------------------------------------------------------------------------------
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!> @brief Rotated elasticity matrices for transformation in Mandel notation
!> ToDo: Completely untested and incomplete and undocumented
!--------------------------------------------------------------------------------------------------
function lattice_C66_trans(Ntrans,C_parent66, &
structure_target, &
CoverA_trans,a_bcc,a_fcc)
use prec, only: &
tol_math_check
use IO, only: &
IO_error
use math, only: &
INRAD, &
MATH_I3, &
math_axisAngleToR, &
math_Mandel3333to66, &
math_Mandel66to3333, &
math_rotate_forward3333, &
math_mul33x33, &
math_tensorproduct33, &
math_crossproduct
implicit none
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integer(pInt), dimension(:), intent(in) :: Ntrans !< number of active twin systems per family
character(len=*), intent(in) :: &
structure_target !< lattice structure
real(pReal), dimension(6,6), intent(in) :: C_parent66
real(pReal), dimension(6,6) :: C_bar66, C_target_unrotated66
real(pReal), dimension(3,3,3,3) :: C_target_unrotated
real(pReal), dimension(6,6,sum(Ntrans)) :: lattice_C66_trans
real(pReal), dimension(3,3,sum(Ntrans)) :: Q,S
real(pReal) :: a_bcc, a_fcc, CoverA_trans
integer(pInt) :: i
!--------------------------------------------------------------------------------------------------
! elasticity matrix of the target phase in cube orientation
if (trim(structure_target) == 'hex') then
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)**2.0_pReal/C_bar66(4,4)
C_target_unrotated66(1,2) = C_bar66(1,2) + C_bar66(1,4)**2.0_pReal/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_pReal/(0.5_pReal*(C_bar66(1,1) - C_bar66(1,2)))
C_target_unrotated66 = lattice_symmetrizeC66(LATTICE_HEX_ID,C_target_unrotated66)
elseif (trim(structure_target) == 'bcc') then
C_target_unrotated66 = C_parent66
else
write(6,*) "Mist"
endif
do i = 1_pInt, 6_pInt
if (abs(C_target_unrotated66(i,i))<tol_math_check) &
call IO_error(135_pInt,el=i,ext_msg='matrix diagonal "el"ement in transformation')
enddo
C_target_unrotated = math_Mandel66to3333(C_target_unrotated66)
call buildTransformationSystem(Q,S,Ntrans,CoverA_trans,a_fcc,a_bcc)
do i = 1, sum(Ntrans)
lattice_C66_trans(1:6,1:6,i) = math_Mandel3333to66(math_rotate_forward3333(C_target_unrotated,Q(1:3,1:3,i)))
enddo
end function
!--------------------------------------------------------------------------------------------------
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!> @brief Non-schmid projections for bcc with up to 6 coefficients
! Koester et al. 2012, Acta Materialia 60 (2012) 38943901, eq. (17)
! Gröger et al. 2008, Acta Materialia 56 (2008) 54125425, table 1
!--------------------------------------------------------------------------------------------------
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function lattice_nonSchmidMatrix(Nslip,nonSchmidCoefficients,sense) result(nonSchmidMatrix)
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use IO, only: &
IO_error
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use math, only: &
INRAD, &
math_tensorproduct33, &
math_crossproduct, &
math_mul33x3, &
math_axisAngleToR
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implicit none
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integer(pInt), dimension(:), intent(in) :: Nslip !< number of active slip systems per family
real(pReal), dimension(:), intent(in) :: nonSchmidCoefficients !< non-Schmid coefficients for projections
integer(pInt), 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(:), allocatable :: &
direction, normal, np
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integer(pInt) :: i
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if (abs(sense) /= 1_pInt) call IO_error(0_pInt,ext_msg='lattice_nonSchmidMatrix')
coordinateSystem = buildCoordinateSystem(Nslip,LATTICE_BCC_NSLIPSYSTEM,LATTICE_BCC_SYSTEMSLIP,&
'bcc',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,'bcc',0.0_pReal) ! Schmid contribution
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do i = 1_pInt,sum(Nslip)
direction = coordinateSystem(1:3,1,i)
normal = coordinateSystem(1:3,2,i)
np = math_mul33x3(math_axisAngleToR(direction,60.0_pReal*INRAD), normal)
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if (size(nonSchmidCoefficients)>0) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) &
+ nonSchmidCoefficients(1) * math_tensorproduct33(direction, np)
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if (size(nonSchmidCoefficients)>1) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) &
+ nonSchmidCoefficients(2) * math_tensorproduct33(math_crossproduct(normal, direction), normal)
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if (size(nonSchmidCoefficients)>2) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) &
+ nonSchmidCoefficients(3) * math_tensorproduct33(math_crossproduct(np, direction), np)
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if (size(nonSchmidCoefficients)>3) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) &
+ nonSchmidCoefficients(4) * math_tensorproduct33(normal, normal)
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if (size(nonSchmidCoefficients)>4) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) &
+ nonSchmidCoefficients(5) * math_tensorproduct33(math_crossproduct(normal, direction), &
math_crossproduct(normal, direction))
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if (size(nonSchmidCoefficients)>5) nonSchmidMatrix(1:3,1:3,i) = nonSchmidMatrix(1:3,1:3,i) &
+ nonSchmidCoefficients(6) * math_tensorproduct33(direction, direction)
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enddo
end function lattice_nonSchmidMatrix
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!--------------------------------------------------------------------------------------------------
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!> @brief Slip-slip interaction matrix
!> details only active slip systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_interaction_SlipSlip(Nslip,interactionValues,structure) result(interactionMatrix)
use IO, only: &
IO_error
implicit none
integer(pInt), dimension(:), intent(in) :: Nslip !< number of active slip systems per family
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real(pReal), dimension(:), intent(in) :: interactionValues !< values for slip-slip interaction
character(len=*), intent(in) :: structure !< lattice structure
real(pReal), dimension(sum(Nslip),sum(Nslip)) :: interactionMatrix
integer(pInt), dimension(:), allocatable :: NslipMax
integer(pInt), dimension(:,:), allocatable :: interactionTypes
select case(structure)
case('fcc')
interactionTypes = LATTICE_FCC_INTERACTIONSLIPSLIP
NslipMax = LATTICE_FCC_NSLIPSYSTEM
case('bcc')
interactionTypes = LATTICE_BCC_INTERACTIONSLIPSLIP
NslipMax = LATTICE_BCC_NSLIPSYSTEM
case('hex','hexagonal') ! ToDo: "No alias policy": long or short?
interactionTypes = LATTICE_HEX_INTERACTIONSLIPSLIP
NslipMax = LATTICE_HEX_NSLIPSYSTEM
case('bct')
interactionTypes = LATTICE_BCT_INTERACTIONSLIPSLIP
NslipMax = LATTICE_BCT_NSLIPSYSTEM
case default
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call IO_error(137_pInt,ext_msg='lattice_interaction_SlipSlip: '//trim(structure))
end select
interactionMatrix = buildInteraction(Nslip,Nslip,NslipMax,NslipMax,interactionValues,interactionTypes)
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end function lattice_interaction_SlipSlip
!--------------------------------------------------------------------------------------------------
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!> @brief Twin-twin interaction matrix
!> details only active twin systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_interaction_TwinTwin(Ntwin,interactionValues,structure) result(interactionMatrix)
use IO, only: &
IO_error
implicit none
integer(pInt), dimension(:), intent(in) :: Ntwin !< number of active twin systems per family
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real(pReal), dimension(:), intent(in) :: interactionValues !< values for twin-twin interaction
character(len=*), intent(in) :: structure !< lattice structure
real(pReal), dimension(sum(Ntwin),sum(Ntwin)) :: interactionMatrix
integer(pInt), dimension(:), allocatable :: NtwinMax
integer(pInt), dimension(:,:), allocatable :: interactionTypes
integer(pInt), dimension(LATTICE_FCC_NTWIN,LATTICE_FCC_NTWIN), parameter :: &
FCC_INTERACTIONTWINTWIN = reshape(int( [&
1,1,1,2,2,2,2,2,2,2,2,2, & ! ---> twin
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 twin
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, &
2,2,2,2,2,2,1,1,1,2,2,2, &
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 &
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],pInt),shape(FCC_INTERACTIONTWINTWIN),order=[2,1]) !< Twin-twin interaction types for fcc
integer(pInt), dimension(LATTICE_BCC_NTWIN,LATTICE_BCC_NTWIN), parameter :: &
BCC_INTERACTIONTWINTWIN = reshape(int( [&
1,3,3,3,3,3,3,2,3,3,2,3, & ! ---> twin
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 twin
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 &
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],pInt),shape(BCC_INTERACTIONTWINTWIN),order=[2,1]) !< Twin-twin interaction types for bcc
!< 1: self interaction
!< 2: collinear interaction
!< 3: other interaction
integer(pInt), dimension(LATTICE_HEX_NTWIN,LATTICE_HEX_NTWIN), parameter :: &
HEX_INTERACTIONTWINTWIN = reshape(int( [&
1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, & ! ---> twin
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 twin
2, 2, 2, 2, 1, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 7, 7, 13,13,13,13,13,13, &
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, &
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, &
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, &
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 &
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],pInt),shape(HEX_INTERACTIONTWINTWIN),order=[2,1]) !< Twin-twin interaction types for hex
select case(structure)
case('fcc')
interactionTypes = FCC_INTERACTIONTWINTWIN
NtwinMax = LATTICE_FCC_NTWINSYSTEM
case('bcc')
interactionTypes = BCC_INTERACTIONTWINTWIN
NtwinMax = LATTICE_BCC_NTWINSYSTEM
case('hex','hexagonal') ! ToDo: "No alias policy": long or short?
interactionTypes = HEX_INTERACTIONTWINTWIN
NtwinMax = LATTICE_HEX_NTWINSYSTEM
case default
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call IO_error(137_pInt,ext_msg='lattice_interaction_TwinTwin: '//trim(structure))
end select
interactionMatrix = buildInteraction(Ntwin,Ntwin,NtwinMax,NtwinMax,interactionValues,interactionTypes)
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end function lattice_interaction_TwinTwin
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!--------------------------------------------------------------------------------------------------
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!> @brief Trans-trans interaction matrix
!> details only active trans systems are considered
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!--------------------------------------------------------------------------------------------------
function lattice_interaction_TransTrans(Ntrans,interactionValues,structure) result(interactionMatrix)
use IO, only: &
IO_error
implicit none
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integer(pInt), dimension(:), intent(in) :: Ntrans !< number of active trans systems per family
real(pReal), dimension(:), intent(in) :: interactionValues !< values for trans-trans interaction
character(len=*), intent(in) :: structure !< lattice structure (parent crystal)
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real(pReal), dimension(sum(Ntrans),sum(Ntrans)) :: interactionMatrix
integer(pInt), dimension(:), allocatable :: NtransMax
integer(pInt), dimension(:,:), allocatable :: interactionTypes
integer(pInt), dimension(LATTICE_FCC_NTRANS,LATTICE_FCC_NTRANS), parameter :: &
FCC_INTERACTIONTRANSTRANS = reshape(int( [&
1,1,1,2,2,2,2,2,2,2,2,2, & ! ---> trans
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 trans
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, &
2,2,2,2,2,2,1,1,1,2,2,2, &
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 &
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],pInt),shape(FCC_INTERACTIONTRANSTRANS),order=[2,1]) !< Trans-trans interaction types for fcc
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if (trim(structure) == 'fcc') then
interactionTypes = FCC_INTERACTIONTRANSTRANS
NtransMax = LATTICE_FCC_NTRANSSYSTEM
else
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call IO_error(137_pInt,ext_msg='lattice_interaction_TransTrans: '//trim(structure))
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end if
interactionMatrix = buildInteraction(Ntrans,Ntrans,NtransMax,NtransMax,interactionValues,interactionTypes)
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end function lattice_interaction_TransTrans
!--------------------------------------------------------------------------------------------------
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!> @brief Slip-twin interaction matrix
!> details only active slip and twin systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_interaction_SlipTwin(Nslip,Ntwin,interactionValues,structure) result(interactionMatrix)
use IO, only: &
IO_error
implicit none
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integer(pInt), dimension(:), intent(in) :: Nslip, & !< number of active slip systems per family
Ntwin !< number of active twin systems per family
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real(pReal), dimension(:), intent(in) :: interactionValues !< values for slip-twin interaction
character(len=*), intent(in) :: structure !< lattice structure
real(pReal), dimension(sum(Nslip),sum(Ntwin)) :: interactionMatrix
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integer(pInt), dimension(:), allocatable :: NslipMax, &
NtwinMax
integer(pInt), dimension(:,:), allocatable :: interactionTypes
integer(pInt), dimension(LATTICE_FCC_NSLIP,LATTICE_FCC_NTWIN), parameter :: &
FCC_INTERACTIONSLIPTWIN = reshape(int( [&
1,1,1,3,3,3,2,2,2,3,3,3, & ! ---> twin
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 slip
3,3,3,1,1,1,2,2,2,3,3,3, &
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 &
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],pInt),shape(FCC_INTERACTIONSLIPTWIN),order=[2,1]) !< 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(pInt), dimension(LATTICE_BCC_NSLIP,LATTICE_BCC_NTWIN), parameter :: &
BCC_INTERACTIONSLIPTWIN = reshape(int( [&
3,3,3,2,2,3,3,3,3,2,3,3, & ! ---> twin
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 slip
2,3,3,3,3,3,3,2,3,3,2,3, &
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 &
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],pInt),shape(BCC_INTERACTIONSLIPTWIN),order=[2,1]) !< 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
integer(pInt), dimension(LATTICE_HEX_NSLIP,LATTICE_HEX_NTWIN), parameter :: &
HEX_INTERACTIONSLIPTWIN = reshape(int( [&
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
1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, & ! |
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
5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, &
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, &
!
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, &
!
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, &
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, &
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, &
!
21,21,21,21,21,21, 22,22,22,22,22,22, 23,23,23,23,23,23, 24,24,24,24,24,24, &
21,21,21,21,21,21, 22,22,22,22,22,22, 23,23,23,23,23,23, 24,24,24,24,24,24, &
21,21,21,21,21,21, 22,22,22,22,22,22, 23,23,23,23,23,23, 24,24,24,24,24,24, &
21,21,21,21,21,21, 22,22,22,22,22,22, 23,23,23,23,23,23, 24,24,24,24,24,24, &
21,21,21,21,21,21, 22,22,22,22,22,22, 23,23,23,23,23,23, 24,24,24,24,24,24, &
21,21,21,21,21,21, 22,22,22,22,22,22, 23,23,23,23,23,23, 24,24,24,24,24,24 &
!
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],pInt),shape(HEX_INTERACTIONSLIPTWIN),order=[2,1]) !< Slip-twin interaction types for hex
select case(structure)
case('fcc')
interactionTypes = FCC_INTERACTIONSLIPTWIN
NslipMax = LATTICE_FCC_NSLIPSYSTEM
NtwinMax = LATTICE_FCC_NTWINSYSTEM
case('bcc')
interactionTypes = BCC_INTERACTIONSLIPTWIN
NslipMax = LATTICE_BCC_NSLIPSYSTEM
NtwinMax = LATTICE_BCC_NTWINSYSTEM
case('hex','hexagonal') ! ToDo: "No alias policy": long or short?
interactionTypes = HEX_INTERACTIONSLIPTWIN
NslipMax = LATTICE_HEX_NSLIPSYSTEM
NtwinMax = LATTICE_HEX_NTWINSYSTEM
case default
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call IO_error(137_pInt,ext_msg='lattice_interaction_SlipTwin: '//trim(structure))
end select
interactionMatrix = buildInteraction(Nslip,Ntwin,NslipMax,NtwinMax,interactionValues,interactionTypes)
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end function lattice_interaction_SlipTwin
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!--------------------------------------------------------------------------------------------------
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!> @brief Slip-trans interaction matrix
!> details only active slip and trans systems are considered
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!--------------------------------------------------------------------------------------------------
function lattice_interaction_SlipTrans(Nslip,Ntrans,interactionValues,structure) result(interactionMatrix)
use IO, only: &
IO_error
implicit none
integer(pInt), dimension(:), intent(in) :: Nslip, & !< number of active slip systems per family
Ntrans !< number of active trans systems per family
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real(pReal), dimension(:), intent(in) :: interactionValues !< values for slip-trans interaction
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character(len=*), intent(in) :: &
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structure !< lattice structure (parent crystal)
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real(pReal), dimension(sum(Nslip),sum(Ntrans)) :: interactionMatrix
integer(pInt), dimension(:), allocatable :: NslipMax, &
NtransMax
integer(pInt), dimension(:,:), allocatable :: interactionTypes
integer(pInt), dimension(LATTICE_FCC_NSLIP,LATTICE_fcc_Ntrans), parameter :: &
FCC_INTERACTIONSLIPTRANS = reshape(int( [&
1,1,1,3,3,3,2,2,2,3,3,3, & ! ---> trans
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 slip
3,3,3,1,1,1,2,2,2,3,3,3, &
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, &
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4,4,4,4,4,4,4,4,4,4,4,4, &
4,4,4,4,4,4,4,4,4,4,4,4, &
4,4,4,4,4,4,4,4,4,4,4,4, &
4,4,4,4,4,4,4,4,4,4,4,4, &
4,4,4,4,4,4,4,4,4,4,4,4, &
4,4,4,4,4,4,4,4,4,4,4,4 &
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],pInt),shape(FCC_INTERACTIONSLIPTRANS),order=[2,1]) !< Slip-trans interaction types for fcc
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select case(structure)
case('fcc')
interactionTypes = FCC_INTERACTIONSLIPTRANS
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NslipMax = LATTICE_FCC_NSLIPSYSTEM
NtransMax = LATTICE_FCC_NTRANSSYSTEM
case default
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call IO_error(137_pInt,ext_msg='lattice_interaction_SlipTrans: '//trim(structure))
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end select
interactionMatrix = buildInteraction(Nslip,Ntrans,NslipMax,NtransMax,interactionValues,interactionTypes)
end function lattice_interaction_SlipTrans
!--------------------------------------------------------------------------------------------------
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!> @brief Twin-slip interaction matrix
!> details only active twin and slip systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_interaction_TwinSlip(Ntwin,Nslip,interactionValues,structure) result(interactionMatrix)
use IO, only: &
IO_error
implicit none
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integer(pInt), dimension(:), intent(in) :: Ntwin, & !< number of active twin systems per family
Nslip !< number of active slip systems per family
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real(pReal), dimension(:), intent(in) :: interactionValues !< values for twin-twin interaction
character(len=*), intent(in) :: structure !< lattice structure
real(pReal), dimension(sum(Ntwin),sum(Nslip)) :: interactionMatrix
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integer(pInt), dimension(:), allocatable :: NtwinMax, &
NslipMax
integer(pInt), dimension(:,:), allocatable :: interactionTypes
integer(pInt), dimension(LATTICE_FCC_NTWIN,LATTICE_FCC_NSLIP), parameter :: &
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FCC_INTERACTIONTWINSLIP = 1_pInt !< Twin-Slip interaction types for fcc
integer(pInt), dimension(LATTICE_BCC_NTWIN,LATTICE_BCC_NSLIP), parameter :: &
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BCC_INTERACTIONTWINSLIP = 1_pInt !< Twin-slip interaction types for bcc
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integer(pInt), dimension(LATTICE_HEX_NTWIN,LATTICE_HEX_NSLIP), parameter :: &
HEX_INTERACTIONTWINSLIP = reshape(int( [&
1, 1, 1, 5, 5, 5, 9, 9, 9, 13,13,13,13,13,13, 17,17,17,17,17,17,17,17,17,17,17,17, 21,21,21,21,21,21, & ! --> slip
1, 1, 1, 5, 5, 5, 9, 9, 9, 13,13,13,13,13,13, 17,17,17,17,17,17,17,17,17,17,17,17, 21,21,21,21,21,21, & ! |
1, 1, 1, 5, 5, 5, 9, 9, 9, 13,13,13,13,13,13, 17,17,17,17,17,17,17,17,17,17,17,17, 21,21,21,21,21,21, & ! |
1, 1, 1, 5, 5, 5, 9, 9, 9, 13,13,13,13,13,13, 17,17,17,17,17,17,17,17,17,17,17,17, 21,21,21,21,21,21, & ! v
1, 1, 1, 5, 5, 5, 9, 9, 9, 13,13,13,13,13,13, 17,17,17,17,17,17,17,17,17,17,17,17, 21,21,21,21,21,21, & ! twin
1, 1, 1, 5, 5, 5, 9, 9, 9, 13,13,13,13,13,13, 17,17,17,17,17,17,17,17,17,17,17,17, 21,21,21,21,21,21, &
!
2, 2, 2, 6, 6, 6, 10,10,10, 14,14,14,14,14,14, 18,18,18,18,18,18,18,18,18,18,18,18, 22,22,22,22,22,22, &
2, 2, 2, 6, 6, 6, 10,10,10, 14,14,14,14,14,14, 18,18,18,18,18,18,18,18,18,18,18,18, 22,22,22,22,22,22, &
2, 2, 2, 6, 6, 6, 10,10,10, 14,14,14,14,14,14, 18,18,18,18,18,18,18,18,18,18,18,18, 22,22,22,22,22,22, &
2, 2, 2, 6, 6, 6, 10,10,10, 14,14,14,14,14,14, 18,18,18,18,18,18,18,18,18,18,18,18, 22,22,22,22,22,22, &
2, 2, 2, 6, 6, 6, 10,10,10, 14,14,14,14,14,14, 18,18,18,18,18,18,18,18,18,18,18,18, 22,22,22,22,22,22, &
2, 2, 2, 6, 6, 6, 10,10,10, 14,14,14,14,14,14, 18,18,18,18,18,18,18,18,18,18,18,18, 22,22,22,22,22,22, &
!
3, 3, 3, 7, 7, 7, 11,11,11, 15,15,15,15,15,15, 19,19,19,19,19,19,19,19,19,19,19,19, 23,23,23,23,23,23, &
3, 3, 3, 7, 7, 7, 11,11,11, 15,15,15,15,15,15, 19,19,19,19,19,19,19,19,19,19,19,19, 23,23,23,23,23,23, &
3, 3, 3, 7, 7, 7, 11,11,11, 15,15,15,15,15,15, 19,19,19,19,19,19,19,19,19,19,19,19, 23,23,23,23,23,23, &
3, 3, 3, 7, 7, 7, 11,11,11, 15,15,15,15,15,15, 19,19,19,19,19,19,19,19,19,19,19,19, 23,23,23,23,23,23, &
3, 3, 3, 7, 7, 7, 11,11,11, 15,15,15,15,15,15, 19,19,19,19,19,19,19,19,19,19,19,19, 23,23,23,23,23,23, &
3, 3, 3, 7, 7, 7, 11,11,11, 15,15,15,15,15,15, 19,19,19,19,19,19,19,19,19,19,19,19, 23,23,23,23,23,23, &
!
4, 4, 4, 8, 8, 8, 12,12,12, 16,16,16,16,16,16, 20,20,20,20,20,20,20,20,20,20,20,20, 24,24,24,24,24,24, &
4, 4, 4, 8, 8, 8, 12,12,12, 16,16,16,16,16,16, 20,20,20,20,20,20,20,20,20,20,20,20, 24,24,24,24,24,24, &
4, 4, 4, 8, 8, 8, 12,12,12, 16,16,16,16,16,16, 20,20,20,20,20,20,20,20,20,20,20,20, 24,24,24,24,24,24, &
4, 4, 4, 8, 8, 8, 12,12,12, 16,16,16,16,16,16, 20,20,20,20,20,20,20,20,20,20,20,20, 24,24,24,24,24,24, &
4, 4, 4, 8, 8, 8, 12,12,12, 16,16,16,16,16,16, 20,20,20,20,20,20,20,20,20,20,20,20, 24,24,24,24,24,24, &
4, 4, 4, 8, 8, 8, 12,12,12, 16,16,16,16,16,16, 20,20,20,20,20,20,20,20,20,20,20,20, 24,24,24,24,24,24 &
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],pInt),shape(HEX_INTERACTIONTWINSLIP),order=[2,1]) !< Twin-twin interaction types for hex
select case(structure)
case('fcc')
interactionTypes = FCC_INTERACTIONTWINSLIP
NtwinMax = LATTICE_FCC_NTWINSYSTEM
NslipMax = LATTICE_FCC_NSLIPSYSTEM
case('bcc')
interactionTypes = BCC_INTERACTIONTWINSLIP
NtwinMax = LATTICE_BCC_NTWINSYSTEM
NslipMax = LATTICE_BCC_NSLIPSYSTEM
case('hex','hexagonal') ! ToDo: "No alias policy": long or short?
interactionTypes = HEX_INTERACTIONTWINSLIP
NtwinMax = LATTICE_HEX_NTWINSYSTEM
NslipMax = LATTICE_HEX_NSLIPSYSTEM
case default
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call IO_error(137_pInt,ext_msg='lattice_interaction_TwinSlip: '//trim(structure))
end select
interactionMatrix = buildInteraction(Ntwin,Nslip,NtwinMax,NslipMax,interactionValues,interactionTypes)
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end function lattice_interaction_TwinSlip
!--------------------------------------------------------------------------------------------------
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!> @brief Schmid matrix for slip
!> details only active slip systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_SchmidMatrix_slip(Nslip,structure,cOverA) result(SchmidMatrix)
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use prec, only: &
tol_math_check
use IO, only: &
IO_error
use math, only: &
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math_trace33, &
math_tensorproduct33
implicit none
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integer(pInt), dimension(:), intent(in) :: Nslip !< number of active slip systems per family
character(len=*), intent(in) :: structure !< lattice structure
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
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integer(pInt), dimension(:), allocatable :: NslipMax
integer(pInt) :: i
select case(structure)
case('fcc')
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NslipMax = LATTICE_FCC_NSLIPSYSTEM
slipSystems = LATTICE_FCC_SYSTEMSLIP
case('bcc')
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NslipMax = LATTICE_BCC_NSLIPSYSTEM
slipSystems = LATTICE_BCC_SYSTEMSLIP
case('hex','hexagonal') ! ToDo: "No alias policy": long or short?
NslipMax = LATTICE_HEX_NSLIPSYSTEM
slipSystems = LATTICE_HEX_SYSTEMSLIP
case('bct')
NslipMax = LATTICE_BCT_NSLIPSYSTEM
slipSystems = LATTICE_BCT_SYSTEMSLIP
case default
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call IO_error(137_pInt,ext_msg='lattice_SchmidMatrix_slip: '//trim(structure))
end select
if (any(NslipMax(1:size(Nslip)) - Nslip < 0_pInt)) &
call IO_error(145_pInt,ext_msg='Nslip '//trim(structure))
if (any(Nslip < 0_pInt)) &
call IO_error(144_pInt,ext_msg='Nslip '//trim(structure))
coordinateSystem = buildCoordinateSystem(Nslip,NslipMax,slipSystems,structure,cOverA)
do i = 1, sum(Nslip)
SchmidMatrix(1:3,1:3,i) = math_tensorproduct33(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_pInt,i,ext_msg = 'dilatational Schmid matrix for slip')
enddo
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end function lattice_SchmidMatrix_slip
!--------------------------------------------------------------------------------------------------
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!> @brief Schmid matrix for twinning
!> details only active twin systems are considered
!--------------------------------------------------------------------------------------------------
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function lattice_SchmidMatrix_twin(Ntwin,structure,cOverA) result(SchmidMatrix)
use prec, only: &
tol_math_check
use IO, only: &
IO_error
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use math, only: &
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math_trace33, &
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math_tensorproduct33
implicit none
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integer(pInt), dimension(:), intent(in) :: Ntwin !< number of active twin systems per family
character(len=*), intent(in) :: structure !< lattice structure
real(pReal), intent(in) :: cOverA !< c/a ratio
real(pReal), dimension(3,3,sum(Ntwin)) :: SchmidMatrix
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real(pReal), dimension(3,3,sum(Ntwin)) :: coordinateSystem
real(pReal), dimension(:,:), allocatable :: twinSystems
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integer(pInt), dimension(:), allocatable :: NtwinMax
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integer(pInt) :: i
select case(structure)
case('fcc')
NtwinMax = LATTICE_FCC_NTWINSYSTEM
twinSystems = LATTICE_FCC_SYSTEMTWIN
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case('bcc')
NtwinMax = LATTICE_BCC_NTWINSYSTEM
twinSystems = LATTICE_BCC_SYSTEMTWIN
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case('hex','hexagonal') !ToDo: "No alias policy": long or short?
NtwinMax = LATTICE_HEX_NTWINSYSTEM
twinSystems = LATTICE_HEX_SYSTEMTWIN
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case default
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call IO_error(137_pInt,ext_msg='lattice_SchmidMatrix_twin: '//trim(structure))
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end select
if (any(NtwinMax(1:size(Ntwin)) - Ntwin < 0_pInt)) &
call IO_error(145_pInt,ext_msg='Ntwin '//trim(structure))
if (any(Ntwin < 0_pInt)) &
call IO_error(144_pInt,ext_msg='Ntwin '//trim(structure))
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coordinateSystem = buildCoordinateSystem(Ntwin,NtwinMax,twinSystems,structure,cOverA)
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do i = 1, sum(Ntwin)
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SchmidMatrix(1:3,1:3,i) = math_tensorproduct33(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_pInt,i,ext_msg = 'dilatational Schmid matrix for twin')
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enddo
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end function lattice_SchmidMatrix_twin
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!--------------------------------------------------------------------------------------------------
!> @brief Schmid matrix for twinning
!> details only active twin systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_SchmidMatrix_trans(Ntrans,structure_target,cOverA,a_bcc,a_fcc) result(SchmidMatrix)
use prec, only: &
tol_math_check
use IO, only: &
IO_error
use math, only: &
math_trace33, &
math_tensorproduct33
implicit none
integer(pInt), dimension(:), intent(in) :: Ntrans !< number of active twin systems per family
real(pReal), intent(in) :: cOverA !< c/a ratio
real(pReal), dimension(3,3,sum(Ntrans)) :: SchmidMatrix
character(len=*), intent(in) :: &
structure_target !< lattice structure
real(pReal), dimension(3,3,sum(Ntrans)) :: devNull
real(pReal) :: a_bcc, a_fcc
! ToDo: Error checking!!!!!!!!!!!!!!!!!!!
call buildTransformationSystem(devNull,SchmidMatrix,Ntrans,cOverA,a_fcc,a_bcc)
end function lattice_SchmidMatrix_trans
!--------------------------------------------------------------------------------------------------
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!> @brief Schmid matrix for cleavage
!> details only active cleavage systems are considered
!--------------------------------------------------------------------------------------------------
function lattice_SchmidMatrix_cleavage(Ncleavage,structure,cOverA) result(SchmidMatrix)
use math, only: &
math_tensorproduct33
use IO, only: &
IO_error
implicit none
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integer(pInt), dimension(:), intent(in) :: Ncleavage !< number of active cleavage systems per family
character(len=*), intent(in) :: structure !< lattice structure
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(pInt), dimension(:), allocatable :: NcleavageMax
integer(pInt) :: i
select case(structure)
case('iso')
NcleavageMax = LATTICE_ISO_NCLEAVAGESYSTEM
cleavageSystems = LATTICE_ISO_SYSTEMCLEAVAGE
case('ort')
NcleavageMax = LATTICE_ORT_NCLEAVAGESYSTEM
cleavageSystems = LATTICE_ORT_SYSTEMCLEAVAGE
case('fcc')
NcleavageMax = LATTICE_FCC_NCLEAVAGESYSTEM
cleavageSystems = LATTICE_FCC_SYSTEMCLEAVAGE
case('bcc')
NcleavageMax = LATTICE_BCC_NCLEAVAGESYSTEM
cleavageSystems = LATTICE_BCC_SYSTEMCLEAVAGE
case('hex','hexagonal') !ToDo: "No alias policy": long or short?
NcleavageMax = LATTICE_HEX_NCLEAVAGESYSTEM
cleavageSystems = LATTICE_HEX_SYSTEMCLEAVAGE
case default
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call IO_error(137_pInt,ext_msg='lattice_SchmidMatrix_cleavage: '//trim(structure))
end select
if (any(NcleavageMax(1:size(Ncleavage)) - Ncleavage < 0_pInt)) &
call IO_error(145_pInt,ext_msg='Ncleavage '//trim(structure))
if (any(Ncleavage < 0_pInt)) &
call IO_error(144_pInt,ext_msg='Ncleavage '//trim(structure))
coordinateSystem = buildCoordinateSystem(Ncleavage,NcleavageMax,cleavageSystems,structure,cOverA)
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do i = 1, sum(Ncleavage)
SchmidMatrix(1:3,1:3,1,i) = math_tensorproduct33(coordinateSystem(1:3,1,i),coordinateSystem(1:3,2,i))
SchmidMatrix(1:3,1:3,2,i) = math_tensorproduct33(coordinateSystem(1:3,3,i),coordinateSystem(1:3,2,i))
SchmidMatrix(1:3,1:3,3,i) = math_tensorproduct33(coordinateSystem(1:3,2,i),coordinateSystem(1:3,2,i))
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enddo
end function lattice_SchmidMatrix_cleavage
!--------------------------------------------------------------------------------------------------
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!> @brief Forest projection (for edge dislocations)
!--------------------------------------------------------------------------------------------------
function lattice_forestProjection(Nslip,structure,cOverA) result(projection)
use math, only: &
math_mul3x3
use IO, only: &
IO_error
implicit none
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integer(pInt), dimension(:), intent(in) :: Nslip !< number of active slip systems per family
character(len=*), intent(in) :: structure !< lattice structure
real(pReal), intent(in) :: cOverA !< c/a ratio
real(pReal), dimension(sum(Nslip),sum(Nslip)) :: projection
real(pReal), dimension(3,3,sum(Nslip)) :: coordinateSystem
real(pReal), dimension(:,:), allocatable :: slipSystems
integer(pInt), dimension(:), allocatable :: NslipMax
integer(pInt) :: i, j
select case(structure)
case('fcc')
NslipMax = LATTICE_FCC_NSLIPSYSTEM
slipSystems = LATTICE_FCC_SYSTEMSLIP
case('bcc')
NslipMax = LATTICE_BCC_NSLIPSYSTEM
slipSystems = LATTICE_BCC_SYSTEMSLIP
case('hex','hexagonal') ! ToDo: "No alias policy": long or short?
NslipMax = LATTICE_HEX_NSLIPSYSTEM
slipSystems = LATTICE_HEX_SYSTEMSLIP
case('bct')
NslipMax = LATTICE_BCT_NSLIPSYSTEM
slipSystems = LATTICE_BCT_SYSTEMSLIP
case default
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call IO_error(137_pInt,ext_msg='lattice_forestProjection: '//trim(structure))
end select
if (any(NslipMax(1:size(Nslip)) - Nslip < 0_pInt)) &
call IO_error(145_pInt,ext_msg='Nslip '//trim(structure))
if (any(Nslip < 0_pInt)) &
call IO_error(144_pInt,ext_msg='Nslip '//trim(structure))
coordinateSystem = buildCoordinateSystem(Nslip,NslipMax,slipSystems,structure,cOverA)
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do i=1_pInt, sum(Nslip); do j=1_pInt, sum(Nslip)
projection(i,j) = abs(math_mul3x3(coordinateSystem(1:3,2,i),coordinateSystem(1:3,3,j)))
enddo; enddo
end function lattice_forestProjection
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!--------------------------------------------------------------------------------------------------
!> @brief Populates reduced interaction matrix
!--------------------------------------------------------------------------------------------------
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function buildInteraction(activeA,activeB,maxA,maxB,values,matrix)
use IO, only: &
IO_error
implicit none
integer(pInt), dimension(:), intent(in) :: &
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activeA, & !< number of active systems as specified in material.config
activeB, & !< number of active systems as specified in material.config
maxA, & !< number of maximum available systems
maxB !< number of maximum available systems
real(pReal), dimension(:), intent(in) :: values !< interaction values
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integer(pInt), dimension(:,:), intent(in) :: matrix !< complete interaction matrix
real(pReal), dimension(sum(activeA),sum(activeB)) :: buildInteraction
integer(pInt) :: &
index_myFamily, index_otherFamily, &
mf, ms, of, os
myFamilies: do mf = 1_pInt,size(activeA,1)
index_myFamily = sum(activeA(1:mf-1_pInt))
mySystems: do ms = 1_pInt,activeA(mf)
otherFamilies: do of = 1_pInt,size(activeB,1)
index_otherFamily = sum(activeB(1:of-1_pInt))
otherSystems: do os = 1_pInt,activeB(of)
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if(matrix(sum(maxA(1:mf-1))+ms, sum(maxB(1:of-1))+os) > size(values)) &
call IO_error(138,ext_msg='buildInteraction')
buildInteraction(index_myFamily+ms,index_otherFamily+os) = &
values(matrix(sum(maxA(1:mf-1))+ms, sum(maxB(1:of-1))+os))
enddo otherSystems; enddo otherFamilies;
enddo mySystems;enddo myFamilies
end function buildInteraction
!--------------------------------------------------------------------------------------------------
!> @brief build a local coordinate system in a slip, twin, trans, cleavage system
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!> @details Order: Direction, plane (normal), and common perpendicular
!--------------------------------------------------------------------------------------------------
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function buildCoordinateSystem(active,complete,system,structure,cOverA)
use IO, only: &
IO_error
use math, only: &
math_crossproduct
implicit none
integer(pInt), dimension(:), intent(in) :: &
active, &
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complete
real(pReal), dimension(:,:), intent(in) :: &
system
character(len=*), intent(in) :: &
structure !< lattice structure
real(pReal), intent(in) :: &
cOverA
real(pReal), dimension(3,3,sum(active)) :: &
buildCoordinateSystem
real(pReal), dimension(3) :: &
direction, normal
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integer(pInt) :: &
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a, & !< index of active system
c, & !< index in complete system matrix
f, & !< index of my family
s !< index of my system in current family
if (trim(structure) == 'bct' .and. cOverA > 2.0_pReal) &
call IO_error(131_pInt,ext_msg='buildCoordinateSystem:'//trim(structure))
if (trim(structure) == 'hex' .and. (cOverA < 1.0_pReal .or. cOverA > 2.0_pReal)) &
call IO_error(131_pInt,ext_msg='buildCoordinateSystem:'//trim(structure))
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a = 0_pInt
activeFamilies: do f = 1_pInt,size(active,1)
activeSystems: do s = 1_pInt,active(f)
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a = a + 1_pInt
c = sum(complete(1:f-1))+s
select case(trim(structure))
case ('fcc','bcc','iso','ort','bct')
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direction = system(1:3,c)
normal = system(4:6,c)
case ('hex')
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direction = [ system(1,c)*1.5_pReal, &
(system(1,c)+2.0_pReal*system(2,c))*sqrt(0.75_pReal), &
system(4,c)*cOverA ] ! direction [uvtw]->[3u/2 (u+2v)*sqrt(3)/2 w*(c/a)])
normal = [ system(5,c), &
(system(5,c)+2.0_pReal*system(6,c))/sqrt(3.0_pReal), &
system(8,c)/cOverA ] ! plane (hkil)->(h (h+2k)/sqrt(3) l/(c/a))
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case default
call IO_error(137_pInt,ext_msg='buildCoordinateSystem: '//trim(structure))
end select
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buildCoordinateSystem(1:3,1,a) = direction/norm2(direction)
buildCoordinateSystem(1:3,2,a) = normal/norm2(normal)
buildCoordinateSystem(1:3,3,a) = math_crossproduct(buildCoordinateSystem(1:3,1,a),&
buildCoordinateSystem(1:3,2,a))
enddo activeSystems
enddo activeFamilies
end function buildCoordinateSystem
!--------------------------------------------------------------------------------------------------
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!> @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_bcc = 0.0 for fcc -> bcc transformation
!--------------------------------------------------------------------------------------------------
subroutine buildTransformationSystem(Q,S,Ntrans,cOverA,a_fcc,a_bcc)
use prec, only: &
dEq0
use math, only: &
math_crossproduct, &
math_tensorproduct33, &
math_mul33x33, &
math_mul33x3, &
math_axisAngleToR, &
INRAD, &
MATH_I3
use IO, only: &
IO_error
implicit none
integer(pInt), 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 structure
a_bcc, & !< lattice parameter a for target bcc structure
a_fcc !< lattice parameter a for parent fcc structure
real(pReal), dimension(3,3) :: &
R, & !< Pitsch rotation
U, & !< Bain deformation
B, & !< Rotation of fcc to Bain coordinate system
ss, sd
real(pReal), dimension(3) :: &
x, y, z
integer(pInt) :: &
i
real(pReal), dimension(3+3,LATTICE_FCC_NTRANS), parameter :: &
LATTICE_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(LATTICE_FCCTOHEX_SYSTEMTRANS))
real(pReal), dimension(4,LATTICE_fcc_Ntrans), parameter :: &
LATTICE_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(LATTICE_FCCTOBCC_SYSTEMTRANS))
integer(pInt), dimension(9,LATTICE_fcc_Ntrans), parameter :: &
LATTICE_FCCTOBCC_BAINVARIANT = reshape(int( [&
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 &
],pInt),shape(LATTICE_FCCTOBCC_BAINVARIANT))
real(pReal), dimension(4,LATTICE_fcc_Ntrans), parameter :: &
LATTICE_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(LATTICE_FCCTOBCC_BAINROT))
if (size(Ntrans) < 1_pInt .or. size(Ntrans) > 1_pInt) print*, 'mist' ! ToDo
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if (a_bcc > 0.0_pReal .and. dEq0(cOverA)) then ! fcc -> bcc transformation
if (a_bcc <= 0.0_pReal) print*, 'mist' ! ToDo
do i = 1_pInt,sum(Ntrans)
R = math_axisAngleToR(lattice_fccTobcc_systemTrans(1:3,i), &
lattice_fccTobcc_systemTrans(4,i)*INRAD)
B = math_axisAngleToR(lattice_fccTobcc_bainRot(1:3,i), &
lattice_fccTobcc_bainRot(4,i)*INRAD)
x = real(LATTICE_fccTobcc_bainVariant(1:3,i),pReal)
y = real(LATTICE_fccTobcc_bainVariant(4:6,i),pReal)
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z = real(LATTICE_fccTobcc_bainVariant(7:9,i),pReal)
U = (a_bcc/a_fcc)*math_tensorproduct33(x,x) &
+ (a_bcc/a_fcc)*math_tensorproduct33(y,y) * sqrt(2.0_pReal) &
+ (a_bcc/a_fcc)*math_tensorproduct33(z,z) * sqrt(2.0_pReal)
Q(1:3,1:3,i) = math_mul33x33(R,B)
S(1:3,1:3,i) = math_mul33x33(R,U) - MATH_I3
enddo
elseif (cOverA > 0.0_pReal .and. dEq0(a_bcc)) then ! fcc -> hex transformation
ss = MATH_I3
sd = MATH_I3
ss(1,3) = sqrt(2.0_pReal)/4.0_pReal
if (cOverA > 1.0_pReal .and. cOverA < 2.0_pReal) &
sd(3,3) = cOverA/sqrt(8.0_pReal/3.0_pReal)
do i = 1_pInt,sum(Ntrans)
x = lattice_fccTohex_systemTrans(1:3,i)/norm2(lattice_fccTohex_systemTrans(1:3,i))
z = lattice_fccTohex_systemTrans(4:6,i)/norm2(lattice_fccTohex_systemTrans(4:6,i))
y = -math_crossproduct(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) = math_mul33x33(Q(1:3,1:3,i), math_mul33x33(math_mul33x33(sd,ss), transpose(Q(1:3,1:3,i)))) - MATH_I3 ! ToDo: This is of interest for the Schmid matrix only
enddo
endif
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end subroutine buildTransformationSystem
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end module lattice