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!--------------------------------------------------------------------------------------------------
!> @author Christoph Kords, Max-Planck-Institut für Eisenforschung GmbH
!> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH
!> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH
!> @brief material subroutine for plasticity including dislocation flux
!--------------------------------------------------------------------------------------------------
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submodule ( phase : plastic ) nonlocal
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use geometry_plastic_nonlocal , only : &
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nIPneighbors = > geometry_plastic_nonlocal_nIPneighbors , &
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IPneighborhood = > geometry_plastic_nonlocal_IPneighborhood , &
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IPvolume = > geometry_plastic_nonlocal_IPvolume0 , &
IParea = > geometry_plastic_nonlocal_IParea0 , &
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IPareaNormal = > geometry_plastic_nonlocal_IPareaNormal0 , &
geometry_plastic_nonlocal_disable
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type :: tGeometry
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real ( pREAL ) , dimension ( : ) , allocatable :: V_0
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integer , dimension ( : , : , : ) , allocatable :: IPneighborhood
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real ( pREAL ) , dimension ( : , : ) , allocatable :: IParea , IPcoordinates
real ( pREAL ) , dimension ( : , : , : ) , allocatable :: IPareaNormal
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end type tGeometry
type ( tGeometry ) , dimension ( : ) , allocatable :: geom
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! storage order of dislocation types
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integer , dimension ( * ) , parameter :: &
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sgl = [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ] !< signed (single)
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integer , dimension ( * ) , parameter :: &
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edg = [ 1 , 2 , 5 , 6 , 9 ] , & !< edge
scr = [ 3 , 4 , 7 , 8 , 10 ] !< screw
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integer , dimension ( * ) , parameter :: &
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mob = [ 1 , 2 , 3 , 4 ] , & !< mobile
imm = [ 5 , 6 , 7 , 8 ] !< immobile (blocked)
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integer , dimension ( * ) , parameter :: &
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dip = [ 9 , 10 ] , & !< dipole
imm_edg = imm ( 1 : 2 ) , & !< immobile edge
imm_scr = imm ( 3 : 4 ) !< immobile screw
integer , parameter :: &
mob_edg_pos = 1 , & !< mobile edge positive
mob_edg_neg = 2 , & !< mobile edge negative
mob_scr_pos = 3 , & !< mobile screw positive
mob_scr_neg = 4 !< mobile screw positive
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! BEGIN DEPRECATED
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integer , dimension ( : , : , : ) , allocatable :: &
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iRhoU , & !< state indices for unblocked density
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iV , & !< state indices for dislocation velocities
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iD !< state indices for stable dipole height
!END DEPRECATED
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real ( pREAL ) , dimension ( : , : , : , : , : , : ) , allocatable :: &
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compatibility !< slip system compatibility between en and my neighbors
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type :: tInitialParameters !< container type for internal constitutive parameters
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real ( pREAL ) :: &
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sigma_rho_u , & !< standard deviation of scatter in initial dislocation density
random_rho_u , &
random_rho_u_binning
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real ( pREAL ) , dimension ( : ) , allocatable :: &
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rho_u_ed_pos_0 , & !< initial edge_pos dislocation density
rho_u_ed_neg_0 , & !< initial edge_neg dislocation density
rho_u_sc_pos_0 , & !< initial screw_pos dislocation density
rho_u_sc_neg_0 , & !< initial screw_neg dislocation density
rho_d_ed_0 , & !< initial edge dipole dislocation density
rho_d_sc_0 !< initial screw dipole dislocation density
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integer , dimension ( : ) , allocatable :: &
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N_sl
end type tInitialParameters
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type :: tParameters !< container type for internal constitutive parameters
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real ( pREAL ) :: &
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V_at , & !< atomic volume
D_0 , & !< prefactor for self-diffusion coefficient
Q_cl , & !< activation enthalpy for diffusion
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atol_rho , & !< absolute tolerance for dislocation density in state integration
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rho_significant , & !< density considered significant
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rho_min , & !< number of dislocations considered significant
w , & !< width of a doubkle kink in multiples of the Burgers vector length b
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Q_sol , & !< activation energy for solid solution in J
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f_sol , & !< solid solution obstacle size in multiples of the Burgers vector length
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c_sol , & !< concentration of solid solution in atomic parts
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p , & !< parameter for kinetic law (Kocks,Argon,Ashby)
q , & !< parameter for kinetic law (Kocks,Argon,Ashby)
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B , & !< drag coefficient in Pa s
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nu_a , & !< attack frequency in Hz
chi_surface , & !< transmissivity at free surface
chi_GB , & !< transmissivity at grain boundary (identified by different texture)
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C_CFL , & !< safety factor for CFL flux condition
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f_ed_mult , & !< factor that determines how much edge dislocations contribute to multiplication (0...1)
f_F , &
f_ed , &
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mu , &
nu
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real ( pREAL ) , dimension ( : ) , allocatable :: &
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i_sl , & !< mean free path prefactor for each
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b_sl !< absolute length of Burgers vector [m]
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real ( pREAL ) , dimension ( : , : ) , allocatable :: &
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slip_normal , &
slip_direction , &
slip_transverse , &
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minDipoleHeight , & ! minimum stable dipole height edge and screw
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peierlsstress , & ! edge and screw
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h_sl_sl , & !< coefficients for slip-slip interaction
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forestProjection_Edge , & !< matrix of forest projections of edge dislocations
forestProjection_Screw !< matrix of forest projections of screw dislocations
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real ( pREAL ) , dimension ( : , : , : ) , allocatable :: &
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P_sl , & !< Schmid contribution
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P_nS_pos , &
P_nS_neg !< combined projection of Schmid and non-Schmid contributions to the resolved shear stress (only for screws)
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integer :: &
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sum_N_sl = 0
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integer , dimension ( : ) , allocatable :: &
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colinearSystem !< colinear system to the active slip system (only valid for fcc!)
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character ( len = : ) , allocatable :: &
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isotropic_bound
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character ( len = pSTRLEN ) , dimension ( : ) , allocatable :: &
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output
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logical :: &
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shortRangeStressCorrection !< use of short range stress correction by excess density gradient term
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character ( len = : ) , allocatable , dimension ( : ) :: &
systems_sl
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end type tParameters
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type :: tNonlocalDependentState
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real ( pREAL ) , allocatable , dimension ( : , : ) :: &
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tau_pass , &
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tau_back
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real ( pREAL ) , allocatable , dimension ( : , : , : , : , : ) :: &
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compatibility
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end type tNonlocalDependentState
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type :: tNonlocalState
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real ( pREAL ) , pointer , dimension ( : , : ) :: &
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rho , & ! < all dislocations
rhoSgl , &
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rhoSglMobile , & ! iRhoU
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rho_sgl_mob_edg_pos , &
rho_sgl_mob_edg_neg , &
rho_sgl_mob_scr_pos , &
rho_sgl_mob_scr_neg , &
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rhoSglImmobile , &
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rho_sgl_imm_edg_pos , &
rho_sgl_imm_edg_neg , &
rho_sgl_imm_scr_pos , &
rho_sgl_imm_scr_neg , &
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rhoDip , &
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rho_dip_edg , &
rho_dip_scr , &
rho_forest , &
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gamma , &
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v , &
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v_edg_pos , &
v_edg_neg , &
v_scr_pos , &
v_scr_neg
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end type tNonlocalState
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type ( tNonlocalState ) , allocatable , dimension ( : ) :: &
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deltaState , &
dotState , &
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state , &
state0
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type ( tParameters ) , dimension ( : ) , allocatable :: param !< containers of constitutive parameters
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type ( tNonlocalDependentState ) , dimension ( : ) , allocatable :: dependentState
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contains
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!--------------------------------------------------------------------------------------------------
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!> @brief Perform module initialization.
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!> @details reads in material parameters, allocates arrays, and does sanity checks
!--------------------------------------------------------------------------------------------------
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module function plastic_nonlocal_init ( ) result ( myPlasticity )
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logical , dimension ( : ) , allocatable :: myPlasticity
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integer :: &
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ph , &
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Nmembers , &
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sizeState , sizeDotState , sizeDependentState , sizeDeltaState , &
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s1 , s2 , &
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s , t , l
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real ( pREAL ) , dimension ( : , : ) , allocatable :: &
a_nS !< non-Schmid coefficients
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character ( len = : ) , allocatable :: &
refs , &
extmsg
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type ( tInitialParameters ) :: &
ini
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type ( tDict ) , pointer :: &
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phases , &
phase , &
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mech , &
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pl
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myPlasticity = plastic_active ( 'nonlocal' )
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if ( count ( myPlasticity ) == 0 ) then
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call geometry_plastic_nonlocal_disable ( )
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return
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end if
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print '(/,1x,a)' , '<<<+- phase:mechanical:plastic:nonlocal init -+>>>'
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print '(/,1x,a)' , 'C. Reuber et al., Acta Materialia 71:333–
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print '( 1x,a)' , 'https://doi.org/10.1016/j.actamat.2014.03.012'
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print '(/,1x,a)' , 'C. Kords, Dissertation RWTH Aachen, 2014'
print '( 1x,a)' , 'http://publications.rwth-aachen.de/record/229993'
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print '(/,1x,a,1x,i0)' , '# phases:' , count ( myPlasticity ) ; flush ( IO_STDOUT )
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phases = > config_material % get_dict ( 'phase' )
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allocate ( geom ( phases % length ) )
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allocate ( param ( phases % length ) )
allocate ( state ( phases % length ) )
allocate ( state0 ( phases % length ) )
allocate ( dotState ( phases % length ) )
allocate ( deltaState ( phases % length ) )
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allocate ( dependentState ( phases % length ) )
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extmsg = ''
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do ph = 1 , phases % length
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if ( . not . myPlasticity ( ph ) ) cycle
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associate ( prm = > param ( ph ) , dot = > dotState ( ph ) , stt = > state ( ph ) , &
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st0 = > state0 ( ph ) , del = > deltaState ( ph ) , dst = > dependentState ( ph ) )
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phase = > phases % get_dict ( ph )
mech = > phase % get_dict ( 'mechanical' )
pl = > mech % get_dict ( 'plastic' )
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print '(/,1x,a,1x,i0,a)' , 'phase' , ph , ': ' / / phases % key ( ph )
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refs = config_listReferences ( pl , indent = 3 )
if ( len ( refs ) > 0 ) print '(/,1x,a)' , refs
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#if defined (__GFORTRAN__)
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prm % output = output_as1dStr ( pl )
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#else
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prm % output = pl % get_as1dStr ( 'output' , defaultVal = emptyStrArray )
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#endif
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plasticState ( ph ) % nonlocal = pl % get_asBool ( 'flux' , defaultVal = . True . )
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prm % isotropic_bound = pl % get_asStr ( 'isotropic_bound' , defaultVal = 'isostrain' )
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prm % atol_rho = pl % get_asReal ( 'atol_rho' , defaultVal = 1.0_pREAL )
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ini % N_sl = pl % get_as1dInt ( 'N_sl' , defaultVal = emptyIntArray )
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prm % sum_N_sl = sum ( abs ( ini % N_sl ) )
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slipActive : if ( prm % sum_N_sl > 0 ) then
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prm % systems_sl = crystal_labels_slip ( ini % N_sl , phase_lattice ( ph ) )
prm % P_sl = crystal_SchmidMatrix_slip ( ini % N_sl , phase_lattice ( ph ) , phase_cOverA ( ph ) )
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if ( phase_lattice ( ph ) == 'cI' ) then
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allocate ( a_nS ( 3 , size ( pl % get_as1dReal ( 'a_nonSchmid_110' , defaultVal = emptyRealArray ) ) ) , source = 0.0_pREAL ) ! anticipating parameters for all three families
a_nS ( 1 , : ) = pl % get_as1dReal ( 'a_nonSchmid_110' , defaultVal = emptyRealArray )
prm % P_nS_pos = crystal_SchmidMatrix_slip ( ini % N_sl , phase_lattice ( ph ) , phase_cOverA ( ph ) , nonSchmidCoefficients = a_nS , sense = + 1 )
prm % P_nS_neg = crystal_SchmidMatrix_slip ( ini % N_sl , phase_lattice ( ph ) , phase_cOverA ( ph ) , nonSchmidCoefficients = a_nS , sense = - 1 )
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else
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prm % P_nS_pos = + prm % P_sl
prm % P_nS_neg = - prm % P_sl
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end if
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prm % h_sl_sl = crystal_interaction_SlipBySlip ( ini % N_sl , pl % get_as1dReal ( 'h_sl-sl' ) , &
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phase_lattice ( ph ) )
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prm % forestProjection_edge = crystal_forestProjection_edge ( ini % N_sl , phase_lattice ( ph ) , &
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phase_cOverA ( ph ) )
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prm % forestProjection_screw = crystal_forestProjection_screw ( ini % N_sl , phase_lattice ( ph ) , &
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phase_cOverA ( ph ) )
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prm % slip_direction = crystal_slip_direction ( ini % N_sl , phase_lattice ( ph ) , phase_cOverA ( ph ) )
prm % slip_transverse = crystal_slip_transverse ( ini % N_sl , phase_lattice ( ph ) , phase_cOverA ( ph ) )
prm % slip_normal = crystal_slip_normal ( ini % N_sl , phase_lattice ( ph ) , phase_cOverA ( ph ) )
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! collinear systems (only for octahedral slip systems in fcc)
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allocate ( prm % colinearSystem ( prm % sum_N_sl ) , source = - 1 )
do s1 = 1 , prm % sum_N_sl
do s2 = 1 , prm % sum_N_sl
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if ( all ( dEq0 ( math_cross ( prm % slip_direction ( 1 : 3 , s1 ) , prm % slip_direction ( 1 : 3 , s2 ) ) ) ) . and . &
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any ( dNeq0 ( math_cross ( prm % slip_normal ( 1 : 3 , s1 ) , prm % slip_normal ( 1 : 3 , s2 ) ) ) ) ) &
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prm % colinearSystem ( s1 ) = s2
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end do
end do
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ini % rho_u_ed_pos_0 = pl % get_as1dReal ( 'rho_u_ed_pos_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_u_ed_neg_0 = pl % get_as1dReal ( 'rho_u_ed_neg_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_u_sc_pos_0 = pl % get_as1dReal ( 'rho_u_sc_pos_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_u_sc_neg_0 = pl % get_as1dReal ( 'rho_u_sc_neg_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_d_ed_0 = pl % get_as1dReal ( 'rho_d_ed_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_d_sc_0 = pl % get_as1dReal ( 'rho_d_sc_0' , requiredSize = size ( ini % N_sl ) )
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prm % i_sl = math_expand ( pl % get_as1dReal ( 'i_sl' , requiredSize = size ( ini % N_sl ) ) , ini % N_sl )
prm % b_sl = math_expand ( pl % get_as1dReal ( 'b_sl' , requiredSize = size ( ini % N_sl ) ) , ini % N_sl )
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allocate ( prm % minDipoleHeight ( prm % sum_N_sl , 2 ) )
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prm % minDipoleHeight ( : , 1 ) = math_expand ( pl % get_as1dReal ( 'd_ed' , requiredSize = size ( ini % N_sl ) ) , ini % N_sl )
prm % minDipoleHeight ( : , 2 ) = math_expand ( pl % get_as1dReal ( 'd_sc' , requiredSize = size ( ini % N_sl ) ) , ini % N_sl )
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allocate ( prm % peierlsstress ( prm % sum_N_sl , 2 ) )
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prm % peierlsstress ( : , 1 ) = math_expand ( pl % get_as1dReal ( 'tau_Peierls_ed' , requiredSize = size ( ini % N_sl ) ) , ini % N_sl )
prm % peierlsstress ( : , 2 ) = math_expand ( pl % get_as1dReal ( 'tau_Peierls_sc' , requiredSize = size ( ini % N_sl ) ) , ini % N_sl )
prm % rho_significant = pl % get_asReal ( 'rho_significant' )
prm % rho_min = pl % get_asReal ( 'rho_min' , 0.0_pREAL )
prm % C_CFL = pl % get_asReal ( 'C_CFL' , defaultVal = 2.0_pREAL )
prm % V_at = pl % get_asReal ( 'V_at' )
prm % D_0 = pl % get_asReal ( 'D_0' )
prm % Q_cl = pl % get_asReal ( 'Q_cl' )
prm % f_F = pl % get_asReal ( 'f_F' )
prm % f_ed = pl % get_asReal ( 'f_ed' )
prm % w = pl % get_asReal ( 'w' )
prm % Q_sol = pl % get_asReal ( 'Q_sol' )
prm % f_sol = pl % get_asReal ( 'f_sol' )
prm % c_sol = pl % get_asReal ( 'c_sol' )
prm % p = pl % get_asReal ( 'p_sl' )
prm % q = pl % get_asReal ( 'q_sl' )
prm % B = pl % get_asReal ( 'B' )
prm % nu_a = pl % get_asReal ( 'nu_a' )
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! ToDo: discuss logic
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ini % sigma_rho_u = pl % get_asReal ( 'sigma_rho_u' )
ini % random_rho_u = pl % get_asReal ( 'random_rho_u' , defaultVal = 0.0_pREAL )
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if ( pl % contains ( 'random_rho_u' ) ) &
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ini % random_rho_u_binning = pl % get_asReal ( 'random_rho_u_binning' , defaultVal = 0.0_pREAL ) !ToDo: useful default?
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! if (rhoSglRandom(instance) < 0.0_pREAL) &
! if (rhoSglRandomBinning(instance) <= 0.0_pREAL) &
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prm % chi_surface = pl % get_asReal ( 'chi_surface' , defaultVal = 1.0_pREAL )
prm % chi_GB = pl % get_asReal ( 'chi_GB' , defaultVal = - 1.0_pREAL )
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prm % f_ed_mult = pl % get_asReal ( 'f_ed_mult' )
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prm % shortRangeStressCorrection = pl % get_asBool ( 'short_range_stress_correction' , defaultVal = . false . )
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!--------------------------------------------------------------------------------------------------
! sanity checks
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if ( any ( prm % b_sl < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' b_sl'
if ( any ( prm % i_sl < = 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' i_sl'
if ( any ( ini % rho_u_ed_pos_0 < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' rho_u_ed_pos_0'
if ( any ( ini % rho_u_ed_neg_0 < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' rho_u_ed_neg_0'
if ( any ( ini % rho_u_sc_pos_0 < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' rho_u_sc_pos_0'
if ( any ( ini % rho_u_sc_neg_0 < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' rho_u_sc_neg_0'
if ( any ( ini % rho_d_ed_0 < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' rho_d_ed_0'
if ( any ( ini % rho_d_sc_0 < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' rho_d_sc_0'
if ( any ( prm % peierlsstress < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' tau_peierls'
if ( any ( prm % minDipoleHeight < 0.0_pREAL ) ) extmsg = trim ( extmsg ) / / ' d_ed or d_sc'
if ( prm % B < 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' B'
if ( prm % Q_cl < 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' Q_cl'
if ( prm % nu_a < = 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' nu_a'
if ( prm % w < = 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' w'
if ( prm % D_0 < 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' D_0'
if ( prm % V_at < = 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' V_at' ! ToDo: in dislotungsten, the atomic volume is given as a factor
if ( prm % rho_min < 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' rho_min'
if ( prm % rho_significant < 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' rho_significant'
if ( prm % atol_rho < 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' atol_rho'
if ( prm % C_CFL < 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' C_CFL'
if ( prm % p < = 0.0_pREAL . or . prm % p > 1.0_pREAL ) extmsg = trim ( extmsg ) / / ' p_sl'
if ( prm % q < 1.0_pREAL . or . prm % q > 2.0_pREAL ) extmsg = trim ( extmsg ) / / ' q_sl'
if ( prm % f_F < 0.0_pREAL . or . prm % f_F > 1.0_pREAL ) &
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extmsg = trim ( extmsg ) / / ' f_F'
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if ( prm % f_ed < 0.0_pREAL . or . prm % f_ed > 1.0_pREAL ) &
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extmsg = trim ( extmsg ) / / ' f_ed'
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if ( prm % Q_sol < = 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' Q_sol'
if ( prm % f_sol < = 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' f_sol'
if ( prm % c_sol < = 0.0_pREAL ) extmsg = trim ( extmsg ) / / ' c_sol'
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if ( prm % chi_GB > 1.0_pREAL ) extmsg = trim ( extmsg ) / / ' chi_GB'
if ( prm % chi_surface < 0.0_pREAL . or . prm % chi_surface > 1.0_pREAL ) &
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extmsg = trim ( extmsg ) / / ' chi_surface'
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if ( prm % f_ed_mult < 0.0_pREAL . or . prm % f_ed_mult > 1.0_pREAL ) &
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extmsg = trim ( extmsg ) / / ' f_ed_mult'
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end if slipActive
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!--------------------------------------------------------------------------------------------------
! allocate state arrays
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Nmembers = count ( material_ID_phase == ph )
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sizeDotState = size ( [ 'rhoSglEdgePosMobile ' , 'rhoSglEdgeNegMobile ' , &
'rhoSglScrewPosMobile ' , 'rhoSglScrewNegMobile ' , &
'rhoSglEdgePosImmobile ' , 'rhoSglEdgeNegImmobile ' , &
'rhoSglScrewPosImmobile' , 'rhoSglScrewNegImmobile' , &
'rhoDipEdge ' , 'rhoDipScrew ' , &
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'gamma ' ] ) * prm % sum_N_sl !< "basic" microstructural state variables that are independent from other state variables
sizeDependentState = size ( [ 'rhoForest ' ] ) * prm % sum_N_sl !< microstructural state variables that depend on other state variables
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sizeState = sizeDotState + sizeDependentState &
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+ size ( [ 'velocityEdgePos ' , 'velocityEdgeNeg ' , &
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'velocityScrewPos ' , 'velocityScrewNeg ' , &
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'maxDipoleHeightEdge ' , 'maxDipoleHeightScrew' ] ) * prm % sum_N_sl !< other dependent state variables that are not updated by microstructure
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sizeDeltaState = sizeDotState
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call phase_allocateState ( plasticState ( ph ) , Nmembers , sizeState , sizeDotState , sizeDeltaState , 0 ) ! ToDo: state structure does not follow convention
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allocate ( geom ( ph ) % V_0 ( Nmembers ) )
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allocate ( geom ( ph ) % IPneighborhood ( 3 , nIPneighbors , Nmembers ) )
allocate ( geom ( ph ) % IPareaNormal ( 3 , nIPneighbors , Nmembers ) )
allocate ( geom ( ph ) % IParea ( nIPneighbors , Nmembers ) )
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allocate ( geom ( ph ) % IPcoordinates ( 3 , Nmembers ) )
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call storeGeometry ( ph )
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if ( plasticState ( ph ) % nonlocal . and . . not . allocated ( IPneighborhood ) ) &
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call IO_error ( 212 , ext_msg = 'IPneighborhood does not exist' )
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st0 % rho = > plasticState ( ph ) % state0 ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
stt % rho = > plasticState ( ph ) % state ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
dot % rho = > plasticState ( ph ) % dotState ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
del % rho = > plasticState ( ph ) % deltaState ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
plasticState ( ph ) % atol ( 1 : 10 * prm % sum_N_sl ) = prm % atol_rho
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stt % rhoSgl = > plasticState ( ph ) % state ( 0 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
dot % rhoSgl = > plasticState ( ph ) % dotState ( 0 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
del % rhoSgl = > plasticState ( ph ) % deltaState ( 0 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
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stt % rhoSglMobile = > plasticState ( ph ) % state ( 0 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
dot % rhoSglMobile = > plasticState ( ph ) % dotState ( 0 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
del % rhoSglMobile = > plasticState ( ph ) % deltaState ( 0 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
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stt % rho_sgl_mob_edg_pos = > plasticState ( ph ) % state ( 0 * prm % sum_N_sl + 1 : 1 * prm % sum_N_sl , : )
dot % rho_sgl_mob_edg_pos = > plasticState ( ph ) % dotState ( 0 * prm % sum_N_sl + 1 : 1 * prm % sum_N_sl , : )
del % rho_sgl_mob_edg_pos = > plasticState ( ph ) % deltaState ( 0 * prm % sum_N_sl + 1 : 1 * prm % sum_N_sl , : )
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stt % rho_sgl_mob_edg_neg = > plasticState ( ph ) % state ( 1 * prm % sum_N_sl + 1 : 2 * prm % sum_N_sl , : )
dot % rho_sgl_mob_edg_neg = > plasticState ( ph ) % dotState ( 1 * prm % sum_N_sl + 1 : 2 * prm % sum_N_sl , : )
del % rho_sgl_mob_edg_neg = > plasticState ( ph ) % deltaState ( 1 * prm % sum_N_sl + 1 : 2 * prm % sum_N_sl , : )
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stt % rho_sgl_mob_scr_pos = > plasticState ( ph ) % state ( 2 * prm % sum_N_sl + 1 : 3 * prm % sum_N_sl , : )
dot % rho_sgl_mob_scr_pos = > plasticState ( ph ) % dotState ( 2 * prm % sum_N_sl + 1 : 3 * prm % sum_N_sl , : )
del % rho_sgl_mob_scr_pos = > plasticState ( ph ) % deltaState ( 2 * prm % sum_N_sl + 1 : 3 * prm % sum_N_sl , : )
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stt % rho_sgl_mob_scr_neg = > plasticState ( ph ) % state ( 3 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
dot % rho_sgl_mob_scr_neg = > plasticState ( ph ) % dotState ( 3 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
del % rho_sgl_mob_scr_neg = > plasticState ( ph ) % deltaState ( 3 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
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stt % rhoSglImmobile = > plasticState ( ph ) % state ( 4 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
dot % rhoSglImmobile = > plasticState ( ph ) % dotState ( 4 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
del % rhoSglImmobile = > plasticState ( ph ) % deltaState ( 4 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
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stt % rho_sgl_imm_edg_pos = > plasticState ( ph ) % state ( 4 * prm % sum_N_sl + 1 : 5 * prm % sum_N_sl , : )
dot % rho_sgl_imm_edg_pos = > plasticState ( ph ) % dotState ( 4 * prm % sum_N_sl + 1 : 5 * prm % sum_N_sl , : )
del % rho_sgl_imm_edg_pos = > plasticState ( ph ) % deltaState ( 4 * prm % sum_N_sl + 1 : 5 * prm % sum_N_sl , : )
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stt % rho_sgl_imm_edg_neg = > plasticState ( ph ) % state ( 5 * prm % sum_N_sl + 1 : 6 * prm % sum_N_sl , : )
dot % rho_sgl_imm_edg_neg = > plasticState ( ph ) % dotState ( 5 * prm % sum_N_sl + 1 : 6 * prm % sum_N_sl , : )
del % rho_sgl_imm_edg_neg = > plasticState ( ph ) % deltaState ( 5 * prm % sum_N_sl + 1 : 6 * prm % sum_N_sl , : )
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stt % rho_sgl_imm_scr_pos = > plasticState ( ph ) % state ( 6 * prm % sum_N_sl + 1 : 7 * prm % sum_N_sl , : )
dot % rho_sgl_imm_scr_pos = > plasticState ( ph ) % dotState ( 6 * prm % sum_N_sl + 1 : 7 * prm % sum_N_sl , : )
del % rho_sgl_imm_scr_pos = > plasticState ( ph ) % deltaState ( 6 * prm % sum_N_sl + 1 : 7 * prm % sum_N_sl , : )
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stt % rho_sgl_imm_scr_neg = > plasticState ( ph ) % state ( 7 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
dot % rho_sgl_imm_scr_neg = > plasticState ( ph ) % dotState ( 7 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
del % rho_sgl_imm_scr_neg = > plasticState ( ph ) % deltaState ( 7 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
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stt % rhoDip = > plasticState ( ph ) % state ( 8 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
dot % rhoDip = > plasticState ( ph ) % dotState ( 8 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
del % rhoDip = > plasticState ( ph ) % deltaState ( 8 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
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stt % rho_dip_edg = > plasticState ( ph ) % state ( 8 * prm % sum_N_sl + 1 : 9 * prm % sum_N_sl , : )
dot % rho_dip_edg = > plasticState ( ph ) % dotState ( 8 * prm % sum_N_sl + 1 : 9 * prm % sum_N_sl , : )
del % rho_dip_edg = > plasticState ( ph ) % deltaState ( 8 * prm % sum_N_sl + 1 : 9 * prm % sum_N_sl , : )
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stt % rho_dip_scr = > plasticState ( ph ) % state ( 9 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
dot % rho_dip_scr = > plasticState ( ph ) % dotState ( 9 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
del % rho_dip_scr = > plasticState ( ph ) % deltaState ( 9 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
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stt % gamma = > plasticState ( ph ) % state ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl , 1 : Nmembers )
dot % gamma = > plasticState ( ph ) % dotState ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl , 1 : Nmembers )
del % gamma = > plasticState ( ph ) % deltaState ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl , 1 : Nmembers )
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plasticState ( ph ) % atol ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl ) = pl % get_asReal ( 'atol_gamma' , defaultVal = 1.0e-6_pREAL )
if ( any ( plasticState ( ph ) % atol ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl ) < 0.0_pREAL ) ) &
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extmsg = trim ( extmsg ) / / ' atol_gamma'
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stt % rho_forest = > plasticState ( ph ) % state ( 11 * prm % sum_N_sl + 1 : 12 * prm % sum_N_sl , 1 : Nmembers )
stt % v = > plasticState ( ph ) % state ( 12 * prm % sum_N_sl + 1 : 16 * prm % sum_N_sl , 1 : Nmembers )
stt % v_edg_pos = > plasticState ( ph ) % state ( 12 * prm % sum_N_sl + 1 : 13 * prm % sum_N_sl , 1 : Nmembers )
stt % v_edg_neg = > plasticState ( ph ) % state ( 13 * prm % sum_N_sl + 1 : 14 * prm % sum_N_sl , 1 : Nmembers )
stt % v_scr_pos = > plasticState ( ph ) % state ( 14 * prm % sum_N_sl + 1 : 15 * prm % sum_N_sl , 1 : Nmembers )
stt % v_scr_neg = > plasticState ( ph ) % state ( 15 * prm % sum_N_sl + 1 : 16 * prm % sum_N_sl , 1 : Nmembers )
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allocate ( dst % tau_pass ( prm % sum_N_sl , Nmembers ) , source = 0.0_pREAL )
allocate ( dst % tau_back ( prm % sum_N_sl , Nmembers ) , source = 0.0_pREAL )
allocate ( dst % compatibility ( 2 , maxval ( param % sum_N_sl ) , maxval ( param % sum_N_sl ) , nIPneighbors , Nmembers ) , source = 0.0_pREAL )
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end associate
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if ( Nmembers > 0 ) call stateInit ( ini , ph , Nmembers )
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!--------------------------------------------------------------------------------------------------
! exit if any parameter is out of range
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if ( extmsg / = '' ) call IO_error ( 211 , ext_msg = trim ( extmsg ) )
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end do
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allocate ( compatibility ( 2 , maxval ( param % sum_N_sl ) , maxval ( param % sum_N_sl ) , nIPneighbors , &
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discretization_nIPs , discretization_Nelems ) , source = 0.0_pREAL )
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! BEGIN DEPRECATED----------------------------------------------------------------------------------
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allocate ( iRhoU ( maxval ( param % sum_N_sl ) , 4 , phases % length ) , source = 0 )
allocate ( iV ( maxval ( param % sum_N_sl ) , 4 , phases % length ) , source = 0 )
allocate ( iD ( maxval ( param % sum_N_sl ) , 2 , phases % length ) , source = 0 )
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do ph = 1 , phases % length
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if ( . not . myPlasticity ( ph ) ) cycle
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phase = > phases % get_dict ( ph )
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Nmembers = count ( material_ID_phase == ph )
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l = 0
do t = 1 , 4
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do s = 1 , param ( ph ) % sum_N_sl
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l = l + 1
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iRhoU ( s , t , ph ) = l
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end do
end do
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l = l + ( 4 + 2 + 1 + 1 ) * param ( ph ) % sum_N_sl ! immobile(4), dipole(2), shear, forest
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do t = 1 , 4
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do s = 1 , param ( ph ) % sum_N_sl
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l = l + 1
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iV ( s , t , ph ) = l
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end do
end do
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do t = 1 , 2
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do s = 1 , param ( ph ) % sum_N_sl
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l = l + 1
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iD ( s , t , ph ) = l
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end do
end do
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if ( iD ( param ( ph ) % sum_N_sl , 2 , ph ) / = plasticState ( ph ) % sizeState ) &
error stop 'state indices not properly set (nonlocal)'
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end do
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end function plastic_nonlocal_init
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!--------------------------------------------------------------------------------------------------
!> @brief calculates quantities characterizing the microstructure
!--------------------------------------------------------------------------------------------------
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module subroutine nonlocal_dependentState ( ph , en )
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integer , intent ( in ) :: &
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ph , &
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en
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integer :: &
no , & !< neighbor offset
neighbor_el , & ! element number of neighboring material point
neighbor_ip , & ! integration point of neighboring material point
c , & ! index of dilsocation character (edge, screw)
s , & ! slip system index
dir , &
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n
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real ( pREAL ) :: &
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FVsize , &
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nRealNeighbors , & ! number of really existing neighbors
mu , &
nu
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integer , dimension ( 2 ) :: &
neighbors
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real ( pREAL ) , dimension ( 2 ) :: &
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rhoExcessGradient , &
rhoExcessGradient_over_rho , &
rhoTotal
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real ( pREAL ) , dimension ( 3 ) :: &
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rhoExcessDifferences , &
normal_latticeConf
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real ( pREAL ) , dimension ( 3 , 3 ) :: &
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invFe , & !< inverse of elastic deformation gradient
invFp , & !< inverse of plastic deformation gradient
connections , &
invConnections
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real ( pREAL ) , dimension ( 3 , nIPneighbors ) :: &
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connection_latticeConf
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real ( pREAL ) , dimension ( 2 , param ( ph ) % sum_N_sl ) :: &
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rhoExcess
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl ) :: &
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rho_edg_delta , &
rho_scr_delta
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: &
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rho , &
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rho0 , &
rho_neighbor0
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , param ( ph ) % sum_N_sl ) :: &
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myInteractionMatrix ! corrected slip interaction matrix
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , nIPneighbors ) :: &
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rho_edg_delta_neighbor , &
rho_scr_delta_neighbor
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real ( pREAL ) , dimension ( 2 , maxval ( param % sum_N_sl ) , nIPneighbors ) :: &
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neighbor_rhoExcess , & ! excess density at neighboring material point
neighbor_rhoTotal ! total density at neighboring material point
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real ( pREAL ) , dimension ( 3 , param ( ph ) % sum_N_sl , 2 ) :: &
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m ! direction of dislocation motion
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associate ( prm = > param ( ph ) , dst = > dependentState ( ph ) , stt = > state ( ph ) )
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mu = elastic_mu ( ph , en , prm % isotropic_bound )
nu = elastic_nu ( ph , en , prm % isotropic_bound )
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rho = getRho ( ph , en )
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stt % rho_forest ( : , en ) = matmul ( prm % forestProjection_Edge , sum ( abs ( rho ( : , edg ) ) , 2 ) ) &
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+ matmul ( prm % forestProjection_Screw , sum ( abs ( rho ( : , scr ) ) , 2 ) )
! coefficients are corrected for the line tension effect
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! (see Kubin,Devincre,Hoc; 2008; Modeling dislocation storage rates and mean free paths in face-centered cubic crystals)
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if ( any ( phase_lattice ( ph ) == [ 'cI' , 'cF' ] ) ) then
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myInteractionMatrix = prm % h_sl_sl &
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* spread ( ( 1.0_pREAL - prm % f_F &
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+ prm % f_F &
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* log ( 0.35_pREAL * prm % b_sl * sqrt ( max ( stt % rho_forest ( : , en ) , prm % rho_significant ) ) ) &
/ log ( 0.35_pREAL * prm % b_sl * 1e6_pREAL ) ) ** 2 , 2 , prm % sum_N_sl )
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else
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myInteractionMatrix = prm % h_sl_sl
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end if
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dst % tau_pass ( : , en ) = mu * prm % b_sl &
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* sqrt ( matmul ( myInteractionMatrix , sum ( abs ( rho ) , 2 ) ) )
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!*** calculate the dislocation stress of the neighboring excess dislocation densities
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!*** zero for material points of local plasticity
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!#################################################################################################
! ToDo: MD: this is most likely only correct for F_i = I
!#################################################################################################
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rho0 = getRho0 ( ph , en )
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if ( plasticState ( ph ) % nonlocal . and . prm % shortRangeStressCorrection ) then
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invFp = math_inv33 ( phase_mechanical_Fp ( ph ) % data ( 1 : 3 , 1 : 3 , en ) )
invFe = math_inv33 ( phase_mechanical_Fe ( ph ) % data ( 1 : 3 , 1 : 3 , en ) )
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rho_edg_delta = rho0 ( : , mob_edg_pos ) - rho0 ( : , mob_edg_neg )
rho_scr_delta = rho0 ( : , mob_scr_pos ) - rho0 ( : , mob_scr_neg )
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rhoExcess ( 1 , : ) = rho_edg_delta
rhoExcess ( 2 , : ) = rho_scr_delta
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FVsize = geom ( ph ) % V_0 ( en ) ** ( 1.0_pREAL / 3.0_pREAL )
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!* loop through my neighborhood and get the connection vectors (in lattice frame) and the excess densities
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nRealNeighbors = 0.0_pREAL
neighbor_rhoTotal = 0.0_pREAL
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do n = 1 , nIPneighbors
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neighbor_el = geom ( ph ) % IPneighborhood ( 1 , n , en )
neighbor_ip = geom ( ph ) % IPneighborhood ( 2 , n , en )
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if ( neighbor_el > 0 . and . neighbor_ip > 0 ) then
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if ( material_ID_phase ( 1 , ( neighbor_el - 1 ) * discretization_nIPs + neighbor_ip ) == ph ) then
no = material_entry_phase ( 1 , ( neighbor_el - 1 ) * discretization_nIPs + neighbor_ip )
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nRealNeighbors = nRealNeighbors + 1.0_pREAL
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rho_neighbor0 = getRho0 ( ph , no )
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rho_edg_delta_neighbor ( : , n ) = rho_neighbor0 ( : , mob_edg_pos ) - rho_neighbor0 ( : , mob_edg_neg )
rho_scr_delta_neighbor ( : , n ) = rho_neighbor0 ( : , mob_scr_pos ) - rho_neighbor0 ( : , mob_scr_neg )
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neighbor_rhoTotal ( 1 , : , n ) = sum ( abs ( rho_neighbor0 ( : , edg ) ) , 2 )
neighbor_rhoTotal ( 2 , : , n ) = sum ( abs ( rho_neighbor0 ( : , scr ) ) , 2 )
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connection_latticeConf ( 1 : 3 , n ) = matmul ( invFe , geom ( ph ) % IPcoordinates ( 1 : 3 , no ) &
- geom ( ph ) % IPcoordinates ( 1 : 3 , en ) )
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normal_latticeConf = matmul ( transpose ( invFp ) , geom ( ph ) % IPareaNormal ( 1 : 3 , n , en ) )
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if ( math_inner ( normal_latticeConf , connection_latticeConf ( 1 : 3 , n ) ) < 0.0_pREAL ) & ! neighboring connection points in opposite direction to face normal: must be periodic image
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connection_latticeConf ( 1 : 3 , n ) = normal_latticeConf * geom ( ph ) % V_0 ( en ) / geom ( ph ) % IParea ( n , en ) ! instead take the surface normal scaled with the diameter of the cell
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else
! local neighbor or different lattice structure or different constitution instance -> use central values instead
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connection_latticeConf ( 1 : 3 , n ) = 0.0_pREAL
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rho_edg_delta_neighbor ( : , n ) = rho_edg_delta
rho_scr_delta_neighbor ( : , n ) = rho_scr_delta
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end if
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else
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! free surface -> use central values instead
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connection_latticeConf ( 1 : 3 , n ) = 0.0_pREAL
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rho_edg_delta_neighbor ( : , n ) = rho_edg_delta
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rho_scr_delta_neighbor ( : , n ) = rho_scr_delta
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end if
end do
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neighbor_rhoExcess ( 1 , : , : ) = rho_edg_delta_neighbor
neighbor_rhoExcess ( 2 , : , : ) = rho_scr_delta_neighbor
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!* loop through the slip systems and calculate the dislocation gradient by
!* 1. interpolation of the excess density in the neighorhood
!* 2. interpolation of the dead dislocation density in the central volume
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m ( 1 : 3 , : , 1 ) = prm % slip_direction
m ( 1 : 3 , : , 2 ) = - prm % slip_transverse
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do s = 1 , prm % sum_N_sl
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! gradient from interpolation of neighboring excess density ...
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do c = 1 , 2
do dir = 1 , 3
neighbors ( 1 ) = 2 * dir - 1
neighbors ( 2 ) = 2 * dir
connections ( dir , 1 : 3 ) = connection_latticeConf ( 1 : 3 , neighbors ( 1 ) ) &
- connection_latticeConf ( 1 : 3 , neighbors ( 2 ) )
rhoExcessDifferences ( dir ) = neighbor_rhoExcess ( c , s , neighbors ( 1 ) ) &
- neighbor_rhoExcess ( c , s , neighbors ( 2 ) )
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end do
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invConnections = math_inv33 ( connections )
if ( all ( dEq0 ( invConnections ) ) ) call IO_error ( - 1 , ext_msg = 'back stress calculation: inversion error' )
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rhoExcessGradient ( c ) = math_inner ( m ( 1 : 3 , s , c ) , matmul ( invConnections , rhoExcessDifferences ) )
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end do
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! ... plus gradient from deads ...
rhoExcessGradient ( 1 ) = rhoExcessGradient ( 1 ) + sum ( rho ( s , imm_edg ) ) / FVsize
rhoExcessGradient ( 2 ) = rhoExcessGradient ( 2 ) + sum ( rho ( s , imm_scr ) ) / FVsize
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! ... normalized with the total density ...
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rhoTotal ( 1 ) = ( sum ( abs ( rho ( s , edg ) ) ) + sum ( neighbor_rhoTotal ( 1 , s , : ) ) ) / ( 1.0_pREAL + nRealNeighbors )
rhoTotal ( 2 ) = ( sum ( abs ( rho ( s , scr ) ) ) + sum ( neighbor_rhoTotal ( 2 , s , : ) ) ) / ( 1.0_pREAL + nRealNeighbors )
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rhoExcessGradient_over_rho = 0.0_pREAL
where ( rhoTotal > 0.0_pREAL ) rhoExcessGradient_over_rho = rhoExcessGradient / rhoTotal
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! ... gives the local stress correction when multiplied with a factor
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dst % tau_back ( s , en ) = - mu * prm % b_sl ( s ) / ( 2.0_pREAL * PI ) &
* ( rhoExcessGradient_over_rho ( 1 ) / ( 1.0_pREAL - nu ) &
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+ rhoExcessGradient_over_rho ( 2 ) )
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end do
end if
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end associate
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end subroutine nonlocal_dependentState
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!--------------------------------------------------------------------------------------------------
!> @brief calculates plastic velocity gradient and its tangent
!--------------------------------------------------------------------------------------------------
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module subroutine nonlocal_LpAndItsTangent ( Lp , dLp_dMp , &
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Mp , ph , en )
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real ( pREAL ) , dimension ( 3 , 3 ) , intent ( out ) :: &
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Lp !< plastic velocity gradient
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real ( pREAL ) , dimension ( 3 , 3 , 3 , 3 ) , intent ( out ) :: &
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dLp_dMp
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integer , intent ( in ) :: &
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ph , &
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en
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real ( pREAL ) , dimension ( 3 , 3 ) , intent ( in ) :: &
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Mp
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!< derivative of Lp with respect to Mp
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integer :: &
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i , j , k , l , &
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t , & !< dislocation type
s !< index of my current slip system
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 8 ) :: &
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rhoSgl !< single dislocation densities (including blocked)
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: &
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rho
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 4 ) :: &
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v , & !< velocity
tauNS , & !< resolved shear stress including non Schmid and backstress terms
dv_dtau , & !< velocity derivative with respect to the shear stress
dv_dtauNS !< velocity derivative with respect to the shear stress
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl ) :: &
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tau , & !< resolved shear stress including backstress terms
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dot_gamma !< shear rate
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real ( pREAL ) :: &
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Temperature !< temperature
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Temperature = thermal_T ( ph , en )
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Lp = 0.0_pREAL
dLp_dMp = 0.0_pREAL
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associate ( prm = > param ( ph ) , dst = > dependentState ( ph ) , stt = > state ( ph ) )
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!*** shortcut to state variables
rho = getRho ( ph , en )
rhoSgl = rho ( : , sgl )
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do s = 1 , prm % sum_N_sl
tau ( s ) = math_tensordot ( Mp , prm % P_sl ( 1 : 3 , 1 : 3 , s ) )
tauNS ( s , 1 ) = tau ( s )
tauNS ( s , 2 ) = tau ( s )
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if ( tau ( s ) > 0.0_pREAL ) then
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tauNS ( s , 3 ) = math_tensordot ( Mp , + prm % P_nS_pos ( 1 : 3 , 1 : 3 , s ) )
tauNS ( s , 4 ) = math_tensordot ( Mp , - prm % P_nS_neg ( 1 : 3 , 1 : 3 , s ) )
else
tauNS ( s , 3 ) = math_tensordot ( Mp , + prm % P_nS_neg ( 1 : 3 , 1 : 3 , s ) )
tauNS ( s , 4 ) = math_tensordot ( Mp , - prm % P_nS_pos ( 1 : 3 , 1 : 3 , s ) )
end if
end do
tauNS = tauNS + spread ( dst % tau_back ( : , en ) , 2 , 4 )
tau = tau + dst % tau_back ( : , en )
! edges
call kinetics ( v ( : , 1 ) , dv_dtau ( : , 1 ) , dv_dtauNS ( : , 1 ) , &
tau , tauNS ( : , 1 ) , dst % tau_pass ( : , en ) , 1 , Temperature , ph )
v ( : , 2 ) = v ( : , 1 )
dv_dtau ( : , 2 ) = dv_dtau ( : , 1 )
dv_dtauNS ( : , 2 ) = dv_dtauNS ( : , 1 )
!screws
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do t = 3 , 4
call kinetics ( v ( : , t ) , dv_dtau ( : , t ) , dv_dtauNS ( : , t ) , &
tau , tauNS ( : , t ) , dst % tau_pass ( : , en ) , 2 , Temperature , ph )
end do
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stt % v ( : , en ) = pack ( v , . true . )
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!*** Bauschinger effect
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forall ( s = 1 : prm % sum_N_sl , t = 5 : 8 , rhoSgl ( s , t ) * v ( s , t - 4 ) < 0.0_pREAL ) &
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rhoSgl ( s , t - 4 ) = rhoSgl ( s , t - 4 ) + abs ( rhoSgl ( s , t ) )
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dot_gamma = sum ( rhoSgl ( : , 1 : 4 ) * v , 2 ) * prm % b_sl
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do s = 1 , prm % sum_N_sl
Lp = Lp + dot_gamma ( s ) * prm % P_sl ( 1 : 3 , 1 : 3 , s )
forall ( i = 1 : 3 , j = 1 : 3 , k = 1 : 3 , l = 1 : 3 ) &
dLp_dMp ( i , j , k , l ) = dLp_dMp ( i , j , k , l ) &
+ prm % P_sl ( i , j , s ) * prm % P_sl ( k , l , s ) &
* sum ( rhoSgl ( s , 1 : 4 ) * dv_dtau ( s , 1 : 4 ) ) * prm % b_sl ( s ) &
+ prm % P_sl ( i , j , s ) &
* ( + prm % P_nS_pos ( k , l , s ) * rhoSgl ( s , 3 ) * dv_dtauNS ( s , 3 ) &
- prm % P_nS_neg ( k , l , s ) * rhoSgl ( s , 4 ) * dv_dtauNS ( s , 4 ) ) * prm % b_sl ( s )
end do
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end associate
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end subroutine nonlocal_LpAndItsTangent
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2014-06-14 02:23:17 +05:30
!--------------------------------------------------------------------------------------------------
!> @brief (instantaneous) incremental change of microstructure
!--------------------------------------------------------------------------------------------------
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module subroutine plastic_nonlocal_deltaState ( Mp , ph , en )
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real ( pREAL ) , dimension ( 3 , 3 ) , intent ( in ) :: &
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Mp !< MandelStress
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integer , intent ( in ) :: &
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ph , &
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en
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integer :: &
c , & ! character of dislocation
t , & ! type of dislocation
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s ! index of my current slip system
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real ( pREAL ) :: &
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mu , &
nu
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: &
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deltaRhoRemobilization , & ! density increment by remobilization
deltaRhoDipole2SingleStress ! density increment by dipole dissociation (by stress change)
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: &
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rho ! current dislocation densities
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 4 ) :: &
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v ! dislocation glide velocity
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl ) :: &
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tau ! current resolved shear stress
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 2 ) :: &
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rhoDip , & ! current dipole dislocation densities (screw and edge dipoles)
dUpper , & ! current maximum stable dipole distance for edges and screws
dUpperOld , & ! old maximum stable dipole distance for edges and screws
deltaDUpper ! change in maximum stable dipole distance for edges and screws
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associate ( prm = > param ( ph ) , dst = > dependentState ( ph ) , del = > deltaState ( ph ) )
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mu = elastic_mu ( ph , en , prm % isotropic_bound )
nu = elastic_nu ( ph , en , prm % isotropic_bound )
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!*** shortcut to state variables
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forall ( s = 1 : prm % sum_N_sl , t = 1 : 4 ) v ( s , t ) = plasticState ( ph ) % state ( iV ( s , t , ph ) , en )
forall ( s = 1 : prm % sum_N_sl , c = 1 : 2 ) dUpperOld ( s , c ) = plasticState ( ph ) % state ( iD ( s , c , ph ) , en )
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rho = getRho ( ph , en )
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rhoDip = rho ( : , dip )
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!****************************************************************************
!*** dislocation remobilization (bauschinger effect)
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where ( rho ( : , imm ) * v < 0.0_pREAL )
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deltaRhoRemobilization ( : , mob ) = abs ( rho ( : , imm ) )
deltaRhoRemobilization ( : , imm ) = - rho ( : , imm )
rho ( : , mob ) = rho ( : , mob ) + abs ( rho ( : , imm ) )
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rho ( : , imm ) = 0.0_pREAL
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elsewhere
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deltaRhoRemobilization ( : , mob ) = 0.0_pREAL
deltaRhoRemobilization ( : , imm ) = 0.0_pREAL
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endwhere
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deltaRhoRemobilization ( : , dip ) = 0.0_pREAL
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!****************************************************************************
!*** calculate dipole formation and dissociation by stress change
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!*** calculate limits for stable dipole height
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do s = 1 , prm % sum_N_sl
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tau ( s ) = math_tensordot ( Mp , prm % P_sl ( 1 : 3 , 1 : 3 , s ) ) + dst % tau_back ( s , en )
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if ( abs ( tau ( s ) ) < 1.0e-15_pREAL ) tau ( s ) = 1.0e-15_pREAL
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end do
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dUpper ( : , 1 ) = mu * prm % b_sl / ( 8.0_pREAL * PI * ( 1.0_pREAL - nu ) * abs ( tau ) )
dUpper ( : , 2 ) = mu * prm % b_sl / ( 4.0_pREAL * PI * abs ( tau ) )
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where ( dNeq0 ( sqrt ( sum ( abs ( rho ( : , edg ) ) , 2 ) ) ) ) &
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dUpper ( : , 1 ) = min ( 1.0_pREAL / sqrt ( sum ( abs ( rho ( : , edg ) ) , 2 ) ) , dUpper ( : , 1 ) )
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where ( dNeq0 ( sqrt ( sum ( abs ( rho ( : , scr ) ) , 2 ) ) ) ) &
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dUpper ( : , 2 ) = min ( 1.0_pREAL / sqrt ( sum ( abs ( rho ( : , scr ) ) , 2 ) ) , dUpper ( : , 2 ) )
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dUpper = max ( dUpper , prm % minDipoleHeight )
deltaDUpper = dUpper - dUpperOld
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!*** dissociation by stress increase
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deltaRhoDipole2SingleStress = 0.0_pREAL
forall ( c = 1 : 2 , s = 1 : prm % sum_N_sl , deltaDUpper ( s , c ) < 0.0_pREAL . and . &
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dNeq0 ( dUpperOld ( s , c ) - prm % minDipoleHeight ( s , c ) ) ) &
deltaRhoDipole2SingleStress ( s , 8 + c ) = rhoDip ( s , c ) * deltaDUpper ( s , c ) &
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/ ( dUpperOld ( s , c ) - prm % minDipoleHeight ( s , c ) )
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forall ( t = 1 : 4 ) deltaRhoDipole2SingleStress ( : , t ) = - 0.5_pREAL * deltaRhoDipole2SingleStress ( : , ( t - 1 ) / 2 + 9 )
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forall ( s = 1 : prm % sum_N_sl , c = 1 : 2 ) plasticState ( ph ) % state ( iD ( s , c , ph ) , en ) = dUpper ( s , c )
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plasticState ( ph ) % deltaState ( : , en ) = 0.0_pREAL
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del % rho ( : , en ) = reshape ( deltaRhoRemobilization + deltaRhoDipole2SingleStress , [ 10 * prm % sum_N_sl ] )
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end associate
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end subroutine plastic_nonlocal_deltaState
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!---------------------------------------------------------------------------------------------------
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!> @brief calculates the rate of change of microstructure
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!---------------------------------------------------------------------------------------------------
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module subroutine nonlocal_dotState ( Mp , timestep , &
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ph , en )
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real ( pREAL ) , dimension ( 3 , 3 ) , intent ( in ) :: &
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Mp !< MandelStress
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real ( pREAL ) , intent ( in ) :: &
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timestep !< substepped crystallite time increment
integer , intent ( in ) :: &
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ph , &
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en
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integer :: &
c , & !< character of dislocation
t , & !< type of dislocation
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s !< index of my current slip system
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: &
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rho , &
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rho0 , & !< dislocation density at beginning of time step
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rhoDot , & !< density evolution
rhoDotMultiplication , & !< density evolution by multiplication
rhoDotSingle2DipoleGlide , & !< density evolution by dipole formation (by glide)
rhoDotAthermalAnnihilation , & !< density evolution by athermal annihilation
rhoDotThermalAnnihilation !< density evolution by thermal annihilation
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 8 ) :: &
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rhoSgl , & !< current single dislocation densities (positive/negative screw and edge without dipoles)
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my_rhoSgl0 !< single dislocation densities of central ip (positive/negative screw and edge without dipoles)
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 4 ) :: &
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v , & !< current dislocation glide velocity
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v0 , &
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dot_gamma !< shear rates
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl ) :: &
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tau , & !< current resolved shear stress
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v_climb !< climb velocity of edge dipoles
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 2 ) :: &
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rhoDip , & !< current dipole dislocation densities (screw and edge dipoles)
dLower , & !< minimum stable dipole distance for edges and screws
dUpper !< current maximum stable dipole distance for edges and screws
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real ( pREAL ) :: &
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D_SD , &
mu , &
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nu , Temperature
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if ( timestep < = 0.0_pREAL ) then
plasticState ( ph ) % dotState = 0.0_pREAL
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return
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end if
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associate ( prm = > param ( ph ) , dst = > dependentState ( ph ) , dot = > dotState ( ph ) , stt = > state ( ph ) )
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mu = elastic_mu ( ph , en , prm % isotropic_bound )
nu = elastic_nu ( ph , en , prm % isotropic_bound )
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Temperature = thermal_T ( ph , en )
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tau = 0.0_pREAL
dot_gamma = 0.0_pREAL
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rho = getRho ( ph , en )
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rhoSgl = rho ( : , sgl )
rhoDip = rho ( : , dip )
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rho0 = getRho0 ( ph , en )
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my_rhoSgl0 = rho0 ( : , sgl )
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forall ( s = 1 : prm % sum_N_sl , t = 1 : 4 ) v ( s , t ) = plasticState ( ph ) % state ( iV ( s , t , ph ) , en )
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dot_gamma = rhoSgl ( : , 1 : 4 ) * v * spread ( prm % b_sl , 2 , 4 )
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! limits for stable dipole height
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do s = 1 , prm % sum_N_sl
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tau ( s ) = math_tensordot ( Mp , prm % P_sl ( 1 : 3 , 1 : 3 , s ) ) + dst % tau_back ( s , en )
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if ( abs ( tau ( s ) ) < 1.0e-15_pREAL ) tau ( s ) = 1.0e-15_pREAL
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end do
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dLower = prm % minDipoleHeight
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dUpper ( : , 1 ) = mu * prm % b_sl / ( 8.0_pREAL * PI * ( 1.0_pREAL - nu ) * abs ( tau ) )
dUpper ( : , 2 ) = mu * prm % b_sl / ( 4.0_pREAL * PI * abs ( tau ) )
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where ( dNeq0 ( sqrt ( sum ( abs ( rho ( : , edg ) ) , 2 ) ) ) ) &
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dUpper ( : , 1 ) = min ( 1.0_pREAL / sqrt ( sum ( abs ( rho ( : , edg ) ) , 2 ) ) , dUpper ( : , 1 ) )
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where ( dNeq0 ( sqrt ( sum ( abs ( rho ( : , scr ) ) , 2 ) ) ) ) &
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dUpper ( : , 2 ) = min ( 1.0_pREAL / sqrt ( sum ( abs ( rho ( : , scr ) ) , 2 ) ) , dUpper ( : , 2 ) )
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dUpper = max ( dUpper , dLower )
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! dislocation multiplication
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rhoDotMultiplication = 0.0_pREAL
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isBCC : if ( phase_lattice ( ph ) == 'cI' ) then
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forall ( s = 1 : prm % sum_N_sl , sum ( abs ( v ( s , 1 : 4 ) ) ) > 0.0_pREAL )
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rhoDotMultiplication ( s , 1 : 2 ) = sum ( abs ( dot_gamma ( s , 3 : 4 ) ) ) / prm % b_sl ( s ) & ! assuming double-cross-slip of screws to be decisive for multiplication
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* sqrt ( stt % rho_forest ( s , en ) ) / prm % i_sl ( s ) ! & ! mean free path
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! * 2.0_pREAL * sum(abs(v(s,3:4))) / sum(abs(v(s,1:4))) ! ratio of screw to overall velocity determines edge generation
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rhoDotMultiplication ( s , 3 : 4 ) = sum ( abs ( dot_gamma ( s , 3 : 4 ) ) ) / prm % b_sl ( s ) & ! assuming double-cross-slip of screws to be decisive for multiplication
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* sqrt ( stt % rho_forest ( s , en ) ) / prm % i_sl ( s ) ! & ! mean free path
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! * 2.0_pREAL * sum(abs(v(s,1:2))) / sum(abs(v(s,1:4))) ! ratio of edge to overall velocity determines screw generation
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endforall
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else isBCC
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rhoDotMultiplication ( : , 1 : 4 ) = spread ( &
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( sum ( abs ( dot_gamma ( : , 1 : 2 ) ) , 2 ) * prm % f_ed_mult + sum ( abs ( dot_gamma ( : , 3 : 4 ) ) , 2 ) ) &
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* sqrt ( stt % rho_forest ( : , en ) ) / prm % i_sl / prm % b_sl , 2 , 4 ) ! eq. 3.26
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end if isBCC
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forall ( s = 1 : prm % sum_N_sl , t = 1 : 4 ) v0 ( s , t ) = plasticState ( ph ) % state0 ( iV ( s , t , ph ) , en )
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!****************************************************************************
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! dipole formation and annihilation
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! formation by glide
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do c = 1 , 2
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rhoDotSingle2DipoleGlide ( : , 2 * c - 1 ) = - 2.0_pREAL * dUpper ( : , c ) / prm % b_sl &
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* ( rhoSgl ( : , 2 * c - 1 ) * abs ( dot_gamma ( : , 2 * c ) ) & ! negative mobile --> positive mobile
+ rhoSgl ( : , 2 * c ) * abs ( dot_gamma ( : , 2 * c - 1 ) ) & ! positive mobile --> negative mobile
+ abs ( rhoSgl ( : , 2 * c + 4 ) ) * abs ( dot_gamma ( : , 2 * c - 1 ) ) ) ! positive mobile --> negative immobile
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rhoDotSingle2DipoleGlide ( : , 2 * c ) = - 2.0_pREAL * dUpper ( : , c ) / prm % b_sl &
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* ( rhoSgl ( : , 2 * c - 1 ) * abs ( dot_gamma ( : , 2 * c ) ) & ! negative mobile --> positive mobile
+ rhoSgl ( : , 2 * c ) * abs ( dot_gamma ( : , 2 * c - 1 ) ) & ! positive mobile --> negative mobile
+ abs ( rhoSgl ( : , 2 * c + 3 ) ) * abs ( dot_gamma ( : , 2 * c ) ) ) ! negative mobile --> positive immobile
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rhoDotSingle2DipoleGlide ( : , 2 * c + 3 ) = - 2.0_pREAL * dUpper ( : , c ) / prm % b_sl &
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* rhoSgl ( : , 2 * c + 3 ) * abs ( dot_gamma ( : , 2 * c ) ) ! negative mobile --> positive immobile
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rhoDotSingle2DipoleGlide ( : , 2 * c + 4 ) = - 2.0_pREAL * dUpper ( : , c ) / prm % b_sl &
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* rhoSgl ( : , 2 * c + 4 ) * abs ( dot_gamma ( : , 2 * c - 1 ) ) ! positive mobile --> negative immobile
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rhoDotSingle2DipoleGlide ( : , c + 8 ) = abs ( rhoDotSingle2DipoleGlide ( : , 2 * c + 3 ) ) &
+ abs ( rhoDotSingle2DipoleGlide ( : , 2 * c + 4 ) ) &
- rhoDotSingle2DipoleGlide ( : , 2 * c - 1 ) &
- rhoDotSingle2DipoleGlide ( : , 2 * c )
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end do
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! athermal annihilation
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rhoDotAthermalAnnihilation = 0.0_pREAL
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forall ( c = 1 : 2 ) &
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rhoDotAthermalAnnihilation ( : , c + 8 ) = - 2.0_pREAL * dLower ( : , c ) / prm % b_sl &
* ( 2.0_pREAL * ( rhoSgl ( : , 2 * c - 1 ) * abs ( dot_gamma ( : , 2 * c ) ) + rhoSgl ( : , 2 * c ) * abs ( dot_gamma ( : , 2 * c - 1 ) ) ) & ! was single hitting single
+ 2.0_pREAL * ( abs ( rhoSgl ( : , 2 * c + 3 ) ) * abs ( dot_gamma ( : , 2 * c ) ) + abs ( rhoSgl ( : , 2 * c + 4 ) ) * abs ( dot_gamma ( : , 2 * c - 1 ) ) ) & ! was single hitting immobile single or was immobile single hit by single
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+ rhoDip ( : , c ) * ( abs ( dot_gamma ( : , 2 * c - 1 ) ) + abs ( dot_gamma ( : , 2 * c ) ) ) ) ! single knocks dipole constituent
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! annihilated screw dipoles leave edge jogs behind on the colinear system
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if ( phase_lattice ( ph ) == 'cF' ) &
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forall ( s = 1 : prm % sum_N_sl , prm % colinearSystem ( s ) > 0 ) &
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rhoDotAthermalAnnihilation ( prm % colinearSystem ( s ) , 1 : 2 ) = - rhoDotAthermalAnnihilation ( s , 10 ) &
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* 0.25_pREAL * sqrt ( stt % rho_forest ( s , en ) ) * ( dUpper ( s , 2 ) + dLower ( s , 2 ) ) * prm % f_ed
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! thermally activated annihilation of edge dipoles by climb
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rhoDotThermalAnnihilation = 0.0_pREAL
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D_SD = prm % D_0 * exp ( - prm % Q_cl / ( K_B * Temperature ) ) ! eq. 3.53
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v_climb = D_SD * mu * prm % V_at &
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/ ( PI * ( 1.0_pREAL - nu ) * ( dUpper ( : , 1 ) + dLower ( : , 1 ) ) * K_B * Temperature ) ! eq. 3.54
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forall ( s = 1 : prm % sum_N_sl , dUpper ( s , 1 ) > dLower ( s , 1 ) ) &
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rhoDotThermalAnnihilation ( s , 9 ) = max ( - 4.0_pREAL * rhoDip ( s , 1 ) * v_climb ( s ) / ( dUpper ( s , 1 ) - dLower ( s , 1 ) ) , &
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- rhoDip ( s , 1 ) / timestep - rhoDotAthermalAnnihilation ( s , 9 ) &
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- rhoDotSingle2DipoleGlide ( s , 9 ) ) ! make sure that we do not annihilate more dipoles than we have
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rhoDot = rhoDotFlux ( timestep , ph , en ) &
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+ rhoDotMultiplication &
+ rhoDotSingle2DipoleGlide &
+ rhoDotAthermalAnnihilation &
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+ rhoDotThermalAnnihilation
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if ( any ( rho ( : , mob ) + rhoDot ( : , 1 : 4 ) * timestep < - prm % atol_rho ) &
. or . any ( rho ( : , dip ) + rhoDot ( : , 9 : 10 ) * timestep < - prm % atol_rho ) ) then
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plasticState ( ph ) % dotState = IEEE_value ( 1.0_pREAL , IEEE_quiet_NaN )
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else
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dot % rho ( : , en ) = pack ( rhoDot , . true . )
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dot % gamma ( : , en ) = sum ( dot_gamma , 2 )
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end if
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end associate
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end subroutine nonlocal_dotState
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!---------------------------------------------------------------------------------------------------
!> @brief calculates the rate of change of microstructure
!---------------------------------------------------------------------------------------------------
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#if __INTEL_COMPILER >= 2020
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non_recursive function rhoDotFlux ( timestep , ph , en )
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#else
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function rhoDotFlux ( timestep , ph , en )
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#endif
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real ( pREAL ) , intent ( in ) :: &
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timestep !< substepped crystallite time increment
integer , intent ( in ) :: &
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ph , &
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en
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integer :: &
ns , & !< short notation for the total number of active slip systems
c , & !< character of dislocation
n , & !< index of my current neighbor
neighbor_el , & !< element number of my neighbor
neighbor_ip , & !< integration point of my neighbor
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neighbor_n , & !< neighbor index pointing to en when looking from my neighbor
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opposite_neighbor , & !< index of my opposite neighbor
opposite_ip , & !< ip of my opposite neighbor
opposite_el , & !< element index of my opposite neighbor
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opposite_n , & !< neighbor index pointing to en when looking from my opposite neighbor
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t , & !< type of dislocation
no , & !< neighbor offset shortcut
np , & !< neighbor phase shortcut
topp , & !< type of dislocation with opposite sign to t
s !< index of my current slip system
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: &
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rho , &
rho0 , & !< dislocation density at beginning of time step
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rhoDotFlux !< density evolution by flux
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 8 ) :: &
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rhoSgl , & !< current single dislocation densities (positive/negative screw and edge without dipoles)
neighbor_rhoSgl0 , & !< current single dislocation densities of neighboring ip (positive/negative screw and edge without dipoles)
my_rhoSgl0 !< single dislocation densities of central ip (positive/negative screw and edge without dipoles)
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 4 ) :: &
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v , & !< current dislocation glide velocity
v0 , &
neighbor_v0 , & !< dislocation glide velocity of enighboring ip
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dot_gamma !< shear rates
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real ( pREAL ) , dimension ( 3 , param ( ph ) % sum_N_sl , 4 ) :: &
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m !< direction of dislocation motion
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real ( pREAL ) , dimension ( 3 , 3 ) :: &
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my_F , & !< my total deformation gradient
neighbor_F , & !< total deformation gradient of my neighbor
my_Fe , & !< my elastic deformation gradient
neighbor_Fe , & !< elastic deformation gradient of my neighbor
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Favg !< average total deformation gradient of en and my neighbor
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real ( pREAL ) , dimension ( 3 ) :: &
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normal_neighbor2me , & !< interface normal pointing from my neighbor to en in neighbor's lattice configuration
normal_neighbor2me_defConf , & !< interface normal pointing from my neighbor to en in shared deformed configuration
normal_me2neighbor , & !< interface normal pointing from en to my neighbor in my lattice configuration
normal_me2neighbor_defConf !< interface normal pointing from en to my neighbor in shared deformed configuration
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real ( pREAL ) :: &
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area , & !< area of the current interface
transmissivity , & !< overall transmissivity of dislocation flux to neighboring material point
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lineLength !< dislocation line length leaving the current interface
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associate ( prm = > param ( ph ) , &
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dst = > dependentState ( ph ) , &
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stt = > state ( ph ) )
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ns = prm % sum_N_sl
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dot_gamma = 0.0_pREAL
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rho = getRho ( ph , en )
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rhoSgl = rho ( : , sgl )
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rho0 = getRho0 ( ph , en )
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my_rhoSgl0 = rho0 ( : , sgl )
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forall ( s = 1 : ns , t = 1 : 4 ) v ( s , t ) = plasticState ( ph ) % state ( iV ( s , t , ph ) , en ) !ToDo: MD: I think we should use state0 here
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dot_gamma = rhoSgl ( : , 1 : 4 ) * v * spread ( prm % b_sl , 2 , 4 )
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forall ( s = 1 : ns , t = 1 : 4 ) v0 ( s , t ) = plasticState ( ph ) % state0 ( iV ( s , t , ph ) , en )
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!****************************************************************************
!*** calculate dislocation fluxes (only for nonlocal plasticity)
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rhoDotFlux = 0.0_pREAL
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if ( plasticState ( ph ) % nonlocal ) then
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!*** check CFL (Courant-Friedrichs-Lewy) condition for flux
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if ( any ( abs ( dot_gamma ) > 0.0_pREAL & ! any active slip system ...
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. and . prm % C_CFL * abs ( v0 ) * timestep &
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> geom ( ph ) % V_0 ( en ) / maxval ( geom ( ph ) % IParea ( : , en ) ) ) ) then ! ...with velocity above critical value (we use the reference volume and area for simplicity here)
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rhoDotFlux = IEEE_value ( 1.0_pREAL , IEEE_quiet_NaN ) ! enforce cutback
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return
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end if
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!*** be aware of the definition of slip_transverse = slip_direction x slip_normal !!!
!*** opposite sign to our t vector in the (s,t,n) triplet !!!
m ( 1 : 3 , : , 1 ) = prm % slip_direction
m ( 1 : 3 , : , 2 ) = - prm % slip_direction
m ( 1 : 3 , : , 3 ) = - prm % slip_transverse
m ( 1 : 3 , : , 4 ) = prm % slip_transverse
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my_F = phase_mechanical_F ( ph ) % data ( 1 : 3 , 1 : 3 , en )
my_Fe = matmul ( my_F , math_inv33 ( phase_mechanical_Fp ( ph ) % data ( 1 : 3 , 1 : 3 , en ) ) )
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neighbors : do n = 1 , nIPneighbors
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neighbor_el = geom ( ph ) % IPneighborhood ( 1 , n , en )
neighbor_ip = geom ( ph ) % IPneighborhood ( 2 , n , en )
neighbor_n = geom ( ph ) % IPneighborhood ( 3 , n , en )
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np = material_ID_phase ( 1 , ( neighbor_el - 1 ) * discretization_nIPs + neighbor_ip )
no = material_entry_phase ( 1 , ( neighbor_el - 1 ) * discretization_nIPs + neighbor_ip )
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opposite_neighbor = n + mod ( n , 2 ) - mod ( n + 1 , 2 )
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opposite_el = geom ( ph ) % IPneighborhood ( 1 , opposite_neighbor , en )
opposite_ip = geom ( ph ) % IPneighborhood ( 2 , opposite_neighbor , en )
opposite_n = geom ( ph ) % IPneighborhood ( 3 , opposite_neighbor , en )
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if ( neighbor_n > 0 ) then ! if neighbor exists, average deformation gradient
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neighbor_F = phase_mechanical_F ( np ) % data ( 1 : 3 , 1 : 3 , no )
neighbor_Fe = matmul ( neighbor_F , math_inv33 ( phase_mechanical_Fp ( np ) % data ( 1 : 3 , 1 : 3 , no ) ) )
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Favg = 0.5_pREAL * ( my_F + neighbor_F )
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else ! if no neighbor, take my value as average
Favg = my_F
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end if
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neighbor_v0 = 0.0_pREAL ! needed for check of sign change in flux density below
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!* FLUX FROM MY NEIGHBOR TO ME
!* This is only considered, if I have a neighbor of nonlocal plasticity
!* (also nonlocal constitutive law with local properties) that is at least a little bit
!* compatible.
!* If it's not at all compatible, no flux is arriving, because everything is dammed in front of
!* my neighbor's interface.
!* The entering flux from my neighbor will be distributed on my slip systems according to the
!* compatibility
if ( neighbor_n > 0 ) then
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if ( mechanical_plasticity_type ( np ) == MECHANICAL_PLASTICITY_NONLOCAL . and . &
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any ( dependentState ( ph ) % compatibility ( : , : , : , n , en ) > 0.0_pREAL ) ) then
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forall ( s = 1 : ns , t = 1 : 4 )
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neighbor_v0 ( s , t ) = plasticState ( np ) % state0 ( iV ( s , t , np ) , no )
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neighbor_rhoSgl0 ( s , t ) = max ( plasticState ( np ) % state0 ( iRhoU ( s , t , np ) , no ) , 0.0_pREAL )
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endforall
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where ( neighbor_rhoSgl0 * IPvolume ( neighbor_ip , neighbor_el ) ** 0.667_pREAL < prm % rho_min &
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. or . neighbor_rhoSgl0 < prm % rho_significant ) &
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neighbor_rhoSgl0 = 0.0_pREAL
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normal_neighbor2me_defConf = math_det33 ( Favg ) * matmul ( math_inv33 ( transpose ( Favg ) ) , &
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IPareaNormal ( 1 : 3 , neighbor_n , neighbor_ip , neighbor_el ) ) ! normal of the interface in (average) deformed configuration (pointing neighbor => en)
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normal_neighbor2me = matmul ( transpose ( neighbor_Fe ) , normal_neighbor2me_defConf ) &
/ math_det33 ( neighbor_Fe ) ! interface normal in the lattice configuration of my neighbor
area = IParea ( neighbor_n , neighbor_ip , neighbor_el ) * norm2 ( normal_neighbor2me )
normal_neighbor2me = normal_neighbor2me / norm2 ( normal_neighbor2me ) ! normalize the surface normal to unit length
do s = 1 , ns
do t = 1 , 4
c = ( t + 1 ) / 2
topp = t + mod ( t , 2 ) - mod ( t + 1 , 2 )
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if ( neighbor_v0 ( s , t ) * math_inner ( m ( 1 : 3 , s , t ) , normal_neighbor2me ) > 0.0_pREAL & ! flux from my neighbor to en == entering flux for en
. and . v0 ( s , t ) * neighbor_v0 ( s , t ) > = 0.0_pREAL ) then ! ... only if no sign change in flux density
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lineLength = neighbor_rhoSgl0 ( s , t ) * neighbor_v0 ( s , t ) &
* math_inner ( m ( 1 : 3 , s , t ) , normal_neighbor2me ) * area ! positive line length that wants to enter through this interface
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where ( dependentState ( ph ) % compatibility ( c , : , s , n , en ) > 0.0_pREAL ) &
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rhoDotFlux ( : , t ) = rhoDotFlux ( 1 : ns , t ) &
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+ lineLength / geom ( ph ) % V_0 ( en ) * dependentState ( ph ) % compatibility ( c , : , s , n , en ) ** 2 ! transferring to equally signed mobile dislocation type
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where ( dependentState ( ph ) % compatibility ( c , : , s , n , en ) < 0.0_pREAL ) &
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rhoDotFlux ( : , topp ) = rhoDotFlux ( : , topp ) &
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+ lineLength / geom ( ph ) % V_0 ( en ) * dependentState ( ph ) % compatibility ( c , : , s , n , en ) ** 2 ! transferring to opposite signed mobile dislocation type
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end if
end do
end do
end if ; end if
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!* FLUX FROM ME TO MY NEIGHBOR
!* This is not considered, if my opposite neighbor has a different constitutive law than nonlocal (still considered for nonlocal law with local properties).
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!* Then, we assume, that the opposite(!) neighbor sends an equal amount of dislocations to en.
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!* So the net flux in the direction of my neighbor is equal to zero:
!* leaving flux to neighbor == entering flux from opposite neighbor
!* In case of reduced transmissivity, part of the leaving flux is stored as dead dislocation density.
!* That means for an interface of zero transmissivity the leaving flux is fully converted to dead dislocations.
if ( opposite_n > 0 ) then
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if ( mechanical_plasticity_type ( np ) == MECHANICAL_PLASTICITY_NONLOCAL ) then
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normal_me2neighbor_defConf = math_det33 ( Favg ) &
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* matmul ( math_inv33 ( transpose ( Favg ) ) , geom ( ph ) % IPareaNormal ( 1 : 3 , n , en ) ) ! normal of the interface in (average) deformed configuration (pointing en => neighbor)
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normal_me2neighbor = matmul ( transpose ( my_Fe ) , normal_me2neighbor_defConf ) &
/ math_det33 ( my_Fe ) ! interface normal in my lattice configuration
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area = geom ( ph ) % IParea ( n , en ) * norm2 ( normal_me2neighbor )
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normal_me2neighbor = normal_me2neighbor / norm2 ( normal_me2neighbor ) ! normalize the surface normal to unit length
do s = 1 , ns
do t = 1 , 4
c = ( t + 1 ) / 2
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if ( v0 ( s , t ) * math_inner ( m ( 1 : 3 , s , t ) , normal_me2neighbor ) > 0.0_pREAL ) then ! flux from en to my neighbor == leaving flux for en (might also be a pure flux from my mobile density to dead density if interface not at all transmissive)
if ( v0 ( s , t ) * neighbor_v0 ( s , t ) > = 0.0_pREAL ) then ! no sign change in flux density
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transmissivity = sum ( dependentState ( ph ) % compatibility ( c , : , s , n , en ) ** 2 ) ! overall transmissivity from this slip system to my neighbor
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else ! sign change in flux density means sign change in stress which does not allow for dislocations to arive at the neighbor
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transmissivity = 0.0_pREAL
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end if
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lineLength = my_rhoSgl0 ( s , t ) * v0 ( s , t ) &
* math_inner ( m ( 1 : 3 , s , t ) , normal_me2neighbor ) * area ! positive line length of mobiles that wants to leave through this interface
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rhoDotFlux ( s , t ) = rhoDotFlux ( s , t ) - lineLength / geom ( ph ) % V_0 ( en ) ! subtract dislocation flux from current type
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rhoDotFlux ( s , t + 4 ) = rhoDotFlux ( s , t + 4 ) &
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+ lineLength / geom ( ph ) % V_0 ( en ) * ( 1.0_pREAL - transmissivity ) &
* sign ( 1.0_pREAL , v0 ( s , t ) ) ! dislocation flux that is not able to leave through interface (because of low transmissivity) will remain as immobile single density at the material point
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end if
end do
end do
end if ; end if
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end do neighbors
end if
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end associate
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end function rhoDotFlux
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!--------------------------------------------------------------------------------------------------
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!> @brief Compatibility update
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!> @detail Compatibility is defined as normalized product of signed cosine of the angle between the slip
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! plane normals and signed cosine of the angle between the slip directions. Only the largest values
! that sum up to a total of 1 are considered, all others are set to zero.
!--------------------------------------------------------------------------------------------------
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module subroutine plastic_nonlocal_updateCompatibility ( orientation , ph , ip , el )
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type ( tRotationContainer ) , dimension ( : ) , intent ( in ) :: &
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orientation ! crystal orientation
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integer , intent ( in ) :: &
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ph , &
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ip , &
el
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integer :: &
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n , & ! neighbor index
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en , &
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neighbor_e , & ! element index of my neighbor
neighbor_i , & ! integration point index of my neighbor
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neighbor_me , &
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neighbor_phase , &
ns , & ! number of active slip systems
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s1 , & ! slip system index (en)
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s2 ! slip system index (my neighbor)
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real ( pREAL ) , dimension ( 2 , param ( ph ) % sum_N_sl , param ( ph ) % sum_N_sl , nIPneighbors ) :: &
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my_compatibility ! my_compatibility for current element and ip
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real ( pREAL ) :: &
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my_compatibilitySum , &
thresholdValue , &
nThresholdValues
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logical , dimension ( param ( ph ) % sum_N_sl ) :: &
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belowThreshold
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type ( tRotation ) :: mis
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associate ( prm = > param ( ph ) )
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ns = prm % sum_N_sl
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en = material_entry_phase ( 1 , ( el - 1 ) * discretization_nIPs + ip )
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!*** start out fully compatible
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my_compatibility = 0.0_pREAL
forall ( s1 = 1 : ns ) my_compatibility ( : , s1 , s1 , : ) = 1.0_pREAL
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neighbors : do n = 1 , nIPneighbors
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neighbor_e = IPneighborhood ( 1 , n , ip , el )
neighbor_i = IPneighborhood ( 2 , n , ip , el )
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neighbor_me = material_entry_phase ( 1 , ( neighbor_e - 1 ) * discretization_nIPs + neighbor_i )
neighbor_phase = material_ID_phase ( 1 , ( neighbor_e - 1 ) * discretization_nIPs + neighbor_i )
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if ( neighbor_e < = 0 . or . neighbor_i < = 0 ) then
!* FREE SURFACE
forall ( s1 = 1 : ns ) my_compatibility ( : , s1 , s1 , n ) = sqrt ( prm % chi_surface )
elseif ( neighbor_phase / = ph ) then
!* PHASE BOUNDARY
if ( plasticState ( neighbor_phase ) % nonlocal . and . plasticState ( ph ) % nonlocal ) &
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forall ( s1 = 1 : ns ) my_compatibility ( : , s1 , s1 , n ) = 0.0_pREAL
elseif ( prm % chi_GB > = 0.0_pREAL ) then
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!* GRAIN BOUNDARY
if ( any ( dNeq ( phase_O_0 ( ph ) % data ( en ) % asQuaternion ( ) , &
phase_O_0 ( neighbor_phase ) % data ( neighbor_me ) % asQuaternion ( ) ) ) . and . &
plasticState ( neighbor_phase ) % nonlocal ) &
forall ( s1 = 1 : ns ) my_compatibility ( : , s1 , s1 , n ) = sqrt ( prm % chi_GB )
else
!* GRAIN BOUNDARY ?
!* Compatibility defined by relative orientation of slip systems:
!* The my_compatibility value is defined as the product of the slip normal projection and the slip direction projection.
!* Its sign is always positive for screws, for edges it has the same sign as the slip normal projection.
!* Since the sum for each slip system can easily exceed one (which would result in a transmissivity larger than one),
!* only values above or equal to a certain threshold value are considered. This threshold value is chosen, such that
!* the number of compatible slip systems is minimized with the sum of the original compatibility values exceeding one.
!* Finally the smallest compatibility value is decreased until the sum is exactly equal to one.
!* All values below the threshold are set to zero.
mis = orientation ( ph ) % data ( en ) % misorientation ( orientation ( neighbor_phase ) % data ( neighbor_me ) )
mySlipSystems : do s1 = 1 , ns
neighborSlipSystems : do s2 = 1 , ns
my_compatibility ( 1 , s2 , s1 , n ) = math_inner ( prm % slip_normal ( 1 : 3 , s1 ) , &
mis % rotate ( prm % slip_normal ( 1 : 3 , s2 ) ) ) &
* abs ( math_inner ( prm % slip_direction ( 1 : 3 , s1 ) , &
mis % rotate ( prm % slip_direction ( 1 : 3 , s2 ) ) ) )
my_compatibility ( 2 , s2 , s1 , n ) = abs ( math_inner ( prm % slip_normal ( 1 : 3 , s1 ) , &
mis % rotate ( prm % slip_normal ( 1 : 3 , s2 ) ) ) ) &
* abs ( math_inner ( prm % slip_direction ( 1 : 3 , s1 ) , &
mis % rotate ( prm % slip_direction ( 1 : 3 , s2 ) ) ) )
end do neighborSlipSystems
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my_compatibilitySum = 0.0_pREAL
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belowThreshold = . true .
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do while ( my_compatibilitySum < 1.0_pREAL . and . any ( belowThreshold ) )
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thresholdValue = maxval ( my_compatibility ( 2 , : , s1 , n ) , belowThreshold ) ! screws always positive
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nThresholdValues = real ( count ( my_compatibility ( 2 , : , s1 , n ) > = thresholdValue ) , pREAL )
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where ( my_compatibility ( 2 , : , s1 , n ) > = thresholdValue ) belowThreshold = . false .
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if ( my_compatibilitySum + thresholdValue * nThresholdValues > 1.0_pREAL ) &
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where ( abs ( my_compatibility ( : , : , s1 , n ) ) > = thresholdValue ) &
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my_compatibility ( : , : , s1 , n ) = sign ( ( 1.0_pREAL - my_compatibilitySum ) / nThresholdValues , &
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my_compatibility ( : , : , s1 , n ) )
my_compatibilitySum = my_compatibilitySum + nThresholdValues * thresholdValue
end do
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where ( belowThreshold ) my_compatibility ( 1 , : , s1 , n ) = 0.0_pREAL
where ( belowThreshold ) my_compatibility ( 2 , : , s1 , n ) = 0.0_pREAL
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end do mySlipSystems
end if
end do neighbors
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dependentState ( ph ) % compatibility ( : , : , : , : , material_entry_phase ( 1 , ( el - 1 ) * discretization_nIPs + ip ) ) = my_compatibility
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end associate
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end subroutine plastic_nonlocal_updateCompatibility
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!--------------------------------------------------------------------------------------------------
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!> @brief Write results to HDF5 output file.
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!--------------------------------------------------------------------------------------------------
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module subroutine plastic_nonlocal_result ( ph , group )
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integer , intent ( in ) :: ph
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character ( len = * ) , intent ( in ) :: group
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integer :: ou
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associate ( prm = > param ( ph ) , dst = > dependentState ( ph ) , stt = > state ( ph ) )
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do ou = 1 , size ( prm % output )
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select case ( trim ( prm % output ( ou ) ) )
case ( 'rho_u_ed_pos' )
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call result_writeDataset ( stt % rho_sgl_mob_edg_pos , group , trim ( prm % output ( ou ) ) , &
'positive mobile edge density' , '1/m²' , prm % systems_sl )
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case ( 'rho_b_ed_pos' )
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call result_writeDataset ( stt % rho_sgl_imm_edg_pos , group , trim ( prm % output ( ou ) ) , &
'positive immobile edge density' , '1/m²' , prm % systems_sl )
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case ( 'rho_u_ed_neg' )
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call result_writeDataset ( stt % rho_sgl_mob_edg_neg , group , trim ( prm % output ( ou ) ) , &
'negative mobile edge density' , '1/m²' , prm % systems_sl )
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case ( 'rho_b_ed_neg' )
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call result_writeDataset ( stt % rho_sgl_imm_edg_neg , group , trim ( prm % output ( ou ) ) , &
'negative immobile edge density' , '1/m²' , prm % systems_sl )
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case ( 'rho_d_ed' )
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call result_writeDataset ( stt % rho_dip_edg , group , trim ( prm % output ( ou ) ) , &
'edge dipole density' , '1/m²' , prm % systems_sl )
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case ( 'rho_u_sc_pos' )
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call result_writeDataset ( stt % rho_sgl_mob_scr_pos , group , trim ( prm % output ( ou ) ) , &
'positive mobile screw density' , '1/m²' , prm % systems_sl )
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case ( 'rho_b_sc_pos' )
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call result_writeDataset ( stt % rho_sgl_imm_scr_pos , group , trim ( prm % output ( ou ) ) , &
'positive immobile screw density' , '1/m²' , prm % systems_sl )
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case ( 'rho_u_sc_neg' )
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call result_writeDataset ( stt % rho_sgl_mob_scr_neg , group , trim ( prm % output ( ou ) ) , &
'negative mobile screw density' , '1/m²' , prm % systems_sl )
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case ( 'rho_b_sc_neg' )
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call result_writeDataset ( stt % rho_sgl_imm_scr_neg , group , trim ( prm % output ( ou ) ) , &
'negative immobile screw density' , '1/m²' , prm % systems_sl )
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case ( 'rho_d_sc' )
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call result_writeDataset ( stt % rho_dip_scr , group , trim ( prm % output ( ou ) ) , &
'screw dipole density' , '1/m²' , prm % systems_sl )
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case ( 'rho_f' )
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call result_writeDataset ( stt % rho_forest , group , trim ( prm % output ( ou ) ) , &
'forest density' , '1/m²' , prm % systems_sl )
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case ( 'v_ed_pos' )
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call result_writeDataset ( stt % v_edg_pos , group , trim ( prm % output ( ou ) ) , &
'positive edge velocity' , 'm/s' , prm % systems_sl )
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case ( 'v_ed_neg' )
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call result_writeDataset ( stt % v_edg_neg , group , trim ( prm % output ( ou ) ) , &
'negative edge velocity' , 'm/s' , prm % systems_sl )
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case ( 'v_sc_pos' )
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call result_writeDataset ( stt % v_scr_pos , group , trim ( prm % output ( ou ) ) , &
'positive srew velocity' , 'm/s' , prm % systems_sl )
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case ( 'v_sc_neg' )
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call result_writeDataset ( stt % v_scr_neg , group , trim ( prm % output ( ou ) ) , &
'negative screw velocity' , 'm/s' , prm % systems_sl )
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case ( 'gamma' )
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call result_writeDataset ( stt % gamma , group , trim ( prm % output ( ou ) ) , &
'plastic shear' , '1' , prm % systems_sl )
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case ( 'tau_pass' )
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call result_writeDataset ( dst % tau_pass , group , trim ( prm % output ( ou ) ) , &
'passing stress for slip' , 'Pa' , prm % systems_sl )
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end select
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end do
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end associate
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end subroutine plastic_nonlocal_result
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!--------------------------------------------------------------------------------------------------
!> @brief populates the initial dislocation density
!--------------------------------------------------------------------------------------------------
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subroutine stateInit ( ini , phase , Nentries )
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type ( tInitialParameters ) :: &
ini
integer , intent ( in ) :: &
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phase , &
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Nentries
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integer :: &
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e , &
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f , &
from , &
upto , &
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s
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real ( pREAL ) , dimension ( 2 ) :: &
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rnd
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real ( pREAL ) :: &
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meanDensity , &
totalVolume , &
densityBinning , &
minimumIpVolume
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associate ( stt = > state ( phase ) )
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if ( ini % random_rho_u > 0.0_pREAL ) then ! randomly distribute dislocation segments on random slip system and of random type in the volume
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totalVolume = sum ( geom ( phase ) % V_0 )
minimumIPVolume = minval ( geom ( phase ) % V_0 )
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densityBinning = ini % random_rho_u_binning / minimumIpVolume ** ( 2.0_pREAL / 3.0_pREAL )
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! fill random material points with dislocation segments until the desired overall density is reached
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meanDensity = 0.0_pREAL
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do while ( meanDensity < ini % random_rho_u )
call random_number ( rnd )
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e = nint ( rnd ( 1 ) * real ( Nentries , pREAL ) + 0.5_pREAL )
s = nint ( rnd ( 2 ) * real ( sum ( ini % N_sl ) , pREAL ) * 4.0_pREAL + 0.5_pREAL )
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meanDensity = meanDensity + densityBinning * geom ( phase ) % V_0 ( e ) / totalVolume
stt % rhoSglMobile ( s , e ) = densityBinning
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end do
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else ! homogeneous distribution with noise
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do f = 1 , size ( ini % N_sl , 1 )
from = 1 + sum ( ini % N_sl ( 1 : f - 1 ) )
upto = sum ( ini % N_sl ( 1 : f ) )
call math_normal ( stt % rho_sgl_mob_edg_pos ( from : upto , : ) , ini % rho_u_ed_pos_0 ( f ) , ini % sigma_rho_u )
call math_normal ( stt % rho_sgl_mob_edg_neg ( from : upto , : ) , ini % rho_u_ed_neg_0 ( f ) , ini % sigma_rho_u )
call math_normal ( stt % rho_sgl_mob_scr_pos ( from : upto , : ) , ini % rho_u_sc_pos_0 ( f ) , ini % sigma_rho_u )
call math_normal ( stt % rho_sgl_mob_scr_neg ( from : upto , : ) , ini % rho_u_sc_neg_0 ( f ) , ini % sigma_rho_u )
stt % rho_dip_edg ( from : upto , : ) = ini % rho_d_ed_0 ( f )
stt % rho_dip_scr ( from : upto , : ) = ini % rho_d_sc_0 ( f )
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end do
end if
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end associate
end subroutine stateInit
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!--------------------------------------------------------------------------------------------------
!> @brief calculates kinetics
!--------------------------------------------------------------------------------------------------
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pure subroutine kinetics ( v , dv_dtau , dv_dtauNS , tau , tauNS , tauThreshold , c , T , ph )
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integer , intent ( in ) :: &
c , & !< dislocation character (1:edge, 2:screw)
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ph
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl ) , intent ( in ) :: &
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tau , & !< resolved external shear stress (without non Schmid effects)
tauNS , & !< resolved external shear stress (including non Schmid effects)
tauThreshold !< threshold shear stress
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real ( pREAL ) , intent ( in ) :: &
T !< T
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl ) , intent ( out ) :: &
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v , & !< velocity
dv_dtau , & !< velocity derivative with respect to resolved shear stress (without non Schmid contributions)
dv_dtauNS !< velocity derivative with respect to resolved shear stress (including non Schmid contributions)
integer :: &
s !< index of my current slip system
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real ( pREAL ) :: &
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tauRel_P , &
tauRel_S , &
tauEff , & !< effective shear stress
tPeierls , & !< waiting time in front of a peierls barriers
tSolidSolution , & !< waiting time in front of a solid solution obstacle
dtPeierls_dtau , & !< derivative with respect to resolved shear stress
dtSolidSolution_dtau , & !< derivative with respect to resolved shear stress
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lambda_S , & !< mean free distance between two solid solution obstacles
lambda_P , & !< mean free distance between two Peierls barriers
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activationVolume_P , & !< volume that needs to be activated to overcome barrier
activationVolume_S , & !< volume that needs to be activated to overcome barrier
activationEnergy_P , & !< energy that is needed to overcome barrier
criticalStress_P , & !< maximum obstacle strength
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criticalStress_S !< maximum obstacle strength
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v = 0.0_pREAL
dv_dtau = 0.0_pREAL
dv_dtauNS = 0.0_pREAL
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associate ( prm = > param ( ph ) )
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do s = 1 , prm % sum_N_sl
if ( abs ( tau ( s ) ) > tauThreshold ( s ) ) then
!* Peierls contribution
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tauEff = max ( 0.0_pREAL , abs ( tauNS ( s ) ) - tauThreshold ( s ) )
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lambda_P = prm % b_sl ( s )
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activationVolume_P = prm % w * prm % b_sl ( s ) ** 3
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criticalStress_P = prm % peierlsStress ( s , c )
activationEnergy_P = criticalStress_P * activationVolume_P
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tauRel_P = min ( 1.0_pREAL , tauEff / criticalStress_P )
tPeierls = 1.0_pREAL / prm % nu_a &
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* exp ( activationEnergy_P / ( K_B * T ) &
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* ( 1.0_pREAL - tauRel_P ** prm % p ) ** prm % q )
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dtPeierls_dtau = merge ( tPeierls * prm % p * prm % q * activationVolume_P / ( K_B * T ) &
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* ( 1.0_pREAL - tauRel_P ** prm % p ) ** ( prm % q - 1.0_pREAL ) * tauRel_P ** ( prm % p - 1.0_pREAL ) , &
0.0_pREAL , &
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tauEff < criticalStress_P )
! Contribution from solid solution strengthening
tauEff = abs ( tau ( s ) ) - tauThreshold ( s )
lambda_S = prm % b_sl ( s ) / sqrt ( prm % c_sol )
activationVolume_S = prm % f_sol * prm % b_sl ( s ) ** 3 / sqrt ( prm % c_sol )
criticalStress_S = prm % Q_sol / activationVolume_S
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tauRel_S = min ( 1.0_pREAL , tauEff / criticalStress_S )
tSolidSolution = 1.0_pREAL / prm % nu_a &
* exp ( prm % Q_sol / ( K_B * T ) * ( 1.0_pREAL - tauRel_S ** prm % p ) ** prm % q )
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dtSolidSolution_dtau = merge ( tSolidSolution * prm % p * prm % q * activationVolume_S / ( K_B * T ) &
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* ( 1.0_pREAL - tauRel_S ** prm % p ) ** ( prm % q - 1.0_pREAL ) * tauRel_S ** ( prm % p - 1.0_pREAL ) , &
0.0_pREAL , &
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tauEff < criticalStress_S )
!* viscous glide velocity
tauEff = abs ( tau ( s ) ) - tauThreshold ( s )
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v ( s ) = sign ( 1.0_pREAL , tau ( s ) ) &
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/ ( tPeierls / lambda_P + tSolidSolution / lambda_S + prm % B / ( prm % b_sl ( s ) * tauEff ) )
dv_dtau ( s ) = v ( s ) ** 2 * ( dtSolidSolution_dtau / lambda_S + prm % B / ( prm % b_sl ( s ) * tauEff ** 2 ) )
dv_dtauNS ( s ) = v ( s ) ** 2 * dtPeierls_dtau / lambda_P
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end if
end do
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end associate
end subroutine kinetics
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!--------------------------------------------------------------------------------------------------
!> @brief returns copy of current dislocation densities from state
!> @details raw values is rectified
!--------------------------------------------------------------------------------------------------
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pure function getRho ( ph , en ) result ( rho )
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integer , intent ( in ) :: ph , en
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: rho
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associate ( prm = > param ( ph ) )
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rho = reshape ( state ( ph ) % rho ( : , en ) , [ prm % sum_N_sl , 10 ] )
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! ensure positive densities (not for imm, they have a sign)
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rho ( : , mob ) = max ( rho ( : , mob ) , 0.0_pREAL )
rho ( : , dip ) = max ( rho ( : , dip ) , 0.0_pREAL )
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where ( abs ( rho ) < max ( prm % rho_min / geom ( ph ) % V_0 ( en ) ** ( 2.0_pREAL / 3.0_pREAL ) , prm % rho_significant ) ) &
rho = 0.0_pREAL
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end associate
end function getRho
!--------------------------------------------------------------------------------------------------
!> @brief returns copy of current dislocation densities from state
!> @details raw values is rectified
!--------------------------------------------------------------------------------------------------
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pure function getRho0 ( ph , en ) result ( rho0 )
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integer , intent ( in ) :: ph , en
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real ( pREAL ) , dimension ( param ( ph ) % sum_N_sl , 10 ) :: rho0
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associate ( prm = > param ( ph ) )
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rho0 = reshape ( state0 ( ph ) % rho ( : , en ) , [ prm % sum_N_sl , 10 ] )
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! ensure positive densities (not for imm, they have a sign)
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rho0 ( : , mob ) = max ( rho0 ( : , mob ) , 0.0_pREAL )
rho0 ( : , dip ) = max ( rho0 ( : , dip ) , 0.0_pREAL )
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where ( abs ( rho0 ) < max ( prm % rho_min / geom ( ph ) % V_0 ( en ) ** ( 2.0_pREAL / 3.0_pREAL ) , prm % rho_significant ) ) &
rho0 = 0.0_pREAL
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end associate
end function getRho0
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subroutine storeGeometry ( ph )
integer , intent ( in ) :: ph
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integer :: ce , co , nCell
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real ( pREAL ) , dimension ( : ) , allocatable :: V
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integer , dimension ( : , : , : ) , allocatable :: neighborhood
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real ( pREAL ) , dimension ( : , : ) , allocatable :: area , coords
real ( pREAL ) , dimension ( : , : , : ) , allocatable :: areaNormal
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nCell = product ( shape ( IPvolume ) )
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V = reshape ( IPvolume , [ nCell ] )
neighborhood = reshape ( IPneighborhood , [ 3 , nIPneighbors , nCell ] )
area = reshape ( IParea , [ nIPneighbors , nCell ] )
areaNormal = reshape ( IPareaNormal , [ 3 , nIPneighbors , nCell ] )
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coords = reshape ( discretization_IPcoords , [ 3 , nCell ] )
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do ce = 1 , size ( material_entry_homogenization , 1 )
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do co = 1 , homogenization_maxNconstituents
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if ( material_ID_phase ( co , ce ) == ph ) then
geom ( ph ) % V_0 ( material_entry_phase ( co , ce ) ) = V ( ce )
geom ( ph ) % IPneighborhood ( : , : , material_entry_phase ( co , ce ) ) = neighborhood ( : , : , ce )
geom ( ph ) % IParea ( : , material_entry_phase ( co , ce ) ) = area ( : , ce )
geom ( ph ) % IPareaNormal ( : , : , material_entry_phase ( co , ce ) ) = areaNormal ( : , : , ce )
geom ( ph ) % IPcoordinates ( : , material_entry_phase ( co , ce ) ) = coords ( : , ce )
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end if
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end do
end do
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end subroutine storeGeometry
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end submodule nonlocal
