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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 ( constitutive : constitutive_plastic ) 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 , &
IPareaNormal = > geometry_plastic_nonlocal_IPareaNormal0
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real ( pReal ) , parameter :: &
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kB = 1.38e-23_pReal !< Boltzmann constant in J/Kelvin
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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 :: &
compatibility !< slip system compatibility between me and my neighbors
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type :: tInitialParameters !< container type for internal constitutive parameters
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
rho_num_significant , & !< number of dislocations considered significant
w , & !< width of a doubkle kink in multiples of the burgers vector length b
Q_sol , & !< activation energy for solid solution in J
f_sol , & !< solid solution obstacle size in multiples of the burgers vector length
c_sol , & !< concentration of solid solution in atomic parts
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p , & !< parameter for kinetic law (Kocks,Argon,Ashby)
q , & !< parameter for kinetic law (Kocks,Argon,Ashby)
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eta , & !< viscosity for dislocation glide in Pa s
nu_a , & !< attack frequency in Hz
chi_surface , & !< transmissivity at free surface
chi_GB , & !< transmissivity at grain boundary (identified by different texture)
f_c , & !< safety factor for CFL flux condition
f_ed_mult , & !< factor that determines how much edge dislocations contribute to multiplication (0...1)
f_F , &
f_ed , &
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mu , &
nu
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real ( pReal ) , dimension ( : ) , allocatable :: &
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d_ed , & !< minimum stable edge dipole height
d_sc , & !< minimum stable screw dipole height
tau_peierls_ed , &
tau_peierls_sc , &
i_sl , & !< mean free path prefactor for each
b_sl !< absolute length of burgers vector [m]
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real ( pReal ) , dimension ( : , : ) , allocatable :: &
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slip_normal , &
slip_direction , &
slip_transverse , &
minDipoleHeight , & ! edge and screw
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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Schmid , & !< Schmid contribution
nonSchmid_pos , &
nonSchmid_neg !< combined projection of Schmid and non-Schmid contributions to the resolved shear stress (only for screws)
integer :: &
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sum_N_sl
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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 = pStringLen ) , 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
nonSchmidActive = . false .
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end type tParameters
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type :: tNonlocalMicrostructure
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real ( pReal ) , allocatable , dimension ( : , : ) :: &
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tau_pass , &
tau_Back
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end type tNonlocalMicrostructure
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type :: tNonlocalState
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real ( pReal ) , pointer , dimension ( : , : ) :: &
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 , &
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 (len Ninstance)
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type ( tNonlocalMicrostructure ) , dimension ( : ) , allocatable :: microstructure
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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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Ninstance , &
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p , i , &
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NipcMyPhase , &
sizeState , sizeDotState , sizeDependentState , sizeDeltaState , &
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s1 , s2 , &
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s , t , l
real ( pReal ) , dimension ( : ) , allocatable :: &
a
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character ( len = pStringLen ) :: &
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extmsg = ''
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type ( tInitialParameters ) :: &
ini
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class ( tNode ) , pointer :: &
phases , &
phase , &
pl
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print '(/,a)' , ' <<<+- plastic_nonlocal init -+>>>'
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myPlasticity = plastic_active ( 'nonlocal' )
Ninstance = count ( myPlasticity )
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print '(a,i2)' , ' # instances: ' , Ninstance ; flush ( 6 )
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if ( Ninstance == 0 ) then
call geometry_plastic_nonlocal_disable
return
endif
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print * , 'Reuber et al., Acta Materialia 71:333–
print * , 'https://doi.org/10.1016/j.actamat.2014.03.012' / / IO_EOL
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print * , 'Kords, Dissertation RWTH Aachen, 2014'
print * , 'http://publications.rwth-aachen.de/record/229993' / / IO_EOL
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allocate ( param ( Ninstance ) )
allocate ( state ( Ninstance ) )
allocate ( state0 ( Ninstance ) )
allocate ( dotState ( Ninstance ) )
allocate ( deltaState ( Ninstance ) )
allocate ( microstructure ( Ninstance ) )
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phases = > config_material % get ( 'phase' )
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i = 0
do p = 1 , phases % length
phase = > phases % get ( p )
if ( . not . myPlasticity ( p ) ) cycle
i = i + 1
associate ( prm = > param ( i ) , &
dot = > dotState ( i ) , &
stt = > state ( i ) , &
st0 = > state0 ( i ) , &
del = > deltaState ( i ) , &
dst = > microstructure ( i ) )
pl = > phase % get ( 'plasticity' )
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phase_localPlasticity ( p ) = . not . pl % contains ( 'nonlocal' )
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#if defined (__GFORTRAN__)
prm % output = output_asStrings ( pl )
#else
prm % output = pl % get_asStrings ( 'output' , defaultVal = emptyStringArray )
#endif
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prm % atol_rho = pl % get_asFloat ( 'atol_rho' , defaultVal = 1.0e4_pReal )
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! This data is read in already in lattice
prm % mu = lattice_mu ( p )
prm % nu = lattice_nu ( p )
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ini % N_sl = pl % get_asInts ( '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 % Schmid = lattice_SchmidMatrix_slip ( ini % N_sl , phase % get_asString ( 'lattice' ) , &
phase % get_asFloat ( 'c/a' , defaultVal = 0.0_pReal ) )
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if ( trim ( phase % get_asString ( 'lattice' ) ) == 'bcc' ) then
a = pl % get_asFloats ( 'nonSchmid_coefficients' , defaultVal = emptyRealArray )
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if ( size ( a ) > 0 ) prm % nonSchmidActive = . true .
prm % nonSchmid_pos = lattice_nonSchmidMatrix ( ini % N_sl , a , + 1 )
prm % nonSchmid_neg = lattice_nonSchmidMatrix ( ini % N_sl , a , - 1 )
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else
prm % nonSchmid_pos = prm % Schmid
prm % nonSchmid_neg = prm % Schmid
endif
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prm % h_sl_sl = lattice_interaction_SlipBySlip ( ini % N_sl , &
pl % get_asFloats ( 'h_sl_sl' ) , &
phase % get_asString ( 'lattice' ) )
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prm % forestProjection_edge = lattice_forestProjection_edge ( ini % N_sl , phase % get_asString ( 'lattice' ) , &
phase % get_asFloat ( 'c/a' , defaultVal = 0.0_pReal ) )
prm % forestProjection_screw = lattice_forestProjection_screw ( ini % N_sl , phase % get_asString ( 'lattice' ) , &
phase % get_asFloat ( 'c/a' , defaultVal = 0.0_pReal ) )
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prm % slip_direction = lattice_slip_direction ( ini % N_sl , phase % get_asString ( 'lattice' ) , &
phase % get_asFloat ( 'c/a' , defaultVal = 0.0_pReal ) )
prm % slip_transverse = lattice_slip_transverse ( ini % N_sl , phase % get_asString ( 'lattice' ) , &
phase % get_asFloat ( 'c/a' , defaultVal = 0.0_pReal ) )
prm % slip_normal = lattice_slip_normal ( ini % N_sl , phase % get_asString ( 'lattice' ) , &
phase % get_asFloat ( 'c/a' , defaultVal = 0.0_pReal ) )
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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
enddo
enddo
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ini % rho_u_ed_pos_0 = pl % get_asFloats ( 'rho_u_ed_pos_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_u_ed_neg_0 = pl % get_asFloats ( 'rho_u_ed_neg_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_u_sc_pos_0 = pl % get_asFloats ( 'rho_u_sc_pos_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_u_sc_neg_0 = pl % get_asFloats ( 'rho_u_sc_neg_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_d_ed_0 = pl % get_asFloats ( 'rho_d_ed_0' , requiredSize = size ( ini % N_sl ) )
ini % rho_d_sc_0 = pl % get_asFloats ( 'rho_d_sc_0' , requiredSize = size ( ini % N_sl ) )
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prm % i_sl = pl % get_asFloats ( 'i_sl' , requiredSize = size ( ini % N_sl ) )
prm % b_sl = pl % get_asFloats ( 'b_sl' , requiredSize = size ( ini % N_sl ) )
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prm % i_sl = math_expand ( prm % i_sl , ini % N_sl )
prm % b_sl = math_expand ( prm % b_sl , ini % N_sl )
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prm % d_ed = pl % get_asFloats ( 'd_ed' , requiredSize = size ( ini % N_sl ) )
prm % d_sc = pl % get_asFloats ( 'd_sc' , requiredSize = size ( ini % N_sl ) )
prm % d_ed = math_expand ( prm % d_ed , ini % N_sl )
prm % d_sc = math_expand ( prm % d_sc , ini % N_sl )
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allocate ( prm % minDipoleHeight ( prm % sum_N_sl , 2 ) )
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prm % minDipoleHeight ( : , 1 ) = prm % d_ed
prm % minDipoleHeight ( : , 2 ) = prm % d_sc
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prm % tau_peierls_ed = pl % get_asFloats ( 'tau_peierls_ed' , requiredSize = size ( ini % N_sl ) )
prm % tau_peierls_sc = pl % get_asFloats ( 'tau_peierls_sc' , requiredSize = size ( ini % N_sl ) )
prm % tau_peierls_ed = math_expand ( prm % tau_peierls_ed , ini % N_sl )
prm % tau_peierls_sc = math_expand ( prm % tau_peierls_sc , ini % N_sl )
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allocate ( prm % peierlsstress ( prm % sum_N_sl , 2 ) )
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prm % peierlsstress ( : , 1 ) = prm % tau_peierls_ed
prm % peierlsstress ( : , 2 ) = prm % tau_peierls_sc
prm % rho_significant = pl % get_asFloat ( 'rho_significant' )
prm % rho_num_significant = pl % get_asFloat ( 'rho_num_significant' , 0.0_pReal )
prm % f_c = pl % get_asFloat ( 'f_c' , defaultVal = 2.0_pReal )
prm % V_at = pl % get_asFloat ( 'V_at' )
prm % D_0 = pl % get_asFloat ( 'D_0' ) !,'dsd0'
prm % Q_cl = pl % get_asFloat ( 'Q_cl' ) !,'qsd'
prm % f_F = pl % get_asFloat ( 'f_F' )
prm % f_ed = pl % get_asFloat ( 'f_ed' ) !,'edgejogs'
prm % w = pl % get_asFloat ( 'w' )
prm % Q_sol = pl % get_asFloat ( 'Q_sol' )
prm % f_sol = pl % get_asFloat ( 'f_sol' )
prm % c_sol = pl % get_asFloat ( 'c_sol' )
prm % p = pl % get_asFloat ( 'p_sl' )
prm % q = pl % get_asFloat ( 'q_sl' )
prm % eta = pl % get_asFloat ( 'eta' )
prm % nu_a = pl % get_asFloat ( 'nu_a' )
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! ToDo: discuss logic
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ini % sigma_rho_u = pl % get_asFloat ( 'sigma_rho_u' )
ini % random_rho_u = pl % get_asFloat ( '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_asFloat ( '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_asFloat ( 'chi_surface' , defaultVal = 1.0_pReal )
prm % chi_GB = pl % get_asFloat ( 'chi_GB' , defaultVal = - 1.0_pReal )
prm % f_ed_mult = pl % get_asFloat ( 'f_ed_mult' )
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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 % eta < = 0.0_pReal ) extmsg = trim ( extmsg ) / / ' eta'
if ( prm % Q_cl < = 0.0_pReal ) extmsg = trim ( extmsg ) / / ' Q_cl'
if ( prm % nu_a < = 0.0_pReal ) extmsg = trim ( extmsg ) / / ' nu_a'
if ( prm % w < = 0.0_pReal ) extmsg = trim ( extmsg ) / / ' w'
if ( prm % D_0 < 0.0_pReal ) extmsg = trim ( extmsg ) / / ' D_0'
if ( prm % V_at < = 0.0_pReal ) extmsg = trim ( extmsg ) / / ' V_at' ! ToDo: in disloTungsten, the atomic volume is given as a factor
if ( prm % rho_num_significant < 0.0_pReal ) extmsg = trim ( extmsg ) / / ' rho_num_significant'
if ( prm % rho_significant < 0.0_pReal ) extmsg = trim ( extmsg ) / / ' rho_significant'
if ( prm % atol_rho < 0.0_pReal ) extmsg = trim ( extmsg ) / / ' atol_rho'
if ( prm % f_c < 0.0_pReal ) extmsg = trim ( extmsg ) / / ' f_c'
if ( prm % p < = 0.0_pReal . or . prm % p > 1.0_pReal ) extmsg = trim ( extmsg ) / / ' p_sl'
if ( prm % q < 1.0_pReal . or . prm % q > 2.0_pReal ) extmsg = trim ( extmsg ) / / ' q_sl'
if ( prm % f_F < 0.0_pReal . or . prm % f_F > 1.0_pReal ) &
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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 ) &
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 ) &
extmsg = trim ( extmsg ) / / ' f_ed_mult'
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endif slipActive
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!--------------------------------------------------------------------------------------------------
! allocate state arrays
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NipcMyPhase = count ( material_phaseAt == p ) * discretization_nIP
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 constitutive_allocateState ( plasticState ( p ) , NipcMyPhase , sizeState , sizeDotState , sizeDeltaState )
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plasticState ( p ) % nonlocal = pl % get_asBool ( 'nonlocal' )
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if ( plasticState ( p ) % nonlocal . and . . not . allocated ( IPneighborhood ) ) &
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call IO_error ( 212 , ext_msg = 'IPneighborhood does not exist' )
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plasticState ( p ) % offsetDeltaState = 0 ! ToDo: state structure does not follow convention
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st0 % rho = > plasticState ( p ) % state0 ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
stt % rho = > plasticState ( p ) % state ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
dot % rho = > plasticState ( p ) % dotState ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
del % rho = > plasticState ( p ) % deltaState ( 0 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
plasticState ( p ) % atol ( 1 : 10 * prm % sum_N_sl ) = prm % atol_rho
stt % rhoSgl = > plasticState ( p ) % state ( 0 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
dot % rhoSgl = > plasticState ( p ) % dotState ( 0 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
del % rhoSgl = > plasticState ( p ) % deltaState ( 0 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
stt % rhoSglMobile = > plasticState ( p ) % state ( 0 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
dot % rhoSglMobile = > plasticState ( p ) % dotState ( 0 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
del % rhoSglMobile = > plasticState ( p ) % deltaState ( 0 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
stt % rho_sgl_mob_edg_pos = > plasticState ( p ) % state ( 0 * prm % sum_N_sl + 1 : 1 * prm % sum_N_sl , : )
dot % rho_sgl_mob_edg_pos = > plasticState ( p ) % dotState ( 0 * prm % sum_N_sl + 1 : 1 * prm % sum_N_sl , : )
del % rho_sgl_mob_edg_pos = > plasticState ( p ) % deltaState ( 0 * prm % sum_N_sl + 1 : 1 * prm % sum_N_sl , : )
stt % rho_sgl_mob_edg_neg = > plasticState ( p ) % state ( 1 * prm % sum_N_sl + 1 : 2 * prm % sum_N_sl , : )
dot % rho_sgl_mob_edg_neg = > plasticState ( p ) % dotState ( 1 * prm % sum_N_sl + 1 : 2 * prm % sum_N_sl , : )
del % rho_sgl_mob_edg_neg = > plasticState ( p ) % deltaState ( 1 * prm % sum_N_sl + 1 : 2 * prm % sum_N_sl , : )
stt % rho_sgl_mob_scr_pos = > plasticState ( p ) % state ( 2 * prm % sum_N_sl + 1 : 3 * prm % sum_N_sl , : )
dot % rho_sgl_mob_scr_pos = > plasticState ( p ) % dotState ( 2 * prm % sum_N_sl + 1 : 3 * prm % sum_N_sl , : )
del % rho_sgl_mob_scr_pos = > plasticState ( p ) % deltaState ( 2 * prm % sum_N_sl + 1 : 3 * prm % sum_N_sl , : )
stt % rho_sgl_mob_scr_neg = > plasticState ( p ) % state ( 3 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
dot % rho_sgl_mob_scr_neg = > plasticState ( p ) % dotState ( 3 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
del % rho_sgl_mob_scr_neg = > plasticState ( p ) % deltaState ( 3 * prm % sum_N_sl + 1 : 4 * prm % sum_N_sl , : )
stt % rhoSglImmobile = > plasticState ( p ) % state ( 4 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
dot % rhoSglImmobile = > plasticState ( p ) % dotState ( 4 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
del % rhoSglImmobile = > plasticState ( p ) % deltaState ( 4 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
stt % rho_sgl_imm_edg_pos = > plasticState ( p ) % state ( 4 * prm % sum_N_sl + 1 : 5 * prm % sum_N_sl , : )
dot % rho_sgl_imm_edg_pos = > plasticState ( p ) % dotState ( 4 * prm % sum_N_sl + 1 : 5 * prm % sum_N_sl , : )
del % rho_sgl_imm_edg_pos = > plasticState ( p ) % deltaState ( 4 * prm % sum_N_sl + 1 : 5 * prm % sum_N_sl , : )
stt % rho_sgl_imm_edg_neg = > plasticState ( p ) % state ( 5 * prm % sum_N_sl + 1 : 6 * prm % sum_N_sl , : )
dot % rho_sgl_imm_edg_neg = > plasticState ( p ) % dotState ( 5 * prm % sum_N_sl + 1 : 6 * prm % sum_N_sl , : )
del % rho_sgl_imm_edg_neg = > plasticState ( p ) % deltaState ( 5 * prm % sum_N_sl + 1 : 6 * prm % sum_N_sl , : )
stt % rho_sgl_imm_scr_pos = > plasticState ( p ) % state ( 6 * prm % sum_N_sl + 1 : 7 * prm % sum_N_sl , : )
dot % rho_sgl_imm_scr_pos = > plasticState ( p ) % dotState ( 6 * prm % sum_N_sl + 1 : 7 * prm % sum_N_sl , : )
del % rho_sgl_imm_scr_pos = > plasticState ( p ) % deltaState ( 6 * prm % sum_N_sl + 1 : 7 * prm % sum_N_sl , : )
stt % rho_sgl_imm_scr_neg = > plasticState ( p ) % state ( 7 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
dot % rho_sgl_imm_scr_neg = > plasticState ( p ) % dotState ( 7 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
del % rho_sgl_imm_scr_neg = > plasticState ( p ) % deltaState ( 7 * prm % sum_N_sl + 1 : 8 * prm % sum_N_sl , : )
stt % rhoDip = > plasticState ( p ) % state ( 8 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
dot % rhoDip = > plasticState ( p ) % dotState ( 8 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
del % rhoDip = > plasticState ( p ) % deltaState ( 8 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
stt % rho_dip_edg = > plasticState ( p ) % state ( 8 * prm % sum_N_sl + 1 : 9 * prm % sum_N_sl , : )
dot % rho_dip_edg = > plasticState ( p ) % dotState ( 8 * prm % sum_N_sl + 1 : 9 * prm % sum_N_sl , : )
del % rho_dip_edg = > plasticState ( p ) % deltaState ( 8 * prm % sum_N_sl + 1 : 9 * prm % sum_N_sl , : )
stt % rho_dip_scr = > plasticState ( p ) % state ( 9 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
dot % rho_dip_scr = > plasticState ( p ) % dotState ( 9 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
del % rho_dip_scr = > plasticState ( p ) % deltaState ( 9 * prm % sum_N_sl + 1 : 10 * prm % sum_N_sl , : )
stt % gamma = > plasticState ( p ) % state ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl , 1 : NipcMyPhase )
dot % gamma = > plasticState ( p ) % dotState ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl , 1 : NipcMyPhase )
del % gamma = > plasticState ( p ) % deltaState ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl , 1 : NipcMyPhase )
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plasticState ( p ) % atol ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl ) = pl % get_asFloat ( 'atol_gamma' , defaultVal = 1.0e-2_pReal )
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if ( any ( plasticState ( p ) % 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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plasticState ( p ) % slipRate = > plasticState ( p ) % dotState ( 10 * prm % sum_N_sl + 1 : 11 * prm % sum_N_sl , 1 : NipcMyPhase )
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stt % rho_forest = > plasticState ( p ) % state ( 11 * prm % sum_N_sl + 1 : 12 * prm % sum_N_sl , 1 : NipcMyPhase )
stt % v = > plasticState ( p ) % state ( 12 * prm % sum_N_sl + 1 : 16 * prm % sum_N_sl , 1 : NipcMyPhase )
stt % v_edg_pos = > plasticState ( p ) % state ( 12 * prm % sum_N_sl + 1 : 13 * prm % sum_N_sl , 1 : NipcMyPhase )
stt % v_edg_neg = > plasticState ( p ) % state ( 13 * prm % sum_N_sl + 1 : 14 * prm % sum_N_sl , 1 : NipcMyPhase )
stt % v_scr_pos = > plasticState ( p ) % state ( 14 * prm % sum_N_sl + 1 : 15 * prm % sum_N_sl , 1 : NipcMyPhase )
stt % v_scr_neg = > plasticState ( p ) % state ( 15 * prm % sum_N_sl + 1 : 16 * prm % sum_N_sl , 1 : NipcMyPhase )
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allocate ( dst % tau_pass ( prm % sum_N_sl , NipcMyPhase ) , source = 0.0_pReal )
allocate ( dst % tau_back ( prm % sum_N_sl , NipcMyPhase ) , source = 0.0_pReal )
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end associate
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if ( NipcMyPhase > 0 ) call stateInit ( ini , p , NipcMyPhase , i )
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plasticState ( p ) % state0 = plasticState ( p ) % state
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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 ) / / '(nonlocal)' )
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enddo
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allocate ( compatibility ( 2 , maxval ( param % sum_N_sl ) , maxval ( param % sum_N_sl ) , nIPneighbors , &
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discretization_nIP , discretization_nElem ) , source = 0.0_pReal )
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! BEGIN DEPRECATED----------------------------------------------------------------------------------
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allocate ( iRhoU ( maxval ( param % sum_N_sl ) , 4 , Ninstance ) , source = 0 )
allocate ( iV ( maxval ( param % sum_N_sl ) , 4 , Ninstance ) , source = 0 )
allocate ( iD ( maxval ( param % sum_N_sl ) , 2 , Ninstance ) , source = 0 )
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i = 0
do p = 1 , phases % length
phase = > phases % get ( p )
if ( . not . myPlasticity ( p ) ) cycle
i = i + 1
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NipcMyPhase = count ( material_phaseAt == p ) * discretization_nIP
l = 0
do t = 1 , 4
do s = 1 , param ( i ) % sum_N_sl
l = l + 1
iRhoU ( s , t , i ) = l
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enddo
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enddo
l = l + ( 4 + 2 + 1 + 1 ) * param ( i ) % sum_N_sl ! immobile(4), dipole(2), shear, forest
do t = 1 , 4
do s = 1 , param ( i ) % sum_N_sl
l = l + 1
iV ( s , t , i ) = l
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enddo
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enddo
do t = 1 , 2
do s = 1 , param ( i ) % sum_N_sl
l = l + 1
iD ( s , t , i ) = l
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enddo
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enddo
if ( iD ( param ( i ) % sum_N_sl , 2 , i ) / = plasticState ( p ) % sizeState ) &
call IO_error ( 0 , ext_msg = 'state indices not properly set (nonlocal)' )
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enddo
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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 plastic_nonlocal_dependentState ( F , Fp , instance , of , ip , el )
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real ( pReal ) , dimension ( 3 , 3 ) , intent ( in ) :: &
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F , &
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Fp
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integer , intent ( in ) :: &
instance , &
of , &
ip , &
el
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integer :: &
no , & !< neighbor offset
neighbor_el , & ! element number of neighboring material point
neighbor_ip , & ! integration point of neighboring material point
neighbor_instance , & ! instance of this plasticity 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 ) :: &
FVsize , &
nRealNeighbors ! number of really existing neighbors
integer , dimension ( 2 ) :: &
neighbors
real ( pReal ) , dimension ( 2 ) :: &
rhoExcessGradient , &
rhoExcessGradient_over_rho , &
rhoTotal
real ( pReal ) , dimension ( 3 ) :: &
rhoExcessDifferences , &
normal_latticeConf
real ( pReal ) , dimension ( 3 , 3 ) :: &
invFe , & !< inverse of elastic deformation gradient
invFp , & !< inverse of plastic deformation gradient
connections , &
invConnections
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real ( pReal ) , dimension ( 3 , nIPneighbors ) :: &
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connection_latticeConf
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real ( pReal ) , dimension ( 2 , param ( instance ) % sum_N_sl ) :: &
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rhoExcess
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl ) :: &
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rho_edg_delta , &
rho_scr_delta
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 10 ) :: &
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rho , &
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rho0 , &
rho_neighbor0
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , param ( instance ) % sum_N_sl ) :: &
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myInteractionMatrix ! corrected slip interaction matrix
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real ( pReal ) , dimension ( param ( instance ) % 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 ( instance ) % sum_N_sl , 2 ) :: &
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m ! direction of dislocation motion
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associate ( prm = > param ( instance ) , dst = > microstructure ( instance ) , stt = > state ( instance ) )
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rho = getRho ( instance , of , ip , el )
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stt % rho_forest ( : , of ) = 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 ( lattice_structure ( material_phaseAt ( 1 , el ) ) == [ LATTICE_bcc_ID , LATTICE_fcc_ID ] ) ) then
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myInteractionMatrix = prm % h_sl_sl &
* spread ( ( 1.0_pReal - prm % f_F &
+ prm % f_F &
* log ( 0.35_pReal * prm % b_sl * sqrt ( max ( stt % rho_forest ( : , of ) , prm % rho_significant ) ) ) &
/ log ( 0.35_pReal * prm % b_sl * 1e6_pReal ) ) ** 2.0_pReal , 2 , prm % sum_N_sl )
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else
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myInteractionMatrix = prm % h_sl_sl
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endif
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dst % tau_pass ( : , of ) = prm % 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 ( instance , of , ip , el )
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if ( . not . phase_localPlasticity ( material_phaseAt ( 1 , el ) ) . and . prm % shortRangeStressCorrection ) then
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invFp = math_inv33 ( Fp )
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invFe = matmul ( Fp , math_inv33 ( F ) )
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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 = IPvolume ( ip , el ) ** ( 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 = IPneighborhood ( 1 , n , ip , el )
neighbor_ip = IPneighborhood ( 2 , n , ip , el )
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no = material_phasememberAt ( 1 , neighbor_ip , neighbor_el )
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if ( neighbor_el > 0 . and . neighbor_ip > 0 ) then
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neighbor_instance = phase_plasticityInstance ( material_phaseAt ( 1 , neighbor_el ) )
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if ( neighbor_instance == instance ) then
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nRealNeighbors = nRealNeighbors + 1.0_pReal
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rho_neighbor0 = getRho0 ( instance , no , neighbor_ip , neighbor_el )
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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 , discretization_IPcoords ( 1 : 3 , neighbor_el + neighbor_ip - 1 ) &
- discretization_IPcoords ( 1 : 3 , el + neighbor_ip - 1 ) )
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normal_latticeConf = matmul ( transpose ( invFp ) , IPareaNormal ( 1 : 3 , n , ip , el ) )
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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
connection_latticeConf ( 1 : 3 , n ) = normal_latticeConf * IPvolume ( ip , el ) / IParea ( n , ip , el ) ! instead take the surface normal scaled with the diameter of the cell
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else
! local neighbor or different lattice structure or different constitution instance -> use central values instead
connection_latticeConf ( 1 : 3 , n ) = 0.0_pReal
rho_edg_delta_neighbor ( : , n ) = rho_edg_delta
rho_scr_delta_neighbor ( : , n ) = rho_scr_delta
endif
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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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endif
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enddo
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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 ) )
enddo
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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enddo
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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 ...
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
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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 , of ) = - prm % mu * prm % b_sl ( s ) / ( 2.0_pReal * PI ) &
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* ( rhoExcessGradient_over_rho ( 1 ) / ( 1.0_pReal - prm % nu ) &
+ rhoExcessGradient_over_rho ( 2 ) )
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enddo
endif
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#ifdef DEBUG
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if ( debugConstitutive % extensive &
. and . ( ( debugConstitutive % element == el . and . debugConstitutive % ip == ip ) &
. or . . not . debugConstitutive % selective ) ) then
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write ( 6 , '(/,a,i8,1x,i2,1x,i1,/)' ) '<< CONST >> nonlocal_microstructure at el ip ' , el , ip
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write ( 6 , '(a,/,12x,12(e10.3,1x))' ) '<< CONST >> rhoForest' , stt % rho_forest ( : , of )
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write ( 6 , '(a,/,12x,12(f10.5,1x))' ) '<< CONST >> tauThreshold / MPa' , dst % tau_pass ( : , of ) * 1e-6
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write ( 6 , '(a,/,12x,12(f10.5,1x),/)' ) '<< CONST >> tauBack / MPa' , dst % tau_back ( : , of ) * 1e-6
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endif
#endif
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end associate
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end subroutine plastic_nonlocal_dependentState
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!--------------------------------------------------------------------------------------------------
!> @brief calculates plastic velocity gradient and its tangent
!--------------------------------------------------------------------------------------------------
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module subroutine plastic_nonlocal_LpAndItsTangent ( Lp , dLp_dMp , &
Mp , Temperature , instance , of , ip , el )
real ( pReal ) , dimension ( 3 , 3 ) , intent ( out ) :: &
Lp !< plastic velocity gradient
real ( pReal ) , dimension ( 3 , 3 , 3 , 3 ) , intent ( out ) :: &
dLp_dMp
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integer , intent ( in ) :: &
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instance , &
of , &
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ip , & !< current integration point
el !< current element number
real ( pReal ) , intent ( in ) :: &
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Temperature !< temperature
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real ( pReal ) , dimension ( 3 , 3 ) , intent ( in ) :: &
Mp
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!< derivative of Lp with respect to Mp
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integer :: &
ns , & !< short notation for the total number of active slip systems
i , &
j , &
k , &
l , &
t , & !< dislocation type
s !< index of my current slip system
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 8 ) :: &
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rhoSgl !< single dislocation densities (including blocked)
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 10 ) :: &
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rho
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real ( pReal ) , dimension ( param ( instance ) % 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 ( instance ) % sum_N_sl ) :: &
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tau , & !< resolved shear stress including backstress terms
gdotTotal !< shear rate
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associate ( prm = > param ( instance ) , dst = > microstructure ( instance ) , stt = > state ( instance ) )
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ns = prm % sum_N_sl
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!*** shortcut to state variables
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rho = getRho ( instance , of , ip , el )
rhoSgl = rho ( : , sgl )
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do s = 1 , ns
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tau ( s ) = math_tensordot ( Mp , prm % Schmid ( 1 : 3 , 1 : 3 , s ) )
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tauNS ( s , 1 ) = tau ( s )
tauNS ( s , 2 ) = tau ( s )
if ( tau ( s ) > 0.0_pReal ) then
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tauNS ( s , 3 ) = math_tensordot ( Mp , + prm % nonSchmid_pos ( 1 : 3 , 1 : 3 , s ) )
tauNS ( s , 4 ) = math_tensordot ( Mp , - prm % nonSchmid_neg ( 1 : 3 , 1 : 3 , s ) )
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else
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tauNS ( s , 3 ) = math_tensordot ( Mp , + prm % nonSchmid_neg ( 1 : 3 , 1 : 3 , s ) )
tauNS ( s , 4 ) = math_tensordot ( Mp , - prm % nonSchmid_pos ( 1 : 3 , 1 : 3 , s ) )
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endif
enddo
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tauNS = tauNS + spread ( dst % tau_back ( : , of ) , 2 , 4 )
tau = tau + dst % tau_back ( : , of )
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! edges
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call kinetics ( v ( : , 1 ) , dv_dtau ( : , 1 ) , dv_dtauNS ( : , 1 ) , &
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tau , tauNS ( : , 1 ) , dst % tau_pass ( : , of ) , 1 , Temperature , instance )
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v ( : , 2 ) = v ( : , 1 )
dv_dtau ( : , 2 ) = dv_dtau ( : , 1 )
dv_dtauNS ( : , 2 ) = dv_dtauNS ( : , 1 )
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!screws
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if ( prm % nonSchmidActive ) then
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v ( : , 3 : 4 ) = spread ( v ( : , 1 ) , 2 , 2 )
dv_dtau ( : , 3 : 4 ) = spread ( dv_dtau ( : , 1 ) , 2 , 2 )
dv_dtauNS ( : , 3 : 4 ) = spread ( dv_dtauNS ( : , 1 ) , 2 , 2 )
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else
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do t = 3 , 4
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call kinetics ( v ( : , t ) , dv_dtau ( : , t ) , dv_dtauNS ( : , t ) , &
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tau , tauNS ( : , t ) , dst % tau_pass ( : , of ) , 2 , Temperature , instance )
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enddo
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endif
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stt % v ( : , of ) = pack ( v , . true . )
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!*** Bauschinger effect
forall ( s = 1 : ns , t = 5 : 8 , rhoSgl ( s , t ) * v ( s , t - 4 ) < 0.0_pReal ) &
rhoSgl ( s , t - 4 ) = rhoSgl ( s , t - 4 ) + abs ( rhoSgl ( s , t ) )
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gdotTotal = sum ( rhoSgl ( : , 1 : 4 ) * v , 2 ) * prm % b_sl
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Lp = 0.0_pReal
dLp_dMp = 0.0_pReal
do s = 1 , ns
Lp = Lp + gdotTotal ( s ) * prm % Schmid ( 1 : 3 , 1 : 3 , s )
forall ( i = 1 : 3 , j = 1 : 3 , k = 1 : 3 , l = 1 : 3 ) &
dLp_dMp ( i , j , k , l ) = dLp_dMp ( i , j , k , l ) &
+ prm % Schmid ( i , j , s ) * prm % Schmid ( k , l , s ) &
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* sum ( rhoSgl ( s , 1 : 4 ) * dv_dtau ( s , 1 : 4 ) ) * prm % b_sl ( s ) &
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+ prm % Schmid ( i , j , s ) &
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* ( prm % nonSchmid_pos ( k , l , s ) * rhoSgl ( s , 3 ) * dv_dtauNS ( s , 3 ) &
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- prm % nonSchmid_neg ( k , l , s ) * rhoSgl ( s , 4 ) * dv_dtauNS ( s , 4 ) ) * prm % b_sl ( s )
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enddo
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end associate
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end subroutine plastic_nonlocal_LpAndItsTangent
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!--------------------------------------------------------------------------------------------------
!> @brief (instantaneous) incremental change of microstructure
!--------------------------------------------------------------------------------------------------
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module subroutine plastic_nonlocal_deltaState ( Mp , instance , of , ip , el )
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real ( pReal ) , dimension ( 3 , 3 ) , intent ( in ) :: &
Mp !< MandelStress
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integer , intent ( in ) :: &
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instance , & ! current instance of this plasticity
of , & !< offset
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ip , &
el
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integer :: &
ph , & !< phase
ns , & ! short notation for the total number of active slip systems
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 ( instance ) % 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 ( instance ) % sum_N_sl , 10 ) :: &
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rho ! current dislocation densities
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 4 ) :: &
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v ! dislocation glide velocity
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl ) :: &
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tau ! current resolved shear stress
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real ( pReal ) , dimension ( param ( instance ) % 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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ph = material_phaseAt ( 1 , el )
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associate ( prm = > param ( instance ) , dst = > microstructure ( instance ) , del = > deltaState ( instance ) )
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ns = prm % sum_N_sl
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!*** shortcut to state variables
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forall ( s = 1 : ns , t = 1 : 4 ) v ( s , t ) = plasticState ( ph ) % state ( iV ( s , t , instance ) , of )
forall ( s = 1 : ns , c = 1 : 2 ) dUpperOld ( s , c ) = plasticState ( ph ) % state ( iD ( s , c , instance ) , of )
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rho = getRho ( instance , of , ip , el )
rhoDip = rho ( : , dip )
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!****************************************************************************
!*** dislocation remobilization (bauschinger effect)
where ( rho ( : , imm ) * v < 0.0_pReal )
deltaRhoRemobilization ( : , mob ) = abs ( rho ( : , imm ) )
deltaRhoRemobilization ( : , imm ) = - rho ( : , imm )
rho ( : , mob ) = rho ( : , mob ) + abs ( rho ( : , imm ) )
rho ( : , imm ) = 0.0_pReal
elsewhere
deltaRhoRemobilization ( : , mob ) = 0.0_pReal
deltaRhoRemobilization ( : , imm ) = 0.0_pReal
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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 % Schmid ( 1 : 3 , 1 : 3 , s ) ) + dst % tau_back ( s , of )
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if ( abs ( tau ( s ) ) < 1.0e-15_pReal ) tau ( s ) = 1.0e-15_pReal
enddo
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dUpper ( : , 1 ) = prm % mu * prm % b_sl / ( 8.0_pReal * PI * ( 1.0_pReal - prm % nu ) * abs ( tau ) )
dUpper ( : , 2 ) = prm % mu * prm % b_sl / ( 4.0_pReal * PI * abs ( tau ) )
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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
deltaRhoDipole2SingleStress = 0.0_pReal
forall ( c = 1 : 2 , s = 1 : ns , deltaDUpper ( s , c ) < 0.0_pReal . and . &
dNeq0 ( dUpperOld ( s , c ) - prm % minDipoleHeight ( s , c ) ) ) &
deltaRhoDipole2SingleStress ( s , 8 + c ) = rhoDip ( s , c ) * deltaDUpper ( s , c ) &
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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 )
forall ( s = 1 : ns , c = 1 : 2 ) plasticState ( ph ) % state ( iD ( s , c , instance ) , of ) = dUpper ( s , c )
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plasticState ( ph ) % deltaState ( : , of ) = 0.0_pReal
del % rho ( : , of ) = reshape ( deltaRhoRemobilization + deltaRhoDipole2SingleStress , [ 10 * ns ] )
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#ifdef DEBUG
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if ( debugConstitutive % extensive &
. and . ( ( debugConstitutive % element == el . and . debugConstitutive % ip == ip ) &
. or . . not . debugConstitutive % selective ) ) then
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write ( 6 , '(a,/,8(12x,12(e12.5,1x),/))' ) '<< CONST >> dislocation remobilization' , deltaRhoRemobilization ( : , 1 : 8 )
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write ( 6 , '(a,/,10(12x,12(e12.5,1x),/),/)' ) '<< CONST >> dipole dissociation by stress increase' , deltaRhoDipole2SingleStress
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endif
#endif
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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 plastic_nonlocal_dotState ( Mp , F , Fp , Temperature , timestep , &
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instance , of , ip , el )
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real ( pReal ) , dimension ( 3 , 3 ) , intent ( in ) :: &
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Mp !< MandelStress
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real ( pReal ) , dimension ( 3 , 3 , homogenization_maxNgrains , discretization_nIP , discretization_nElem ) , intent ( in ) :: &
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F , & !< elastic deformation gradient
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Fp !< plastic deformation gradient
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real ( pReal ) , intent ( in ) :: &
Temperature , & !< temperature
timestep !< substepped crystallite time increment
integer , intent ( in ) :: &
instance , &
of , &
ip , & !< current integration point
el !< current element number
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integer :: &
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ph , &
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ns , & !< short notation for the total number of active slip systems
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 ( instance ) % 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 ( instance ) % 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 ( instance ) % sum_N_sl , 4 ) :: &
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v , & !< current dislocation glide velocity
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v0 , &
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gdot !< shear rates
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl ) :: &
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tau , & !< current resolved shear stress
vClimb !< climb velocity of edge dipoles
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real ( pReal ) , dimension ( param ( instance ) % 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
real ( pReal ) :: &
selfDiffusion !< self diffusion
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ph = material_phaseAt ( 1 , el )
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if ( timestep < = 0.0_pReal ) then
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plasticState ( ph ) % dotState = 0.0_pReal
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return
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endif
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associate ( prm = > param ( instance ) , &
dst = > microstructure ( instance ) , &
dot = > dotState ( instance ) , &
stt = > state ( instance ) )
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ns = prm % sum_N_sl
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tau = 0.0_pReal
gdot = 0.0_pReal
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rho = getRho ( instance , of , ip , el )
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rhoSgl = rho ( : , sgl )
rhoDip = rho ( : , dip )
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rho0 = getRho0 ( instance , of , ip , el )
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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 , instance ) , of )
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gdot = rhoSgl ( : , 1 : 4 ) * v * spread ( prm % b_sl , 2 , 4 )
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#ifdef DEBUG
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if ( debugConstitutive % basic &
. and . ( ( debugConstitutive % element == el . and . debugConstitutive % ip == ip ) &
. or . . not . debugConstitutive % selective ) ) then
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write ( 6 , '(a,/,10(12x,12(e12.5,1x),/))' ) '<< CONST >> rho / 1/m^2' , rhoSgl , rhoDip
write ( 6 , '(a,/,4(12x,12(e12.5,1x),/))' ) '<< CONST >> gdot / 1/s' , gdot
endif
#endif
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!****************************************************************************
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!*** limits for stable dipole height
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do s = 1 , ns
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tau ( s ) = math_tensordot ( Mp , prm % Schmid ( 1 : 3 , 1 : 3 , s ) ) + dst % tau_back ( s , of )
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if ( abs ( tau ( s ) ) < 1.0e-15_pReal ) tau ( s ) = 1.0e-15_pReal
enddo
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dLower = prm % minDipoleHeight
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dUpper ( : , 1 ) = prm % mu * prm % b_sl / ( 8.0_pReal * PI * ( 1.0_pReal - prm % nu ) * abs ( tau ) )
dUpper ( : , 2 ) = prm % mu * prm % b_sl / ( 4.0_pReal * PI * abs ( tau ) )
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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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!****************************************************************************
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!*** dislocation multiplication
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rhoDotMultiplication = 0.0_pReal
isBCC : if ( lattice_structure ( ph ) == LATTICE_bcc_ID ) then
forall ( s = 1 : ns , sum ( abs ( v ( s , 1 : 4 ) ) ) > 0.0_pReal )
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rhoDotMultiplication ( s , 1 : 2 ) = sum ( abs ( gdot ( s , 3 : 4 ) ) ) / prm % b_sl ( s ) & ! assuming double-cross-slip of screws to be decisive for multiplication
* sqrt ( stt % rho_forest ( s , of ) ) / 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 ( gdot ( s , 3 : 4 ) ) ) / prm % b_sl ( s ) & ! assuming double-cross-slip of screws to be decisive for multiplication
* sqrt ( stt % rho_forest ( s , of ) ) / 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
endforall
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else isBCC
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rhoDotMultiplication ( : , 1 : 4 ) = spread ( &
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( sum ( abs ( gdot ( : , 1 : 2 ) ) , 2 ) * prm % f_ed_mult + sum ( abs ( gdot ( : , 3 : 4 ) ) , 2 ) ) &
* sqrt ( stt % rho_forest ( : , of ) ) / prm % i_sl / prm % b_sl , 2 , 4 )
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endif isBCC
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forall ( s = 1 : ns , t = 1 : 4 ) v0 ( s , t ) = plasticState ( ph ) % state0 ( iV ( s , t , instance ) , of )
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!****************************************************************************
!*** calculate dipole formation and annihilation
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!*** formation by glide
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 ( gdot ( : , 2 * c ) ) & ! negative mobile --> positive mobile
+ rhoSgl ( : , 2 * c ) * abs ( gdot ( : , 2 * c - 1 ) ) & ! positive mobile --> negative mobile
+ abs ( rhoSgl ( : , 2 * c + 4 ) ) * abs ( gdot ( : , 2 * c - 1 ) ) ) ! positive mobile --> negative immobile
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rhoDotSingle2DipoleGlide ( : , 2 * c ) = - 2.0_pReal * dUpper ( : , c ) / prm % b_sl &
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* ( rhoSgl ( : , 2 * c - 1 ) * abs ( gdot ( : , 2 * c ) ) & ! negative mobile --> positive mobile
+ rhoSgl ( : , 2 * c ) * abs ( gdot ( : , 2 * c - 1 ) ) & ! positive mobile --> negative mobile
+ abs ( rhoSgl ( : , 2 * c + 3 ) ) * abs ( gdot ( : , 2 * c ) ) ) ! negative mobile --> positive immobile
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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 ( gdot ( : , 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 ( gdot ( : , 2 * c - 1 ) ) ! positive mobile --> negative immobile
rhoDotSingle2DipoleGlide ( : , c + 8 ) = abs ( rhoDotSingle2DipoleGlide ( : , 2 * c + 3 ) ) &
+ abs ( rhoDotSingle2DipoleGlide ( : , 2 * c + 4 ) ) &
- rhoDotSingle2DipoleGlide ( : , 2 * c - 1 ) &
- rhoDotSingle2DipoleGlide ( : , 2 * c )
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enddo
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!*** athermal annihilation
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 ( gdot ( : , 2 * c ) ) + rhoSgl ( : , 2 * c ) * abs ( gdot ( : , 2 * c - 1 ) ) ) & ! was single hitting single
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+ 2.0_pReal * ( abs ( rhoSgl ( : , 2 * c + 3 ) ) * abs ( gdot ( : , 2 * c ) ) + abs ( rhoSgl ( : , 2 * c + 4 ) ) * abs ( gdot ( : , 2 * c - 1 ) ) ) & ! was single hitting immobile single or was immobile single hit by single
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+ rhoDip ( : , c ) * ( abs ( gdot ( : , 2 * c - 1 ) ) + abs ( gdot ( : , 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 ( lattice_structure ( ph ) == LATTICE_fcc_ID ) &
forall ( s = 1 : ns , prm % colinearSystem ( s ) > 0 ) &
rhoDotAthermalAnnihilation ( prm % colinearSystem ( s ) , 1 : 2 ) = - rhoDotAthermalAnnihilation ( s , 10 ) &
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* 0.25_pReal * sqrt ( stt % rho_forest ( s , of ) ) * ( dUpper ( s , 2 ) + dLower ( s , 2 ) ) * prm % f_ed
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!*** thermally activated annihilation of edge dipoles by climb
rhoDotThermalAnnihilation = 0.0_pReal
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selfDiffusion = prm % D_0 * exp ( - prm % Q_cl / ( kB * Temperature ) )
vClimb = prm % V_at * selfDiffusion * prm % mu &
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/ ( kB * Temperature * PI * ( 1.0_pReal - prm % nu ) * ( dUpper ( : , 1 ) + dLower ( : , 1 ) ) )
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forall ( s = 1 : ns , dUpper ( s , 1 ) > dLower ( s , 1 ) ) &
rhoDotThermalAnnihilation ( s , 9 ) = max ( - 4.0_pReal * rhoDip ( s , 1 ) * vClimb ( s ) / ( dUpper ( s , 1 ) - dLower ( s , 1 ) ) , &
- rhoDip ( s , 1 ) / timestep - rhoDotAthermalAnnihilation ( s , 9 ) &
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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 ( F , Fp , timestep , instance , of , ip , el ) &
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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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#ifdef DEBUG
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if ( debugConstitutive % extensive ) then
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write ( 6 , '(a,i5,a,i2)' ) '<< CONST >> evolution rate leads to negative density at el ' , el , ' ip ' , ip
write ( 6 , '(a)' ) '<< CONST >> enforcing cutback !!!'
endif
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#endif
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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 ( : , of ) = pack ( rhoDot , . true . )
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dot % gamma ( : , of ) = sum ( gdot , 2 )
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endif
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end associate
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end subroutine plastic_nonlocal_dotState
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!---------------------------------------------------------------------------------------------------
!> @brief calculates the rate of change of microstructure
!---------------------------------------------------------------------------------------------------
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function rhoDotFlux ( F , Fp , timestep , instance , of , ip , el )
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real ( pReal ) , dimension ( 3 , 3 , homogenization_maxNgrains , discretization_nIP , discretization_nElem ) , intent ( in ) :: &
F , & !< elastic deformation gradient
Fp !< plastic deformation gradient
real ( pReal ) , intent ( in ) :: &
timestep !< substepped crystallite time increment
integer , intent ( in ) :: &
instance , &
of , &
ip , & !< current integration point
el !< current element number
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integer :: &
ph , &
neighbor_instance , & !< instance of my neighbor's plasticity
ns , & !< short notation for the total number of active slip systems
c , & !< character of dislocation
n , & !< index of my current neighbor
neighbor_el , & !< element number of my neighbor
neighbor_ip , & !< integration point of my neighbor
neighbor_n , & !< neighbor index pointing to me when looking from my neighbor
opposite_neighbor , & !< index of my opposite neighbor
opposite_ip , & !< ip of my opposite neighbor
opposite_el , & !< element index of my opposite neighbor
opposite_n , & !< neighbor index pointing to me when looking from my opposite neighbor
t , & !< type of dislocation
no , & !< neighbor offset shortcut
np , & !< neighbor phase shortcut
topp , & !< type of dislocation with opposite sign to t
s !< index of my current slip system
real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 10 ) :: &
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 ( instance ) % sum_N_sl , 8 ) :: &
rhoSgl , & !< current single dislocation densities (positive/negative screw and edge without dipoles)
neighbor_rhoSgl0 , & !< current single dislocation densities of neighboring ip (positive/negative screw and edge without dipoles)
my_rhoSgl0 !< single dislocation densities of central ip (positive/negative screw and edge without dipoles)
real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 4 ) :: &
v , & !< current dislocation glide velocity
v0 , &
neighbor_v0 , & !< dislocation glide velocity of enighboring ip
gdot !< shear rates
real ( pReal ) , dimension ( 3 , param ( instance ) % sum_N_sl , 4 ) :: &
m !< direction of dislocation motion
real ( pReal ) , dimension ( 3 , 3 ) :: &
my_F , & !< my total deformation gradient
neighbor_F , & !< total deformation gradient of my neighbor
my_Fe , & !< my elastic deformation gradient
neighbor_Fe , & !< elastic deformation gradient of my neighbor
Favg !< average total deformation gradient of me and my neighbor
real ( pReal ) , dimension ( 3 ) :: &
normal_neighbor2me , & !< interface normal pointing from my neighbor to me in neighbor's lattice configuration
normal_neighbor2me_defConf , & !< interface normal pointing from my neighbor to me in shared deformed configuration
normal_me2neighbor , & !< interface normal pointing from me to my neighbor in my lattice configuration
normal_me2neighbor_defConf !< interface normal pointing from me to my neighbor in shared deformed configuration
real ( pReal ) :: &
area , & !< area of the current interface
transmissivity , & !< overall transmissivity of dislocation flux to neighboring material point
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lineLength !< dislocation line length leaving the current interface
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ph = material_phaseAt ( 1 , el )
associate ( prm = > param ( instance ) , &
dst = > microstructure ( instance ) , &
dot = > dotState ( instance ) , &
stt = > state ( instance ) )
ns = prm % sum_N_sl
gdot = 0.0_pReal
rho = getRho ( instance , of , ip , el )
rhoSgl = rho ( : , sgl )
rho0 = getRho0 ( instance , of , ip , el )
my_rhoSgl0 = rho0 ( : , sgl )
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forall ( s = 1 : ns , t = 1 : 4 ) v ( s , t ) = plasticState ( ph ) % state ( iV ( s , t , instance ) , of ) !ToDo: MD: I think we should use state0 here
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gdot = 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 , instance ) , of )
!****************************************************************************
!*** calculate dislocation fluxes (only for nonlocal plasticity)
rhoDotFlux = 0.0_pReal
if ( . not . phase_localPlasticity ( material_phaseAt ( 1 , el ) ) ) then
!*** check CFL (Courant-Friedrichs-Lewy) condition for flux
if ( any ( abs ( gdot ) > 0.0_pReal & ! any active slip system ...
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. and . prm % f_c * abs ( v0 ) * timestep &
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> IPvolume ( ip , el ) / maxval ( IParea ( : , ip , el ) ) ) ) then ! ...with velocity above critical value (we use the reference volume and area for simplicity here)
#ifdef DEBUG
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if ( debugConstitutive % extensive ) then
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write ( 6 , '(a,i5,a,i2)' ) '<< CONST >> CFL condition not fullfilled at el ' , el , ' ip ' , ip
write ( 6 , '(a,e10.3,a,e10.3)' ) '<< CONST >> velocity is at ' , &
maxval ( abs ( v0 ) , abs ( gdot ) > 0.0_pReal &
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. and . prm % f_c * abs ( v0 ) * timestep &
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> IPvolume ( ip , el ) / maxval ( IParea ( : , ip , el ) ) ) , &
' at a timestep of ' , timestep
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print * , '<< CONST >> enforcing cutback !!!'
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endif
#endif
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rhoDotFlux = IEEE_value ( 1.0_pReal , IEEE_quiet_NaN ) ! enforce cutback
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return
endif
!*** be aware of the definition of slip_transverse = slip_direction x slip_normal !!!
!*** opposite sign to our t vector in the (s,t,n) triplet !!!
m ( 1 : 3 , : , 1 ) = prm % slip_direction
m ( 1 : 3 , : , 2 ) = - prm % slip_direction
m ( 1 : 3 , : , 3 ) = - prm % slip_transverse
m ( 1 : 3 , : , 4 ) = prm % slip_transverse
my_F = F ( 1 : 3 , 1 : 3 , 1 , ip , el )
my_Fe = matmul ( my_F , math_inv33 ( Fp ( 1 : 3 , 1 : 3 , 1 , ip , el ) ) )
neighbors : do n = 1 , nIPneighbors
neighbor_el = IPneighborhood ( 1 , n , ip , el )
neighbor_ip = IPneighborhood ( 2 , n , ip , el )
neighbor_n = IPneighborhood ( 3 , n , ip , el )
np = material_phaseAt ( 1 , neighbor_el )
no = material_phasememberAt ( 1 , neighbor_ip , neighbor_el )
opposite_neighbor = n + mod ( n , 2 ) - mod ( n + 1 , 2 )
opposite_el = IPneighborhood ( 1 , opposite_neighbor , ip , el )
opposite_ip = IPneighborhood ( 2 , opposite_neighbor , ip , el )
opposite_n = IPneighborhood ( 3 , opposite_neighbor , ip , el )
if ( neighbor_n > 0 ) then ! if neighbor exists, average deformation gradient
neighbor_instance = phase_plasticityInstance ( material_phaseAt ( 1 , neighbor_el ) )
neighbor_F = F ( 1 : 3 , 1 : 3 , 1 , neighbor_ip , neighbor_el )
neighbor_Fe = matmul ( neighbor_F , math_inv33 ( Fp ( 1 : 3 , 1 : 3 , 1 , neighbor_ip , neighbor_el ) ) )
Favg = 0.5_pReal * ( my_F + neighbor_F )
else ! if no neighbor, take my value as average
Favg = my_F
endif
neighbor_v0 = 0.0_pReal ! needed for check of sign change in flux density below
!* FLUX FROM MY NEIGHBOR TO ME
!* This is only considered, if I have a neighbor of nonlocal plasticity
!* (also nonlocal constitutive law with local properties) that is at least a little bit
!* compatible.
!* If it's not at all compatible, no flux is arriving, because everything is dammed in front of
!* my neighbor's interface.
!* The entering flux from my neighbor will be distributed on my slip systems according to the
!* compatibility
if ( neighbor_n > 0 ) then
if ( phase_plasticity ( material_phaseAt ( 1 , neighbor_el ) ) == PLASTICITY_NONLOCAL_ID . and . &
any ( compatibility ( : , : , : , n , ip , el ) > 0.0_pReal ) ) then
forall ( s = 1 : ns , t = 1 : 4 )
neighbor_v0 ( s , t ) = plasticState ( np ) % state0 ( iV ( s , t , neighbor_instance ) , no )
neighbor_rhoSgl0 ( s , t ) = max ( plasticState ( np ) % state0 ( iRhoU ( s , t , neighbor_instance ) , no ) , 0.0_pReal )
endforall
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where ( neighbor_rhoSgl0 * IPvolume ( neighbor_ip , neighbor_el ) ** 0.667_pReal < prm % rho_num_significant &
. or . neighbor_rhoSgl0 < prm % rho_significant ) &
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neighbor_rhoSgl0 = 0.0_pReal
normal_neighbor2me_defConf = math_det33 ( Favg ) * matmul ( math_inv33 ( transpose ( Favg ) ) , &
IPareaNormal ( 1 : 3 , neighbor_n , neighbor_ip , neighbor_el ) ) ! normal of the interface in (average) deformed configuration (pointing neighbor => me)
normal_neighbor2me = matmul ( transpose ( neighbor_Fe ) , normal_neighbor2me_defConf ) &
/ math_det33 ( neighbor_Fe ) ! interface normal in the lattice configuration of my neighbor
area = IParea ( neighbor_n , neighbor_ip , neighbor_el ) * norm2 ( normal_neighbor2me )
normal_neighbor2me = normal_neighbor2me / norm2 ( normal_neighbor2me ) ! normalize the surface normal to unit length
do s = 1 , ns
do t = 1 , 4
c = ( t + 1 ) / 2
topp = t + mod ( t , 2 ) - mod ( t + 1 , 2 )
if ( neighbor_v0 ( s , t ) * math_inner ( m ( 1 : 3 , s , t ) , normal_neighbor2me ) > 0.0_pReal & ! flux from my neighbor to me == entering flux for me
. and . v0 ( s , t ) * neighbor_v0 ( s , t ) > = 0.0_pReal ) then ! ... only if no sign change in flux density
lineLength = neighbor_rhoSgl0 ( s , t ) * neighbor_v0 ( s , t ) &
* math_inner ( m ( 1 : 3 , s , t ) , normal_neighbor2me ) * area ! positive line length that wants to enter through this interface
where ( compatibility ( c , : , s , n , ip , el ) > 0.0_pReal ) &
rhoDotFlux ( : , t ) = rhoDotFlux ( 1 : ns , t ) &
+ lineLength / IPvolume ( ip , el ) * compatibility ( c , : , s , n , ip , el ) ** 2.0_pReal ! transferring to equally signed mobile dislocation type
where ( compatibility ( c , : , s , n , ip , el ) < 0.0_pReal ) &
rhoDotFlux ( : , topp ) = rhoDotFlux ( : , topp ) &
+ lineLength / IPvolume ( ip , el ) * compatibility ( c , : , s , n , ip , el ) ** 2.0_pReal ! transferring to opposite signed mobile dislocation type
endif
enddo
enddo
endif ; endif
!* FLUX FROM ME TO MY NEIGHBOR
!* This is not considered, if my opposite neighbor has a different constitutive law than nonlocal (still considered for nonlocal law with local properties).
!* Then, we assume, that the opposite(!) neighbor sends an equal amount of dislocations to me.
!* So the net flux in the direction of my neighbor is equal to zero:
!* leaving flux to neighbor == entering flux from opposite neighbor
!* In case of reduced transmissivity, part of the leaving flux is stored as dead dislocation density.
!* That means for an interface of zero transmissivity the leaving flux is fully converted to dead dislocations.
if ( opposite_n > 0 ) then
if ( phase_plasticity ( material_phaseAt ( 1 , opposite_el ) ) == PLASTICITY_NONLOCAL_ID ) then
normal_me2neighbor_defConf = math_det33 ( Favg ) &
* matmul ( math_inv33 ( transpose ( Favg ) ) , IPareaNormal ( 1 : 3 , n , ip , el ) ) ! normal of the interface in (average) deformed configuration (pointing me => neighbor)
normal_me2neighbor = matmul ( transpose ( my_Fe ) , normal_me2neighbor_defConf ) &
/ math_det33 ( my_Fe ) ! interface normal in my lattice configuration
area = IParea ( n , ip , el ) * norm2 ( normal_me2neighbor )
normal_me2neighbor = normal_me2neighbor / norm2 ( normal_me2neighbor ) ! normalize the surface normal to unit length
do s = 1 , ns
do t = 1 , 4
c = ( t + 1 ) / 2
if ( v0 ( s , t ) * math_inner ( m ( 1 : 3 , s , t ) , normal_me2neighbor ) > 0.0_pReal ) then ! flux from me to my neighbor == leaving flux for me (might also be a pure flux from my mobile density to dead density if interface not at all transmissive)
if ( v0 ( s , t ) * neighbor_v0 ( s , t ) > = 0.0_pReal ) then ! no sign change in flux density
transmissivity = sum ( compatibility ( c , : , s , n , ip , el ) ** 2.0_pReal ) ! overall transmissivity from this slip system to my neighbor
else ! sign change in flux density means sign change in stress which does not allow for dislocations to arive at the neighbor
transmissivity = 0.0_pReal
endif
lineLength = my_rhoSgl0 ( s , t ) * v0 ( s , t ) &
* math_inner ( m ( 1 : 3 , s , t ) , normal_me2neighbor ) * area ! positive line length of mobiles that wants to leave through this interface
rhoDotFlux ( s , t ) = rhoDotFlux ( s , t ) - lineLength / IPvolume ( ip , el ) ! subtract dislocation flux from current type
rhoDotFlux ( s , t + 4 ) = rhoDotFlux ( s , t + 4 ) &
+ lineLength / IPvolume ( ip , el ) * ( 1.0_pReal - transmissivity ) &
* sign ( 1.0_pReal , v0 ( s , t ) ) ! dislocation flux that is not able to leave through interface (because of low transmissivity) will remain as immobile single density at the material point
endif
enddo
enddo
endif ; endif
enddo neighbors
endif
end associate
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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 , instance , i , e )
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type ( rotation ) , dimension ( 1 , discretization_nIP , discretization_nElem ) , intent ( in ) :: &
orientation ! crystal orientation
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integer , intent ( in ) :: &
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instance , &
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i , &
e
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integer :: &
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n , & ! neighbor index
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neighbor_e , & ! element index of my neighbor
neighbor_i , & ! integration point index of my neighbor
ph , &
neighbor_phase , &
ns , & ! number of active slip systems
s1 , & ! slip system index (me)
s2 ! slip system index (my neighbor)
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real ( pReal ) , dimension ( 2 , param ( instance ) % sum_N_sl , param ( instance ) % sum_N_sl , nIPneighbors ) :: &
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my_compatibility ! my_compatibility for current element and ip
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real ( pReal ) :: &
my_compatibilitySum , &
thresholdValue , &
nThresholdValues
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logical , dimension ( param ( instance ) % sum_N_sl ) :: &
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belowThreshold
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type ( rotation ) :: mis
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ph = material_phaseAt ( 1 , e )
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associate ( prm = > param ( instance ) )
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ns = prm % sum_N_sl
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!*** start out fully compatible
my_compatibility = 0.0_pReal
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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 , i , e )
neighbor_i = IPneighborhood ( 2 , n , i , e )
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neighbor_phase = material_phaseAt ( 1 , neighbor_e )
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if ( neighbor_e < = 0 . or . neighbor_i < = 0 ) then
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!* FREE SURFACE
!* Set surface transmissivity to the value specified in the material.config
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forall ( s1 = 1 : ns ) my_compatibility ( : , s1 , s1 , n ) = sqrt ( prm % chi_surface )
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elseif ( neighbor_phase / = ph ) then
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!* PHASE BOUNDARY
!* If we encounter a different nonlocal phase at the neighbor,
!* we consider this to be a real "physical" phase boundary, so completely incompatible.
!* If one of the two phases has a local plasticity law,
!* we do not consider this to be a phase boundary, so completely compatible.
if ( . not . phase_localPlasticity ( neighbor_phase ) . and . . not . phase_localPlasticity ( ph ) ) &
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forall ( s1 = 1 : ns ) my_compatibility ( : , s1 , s1 , n ) = 0.0_pReal
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elseif ( prm % chi_GB > = 0.0_pReal ) then
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!* GRAIN BOUNDARY !
!* fixed transmissivity for adjacent ips with different texture (only if explicitly given in material.config)
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if ( any ( dNeq ( material_orientation0 ( 1 , i , e ) % asQuaternion ( ) , &
material_orientation0 ( 1 , neighbor_i , neighbor_e ) % asQuaternion ( ) ) ) . and . &
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( . not . phase_localPlasticity ( neighbor_phase ) ) ) &
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forall ( s1 = 1 : ns ) my_compatibility ( : , s1 , s1 , n ) = sqrt ( prm % chi_GB )
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else
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!* 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.
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mis = orientation ( 1 , i , e ) % misorientation ( orientation ( 1 , neighbor_i , neighbor_e ) )
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mySlipSystems : do s1 = 1 , ns
neighborSlipSystems : do s2 = 1 , ns
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my_compatibility ( 1 , s2 , s1 , n ) = math_inner ( prm % slip_normal ( 1 : 3 , s1 ) , &
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mis % rotate ( prm % slip_normal ( 1 : 3 , s2 ) ) ) &
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* abs ( math_inner ( prm % slip_direction ( 1 : 3 , s1 ) , &
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mis % rotate ( prm % slip_direction ( 1 : 3 , s2 ) ) ) )
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my_compatibility ( 2 , s2 , s1 , n ) = abs ( math_inner ( prm % slip_normal ( 1 : 3 , s1 ) , &
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mis % rotate ( prm % slip_normal ( 1 : 3 , s2 ) ) ) ) &
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* abs ( math_inner ( prm % slip_direction ( 1 : 3 , s1 ) , &
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mis % rotate ( prm % slip_direction ( 1 : 3 , s2 ) ) ) )
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enddo neighborSlipSystems
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my_compatibilitySum = 0.0_pReal
belowThreshold = . true .
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do while ( my_compatibilitySum < 1.0_pReal . and . any ( belowThreshold ) )
thresholdValue = maxval ( my_compatibility ( 2 , : , s1 , n ) , belowThreshold ) ! screws always positive
nThresholdValues = real ( count ( my_compatibility ( 2 , : , s1 , n ) > = thresholdValue ) , pReal )
where ( my_compatibility ( 2 , : , s1 , n ) > = thresholdValue ) belowThreshold = . false .
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if ( my_compatibilitySum + thresholdValue * nThresholdValues > 1.0_pReal ) &
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where ( abs ( my_compatibility ( : , : , s1 , n ) ) > = thresholdValue ) &
my_compatibility ( : , : , s1 , n ) = sign ( ( 1.0_pReal - my_compatibilitySum ) / nThresholdValues , &
my_compatibility ( : , : , s1 , n ) )
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my_compatibilitySum = my_compatibilitySum + nThresholdValues * thresholdValue
enddo
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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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enddo mySlipSystems
endif
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enddo neighbors
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compatibility ( : , : , : , : , i , e ) = my_compatibility
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end associate
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end subroutine plastic_nonlocal_updateCompatibility
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!--------------------------------------------------------------------------------------------------
!> @brief writes results to HDF5 output file
!--------------------------------------------------------------------------------------------------
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module subroutine plastic_nonlocal_results ( instance , group )
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integer , intent ( in ) :: instance
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character ( len = * ) , intent ( in ) :: group
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integer :: o
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associate ( prm = > param ( instance ) , dst = > microstructure ( instance ) , stt = > state ( instance ) )
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outputsLoop : do o = 1 , size ( prm % output )
select case ( trim ( prm % output ( o ) ) )
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case ( 'rho_u_ed_pos' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_mob_edg_pos , trim ( prm % output ( o ) ) , &
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'positive mobile edge density' , '1/m²' )
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case ( 'rho_b_ed_pos' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_imm_edg_pos , trim ( prm % output ( o ) ) , &
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'positive immobile edge density' , '1/m²' )
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case ( 'rho_u_ed_neg' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_mob_edg_neg , trim ( prm % output ( o ) ) , &
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'negative mobile edge density' , '1/m²' )
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case ( 'rho_b_ed_neg' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_imm_edg_neg , trim ( prm % output ( o ) ) , &
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'negative immobile edge density' , '1/m²' )
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case ( 'rho_d_ed' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_dip_edg , trim ( prm % output ( o ) ) , &
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'edge dipole density' , '1/m²' )
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case ( 'rho_u_sc_pos' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_mob_scr_pos , trim ( prm % output ( o ) ) , &
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'positive mobile screw density' , '1/m²' )
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case ( 'rho_b_sc_pos' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_imm_scr_pos , trim ( prm % output ( o ) ) , &
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'positive immobile screw density' , '1/m²' )
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case ( 'rho_u_sc_neg' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_mob_scr_neg , trim ( prm % output ( o ) ) , &
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'negative mobile screw density' , '1/m²' )
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case ( 'rho_b_sc_neg' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_sgl_imm_scr_neg , trim ( prm % output ( o ) ) , &
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'negative immobile screw density' , '1/m²' )
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case ( 'rho_d_sc' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_dip_scr , trim ( prm % output ( o ) ) , &
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'screw dipole density' , '1/m²' )
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case ( 'rho_f' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % rho_forest , trim ( prm % output ( o ) ) , &
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'forest density' , '1/m²' )
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case ( 'v_ed_pos' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % v_edg_pos , trim ( prm % output ( o ) ) , &
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'positive edge velocity' , 'm/s' )
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case ( 'v_ed_neg' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % v_edg_neg , trim ( prm % output ( o ) ) , &
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'negative edge velocity' , 'm/s' )
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case ( 'v_sc_pos' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % v_scr_pos , trim ( prm % output ( o ) ) , &
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'positive srew velocity' , 'm/s' )
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case ( 'v_sc_neg' )
if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % v_scr_neg , trim ( prm % output ( o ) ) , &
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'negative screw velocity' , 'm/s' )
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case ( 'gamma' )
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if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , stt % gamma , trim ( prm % output ( o ) ) , &
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'plastic shear' , '1' )
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case ( 'tau_pass' )
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if ( prm % sum_N_sl > 0 ) call results_writeDataset ( group , dst % tau_pass , trim ( prm % output ( o ) ) , &
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'passing stress for slip' , 'Pa' )
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end select
enddo outputsLoop
end associate
end subroutine plastic_nonlocal_results
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!--------------------------------------------------------------------------------------------------
!> @brief populates the initial dislocation density
!--------------------------------------------------------------------------------------------------
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subroutine stateInit ( ini , phase , NipcMyPhase , instance )
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type ( tInitialParameters ) :: &
ini
integer , intent ( in ) :: &
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phase , &
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NipcMyPhase , &
instance
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integer :: &
e , &
i , &
f , &
from , &
upto , &
s , &
phasemember
real ( pReal ) , dimension ( 2 ) :: &
noise , &
rnd
real ( pReal ) :: &
meanDensity , &
totalVolume , &
densityBinning , &
minimumIpVolume
real ( pReal ) , dimension ( NipcMyPhase ) :: &
volume
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associate ( stt = > state ( instance ) )
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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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do e = 1 , discretization_nElem
do i = 1 , discretization_nIP
if ( material_phaseAt ( 1 , e ) == phase ) volume ( material_phasememberAt ( 1 , i , e ) ) = IPvolume ( i , e )
enddo
enddo
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totalVolume = sum ( volume )
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minimumIPVolume = minval ( volume )
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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
meanDensity = 0.0_pReal
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do while ( meanDensity < ini % random_rho_u )
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call random_number ( rnd )
phasemember = nint ( rnd ( 1 ) * real ( NipcMyPhase , pReal ) + 0.5_pReal )
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s = nint ( rnd ( 2 ) * real ( sum ( ini % N_sl ) , pReal ) * 4.0_pReal + 0.5_pReal )
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meanDensity = meanDensity + densityBinning * volume ( phasemember ) / totalVolume
stt % rhoSglMobile ( s , phasemember ) = densityBinning
enddo
else ! homogeneous distribution with noise
do e = 1 , NipcMyPhase
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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 ) )
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do s = from , upto
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noise = [ math_sampleGaussVar ( 0.0_pReal , ini % sigma_rho_u ) , &
math_sampleGaussVar ( 0.0_pReal , ini % sigma_rho_u ) ]
stt % rho_sgl_mob_edg_pos ( s , e ) = ini % rho_u_ed_pos_0 ( f ) + noise ( 1 )
stt % rho_sgl_mob_edg_neg ( s , e ) = ini % rho_u_ed_neg_0 ( f ) + noise ( 1 )
stt % rho_sgl_mob_scr_pos ( s , e ) = ini % rho_u_sc_pos_0 ( f ) + noise ( 2 )
stt % rho_sgl_mob_scr_neg ( s , e ) = ini % rho_u_sc_neg_0 ( f ) + noise ( 2 )
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enddo
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stt % rho_dip_edg ( from : upto , e ) = ini % rho_d_ed_0 ( f )
stt % rho_dip_scr ( from : upto , e ) = ini % rho_d_sc_0 ( f )
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enddo
enddo
endif
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 , Temperature , instance )
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integer , intent ( in ) :: &
c , & !< dislocation character (1:edge, 2:screw)
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instance
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real ( pReal ) , intent ( in ) :: &
Temperature !< temperature
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real ( pReal ) , dimension ( param ( instance ) % 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 ) , dimension ( param ( instance ) % 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 :: &
ns , & !< short notation for the total number of active slip systems
s !< index of my current slip system
real ( pReal ) :: &
tauRel_P , &
tauRel_S , &
tauEff , & !< effective shear stress
tPeierls , & !< waiting time in front of a peierls barriers
tSolidSolution , & !< waiting time in front of a solid solution obstacle
vViscous , & !< viscous glide velocity
dtPeierls_dtau , & !< derivative with respect to resolved shear stress
dtSolidSolution_dtau , & !< derivative with respect to resolved shear stress
meanfreepath_S , & !< mean free travel distance for dislocations between two solid solution obstacles
meanfreepath_P , & !< mean free travel distance for dislocations between two Peierls barriers
jumpWidth_P , & !< depth of activated area
jumpWidth_S , & !< depth of activated area
activationLength_P , & !< length of activated dislocation line
activationLength_S , & !< length of activated dislocation line
activationVolume_P , & !< volume that needs to be activated to overcome barrier
activationVolume_S , & !< volume that needs to be activated to overcome barrier
activationEnergy_P , & !< energy that is needed to overcome barrier
activationEnergy_S , & !< energy that is needed to overcome barrier
criticalStress_P , & !< maximum obstacle strength
criticalStress_S , & !< maximum obstacle strength
mobility !< dislocation mobility
associate ( prm = > param ( instance ) )
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ns = prm % sum_N_sl
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v = 0.0_pReal
dv_dtau = 0.0_pReal
dv_dtauNS = 0.0_pReal
do s = 1 , ns
if ( abs ( tau ( s ) ) > tauThreshold ( s ) ) then
!* Peierls contribution
!* Effective stress includes non Schmid constributions
!* The derivative only gives absolute values; the correct sign is taken care of in the formula for the derivative of the velocity
tauEff = max ( 0.0_pReal , abs ( tauNS ( s ) ) - tauThreshold ( s ) ) ! ensure that the effective stress is positive
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meanfreepath_P = prm % b_sl ( s )
jumpWidth_P = prm % b_sl ( s )
activationLength_P = prm % w * prm % b_sl ( s )
activationVolume_P = activationLength_P * jumpWidth_P * prm % b_sl ( s )
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criticalStress_P = prm % peierlsStress ( s , c )
activationEnergy_P = criticalStress_P * activationVolume_P
tauRel_P = min ( 1.0_pReal , tauEff / criticalStress_P ) ! ensure that the activation probability cannot become greater than one
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tPeierls = 1.0_pReal / prm % nu_a &
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* exp ( activationEnergy_P / ( kB * Temperature ) &
* ( 1.0_pReal - tauRel_P ** prm % p ) ** prm % q )
if ( tauEff < criticalStress_P ) then
dtPeierls_dtau = tPeierls * prm % p * prm % q * activationVolume_P / ( kB * Temperature ) &
* ( 1.0_pReal - tauRel_P ** prm % p ) ** ( prm % q - 1.0_pReal ) * tauRel_P ** ( prm % p - 1.0_pReal )
else
dtPeierls_dtau = 0.0_pReal
endif
!* Contribution from solid solution strengthening
!* The derivative only gives absolute values; the correct sign is taken care of in the formula for the derivative of the velocity
tauEff = abs ( tau ( s ) ) - tauThreshold ( s )
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meanfreepath_S = prm % b_sl ( s ) / sqrt ( prm % c_sol )
jumpWidth_S = prm % f_sol * prm % b_sl ( s )
activationLength_S = prm % b_sl ( s ) / sqrt ( prm % c_sol )
activationVolume_S = activationLength_S * jumpWidth_S * prm % b_sl ( s )
activationEnergy_S = prm % Q_sol
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criticalStress_S = activationEnergy_S / activationVolume_S
tauRel_S = min ( 1.0_pReal , tauEff / criticalStress_S ) ! ensure that the activation probability cannot become greater than one
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tSolidSolution = 1.0_pReal / prm % nu_a &
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* exp ( activationEnergy_S / ( kB * Temperature ) * ( 1.0_pReal - tauRel_S ** prm % p ) ** prm % q )
if ( tauEff < criticalStress_S ) then
dtSolidSolution_dtau = tSolidSolution * prm % p * prm % q * activationVolume_S / ( kB * Temperature ) &
* ( 1.0_pReal - tauRel_S ** prm % p ) ** ( prm % q - 1.0_pReal ) * tauRel_S ** ( prm % p - 1.0_pReal )
else
dtSolidSolution_dtau = 0.0_pReal
endif
!* viscous glide velocity
tauEff = abs ( tau ( s ) ) - tauThreshold ( s )
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mobility = prm % b_sl ( s ) / prm % eta
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vViscous = mobility * tauEff
!* Mean velocity results from waiting time at peierls barriers and solid solution obstacles with respective meanfreepath of
!* free flight at glide velocity in between.
!* adopt sign from resolved stress
v ( s ) = sign ( 1.0_pReal , tau ( s ) ) &
/ ( tPeierls / meanfreepath_P + tSolidSolution / meanfreepath_S + 1.0_pReal / vViscous )
dv_dtau ( s ) = v ( s ) ** 2.0_pReal * ( dtSolidSolution_dtau / meanfreepath_S + mobility / vViscous ** 2.0_pReal )
dv_dtauNS ( s ) = v ( s ) ** 2.0_pReal * dtPeierls_dtau / meanfreepath_P
endif
enddo
end associate
end subroutine kinetics
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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 ( instance , of , ip , el )
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integer , intent ( in ) :: instance , of , ip , el
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 10 ) :: getRho
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associate ( prm = > param ( instance ) )
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getRho = reshape ( state ( instance ) % rho ( : , of ) , [ prm % sum_N_sl , 10 ] )
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! ensure positive densities (not for imm, they have a sign)
getRho ( : , mob ) = max ( getRho ( : , mob ) , 0.0_pReal )
getRho ( : , dip ) = max ( getRho ( : , dip ) , 0.0_pReal )
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where ( abs ( getRho ) < max ( prm % rho_num_significant / IPvolume ( ip , el ) ** ( 2.0_pReal / 3.0_pReal ) , prm % rho_significant ) ) &
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getRho = 0.0_pReal
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 ( instance , of , ip , el )
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integer , intent ( in ) :: instance , of , ip , el
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real ( pReal ) , dimension ( param ( instance ) % sum_N_sl , 10 ) :: getRho0
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associate ( prm = > param ( instance ) )
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getRho0 = reshape ( state0 ( instance ) % rho ( : , of ) , [ prm % sum_N_sl , 10 ] )
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! ensure positive densities (not for imm, they have a sign)
getRho0 ( : , mob ) = max ( getRho0 ( : , mob ) , 0.0_pReal )
getRho0 ( : , dip ) = max ( getRho0 ( : , dip ) , 0.0_pReal )
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where ( abs ( getRho0 ) < max ( prm % rho_num_significant / IPvolume ( ip , el ) ** ( 2.0_pReal / 3.0_pReal ) , prm % rho_significant ) ) &
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getRho0 = 0.0_pReal
end associate
end function getRho0
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end submodule plastic_nonlocal
