2 NSLIP, NSLPTL, NSET, H, PROP, ND) CFIXB

C----- This subroutine calculates the current self- and latent-

C hardening moduli for all slip systems in a rate-dependent single C crystal. Two kinds of hardening law are used here. The first

C law, proposed by Asaro, and Pierce et al, assumes a HYPER SECANT C relation between self- and latent-hardening moduli and overall C shear strain. The Bauschinger effect has been neglected. The C second is Bassani's hardening law, which gives an explicit C expression of slip interactions between slip systems. The C classical three stage hardening for FCC single crystal could be C simulated.

C----- The hardening coefficients are assumed the same for all slip C systems in each set, though they could be different from set to C set, e.g. <110>{111} and <110>{100}.

C----- Users who want to use their own self- and latent-hardening law C may change the function subprograms HSELF (self hardening) and C HLATNT (latent hardening). The parameters characterizing these C hardening laws are passed into HSELF and HLATNT through array C PROP.

C----- Function subprograms: C

C HSELF -- User-supplied self-hardening function in a slip C system C

C HLATNT -- User-supplied latent-hardening function

C----- Variables: C

C GAMMA -- shear strain in all slip systems at the start of time C step (INPUT)

C TAUSLP -- resolved shear stress in all slip systems (INPUT) C GSLIP -- current strength (INPUT)

CFIX GMSLTL -- total cumulative shear strains on each individual slip system CFIX (INPUT)

C GAMTOL -- total cumulative shear strains over all slip systems C (INPUT)

C NSLIP -- number of slip systems in each set (INPUT)

C NSLPTL -- total number of slip systems in all the sets (INPUT)

C NSET -- number of sets of slip systems (INPUT) C

C H -- current value of self- and latent-hardening moduli C (OUTPUT)

C H(i,i) -- self-hardening modulus of the ith slip system C (no sum over i)

C H(i,j) -- latent-hardening molulus of the ith slip

C system due to a slip in the jth slip system C (i not equal j) C

C PROP C C

C C C C C C C C C C C C C

C C C C C C C C C C C C C C C C C C -- material constants characterizing the self- and latent- hardening law (INPUT) For the HYPER SECANT hardening law

PROP(1,i) -- initial hardening modulus H0 in the ith set of slip systems

PROP(2,i) -- saturation stress TAUs in the ith set of slip systems

PROP(3,i) -- initial critical resolved shear stress TAU0 in the ith set of slip systems PROP(9,i) -- ratio of latent to self-hardening Q in the ith set of slip systems

PROP(10,i)-- ratio of latent-hardening from other sets of slip systems to self-hardening in the ith set of slip systems Q1 For Bassani's hardening law

PROP(1,i) -- initial hardening modulus H0 in the ith set of slip systems

PROP(2,i) -- stage I stress TAUI in the ith set of

slip systems (or the breakthrough stress where large plastic flow initiates) PROP(3,i) -- initial critical resolved shear stress TAU0 in the ith set of slip systems

PROP(4,i) -- hardening modulus during easy glide Hs in the ith set of slip systems

PROP(5,i) -- amount of slip Gamma0 after which a given interaction between slip systems in the ith set reaches peak strength

PROP(6,i) -- amount of slip Gamma0 after which a given interaction between slip systems in the ith set and jth set (i not equal j) reaches peak strength

PROP(7,i) -- representing the magnitude of the strength

C of interaction in the ith set of slip C system

C PROP(8,i) -- representing the magnitude of the strength C of interaction between the ith set and jth C set of system

C PROP(9,i) -- ratio of latent to self-hardening Q in the C ith set of slip systems

C PROP(10,i)-- ratio of latent-hardening from other sets C of slip systems to self-hardening in the C ith set of slip systems Q1 C

C ND -- leading dimension of arrays defined in subroutine UMAT C (INPUT)

C----- Use single precision on cray C

IMPLICIT REAL*8 (A-H,O-Z) EXTERNAL HSELF, HLATNT CFIXA

DIMENSION GAMMA(NSLPTL), TAUSLP(NSLPTL), GMSLTL(NSLPTL), 2 GSLIP(NSLPTL), NSLIP(NSET), PROP(16,NSET), 3 H(ND,NSLPTL) CFIXB

CHECK=0. DO I=1,NSET DO J=4,8

CHECK=CHECK+ABS(PROP(J,I)) END DO END DO

C----- CHECK=0 -- HYPER SECANT hardening law C otherwise -- Bassani's hardening law

ISELF=0

DO I=1,NSET ISET=I

DO J=1,NSLIP(I) ISELF=ISELF+1

DO LATENT=1,NSLPTL

IF (LATENT.EQ.ISELF) THEN CFIXA

H(LATENT,ISELF)=HSELF(GAMMA,GMSLTL,GAMTOL,NSLPTL, 2 NSET,NSLIP,PROP(1,I),CHECK, 3 ISELF,ISET) CFIXB

ELSE CFIXA

H(LATENT,ISELF)=HLATNT(GAMMA,GMSLTL,GAMTOL,NSLPTL, 2 NSET,NSLIP,PROP(1,I),CHECK, 3 ISELF,ISET,LATENT) CFIXB

END IF END DO

END DO END DO

RETURN END

C-----------------------------------

C----- Use single precision on cray CFIXA

REAL*8 FUNCTION HSELF(GAMMA,GMSLTL,GAMTOL,NSLPTL,NSET, 2 NSLIP,PROP,CHECK,ISELF,ISET) CFIXB

C----- User-supplied self-hardening function in a slip system

C----- Use single precision on cray C

IMPLICIT REAL*8 (A-H,O-Z) CFIXA

DIMENSION GAMMA(NSLPTL), NSLIP(NSET), PROP(16), 2 GMSLTL(NSLPTL) CFIXB

IF (CHECK.EQ.0.) THEN

C----- HYPER SECANT hardening law by Asaro, Pierce et al

TERM1=PROP(1)*GAMTOL/(PROP(2)-PROP(3))