11.3.1 General Curve USRCRV

If USRCRV is specified in table 'MATERI' then subroutine USRCRV must be supplied to set up the yield stress $\sigma_{{\mathrm{y}}}^{}$ or the cohesion c depending on the plasticity model. This subroutine is particularly used to specify a user-defined hardening curve. It may also be used to specify an ambient or time dependency of the yield stress or the hardening curve.

    (Fortran)

      SUBROUTINE USRCRV( parnam, usrpar, kappa,
     $                   te, co, ma, ti, sy, hc )

parnam

is the name of the hardening curve which depends on the yield criterion: 'SQVCRV' for the Tresca and Von Mises criteria, 'COHCRV' for the Mohr-Coulomb and Drucker-Prager criteria, 'TENCRV' for the Rankine criterion, and 'COMCRV' for the compressive equivalent plastic strain, equivalent stress, of the Von Mises part of the Rankine/Von Mises yield criterion.

In case of the Rankine/Drucker-Prager yield criterion, the name of the hardening curves are 'TENCRV' and 'COHCRV' for the Rankine and the Drucker-Prager yield criterion respectively. This can be used to model various functions within one subroutine.

usrpar

are the user-specified parameters of the hardening curve which are input in the table 'MATERI' with the input item USRPAR.

sy

is the yield stress $\sigma_{{\mathrm{y}}}^{}$ , applied with the Tresca and Von Mises yield criteria [§5.1.1.2] or the cohesion c , applied with the Mohr-Coulomb and Drucker-Prager yield criteria [§5.1.2.2].

hc

is the hardening modulus d$\sigma$/d$\kappa$ . If no hardening is used, hc must be set to zero.

Example.

A simple example of the user-supplied hardening curve USRCRV in a non-linear analysis is the following saturation hardening law of the exponential type for a Von Mises yield criterion. The formulation of the curve is given by

$$\bar{{\sigma }} \kappa \bar{{\sigma }}_{{0}}^{} \kappa \bar{{\sigma }}_{{\infty}}^{} \bar{{\sigma }}_{{0}}^{} \gamma \kappa$$ (11.12)

with the initial equivalent stress $\bar{{\sigma }}_{{0}}^{}$ , the hardening modulus E_p , the ultimate equivalent stress $\bar{{\sigma }}_{{\infty}}^{}$ (at a hardening modulus equal to zero) and the decaying factor $\gamma$ [Fig.11.1].

Figure 11.1: Saturation

The derivative of the equivalent stress with respect to the hardening parameter

$$\bar{{\sigma }}^{{\prime}}_{} \gamma \bar{{\sigma }}_{{\infty}}^{} \bar{{\sigma }}_{{0}}^{} \gamma \kappa$$ (11.13)

The implementation of the user-supplied subroutine USRCRV is given below.

    (usrcrv.f)

      SUBROUTINE USRCRV( PARNAM, USRPAR, KAPPA, TE, CO, MA, TI, SY, HC )
C
C...     USER-SUPPLIED HARDENING CURVE.
C...     RETURN THE EQUIVALENT STRESS AND DERIVATIVE WITH RESPECT TO
C...     THE EQUIVALENT PLASTIC STRAIN.
C
      DOUBLE PRECISION USRPAR(*), KAPPA, TE, CO, MA, TI, SY, HC
      CHARACTER*6      PARNAM
C
      DOUBLE PRECISION EHAR, GM, SIG0, SIGF
C
      IF ( PARNAM .EQ. 'SQVCRV' ) THEN
         SIG0 = USRPAR(1)
         SIGF = USRPAR(2)
         GM   = USRPAR(3)
         EHAR = USRPAR(4)
C
         SY = SIG0
     $      + KAPPA * EHAR
     $      + ( SIGF - SIG0 ) * ( 1.D0 - EXP( -GM * KAPPA ) )
C
         HC = EHAR
     $      + GM * ( SIGF - SIG0 ) * EXP( -GM * KAPPA )
      ELSE
         PRINT *, 'SUBROUTINE USRCRV CANNOT BE USED FOR HARDENING'
         PRINT *, 'CURVE ', PARNAM
         CALL PRGERR( 'USRCRV', 1 )
      END IF
C
      RETURN
      END

The user-supplied material parameters which are given in the input data item USRPAR are the initial equivalent stress, $\bar{{\sigma }}_{{0}}^{}$ , the the ultimate equivalent stress $\bar{{\sigma }}_{{\infty}}^{}$ (at a hardening modulus equal to zero), the decaying factor $\gamma$ [Fig.11.1], and the hardening modulus E_p . The following could be the data file for this example.

    (file.dat)

TEST USER-SUPPLIED HARDENING CURVE
'COORDINATES' DI=2
   1      0.     0.
   2      1.     0.
   3      1.     1.
   4      0.     1.
'ELEMENTS'
CONNECTIVITY
    1 Q8MEM  1  2  3  4
MATERIALS
   1     1
GEOMETRY
   1     1
DATA
   1     1
'MATERIALS'
1    YOUNG    7.0E+04
     POISON   0.2
     YIELD    VMISES
     SQVCRV   USRCRV
     USRPAR   243.0 243.0 0.0 2240.0
'GEOMETRY'
1     THICK  1.0
      XAXIS  1.0 0.0 0.0
'DATA'
1     NINTEG  2 2
'DIRECTIONS'
   1    1.   0.  0.
   2    0.   1.  0.
   3    0.   0.  1.
'SUPPORTS'
/ 1-4 / TR 1
/ 1-2 / TR 2
'TYINGS'
EQUAL TR 2
  3      4
'LOADS'
CASE 1
DEFORM
/ 2-3 / TR  1    0.0034714286
'END'

The following command file runs a DIANA calculation with the user-supplied subroutine usrcrv.

    (file.dcf)

*FILOS
INITIA
*INPUT
*FORTRAN
TAKE "usrcrv.f"
*NONLIN
TYPE PHYSIC
commands