# 15.5. BCJ Model

    BEGIN PARAMETERS FOR MODEL BCJ
      #
      # Elastic constants
      #
      YOUNGS MODULUS = <real>
      POISSONS RATIO = <real>
      SHEAR MODULUS  = <real>
      BULK MODULUS   = <real>
      LAMBDA         = <real>
      TWO MU         = <real>
      #
      #
      #
      C1 = <real>c1
      C2 = <real>c2
      C3 = <real>c3
      C4 = <real>c4
      C5 = <real>c5
      C6 = <real>c6
      C7 = <real>c7
      C8 = <real>c8
      C9 = <real>c9
      C10 = <real>c10
      C11 = <real>c11
      C12 = <real>c12
      C13 = <real>c13
      C14 = <real>c14
      C15 = <real>c15
      C16 = <real>c16
      C17 = <real>c17
      C18 = <real>c18
      C19 = <real>c19
      C20 = <real>c20
      DAMAGE EXPONENT = <real>damage_exponent
      INITIAL ALPHA_XX = <real>alpha_xx
      INITIAL ALPHA_YY = <real>alpha_yy
      INITIAL ALPHA_ZZ = <real>alpha_zz
      INITIAL ALPHA_XY = <real>alpha_xy
      INITIAL ALPHA_YZ = <real>alpha_yz
      INITIAL ALPHA_XZ = <real>alpha_xz
      INITIAL KAPPA = <real>initial_kappa
      INITIAL DAMAGE = <real>initial_damage
      YOUNGS MODULUS FUNCTION = <string>ym_function_name
      POISSONS RATIO FUNCTION = <string>pr_function_name
      SPECIFIC HEAT = <real>specific_heat
      THETA OPT = <integer>theta_opt
      FACTOR = <real>factor
      RHO    = <real>rho
      TEMP0  = <real>temp0
    END [PARAMETERS FOR MODEL BCJ]

The BCJ plasticity model is a state-variable model for describing the finite deformation behavior of metals. It uses a multiplicative decomposition of the deformation gradient into elastic, volumetric plastic, and deviatoric parts. The model considers the natural configuration defined by this decomposition and its associated thermodynamics. The model incorporates strain rate and temperature sensitivity, and damage, through a yield-surface approach in which state variables follow a hardening-minus-recovery format.

Because the BCJ model has such an extensive list of parameters, we will not present the usual synopsis of parameter names with command lines. As with most other material models, the `thermal strain option` is used to define thermal strains. See the Sierra/SM User Manual for further information on defining and activating thermal strains. In addition, only two of the five elastic constants are required. The user should consult [[1], [2], [3]] for a description of the various parameters. The parameters for the `SPECIFIC HEAT`, `THETA OPT`, `FACTOR`, `RHO`, and `TEMP0` command lines are used to accommodate changes to the model for heat generation due to plastic dissipation. For coupled solid/thermal calculations, the plastic dissipation rate is stored as a state variable and passed to a thermal code as a heat source term. For uncoupled calculations, temperature is stored as a state variable, and temperature increases due to plastic dissipation are calculated within the material model.

If temperature is provided from an external source, `theta_opt` is set to 1. If the temperature is calculated by the BCJ model, `theta_opt` is set to 1.

Output variables available for this model are listed in Table 15.3.

Table 15.3 State Variables for BCJ Model
Name | Description
--- | ---
`BACK_STRESS_XX` | back stress tensor - xx component
`BACK_STRESS_YY` | back stress tensor - yy component
`BACK_STRESS_ZZ` | back stress tensor - zz component
`BACK_STRESS_XY` | back stress tensor - xy component
`BACK_STRESS_YZ` | back stress tensor - yz component
`BACK_STRESS_ZX` | back stress tensor - zx component
`KAPPA` | hardening scalar
`DAMAGE` | damage term
`DAMAGE_RATE` | rate of change of damage term
`EQPS` | equivalent plastic strain
`THETA` | temperature for adiabatic heating
`HEAT` | rate of heating due to plastic dissipation
`YM` | Young’s modulus
`PR` | Poisson’s ratio

Warning

Strongly rate-dependent models may fare poorly in implicit quasistatic solution. In implicit analyses the rate term used to evaluate the current load step is the rate seen by the model in the previous load step. This may cause the solution to oscillate between high- and low-rate equilibrium states from step to step.

[1]

D.J. Bammann, M.L. Chiesa, and G.C. Johnson. Modelling large deformation and failure in manufacturing processes. In T. Tatsumi, E. Watanabe, and T. Kambe, editors, Proceedings of the 19th International Congress of Theoretical and Applied Mechanics, 359–376. Amsterdam, 1997. Elsevier Science Publishers.

[2]

D.J. Bammann, M.L. Chiesa, M.F. Horstemeyer, and L.E. Weingarten. Failure in ductile materials using finite element methods. In N. Jones and T. Wierzbicki, editors, Structural Crashworthiness and Failure, 1–53. London, 1993. Elsevier Applied Science.

[3]

D.J. Bammann. Modeling temperature and strain dependent large deformations in metals. Applied Mechanics Reviews, 43(5):312–319, 1990. doi:10.1115/1.3120834.