# 4.37.5. Modular BCJ Failure

The modular BCJ (Bammann-Chiesa-Johnson [[1]]) failure model develops the damage variable through a micromechanical representation of the void evolution process. In this model, damage is defined as void volume fraction as function of void size \(v_v\) and count \(f\) in a nominal unit volume \(V_0\) [[2]]:

\[d = \frac{v_v f V_0}{V_0 + v_v f V_0} = \frac{v_v f}{1 + v_v f} .\]

Void evolution is determined by incrementation of void nucleation and void growth models. Implemented void nucleation models include Horstemeyer-Gokhale (HG) [[3]] and Chu-Needleman strain distribution (CNSD) [[4]]:

(4.262)\[(\dot{f})^{\text{HG}} = f \dot{\bar{\varepsilon}}^p N_4 \left( N_1 \left[ \frac{4}{27} - \frac{{J_3}^2}{{J_2}^3} \right] + N_2 \left[ \frac{J_3}{{J_2}^{\frac32}} \right] + N_3 \left[ \frac{|p|}{\sigma_{vm}} \right] \right),\]

where \(\{N_1, N_2, N_3, N_4\}\) are fitting constants, \(\{J_2, J_3\}\) are deviatoric stress tensor invariants, \(p\) is the mean hydrostatic stress, and \(\sigma_{vm}\) is the von-Mises stress. The \(N_1\) term premultiplies a term indicating a shear-dominated stress state (maximal in torsion); the \(N_2\) term premultiplies a term indicating uniaxial stress state (maximal in uniaxial tension); the \(N_3\) term premultiplies a term indicating stress triaxiality (maximal in hydrostatic tension/compression).

\[(\dot{f})^{\text{CNSD}} = \frac{1}{(1-d)^2} \left[ \frac{\psi}{s v_0 \sqrt{2\pi}} \right] \exp\left( -\frac12 \left( \frac{\bar{\varepsilon}^p-\varepsilon_N}{s} \right)^2 \right),\]

where \(\psi\) is the void amplitude, \(\varepsilon_N\) is the mean of the nucleation strain distribution, and \(s\) is the standard deviation of the nucleation strain distribution.

The Cocks-Ashby (CA) model [[5]] is the only growth model implemented to date:

(4.263)\[(\dot{v}_v)^{\text{CA}} = \sqrt{\frac23} \dot{\bar{\varepsilon}}^p \left( \frac{1 - (1-d)^{m+1}}{(1-d)^m} \right) \sinh\left[ \frac{2(2m-1)}{2m+1} \frac{p}{\sigma_{vm}} \right] ,\]

where \(m\) is the damage growth exponent.

The void nucleation and growth models are combined to develop the full expression for void evolution,

(4.264)\[\dot{d} = \dot{v_v} w_1^G(\dot{\bar{\varepsilon}}^p) w_2^G(T) + (1-d)^2 \dot{f} v_0 w_1^N(\dot{\bar{\varepsilon}}^p) w_2^N(T) ,\]

where optional functions \(w_i^X\) are introduced to account for strain-rate and temperature effects. Implemented functions provide for strain-rate and temperature effects using the \(D_4\) and \(D_5\) terms of the Johnson-Cook failure model (4.259).

Note that the void evolution equation is defined implicitly, as void nucleation and growth expressions may each depend on damage. A model parameter \(\beta\) exposes the option to control the temporal integration of (4.264), with \(\beta=0\) representing forward Euler integration (explicit), \(\beta = 1\) representing backward Euler integration (implicit), and \(\beta = 0.5\) (default) representing trapezoidal integration (implicit). Other common parameters include the initial damage \(d_0\) and initial void size \(v_0\).

## 4.37.5.1. User Guide

    #
    # MODULAR_BCJ_FAILURE Failure model definitions
    #
    INITIAL DAMAGE    = <real>
    INITIAL VOID SIZE = <real>
    DAMAGE BETA       = <real> (0.5)
    GROWTH MODEL      = COCKS_ASHBY | NO_GROWTH (NO_GROWTH)
    NUCLEATION MODEL  = HORSTEMEYER_GOKHALE | CHU_NEEDLEMAN_STRAIN
                        | NO_NUCLEATION (NO_NUCLEATION)
    #
    GROWTH RATE FAIL MULTIPLIER        = JOHNSON_COOK | USER_DEFINED
                                         | RATE_INDEPENDENT
                                         (RATE_INDEPENDENT)
    GROWTH TEMPERATURE FAIL MULTIPLIER = JOHNSON_COOK | USER_DEFINED
                                         | TEMPERATURE_INDEPENDENT
                                         (TEMPERATURE_INDEPENDENT)
    #
    NUCLEATION RATE FAIL MULTIPLIER        = JOHNSON_COOK | USER_DEFINED
                                             | RATE_INDEPENDENT
                                             (RATE_INDEPENDENT)
    NUCLEATION TEMPERATURE FAIL MULTIPLIER = JOHNSON_COOK | USER_DEFINED
                                             | TEMPERATURE_INDEPENDENT
                                             (TEMPERATURE_INDEPENDENT)
    #
    # Definitions for individual growth and nucleation models
    #
    GROWTH MODEL = COCKS_ASHBY
    DAMAGE EXPONENT = <real> (0.5)
    #
    NUCLEATION MODEL      = HORSTEMEYER_GOKHALE
    NUCLEATION PARAMETER1 = <real> (0.0)
    NUCLEATION PARAMETER2 = <real> (0.0)
    NUCLEATION PARAMETER3 = <real> (0.0)
    #
    NUCLEATION MODEL          = CHU_NEEDLEMAN_STRAIN
    NUCLEATION AMPLITUDE      = <real>
    MEAN NUCLEATION STRAIN    = <real>
    NUCLEATION STRAIN STD DEV = <real>
    #
    # Definitions for rate and temperature fail multiplier
    #   Note: only showing definitions for growth.
    #    Nucleation terms are the same just with NUCLEATION instead
    #    of GROWTH
    #
    GROWTH RATE FAIL MULTIPLIER = JOHNSON_COOK
    GROWTH JOHNSON COOK D4      = <real>
    GROWTH REFERENCE RATE       = <real>
    #
    GROWTH RATE FAIL MULTIPLIER          = USER_DEFINED
    GROWTH RATE FAIL MULTIPLIER FUNCTION = <string> growth_rate_fail_mult_func
    #
    GROWTH TEMPERATURE FAIL MULTIPLIER = JOHNSON_COOK
    GROWTH JOHNSON COOK D5             = <real>
    GROWTH REFERENCE TEMPERATURE       = <real>
    GROWTH MELTING TEMPERATURE         = <real>
    #
    GROWTH TEMPERATURE FAIL MULTIPLIER          = USER_DEFINED
    GROWTH TEMPERATURE FAIL MULTIPLIER FUNCTION = <string> temp_fail_mult_func
    #

[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]

Kyle N. Karlson, Coleman Alleman, James W. Foulk III, Kevin L. Manktelow, Jakob T. Ostien, Michael E Stender, Andrew J. Stershic, and Michael G. Veilleux. Sandia Fracture Challenge 3: detailing the Sandia Team Q failure prediction strategy. International Journal of Fracture, 218(1):149–170, 2019.

[3]

Mark F. Horstemeyer and Arun M. Gokhale. A void–crack nucleation model for ductile metals. International Journal of Solids and Structures, 36(33):5029–5055, 1999.

[4]

C. C. Chu and A. Needleman. Void nucleation effects in biaxially stretched sheets. Journal of Engineering Materials and Technology, 102(3):249–256, 07 1980.

[5]

A. Cocks and M. Ashby. Intergranular fracture during power-law creep under multiaxial stresses. Metal Sci., 14(8–9):395–402, 1980.