FLAC3D Theory and Background ? Constitutive Models

Single-Zone JWL Pressurization

Note

The project file for this example is available to be viewed/run in FLAC3D. The project’s main data file is shown at the end of this example.

Problem Statement

This example illustrates the application of the Jones-Wilkins-Lee (JWL) equation of state to simulate the post-detonation response of ANFO using a simplified single-zone model. The single zone idealizes the explosive as a material confined within a cylindrical charge geometry. Simulations are conducted in both small and large-strain formulations. The large strain model employs a zone geometry update each step (as opposed to the default every 10 steps) by using the FLAC3D command zone geometry update-interval. As the surrounding rock mass is not explicitly modeled, radial expansion is approximated by imposing a constant velocity boundary condition on the \(+x\) face, serving as a proxy for crushing and displacement of the adjacent rock that typically results in permanent deformation (Figure 1). The material properties of the explosive used in this example are:

Table 1: JWL model constants.

Property	Value	Unit
constant \(A\)	203.582	GPa
constant \(B\)	2.973	GPa
constant \(C\)	0.389	GPa
constant \(R_1\)	6.651
constant \(R_2\)	1.127
constant \(\omega\)	0.390
initial density \(\rho_0\)	800	kg/m3

JWL model parameters for a range of explosives are available in Sanchidrian et al. (2015). For the purposes of this simplified model, energy release is controlled by a constant burn fraction set to unity for the model duration, implying instantaneous and complete detonation at time \(t=0\) s. The total model run time is 1 ms, where radial expansion reaches twice the initial length of the FLAC3D zone.

Figure 1: Single zone model of an explosive material with a prescribed velocity boundary condition.

Results

Figures 2, 3, and 4 show the time evolution of pressure, relative volume, and density within the explosive during the pressurization and expansion phases. Immediately following detonation, the pressure rapidly increases to approximately 1.6 GPa. The magnitude of peak pressure is controlled by explosive type and the associated JWL equation of state model parameters. As radial expansion progresses, pressure decreases and relative volume increases, reflecting rapid unloading from the initial peak. In this simplified model, confinement by the surrounding rock is approximated using a prescribed velocity boundary condition to simulate crushing. In practical applications, the explosive-rock interaction would result in a stabalized expanded borehole diameter and an equilibrium pressure typically on the order of several hundred MPa. However, neither are captured in this idealized model due to the absence of explicit rock interaction and energy dissipation mechanisms.

Figure 2: Pressure (\(P=-\sigma_{kk}/3\) in FLAC3D) vs time.

Figure 3: Relative volume (\(V_r=V/V_0\)) vs time.

Figure 4: Explosive density (\(\rho\)) vs relative volume.

Reference

Sanchidrian et al. “Determination of the JWL constants for ANFO and emulsion explosives from cylinder test data,” Central European journal of energetic materials, 12.2, (2015).

Data Files

pressurize.dat

fish define set_jwl_params
    global a, b, c, r1, r2, omega, dens
    a = 203.582e9
    b = 2.973e9
    c = 0.389e9
    r1 = 6.651
    r2 = 1.127
    omega = 0.39
    dens = 800
end
[set_jwl_params]

zone cmodel assign jones-wilkins-lee
zone property density [dens] initial-density [dens] ...
    a [a] b [b] c [c] r-1 [r1] r-2 [r2] omega [omega] ...
    burn-fraction 1.0
    
zone face apply velocity-normal 0.0 range group "East" not
zone face apply velocity-normal 1e2 range group "East"

model dynamic timestep fix 5e-7
model save "init.sav"

; Small strain
model large-strain off
model solve-dynamic time-total 1e-3
model save "small_strain.sav"

; Large strain
model restore "init.sav"
model large-strain on
zone geometry update-interval 1
model solve-dynamic time-total 1e-3
model save "large_strain.sav"