Thermomechanics of transient oblique compaction shock reflection from a rigid boundary

Abstract

Transient oblique reflection of resolved compaction shocks in porous material from a rigid planar boundary is computationally examined to characterize how spatial reflection structures vary with reflection angle. The material response is described by a hydrodynamic theory that accounts for both elastic and inelastic volumetric deformation. The mathematical model, expressed in terms of curvilinear coordinates, is numerically integrated using a high-resolution technique. Emphasis is placed on characterizing the relative importance of compression and compaction work as heating mechanisms. Spatially continuous structures are predicted that propagate at speeds below the ambient sound speed of the solid component which are analogous to discontinuous structures for oblique reflection of gas shocks. An analogous transition from a von Neumann reflection to a Mach reflection to a regular reflection is predicted with increasing reflection angle, with high dissipative heating induced by configurations possessing a stem-like structure. Compression and dissipation by rate-dependent compaction are shown to be primary heating mechanisms, whereas dissipation by inelastic compaction is of secondary importance.

Appendix: Constitutive relations for granular HMX

Constitutive relations used in this study describe the state behavior of the solid component HMX and the compaction behavior of granular HMX.

A Mie-Grüneisen equation of state is used to describe the thermodynamics:

$$\begin{aligned} P_\mathrm{s} = P_H + \frac{\varGamma }{\nu _{s0}}\left( e_\mathrm{s} - e_H\right) , \end{aligned}$$

(57)

where $\nu _{s0} = 1/\rho _{s0}$ is the initial mass-specific volume and $\varGamma$ is a constant Grüneisen coefficient. The functions $P_H(\nu _\mathrm{s})$ and $e_H(\nu _\mathrm{s})$ are defined by

$$\begin{aligned} P_H&\equiv \left[ \frac{\omega }{\nu _{s0} - s\left( \nu _{s0} - \nu _\mathrm{s}\right) }\right] ^2 \left( \nu _{s0} - \nu _\mathrm{s}\right) , \nonumber \\ e_H&\equiv \frac{1}{2}\left[ \frac{\omega \left( \nu _{s0} - \nu _\mathrm{s}\right) }{\nu _{s0} - s\left( \nu _{s0} - \nu _\mathrm{s}\right) }\right] ^2, \end{aligned}$$

(58)

where $\nu _\mathrm{s} = 1/\rho _\mathrm{s}$ is the local mass-specific volume. This incomplete equation of state is compatible with the shock Hugoniot $D = \omega + s U_p$, where D is the shock speed, $U_p$ is the particle velocity behind the shock, and $\omega$ and s are empirically determined constants [9]. The equation of state is completed by the caloric equation $e_\mathrm{s} = c_v \left( T_\mathrm{s} - T_{0}\right)$, where $c_v$ is the constant volume specific heat and $T_{0}$ is the initial temperature. Values of the constant parameters used for the equation of state are listed in Table 2.

Table 2 Values of the parameters used with the Mie-Grüneisen equation of state for HMX

The sound speed of HMX is needed for use with the shock-capturing numerical technique. It is defined by

$$\begin{aligned} {c}^2 \equiv \left. \frac{\partial P_\mathrm{s}}{\partial \rho _\mathrm{s}}\right| _\eta = \left. \frac{\partial P_\mathrm{s}}{\partial \rho _\mathrm{s}}\right| _{e_\mathrm{s}} + \frac{P_\mathrm{s}}{\rho _s} \varGamma , \end{aligned}$$

(59)

where $\eta$ is the solid entropy. Based on the Mie-Grüneisen equation of state

$$\begin{aligned} \left. \frac{\partial P_\mathrm{s}}{\partial \nu _\mathrm{s}}\right| _{e_\mathrm{s}} = \frac{\mathrm{d}P_H}{\mathrm{d}\nu _\mathrm{s}} - \frac{\varGamma }{\nu _{s0}} \frac{\mathrm{d}e_H}{\mathrm{d}\nu _\mathrm{s}}, \end{aligned}$$

(60)

where

$$\begin{aligned} \frac{\mathrm{d}P_H}{\mathrm{d}\nu _\mathrm{s}}= & {} - \left[ \frac{\omega }{\nu _{s0} - s \left( \nu _{s0}-\nu _\mathrm{s}\right) }\right] ^2 \left[ 1+ \frac{2s\left( \nu _{s0}-\nu _\mathrm{s}\right) }{\nu _{s0}-s\left( \nu _{s0}-\nu _\mathrm{s}\right) }\right] , \nonumber \\\end{aligned}$$

(61)

$$\begin{aligned} \frac{\mathrm{d}e_H}{\mathrm{d}\nu _\mathrm{s}}= & {} -\left( \nu _{s0}-\nu _\mathrm{s}\right) \left[ \frac{\omega }{\nu _{s0}-s \left( \nu _{s0}-\nu _\mathrm{s}\right) }\right] ^2 \nonumber \\&\times \left[ 1 + \frac{s\left( \nu _{s0} - \nu _\mathrm{s}\right) }{\nu _{s0}-s \left( \nu _{s0}-\nu _\mathrm{s}\right) }\right] . \end{aligned}$$

(62)

Expressions for the compaction of granular HMX are based on quasi-static data [5, 6]. The intergranular stress $\beta$ and the yield surface for onset of inelastic volumetric deformation f are given by

$$\begin{aligned}&\beta (\rho _\mathrm{s},\phi ,\tilde{\phi }) = - \beta _c \frac{\rho _\mathrm{s}}{\rho _{s0}} \phi \left( \phi -\tilde{\phi }\right) \frac{\ln \left( \kappa -\left( \phi -\tilde{\phi }\right) \right) }{\kappa - \left( \phi -\tilde{\phi }\right) }, \end{aligned}$$

(63)

$$\begin{aligned}&f(\phi ) = \phi _{fp} + c \left( \phi - \phi _{fp}\right) , \end{aligned}$$

(64)

where $\beta _c = 6.0$ MPa, $c = 0.913$, $\kappa = 0.03$, and $\phi _{fp} = 0.655$ [10]. This expression for $\beta$ is a monotonically increasing function of the elastic component of volume fraction $\phi -\tilde{\phi }$ and contains a linear dependence on solid density as required by thermodynamic constraints [14]. The expression for f is a monotonically increasing function of $\phi$ which is indicative of work hardening. Using (63), the following expression for the compaction potential energy $B = \int _0^{(\phi -\tilde{\phi })} \frac{\beta }{\rho _\mathrm{s} \phi }\, d(\phi - \tilde{\phi })$ is obtained:

$$\begin{aligned} B(\phi -\tilde{\phi }) =&\frac{\beta _c}{\rho _{s0}} \biggl \{\frac{\kappa }{2}\left[ \left( \ln \left( \kappa - \left( \phi -\tilde{\phi }\right) \right) \right) ^2 - \left( \ln \kappa \right) ^2\right] \biggr . \nonumber \\&-\biggl .\left( \kappa - \left( \phi -\tilde{\phi }\right) \right) \biggr . \nonumber \\&\times \biggl . \left[ \ln \left( \kappa -\left( \phi -\tilde{\phi }\right) \right) - 1\right] + \kappa \left( \ln \kappa -1\right) \biggr \}. \end{aligned}$$

(65)