Disipación de ondas de compactación en explosivos granulares de baja densidad

La microestructura de explosivos granulares puede afectar el efecto de disipación del calentamiento que se produce a causa de las ondas de choque dentro de ellos y pueden desencadenar tanto una combustión e iniciar una detonación. Debido a que los fenómenos de iniciación ocurren sobre distancias que son más grandes que el tamaño medio de partícula, la teoría homogeneizada, que trata los fenómenos de la detónica en la macroescala, y es un riguroso método de promediar basado en extensiones de la perturbación en múltiples escalas, a menudo se utilizan para describir estados termodinámicos locales dentro y detrás de la onda de choque, tales que son considerados como la manifestación promedio de campos termodinámicos en la escala de la partícula.

with a narrow distribution. The average initial solid volume fraction of the ensemble is ϕ¯0=0.84 and all inter-particle pores are massless. Though the assumptions of 2D plane strain and circular particle geometry are restrictive, they enable leading-order effects of packing density on compaction shock dissipation to be examined based on well-resolved computations; they also form a basis for systematically examining the effects of other microstructural features and three-dimensionality in the future. Periodic conditions are imposed along transverse boundaries of the computational domain, and a free condition is imposed at the far-field axial boundary. A rigid piston impacts the material with constant speed Up causing the rapid development and subsequent propagation of a quasi-steady compaction shock having speed D. Shock strength is characterized in terms of effective pressure P¯¯¯ which depends on both ϕ¯0 and Up. Figure 2(a) illustrates a computed piston supported shock having speed D = 2136 m/s for ϕ¯0=0.84 and Up  = 500 m/s. The shock quickly develops into a quasi-steady wave following impact, as indicated by its position-time curve in Fig. 2(b), and its pressure P¯¯¯=1.6 GPa

is sufficiently high to eliminate porosity. The shock speed is lower than the ambient sound speed of solid HMX (≈2740 m/s) due to energy dissipation and dispersion associated with pore collapse.

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FIG. 1.Representative computational domain and boundary conditions used for the mesoscale simulations.

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FIG. 2.Development of a quasi-steady compaction shock corresponding to Up = 500 m/s and ϕ0=0.84

As discussed later in this paper, quasi-steady compaction shock profiles are obtained by first filtering thermomechanical fields at fixed times and then performing ensemble averaging of the resulting spatially filtered profiles, as illustrated in Fig. 3 for the case shown in Fig. 2. This procedure is necessary to obtain the relation between effective packing density, shock pressure, and dissipated work, and to obtain effective profiles that can be directly compared with those given by the macroscale theory. Fluctuations in spatially filtered profiles are relatively small for ϕ¯0=0.84

, but become more pronounced as ϕ¯0

decreases.

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FIG. 3.Development of a quasi-steady compaction shock corresponding to Up = 500 m/s and ϕ0=0.84

The plan of this paper is as follows. First, the mesoscale and macroscale theories are summarized in Sections II and III, respectively, where their mathematical and physical descriptions of compaction induced dissipation are highlighted. Next, predictions are given in Section IV that illustrate how effective shock profiles vary with ϕ¯0

, Up, and P¯¯¯

where ρ is the local density, u is the displacement vector, σ is the Cauchy stress tensor, r is the deformation induced heating, q is the heat flux, cv is the constant volume specific heat, and ∇≡∂(∙)/∂x is the spatial gradient operator. The time derivative of a variable is denoted by a “dot” above it. Particles are assumed to be initially stationary and stress free.

Mechanical and thermal contact conditions imposed along particle boundaries Γ are given by

where n is the local unit normal vector to Γ, tc is the local contact traction vector, qf is the local frictional heat flux, qc is the local heat flux needed to impose ideal thermal contact, and k is the thermal conductivity. The stress–strain behavior of the explosive material is modeled by a hyperelastic, multiplicative, finite strain constitutive theory with a Von-Mises yield criterion and Perzyna viscoplastic flow rule. Friction is modeled using an Amontons–Coulomb stick-slip theory. Details of the constitutive theory are given in Refs. 6, 19, and 20. Thermomechanical properties for HMX used in the simulations are listed in Table I.

TABLE I.

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TABLE I.HMX material properties used for the mesoscale simulations.

Because a key focus of this study is to examine dissipation within compaction shocks, the energetics described by the model is briefly highlighted. The total mass-specific deformation power is locally given by P=σ:d/ρ

, where d≡(Ḟ F−1+(Ḟ F−1)⊤)/2 is the strain rate tensor, F=I+∇0u is the deformation gradient tensor, I is the second order identity tensor, and ∇0≡∂(⋅)/∂X is the Lagrangian gradient operator. By partitioning d into elastic and plastic components (i.e., d=de+dp), and partitioning σ into volumetric and deviatoric components (i.e., σ=PI+τ, where P is pressure and τ

is the deviatoric stress tensor), the mass-specific deformation power may be expressed by

where ρ=ρ0det(F) and ρ0 is the initial density in the undeformed configuration. Terms on the right side of this equation represent shear, compression, and plastic work rates, respectively.

Within the context of this model, shear work results in a non-thermal change of stored strain energy, whereas compression and plastic work result in deformation induced heating [represented by r in Eq. (3)]. As shown in Refs. 20 and 21, only plastic work is dissipative in that it causes an irreversible production of entropy, with the local dissipation power given by

Here, η is the mass-specific entropy. Friction work occurring along inter-particle contact surfaces is also dissipative when slip occurs; the frictional dissipation power is locally given by Tη̇ f=tc⋅vr, where vr is the relative slip velocity. Because frictional dissipation is localized along contact surfaces, it does not appreciably contribute to the effective (or bulk) mass-specific dissipated energy. Therefore, dissipated energy by plastic work dominates the effective response, though frictionally induced tangential surface tractions can indirectly contribute to Eq. (7) by enhancing plastic deformation within particles. It is noted that frictional dissipation can, however, produce small mass regions of elevated temperature along inter-particle contact surfaces (hot-spots) that may influence the shock sensitivity of reactive solids. Comprehensive discussions and analysis of hot-spots are given in Refs. 6, 9, and 22.

Particle fracture is not modeled in this study despite evidence indicating that it can cause changes in microstructure during quasi-static compaction of granular HMX.^17,23 Because HMX has a low fracture surface energy (approximately γf ≈ 0.05 J/m²),²⁴ it is energetically inconsequential compared with the effective dissipated work induced by shocks. For example, if the average particle size of HMX decreases from approximately d1 = 60 μm to d2 = 10 μm due to fracture of spherical particles, then the change in particle surface area is approximately given by ΔA≈6(1/d2−1/d1)/ρ0=263 m2/kg

. The mass-specific fracture energy is approximately ef≈γfΔA=13 J/kg

, a window having area Atotal is locally positioned at x¯ which represents a rectangle of length δx and width H, and an appropriate average is computed based on all finite elements contained within the window. Mass weighted averages are computed for variables that represent mass specific quantities, such as velocity (mass-specific momentum) and mass-specific power and dissipation rate [given by Eqs. (6) and (7)]. Variables that represent volume specific quantities, such as mass density (volume specific mass) and stress (volume specific energy), are computed using area weighted averages. For a variable Φ, at an axial position x=x¯i

, mass and area weighted averages are, respectively, defined by

where As is the total area of explosive mass contained within the averaging window. Discrete representations of these averages are given by

where Φ^(j) is the value of the variable at the centroid of a finite element, A^(j) is its current area, ρ^(j) is its current density, and Na(i) is the number of finite elements within the window. The smoothness of average profiles depends on both the number of axial positions n for which the running average is computed and the window width δx, and is most sensitive to the choice of δx. High-frequency fluctuations in variables result if δx is too small, whereas excessively smeared profiles result if it is too large. Therefore, it is necessary to establish a rationale for determining an appropriate window size that will provide meaningful comparisons with the macroscale theory, as discussed below.

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FIG. 4.Schematic of the transverse averaging window used to obtain effective 1-D profiles from mesoscale fields.

Second, spatially averaged quasi-steady profiles obtained at different times for a given simulation were superposed and mathematically averaged to obtain an effective (mean) profile; this process is referred to as ensemble averaging in this study. Approximately 50 quasi-steady profiles were averaged for each simulation. Figure 5 illustrates bounds on the computed variation in quasi-steady profiles for solid pressure and mass-specific dissipation rate [given by Eq. (7)] for ϕ¯0=0.68

and ϕ¯0=0.84

at Up  = 300 m/s. Corresponding ensemble average profiles are indicated in red in the figure. Larger spatial fluctuations are predicted for materials having lower packing density. The peak dissipation rate within a quasi-steady shock can vary by nearly 100% which may significantly affect the material’s local heating response resulting in stochastic ignition behavior.⁹ Because this study is concerned with effective dissipation within compaction shocks, profiles presented in this paper represent averages in space and time.

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FIG. 5.Ensemble averages used to obtain quasi-steady shock pressure and dissipation rate profiles predicted by mesoscale simulations for Up = 300 m/s: (a) ϕ¯0=0.68

and (b) ϕ¯0=0.84

Figure 6 illustrates how the effective pressure and dissipation rate profiles vary with δx for ϕ¯0=0.68

and Up = 300 m/s, where 10 ≤ δx ≤ 200 μm. The dissipation rate is expressed by ẇ p¯=Tη̇ ¯ using Eq. (7). High frequency fluctuations result for approximately δx < 30 μm, whereas smeared profiles result for δx > 140 μm. Pressure profiles are less sensitive to δx than dissipation rate profiles. Only minor variations in shock width result for δx < 50 μm, which is easily seen in the dissipation rate profiles which vanish in the equilibrium states surrounding the shock. In this case, δx = 50 μm is chosen for purposes of comparison with the macroscale theory because it results in a nearly converged shock width (≈0.4 mm) and minimal fluctuations. A similar rationale is used to identify appropriate window sizes for other values of ϕ¯0

and Up.

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FIG. 6.Mesoscale predictions for the variation in shock profiles with averaging window width for ϕ¯=0.68

where the conserved variable, flux, and source vectors are given by

and Λ is defined by

Vector components in Eqs. (11)–(15) describe the evolution of mass, momentum, total energy, solid volume fraction, and inelastic volume fraction of the material, respectively. Independent variables are time t and position x. Dependent variables, which implicitly represent average quantities, are: solid phase density ρs; solid volume fraction ϕ; no-load or inelastic component of solid volume fraction ϕ˜; velocity u; solid phase pressure Ps; and mass-specific total energy E=e+(u2/2), where e=es+B, es is the mass-specific internal energy of the solid phase, and B=∫(ϕ−ϕ˜)0βρsϕ d(ϕ−ϕ˜) is the mass-specific compaction potential energy that depends on the elastic component of solid volume fraction ϕ−ϕ˜. The configurational (or interparticle) stress β is empirically determined and implicitly accounts for contact forces between particles that collectively resist compaction. In Eq. (15), f(ϕ) is the compaction yield function. Compaction is driven by an imbalance between Ps and β. The parameters μ and μ˜ in the evolution equations of ϕ and ϕ˜ establish the rates of stress equilibration ( Ps→β) and relaxation to the yield surface ( ϕ˜→f). The viscosity μ controls the compaction shock thickness and rise time, as discussed in Section IV.

The system of equations is mathematically closed by prescribing the relations Ps(ρs,es), β(ρs,ϕ)

, and f(ϕ)

. To this end, a Mie–Gruneisen equation of state is used to describe the thermodynamics of HMX

where

Here, νs=1/ρs is the solid phase specific volume and Γ is the Gruneisen coefficient. This equation of state is compatible with the Hugoniot relation D=ω+su, where D is the shock speed, and ω and s are empirically determined constants.²⁹ The configurational stress and compaction yield function are described by

and

where Λ, k, and c are constants, and ϕf is the free-pour solid volume fraction of the virgin material. Because compaction experiments indicate fast relaxation to the yield surface, it is assumed that μ˜→0 in Eq. (15) resulting in the equilibrium condition ϕ˜=f(ϕ). Parameter values used for HMX are listed in Table II. These constitutive relations are discussed in more detail in Ref. 25.

TABLE II.

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TABLE II.HMX material properties used for the macroscale compaction theory.

Of particular importance are the effective heating rates induced by compaction shocks. The mass-specific deformation power is given by

where

and

Here, d(∙)/dt≡∂(∙)/∂t+v⋅∇(∙) is the Lagrangian derivative. Terms on the left of the second equality in Eq. (19) are, respectively, defined by corresponding terms on the right. Equation (21) is obtained by taking μ˜→0 in Eq. (15), with f‘≡df/dϕ=c prescribed by the compaction yield function. In Eq. (19), Ḃ  implicitly accounts for the rate of stored compaction energy at the particle scale due to shear, ė ρ accounts for the compression work rate, and ė c=ė ϕ+ė ϕ˜ accounts for the compaction work rate. The term ė c consists of both a rate-dependent component ė ϕ, which vanishes in the slow compaction limit (i.e., Ps→β), and an inelastic component ė ϕ˜. These contributions to the deformation power are analogous to the integrated contributions of those given in Eq. (6) for the mesoscale description. Combining Eqs. (11)–(15) with the second law of thermodynamics gives the strong form of the dissipation inequality

where η=ηs is the mass-specific granular solid entropy. The first term on the right side of this equation is non-negative, as is the second term because β≥0 and dϕ˜/dt≥0 by construction. This thermodynamic description is compatible with a Helmholtz free energy of the form ψ(ρs,Ts,ϕ,ϕ˜)=ψs(ρs,Ts)+B(ϕ−ϕ˜), where ψs is the mass-specific solid phase free energy. Here, ψs is the thermal component of free energy. Rate-dependent compaction is the dominant dissipation mechanism for compaction shocks because ė ϕ≫ė ϕ˜. A detailed discussion about the role of ϕ˜ in the mechanics of strain hardened granular material is given in Ref. 25.

The macroscale model is posed as an Initial Value Problem (IVP) that can be integrated to obtain steady spatial shock structures. To this end, Eqs. (11)–(15) are expressed in a right propagating shock-attached frame using the transformation ζ=x−Dt

and w=u−D, where ζ and w are position and velocity measured in this frame. Substituting the derivatives ∂∂t∣∣x=−D∂∂ζ∣∣t+∂∂t∣∣ζ and ∂∂x∣∣t=∂∂ζ∣∣t into these equations, requiring ∂∂t≡0, and simplifying the result, gives a system of Ordinary Differential Equations (ODEs) of the form dfdζ=s, with particle velocity given by w. Conditions at ζ = 0 in front of the shock represent the stress-free ambient state: ϕ(0)=ϕ˜(0)=ϕ0, Ps(0)=0, ρs(0)=ρs0, w(0)=−D

. The homogeneous ODEs corresponding to the mass, momentum, and energy equations can be directly integrated, with initial conditions applied, and combined to obtain

Equations (24) and (25), combined with the equation of state, define the Rayleigh line R and Hugoniot curve H for the granular solid. Equation (21) can be directly integrated with ϕ˜0=ϕ0 to give

The remaining ODE for ϕ in terms of the wave-coordinate ζ is given by

Compaction shock profiles are obtained by numerically integrating Eq. (27) using an explicit solver contained in the package LSODE, subject to Eqs. (23)–(25), the initial condition ϕ(0)=ϕ0

, and the constitutive relations given by Eqs. (16)–(18). The integration procedure is started by taking the initial value of solid volume fraction to be ϕ∗=ϕ0+Δϕ, where Δϕ≈10−6, which is sufficient to perturb the solution off the ambient equilibrium state with minimal error. Compaction shock equilibrium end states and shock speed can be directly determined by combining the equilibrium conditions Ps=β and ϕ˜=ϕ0 for ϕ0≤ϕ≤f−1(ϕ0), or ϕ˜=f(ϕ) for ϕ>f−1(ϕ0), with Eqs. (23)–(25) and Eqs. (16)–(18). The ambient state of the material is unstressed and at standard temperature. The initial solid volume fraction ϕ0 is taken equal to ϕ¯0

in the mesoscale description.

Representative shock profiles for compaction dissipation power, given by Eq. (22), are shown in Fig. 7 for Up  = 300 m/s. A comparison of these profiles with those predicted by the mesoscale theory is given in the following section. Highlighted in the figure is the dissipation power induced by inelastic compaction ė ϕ˜

and the total dissipation power ė c. Dissipation by inelastic compaction represents a significant fraction of total dissipation (≈15%), particularly for low strength shocks, and its relative significance marginally increases with ϕ0. Most conventional macroscale theories applied to granular explosives do not account for dissipation by inelastic compaction;^13,30 consequently, these theories are only formally predictive for strong shocks. Dissipation power increases with ϕ0 for fixed Up due to an increase in shock impedance of the material which causes faster shocks, whereas spatial shock width is largely insensitive to variations in ϕ0

.

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FIG. 7.Shock profiles for the (a) inelastic dissipation power ( ė ϕ˜

) and (b) total dissipation power ( ė c) predicted by the compaction theory for Up = 300, and ϕ0=0.68

and is computed by averaging Eq. (7) using Eq. (9); dissipated work w¯¯¯d

is computed by integrating Eq. (7) in time and averaging the result. Dissipation power for the macroscale theory is directly computed using Eq. (22), and dissipated work is computed by integrating this result in time. Dissipation power also provides a good indication of shock width because it vanishes in both the ambient and shock end state.

Materials used in study have an initial solid volume fraction within the range 0.68≤ϕ¯s≤0.84

and a mean particle size of d¯=60 μm with a narrow distribution. Figure 8 shows particle ensembles having ϕ¯s=0.68

and 0.84. Significant local fluctuations in packing density exist within low density material that can influence shock propagation and dissipation, as discussed below. Simulations are performed for piston speeds within the range 100 ≤ Up  ≤ 500 m/s because of their relevance to deflagration and shock-to-detonation transition. The mesoscale algorithm used in this study, implemented using FORTRAN 90, was executed on 64, 2.66 GHz Dual Core Xeon 64-bit workstations having 4 GB RAM each. One-dimensional parallelization was accomplished by dividing the spatial domain into transverse sectors containing 50–60 particles each, and by assigning each sector to a specific processor. Message Passing Interface (MPI) libraries were used to exchange information between processors at sector boundaries. A typical run time for a single mesoscale simulation was approximately 3 days.

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FIG. 8.Representative initial particle ensembles generated by the pseudo-gravity settling algorithm.

Hugoniot planes, where D is the steady shock speed and P¯¯¯=P¯¯¯ϕ¯=Psϕ is the effective pressure behind the shock predicted by both mesoscale simulations and macroscale compaction theory. Effective pressure is a commonly used measure of shock strength. Also shown in the figure are data for granular HMX having ϕ¯0=0.66¹⁵ and ϕ¯0=0.99.³¹ For fixed Up, both D and P¯¯¯ increase with increasing ϕ¯0

due to higher acoustic impedance of the material and enhanced stress bridging between particles within shocks.

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FIG. 9.Effective shock end states (Hugoniots) predicted by the mesoscale simulations and compaction theory for different initial packing densities ( 0.68≤ϕ¯s≤0.84

): (a) Up–D plane and (b) Up–P plane. Mesoscale simulation predictions are connected by solid lines, compaction theory predictions are connected by dashed lines, and experimental data are connected by dotted lines. Red, black, and green markers correspond to ϕ¯0=0.68

for Up  = 300 m/s and 500 m/s. These profiles connect the ambient material state to the quasi-equilibrium state behind the shock. Shown in the figures are profiles for effective pressure, dissipation power, and dissipated work, respectively, expressed in a shock-attached reference frame. Mesoscale simulations fundamentally resolve shock width based on interactions between particles whereas it is modeled by the value of viscosity μ in the compaction theory.

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FIG. 10.Shock pressure profiles predicted by the mesoscale simulations and compaction theory: (a) Up = 300 m/s and (b) Up  = 500 m/s. Mesoscale simulation profiles are given by solid lines, and compaction theory profiles are given by dashed lines. Red, black, and green lines correspond to ϕ¯0=0.68

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FIG. 11.Dissipation rate profiles predicted by the mesoscale simulations and compaction theory: (a) Up = 300 m/s and (b) Up  = 500 m/s. Mesoscale simulation profiles are given by solid lines, and compaction theory profiles are given by dashed lines. Red, black, and green lines correspond to ϕ¯0=0.68

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FIG. 12.Dissipated work profiles predicted by the mesoscale simulations and compaction theory: (a) Up = 300 m/s and (b) Up  = 500 m/s. Mesoscale simulation profiles are given by solid lines, and compaction theory profiles are given by dashed lines. Red, black, and green lines correspond to ϕ¯0=0.68

Several observations are noteworthy regarding shock width and dissipation. First, shock widths predicted by mesoscale simulations and compaction theory are comparable suggesting that the value of viscosity μ = 40 kg/(m s) used with the compaction theory is compatible with a microstructure having a nearly uniform particle size of d¯=60 μm

. Dissipation rate profiles indicate that shock width is weakly dependent on initial packing density for ranges considered in this study, and that it appreciably decreases with increasing shock strength.

Second, computed dissipation power profiles shown in Fig. 11 qualitatively agree, but quantitative discrepancies exist. Bars are shown on mesoscale profiles indicating the range of fluctuations in effective peak dissipation power within compaction shocks due to local spatial variations in density. The fluctuations are substantial, and are approximately 50% of their ensemble average values. This sensitivity is more pronounced for dissipation power than other integrated quantities because it represents a time rate of change in dissipated work. Peak dissipation power increases with shock strength, and discrepancies between the compaction theory and mesoscale simulations become more pronounced due to differences in their equations of state. Both descriptions indicate that the effective peak dissipation power increases with ϕo

for fixed Up. This result is important because experiments on the shock loading of low density explosives indicate a reduction in shock sensitivity with increased packing density; therefore, dissipation power by itself is not an accurate measure of shock sensitivity. Dissipated work profiles, obtained by integrating dissipation power across shocks, are shown in Fig. 12. Though peak dissipation power increases with ϕ0

for fixed Up, dissipated work decreases. An increase in dissipated work increases the effective temperature rise which is compatible with enhanced sensitivity; consequently, it represents a plausible means to characterize shock induced ignition and burn in that it implicitly accounts for hot-spot formation. Moreover, dissipated work can principally account for shock desensitization because it is influenced by shock loading history.³³

Shock impedance increases with packing density resulting in faster and stronger shocks for fixed Up. It is also insightful to examine how dissipated work is affected by packing density for equal strength shocks. To this end, Up−P¯¯¯

Hugoniot relations are used to express Up in terms of P¯¯¯. Figure 13 shows the variation in effective dissipated work with shock pressure and packing density predicted by both the compaction theory and mesoscale simulations. Predictions indicate that the relation between dissipated work and pressure is well-described by the power-law w¯¯¯d=aP¯¯¯ns, where both the prefactor a and the exponent n=∂ log w¯¯¯d/∂ log P¯¯¯s depend on ϕ¯0. The value of n is predicted to monotonically increase from n = 1.13 for ϕ¯0=0.68 to n = 1.20 for ϕ¯0=0.84 which is indicative of enhanced pressure sensitivity. A consequence of this prediction is that differences in shock dissipation with packing density decrease, albeit slowly, as shock pressure increases. It is interesting that the measured variation in run distance to detonation with input shock pressure (referred to as a Pop-plot curve) for low density HMX having comparable packing densities to those studied here indicates a similar trend. Measurements show that the run distance l∼P¯¯¯−ms, where the constant exponent m=−∂ log l/∂ log P¯¯¯s slightly increases with packing density.³⁴ If it is assumed that l∼w¯¯¯−1d to leading-order, then n∼m

. Though speculative due to complex hot-spot formation, chemistry, and multiphase transport associated with shock initiation, the similar dependency of shock dissipation and Pop-plot curves on packing density warrants additional study.

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FIG. 13.Variation in dissipated work with shock pressure predicted by the mesoscale simulations and compaction theory. Mesoscale simulation data are given by square markers, and compaction theory data are given by circular markers. The dashed lines represent best fit power-laws. Red, black, and green markers correspond to ϕ¯0=0.68

are not significant and are included for ease of presentation. For consistency with experiments, rise time is estimated as the time between the 5% and 95% of maximum particle velocity levels.¹⁵

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FIG. 14.Particle velocity histories predicted by the mesoscale simulations and compaction theory: (a) Up = 300 m/s and (b) Up  = 500 m/s. Mesoscale simulation profiles are given by solid lines, and compaction theory profiles are given by dashed lines. Red, black, and green lines correspond to ϕ¯0=0.68

The variation in computed shock rise time with Up is shown in Fig. 15 for different values of ϕ¯0

. Also shown in the figure are data reported by Sheffield and co-workers¹⁵ for low density HMX. Data are shown that highlight variations due to packing density and particle size: here, “coarse” HMX refers to material having a mean particle size of ≈120 μm; and “fine” HMX refers to material having a mean size of ≈10–15 μm but also having some larger particles of ≈50 μm. No information about the particle size distribution is provided in Ref. 15. The data indicate that rise time decreases with increasing Up and is more sensitive to particle size than packing density. Measured rise times for fine HMX are 0.1 μs or less for the entire range of Up, whereas rise time increases with decreasing piston speed for coarse HMX reaching a value of ≈0.5 μs for Up ≈ 300 m/s. Because the mesoscale simulations performed in this study are for ensembles having a mean particle size of 60 μm with a narrow distribution, they are more representative of coarse HMX than fine HMX. Both the compaction theory and mesoscale simulations give rise time estimates that qualitatively agree with data for coarse HMX, though greater sensitivity to ϕ¯0

is predicted than indicated by the data. Because particle size influences rise time, quantitative discrepancies between computed estimates and the data may be partly due to variations in particle size distribution; a systematic study of such effects represents an ongoing topic of our research.

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FIG. 15.Variation in shock rise time with Up predicted by the mesoscale simulations and compaction theory. Also shown in the figure are data reported by Sheffield et al.¹⁵ Mesoscale simulation data are given by square markers and are connected by solid lines, and compaction theory data are given by circular markers and are connected by dashed lines. Red, black, and green markers correspond to ϕ¯0=0.68

It is insightful to examine how rise time is affected by packing density for equal strength shocks. Shown in Fig. 16 are the computed estimates and data given in Fig. 15 expressed as a function of effective shock pressure. To this end, Up−P¯¯¯

Hugoniot data given in Fig. 9(a) for ϕ¯0=0.65 were used to express Up in Fig. 16 in terms of P¯¯¯. The compaction theory and mesoscale simulations give similar results that exhibit a power-law relation τr=bP¯¯¯−z, where the exponent z ≈ 1 and the prefactor b has a weak dependence on ϕ¯0

. Agreement in computed rise times suggests that a constant value of viscosity may be reasonably assumed if the material has a narrow particle size distribution.

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FIG. 16.Variation in shock rise time with shock pressure predicted by the mesoscale simulations and compaction theory. Estimates given by Eq. (28) were obtained using μ = 40 kg/(m s) for coarse particles and μ  = 10 kg/(m s) for fine particles. Mesoscale simulation data are given by square markers and are connected by solid lines, and compaction theory data are given by circular markers and are connected by dashed lines. Rise time estimates are given by solid lines. Red, black, and green markers correspond to ϕ¯0=0.68

A simple estimate for rise time can be obtained from Eq. (20). Because most shocks considered in this study are pressure dominated resulting in the elimination of porosity, it is reasonable to assume that Ps≫β

. If it is further assumed that Ps≈

constant, which is compatible with a strong shock approximation, then Eq. (20) gives

where τ is time. This expression implies that ϕ→1 asymptotically as τ→∞. An estimate for rise time τr can be obtained by evaluating τ(ϕ=0.99). This estimate indicates that τ∼μP−1s and that it weakly depends on ϕ¯0 in agreement with data. The viscosity μ may be assumed to monotonically decrease with particle size in that smaller particles will cause faster stress equilibration rates due to higher frequency pressure reverberations within and between particles.

Also shown in Fig. 16 are estimates based on Eq. (28) for ϕ¯0=0.68

, 0.77, and 0.84. Two values of viscosity are used: μ = 40 kg/(m s), the value used throughout this study, which is representative of coarse HMX, and μ = 10 kg/(m s) which is representative of fine HMX. Equation (28) properly characterizes the shock pressure dependence z≈1

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