Shock initiation of low density granular explosives (GXs) is an inherently macroscale phenomenon that occurs over distances that are appreciably larger than the mean particle size. Because GXs contain high inter-particle porosity (>25% by volume), dispersed compaction shocks can induce intense thermomechanical fluctuations at the particle scale due to inelastic pore collapse, even for relatively mild shocks. The integrated effect of dissipative heating within compaction shocks is important because it can locally trigger combustion that can spread and subsequently lead to detonation under suitable confinement. Unlike GXs, plastic-bonded explosives (PBXs) contain much less porosity (<5%); therefore, comparatively stronger shocks are required to trigger combustion, and possibly by different mechanisms, such as friction work occurring at deconsolidated explosive-binder surfaces. Mesoscale Modeling and Simulation (M&S) is currently being used to examine the evolution of thermomechanical fluctuations within and behind shocks in GXs and PBXs,1\u201311<\/sup><\/a><\/span> but the relative importance of dissipation mechanisms, their dependence on microstructure (e.g., particle size, shape, and packing distributions, and intra-particle defects) and loading conditions, and their influence on macroscale behavior remain largely unresolved issues. In addition to enhancing fundamental understanding of such phenomena, this information is useful in developing microstructure-dependent theories for shock induced ignition and combustion of these materials at the macroscale. It is tacitly assumed that mesoscale modeling applied to these materials is compatible with commonly used macroscale theories, though it is not often rigorously examined, particularly as it pertains to compaction shocks in GXs.<\/p>\n The primary objective of this study is to computationally examine dissipation within resolved planar compaction shocks in a GX and to establish a functional relationship between dissipated work and shock strength, and its dependency on particle packing density. Dissipated work is important because it is sensitive to the shock loading history of the material and it represents a key macroscale measure of hot-spot formation at the particle scale. Models used to describe compaction of granular materials, including GXs, often relate changes in effective porosity to pressure,12<\/sup><\/a><\/span> which can result in significant dissipation and hot-spots provided that the compaction rate is sufficiently high. Such a relation can also facilitate the development of macroscale theories that describe history-dependent phenomena, such as shock desensitization. This paper summarizes predictions given by both mesoscale M&S and an independent macroscale compaction theory applied to HMX (C4<\/span>H8<\/span>N8<\/span>O8<\/span>). The compaction theory is based on principles of mixture theory and is an extension of the theories formulated by Baer and Nunziato13<\/sup><\/a><\/span> and Powers et al<\/span>.14<\/sup><\/a><\/span> The explosive HMX is chosen because its compaction behavior is empirically well-characterized and it is commonly used in applications.15\u201318<\/sup><\/a><\/span> Only simple microstructures are considered to isolate the effects of initial packing density from those due to complex variations in particle size and shape.<\/p>\n A secondary objective of this study is to examine the extent to which mesoscale predictions of compaction shock profiles agree with those given by the macroscale compaction theory. No attempt is made to adjust or fit macroscale model parameters to obtain best agreement with mesoscale predictions; rather, emphasis is placed on validating the macroscale theory based on resolved mesoscale computations for quasi-steady shocks. Such examinations can also provide rigorous mesoscale interpretations of macroscale predictions, particularly as it pertains to the relation between microstructure, shock dissipation, and hot-spot formation. An analysis of hot-spots based on predicted temperature fields within particles (as done in Ref. 9<\/a><\/span>) is beyond the scope of this study; however, it is possible to formulate a comprehensive relation that includes hot-spot information, such as intensity, size, and spatial proximity, for use with macroscale theories. This issue constitutes an ongoing topic of research.<\/p>\n The paradigm for this study is illustrated in Figs. 1<\/a><\/span> and 2<\/a><\/span>. The mesoscale analysis describes two-dimensional (2D), plane strain loading of a large ensemble of deformable HMX particles. The representative ensemble shown in Fig. 1<\/a><\/span> consists of approximately 4000 randomly packed, initially circular particles having an average size of d<\/span>\u00af<\/span><\/span>=<\/span>60<\/span>\u2009<\/span>\u03bc<\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n with a narrow distribution. The average initial solid volume fraction of the ensemble is \u03d5<\/span>\u00af<\/span><\/span><\/span>0<\/span><\/span>=<\/span>0.84<\/span><\/span><\/span><\/span><\/span><\/span> 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<\/span><\/span> causing the rapid development and subsequent propagation of a quasi-steady compaction shock having speed D<\/span>. Shock strength is characterized in terms of effective pressure P<\/span>\u00af\u00af\u00af<\/span><\/span><\/span><\/span><\/span><\/span><\/span> which depends on both \u03d5<\/span>\u00af<\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span> and Up<\/span><\/span>. Figure 2(a)<\/a><\/span> illustrates a computed piston supported shock having speed D<\/span>\u2009=\u20092136\u2009m\/s for \u03d5<\/span>\u00af<\/span><\/span><\/span>0<\/span><\/span>=<\/span>0.84<\/span><\/span><\/span><\/span><\/span><\/span> and Up<\/span><\/span>\u2009 =\u2009500\u2009m\/s. The shock quickly develops into a quasi-steady wave following impact, as indicated by its position-time curve in Fig. 2(b)<\/a><\/span>, and its pressure P<\/span>\u00af\u00af\u00af<\/span><\/span>=<\/span>1.6<\/span>\u2009<\/span>GPa<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n is sufficiently high to eliminate porosity. The shock speed is lower than the ambient sound speed of solid HMX (\u22482740 m\/s) due to energy dissipation and dispersion associated with pore collapse.<\/p>\n Click to view<\/p>\n<\/div>\n Click to view<\/p>\n<\/div>\n : (a) pressure contours at t<\/span>\u2009=\u20094.5\u2009\u03bc<\/span>s and (b) shock position as a function of time indicating a quasi-steady shock speed of D<\/span> \u2248 2136\u2009m\/s.<\/p>\n<\/div>\n 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<\/a><\/span> for the case shown in Fig. 2<\/a><\/span>. 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 \u03d5<\/span>\u00af<\/span><\/span><\/span>0<\/span><\/span>=<\/span>0.84<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n , but become more pronounced as \u03d5<\/span>\u00af<\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n decreases.<\/p>\n Click to view<\/p>\n<\/div>\n : (a) superposed quasi-steady profiles highlighting local fluctuations; and (b) effective quasi-steady pressure profiles at advancing times.<\/p>\n<\/div>\n The plan of this paper is as follows. First, the mesoscale and macroscale theories are summarized in Sections II<\/a><\/span> and III<\/a><\/span>, respectively, where their mathematical and physical descriptions of compaction induced dissipation are highlighted. Next, predictions are given in Section ![\"\"](\"http:\/\/scitation.aip.org\/docserver\/ahah\/fulltext\/aip\/journal\/jap\/119\/22\/1.4953650.figures.online.f1_thmb.gif\")
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