DEFLAGRATION AND DETONATION MODELING OF HETEROGENEOUS CONDENSED PI-IASE EXPLOSIVES. Joseph Ryan Peterson

Transcription

1 DEFLAGRATION AND DETONATION MODELING OF HETEROGENEOUS CONDENSED PI-IASE EXPLOSIVES by Joseph Ryan Peterson A Senior Honors Thesis Submitted to the Faculty of The University of Utah In Partial Fulfillment of the Requirements for the Honors Degree in Bachelor of Science In Chemisty Approved: Dr. Charles A. Wight Supervisor Dr.'Henry S. White Chair, Department of Chemistry Department Honors Advisor Dean, Flonors College May 2012

2 ABSTRACT A modeling approach to the deflagration and detonation phenomena is employed to study a variety of explosive scenarios. An engineering model for macroscale reaction of energetic materials over the wide range of explosive reaction from deflagration to detonation is developed based upon solid foundation of two previously developed reactive models. The model is calibrated for the representative explosive PBX-9501 with a number of standard experiments including the strand burner, flyer plate, and rate stick tests. The model is then validated against a number of more complex experiments including the cylinder test and Steven test. Resolution effects for both deflagration and detonation models are explored. Upper bound resolutions are identified for both reaction regimes and implications for large-scale modeling are presented. Work continues in extending the model to represent heterogeneous explosive configurations as homogenous materials. This necessitates implementation of a bulk compaction model, which has yet to be completed. Validation of the bulk compaction model in various scenarios and the resulting thermodynamic state necessitated comparisons with mesoscale simulations. A mesoscale model is implemented from various thermodynamic and material models posed in literature. Validation of the mesoscale models against experiments led to very favorable agreement. Results from varied density granular simulations lead to insight into initiation dependence on initial density, leaving the platform good for comparing bulk scale scenarios for experimentally intractable problems.

3 TABLE OF CONTENTS ABSTRACT ii I. INTRODUCTION 1 A. DEFLAGRATION 3 B. DETONATION 7 C. MODELING 10 II. METHODS 15 A. SIMULATION METHODOLOGY 15 B. BULK SCALE MODELING DDT1 FORMULATION VALIDATION EXPERIMENTS 21 C. MESOSCALE MODELING 23 III. RESULTS 24 A. BULK SCALE MODELS STRAND BURNER TEST POP-PLOT TEST RATE STICK TEST CYLINDER TEST STEVEN TEST SLOW COOK-OFF TEST 28 B. MESOSCALE MODELS SINGLE CRYSTAL EXPERIMENTS 29 iii

4 I j 2. GRANULAR HMX STRESS GAUGE EXPERIMENTS VARYING DENSITY GRANULAR BEDS 30 IV. DISCUSSION 31 A. BULK SCALE MODELS 31 B. MESOSCALE MODELS 36 V. CONCLUSIONS 38 VI. REFERENCES 40 A. FIGURES 45 iv

5 1 I. INTRODUCTION A great push occurred in the last century towards understanding the energetics of explosive materials largely due to the desire to create weapons that are more powerful, more reliable and safer to handle. What captivated the researchers of explosions was the rate at which high energetic (explosive) materials react and the ensuing damage from the energy release. Furthermore, the sensitivity of the explosives could vary significantly with formulation, temperature, pressure and age. One of the greatest curiosities is that many explosives can react at two very different rates. The two regimes are separated by the Mach 1 point, and named deflagration and detonation for reaction in subsonic and supersonic flow regimes respectively. The shock physics nature of detonation interested such great scientists as Taylor and ZePDovich, von Neumann and Doring, and great strides where made in understanding the detonation phenomena [21,55,61]. At the other end of the spectrum, those such as Kuo and Beckstead applied the principles of gas flame and thermodynamics of various phases of materials to understand the low rate combustion of explosives thus brining fine control to practical applications [7]. Of particular interest over the past half century has been the safety of explosives. Inevitably, the explosive sciences and technologies were used to generate great weapons. The safety of these weapons in storage and transport has been of great interest as a number of situations have caused loss of life and property. A number of accidents involving rocket motors and mining explosives have resulted from what conventionally was thought would be a safe operating conditions. In general, some thermal or mechanical insult occurred and the explosive transitioned from the slow reaction to a detonation. The rate of energy release is much larger in detonations resulting in more

6 2 damage and increased danger to human. Studies have identified a number of different types of transitions from slow reaction to detonation including deflagration to detonation transition (DDT), slow cook-off to detonation (XDT) and shock to detonation transition (SDT) depending on the type and duration of the initiation. A difficulty exists in designing, implementing and accurately analyzing large accident scenarios experimentally. Generally, small tests have been performed on the order of a meter at maximum. Furthermore, the tests generally involve one or just a handful of the explosive devices. These tests cannot adequately address the sympathetic explosion seen during real world shipping and storage accidents with any economic feasibility. This fact has driven the field of explosives modeling. Computational models provide the potential for a safe and economically feasible means of exploring these real life scenarios. Advances in explosives modeling, material modeling and simulation capabilities in parallel with increase in computational capability in the last two decades has put the predictive simulation within grasp. What remains is development of bulk scale models capable of capturing relevant physics. The aim of this paper is to describe the advancements made at the University of Utah towards these ends. The introduction continues with discussion of subsonic reaction known as deflagration, followed by discussion of supersonic reaction known as detonation and concludes with a discussion of general thermodynamics modeling and a brief discussion of the paradigms chosen in these modeling efforts. Simulation methods are presented in Section II with discussions of the reaction model developed along with discussion of the material models and material parameters chosen for various materials in the validation experiments. Section III begins with results from the calibration experiment results,

7 3 continues with discussion of validation experiments for the bulk scale reaction model, and concludes with results from more resent mesoscale modeling of granular compaction. Section IV concludes with a discussion of the results and implications for current capabilities. Finally, a critical review of the necessary model advancements that still must be accurately modeled in order to simulate the large accidents is presented. A. DEFLAGRATION Deflagration is a term used to describe subsonic combustion of a material. The subsonic nature of the reaction is due to the time scale of thermal activation. The actual burning rate of energetic material is generally temperature and pressure dependent due to the competition of hot product gases away from the reaction surface and the thermal diffusion from those products back to the unreacted starting material. Usually, two modes of deflagration are identified, due to the large discrepancy in the rate of reaction propagation normal to the surface of the explosive of interest. Conductive burning is the slower of the two, generally consuming material at rates between a few millimeters per second and a few centimeters per second at reasonable pressure and temperature. At the other end of the spectrum is convective burning which is characterized by rates between a few hundred meters per second to perhaps one or two thousand meters per second that are usually induced via some mechanical insult or reactant confinment The nature of conductive burning can easily be understood by striking a match and watching the flame burn down the surface of the match staff. The propagation of the flame down the staff can be seen to be slow, and the flame itself is seen to be sheet-like, or in computational fluid dynamic language the flame is said to be laminar. Conductive

8 burning can then be thought of as surface burning. Convective burning is a little bit more difficult to visualize, but one can easily think of a scenario that embodies this type of burning. For example, if one imagines a dust cloud of energetic particles that are burning it is easy to imagine the flame flowing through this material rapidly due to the large amount of low-density space where hot gas can diffuse. The increased surface area allows the reaction front to move rapidly through and thus consume more of the body of explosive in a given timeframe. In this way, convective burning can be thought of as the process of pushing hot gas into a material, which differs from surface burning where hot gas is pushed way form the material surface. Before further discussing conductive and convective burning, and the transition between the two, it is necessary to understand the chemical and physical processes involved in deflagration. Combustion is a thermally activated oxidation reaction between some fuel and an oxidizer. Solid materials generally have combustion character similar to the schematic seen in Figure 1. The solid is heated by conduction or radiation until reaching its melting point or in some cases thermal decomposition temeprautre, after which combustion generally begins to occur, causing the large molecules making up the solid to split into smaller reactive intermediates. These reactive intermediates generally foam out of the melt layer in bubbles. Beyond this melt layer, exists a dark-zone which is characterized by a stand-off distance. This dark zone is generally a region in which the reactive intermediates absorb more energy and eventually either react into smaller molecules or act as radical catalysts for further reaction. Following the decomposition of large molecules, smaller diatomic and triatomic molecules are formed releasing energy that causes the luminous flame seen above a burning surface. In high energetic

9 5 materials, or those that are used as primary, secondary or tertiaiy explosives, this trend of burning is followed, with some variation in flame nature. For example, some materials begin to react before they melt or some materials have a double flame structure where a temperature plateau is hit, prior to further reaction of high activation energy reactants that increases the temperature to a second plateau. An example of the first is ammonium perchlorate (AP), a common rocket propellant. These double plateau combustibles are generally a mixture of two different fuels and are often called double base propellants for this reason. The explosives studied here are generally made of organic molecules that show the single plateau nature. Furthermore, the explosives studied are generally mixed with a small percentage of plastic bonding agent and are known as plastic bonded explosives (PBX). The ideal nature of these explosives, or the tendency to detonate completely with little effect from inertial confinement, makes them useful in a number of applications. In addition, this property makes them dangerous thus desirable materials for computational study. The organic molecule studied most commonly, and the focus of this study, is known as octahydro-l,3,5,7-tetranitro-l,3,5,7-tetrazocine (HMX), and can be seen in Figure 2 along with other common organic high energetics such as pentaerythritol tetranitrate (PETN), 1,3,5-trinitroperhydro-1,3,5-triazine (RDX), triaminotrinitrobenzene (TATB), 2,4,6-trinitrophenylmethylnitramine (Tetryl) and trinitrotoluene (TNT) (as well as AP for comparison). The common feature of these explosives is the ring like structure, and the abundance of nitro groups. The nitro groups are the oxidizer, while the carbon linkages are the reducing agent. Each of these molecules is itself a complete reaction source, and thus finds use as propellants, fracking equipment and mining explosives.

10 Many studies have been performed attempting to elucidate the reaction of products through the reaction zone, and modeling studies have been performed. Many of the intermediates are not known but the major players have been identified using mass spectrometry and give insight into the main reaction pathway [7]. While some total kinetic models formulated with over 40 species and over a hundred reactions, the concern here will be on the major intermediates and products, and their general behaviors. Cleavage of the N-N bond between the nitro group and the ring for reactants such as RDX and HMX, and the cleavage of the C-N bond in others such as TATB and TNT are generally accepted as the initiation step [11]. The final products of combustion reactions of these organic molecules are the simple diatomic and triatomic molecules H2, CO, CO2, N2, NO and NO2. A schematic of the degradation pathway for HMX can be seen in Figure 3. There are two pathways from initiation. The first is the accepted pathway for N-N bond schism with concerted bond formation of the nitro group with the neighboring axial hydrogen forming the HONO intermediate [11]. Another is the breaking of one of the C-N bonds, which is thought to rapidly cascade around the ring breaking every other C-N bond forming a number of CH2N2O2 intermediates that then react to simpler combustibles. HONO and CH2N2O2 rapidly react in the dark zone to the final products as the molecules get closer to the luminous flame [34]. With this knowledge of combustion, it is time to readdress both modes of deflagration. Both are thought to generally follow the same reaction pathway but the observed behaviors are radically different. Belayev presented a theory for this fact based on the stand-off distance of the flame [8]. reaction rate and the rate of diffusion. The stand-off distance is affected by the Thus, the distance is a function of both

11 7 temperature and pressure, with temperature tending to increase and pressure tending to decrease this stand-off. Since the dark zone is generally endothermic, in order for a flame to be able to sustain itself above some surface, there must be enough space for the molecules to absorb the enetgy required to react to final products and form the flame prior to diffusing away from the flame front. Since convective burning can thought of as the burning of explosives inside of defects such as pores, cracks and other surface features, it becomes a function of the pressure of the gas above a given surface. What Belayev noted was that if the pressure was large enough, the luminous, heat-releasing zone of the flame could penetrate into surface defects and accelerate burning normal to the laminar bum front. However, this increased surface area tends to increase the convection speed and hot reacting gases pressurize the defect causing both damage and ignition of larger portions of the material, A great example of mesoscale modeling of this phenomenon can be seen in works of Beckstead et al. [7]. The transition between conductive and convective burning is a function of many variables, including but not limited to: material damage, material temperature, device confinement, porosity, gas pressure and reaction timescales. Therefore the process is difficult to model with many efforts having been made over the years. B. DETONATION Supersonic reactions in high energetics are also known. Called detonation, this reaction is characterized by a reactive shock wave that propagates through the material faster than the speed of sound. In these reactions, the shock wave heats the material at the reaction front quickly enough to induce reaction on the time scale of nanoseconds.

12 The reaction energy is partitioned between the enthalpy and work in such a way that the expansion of products sustains the high pressure of the shock front and ultimately the reaction. The reactive shock wave is well studied both analytically and experimentally in gases [13,31], and the theory is generally applied to condensed phase reactions [25,46,50,52]. And while the empirical models designed for condensed explosives give reasonable results, the actual physical validity of the theory in condensed explosives is still subject to question. The basis for detonation modeling is in shock physics. Models were largely based on the conservation equations for mass, momentum and energy that will be presented in the next section. Theories created around the turn of the century by those such as Chapman and Jouguet formed the bases of early reaction models and relied on the Hugoniot curve and the Rayleigh line [13,31]. The Hugoniot is the curve showing the progress of the shock compression as the wave passes a region of material and the Rayleigh line relates shock pressure, detonation velocity and the initial state of an equilibrium mixture. The shock moves up this curve as it passes, and expands down an adiabat that is determined by any entropy or energy change in the system due to melting, reaction, damage or other dissipative mechanisms. A schematic of the Chapman-Jouguet (CJ) theory can be seen in Figure 4, with interesting features such as the Rayleigh line, which identifies a minimum of Gibbs free energy at the end of the reaction [16], and the intermediate reaction Hugoniots which can be thought of as a mixture curve between reactants and products. In CJ theory, the curve moves up the Rayleigh line towards the CJ point, the tangent between the product Flugoniot and line from the starting state, until it reaches the point. This point of reaction is known as the CJ point, and is where the

13 Gibbs energy is minimum and the reaction is complete. These reactions are generally modeled as first order decomposition. A major deficiency with this theory was identified near the middle of the last century, namely that the reaction occurs instantly, and motivated new work in the field. Three scientists, Zel dovich from Russia, von Neumann from Germany and Doring from America independently discovered the new theory. The theory is known as ZePdovich-von Neumann-Doring (ZND) theory, and is based on the idea that the explosive reacts over a finite timescale [21,55,61] The theory can be understood by examining Figure 4. The consequences of the finite reaction time is that the shock must compress the material up to some huge pressure, known as the von Neumann (VN) spike to induce reaction, upon which the molecules have enough energy to react. At the extreme the reaction can happen as quickly as a few nanoseconds. Two possibilities in the way this process proceeds have been postulated. The first is that the reaction occurs quickly and the actual CJ point is reached at the end of the shock wave. The other is that the reaction acts slowly, and the product Hugoniot is met far down stream (see Figure 5 for an explanation of a shock propagating) beyond the sonic plane. The distinction is important, as the sonic plane, which is defined by the equation: Us = Up + C (1) where Us is the shock speed, Up is the material velocity (also called the particle velocity) and C is the speed of sound in the shocked material, defines the point at which reaction can affect the propagation of the reactive wave. Therefore, two types of detonations can be seen, those that react entirely before the CJ point, and those that do not. This cut-off acts as the distinction between high and secondary explosives. High explosives are those

14 such as HMX, RDX, TNT and PETN and secondary explosives are those such as TATB and ammonium nitrate fuel oil (ANFO). The high explosive class is of interest as the explosives have similar characteristics to gas phase detonations, and are more easily described with first-order kinetics using good Hugoniot fits [38]. Typical pressures seen in detonation are in the gigapascal (GPa) range, with VN spikes as high as perhaps 60 GPa and CJ points for high explosives between 25 and 35 GPa [42]. A state of the art schematic of the whole detonation process was presented by Chidester et al. [14], which lends great insight into the progress of the system behavior as the reactive wave passes over explosive material. Vibrational energy is imparted as the shock passes. This energy redistributes and is consumed in the bond breaking process. Bond breaking results in exothermic reaction and small reactive products form with high intermolecular vibrational energy. Finally, these reaction products expand towards chemical equilibrium and the CJ state is reached. These temporal and spatial scales are impractical for any bulk simulation and thus simple models have been employed. Discussion of these models as well as general discussion of computer modeling of thermodynamics and kinetics will be presented in the next section. In general, the process of detonation is understood well enough on the length and timescales that are of interest in real scenarios. C. MODELING Computer models require a set of conservation equations as well as a discretization strategy for solving the partial differential equations that arise in the formulation of these equations. Models vary in complexity depending on the desired

15 length scale (also known as resolution). Consequently, the demand for experimental fitting data at a desired resolution scales at least linearly with model complexity. The result is that the predictability of the model outside of the range for which it was fit is suspect. This point will be discussed shortly. The discretization strategy also imparts a degree of freedom into the solution resulting in increased uncertainty. The implications are explored extensively in this work however; a high level description of the simulation methodologies adopted in order to solve the conservation equations must first be presented. The conservation equations (also known as governing equations) of continuum mechanics generally include terms for mass, momentum and energy as well as a scheme for advancing the system in time. Examples of the balance equations can be written in the Lagrangian from: f *0 a * * # where dots indicated derivatives, arrows indicate vector quantities, T indicates a transpose and the V symbol implies the gradient of that quantity (or the first spatial derivative in each of the cardinal coordinates). Here p is the density, v is the velocity vector, b is the forces vector at the point of the body, o is the Cauchy stress tensor, e is the internal energy, q is the heat flux term and s includes a source of energy (for instance in a reaction). Each of the vector quantities is a function of both space and time. These equations allow a closure over the system that can be solved using partial differential computational techniques.

16 There are two formulations of continuum mechanics that are popular, namely Lagrangian and Eulerian. Eulerian mechanics is generally used in grid based models and are useful for fluid mechanics where advection of mass, momentum and energy between neighboring cells in the grid approximates the motion of the system. The downfall of the formulation is that the grid remains fixed, and time and spatial location based history of the system cannot be saved. Lagrangian mechanics is formulated to rectify this problem. The materials are discretized not into cells but into a set of finite points that approximate the material. Solid materials are often better represented with the Lagrangian formulation, as history of the stresses in the material as well as the extent of plasticity or damage, or other spatially localized quantities of interest may be tracked. In these formulations, the Lagrangian particles are moved around the domain, and the history the material is stored. Both find their place in simulations of explosives and both impart a different type of uncertainty due to approximation. In the case of the Eulerian formulation, the cell size is the limiting factor on convergence. Convergence is the correct solution of the governing equations. In the case of Lagrangian formulation, the particle density, or the total number of particles chosen to represent the material is the limiting factor on convergence. Care must be taken to minimize resolution effects, while maintaining simulation sizes that are feasible on available computational resources. On top of these formulations, more specialized models are generally chosen to better represent specific material behaviors in the domain. These include reaction models, material models, equation of state (EOS), melt temperature, specific heat equations and a host of others. The complexity of these models, especially from the

17 standpoint of predictability of results, largely depends on the extent to which they may be formulated a priori (from first principles). Specifically for reaction, empirical, semi- empirical and first principles based models have been formulated. While empirical models exist that do decently well in both the deflagration and detonation regimes, for example the Vielle s power law model and the Ignition and Growth (I&G) model [52], they have little predictability outside the temperature or pressure range for which they were calibrated. At the other end of the spectrum, species based models have been developed for deflagration with more than 40 species and 150 reactions [7]. However, the tractability of these problems on any length or timescales is impossible. Hence, we attempt to focus on the semi-empirical equations that are mostly based on physics with a few fitting terms. Semi-empirical equations have found more success in deflagration than detonation, with correct temperature and pressure dependence attained [56]. In the case of detonation, recent work in reactive flow modeling has extended some of the more empirical equations to include entropy dependence [25]. While the fitting form, known as CREST, is still empirically based, the inclusion of physically based entropy dependence allows the model to work outside of the range for which it was fit with good fidelity [60]. Existing models for reaction are largely formulated for a specific scenario, for example WSB is for surface deflagration, I&G and CREST are for shock-to-detonation transition (SDT), and other models like those of Bdzil et al. are for deflagration-to- detonation transition (DDT) [6]. This means the Holy Grail for deflagration and detonation modeling is a general, semi-empirical model containing enough of the physics to reasonably reproduce the behavior of deflagration, detonation, DDT and SDT in both

18 granular and solid explosives on a large range of temporal and spatial scales, while running quickly enough on available resources to prove useful. The following pages describe progress towards these ends, a model entitled DDT1.

19 II. METHODS A. SIMULATION METHODOLOGY The Uintah Computational Framework 1 (Uintah) was chosen for the implementation of DDT1 a high energetic reaction model. Uintah includes model types such as constitutive models, reaction models, equations of state and component models. Component models are such things as implicit continuous-fluid Eulerian (ICE) [26] or material point method (MPM) [51], in which the basic physics of a system are solved. ICE is an Eulerian grid-based model on which thermodynamic properties including pressure, volume and temperature are solved iteratively, MPM is a particle-incell model consisting of Lagrangian points, implemented as a quasi-meshless method. The GIMP particle-to-grid interpolator is used for its accuracy to performance ratio [23,32]/. DDT1 requires both MPM materials and ICE materials and as such is implemented in the MPMICE component. A variety of explosive scenarios have already been simulated successfully with the MPMICE component [22]. The DDT1 model balances sources and sinks for mass (from one material to another) and energy (representing exothermic or endothermic reactions). DDT1 is designed to solve the appropriate rate equations for the reactions and update these sources and sinks. A parallel task graph ensures that tasks are executed in the correct order to make necessary data available to each module in a consistent manner and that the material physics are modeled by the simulation component. 1 w w.uintah.utah.edu

20 The plug-and-play style model interface employed by Uintah allows for a high degree of complexity in problem formulation, while at the same time facilitating fast prototyping of new simulations. Once a good problem setup is found, activating more advanced models entails implementing sources/sinks, writing an input specification and adding the input parameters to the input file. Using a material model already implemented merely requires having correct input parameters. For example steel, copper and aluminum already have well validated hypo-elastic behavior and parameterization available in Uintah [2]. Each validation simulation performed herein went through a variety of stages, consisting of increased complexity at each iteration step. The following subsections describe the models used for the final validation experiments. Of particular importance to modeling are the spatial resolution dependences of the various fundamental metrics of interest. Because MPMICE is a mixed Eulerian/Lagrangian code, studies over cell and particle convergence spaces are required. Quantification of resolution dependences allows assignment of reasonable uncertainty outside the ranges of model calibration. Resolution studies were performed for all simulated observables, including detonation velocity, deflagration velocity, CJ pressure and case expansion velocities. B. BULK SCALE MODELING Bulk scale modeling came in two phases, calibration of the reaction models, followed by validation experiments. During calibration, a consistent set of EOSs and reaction parameters, as well as a material model were selected for the explosive. The explosive chosen for all simulations was the HMX based plastic bonded explosive (PBX)

21 17 known as PBX 9501, which contains 5 percent plasticized Estane. These were then used in validation experiments, with a number of other materials such as aluminum, steel and copper. Material models for non-reactive materials will be mentioned in name only. For a complete description of the models used for bulk scale modeling as well as comprehensive set of citations, see the paper by Peterson and Wight [41]. 1. DDT1 FORMULATION Three EOSs were used for representing the high energetic reactant and product states. Two of these are JWL equations, the standard equation of state chosen for SDT simulations by a number of scientific groups [14,46,48]. The JWL EOS takes the relative volume, v, the temperature, T, and specific heat, Cv, as well as five fitting parameters, A, B, Ri, R 2 and to, and calculates the bulk average pressure. Numerous fits for product and reactant materials axe available in literature. Most of which have standardized Ri, R 2 and 0) parameters, making the use of the JWL EOS efficient when prototyped with a different explosive material. EOS parameters for detonation products and reactant behavior of PBX 9501 were taken from Vandersall et al. [54], A second product equation of state is necessary for representing products from surface and bulk burning, as they have fundamentally different behavior than detonation products. The TST EOS, F m {f «!)CrF/(K J was used for its!> particularly good representation of species relevant to combustion [53] Parameters used for product gases are Pa m3 for a, xlo"4 m3 for b and y is Variable V, T and Cv are the specific volume, temperature and specific heat respectively.

22 Mass and energy is either transferred to the burn products or detonation products depending on the mode of combustion. Surface burning and convective burning products were represented by the TST EOS and detonation products were represented by the JWL EOS. Regardless of the reaction mode, the reactants were represented by the JWL reactant EOS. Bennett presented a model for statistical treatment of cracks in a bulk volume of viscoelastic material and called it ViscoSCRAM [9]. The JWL EOS was implemented in the ViscoSCRAM material model. Advantages of this model include the ability to evolve average cracking without predefined boundaries or advancement of the state of such boundaries as other crack models do, making it suitable for use in Uintah where boundaries are not well defined. ViscoSCRAM takes an initial crack size, maximum crack growth rate, several other crack rate parameters and five Maxwell elements, each including a modulus and related relaxation time. From these, average crack size is evolved from a balance between crack coalescence and separation. PBX 9501 calibrated parameters from Bennet et al were used [9]. Average crack size is used in the threshold criteria for onset of convective burning into cracks, as will be discussed shortly. A bulk modulus of 11.4 GPa was used as it is an often cited literature value [91]. Two previously validated models for the two limiting modes of combustion were used in DDT1. Combustion in both surface burning and bulk convective burning regimes were represented by the WSB reaction model [56]. This model had previously been implemented in Uintah as a 3D model, named Steady Burn, by including a surface area calculation based on the direction of density gradient in the cells and the total mass of reactant in that cell [59]. Surface detection for surface burning and density gradient require MPM materials, hence the reason MPMICE is used. PBX 9501 parameters can

23 19 be found in literature [56,59]. Transition from surface burning, where only the top cell of the surface is allowed to burn, to convective burning, where the burning front penetrates into the material, is determined by Eq. (3), a relationship described and fit by Berghout et al. and proposed by Belyaev et al. [8,10]. m IQ* (3) An extent of cracking in the cell, w, is calculated by ViscoScram. The critical pressure for penetration of combustion, pc, is then calculated. When the pressure in a cell exceeds the critical pressure, pc, bulk burning occurs. In addition to this requirement, the gas in the cell must be above the melting and thermal runaway temperature for the explosive, 550 K in the case of PBX 9501 [27]. This allows burning in damaged material (i.e. cracked or porous) ahead of the deflagration wave without any sub-grid scale models for hot spots or shear heating. Good agreement has been obtained with a variety of experiments that exhibit convective burning. This threshold is applicable in cook-off scenarios, where the material is heated to its decomposition point over large areas of the bulk. Detonation is carried out by a modified version of the JWL++ model that was originally formulated by Souers et al. [46]. Model fitting simplicity is attained by using the first-order, single-term rate model, seen in Equation (4), as only two parameters need to be fit. -exa-fjxi * a (4)

24 The pressure exponent was chosen to be 1.2 and the rate coefficient 2.33 jxpa 1,2s-1. As described elsewhere, these parameters control the size effect curvature and infinite radius detonation velocity, respectively [45,46,48]. Two differences exist between the JWL++ implementation by Souers et al and DDT1. First, DDT1 uses a JWL EOS for reactants instead of the Murnaghan EOS. The decision to use the JWL equation of state was made as a consequence of the zero experienced by the Murnaghan EOS at relative volume equal to 1.0 and subsequent negative pressure during expansion beyond initial state a common occurrence for viscoelastic materials undergoing large relaxation. Negative pressures cause exceptions to be thrown in Uintah. As mentioned, parameters from Vandersall et al were used as they had been found to give good detonation and CJ behavior with Ignition and Growth [54]. Second, instead of additive pressure, the built in iterative pressure solve in ICE is used to correct for the simple Dalton's law assumption in its original formulation. A transition mechanism in the form of a pressure threshold acts as the link between burning and detonating. The models are mutually exclusive, only burning or detonating is allowed in any given cell at a time. Furthermore, burning is prevented in cells adjacent to detonation, as this sort of burning was found to accelerate the shock wave non-physically. Once the pressure threshold is exceeded, the detonation begins. Equilibrium cell pressure has mixed contributions from stresses of all MPM materials and the cell centered pressures for each ICE material. However, the primary functionality of ViscoSCRAM is turned off in a cell undergoing detonation to minimize computation,

25 leaving stress to be simply calculated from the reactant JWL equation of state. For PBX 9501, 5.3 GPa was chosen as the pressure threshold to match Pop-plot data. The deflagration parameters were chosen to match the strand burner tests done by a number of experimenters [1,56]. This required validation against both pressure and temperature dependence, as well as a thorough assessment of errors. Detonation parameters were chosen to match the size-effect curve [47]. The detonation velocity is known to decrease in small explosive samples, due to reduced inertial effects, causing larger transverse motion. This effect is also seen in curved explosive devices and hence the need to calibrate against. Schematics of these experiments can be seen in Figure VALIDATION EXPERIMENTS Three validation tests for the models general performance were chosen, including the confined cylinder test [29,42], the modified Steven test [30] and an annular confined slow cook-off test [20]. Schematics of these experiments can be seen in Figure 7. The first is to test the detonation properties, including reaction zone length and product expansion behavior by comparing the velocity of the expanding case, post-detonation. The second is for low velocity impact where a detonation transition occurred on timescales longer than a normal SDT or DDT and tests both the reaction pressure threshold and the deflagration behavior. The final test is of reaction in a heated sample that eventually undergoes a transition to detonation. Models used to represent the nonreactive materials in these simulations are discussed below. Generally, the best available models available in Uintah were chosen to reduce uncertainty due to reactive material interactions with non-reactive materials.

26 22 Oxygen-free, high conductivity copper was used as the encasing material in the Cylinder tests. Since the velocity of the copper is the data of interest in these simulations, accurate representation of the material is paramount. A compressible Neo-Hookean stress relationship was used, with standard bulk and shear moduli. A yield stress of 70 MPa and a hardening modulus of 4.38 p.pa were used along with a thermal conductivity of 400 W/tnK and a specific heat of 386 J/kgK. During Pop-plot simulations, the impactor and cover plate was made of Aluminum 6061-T6, and was the only non-reactant material type of relevance in the simulation. Standard shear and bulk moduli were used along with a Mie-Griineisen EOS with common parameters. A shear modulus model developed by Nadal-LePoac (NP) was used. Melting was modeled with the Steinberg-Cochran-Guinan (SCG) model. A Hancock-MacKenzie damage model in conjunction with a Gurson yield condition allowed plastic flow of the material post failure. The final material used in simulating the validation experiments was steel. A hypoelastic-plastic constitutive model with a Mie-Griineisen EOS was used with common parameters. A John son-cook plasticity model was used in conjunction with a Johnson- Cook damage model, and a Gurson yield condition with the same parameters as those for copper. The specific heat model described by Lederman et al was used. A full description of the steel, copper and aluminum material representations as well as citations for material parameters with assessment of accuracy and error can be found in the works of Banerjee [2]. Finally, a friction model developed for MPM was used for interaction between materials, generally with a frictional coefficient of 0.3.

27 23 C. MESOSCALE MODELING Mesoscale simulations of the explosives have become desirable while model development on DDT1 continues. Specifically, it will be necessary to validate bulk-scale models against finely resolved simulations, as experimental data is difficult to obtain on the length scales of interest. As such, a model has been implemented in Uintah and validated against experiments for granular compaction. The HMX material was represented by the SCG viscoelasticity model with parameters from literature [15]. Melting behavior and specific heat behavior was modeled using models of Menikoff [39]. Validation simulations were based on rise time measurements from Sheffield [43], using randomly generated granular beds with the appropriate grain size distribution [4,18], and single crystal measurements [19]. Schematics of these experiments can be seen in Figure 8. Some of the experimental cases from Sheffield indicated reaction of the material, and both the WSB model and a Prout-Tompkins model [49] have been applied to attempt to elucidate the process of ignition and reaction in granular beds that have been shocked. The models were then applied to impact scenarios with varying initial granular bed densities from 42 to 77 percent theoretical max density (TMD). Wave profiles and temperature profiles were averaged transverse to the impact. Results were compared qualitatively with the works of Barua [5]. Randomly generated spheres from measured particle size distributions where used [18].

28 24 III. RESULTS A. BULK SCALE MODELS 1. STRAND BURNER TEST Surface regression rate normal to the gas/solid interface is used as the primary metric for validation of combustion in both the convective burning and surface burning regimes. Surface regression rates were found to generally agree in value and trend for pressures greater than 1 MPa and less than 70 MPa. As seen in Figure 9, outside the 2.3 to 9.2 MPa range, the simulated regression rate was found to deviate by as much as 2 times, though this large a magnitude difference is only seen at very low pressures. At large pressures, simulations overestimate the burn rate by approximately 10%. Simulated burn rate dependence on bulk solid temperature, seen in Figure 10, is overestimated for low and high temperatures. Overestimated rates are especially pronounced at high pressures. Convergence results for the burn rate with varied cell resolution can be seen in Figure 11. A representative mid-range simulation was used for convergence, that of the 6.21 MPa pressure and 298 K bulk temperature. The bum rate converges at higher zonings per millimeter. As can be seen in the figure, the convergence is fast for resolutions higher than 1 zone/mm, though the converged value is about 7% larger than the averaged experimental value of 1.089±0.077 cm/s. Relaxation to steady burning state is a function of the cell size. Figure 11 shows time to steady state as a function of resolution. Particle density was varied from 2 cubed to 6 cubed per cell and had negligible effect on the burn rate. Because the Steady Burn implementation only uses cell centered values for computation of the actual burn rate, no

29 25 paxticle density dependence was expected. At 2, 3, 4 and 6 cubed particles per cell, the burn rates differed by a maximum of 0.02% over a factor of 27 times higher particle resolution. A weighted average of the uncertainties in Figure 10 gives a value of 6.04%. At low bulk temperatures reasonable error is seen, with reasonable error being quantified as less than 10%. Errors calculated over the full pressure range at 273, 298 and 423 K were computed with a weighted average of 4.42% error. These results indicate that simulations performed with bulk temperature near room temperature should give less than 7% uncertainty during burning. 2. POP-PLOT TEST Figure 12 shows the Pop-plot for simulations and experiments. Though the simulations were run with g/cm3 density, the Pop trend is found to lie closer to the g/cm3 experiments from LASL in magnitude [42]. Of particular note, is the slope of the trend, which is in between the two density trends cited for experiments. Detonation characteristics were determined from the Pop-plot simulations. 3. RATE STICK TEST Parameters for Equation (4) were found that match both detonation velocity and detonation pressure at the resolution used for calibration with values of 1.2 for b and 2.33 HPa"1,2s-1 for G. The detonation velocity at this resolution was found to be 8843 m/s and the detonation pressure was found to be 37.2 GPa. Resolution effects were explored for the same resolutions studied for the Strand Burner simulations. Particle resolution

30 26 dependence was studied at 4 zones/mm cell resolution, as this was the calibration resolution. Results are presented in Figure 13. Detonation pressure error was found to increased with greater particle resolution, relative to the 37.2 GPa at 8 particles per cell. However this effect is small, with the error decreasing by only a factor of two over 27 times higher particle density. Detonation velocity error, however, increased with higher particle resolution relative to the mm/jus cited in literature. In fact, increasing the density of particles by 27 times increases the error from about 0.04 mm/ us to as high as mm/ iis. The relative error in all cases studied for both velocity and pressure were less than 6%. Cell resolution has a considerably more noticeable effect on detonation pressure and velocity as demonstrated in Figure 14. Convergence in detonation velocity was found to be strong, while convergence of detonation pressure less so. The values are converged to within 5% relative simulation error at resolutions lower than the calibration resolution 4 zone/mm but greater than 0.5 zones/mm. Detonation velocities quickly diverge at coarser zoning. Chapman-Jouguet pressure for the explosions was found to follow a less favorable trend, with unexpected behavior at low zoning. At low resolution, the pressures and velocities were underestimated, with normal convergence behavior found at finer zoning. Simulated detonation velocity versus inverse radius can be seen in Fig. 15 extending to infinite radius detonation velocity already discussed. The general behavior of explosives has been found to follow: D(f) ** ^ where A and rc are fit parameters. A fit to the simulation data yields mm/fxs for D(r), mm±0.009mm for A and 0.29 mm±0.19 mm for rc. Extrapolating to intersection with the inverse radius axis gives a failure diameter (df) of 0.70 mm. This is consistent with the

31 27 experimental value cited in LASL Explosive Property Data, df smaller than 1.52 mm, which is of little certainty, as they never achieved failure with PBX 9501 charges [42]. Experimental fit parameters are D(oo) equal to mm/us, A equal to mm±0.001 mm and rc equal to 0.48 mm±0.02 mm and lie on the edge of uncertainty of the simulation fit parameters. Error between experimental and simulated had a standard deviation of 0,063 mm/ AS, which is equivalent to about 0.7% total velocity. 4. CYLINDER TEST Case expansion velocity as a function of case expansion distance from initial radial position is the primary metric. Measurements were taken at least 9 times the diameter down the axis of the tube from the initiation point, as suggested by Souers et al [47]. Case expansion velocity at 2 zones/mm resolution was used as a benchmark and a screenshot of the test can be seen in Fig. 16. This simulation consisted of 5.3 million grid cells and 12 million particles. During simulation the CJ pressure at the front reached GPa, in good agreement with detonation pressure presented in the previous section. Comparison of simulated data and an experimental data set from Gibbs et al are seen in Figure 17 [42]. Standard deviation between experimental and simulated velocity profiles was between 6.3 and 7.8%. The detonation velocity varied by only about 0.2% and the detonation pressure varied by about only 1.3%. Case expansion velocity varied by a maximum of 3.7% over the range of particle resolutions studied.

32 5. STEVEN TEST In simulation, pressure in the vessel never exceeds 5.3 GPa indicating detonation was not achieved, in good agreement with analysis of experiments where explosive energy releases are much lower than fully formed detonation [30], Experiments show a threshold for large, 1-inch thick targets of approximately m/s with cracking occurring at lower velocities, and target consumption at higher velocities than the threshold. A stress threshold of approximately 250 MPa follows from 75 m/s experimental case for rapid reaction, indicating a lower threshold to dissipation of energy. Figure 18 shows simulated versus experimental go/no-go results. Good agreement was found between DDT1 and experiments, indicating decent burn initiation behavior. Pinducer traces of the experiments qualitatively match simulated pressure profiles at the same point in the explosive as can be seen in Figure 19. This is especially true with regards to timescales of material response, but stress magnitudes differ by as much as three times. Inconsistencies have been attributed to MPMICE's failure to handle negative pressures. Figure 20 shows reaction beginning in the 75 m/s case. 6. SLOW COOK-OFF TEST A five-millimeter thick annularly confined disk of PBX 9501 heated to 573 K was simulated. The experimental setup can be seen in Figure 8. Of particular interest is the existence of detonation, as well as the time to detonation. In the experiments [20], one case was thought to have detonated, due to the luminosity measured. Of interest was the long timescale of reaction prior to detonation. It is estimated from the frames taken in the experiment that the explosive underwent detonation somewhere between 15 and 20

33 29 microseconds. A simulation of the same scenario transitioned to detonation in roughly 25 microseconds, which shows very favorable results with compared with experiments. A frame of the detonated explosive can be seen in Figure 21. Also, the material damage, represented as cracking in simulation, looks similar to that in experiments as evidenced by Figure 22. B. MESOSCALE MODELS 1. SINGLE CRYSTAL EXPERIMENT Comparison of single crystal shock experiments with simulations can be seen in Figure 23. The magnitudes can be seen to be similar, however, due to the fact that an interface was not simulated, no elastic precursor is seen at the points of measurement in experiment. These results can be seen to be generally within about 10% of the value. 2. GRANULAR HMX STRESS GAUGE EXPERIMENTS In granular bed experiments of large diameter HMX crystals, generally good agreement was found between models and experiments. Figure 24 how stress propagates through the bed as well as how the temperature is localized in boundaries and in plastic flow regions. Figures 25 and 26 shows the comparison of stress and velocity profiles respectively. Front gauge measurements are in very good agreement, while back gauge measurements differ in event initiation. An elastic precursor wave appears to traverse the bed too quickly and prematurely cause stress to be felt at the back boundary.

34 3. VARYING DENSITY GRANULAR BEDS Varying the initial density of the explosive was seen to affect both compaction rate and average temperature. Temperature minimum and maximums were similar in each case. Mean temperature through the compaction zone was found to increase with porosity. The standard deviation highlights this trend, as can be seen in Figure 27.

35 31 IV. DISCUSSION A. BULK SCALE MODELS Due to overestimation in bum rates at high temperatures, slow cook-off scenarios will be accelerated. In room temperature scenarios, such as those in which DDT experiments are performed, the agreement in trend is decent over a large pressure range [58]. The agreement is especially good over the range of pressures that surface burning transitions to convective burning in damaged materials which is imperative for the convective burning, crack-size switching criteria. At low zoning per millimeter the trend of increased error in the burn rate could cause a DDT timescales to be skewed due to overestimated burn rate during ignition and subsequent move towards stabilization. An effect that may require the model pressure threshold for transition to detonation to be larger than those experimentally determined [44]. However, as long as the resolution is close to convergence the error is dominated by the temperature dependence. Run distance to detonation is an important metric of shock sensitivity that is generally studied using a flyer plate test [42]. Buildup to detonation is due largely to nucleation and growth of hotspots, as well as other dissipative mechanisms such as frictional heating between cracks and resultant bulk decomposition due to expansion of hot product gasses through cracks; effectively increasing burning surface area and in turn the rate of reaction. Support and strengthening of the lead front results, eventually providing enough available energy for rapid transition to detonation. A pressure threshold of 5.3 GPa was used for transition from deflagration to detonation in simulations, a value that roughly represents the amount of work necessary for rapid reaction in the bulk explosive. The results of the Pop-plot are favorable, however, the

36 32 slope of line is incorrect indicating the density dependence of the model is incorrect. Detonation characteristics where also studied using the flyer-plate tests. The converged CJ pressure is overestimated; however this has already been identified by Menikoff as a consequence of the EOS fits [37]. Particle resolution dependences of the pressure and velocity have opposite behaviors. Therefore, the particle resolution should be limited to between 2 cubed and 4 cubed particles per cell for the reactant. Simulations run must be at least 4 zone/mm resolution for accurate detonation. Size effect curves are indicative of explosive performance as they give an idea of the reaction zone length. Larger reaction zones cause failure on larger length scales (of the order of millimeters to centimeters). On length scales slightly larger than failure, the detonation velocity is greatly reduced due to curvature of the reaction front, essentially; the explosive violence is dampened due to inertial effects in the case of unconfined explosives. The size effect was well represented by the parameters chosen for Equation (4). The difference in infinite radius fit parameters, D(o ), accounts for much of the difference in lit parameters. The first of the validation tests, the cylinder test, gave good agreement, with a few exceptions. Of note is the overestimation of case expansion at low expansion volumes (early times), potentially due to the over estimation in CJ pressure. This is consistent with the overestimation of CJ pressure due to a fitting form that is too stiff for PBX 9501 [37]. The agreement increases with radial distance. Overestimation of explosive violence should be expected in packed arrays of explosives and other sympathetic scenarios. Slow cook-off simulations proved to be qualitatively correct. The favorable comparison of cracking behavior is encouraging, as is the fact that detonation was

37 achieved for the experimental case that was suspected to undergo the transition [20]. Simulations were only performed on the 5-mm, copper confined, thick-disk experiment, and should not be considered conclusive. Differences should be studied with thick and thin confinements, as well as thick and thin disks in order to fully quantify slow cook-off behavior. The Steven test is perhaps the most complicated experiment designed for explosive model validation [30]. Similarly, in simulation, this is the most expensive test, consisting of 66.5 million cells and 15.6 million particles. A lightly confined disk of PBX 9501 is impacted with a round nosed projectile and stress histories are measured. From this test, a velocity threshold is determined for go/no-go behavior where go implies rapid, sustained reaction. Go/no-go is a basic metric for explosive impact assigned based on reaction of the sample. Sometimes another set of terms: nonviolent, semi-violent and violent, are used to describe the test behavior. Cracking and damage to the material are exhibited in all velocity impacts and are not used as the go/no-go criteria. Instead, consumption of large amounts of target material along with melting and scorching are used to assign the go/no-go label. Simulations of the go/no-go threshold was in good agreement within a 10 m/s window, and the stress measurements prior to ignition were shown to be at least qualitatively correct. This indicates that a good resolution was chosen and that the reactant material model adequately represents damage. Furthermore, the fact that the device did not undergo detonation helps affirm that the behavior of our burn model and the transition to detonation pressure are reasonable. Of largest uncertainty in the model is the pressure threshold chosen to transition from fast deflagration to detonation. Thermodynamically, the activation of ITMX

38 34 reaction requires average energy in the range of 140 to 165 kj/mol [12]. Bulk reaction should occur when the entire material within a given volume has exceeded this energy, which will result in fast reaction and large energy buildup. The pressure threshold of 5.3 GPa in this model was chosen for two reasons. First, the threshold was chosen so that the 351 m/s aluminum impact experiment gave a reasonable run distance to detonation. Overestimated burn rate likely increases the threshold pressure above what it might be in reality. Second, a 2003 paper by Esposito et al shows an increase in the pressure dependence (e,g, Vielle s Law and JWL++) of HMX burn rate from to 1.27 at about 5 GPa. WSB has a pressure dependence of approximately 0.9 at low pressures and the pressure dependence of the detonation model calibrated herein show reasonable size effect at a pressure dependence of 1.2, indicating that there is good agreement with this experimental data. A simple mathematical analysis of the energy available under compression to 5.3 GPa leads to further understanding. Using transition state theory and assuming reversible work a value for the energy imparted can be calculated. Using a simple expression P=IC(l/v - v)/2, where v is the relative volume, the bulk modulus, K, is 11.4 GPa and FIMX has an approximate molar volume of 6200 mols/m, one comes to an average energy of compression of kj/mol. Surprisingly this value is close to the experimental activation energies. Using an upper bound of experimental bulk moduli for HMX reported in literature of 15.7 GPa the same analysis leads to a slightly lower value of kj/mol available energy [33]. Since the bulk modulus of PBX 9501 is lower than HMX due to the plasticized binder, the higher energy result first discussed may be more reasonable in light of the energy uptake during plastic flow. As a check we apply

39 35 the same analysis with the shock EOS. Using the JWL equation of state for reactants, which fits the experimental data up to relative volume of 0.8 fairly well, and the aforementioned molar volume, the available energy is much lower, around 45.2 kj/mol. Some drawbacks and uncertainties are associated with using DDT1. Firstly, the calibration of JWL++ rate parameters are required after a factor of 8 coarser zoning, as the reaction zone becomes over-represented and the errors in detonation pressure and velocity are greater than 5%, which seems to be the reasonable value for simulations. Souers et al has already covered this issue in several of their papers [46,48]. The recommended resolution for correct detonation is 2 times the converged resolution, or 0.5 zones/mm for PBX Secondly, a constant value for a pressure threshold is subject to much uncertainty, due in part to the fact that this initiation pressure is dependent on explosive density [44], A third drawback comes from the WSB model. Few explosives parameters are available in literature. Such values as chemical heat release and the specific heat of both gas and solid, and condensed phase activation energy have been identified as the sensitive parameters of the model and thus need to be known to high certainty from experiments [56]. In addition, the time to steady state burning likely has a large effect on transition to detonation. The initial burn rate instability is due to pressure oscillations in the domain. These are inherent to the ignition means and the boundary conditions. Along a similar note, the final drawback is the limited parameters in literature for the ViscoSCRAM model cracking behavior. Recent work in some LX explosives and RX explosives (two PBX formulations from different national laboratories) could allow the parameterization of the mechanical model for of explosives [57], but to the author s

40 36 knowledge no data regarding propagation of surface flames into cracks is available for explosives other than PBX 9501 [56]. Fortunately, flame stand-off and crack penetration are probably much more similar between explosives than burning parameters, and thus the models should be applicable within a quantitative sense, for other explosives. However, lack of complete sets of ID, 2D and 3D experiments for many explosives other than PBX 9501 make validating the model for other explosives difficult. B. MESOSCALE MODELS The use of the SCG model for the HMX is suspect, as it was originally designed to be a material model for metals. The oscillations seen in the flow stress level are likely due to this fact. However, the parameters seem to be give reasonable behavior, especially when put into the granular simulation, as was evidenced by the agreement in Figures 25 and 26. Of largest suspect in these simulations is the early arrival of the wave at the back boundary. Two things contribute to this simulation trace. The first is that the stress traces were taken in the HMX grains instead of the receiving material. This preferentially weights any precursor waves that might have averaged to a lower value when considering the actual surface area of HMX that is contacting the receiving material. Secondly, no cut-off distance was used in the frictional contact algorithm, which could increase the elastic wave s speed through the bed as nearby grains feel the stresses more quickly [3]. Temperature comparisons of the model with other modeling efforts showed similar averages and maximums [5]. Additionally, all features that are expected in granular compaction can be seen in the temperature field demonstrated in Figure 24.

41 These include frictional heating at contact surfaces, heating in the plastic flow region, as well as in fracture paths. Assuming correct behavior, it then becomes interesting that the distribution of temperatures is wider in the low-density case. This can be understood to be due to the decreased mass to shock energy ratio. While there are fewer contact points in the lower packing, once compaction yields to flow, there is a smaller amount of overall mass in which the energy is localized. This supports the general observation that shock initiation pressure decreases with increased porosity as evidenced in a compilation of experimental data [44]. Energy localization, along with the slower compaction wave likely contributes to the observation that run-distance to detonation in DDT of granular beds decreases with increased porosity. Further study is needed in the area to apply trends seen on the mesoscale that may be applied as sub-grid scale models in the bulk scale.

42 38 V, CONCLUSIONS While the goal of macroscale simulations of hazard scenarios has not yet been achieved, great strides have been made. A generalized engineering model capable of deflagration and detonation, and the transition from one to the other in energetic materials at a variety of length scales has been created, and validated against a number of wellposed experiments. These include the cylinder test, the Steven test and the annularly confined slow cook-off test. In addition, models have been implemented that allow simulation of mesoscale simulation scenarios with high fidelity. This creates the potential to validate via simulation macroscale models of experimentally intractable problems posed on the micrometer length scale. However, the road to predictive simulation is long. In order to move to larger length scales, sub-grid scale models based on multiscale modeling results will need to be incorporated to capture enough of the physics in length scales associated with accident scenarios. If the current work is carried forward, a few key steps must be taken on the road to predictive simulation. The first of those steps is the implementation of a reaction model in the mesoscale simulations. The WSB model would likely suffice, and could easily be compared with experimental reactive granular results. The next step would be the inclusion of plastic binder in mesoscale simulations, as well as inclusion of interface debonding models. These would allow simulation of PBXs over a range of length scales. Heterogeneous repeatable scenarios such as a box full of explosive rocket motors (modeled as cylinders) must be modeled via homogenous mixture models with sub-grid scale behaviors inspired by results from resolved mesoscale simulations. Sub-grid scale models will likely need to

43 incorporate surface area effects, permeability effects, and damage effects [40]. Only then can real predictive simulations of accident scenarios be simulated. Finally, it becomes necessary to actually extend the model to work for other explosives for it to truly achieve the goal of being a general engineering model.

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46 Handley, C.A. "THE CREST REACTIVE BURN MODEL." 13th Int. Detonation Symp. Norfolk: Proceedings of 13th Detonation Symposium, Harlow, F.H., Amsden, A. A. "Numerical calculation of almost incompressible flow." J. Comp. Phys. 3, no. 1 (1968): Henderson, B.F., Smilowitz, L., Asay, B.W., Sandstrom, M.M. and Romero, J J. "An Ignition Law For PBX 9501 From Thermal Explosion to Detonation." 13th Int. Symp. on Detonation. Norfolk, Virginia, USA: 13th Int. Symp. on Detonation, Henson, B.F. et al. "A thermal decomposition model of HMX and PBX 9501." JANNAF 22nd Propulsion Safty Hazards Subcommittee Meeting. Charleston, South Carolina: JANNAF, Hill, L.G., and Catanach, R.A. W-76 PBX9501 Cylinder Tests. LA MS, Los Alamos, New Mexico: Los Alamos National Laboratory, 1998, Idar, D.J., Lucht, R.A., Straight, J.W., Scammon, R.J., Browning, R.V., Middleditch, J., Dienes, J.K., Skidmore, C.B. and Buntain, G. "Low Amplitude Insult Project: PBX 9501 High Explosive Violent Reaction Experiments." Proceedings o f the 11th International Detonation Symposium. Snowmass, Colorado: 11th Int. Detonationa Symp., Jouguet, E. "On the propagation of chemical reacitons in gases." J. de Mathematiques Pures et Appliquees 1 (1906): Kober, S.G. Bardenhagen and E.M. "The Generalized Interpolation Material Point Method." Comp Model Eng. Sci. 5, no. 6 (2004): Li, M., Tan, W.J., Kang, B., Xu, R.J., and Tang, W. "The Elastic Modulus of beta- HMX Crystals Determined by Nanoindentation." Propel Explos. Pyrot. 35 (2010): Lin, C.Y., et al. "A shock-tube study of the CIT20+N02 reaction at high temperatures." Int. J, Chem. kinet. 22 (1990): Lin, M.C., et al. "Implications of the HCN YL ITNC process to high-temerature nitrogen-containing fuel chemistry." Int. J. Che,. Kinet. 24 (1992): Menikoff, R. and Shaw, M.S. "Reactive Burn Models and Ignition & Growth Concept." 8th Bienniel Int. Conf. on New Models and Hydrocodes fo r Shock Waves. Paris, France, Menikoff, R. Comparison o f Constitutive Models fo r PBX LA-UR , Las Alamos, New Mexico: Las Alamos National Laboratory, 2006.

47 Menikoff, R. Detonation Waves in P B X LA-UR , Las Alamos: Las Alamos National Laboratory Menikoff, R., and Sewell, T.D. Constituent Properties o f HMX Needed for Meso- Scale Simulations. LA-UR rev, Las Alamos, New Mexico: Las Alamos National Laboratory, 2001, Parker, G.R., and Rae, P.J. Mechanical and Thermal Damage. Vol. 5, in Non-Shock Initiation o f Explosives, by B.W. Asay, Berlin: Springer-Verlag, Peterson, J.R., and Wight, C.A. "An Eulerian-Lagrangian computational model for deflagration and detonation of high explosives." Combustion and Flame, 2012: in Press. 42. Popolato, T.R., and Gibbs, A. LASL Explosive Property Data. Berkeley: University of California Press, Sheffield, S.A., Gustavsen, R.L. and Alcon, R.R. "Shock Initiation Studios of Low Density HMX Using Electromagnetic Particle Velocity and PVDF Stress Gauges." 10th Int. Detonation Symp. Boston, Massachusetts: 10th Int. Detonation Symp., Souers, P.C., and Vitello, P. "Initiation Pressure Thresholds from Three Sources." Propell. Explos. Pyrot. 32, no. 4 (2007): Souers, P.C., Anderson, S., McGuire, E., Murphy, M.J., Vitello, P. "Reactive Flow and the Size Effect." Propel. Explos. Pyrot. 26, no. 1 (2001): Souers, P.C., Anderson, S., Mercer, J., McGuire, E., and Vitello, P. Propell. Explos. Pryot. 25 (2000): Souers, P.C., Forbes, J.W., Fried, L.E., Howard, W.H., Anderson, S., Dawson, S. "Detonation Energies from the Cylinder Test and CHEETAH V3.0." Propell Explos. Pyrot. 26 (2001): Souers, P.C., Garza, R., and Vitello, P. "Ignition & Growth and JWL++ Detonation Models in Coarse Zones." Propell Explos. Pyro. 27 (2002): Springer, H.K., Glascoe, E.A., Reaugh, J.E., Kercher, J.R., and Maienschein, J.L. "Mesoscale Modeling of Deflagration-Induced Deconsolidation in Polymer-Bonded Explosives." 17th APS SCCM Conference. Chicago, Illinois: APS, Stewart, D.S., and Bdzil, J.B. "The shock dynamics of stable multidimensional detonation." Combust, and Flame 72 (1988):

48 Sulsky, D., Zhou, S.-Jian9and Schreyer, H.L. "Application of a particle-in-cell method to solid mechanics." Computer Physics Communications 87, no. 1-2 (1995): Tarver, E.L. Lee and C.M. "Phenomenological model of shock initiation in heterogeneous explosives." Phys. Fluids 23 (1980): Twu, C.H., Tassone, V., Sim, W.D. and Watanasiri, S. "Advanced equation of state method for modeling TEG-water for glycol gas dehydration." Fluid Phase Equilibr (2005): Vandersall, K.S., Tarver, C.M., Garcia, F., and Chidester, S.K. "On the low pressure shock initiation of octahydro-1,3,5,7-tegranitro-1,3,5,7-tetrazocine based plastic bonded explosives." J. Appl. Phys. 107, no (2010). 55. von Neumann, J. Theory of detonaiton waves. Progress Report to the National Defense Research Committee Div. B. OSRD-549, Vol. 6, in John von Neumann: Collected Works, , by A.H, Taub, New York: Pergamon Press, Ward, M.J., Son, S.F., and Brewster, M.Q. "Steady Deflagration of HMX With Simple Kinetics: A Gas Phase Chain Reaction Model." Combustion and Flame 114 (1998): Weese, R.K., Burnham, A.K., Turner, H.C., and Tran, T. "Physical Characterization of RX-55-AE-5A formulations of 97.5% 2,6-diamino-3,5-dinitropyrazine-l-oxide (LLM 105) and 2.5% Viton A." JOWOG. Reading, United Kingdom, Wiegand, D.A. "The influence of confinement on the mechanical properties of energetic materials." Edited by M.D., Chabildas, L.C., and Hixson, R.S. Furnish. Shock Compression o f Condensed Matter (APS), 1999: Wight, C.A., and Eddings, E.G. "SCIENCE-BASED SIMULATION TOOLS FOR HAZARD ASSESSMENT AND MITIGATION." Advancements in Energetic Materials and Chemical Propulsion 114 (2008): Witworth, N. Mathematical and Numerical Modelling o f Shock Initiation in Heterogeneous Solid Explosives. PhD Thesis, ENGINEERING SYSTEMS DEPARTMENT, Cranfield: CRANFIELD UNIVERSITY DEFENCE COLLEGE OF MANAGEMENT AND TECHNOLOGY, Zel'dovich, Ya.B. "On the theory of deflagration and detonations of gaseous system." Zh. Eksp. Teor. Fiz. 10 (1940):

49 45 APPENDIX A. FIGURES Figure 1. A temperature profile through the reaction zone in deflagration of a solid explosive is depicted. Between the initial temperature, To, and the melt temperature, Tm, is an exponential temperature profile. In the melt (also called foam zone) the temperature increases slightly to the surface temperature, Ts. Beyond this exists a zone of endothermic reaction where the N-N bond cleavage occurs and reactive radicals are formed [28]. Exothermic reaction increases, and near the luminous flame, which begins roughly at the stand-off distance, Xg, the reaction tends to completion.

50 PETN RDX T etryl OpN> N 0I N+ N- \ r \ /' N+ N N---N+ / \ TATB H M X Figure 2. Chemical structures of common high explosives are depicted. The common motif of fuel and oxidizer built into the same molecule can be seen in the high oxygen to carbon ratios.

51 47 A. C4N80 8H ^ 4(CH2NN02) il u 4HONO + 4HCN CH20 + N20 B. c h 2o + n o 2 n 2o + h _ HONO HCO + HONO - * - N2 + OH >- NO + OH C. H CN HNC HNC + O _ ^ NCO + H NO + N N2 + O NO + NH n + OH Figure 3. Commonly cited main decomposition steps for HMX are depicted [11,28]. A. Solid phase reactions. B. Endothermic gas phase reactions. C. Exothermic gas phase reactions. The rate-limiting step is thought to be the isomerization of HCN to HNC [34].

52 48 Figure 4. A representation in the pressure-volume plane of the equilibrium state of reaction in both the deflagration and detonation regimes is depicted. The lower and upper curves represent the locus of states for the Hugoniots of the reactants and products respectively. The line tangent to the each product curve is known as the upper Rayleigh line and represents the equilibrium state for reaction. The CJ point and the VN point are both marked 0 11 the Rayleigh line. The zones denoted in numerals are: I) strong detonation, II) weak detonation, III) forbidden zone, IV) weak deflagration, and V) strong deflagration. Of particular note is the forbidden zone, which has a nonphysical solution to the entropy constraint. V

53 M, W o Vo v. Po P, To T, Figure 5. A diagram showing a shock wave from the material frame of reference. The variables M, v, P and T are the mass, specific volume, pressure and temperature of the material in the given state. The 0 and 1 denote material states upstream and downstream of the shock wave. The shock wave is shown with velocity Us.

54 j-« or 90 mm p - 10 mm Impactor ^ W /////////////M Gas N <l> 1 «- 5 mm Cover Plate N Gas S i S S R J N 1 S Gas a xR Figure 6. Schematics of the three simulation setups run in code. The top shows the strand burner simulation setup. In the middle is a ID simulation of an explosive cylinder either 45 or 90 mm in length. On top a 5 mm thick cover plate provides inertial confinement (experiments also have a 5 mm bottom cover plate that was not included in the simulations). A 10 mm aluminum impactor contacts the cover plate at the beginning of the simulation with ail initial velocity of interest. The bottom simulation is used for the rate stick test, and is the same as the cylinder test minus the confinement. S denotes a symmetry boundary and N 5denotes a Neumann boundary.

55 inra- N _25.4 mm_ HMX 5 mm Cu Confinement Figure 7. Validation experiments schematics are depicted. The top simulation setup is that for the Cylinder test, where a copper confinement sits around a cylinder with a 1-inch radius. The middle shows the Steven test with the steel impactor. The third shows the setup for the annularly confined slow cook-off test. S denotes a symmetry boundary and N* denotes a Neumann boundary.

56 52 j N Gas Kel-F o o o o o o o o o o o o o o o o o o o o oz. o "o o o o o o o o o o o o o o o 0o0o 0n0n o o o o o G r a n u la n 0 o0 o o o o'h M X >gogo o o o o o _ O o _ O o O o o o o o o o o o o o o^o o o):o o O O. O O..Q. Q - Q. Q :TPX N Figure 8. Granular compaction schematic is depicted. The granular bed was about 4 mm thick and the TPX and Kel-F 800 inert materials varied in thickness depending on the simulation. S denotes a symmetry boundary and N denotes a Neumann boundary.

57 53 10 >*#A A -A j * i v> E & 03 cd t r c 3 QQ $ A f M Exp. 273 K Exp. 298 K 1 & ' Exp. 423 K -...*... < Sim. 273 K «... Sim. 298 K Sim, 423 K... *... A...,f,... I... *».,,li.,l, A Pressure (MPa) Figure 9. Simulated surface regression rates compared with experimental values. Error bars are those assigned by Atwood et al [1]. Note the range of good agreement are those typical of convective burning. This suggests application of the model to those scenarios is reasonable. Values for both experimental and simulations at high and low temperatures have been multiplied and divided by 4 respectively, for clarity.

58 w 5 EC c 3 CO MPa Experimental 5.52 MPa Experimental 6.21 MPa Experimental MPa Experimental 0,3...fi I L Bulk Temperature (K) f!#. Figure 10. Simulated temperature dependence is shown by connected points. Error bars are those assigned by Atwood et al. [1]. Note that too high a bulk temperature will generally cause bum rate to be overestimated in simulation, meaning that cook-off transition to detonation occurs too early.

59 g 6.2 w S 8.0 OJ E " ' Zones/mm Figure 11. Burn rates showing convergence at finer resolution. The bum rate converges to an error near 6.5% larger than the experimental value. On the right axis is shown the time to steady state burning for the different resolutions. The right axis is plotted on a log scale.

60 56 E E C o t3 cḏj-j CP Q O t Vandersall/Ghidester Experiments Other Experimental o 0 'V X : $s, N ' \ \ i Simulated RTD LASL 1833 kg/mj? Fit LASL1844 kg/m Fit C 05 Vi b c V s. < \:.....Xvo DC Pressure (GPa) Figure 12. Pop plot for simulation of g/cc PBX Note general trend is close to experimental results over a wide range of pressures. Experimental data comes from Vandersall et al. [54]. Other experimental data and fits from Gibbs et al and Gustavsen et al. [42, 24].

61 57 sd O"'' e? LU Q 4 > 0 Du Particle Density Figure 13. Particle density dependence of CJ pressure and detonation velocity depicted as a function of particle density. Particle resolution was varied from 8 to 216 particles per cell. Both axes are logarithmic scale.

62 58 Detonation Velocity (mm/ ns) Zones/mm Figure 14. Cell Resolution dependence of the detonation velocity shows that 2 to 4 zones/mm are close to the edge of convergence for the JWL++ model.

63 59 Detonation Velocity {mm/jis) Os* Experimental Data Fit to Experimental Data - o Simulated Fit to Simulated Data , /R (mm"1) Figure 15. Simulated and experimental size effect curves for PBX 9501 plotted against inverse radius. Data are from the Las Alamos Explosive Property manuscript [42].

64 60 Figure 16. An image from the simulation of the 1-inch Cylinder test shows case expansion in gray. Pressure is cutoff below 1 GPa with values represented by the scale bar. The curved reaction front can also be seen, at the interface between MPM reactant material and ICE product material.

65 61 JL E E oo 0) > Experimental Simulated Expansion Radius (mm) 30 Figure 17. Case expansion comparing simulated and experimental data [42]. Agreement is good for large case expansion, while overestimated near beginning of expansion due to the stiff EOS fit.

66 62 gexperimental hhevr e v r 1 r Simulated HEVR...* Reaction ' I*#*# WW WV*. ItHH WftifM* **W«t«w Xkttf WWHm WW SWK nt# #TJKws^ff HfJfi MCfl*WWlNLp m i i i Velocity (m/s) Figure 18. Go/no-go simulated and experimental results. Violent reaction as defined is used to determine a go and represented by a 1 in the plot.

67 Time (jis) Figure 19. Comparison of stress profiles shows similar time scales of pinducer stress profiles, although they may deviate in magnitude by as much as three times and negative pressures are not represented. Experimental traces are from Idar et al. [30].

68 Figure 20. A steel enclosure, with a hockey puck shaped cavity containing a disk of PBX 9501, is impacted by a round nosed steel projectile at 75 m/s. Reaction begins at roughly a 45-degree angle from the centerline, and grows as a set of reacting pockets or hot

69 65 Figure 21. A frame from the simulation of the annularly confined explosive puck postdetonation.

70 66 Figure 22. A comparison of cracking between simulations shown on the right, and experiments shown on the left is depicted. Similar cracking behavior is seen, albeit, the cracking has a more random nature in the experiments. Extent of damage in the simulated explosion is colored with the warmness of the color. The experimental images are from Dickson et al. [20]

71 67 SCG HMX Model Validation Time (&) Exp. Shot 1067 Sim* Shot 1180 Sim. Shot Exp, Shot 1182 Exp. Shot Sim. Shot 1182 Figure 23. Comparison of single crystal impact experiments with SCG model parameters for HMX give good agreement in magnitude of the wave. Too much oscillatory behavior can be seen, but does not affect the overall error significantly. Experimental data are from single crystal experiments [19].

72 68 Figure 24. A representative simulation of granular compaction of the simulation setup used for granular compaction validation showing impact at 288 m/s. The color scales counter clock-wise from the top are stress of the Kel-F 800 impactor, stress of the FIMX grains and temperature of the HMX. The temperature scale is capped to the ignition temperature of FIMX to show the general trend. The compaction front is denoted by C and shows the stress fingers due to bridging of stress across grains. Following this