dislocation mechanics aspects of energetic material composites

13Dislocation mechanics aspects of energetic material composites

© 2009 Advanced Study Center Co. Ltd.

Rev.Adv.Mater.Sci. 19(2009) 13-40

Corresponding author: R.W. Armstrong, e-mail: [email protected]

DISLOCATION MECHANICS ASPECTS OF ENERGETICMATERIAL COMPOSITES

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�� University of Maryland,College Park, MD 20742, U.S.A.

Received: November 19, 2008

Abstract. The dislocation mechanics based properties of solid energetic materials, particularly, ofhigh explosives, are of particular interest in connection with issues of intrinsic chemical stabilityand with their fast chemical decomposition when employed as propellants or in explosive formula-tions. The ballistic impact and shock-associated plasticity responses of such materials presentgreat experimental and model challenges for establishment of predictable performances. As dem-onstrated in the present report, much has been learned through direct investigation with a fullrange of scientific tools of the individual crystal and composite material properties and, also, throughtheir comparison with relevant inert ionic and metallic material behaviors. Thus, in relation to othersolid material structures, energetic crystals are elastically compliant, plastically strong, and proneto cleavage fracture. Somewhat surprisingly perhaps for such materials, individual dislocationself-energies are indicated to be relatively large while the intrinsic crystal-determined dislocationmobility is restricted because of the complicated and rather dense molecular packing of awk-wardly-shaped molecules that are self-organized within lower-symmetry crystal structures. Be-cause crack surface energies are low, cleavage is able to be initiated by relatively small dislocationpile-ups and, with the restricted dislocation mobility, there is little additional plastic work require-ment associated with cleavage crack propagation. Nevertheless, when compared with indentationfracture mechanics prediction, crack propagation appears to be controlled by the behavior of verylimited dislocation activity at the crack tip. Adiabatic heating associated with dislocation pile-upavalanches provides an important mechanism for the thermal “hot spot” model of explaining theinitiation of rapid chemical decompositions and relates directly to predicted influence of crystal (orparticle) sizes. On such basis, desired characteristics of greater mechanical insensitivity to initia-tion but afterwards greater power dissipation are predicted to occur for energetic composite formu-lations made from smaller particle-sized ingredients.

1. INTRODUCTION

Solid energetic composite materials are includedwithin the category of advanced materials, first,because complete information remains to be de-termined both for the individual component behav-iors and for their interactions in relatively complexformulations; and, secondly, because there is theneed for understanding of the composite perfor-mances, for example, under mechanics-based highrate loading conditions involving very substantial

rates of energy decompositions occurring within ashock wave associated environment. Beginningfrom the late 19th century until the present time,much chemical- and physics-based information hasbeen gained on the component and formulatedenergetic material properties and uses. More re-cently, additional information has been gainedthrough a material science approach of establish-ing microstructurally-based relations between thematerial compositions and their cumulative prop-erties [1-4]. The multifunctional topic of reference

14 R.W. Armstrong

Fig. 1. Schematic picture of structural elements associated with propellant combustion.

[4] relates, in part, to the possible combination ofenergetic and structural load-bearing properties ofa composite material being built into a single de-sign package. As will be seen in the present re-port, the possibility is based in part on the achieve-ment of advantageous properties on both consid-erations at nanometric dimensions of the compos-ite ingredients [5]. In that regard, it’s interesting thatresearches on the energetic materials themselvesare often intertwined with studies of their detona-tion influences on the dynamic deformation prop-erties of structural materials [6]. Such connection

with the high strain rate properties of metals hasbeen recently reviewed [7]. An historical note inthat regard comes from B. Hopkinson, son ofJ. Hopkinson of split Hopkinson pressure bar(SHPB) fame, in contributing an early article on thetopic of measuring the pressures generated in thedetonation of high explosives or by the impact ofbullets [8].

The occurrence of local thermal “hot spots” pro-vides a historically-established mechanism for ex-plaining the initiation of fast chemical decomposi-tion within the various energetic materials; see, for


Fig. 2. Scanning electron microscope image ofcyclotrimethylenetrinitramine (RDX) crystals.

example, reference [9]. The task of tracking suchproposed hot spots is made more difficult by hav-ing to ferret through the complicated environmentof energetic crystal responses occurring within atypical propellant formulation, as schematicallydepicted in Fig. 1, or within the more heavily filledconstitution of an explosive formulation; see forexample references [10-12], including in the lastreference the formulation of a reactive metal sys-tem. And, for whatever energetic composite de-scription is of interest, including that of a moreheavily-filled plastically-bonded explosive (PBX),the decomposition behaviors, whether from directthermal stimulus or under mechanical loading atthe highest applied rates, including shock waveloading, are connected generally with variouslymodified Arrhenius-type thermal activation equa-tion descriptions for either the measured or pre-dicted temperature dependencies of the hot spotignitions and their growth [13-20]. A principal con-cern of the present report is to relate to such con-siderations on two counts: (1) a dislocation defectrole in providing localized sources for potential hotspot development; and, (2) the action of disloca-tion pile-up avalanches under mechanical loadingin providing an important mechanism for initiationof fast chemical decomposition [21]. On this basis,special importance is attributed both to the role ofdislocations in the original production of an ener-getic crystal as well as to the subsequent defor-mation-induced crystal responses.

2. CRYSTAL INGREDIENTS/STRUCTURES

There is an important role of structural chemistryin understanding the nature of energetic crystalproperties. Fig. 3 shows the crystal unit cell struc-ture of cyclotrimethylenetrinitramine that is knownby the substitute designation RDX and has an indi-vidual molecular formula [CH

2·N·NO

2]3. The brack-

eted manner of specifying the formula relates tothe configuration of the molecule. Eight such RDXmolecules, with individual covalent bonding be-tween the carbon, C, hydrogen, H, nitrogen, N, andoxygen, O, elements, are relatively closely packedthen within a molecularly-bonded orthorhombic unitcell with a Pbca crystallographic space group des-ignation. The crystal lattice parameters are a = 1.3182 nm, b = 1.1574 nm, and c = 1.0709 nm sohaving a unit cell volume of ~ 1.63 nm3 containing168 atoms [20]. Identification in the reported struc-tural analysis of any smaller length for a typicalcovalent intramolecular bond distance, for example,for a labeled C3 – H5 bond length of 0.11 nm maybe compared with a larger intermolecular bond dis-tance, say, O1 – H2 of 0.25 nm length.

Two other energetic crystal materials to be es-pecially referenced in the present article are: chemi-cally-related to RDX, tetramethylenetetranitramine(HMX), [CH

2·N·NO

2]

4; and, pentaerythritol

tetranitrate (PETN), [C·(CH2·O·NO

2)

4]. The

Arrhenius-based hot spot model assessment men-tioned above [9] lists RDX as being relatively stablethermally when compared to HMX and PETN that,in turn, are individually about equally less so butwith PETN showing a stronger dependence of itscritical hot spot size on temperature. The energeticcrystal triaminotrinitrobenzine, (TATB),[(NH

2)·C·C·(NO

2)]

3, that is relatively more stable

mechanically in comparison to RDX, is an impor-tant ingredient of certain PBX formulations. Addi-tional newer energetic crystals whose propertiesare being researched are CL-20, 2[C·N·NO

2]3, see

[23], and FOX-7, [(NH2) ·C·C·(NO

2)

2], see [24,25].

Research continues on the design and synthesisof new energetic crystal structures and theirthermo-mechanical response on the molecularscale [26-29].

Included among a number of other energeticcomposite ingredients whose properties are of in-terest, as to be described here, are so-called oxi-dizer crystals such as ammonium perchlorate (AP),[NH

4·Cl

4], and the newer ingredient, ammonium

dinitramide (ADN), [NH4·N·(NO

2)

2]. Aluminum, in

polycrystalline form, is also an important additional

16 R.W. Armstrong

Fig. 3. RDX orthorhombic unit cell.

composite ingredient for energy release by oxida-tion. Recent research effort has been directed tothe increased burning rate achieved for aluminumat nanoparticle sizes [30,31]. For comparison withthe larger RDX unit cell volume, the well-knownaluminum face-centered cubic (fcc) unit cell vol-ume is ~ 0.064 nm3 and contains four atoms. Onbalance, the mixture of covalent and molecularbonding among the elements in the RDX unit cellproduces a lower material density of 1.8 g/cm3 ascompared with 2.7 g/cm3 for aluminum. In general,concentrated mixtures of formulated ingredientsare cast or pressed within a polymer binder matrixsuch as indicated in Fig. 1. The figure presents aschematic representation of the range in dimen-sional scales from the macroscopic level to themolecular scale within a hypothetical propellantmaterial. As shown at the modeled burning inter-face of the polygonal energetic crystal in the fig-ure, chemical decomposition occurs after meltingensues; and, this is experimentally demonstratedrecently over a full range of compositions measuredin the determination of an HMX-RDX phase dia-gram [32].

3. CRYSTAL CHARACTERIZATIONS

The energetic crystals themselves are generallyproduced by growth from supersaturated solventsolutions [33,34]. Fig. 3 shows an example batchof RDX crystals produced in the laboratory after asecond crystallization from acetone solution of re-dissolved first-crystallized material [35]. The solu-tion growth process is a standard method of crys-tal production, for example, as has been employedalso in the obtainment of sucrose (sugar) crystalswhose hardness properties were investigated as acomparative inert, or possibly energetic, materialitself [36]. With regard to the later description to begiven of the compaction properties of energeticmaterial particle systems, mention is made thatsucrose is also a model crystalline material that isemployed in comparative pharmaceutical powdercompaction studies.

A particular advantage of the solution-growthmethod is that different crystal morphologies maybe produced with different solvents. The techniquedoesn’t easily lend itself to the production of ultrafinecrystal sizes except possibly by rapid expansion ofsupercritical solutions [37]. Other crystal growthmethods of achieving finer-scale energetic crystalsizes are being investigated, for example, of mem-brane-assisted centrifugal crystallization or ofevaporative crystallization [38]. Otherwise, finer

crystal sizes are achieved by mechanical commi-nution under protective environmental conditions[39]. And powder X-ray diffraction [40] is employedin most cases to evaluate the produced crystals sofar as internal (dislocation) strains and porosity (alsoassessed through accurate density determinations)are important variables to be correlated with thecrystal performances [41]. Association of voids andmechanically-induced hot spots is a long-standingresearch topic as exemplified in a recent report ofnecessarily ultra-small angle X-ray scattering be-ing employed to monitor pore size distributions inTATB-based PBX-9502 materials and related for-mulations after being subjected to temperaturecycling [42].

For research laboratory purposes of trackingindividual or dislocation group characteristics, etch-pitting and X-ray diffraction topography methodshave proved to be useful [21]. Electron transmis-sion microscopy is generally not a useful methodbecause of causing chemical decompositions. Figs.4a – 4d show two constructions of modeled crystalgrowth configurations [43] and their match with op-tically etched [44] and X-ray (transmission) topo-graphic measurements [45,46], respectively, ob-tained for sectioned RDX crystal specimens. Theoptically-identified traces of different {111} and {102}planes intersecting the viewed (001) planer sur-face of the RDX crystal in Fig. 4c relate directly tothe crystal habit planes shown in Fig. 2. For ex-ample, the tabular crystal shown near the bottom-center of Fig. 2 exhibits four pyramidal planes of{111} type meeting at the planar (001) top facet


Fig. 4. RDX crystal growth sector and dislocation line structures.

and there is indication of a {102}-type facet traceat the top left crystal corner. The same crystal ge-ometry is indicated for the edge-on view of thelarger tabular crystal shown in the top-left cornerof Fig. 2. Thus the identified planar traces in Fig.4c are indicative of the sequential surface positionsoccupied by the RDX crystal during its growth andare distinguished apparently in the crystal etchingprocess of Fig. 4c because of extremely small varia-tions in solute concentrations incorporated withinthe crystal during growth.

Very often, the initial crystal nucleation is knownto occur on an inclusion or intentional “seed” par-ticle, as indicated schematically in Fig. 4a. Fig. 4cgives such indication within the RDX crystal. Theinternal strains sometimes generated by such seed-ing mechanism have been imaged in several RDXX-ray topographic images [47]. Also of interest in

relation to the dislocation line structures shown inFig. 4d are the crystal defect indications exhibitedat the top surface and corner of the larger tabularcrystal viewed edge-on in Fig. 2. The steeply in-clined (001) planar top is seen to involve a partialtwist about its normal [001] direction that must cor-respond to an internal defect with displacementvector along the same [001] direction, that is thesmallest repeat distance for a dislocation Burgersvector in the crystal lattice. Generally but not al-ways, such smaller vectors are associated with thedefect line characteristics modeled in Fig. 4b fordislocation-associated crystal growth. The mis-match of growth facets at one corner of the largercrystal is seen to be associated with the formationof another crystal appendage. Other X-ray topo-graphic results reported for dislocation distributionswithin solution-grown RDX crystals have indicated

18 R.W. Armstrong

Fig. 5. Intermolecular blockages for shear in a [100] direction across the (02-1) plane in RDX.

that the crystals contain relatively low dislocationdensities and have a variety of dislocation line di-rections and Burgers vectors [48,49], perhaps notunexpected for a relative weakly-bonded molecu-lar crystal structure.

4. EXPERIMENTAL MECHANICS

The relative brittleness of energetic crystals hasbeen a main factor leading to the use of indenta-tion hardness testing as a method of evaluatingtheir plastic deformation behaviors and, likewise,has led to the use of indentation fracture mechan-ics (IFM) as a method of assessing the materialcracking behaviors. The susceptibility of the mate-rials to initiation of fast chemical decompositionunder impact has been established in laboratorytests by use of calibrated drop-weight tests, gen-erally, of loose piles of the crystals. Higher loading

rate shock wave testing, via explosive loading oraccelerated plate impacts or by means of impact-ing with laser beams, has been applied to crystalswhile also their decompositions are monitored witha variety of physico-chemical test diagnostics; seefor example [50,51]. The material deformations andcracking behaviors when subjected to granularcompaction under controlled conditions are alsoof interest for both research and manufacturingpurposes.

4.1. Indentation hardness properties

In laboratory experiments, dislocation etch-pittingapplied to microindentation hardness impressionsput into RDX crystal surfaces led to the observa-tion of very limited spreading of the dislocationshaving occurred in comparison with the well-es-tablished, more extensive, patterns of strain ro-


Fig. 6. Hardness stress – strain description for elastic, plastic, and cracking behaviors.

settes observed at etched microindentations in LiF[52]. Also, the observation of slip on {021} planeswas identified at the RDX indentation sites [53] inextension of other pioneering etch pit observations[33]. The {021} slip system result was confirmed inaccompaniment of the reported X-ray topographicresults mentioned above [49] and was included ina further analysis of operative slip systems at RDXhardness indentations [54]. On the molecular lat-tice scale, mutual blockages to slip across the {021}planes in an [001] slip direction were shown to becaused by interactions of out-cropping molecularappendages, as modeled for RDX in Fig. 5 at thesites of the partially labeled atomic elements. Theinteractions were associated with reported nitroso-compound formations that have been detected indrop-weight impacted crystals tested just below theinitiation drop-height [55] and also were detectedin combustion residues [56]. Thus, the etch-pitting

observations made on RDX crystals provided a firstindication of dislocation displacements being hin-dered in a particular manner characteristic of com-plex molecular interactions occurring within therather more complicated energetic crystal lattices.

Fig. 6 shows a hardness stress-strain methodof analysis that provides for further comparison ofelastic, plastic and cracking behaviors among ener-getic and other material crystal types [57]. In thefigure, the plastic hardness stress, σ

H, is specified

for a ball indentation as the indenter load, W, di-vided by the surface projected contact area of theresidual indentation having a real or effective cra-ter diameter, d. The hardness strain is the indenta-tion diameter divided by the ball diameter, D. Thedashed lines sloping steeply upwards in Fig. 6 fromthe lower left edges are Hertzian elastic line de-pendencies for which the d values are taken asthe elastic contact diameter in the expression

20 R.W. Armstrong

σ πE b b

s s

v E

v E d D

= − +

−−

4 3 1

1

2

2 1

/ /

/ / .

� � � ��

� �� (1)

In Eq. (1), vb, E

b and v

s, E

s are the Poisson’s

ratio and Young’s modulus for the ball and speci-men, respectively. The reciprocal factor in squarebrackets on the right side of Eq. (1) is normallyreplaced by an effective modulus, Er. Separate fromthe elastic predictions, a number of crystal plastichardness values are shown in the figure as filledcircle points vertically distributed along the ordinatescale at an effective value of (d/D) = 0.375, as de-termined for diamond pyramid or Vickers hardness(test) numbers (VHN). The crystal materials thatare listed for these hardness values include, inaddition to the crystals identified thus far, MgO,NaCl, ice, and anthracene. As shown in the figure,an increasing hardness is determined for the ma-terials of present interest in the order of AP, PETN,RDX, and HMX. For NaCl, the solid elastic loadingline and connected plastic flow curve were deter-mined in a continuous loading test. Such elasticloading behavior is more easily demonstrated inmodern nanoindentation hardness tests, for ex-ample, as reported for sucrose crystals that wereinvestigated for comparison with energetic crystalproperties [58]. The series of open square pointsshown for an RDX crystal were measured bymeans of a microindentation hardness tester fittedwith a 1.59 mm diameter steel ball indenter.

In Fig. 6, the open circle points specified at par-ticular D values along the dashed Hertzian linesare modeled hardness stresses for cracking, σ

C,

computed in accordance with the indentation frac-ture mechanics (IFM) expression:

σ γ π

κ κ

C s sE D v

d D

= − ⋅

+ −

4 1 2

1

2

2

21 2 1 2

/

/ ,/ /

� ��

� � � � � (2)

in which γ is the true surface energy and the nu-merical factor (κ

12 + κ

22) = 2.5·10-5 was taken as

originally reported [59]. The highest average hard-ness (closed circle) point in Fig. 6 was determinedfor a number of MgO crystal indentations made on(001) plane surfaces. The D = 0.124 mm label atthe highest value of σ

C on the dashed MgO elastic

loading line was determined as the equivalent (av-erage) ball size for the actual diamond pyramidindentation measurements that are shown. At suchindentations on (001) surfaces of MgO crystals,dislocation pile-ups are known to produce distinc-tive, crystallographically-defined, cracking patterns[60]. Very importantly, the dislocation pile-ups are

shown in the figure to initiate cracking at a signifi-cantly lower hardness stress than that predictedon an IFM basis and, even so, the cracking occurson {110} cleavage planes that require a higher sur-face energy than the otherwise favored (001)planes. The lowest σ

C value marked at D = 6.35

mm on the MgO Hertzian line, nevertheless, dem-onstrates the reduction in predicted IFM crackingstress occurring in accordance with the inversesquare root of D factor in the coefficient of the (d/D)-1/2 strain dependence in Eq. (2).

The cracking specified to occur for RDX at thelarger strain, open square, measurements in Fig.6 are also positioned below the predicted σ

C value

for a D = 1.59 mm ball as designated at the termi-nal point of the Hertzian elastic line. Here too, themeasured hardness for cracking is lower becauseof a same dislocation pile-up-induced cracking rea-son although very probably needing only a smallnumber of dislocations in the pile-ups. Thus, thedifference in RDX stress levels between the crack-ing-associated plastic hardness and the predictedIFM-determined cracking stress is shown to be sig-nificantly smaller than that shown for the corre-sponding separation of stresses for NaCl, which isnot known to be an exceptionally ductile material.Such observation is indicative of a lesser range instress being available in RDX for any influence ofdynamic loading. A further comparison of the RDXand NaCl hardness results shows that NaCl is elas-tically stiffer, as expected on the basis of its stron-ger ionic bonding, while RDX is plastically stron-ger, because of its lattice hindrance to slip and,also, is more cleavage prone because of its rela-tively lower surface energy [21]. Other comparisonshave been made between RDX and MgO crystalsin terms of an IFM predicted c3/2 crack size depen-dence of σ

H, or c2 dependence in accordance with

cracking being controlled by the hardness value[61]. The indication for RDX of the measurementsfitting the latter dependence is in agreement withcrack growth being controlled by the limited extentof plasticity occurring at the crack tip. The sameconclusion had been reached in an earlier assess-ment of the IFM measurements reported for theRDX cracking behavior [62].

Additional evidence for plastic flow being a prin-cipal cause of crack formation in both energeticand ionic crystals was demonstrated for plasticity-induced cracking observations made for diamondpyramid hardness indentations put into AP crystalsurfaces [63,64]. The AP results in [63] includedX-ray topographic evidence, analogous to the casefor {110} cracking in MgO, of substantial internal


Fig. 7. Schematic drop-weight impact test of RDX crystals and hot spot generations.

lattice strains associated with locally-identified slipband stress concentrations that produced crack-ing. In [63], a hardness measurement of 265 MPawas obtained for the newer ADN oxidizer crystal.The value is greater than that for AP shown in Fig.6 and is nearer to the level shown for the RDXmeasurements. In [64], strikingly imaged terracedstructures are shown at dislocation etch pits im-aged by atomic force microscopy on the (210) sur-face of an AP crystal. The terrace structures arevery likely a finer scale manifestation of the lay-ered-type growth structure features revealed in Fig.4c for an RDX crystal [44].

4.2. Impact properties

A schematic representation of a drop-weight im-pact test is shown in Fig. 7 as applied to measur-ing the impact sensitivity of loose piles of energetic

crystals. The figure includes a sample view of RDXcrystals and, also, a model interpretation of crystallattice shearing for development of hot spots lead-ing to initiation of the impacted crystals [65]; notethe 1-2 mm size of the crystals in Fig. 7 comparedto the ~50 micron-sized crystals shown in Fig. 2.More detailed descriptions of drop-weight impacttesting machines have been reported [66,67]; inthe first case involving optical recording of thematerial deformation and cracking behaviors and,in the second case, also including results obtainedon a number of explosive materials tested in a bal-listic impact chamber (BIC). Fig. 8 shows a compi-lation of early results reported for the drop-weightimpact height for 50% probability of initiation, H50,as a function of the average size of individual crys-tals within the pile [68,69]. As observed for the dataincluded within the limits of the dashed inset in the

22 R.W. Armstrong

Fig. 8. Crystal size dependence for initiation of RDX and CL-12 in drop-weight impact tests.

figure, a strong dependence of H50

on crystal sizeis obtained; for example, an order of magnitudeincrease in H

50 from 10 to 100 cm is produced for

a crystal average size reduction from 1.0 to 0.01mm, while also indicating a reciprocal square rootdependence of H

50 on the crystal size. The larger

dimensional extension of the graph comprising Fig.8 was constructed for connection with current re-search emphasis on measuring energetic crystalproperties at nanometric crystal sizes. Determina-tion of a drop-weight height in excess of 100 cmhas been reported for RDX crystal diameters inthe range of 110 to 300 nm, as referenced aboveto be obtained by the method of rapid expansion ofa supercritical solution of RDX in carbon dioxide[37].

Other methods of researching the impact-initi-ated properties of formulated composite materialsinclude: Taylor-type cylinder impact tests [70], as

employed for characterizing the strength proper-ties of structural metals and alloys; split-Hopkinsonpressure bar (SHPB) or hybrid-SHPB tests [71];ballistic penetrations [72], as mentioned above forthe historical article [8] of Hopkinson; and, espe-cially, shock-induced impact tests achieved by ex-plosive detonation-type gap tests or gas-gun- orlaser-launched projectiles [73]. For example, thelast-mentioned testing procedures of gap tests andlaser launched flyer plates were applied to evalua-tion of the shock sensitivity for PETN crystals of~1 and 180 micron sizes pressed to 90% theoreti-cal maximum density. Opposite size-dependentcrystal sensitivity results were obtained with the twotest methods. The smaller particles were moreshock sensitive in the laser impact results and lessshock sensitive in the gap tests. The results ap-pear to be particularly important because of thesuggested relationship made here of the neat crys-


Fig. 9. Schematic size dependence of fine (F)coarse (C), and very coarse (VC) crystal ingredi-ent influences on the run-to-detonation distancesas a function of required shock-induced decompo-sition pressures.

tal results being analogous to pioneering flyer platetest results made on cast 70% RDX/30% polyure-thane PBX formulations containing individualmonomodal RDX particle sizes ranging between 6and 428 microns; see the schematic Fig. 9 for thedifferent shock pressure dependencies on the run-to-detonation distances for different fine (F), coarse(C), and very coarse (VC) crystal sizes [74]. Asargued in the earlier PBX case for discriminatingbetween proposed control by ignition of hot spotswithin individual particles at low pressures as com-pared with control at high pressures by growth ofreaction through hot spot coalescence, the differ-ence in PETN results were attributed to the re-stricted growth of hot spots in the narrower pulse(and higher pressure) laser tests as compared toinitiation control in the longer pulse (and lower pres-sure) gap tests. The (crystal or) particle size effectdescribed in Fig. 9 has been investigated morerecently in development of an “extended statisticalhot spot model” description leading to predicted hotspot ignition and growth dependences for a neatparticle-sized material in which the hot spots areinitiated at particle-to-particle contact points andthe thickness of a coalesced hot spot region istaken into account for the hot spot growth mecha-

Fig. 10. Experimental and modeled granular com-paction pressures for two commercial powdermaterials as a function of solid fraction (SF) as mod-eled with a specified fraction of particles in a simplecubic lattice, f

sc, and a mean-hardness-determined

particle-to-particle contact pressure, py.

nism [75]. Easier initiation in gas-gun experimentsof reactive Al/PTFE (polytetrafluoroethylene) for-mulations has been quantitatively demonstrated forthe composite material containing smaller alumi-num particles [76].

The employment of laser-induced initiation be-haviors deserves particular attention for analogoususe of the method to produce local melting/dam-age sites for examination in individual crystal stud-ies, for example, of AP crystals [63,77]. Aquarium-type shock-induced chemistry results have beenobtained after pre-test microindentations had beenput into the crystal surfaces to serve as potentialdamage sites [78-80]. For an AP crystal, both nano-second and picosecond laser-pulse-induced cleav-age-type cracking observations were reported inassociation with sub-surface chemical decompo-sition sites produced below an (001) crystal sur-

24 R.W. Armstrong

Fig. 11. Reciprocal square root of crystal diameter dependence of piston speed modeled to assess thesensitivity for combustion of HMX crystals.

face impacted with a Nd/YAG-generated beam.Sufficiently high temperatures were reached at thelaser-impacted crystal regions to cause the APorthorhombic-to-rocksalt structure-type phasetransformation and to cause crystallographically-defined cracking on the (210) and {100} planes ofthe respective structures. X-ray photoelectron spec-troscopy was applied to detect chlorine as a de-composition product. Scanning electron and atomicforce microscopy observations led to associationof chemical decomposition with the micron-scale,sub-surface, hot spot sites. A dislocation reactionmodel basis was developed to show local disloca-tion-generated stress enhancement for chemicalreaction. With regard to the role of cracking in cer-tain energetic crystal initiations, a statistical crackmechanics approach, based on the well-estab-lished theory of penny-shaped cracks, has beenproposed for wider application to explaining theshock-induced impact sensitivity behaviors of bothpropellant and high explosive materials [81].

4.3. Granular compaction

As might be imagined from the mention in Section3 above of mechanical pressing operations doneto manufacture energetic material formulations andfor the shock-induced hot spot behavior of pressed

PETN crystals [73] mentioned in the immediatelyabove Section 4.2, crystal compaction is of inter-est both for practical and research-driven purposes.Considerable research effort has been devoted toadapting the knowledge gained from powder met-allurgical [82] and pharmaceutical [83] experienceto that for compacting energetic materials but withadded concern, connecting to the latter industry,for the material brittle cracking behaviors. Suchpharmaceutical connection has been mentioned inobtainment of recent nanoindentation test resultsreported for sucrose as a model brittle molecularcrystal [58]. Elegant nanoindentation test resultswere reported also for AP in [64]. The hardness ofsuch materials might be imagined to be intimatelyconnected with their compaction behaviors as well.Fig. 10 shows compaction results reported for twocommercial energetic material formulations and asfitted in each case with a model description of thecompaction process; FLUID A is a solid material[84]. In the figure, the compaction model param-eter, f

sc, is determined as the mass fraction of

simple cubic lattice site positions distributed amongpresumed simple cubic and face-centered cubicparticle organizations; and, p

y is the mean hard-

ness stress applicable at the deforming particle-to-particle contacts.


Fig. 12. Schematic picture of a [-100] edge dislocation on the (040) plane in an RDX crystal projectionhaving a (002) planar depth as shown for the framed rectangular unit cell.

In [84], experimental results were reported alsofor the quasi-static compression of different crys-tal sized HMX particle beds. In another investiga-tion employing the same results, a model descrip-tion was developed for the particle size influenceon combustion through plasticity-induced hot spotsdeveloping at the particle-to-particle contacts andrelating to the material contact hardness proper-ties [85]. Fig. 11 shows the modeled results as lateranalyzed on a reciprocal square root of particle di-ameter basis [57] consistent with the display of thecrystal sensitivity measurements shown in Fig. 8[69]. In construction of the figure, the modeled pis-ton velocity was taken to be proportional to the loga-rithm of the plastic strain rate, as will be describedbelow for conventional dislocation mechanicsbased constitutive equation behaviors. The breakin the linear dependence, distinguishing betweensustained combustion and extinction, is seen tooccur approximately at the point of full plastic yield-ing behavior of the particle bed. The results dem-onstrate again a greater resistance to initiation ofdecomposition, this time for combustion, at smallerparticle sizes.

5. DISLOCATION MECHANICS

Thus far in the information covered above, corre-lations have been indicated for a number of de-scribed energetic material properties and variousaspects of the material dislocation behaviors, forexample, as centered on the several levels of: (1)individual dislocation characterizations; or (2) thedislocation pile-up avalanche model proposed toexplain hot spot formations and crystal (particle)size effects; or (3) the related consideration of dis-location pile-up-induced cracking behaviors.

5.1. Dislocation characterizations

Fig. 12 shows a schematic (001) plane represen-tation of an edge dislocation with [100] Burgersvector on an (040) slip plane in the orthorhombicRDX crystal lattice [86]. The (010) slip plane hadbeen identified in the referenced pioneering etchpit study [33]. The dislocation is naturally fully ca-pable of cross-slipping onto the (021) plane thathas been described with respect to promoting thenitroso-compound formation detected in drop-weighted impact specimens falling short of full ini-

26 R.W. Armstrong

Table 1. Comparison of dislocation and cracking parameters.

tiation behavior [55]. From a most basic disloca-tion mechanics evaluation, the total dislocationenergy, E

T, is specified for unit molecular length,

∆ξ, of dislocation line as

E Gb R r ET C= +2

04∆ξ / ln / .πα� � � � (3)

In Eq. (3), G is the shear modulus, b is the disloca-tion Burgers vector (displacement), α is (1 – v) foran edge dislocation or 1.0 for a screw dislocationor a (reciprocal) average value for a mixed dislo-cation, R is an outer cut-off radius, r

0 is the dislo-

cation core radius, and EC is the non-linear elastic

dislocation core energy.Table 1 provides in the first column after the

material identifications a comparison among sev-eral materials of the bracketed coefficient of thelogarithmic factor in Eq. (3). And the adjacent col-umn presents the same coefficient divided by theheat of formation per molecule. Comparison of thedifferent material numbers shows that dislocationsin RDX and PETN have relatively higher self-ener-gies, consistent with relatively low dislocation den-sities being measured by etch-pitting and X-raytopography. The result is a consequence of therelatively large Burgers vector and unit line lengthfactors out-weighing a relatively lower shear modu-lus. The dislocation line energy in Eq. (3), particu-larly including the anisotropy of G values, is pro-posed to be an important factor along with b in

determining the line orientations of dislocationsparticipating in such crystal growth processes asindicated in Figs. 4b and 4d [43].

The dislocation core energy and radius, r0 in

Eq. (3), are associated with another interesting fea-ture for molecular crystals and other materials ex-hibiting large Burgers vectors. For them, an earlydislocation model computation showed that a lowerenergy condition was favored by having a cylindri-cal hole or liquid core running along the dislocationline [87]. Recent calculation has been given for thehole possibly having an elliptical shape [88]. Thepresence of water in the dislocation cores of su-crose crystals has been employed to explain anenhancement of their electrical conductivity [89].An early report was made of solute-containing in-clusions occurring during the growth of RDX crys-tals [90] and, more recently, gas bubbles, some-times elongated into channels, have been reportedin a careful study of the growth characteristics of(organic) benzil crystals [91]. Hollow “micropipes”have been imaged at surface intersections for sili-con carbide crystals and have been proved via X-ray topographic examination to be centered on dis-location lines [92]. Related nanopipe influenceshave been described for epitaxial layers of galliumnitride grown onto sapphire crystal surfaces [93].Such special dislocation core considerations areparticularly important for energetic crystals because

Material Dislocation Peierls- (τp/G) Dislocation (k

s/G), (g/Gb)1/2

Strain Energy

E

H

p

f m

⊥�

��

/

/∆ Nabarro Stress

mm1/2

(E p⊥ /), J Stress (τ

p), Intensity (k

s),

N/mm2 N/mm3/2

RDX 6.3·10-19 12 580 0.077

7.7 10·10-4 0.066

PETN 3.3·10-19 4.0 394

0.073 5.9 11·10-4 0.070

AP 1.2·10-19 ——

1200

0.162 6.3 8.5·10-4 0.088

LiF 0.9·10-19

2.1

3100

0.050

29 4.6·10-4

0.14

MgO 2.4·10-19 1.9

8000

0.054

70 4.7·10-4

0.17

E Gb

G d b

k Gb

p

p p N

s

⊥

⋅

=

= = − − ⋅

=

/

/

/

/ exp /

/

2

1 2

4

2 1 1

4

∆ξ πα

τ τ ν π ν

π α

� ��


Fig. 13. The monoclinic unit cell of β-HMX (cyclotetramethylenetetranitramine).

of their relatively low surface energies, as has beenmentioned [62] and will be discussed with regardto the cleavage cracking behaviors too.

The Peierls-Nabarro (P-N) stress evaluationsidentified in Table 1 are those defined on a con-tinuum mechanics basis for which d is the slip planespacing. On such basis, that can only be consid-ered very approximate because of the molecularcomplexity pointed out for the modeled shear-typedisplacements occurring for slip in an energeticcrystal lattice, for example, for RDX in Fig. 5, bothit and PETN have relatively higher P-N stresseswhen normalized with regard to their shear moduli.A pioneering molecular dynamics calculation hasbeen described [94] for an edge dislocation in an-thracene that is plastically relatively soft, as indi-cated on the basis of the plastic hardness com-parison demonstrated in Fig. 6. Other analysis ofthe relative difficulty presented by intermolecularinteractions of the type mentioned for Fig. 5 hadbeen described earlier for different slip systems inanthracene [95]. More recently, a molecular dynam-ics description has been developed for anthracene

so as to distinguish on a model basis between hy-drostatic and non-hydrostatic influences in com-pression tests [96].

The deformation behavior of HMX, though notincluded in Table 1, is of special interest becauseof twinning being the main response in this case tomechanical loading. Fig. 13 is an illustration of theHMX monoclinic unit cell for the structure of theambient temperature β-polymorph, among threeother polymorphs occurring at higher temperaturesand pressures [97]; and, Fig. 14 provides a modeldescription of the (101)[10-1] twinning shear of lat-tice points. In general, the fixed shear strain char-acteristic for twin nucleation requires that localstress concentrations from micro-slip of disloca-tion pile-ups be involved; and, consequently, forpolycrystalline materials, a relatively strong grainsize dependence is observed for the applied twin-ning stress [98]. Such pre-twinning slip is indicatedto be a minor consideration for HMX, however,because of the twin occurrence being reversible atsmall strains [99], as is documented for the classiccase of deformation twinning in calcite crystals. The

28 R.W. Armstrong

Fig. 14. (101)[10-1] deformation twinning of HMX on a lattice point basis.

comparison of slip being dominant in RDX as com-pared to twinning being dominant for its chemicallyrelated counterpart has been attributed to the flex-ibility of the larger HMX molecule [100], in agree-ment also with the changed shape of the HMXmolecule in its different polymorphs [97].

5.2. Pile-up avalanche model

The dislocation pile-up avalanche model for me-chanically-induced hot spot generation is shownschematically in Figs. 15a-15d. The sequential pic-tures step through the following stages: (a) isother-mal development of a pile-up under a shear stress,τ

1, and blocked, for example, at a polycrystal grain

boundary or at an internal crystal boundary or atcrystal-to-crystal contacts produced in earlier de-formation; (b) consequent achievement at largerdislocation number, n

2, and higher shear stress,

τ2, of a critical stress concentration, τ

C*, correspond-

ing to collapse of the vobstacle; and, (c) dynamicrelease of the pile-up to form a localized hot spot

[65]; see also the earlier presented model hot spotdepictions shown in Fig. 7. Fig. 15d is included toindicate association of a discontinuous stress dropwith the occurrence of significant hot spot genera-tion, as had been discriminated among stressrecords compiled from calibrated drop-weight testsconducted on a number of energetic and relatedcrystals [66]. The dislocation pile-up avalanchemodel in Fig. 15 builds onto an earlier continuummechanics description for characterization of therelease of a pile-up [101] and, also, builds onto anumerical model description that was given for dis-location pile-ups “breaking through” a polycrystalgrain boundary region offering viscous resistancewithin a limited width of the boundary [102].

The temperature rise, ∆T, for pile-up releasewas modeled from that reported earlier for the ther-mal dissipation of work associated with individualdislocation motions [103] but taken for the pile-upas a multiple Burgers vector dislocation of strength,nb, in the avalanche. With n being the number of


Fig. 15. Schematic model for a dislocation pile-up avalanche.

dislocations in the released pile-up, then in oneapproximation [65]

∆T k v c bKs

= l1 2 1 216 2/ /

/ / * .π� �� (4)

In Eq. (4), that applies for the combined materialparameters (2K/c*vb) < 1.0, k

s is the microstruc-

tural shear stress intensity generated by the pile-up sufficient to cause pile-up breakthrough, l is thegrain or crystal diameter, v is the dislocation veloc-ity, c* is the specific heat at constant volume, andK is the thermal conductivity. A largest k

s occurs

for shear induced cleavage cracking, as has beenestimated for a number of materials [104]; and, theparticular ks values listed in the fourth and fifth col-umns of Table 1 show that (k

s/G) is relatively large

for RDX, PETN and AP. An immediate result wor-thy of notice in Eq. (4) is that a higher temperature

rise is potentially generated for a larger crystal di-ameter.

Eq. (4) has been applied to explaining the crys-tal size dependence of the drop-weight impact re-sults shown in Fig. 8 [63]. The influence of mate-rial loading rate was estimated from the disloca-tion velocity parameter, v, expressed as

v v G bA d kTth= − − �0 0exp * / .τ� � (5)

In Eq. (5), v0 is the reference dislocation veloc-

ity limited by the shear wave speed, G0 is a refer-ence Gibbs free energy, A* is the dislocation ther-mal activation area, and k is Boltzmann’s constant.This equation, that also forms the starting point forderivation of the so-called Z-A constitutive equa-tions which have been applied to material dynam-ics calculations of structural face-centered cubic

30 R.W. Armstrong

Fig. 16. Comparative dislocation pile-up avalanche and thermal decomposition temperatures for RDXand PETN crystals of different sizes; with the avalanche hot spot size determined by the dislocationseparation at the pile-up tip just prior to release.

(fcc) and body-centered cubic (bcc) metal defor-mation behaviors [7], is seen to have a stress-modi-fied Arrhenius form not unrelated to the thermalinitiation equation analysis employed for the criti-cal temperature dependence on hot spot size [9]and involved in the subsequently listed references[13-20]. With the several conditions: first, of thedislocation velocity being connected to the mate-rial shear strain rate in the normal thermally-acti-vated manner and A* taken inversely proportionalto τ

th; and, secondly, the velocity taken to have a

power exponent dependence on the material drop-weight sensitivity, then the result was obtained [68]

log ~ log /

log , ,... .

*

/

H H kT mW

f T T

50 50 0

1 2

+ ⋅−

� �� ∆ l

(6)

In Eq. (6), m is the power exponent for an as-sumed dependence of τ

th on H

50, W

0 is the con-

stant product of bA*τth and f{(∆T, T, ...} represents

a combination of the other factors in Eq. (4). Thus,for a fixed ∆T presumably being needed for a criti-cal hot spot, Eq. (6) gives a reasonable explana-tion of the drop-weight height dependence on theinverse square root of the crystal size [21,69,104].With the hot spot size taken proportional to the spac-ing of the leading dislocations at the pile-up tip priorto release, see Fig. 15, further comparison wasmade for RDX and PETN to show in Fig. 16 inter-section of the mechanical and thermal model pre-dictions for initiation of decomposition, as illustratedalso for different crystal sizes. The crystal size ef-fect is consistent with that shown in Fig. 8. In Fig.16, RDX is seen to be relatively less sensitive tomechanical initiation because of the lesser thermalstability of PETN; an observation in general agree-ment with variously tabulated drop-weight measure-ments comparing material sensitivities.


Fig. 17. The ratio of pile-up shear stress intensity for cracking, ks, and thermal conductivity, K, as a

criterion of susceptibility to hot spot development.

An important consideration relating to the pile-up avalanche model concerns the role of k

s among

other factors in the pile-up avalanche model de-scription. For the combination of parameters (2K/c*vb) ≥ 1.0, as is estimated to apply for metallicand ionic solids, Eq. (4) is replaced by

∆T k v K K c vbs= l1 2 16 2/ / ln / * .π� � � � (7)

On the basis of Eq. (7), the important materialparameters controlling ∆T are k

s, K, and v. In order

to make an estimation of energetic, ionic, and me-tallic material susceptibilities to hot spot genera-tion at constant v, a plot of ks against K was madeas shown in Fig. 17. An increasing slope valuedrawn for any line to the variously plotted (opencircle) points is indicative of an increasing suscep-tibility to hot spot initiation. To include the energetick

s and K parameters along with the metallic and

ionic ones in the figure, both the abscissa and or-dinate scales were reduced by an order of magni-tude, as shown by the inset rectangular graph in-

cluding the data for RDX and PETN. Also, on theright hand ordinate scale of the larger figure frame,the product K∆T is computed for the (filled circle)points plotted by assuming an upper-limiting shearwave speed for v. Thus, on both (k

s/K) and over-

estimated (principally because of v) ∆T bases, then,the energetic materials are shown among all of thematerials to be relatively more susceptible to hotspot initiation. A very recent molecular dynamics(MD) model investigation of plasticity at ananoindentation put into an RDX crystal has shownan appreciable temperature rise accompanying theprocess, also done at a relatively high loading rate[105]. Furthermore, the order of increasing slopesshown in Fig. 17 in the order of α-iron, α-titanium,and MgO as compared with a lowest slope for alu-minum, are in line with increasingly greater shearbanding susceptibilities being previously reportedfor these materials [106]. More recently, numericalcomputations have been reported [107] for the tem-perature rise associated with an avalanche break-

32 R.W. Armstrong

through of the viscous obstacle model given in[102].

Lastly here, relating to the furthest right-handside column of Table 1, is the consideration of cleav-age susceptibility, that can be taken as a measureof the generation of very localized plastic flow at acrack tip and, hence, be reflective of the localizedgeneration of hot spots [81]. The parameter(γ/Gb)1/2 whose numerical values are tabulated wasobtained from a model estimation of the ratio ofshear stresses needed either to propagate a crackby the Griffith mechanism or by a dislocation gen-eration mechanism [107]. The listed numerical val-ues for the ratio are consistent with the energeticmaterials being cleavage prone. They are certainlymore so than is indicated by the higher values givenfor the ionic materials. The results are in line withthe RDX vs. NaCl crystal comparison made on anIFM basis in Fig. 6. The dislocation generation partof the calculation was refined by more detailedevaluation of the dislocation nucleation mechanismbut the improved criterion for brittleness appearsto be approximately the same [109]. Even morerecently, the subject has been reviewed with re-gard to better understanding the ductile-to-brittletransition in behavior that may occur on an indi-vidual dislocation basis, for example, when a ma-terial is subjected to dynamic loading [110]. And,there is the added issue of cleavage initiation be-ing accomplished via dislocation pile-ups, as givenemphasis here both for energetic materials and fortheir structural material counterparts too. A princi-pal consideration in that regard is the dislocationpile-up explanation for the inverse square root ofgrain size dependence of the strength propertiesof structural materials [7] now having been shownto carry over to providing a better understanding ofenergetic crystal initiation behaviors, particularly,involving crystal size effects both in particle initia-tion behaviors and in combustion behavior associ-ated with granular compaction.

5.3 Shock-induced initiations

Porosity, cracking, and plasticity influences are of-ten intertwined in modeling the shock inducedchemical decomposition properties of energeticcrystal and composite materials [111]. MD simula-tions have proved to be an especially important toolin modeling of the shock-induced material behav-iors for both energetic [112,113] as well as struc-tural material [7,114,115] performances. A role forporosity in energetic material characterizations,which was carried over initially from the demon-

strated sensitizing influence of gas bubbles in liq-uid explosives, as recently reviewed [116], has beenand continues to be researched by the MD methodapplied at ever larger dimensions beginning fromthe nanoscale level for an isolated pore. From asolid mechanics viewpoint at the limiting largercontinuum scale, the applied stress is known to bemagnified locally by a factor of three times for aspherical void but this factor is magnified manytimes over in proportion to the aspect ratio of lengthto thickness in transforming the void into a crackof negligible thickness [117]. At such atomic levelthickness, the crack is modeled equivalently on acontinuum mechanics basis as a dislocation pile-up [118] consistent with the inverse square root ofcrack size dependence of strength properties de-scribed in the present report.

The preceding issue of dimensional scale atwhich porosity, cracking, and plasticity influencesare to be accounted for may be traced by begin-ning from the smallest scale modeling involved inthe ever increasing pursuit of more reliable MDcomputations. Early results for atomic- or molecu-lar-scale pore collapses were obtained in relationto energetic material properties by modeling thestress-induced collapse of simple lattice point struc-tures; see for example references [119,120]. In[113], beginning from an earlier hot spot associ-ated Arrhenius-type description [13] and relatingto pioneering orientation dependent PETN shockinitiation measurements [121], not unrelated to thatsame consideration described for Fig. 5, reactivedirection-dependent collisions among PETN orRDX molecules were investigated in relation to theirrespective crystal structures.

A latest investigation of porosity influence at thelarger micron dimensional scale involves a crystalmechanics model of dynamic sub-surface porecollapse in an HMX crystal [122]. A constitutiveequation incorporating dislocation drag was em-ployed for tracking the role of dislocations duringdynamic pore collapse; and, the crystal mechan-ics model was calibrated using a combination ofMD and single crystal experimental results. Crack-ing and/or plastic shear under associated meltingand non-melting conditions were investigated. Plas-ticity-associated jetting of material was demon-strated at the collapsing pore, without melting, butwith occurrence of a relatively high dislocation den-sity of ~8·109 cm-2, which is proposed to be typicalfor shock loading; however, see the above Eq. (3)and Table 1 consequences. The results suggestthat pore collapse details depend on the interplayamong pore size, wave structure as supported by


the initial dislocation density and dislocation kinet-ics, and other factors. Another forward-looking in-vestigation [123], also referenced as an exampleof bridging length scales, has described a disloca-tion mechanics role involving shear on the[100](021) lattice-modeled system, as in Fig. 5, andas incorporated within a hydrocode model descrip-tion of bullet penetration into a high explosive ma-trix. The study is remindful of the historical refer-ence [8] and such modern practical assessmentssuch as made in reference [72].

Dislocation density generations as comparedwith dislocation velocity-associated phonon draginfluences are also of current research concern forstructural materials with respect to shock-induceddeformations and the relatively more recent methodof testing at equivalent or higher imposed stresslevels in a shock-less isentropic compression ex-periment (ICE). At relatively high shock pressures,the shear strains at all points within thenanometrically-scaled width of the shock front arenot able to be relaxed by the remote displacementsof an initially resident dislocation density behind theshock front nor by shear stress action applicableto the relatively few dislocation line intersectionswith the front [7,86,124]. Consequently, a nanoscaledislocation density is generated within a high pres-sure shock front and controls the shock-inducedplastic strain rate. For the very differently imposedshock-less ICE-type loading, an opposite disloca-tion mechanism operates in that the uniformly in-creasing stress activates mobile dislocations fromwithin the initially resident defect density but therelatively few dislocations, compared to the shockcase, have to move at velocities approaching theshear wave speed and, hence, the constitutivedeformation behavior is taken over by drag-con-trol of the dislocation velocities [7,125]. The recentresults reported for the comparison of high rateshock and ICE deformations of fcc copper and bcciron materials [126] appear to be followed by analo-gous recent results reported for RDX crystals [127].

6. DISCUSSION

Explicitly indicated in Fig. 1 is the long (structural)“path” to be traveled in proceeding from the majordescriptors for a composite material employed asa propellant (or explosive formulation) and themagnified view that is shown at expanded scalefor an (010)[100] edge dislocation in its encompass-ing crystal lattice environment. Support for the con-siderable research effort that has been expendedon energetic materials was applied with the pur-

pose in mind of bridging the gap between the mo-lecular and composite material properties [128].And several of the listed article references givenhere have indicated that the composite materialbehavior could be tracked in a number of cases tothe individual crystal properties. As indicated above,an extensive array of research diagnostics is be-ing applied to achieving a better understanding ofthe shock-induced decomposition behavior for afull range of composite systems, especially, withregard to tracking the relevant hierarchical level atwhich critical events are controlled [111,129].

An additional case to illustrate the point of mak-ing a connection with the overall composite behav-ior is the search for an explanation of the onset ofan unstable burn rate that may seemingly occurunpredictably during propellant combustion. Theworry is for a deflagration-to-detonation (DDT) tran-sition in behavior, which subject has been and con-tinues to be a heavily researched topic [130]. Apioneering research description of internal stresseffects associated with sub-surface cracking ofcrystals during combustion of HMX-based rocketpropellant formulations was concerned with crys-tal size effects importantly suggested to be basedin the β - δ solid-state polymorphic transformationoccurring at the higher pressures and temperaturesin service [97]. Recently further connection hasbeen proposed for the correlation of IFM crackingmeasurements made on HMX crystals and ob-served burning rate behaviors of HMX formulations[131].

The use of comparative inert material behav-iors in composite formulations, for example, of su-crose extending from the individual crystal studies[36,58], and incorporated as a “PBS”, bonded withhydroxy-terminated-polybutadiene (HTPB) as fora PBX formulation, has been employed for the de-velopment of SHPB test procedures to be ap-plied to energetic formulations, including observa-tions of brittle fracture of the polymer binder at be-low glass transition temperatures [132,133]. Pre-vious comment has been made in the present re-port concerning the importance of particle-to-par-ticle contacts among the relatively hard energeticparticles bonded within a softer polymer matrix[75,84]. In that regard, Fig. 18 is shown to indicatepossible relevance of the material contiguity, C,parameter as a measure of such hard particle con-tacts and known to have significant influence onthe mechanical behaviors of tool-cutting tungstencarbide-cobalt composite materials [118,134]. Thefigure is useful also to show relationship betweenthe size and volume fraction dependencies of crys-

34 R.W. Armstrong

Fig. 18. The hardness as a function of volume fraction of tungsten carbide, VWC

, in WC-Co compositematerials as dependent on the contiguity, C, measurement of particle-to-particle contacts and the sizes ofcarbide particles and the mean free path within the cobalt binder phase.

tal hardness for both particles and binder within acomposite material.

In Fig. 18, the Vickers (diamond pyramid) hard-ness, H, is shown to depend on two volume frac-tion characteristics of the composite: the straight-forward determination of V

WC, for the relative vol-

ume of tungsten carbide (WC) crystals; and, (VWC

C)for an effective volume fraction of particle-to-par-ticle contacted material. The plotted cross, opencircle, and filled triangle point measurements shownfor the composite hardness measurements[135,136] are plotted on the abscissa V

WC scale

and show greater hardness values, beyond the VWC

dependence, at smaller carbide particle sizes.Quantitative stereological investigation of the par-

ticle size effect led to the composite hardness ex-pression

H H V C H V CWC WC m WC

= + −1� �, (8)

in which HWC

and Hm are the separately determined

hardness values of the individual carbide and co-balt matrix phases, respectively. A linear hardnessdependence on the inverse square root of the par-ticle size or grain size was determined separatelyfor the tungsten carbide and cobalt constituents.Also, the size dependent hardness values deter-mined for the two cross and one open circle mea-surement values marked in the figure by the hori-zontal arrows pointing leftwards were shifted to the


effective volume fraction points, VWC

C, as indicatedfor Eq. (8). As a first step for illustration of Eq. (8)application, the individual size-dependent hardnessmeasurements applicable for both constituents ofthe three composite formulations were plotted asthe filled square points shown on the terminal ordi-nate axes of the constituents. Then the linear de-pendence of the composite hardness on the effec-tive volume fraction, V

WCC, was demonstrated for

Eq. (8) by plotting the intermediate filled squarepoints computed for the shifted measurement po-sitions. Additional results on the cracking proper-ties of this particular composite system have beenreported also on a fracture mechanics basis [137].It should be interesting to investigate whether simi-lar behavior would be obtained for an appropriateenergetic composite material especially as the con-tiguity factor, C, is indicated as well from the refer-ences, for example [75,84,85], made to the impor-tance of particle-to-particle contacts being inti-mately involved in determining the composite ma-terial hot spot behaviors.

7. SUMMARY

A description has been given of experimental andmodel dislocation mechanics aspects of solid en-ergetic crystal and composite material strength andchemical decomposition behaviors and also involv-ing, as well, consideration in a number of instancesof reference inert material properties. Dislocationsare present at the beginning origination of any en-ergetic crystal or composite formulation; and, thenature of the dislocations and their behaviors areshown here to be tied to the molecular bonding ofthe relatively complex energetic crystal lattices. Oncombined dislocation and crystal lattice bases, en-ergetic crystals are shown to be elastically compli-ant, plastically strong, and especially susceptibleto dislocation-assisted cleavage cracking. Inden-tation hardness, drop-weight impact, and stresswave loading techniques, in the latter case begin-ning with SHPB testing and extending to shockwave and ICE results, are shown collectively toprovide important information about the strengthand energy release properties of the materials,especially, as interpreted through the initiation andgrowth mechanisms described for localized hotspot developments. Emphasis is placed through-out the article on the crystal size dependenciesobserved for material strength properties and thesensitivities of the crystals to mechanically-induceddecompositions. The total results point to advan-tages of greater strength and power dissipation

being achieved for composite formulations fabri-cated from smaller particle-sized ingredients.

ACKNOWLEDGMENTS

Professor Ilya Ovid’ko is thanked for inviting thepresent article. Appreciation is expressed to a num-ber of colleagues: Herman Ammon and Zhu YuDu, University of Maryland, for previous collabora-tion and providing the model crystal lattice pictures;to Wayne Elban, Loyola University Maryland, forcollaboration on achievement of many of the re-sults that are described in the present article; andalso, to Frank Zerilli and now deceased StephenCoffey, among other helpful scientists at the U.S.Naval Surface Warfare Center, Indian Head Divi-sion, for a number of collaborations too. NathanBarton has provided helpful description of the re-sults in Ref. 122. Valuable research periods withcolleague associations were spent at theCavendish Laboratory, University of Cambridge,U.K. and at the Energetic Materials Branch, Muni-tions Directorate, Eglin AFB, FL.

Note added in proof: A valuable description oflaser ignition and initiation of explosives is given inN.K. Bourne // Proc. Roy. Soc. Lond. A 457 (2001)1401.

REFERENCES

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