An alternative to the use of plasticity theory to characterize the inelastic behavior of solids is to represent the flaws by statistical methods. We have taken such an approach to study fragmentation because it offers a number of advantages. Foremost among these is that, by considering the effects of flaws, it becomes possible to address the underlying physics directly. For example, we have been able to explain why rocks exhibit large strain-rate effects (a consequence of the finite growth rate of cracks), why a spherical explosive imbedded in oil shale produces a cavity with a nearly square section (opening of bedding cracks) and why propellants may detonate following low-speed impact (a consequence of frictional hot spots).

In a previous article in this journal, it was shown that vorticity is only an approximation to the rate of rotation of an element of material, and that an exact expression for the rate of rotation consists of the vorticity plus an added term, denoted by H, but not then given a proper name. It was shown that the new formulation led rigorously to the so-called Green-Naghdi stress rate, which has been previously reported but not actually derived. In an example it was shown that Green-Naghdi stress rate leads to a stable deformation in a calculation of simple shear, while the more traditional Jaumann-Noll stress rate leads to a late-stage instability, though they are very nearly equal for small strains. Here we name the term to be added to the vorticity, the swirl, and derive an alternate, but simpler, algebraic expression. In addition, we provide examples of its use with the goal of providing added credibility. The first concerns the dynamics of a classical vortex, showing that the rate of rotation is not zero as classically claimed but is consistent with the macroscopic rate of rotation. The second discusses a formula for a test known as simple strain that is compared with an experiment by Moreira and Nunes. The analysis follows standard usage in continuum mechanics, but the special use of polar decomposition allows for a very convenient derivation of the algebraic expression for the swirl.

The impact tests for propellant sensitivity reported by Jensen, Blommer and Brown (JBB) were anomalous in several respects. First, 12 out of 50 impacts at moderate speeds led to especially violent explosions (XDT) while the remaining 38 produced only mild deflagrations (DEF). XDT generally occurred following impact at speeds lower than those leading to shock-to-detonation (SDT), which occurred in the 22 shots at speeds above 2500fps. Moreover, the violent reactions occurred at relatively late times, in excess of 30 microseconds after impact, rather than the 5 microseconds typically associated with SDT. Finally, the blast pressure associated with the violent explosions was even higher than in SDT and the cratering was more profound. Since these results defied explanation by the mechanisms associated with classical detonation theory, the process was termed XDT. In XDT shock heating was considered negligible, as further evidenced by the late reaction time.This report summarizes some mechanisms postulated by the author during the last 30 years to explain the observations of JBB and other violent but non-repeatable reactions in propellants and explosives. The fundamental assumption herein is that defects occur at random and that heating of these defects following impact may lead to reactive hot spots. More specifically, it is postulated that the defects are shear cracks in which interfacial sliding leads to frictional heating that, in turn, causes vigorous reactions that may culminate in violent explosions. Our analysis shows, however, that a single hot spot will not result in violent explosion since rapid expansion of the gas-filled cavities reduces the gas pressure enough to quench the reaction. Thus, we conclude that interactions of defects is responsible for the observed violence.Three types of interactions are thought to contribute to the suddenness of the violence. In one, pulses from reactions in one defect enhance the pressure in adjoining defects and this feeds back to the first defect, initiating a divergent oscillation. Such interactions have been observed in special simulations. Further, growing cracks can intersect to form larger, more unstable, cracks, and a large enough number of intersections can result in a network of cracks that supports violent reactions. This is thought to occur when the concentration of cracks exceeds the percolation threshold, causing a violent reaction. Finally, we hypothesize that radiation between connected cracks can cause the heat produced by chemical reactions to spread extremely rapidly, leading to violent explosions.

This paper presents a study of anisotropic damage and cracking in a hot isostatically pressed assembly of titanium alloy encapsulated AD-995 ceramic under ballistic impact using the statistical crack mechanics approach. Anisotropy of crack growth in the ceramic is illustrated numerically by examining the growth in crack sizes along three orientations. Comparisons with the experimental measurements of the predicted backsurface profile and the damage (cracking) in the ceramic suggest that the model predictions are consistent with the experimental data. Numerical simulation also indicates that a prestress of roughly 2kbar (200MPa) compensates for about 1% of initial porosity in the ceramic. A comparison is made to the Rajendran–Grove ceramic model in EPIC which assumes an isotropic crack distribution.

Brittle polycrystalline materials such as rocks, ceramics, and certain metals contain microcracks that can grow and coalesce Linder sufficiently high stress, resulting in failure and, possibly, fragmentation. Such processes are idealized in this paper by treating the cracks as circular disks whose coalescence forms clusters and can terminate growth after a number of intersections. A non-linear recurrence relation for the probability of cluster size is developed and solved by means of a generating function, providing information on the mean size of the crack clusters and the standard deviation. This solution leads to a simple expression for the percolation threshold. The probability of clusters of size it is also determined. Above the percolation threshold the probabilities of finite and infinite clusters are treated separately. Explicit expressions for the probabilities can be approximated by taking a Laplace transform in a simple case, thus clarifying the behaviour of the solution. Appendices show the relation of the theory to practical problems, Monte Carlo approaches, the probability of infinite Clusters, and discuss the uniqueness of solutions to such geometrical problems, i.e. Bertrand's Paradox. Copyright (c) 2005 John Wiley & Sons, Ltd.

A statistical approach has been developed for modeling the dynamic response of brittle materials by superimposing the effects of a myriad of microcracks, including opening, shear, growth and coalescence, taking as a starting point the well-established theory of penny-shaped cracks. This paper discusses the general approach, but in particular an application to the sensitivity of explosives and propellants, which often contain brittle constituents. We examine the hypothesis that the intense heating by frictional sliding between the faces of a closed crack during unstable growth can form a hot spot, causing localized melting, ignition, and fast burn of the reactive material adjacent to the crack. Opening and growth of a closed crack due to the pressure of burned gases inside the crack and interactions of adjacent cracks can lead to violent reaction, with detonation as a possible consequence.This approach was used to model a multiple-shock experiment by Mulford et al. [1993. Initiation of preshocked high explosives PBX-9404, PBX-9502, PBX-9501, monitored with in-material magnetic gauging. In: Proceedings of the 10th International Detonation Symposium, pp. 459-467] involving initiation and subsequent quenching of chemical reactions in a slab of PBX 9501 impacted by a two-material flyer plate. We examine the effects of crack orientation and temperature dependence of viscosity of the melt on the response. Numerical results confirm our theoretical finding [Zuo, Q.H., Dienes, J.K., 2005. On the stability of penny-shaped cracks with friction: the five types of brittle behavior. Int. J. Solids Struct. 42, 1309-1326] that crack orientation has a significant effect on brittle behavior, especially under compressive loading where interfacial friction plays an important role. With a reasonable choice of crack orientation and a tempera tu re-dependent viscosity obtained from molecular dynamics calculations, the calculated particle velocities compare well with those measured using embedded velocity gauges. (c) 2006 Elsevier Ltd. All rights reserved.

We compute the temperature profile near a laterally growing uniformly sheared viscous layer, a melt of the host material. A similarity variable is identified and an analytical solution is found for the temperature field and phase boundary trajectory for the case of uniform sliding. A numerical method for the iterative solution of this non-linear Stefan problem is implemented. In the case of a chemically reactive material with an Arrhenius dependence of the reaction rate on temperature we compute the ignition times. The dependence of the ignition time on the sliding velocity and latent heat of the material is determined numerically.

The theory of penny-shaped cracks has been the subject of numerous investigations because of its conceptual simplicity and the feasibility of obtaining mathematical solutions. The simplicity of many of the final results makes the theory useful in many applications. This paper addresses a gap in the theory, the effect of interfacial friction in closed cracks and, in particular, its influence on the most unstable crack orientation and, hence, the compressive strength of materials. When friction is accounted for it is found that five types of brittle behavior are possible: (a) mode-I opening, (b) mixed opening and shear, (c) pure-shear without friction, and (d) shear with interfacial friction. A fifth type of behavior (e) which corresponds to a stable material response occurs when the compressive traction on the crack is so large that friction inhibits crack growth. The first four types, namely, (a), (b), (c), (d) result in incipient material failure. The range of stress states for which each of the failure (incipient) types applies is given explicitly. Failure of a brittle material under triaxial test conditions is considered in detail to illustrate the results. An experiment performed by Howe et al. [Howe, P.M., Gibbons, G.G., Webber, P.E., 1985. An experimental investigation of the role of shearing initiation of detonation. In: Short, J.M. and Deal, W.E. (Eds.), Proceedings of the 8th International Symposium on Detonation, Albuquerque, NM] showing the response of a brittle material (TNT) to impact illustrates (perhaps surprisingly) behavior of types (c) and (d). The chemical sensitivity of the TNT allows us to observe the effect of friction better than would be possible in a non-reactive material. The conditions that allow crack growth within the crack plane are discussed briefly.

rate-dependent, continuum damage model is developed for brittle materials under dynamic loading. This model improves on the approach (ISOSCM) of [Addessio, F.L., Johnson, J.N., 1990. A constitutive model for the dynamic response of brittle materials. Journal of Applied Physics 67, 3275-3286] in several respects. (1) A new damage surface is found by applying the generalized Griffith instability criterion to the dominant crack (having the most unstable orientation), rather than by averaging the instability condition over all crack orientations as done previously. The new surface removes a discontinuity in the damage surface in ISOSCM when the pressure changes sign. (2) The strain due to crack opening is more consistent with crack mechanics, with only the tensile principal stresses contributing to the crack opening strain. This is achieved by incorporating a projection operator in the equation for the crack opening strain. One consequence of incorporating the projection operator is a prediction of shear dilatancy, which is not accounted for in ISOSCM. (3) The evolution of damage, which is based on the energy-release rate for the dominant crack, has a physical basis, whereas in the previous approach the damage growth rate was assumed to be an exponential function of the distance from the stress state to the damage surface without specific physical justification.An implicit algorithm has been developed so that a larger time step can be used than with the explicit algorithm used in ISOSCM. The numerical results of a silicon carbide (SiC) ceramic under several loading paths (hydrostatic tension/ compression, uniaxial strain, uniaxial stress, and shear) and strain rates are presented to illustrate the main features of the model. (c) 2005 Elsevier Ltd. All rights reserved.

Many materials contain a large number of microcracks that can propagate under sufficiently high stress, but their stability is sensitive to crack orientation. We have explored this sensitivity using classical fracture mechanics with the added feature that interfacial friction is accounted for in the behavior of compression cracks. Our analysis shows that four types of unstable crack growth are possible for a penny-shaped crack under a general state of stress, depending on crack orientation: opening without shear, mixed opening and shear, pure shear without friction, and shear with interfacial friction. In addition, interfacial friction prevents crack growth at all stress intensities in a certain range of compressive stress. It will be shown that these analytic results are captured by the SCRAM brittle-failure algorithm, and that friction strongly affects the orientation of the most unstable shear crack as well as the range of unstable orientations. A second study examines the variations in material response as a function of the number of orientations represented. This is done by computing the dynamic response of an axisymmetric thick ring to internal pressure. With the traditional 9 crack orientations the fluctuation in porosity is about 28%, while with 480 orientations the fluctuation drops to just over 2%.

The theory of large deformations developed here is closely related to continuum mechanics but it differs in several major respects, especially in considering the deformation associated with various types of physical behavior, making it possible to synthesize a general approach to formulating constitutive laws. One goal is to derive general concepts of strain, strain rate, stress, and stress rate that are somewhat more physics-based than in most standard works on continuum mechanics, and to demonstrate some new relations between these quantities. With these concepts it is possible to develop a generalized principle of superposition of strain rates (GSSR) that accounts for damage as well as plastic flow. The traditional superposition of strain rates allows for addition of elastic and plastic strain rates and is commonly thought to be valid only for small strains. The GSSR allows us to compute deformations involving plastic flow and, in addition, brittle failure, fragmentation, high-pressure effects and other types of behavior as necessary, and the theory is valid for arbitrarily large deformations. In fact, GSSR is derived from more basic ideas and has broader application than the standard superposition of strain rates. The physical basis for calculations of complex material response is developed in amore » separate report. The implementation into the SCRAM computer program is documented separately. The polar decomposition theorem is taken as a starting point for the theory of large deformation, an approach somewhat different from that usually taken in continuum mechanics. Two sets of orthogonal axes are distinguished, space axes that are fixed in ambient space, and polar axes that are related to material deformation. This clarifies several concepts; for example, it is shown that the Signorini and Green-St. Venant strains are actually measures of the same physical entity, one in space axes and the other in polar axes. It follows that they are not competing measures, as is often implied in traditional continuum mechanics. It also follows that Piola stress is a measure in polar axes, while Cauchy stress is a measure in space axes. Another consequence of polar decomposition is a proof that vorticity is not a measure of the rate of material rotation (as is often stated in the hydrodynamics literature) but that they are related. This allows us to develop an exact approach to computing rates of tensor quantities, called polar rates, that account for material rotation in an exact way. This leads to a simple relation between Signorini strain rate and stretching (the symmetric part of the velocity gradient). It also follows that the polar stress rate is the appropriate measure for the rate of change of Cauchy stress, and that the more traditional stress rate of Zaremba, Jaumann, and Noll is only an approximation, valid at small strains. Examples are described for materials undergoing simple shear, vortex motion, and torsion.« less

We have developed a general theory for the formation of hot spots from defects in explosives and propellants, and applied the theory to a variety of issues concerning the sensitivity of reactive materials. The defects of greatest concern in PBXs are cracks formed in the explosive grains, which are normally brittle. The theory accounts for the opening, shear, growth, and coalescence of cracks. In addition, the theory accounts for the heating caused by interfacial friction in closed (shear) cracks and the ignition process that results. Heat conduction and chemical reactions are treated on a smaller spatial scale than the overall continuum response; this is accomplished in the numerical (FEA) simulation with a sub-grid model. In previous work we have shown the feasibility of using this approach to model explosions that result from relatively mild insults, where many other hot-spot mechanisms fail. This paper addresses some of the complications that arise as mechanical failure and heating are examined in greater detail, including the effects of crack orientation, friction, melting, viscosity in molten regions, radial crack formation via a new approach to percolation theory, and 3-D effects.

Cracks that are subjected to a sudden, sufficiently large, increase in stress respond with a combination of opening and unstable growth. This behavior is idealized here by representing the opening and growth as two generalized coordinates and determining the corresponding Lagrangean. Then the equation of motion are established for an arbitrary time-dependent driving pressure. Lagrange's equations are especially appropriate for systems involving dissipation, nonholonomic constraints, and other such processes that can not be accounted for in a Hamiltonian formulation. It is shown that the standard results for crack opening and stability are consequences of the Lagrange formulation, as is an algebraic expression for crack speed. Of course, the solution for the crack dynamics is approximate, but solving the full PDEs for the unstable cracks in a brittle structure (∼10,000 cracks/cc) is not practical, nor would it be appropriate since the details of the cracks vary between samples. Thus, the Lagrange formulation makes it possible to analyze brittle materials containing an ensemble of cracks in an efficient manner. In an example the ODEs are integrated numerically and it is shown that the results are consistent with analytic results. In that work the surface energy is considered constant, but when the stresses are modest, crack growth is governed by creep processes. It is shown that this slow growth can be accounted for by letting the surface energy depend on the stress-intensity factor in a manner that is based on crack-speed data.

The growth of cracks and their role in the formation of hot spots in explosives are addressed in a three-part discussion. First the method of generalized coordinates is used to represent crack dynamics with two ordinary differential equations for crack opening and growth. These account for both stable and unstable behavior. Second, the behavior of burning cracks is addressed by coupling those differential equations to a burn model to show that burning cracks can exhibit either mild or violently unstable behavior. Finally, in the third part, it is shown that the burning crack algorithm in combination with SCRAM (which deals with Statistical CRAck Mechanics) can account for the reactive behavior of a plastic-bonded explosive (PBX 9501) subjected to multiple shocks. SCRAM allows for representation of brittle behavior in a variety of explosives, propellants, ceramics and geological materials.

A hot isostatically pressed (HIP) assembly of titanium alloy encapsulated AD995 ceramic, subjected to ballistic impact, is studied in detail. The crack behavior of ceramic confined in this way is studied in a combined experimental and computational effort. Fabrication of the HIP assembly is described. An experiment in which an assembly was impacted with a Lexan impactor at 1560 m/s is discussed, and the resulting deformation of the assembly and cracking of the ceramic are characterized. The Statistical Crack Mechanics (SCM) model for brittle materials is briefly described. The implementations of this model into the Lagrangian code PRONTO and into the Eulerian code CTH are described and used to further study the response of the ceramic during the experiment.

In rock mechanics it is often assumed that the number of cracks whose size exceeds c is given by the exponential N(0)e(-c/(c) over bar). It is difficult, however, to examine the cracks in a three-dimensional body to verify this exponential variation, and one is normally limited to observations on an outcropping, a cut, or a plane obtained by sectioning a sample. In this paper, we consider two mathematical problems. The direct problem is to find the distribution of the line segments in a plane section when the three-dimensional distribution of cracks is homogeneous, isotropic, and exponential. It will be shown that this distribution can be expressed by means of a Hankel function and that the distribution in a plane section is qualitatively different from the three-dimensional exponential distribution in having a peak at a finite value of segment length. It is found that the mean segment size in the plane is pi/2 times the mean crack diameter in these dimensions. This is consistent with the well-known observation that small cracks have a lower probability of being intercepted by a plane than larger cracks.The indirect problem is to infer the three-dimensional distribution of cracks from the distribution on a plane section. This problem is solved by deriving an integral equation relating the three-dimensional distribution of cracks to the distribution of line segments in a plane and showing that it can be solved for an arbitrary distribution of segments, The special case of the Hankel distribution in the plane leads to an exponential distribution in three dimensions, verifying the general solution of the indirect problem, (C) 2000 Academic Press.

We calculate the stress-strain relation for elastomeric foam from anab initio theory, which shows that the “plateau” and “densification” regions should be described by a hyperbola. The theory seems to agree reasonably well with experiment.