## Abstract

The response of polycrystalline ceramics to shock loading has been studied for the past 30 years. Yet formulation of continuum models describing their response has been only partially successful. In uniaxial-strain loading, failure mechanisms have been noted that proceed behind the shock front from the impact face. It has become clear that operating mechanisms have not been completely described, and so the technique of mesoscale simulation has been applied to understand composite behaviour at the grain scale to explain processes operating at the continuum. The aluminas chosen for the study have been experimentally investigated by using the plate impact shock loading. The microstructure has been investigated using transmission electron microscopy. This microstructure has been converted into an input file for computer simulation of the experimental impacts assuming that each phase has known properties. A view of the mesoscale response is presented and the operating mechanisms are highlighted. A connection with the macroscopic response is made to illustrate the features measured at the continuum that originate at the mesoscale.

## 1. Introduction

Advances in the understanding of the response of brittle materials to high-rate loading have not been reflected in advance in constitutive model development. This results from an incomplete knowledge of operating mechanisms that consequently are not reflected in global models. To model heterogeneous materials for numerical packages requires macroscale description of behaviour based on continuum theory. Other works have addressed atomic length-scales using molecular dynamics methods (Holian 1998; Belak 2000). However, these approaches cannot completely describe processes associated with operating mechanisms in heterogeneous media, where the phases interact at a scale of order microns. For these materials, the mesoscale has not been extensively studied. However, it remains the key link that bridges the continuum to the atomistic length-scales.

The dynamic response of strong ceramics to shock wave loading has been addressed by the shock physics community over the past 30 years. During that time much has been learnt, but vital questions remain still unresolved (Rosenberg 1994). These materials are characterized by very high Hugoniot elastic limits (HELs) (6–20 GPa) and extremely low spall strengths (0.3–0.6 GPa). One feature of their response is that damage is introduced, even when shocked to relatively low stresses (about half the HEL). Various failure criteria, which are used in numerical codes, have been discussed in an attempt to account for their advantages and discrepancies. A summary of relevant properties includes

the high HEL of shock loaded ceramics,

the observation that the shear strength increases with shock pressure,

that the general shapes of the measured stress and particle-velocity histories of these materials show a rapid elastic rise, followed by a convex curved region when loaded to about 1.5 times the HEL,

at higher speeds this convex region turns back to give a very rapid rise at higher amplitudes.

This work will focus upon the response of alumina to shock loading. A full microstructural and experimental programme has been conducted and reported for a series of well-pedigreed materials, and these results and insights will be expanded upon from a new perspective in this work (Murray *et al*. 1996; Murray 1997; Bourne *et al*. 1998; Murray *et al*. 1998*a*,*b*). In particular, this paper will address compression loading with the hope of explaining the experimental observations. One of these is the complex wave shape observed as a step-loading wave at the contact face that disperses in plate impact experiments. This will necessarily lead to interpretation of the Hugoniot state in terms of mixture theory.

The mesoscale modelling approach has been adopted for materials where localization allows small regions to achieve critical thresholds that are not reached using continuum numerical descriptions. One field where this is of great importance is the triggering of reaction at local, high temperature points within an energetic material. In this case, ignition may quickly progress to violent reaction with consequent safety implications. Recent advance in constructing more applicable continuum models has used mesoscale simulation of heterogeneous materials represented as ordered and disordered arrays of crystal grains (Baer 1997; Baer 2000; Espinosa & Zavattieri 2000). Numerical simulations reveal fluctuating stress states and localizations of energy that evolve in three-dimensional analysis with appropriate equation of state and elastic–plastic material strength descriptions. To create three-dimensional, randomly oriented particle configurations and high packing density, these authors used Monte Carlo and molecular dynamic methods (Baer *et al*. 2000; Baer & Trott 2002). In this work, real microstructures will be read directly into the code as input files for simulation to connect with continuum measurements on the same materials.

At present, engineering models for ceramics are generally empirical. The simplest descriptions are plasticity-based adaptations of metals' models (Steinberg 1991, 1992). The most widely used continuum extension of elasto-plastic models is the Johnson–Holmquist description that has evolved into a series of forms to cover a variety of responses (Johnson & Holmquist 1992, 1994, 1999; Johnson *et al*. 2003). In these models, a material element, brought to a yield envelope, is then automatically moved to the failure curve which lies below the envelope (decreased strength) by a certain factor. A second route is to explicitly assume fracture mechanics operates on the materials and consider failure criteria similar to that of Griffith for brittle solids (Margolin 1984; Addessio & Johnson 1990; Dienes 2001). In recent years, advance in numerical treatment has had some success in overcoming the weaknesses of regular, polygonal meshes. Techniques such as smooth particle hydrodynamics have allowed arbitrary material failure that can better handle localized phenomena (like fracture) than continuum codes (Monaghan & Gingold 1983; Mandell & Wingate 1996). Future developments such as the method of cells offer the promise of relating numerical schemes to necessary physical length-scales (Clements *et al*. 1998).

To connect the mesoscale to the continuum requires some homogenization scheme. Further, one must validate the output of simulations at the mesoscale by comparing with the global response at the continuum. The approach taken here divides the problem into a series of units. The definition of the microstructural features of the two ceramics, the selection of a representative unit-cell for computation, and the response of that cell, have been covered in a previous paper (Bourne 2006). In this work, the amplification of this to a larger length-scale, and the quantitative validation against experiment are addressed.

## 2. Materials

The materials selected for the present work are two aluminas (880 and 999) prepared by differing routes. The 880 was pressureless sintered but the 999 was hot, isostatically pressed (HIPed). Their detailed properties and microstructure have been addressed in a previous paper (Bourne 2006). The quasi-static, high-rate and shock continuum response have been studied in detail and presented in previous works (Murray *et al*. 1996; Murray 1997; Bourne *et al*. 1998; Murray *et al*. 1998*b*). The properties of these materials are covered in tables 1–3 of another paper (Bourne 2006).

The grain distributions in the two materials are different. The 880 material has grain size variation and contains elonged grains. It has considerable porosity. The 999 material has many more uniform grains and much less porosity. In both materials there is evidence of some residual strain within the grains.

## 3. Numerical

As in the previous paper (Bourne 2006), the calculations described here were performed using a multi-material Eulerian hydrocode Eden (FGE Ltd; Milne 2004). It was realized that a successful description required capture of reflection and transmission at boundaries that suggested Eulerian hydrodynamics for the simulations. Since the microstructure was isotropic, it was hoped that two-dimensional simulations might be used. This meant that a degree of freedom for deformation of the material was constrained. This is clearly not the case in the real situation, yet this simplifying assumption will be shown to effectively capture the continuum response suggesting that for these materials lateral motion does not contribute greatly. This will be further commented upon in a later section. Further, the equations of state and constitutive models for the materials were kept simple. The material constitutive descriptions were elastic– perfectly plastic, isotropic models. The pressure was modelled using a Murnaghan model constructed using experimentally derived parameters (Barker & Hollenbach 1970; Bourne & Rosenberg 1996; Millett *et al*. 1997). Table 4 in Bourne (2006) presents the constants used in the models employed. The microstructure was digitized directly from micrographs where some grain pull out had occurred. This in fact made the 999 composite 98% alumina and 1% voids (not precisely the properties of the continuum). In the case of the 880, the digitized micrographs showed only 80% alumina, 19% glass and 1% voids. As with the 999, these values are retained in the calculations to observe resulting effects as they result from the units digitized.

The mesh resolution was checked to ensure convergence and it was found that 32 cells across the base of the simulation were required to capture the grain morphology. Figure 1*a* shows a representation of the microstructure of 999 alumina to the left, and of 880 to the right, achieved after digitization. The glass phase bonding them together has been omitted. In previous simulations, pores and fractures were modelled to assess their effect. It was found that their presence did not affect the continuum compressive response but rather set up sites for later failure. They did, however, delay the shock front. Thus, they are omitted in the majority of the simulations conducted here. In those discussed below, the material was represented in a uniform rectangular mesh with the number of elements chosen so that each mesh element spanned 250 nm.

The link from mesoscale to continuum was drawn by extending the simulation to run down many rows of the unit structure of figure 1*a*. In one simulation, the maximum lateral motion that occurred within the lattice was calculated to be less than 5% of the lateral dimension (less than 400 nm). It was thus assumed that a single structure could be stacked (with rigid lateral boundaries) to minimize the number of cells necessary. Using the same cell, and stacking regularly however, gave periodic effects that resulted in a superposed ringing on the recorded histories corresponding to waves bouncing within the regular structure. To combat this effect, the microstructure was used to generate four states by reflection and by rotation. These states were assigned by generating a random number sequence (figure 1*b*) from which the configuration of the stacked cells was fixed. The result is shown in figure 1*c* where a sample column from a simulation is reproduced.

The segment in figure 1*c* is 40 μm in height and includes four units of the 999 microstructure. The base of the segment is 300 μm from the impact point and the shock wave is around two-thirds of its way up the column at this time. The shades are contours of pressure in the stack. Each of the four units of microstructure has a different rotation, but all have a straight upper and lower boundary so that they may be stacked with no difficulty. One of the alumina grains is highlighted in white through the column to illustrate its rotation and reflection.

The composite microstructure is a three-dimensional assemblage of alumina grains, bonded by glass matrix. Single crystal (sapphire) elastic properties are anisotropic, but the simple models used here take no account of directional strength differences at the grain scale, even though alumina grains are under elastic compression at the lower impact velocities (Graham & Brooks 1971). Since the composite is isotropic, directional models were not used since there were many randomly oriented grains (thousands in each simulation). Two dimensions make representations of grains columnar, and interaction between these and other microstructural entities will be stronger in three dimensions since there is an extra degree of freedom. Nevertheless, in this work, under uniaxial compression, it is assumed that these effects are small and results suggest this to be the case. The glass phase was chemically similar to soda-lime (SL) glass and its dynamic properties were employed. The horizontal boundary between cells has no glass within it and because of its orientation does not perturb the shock in any way.

There must be some means of verifying the performance of this approach in reproducing the behaviour noted for the continuum response of polycrystalline ceramics subject to shock loading. The simplest link is to compare the behaviour (averaged over a suitable area) with that observed by continuum sensors. The limitations of the performance of the desktop machine used here did not allow as large a simulation as would have been desired. However, it will be shown that what could be achieved allowed links to sensor records observed in experiment.

## 4. Results and discussion

In the following sections, the experimental results for impact on the two aluminas will be reviewed. Then, simulations for impact onto the unit cell configurations of each material will be presented. With this background, the consideration of the quantitative aspects of the response will be considered. The phase equilibration at the impact face of a cell will be addressed to define a response time for the microstructure. The utility of this figure will be addressed as a measure of the required resolution of experimental instrumentation for investigation of these phenomena. This will then be followed by presentation of the measurements station histories recorded in simulation. First, these will be observed for a single cell, then for an assemblage of them. Finally, these will be compared to the experimentally determined stress histories.

Figure 2*a*,*b* shows the impacts of unit cells of 880 and 999 aluminas onto a rigid boundary at 500 m s^{−1}. In this reverse ballistic simulation, the wave velocity is slower than in the case of impact onto a stationary target by an amount equal to the impact velocity. The simulations conducted here show that the deformation of both materials, and thus the elastic limit (HEL) of the material, is driven by the plasticity within the grains. It will be seen that there is a dispersive wavefront, particularly in 880 (the lower purity material), but that there were no tensile stresses created within the alumina grains, even around inhomogeneities. There is an inhomogeneous pressure field behind the front, and the 880 material is more dispersive since the pressure here is *ca*. two-thirds of that in the 999. There is no indication that the glass binder phase reaches conditions where it fails. Rather, the composite microstructure, and the release effects of other phases control the yield. Indeed experimentally, the decay of the elastic precursor wave was greatest in the lowest purity alumina, which suggests that these effects were dominant. The calculated bulk stress, and pressures for these impacts are tabulated in table 1.

Porosity has a major influence at this scale, since voids disperse the shock. When favourably oriented, they may also cause tensile zones. This view sees the shocked region as one of local wave focusing and release, leading to inhomogeneous pressure fields acting through the microstructure. The pores do not collapse at lower impact velocities since the alumina grains may be within their elastic region. However, glass boundaries between grains may fail to allow comminution of the ceramic to occur. Qualitatively similar features are observed for the two higher impact velocities.

### (a) Impact of a microstructural unit

Table 1 shows the calculated conditions within the bulk of an 880 or 999 target using the derived Hugoniots for the materials and measured shear strengths from experiment (Murray 1997; Bourne *et al*. 1998; Millett & Bourne 2001). The pressure at the HEL, *P*_{HEL}, is calculated using the average of the three principal stresses (*σ*_{ii}) thus,(4.1)Further, one may use the measured strength of the material (Bourne *et al*. 1998; Bourne & Millett 2000) to convert to the hydrodynamic pressure in the material using(4.2)where *σ*_{x} is the longitudinal stress, *P* is the hydrodynamic pressure and *τ* is the shear strength.

### (b) Phase equilibration at the impact face

For one simulation, where the impact velocity was 1000 m s^{−1}, a station was placed in an alumina grain, and within a glass matrix region at the impact face. The pressures were noted at each of these points and the time taken for the pressure state behind the shock to attain equilibrium was noted. The pressure histories are shown in figure 3. The pressure in the alumina rises quickly and reaches *ca*. 45 GPa, while that in the glass reaches *ca*. 15 GPa. The pressure in the alumina then drops and that in the glass rises until they reach a constant value of *ca*. 30 GPa over a time-interval of *ca*. 2 ns.

It is possible with experimental Hugoniot data to determine the state of the material when impacted (equation (4.2)). Using the measured Hugoniots for sapphire and SL glass, and correcting for the measured shear strength to recover the pressure, one expects 43 and 12 GPa pressure for pure sapphire or glass at a rigid boundary (Barker & Hollenbach 1970; Bourne & Millett 2000). This is in agreement with observed pressure before they step up or down to equilibrate. In this microstructure, the composite has an average grain diameter of *ca*. 3 μm and waves must travel this distance to equilibrate the pressure state (Bourne 2006). Three bounces within the microstructural element are sufficient to raise/lower the pressure to ambient values and this gives a time of order 2 ns for equilibration as observed in the simulation.

The implication of such calculations is to reinforce the inhomogeneous nature of the states within the shock. The situation modelled here only includes glass and alumina grains, but in the real microstructure there will also be voids which must close (Bourne 2006). Further, unless mesoscale sensors can be fabricated, it is not valuable to track at high sampling rates greater than *ca*. 500 MHz (for this material) without appreciation of its composite nature. Further, since the equilibration time depends upon grain size, there is a kinetic, as well as a strength effect in reducing grain size.

The digitized microstructural units (figure 1*a*) have 15 station points that are indicated in each schematic with black discs. There are two rows of five stations that start 1 μm from the impact plane, and are then spaced at intervals of 1.5 μm. There is a lateral row of stations at 10 μm from this point with 2 μm between each in the structure in figure 1*a*. In figure 1*b*, the row is at 7.5 μm with 1.5 μm between the stations. At each of these station points, a series of state variables is monitored to track the progress of the impact. Figure 4 presents pressure histories recorded at these station points. To the left are the five stations tracking the progress of the shock and to the right are the five stations across a line sampling the arrived front.

In figure 4*a*, histories for the pressure determined at two columns of station points are presented for a unit of 880 impacting a rigid plane at 500 m s^{−1}. Before commenting upon the observed response it is useful to calculate the expected pressure levels for elastic and the plastic parts of the calculated history. The pressure in the elastic region is the mean of the principal stresses (equation (4.1)) that for 880 gives 2.9 GPa assuming the measured HEL in table 2. For an impact at 500 m s^{−1}, the pressure is expected to reach 11 GPa using the Murnaghan equation of state with the constants defined in Bourne (2006). The same parameters for 999 are 6.0 and 16 GPa, respectively. The histories at points moving away from the impact face are on the left-hand side of figure 4 and an impact at 500 m s^{−1}, the results from each of the two columns of stations are presented as either solid or dotted lines to distinguish the two locations. For 880 (right-hand side of figure 4*a*), the first observation is that there is a scatter in the values recorded, and that the histories themselves are noisy. Although stations are at the same distance from the impact plane, the times of arrival of the waves are also scattered. However, qualitatively one can see features of the flow that are reminiscent of observations made at the continuum scale.

Looking at the stations on the left-hand side, the wave arrival times are reasonable, and one may recover the bulk sound speed from the elastic arrival of the pulse. It is difficult over such short run distances to discuss shock velocities. The first two histories show breaks in the rise at above the expected elastic limit, but the environment of the measurement station results in other positions seeing differing values. The scatter of the behaviours at the two tracked lines in the microstructure makes further conclusions difficult to draw. To the right-hand side of each figure, five stations along a line equidistant from the impact face are present. The stations along a line 10 μm from impact to the right-hand side of figure 4*a* show similar pulse shapes but of different amplitudes. One (the lowest) was discounted as the station moves across the boundaries of a moving grain within the microstructure. The solid line is an average of the histories at each station and it rises to around the correct elastic level but is low behind the plastic front. It should be emphasized that there is scatter of *ca*. 50% of the pulse amplitude at these stations along a line of stations.

Conversely, to the left-hand side of figure 4*b*, the 999 microstructure allows much more regular and predictable behaviour. The elastic part of the pulse rises to around 8 GPa at the first break in slope and then tops at the expected value for the dotted trace, but overshoots that value for the solid one. At the second station, both locations record pressures below those expected. However, by the third, the pulse rises sharply, shows a convex curved aspect and then rises up a well-defined shock to the expected peak pressure. At later locations, the dotted histories do not reach the expected pressure values. To the right-hand side, the five stations at 7.5 μm from the impact face show broadly similar forms. The averaged value gives a rise to an elastic break at *ca*. 5 GPa and then moves up to 15 GPa by the end of the pulse. These amplitudes are broadly what was expected of this material. There is some scatter in the station histories about this mean; the peak pressure is 15±3 GPa at 0.6 ns.

Comparing the effect upon the response of two microstructures, there are several features worthy of comment. Firstly, the histories are much cleaner for the 999 than the 880. Secondly, the rise-time at distance from the impact face is much faster for 999 than 880. The shock fronts are better defined, and the scatter in the data from different positions is markedly less for the purer material. Both aluminas show higher elastic limits near the impact face than further away, consistent with the equilibration necessary in the microstructure to accommodate inhomogeneities in the pressure state behind the shock. This is reminiscent of the observed precursor decay at the continuum scale.

### (c) Homogenization

The simulations outlined above, allowing a shock to disperse over only *ca*. 10 μm, do not approach conditions at the continuum scale. However, it may be noted that for the 999 at least, the results have features that can be recognized from laboratory experiments on millimetre-scale targets. In order to make a closer connection, a means of scaling up the calculations was described above. A randomly generated column of microstructural units in various rotations was stacked until a column of height *ca*. 500 μm had been constructed. Rows of five stations were positioned at 10, 110, 210, 310 and 410 μm from the impact face in each simulation. To speed the cell generation and the calculation time, the porosity was omitted from these calculations. Thus, the fractions of alumina and glass in the calculations described below for 880 were 80% alumina, 20% glass, and for 999 were 98% alumina, 2% glass.

The results of impacts up to 1500 m s^{−1} have been described above and relate to stresses up to *ca*. 50 GPa. In the experiments conducted, it was not possible to reach such amplitudes. Thus, a range of impact velocities of 250, 500 and 750 m s^{−1} were chosen (again onto the rigid boundary) so as to induce stresses up to *ca*. 25 GPa. A summary of shot conditions with calculated pressures is given in table 2. In all the simulations, the left and right boundaries in the reverse ballistic impact were also rigid.

Figure 5 shows results of an impact at 750 m s^{−1} onto a rigid boundary. The results are presented pictorially in terms of distance–time diagrams for the impact. Each figure consists of a left- and a right-hand part of the frame that show an overview, and a region, of the detailed distance–time representation of the behaviour in the impact. The colours refer to the pressures generated in the materials; the left-hand images show no grain boundaries while those on the right-hand side have them superimposed. The ambient conditions are represented as a green background colour. In this region, the material is flowing towards a rigid boundary at *x*=0, and this rearward velocity may be seen in the images to the right-hand side to give backward-angled, linear features. After impact the shocked material (which rises up to a yellow colour) is stationary. Thus, velocities calculated back from these representations must be adjusted for the impact speed. The same is true for the images on the right, but the lack of grain boundaries means that there is no feature present that allows observation of the effect.

In these simulations, the elastic front can be seen to rise over a greater spatial distance for 880 than 999. This wave dispersion in the diffuse 880 material has been noted earlier in the simulations of the microstructural cell. Nevertheless, it is possible to analyse these sequences to recover the wave speeds for the elastic and the plastic fronts. These can be more easily recovered from the station histories that will be presented below. The contours of pressure behind the shock show clearly that there are inhomogeneities within the flow on the macroscopic scale from which failure can propagate. These will send waves to the compression front. However, the simulations show that the front itself travels with constant velocity without any velocity fluctuations at these resolutions.

Figure 6 presents the station histories for impacts at the three velocities onto the two aluminas. Each of the histories represents the averaged pressure at the distances 10, 110, 210, 310 and 410 μm from the impact face, and each is the average of 10 station points at these locations. This removes much of the variability noted in the simulations of figure 4 at stations on the same plane from impact. However, there is only averaging over a distance range of 10 μm, whereas sensors in experiment are of order one hundred times greater. That being said, this does appear to define a region of continuum response for the microstructure. Both alumina materials exhibit stress histories of similar forms. The three impact velocities chosen for simulation are plotted together. The lowest impact shows a rise, followed by a convex region up to the Hugoniot stress. This is similar to the observed behaviour of polycrystalline ceramics. The intermediate level rises as above, and then the pulse steepens as it rises to the maximum stress. The highest impact speed rises further to reach the maximum stress level.

The 999 histories (figure 6*b*) are much cleaner than those of the 880 (figure 6*a*) that mirrors the dispersion in the microstructure seen earlier. The pulses rise faster and show less superposed noise for 999 alumina. The 880 pulses rise to 4.5, 4.0, 3.3 and 3.0 GPa at the first four stations. The full rise is not captured at the fifth. This is in accord with the decay of the elastic precursor observed at the continuum level for this material (Murray *et al*. 1996). The limit is decaying less as propagation distance extends and the pressure value has reached the value of *ca*. 3 GPa by 400 μm, which is close to the calculated pressure of 2.9 GPa (table 2). Thus, the mesoscale simulation has recovered close to the measured elastic limit. In the case of 999 (in figure 5*b*), the first breaks in slope of the elastic rise are at 7.0, 6.0, 5.5, 5.0 and 5.0 GPa at each of the stations.

Again the limit shows precursor decay and finally achieves a pressure close to the calculated value of 5.9 GPa (table 2). The clearer pulses show other steadiness effects within the waves. The fourth station shows elastic precursor amplitudes that are different for each of the stress levels with the highest amplitude corresponding to the greatest stress. This behaviour is consistent with continuum measurements of the same effect (Murray *et al*. 1998*a*).

The rise-time of the elastic pulse is a feature of both histories. In each case, there is a region close to the impact face where the rate of rise is decaying with distance, however, this has stabilized to constant values by the second or third station. Thereafter it is *ca*. 2 ns for the 880 alumina and 250 ps for the 999 material. This is a reflection of the equilibration time of the three phases in the composite. This has been discussed above and a time of 2 ns was recovered. In the case of 999, glass phase is found in fine orifices less than *ca*. 500 nm which determines the short rise times recovered from simulation. Shock front widths, on the other hand, are numerically approximated with viscosities at the continuum scale, but here a physical explanation relates to the zone widths that can be seen in microstructural simulation such as those in figure 2. Here, shock widths were determined by grain morphology and porosity dispersing the wavefront, and widths of 6 μm for 880 and 2 μm for 999 were recovered corresponding to rise times of *ca*. 750 ps for 999 and 2.25 ns for 880.

The shock pressures for the 880 are calculated to be 6, 13 and 21 GPa, which compare favourably with the measured 7, 14 and 23 GPa. It will be noted that there is some jitter in the values of pressure achieved. Indeed, this amounts to 2 GPa for the pressure at the third station. This is the result of averaging over a region of only 10 μm. Those for the 999 material are 7, 17 and 29 GPa in the simulations, which compares with the measured 8, 16.6 and 26 GPa. To conclude, the derived measurements of elastic limit and of the peak state achieved agree well with the measured continuum measurements. Note that the input variables for the calculations are the numerical descriptions of individual phases (Bourne 2006). Comparisons drawn in the discussion above are with the experimental values not with comparisons with derived numerical constants for the composites. A discussion of the derivation of the material descriptions from the constituent phases will be given below.

The arrival times of the waves give valuable information concerning the derived shock constants for the material. The experimental data is most complete for the 880 material and so this was chosen as the candidate to best illustrate the analysis: 999 shows similar results and levels of agreement. The arrival of the shock pulses is taken to be at the point at which the maximum (Hugoniot) stress is achieved. This value is enumerated for the highest impact velocity (750 m s^{−1}). The times of arrival, the known positions of the measurement stations and the fixed initial velocity resulting from reverse impact can be used to infer a shock velocity of 8.75 mm μs^{−1}.

### (d) Mixture rules

The first means of deriving the parameter *c*_{0} (the intercept at zero pressure of the shock velocity particle relation) is to equate it with the bulk sound speed of the material thus,(4.3)where *K* is the bulk modulus and *ρ* the density of the material and *c*_{L} and *c*_{S} are its longitudinal and shear wave speeds. This yields a value of 6.6 mm μs^{−1} which is clearly too low for such a ceramic. A linear mixture rule for the calculation of *c*_{0} and *S* from the data yields(4.4)where *α* and *β* are the volume fractions of alumina and glass, respectively. Application of such a rule yields 9.8 mm μs^{−1} for *c*_{0} and 1.0 for *S* using the values for sapphire and glass (Bourne 2006). The value obtained is clearly too high to explain the results. Thus, a *reduced c*_{0} is calculated where(4.5)that has been shown to successfully describe the behaviour when the response is hydrodynamic (Milne *et al*. submitted). Here, *γ* now describes the volume fraction of porosity within the microstructure. Setting *γ*=0 (as in simulation) yields 8.4 mm μs^{−1} for *c*_{0}. The calculated shock velocity for this impact speed is thus 9.1 mm μs^{−1}. This is in better agreement with the velocity from the simulation. Thus, the reduced wave speed approach is suggested as a successful analytical form to recover composite shock velocity in the more complex simulation. Recall that *γ*=0 in the simulation and so putting in a value for the porosity reduces *c*_{0} further to 7.3 mm μs^{−1}.

Continuum measurements have determined the value of *c*_{0} and *S* for 880. These values were 7.7 mm μs^{−1} and 1.2, respectively. The data used to deduce these values were, however, not from experiments designed to recover Hugoniot data, so there is some error in these derived constants. However, the value of *c*_{0} recovered by this means, using purely constituent phase data (Bourne 2006), is in the range of that obtained by adding a porosity correction to the reduced value. Using manufacturers' data (Bourne 2006) for the composite alumina (which results from data gathered on the impact phases put into the processing and on density measurements after the processing) gives values for *c*_{0} of 10.0 using the simple rule of mixtures (equation (4.4)), and 7.5 using the reduced value approach (equation (4.5)). The latter yields the best agreement with experiment and is within the uncertainties in the measurement. It is clear that adequate agreement must take into account the porosity. In order to observe the importance of strength, one simulation was run with none. The pressures and wave arrival times were the same for the inelastic component of the wave. Finally, it is noted that the mesoscale calculation predicts a shock wave speed of 8.75 that compares with an experimental value of 8.7 mm μs^{−1}. These comments have application to derivation of shock parameters for other composite systems.

Applying equation (4.5) to 999 for this pressure (close to the HEL) yields a value of 11 mm μs^{−1} that is high (greater than the longitudinal wave speed for this material). Thus, the form of the rule needs to be further developed to take into account the different operating mechanisms in these materials. In particular, the use of the equations is best suited to the hydrodynamic regions where strength is not an issue, whereas at these speeds in 999 some of the grains are still deforming purely elastically. Also not that for this simulation, porosity was absent. However, one clear result is that for such HIPed materials, the inclusion or not of glass into the intergranular regions makes little difference to the calculated compressive wave speeds. To verify this numerically, a further simulation was undergone in which a target with no glass content but with void between the grains was impacted, and both the pressures and wave speeds were identical. This behaviour under compression would clearly differ from that observed under tension. Removing the glass phase from 880 to yield a granular, loose-packed alumina did delay the shock and further lowered the pressure as expected.

## 5. Conclusions

There is a range of observed behaviours in the behaviour of polycrystalline ceramics that require greater understanding. It is suggested that mesoscale mechanistic detail may help to explain this continuum response. In particular, it is necessary to advance continuum models beyond the semi-empirical descriptions that are most widely used at present. One lesson may be drawn from the study of composite explosives, where localized high temperatures lead to chemical reaction. There mesoscale simulation has demonstrated the inhomogeneous states that occur in an effort to make continuum models more physically based.

This work has addressed shock-loading in compression of two aluminas. No release has been allowed from free surfaces, so that there has been no tension introduced unless it has propagated from microstructural defects such as pores introduced during processing. A methodology has been adopted which is based around the concept of the microstructural unit. The size of the unit and its composition must be carefully chosen to correctly reproduce continuum behaviour. Similarly, the mesh resolution chosen, to adequately capture details of the important phenomena, must be matched with the physical dimensions of the controlling microstructural elements. In this work, the cells have a maximum dimension of 250 nm, and are usually smaller to capture details in the flow.

The simulation method has used randomly stacked unit cells between rigid boundary walls, and thereby illustrated several features of the mesoscale flow that relate to the continuum. The three phases of the composite investigated, and their morphology, define equilibration times within the microstructure. These are on a scale of order 2 ns which defines a minimum, useful sampling rate for sensors in the flow. At the mesoscale, porosity has a major influence. However, removing it for continuum simulations did not affect response in compression for these low porosity levels. Even the removal of the glass phase had little effect for the HIPed material. Nevertheless, these phases set the three-dimensional flow at the mesoscale. They result in the shock fronts being dispersed, with purity affecting the magnitude of the effect. But regardless of purity and front thickness, the propagating boundary settles to constant velocity in either material.

Various simple mixture rules have been investigated, and unsurprisingly they apply better to the hydrodynamic than the strength-dominated regimes. Nevertheless, the simulations do recover some of the operating elastic observations such as precursor decay. Further, features of the pulse, such as the rounding above the elastic limit, are also captured. It is believed that the start of rounding in the pulse corresponds to the yield of the weakest phase, and the end of this region corresponds to yield of the stronger one. The stress levels at which these are seen in pulse shapes depend on the kinetics of the processes occurring, and phenomena such as precursor decay will be prevalent in impure microstructures as observed (Murray *et al*. 1996). Nevertheless, further study of the strength laws for composites is necessary.

It is undoubtedly the case that the poor tensile strength of brittle materials results from micro-fracture in the bulk of the material, but it seems unlikely that this is dominant during the compressive portion of the loading. The HEL has been hypothesized to correspond to a condition at which damaged zones might join, interact and subsequently lead to failure of the material. This work has indicated that the HEL is a consequence of grain plasticity, and this has been supported in recent shock recovery work on AD995 where intergranular plasticity was observed (Chen *et al*. in press). Nevertheless, in tensile loading, the lower strength glass/porous matrix will fail more easily than pure alumina grains and grain boundaries will be the favoured routes down which these fractures might extend.

The success of the simulations in capturing the correct pulse amplitudes for both the elastic and inelastic regimes, and the observed pulse shape for these materials, is encouraging and opens the way for further investigation. The inputs to the model are simple, and the numerical technique does not include friction or failure, and yet the behaviour, at least in compression, is well reproduced. Thus, the range of mechanisms necessary to include in an improved continuum constitutive model appears small. However, compression and tension result in very different responses, and the next phase of this work will be to include release to capture mesoscale effects on tensile failure models.

## Acknowledgements

This work could not have progressed without the materials' study and continuum measurements of Dr Natalie Murray. Dr Alec Milne inspired the mesoscale simulations, the numerical platform and the multiphase flow insights.

## Footnotes

- Received July 14, 2005.
- Accepted February 27, 2006.

- © 2006 The Royal Society