Size Reduction

Summary

This report describes a twodimensional computer model that is suitable for studying flows of dry particles in which the particles are allowed to break. The model is based on discrete particle computer simulations. Here, macroscopic polygonal particles are constructed by ugluingn together small elements (hereafter referred to as “elementsn). Depending on the stress conditions the glued bonds can respond elastically, undergo plastic failure or break, allowing cracks to propagate across the macroscopic particle along the boundaries between their microscopic constituents.

In essence, this process creates a simulated material. The manner in which the microscopic elements are organized has an effect on the mechanical behavior in much the same way that the organization of molecules affects the behavior of a real solid. For example, the material must have internal, crystal-like slip planes in order for the resultant material to exhibit plastic behavior. Similarly, the elastic behavior of the bulk material depends on both on the the microscopic element shape and on the spring constants used to model the interactions of the microscopic elements. Consequently, many element shapes and arrangements of elements have been investigated.

Some examples are presented, including compression failure of a rectangular sample, the impact of particles with a plate or binary impacts of particles. Some preliminary simulations of the Utah ball-drop experiments have also been performed that show good qualitative agreement.

Introduction

Attrition is a ubiquitous problem in processing and handling of particulate solids. It causes dust formation, and has a detrimental effect on product quality and the reliable operation of process equipment. The research coordination of IFPRI identified the need for a better understanding of the various mechanisms involved in attrition in order to provide a fundamental base on which to address the problem. The objective of our research programme is therefore to elucidate the mechanisms of attrition of particulate solids, and to relate the rate of attrition to the material properties and to the loading conditions to which the particles are subjected.

A predictive model of impact attrition of particulate solids having a semi-brittle failure mode was developed in the previous IFPRI research programme. The model has been applied to the analysis of attrition propensity for several species of ionic crystal. This report summarises the results obtained in the past year on the effect of particle size on the attrition rate. Furthermore, the approach developed for the analysis of impact attrition has been extended to modelling wear of single particles. A summary of the literature survey is also included in this report.

The mechanism of attrition considered here is a chipping process, where small quantities of material are removed from the surfaces around the comers and edges of the particle. Our previous work has shown this to be an important process in the impact attrition of ionic crystals with a semi-brittle failure mode in the velocity range up to about 40 m/s. Fragmentation of the whole particle occurs at higher impact velocities or for materials with a relatively low value of toughness. However, this mechanism has not been addressed so far in our work.

In the chipping process, material removal is caused by the initiation and propagation of sub-surface lateral cracks. These cracks form readily during the unloading stage of an elastic-plastic deformation, and are driven by the residual tensile stresses produced by the plastic flow. Therefore, the analysis of impact attrition is based on the fracture mechanics of sub-surface lateral cracks. A dimensionless parameter has been derived from this analysis, which represents the volume fraction of material lost from a single particle by the formation of such cracks:

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where H is the hardness, p is the density, U is the impact velocity, 1 is the linear dimension-of particle, Kc is the critical stress intensity factor, and + is the constraint factor defined as the ratio of the hardness to the yield stress. The parameter IJ quantifies the attrition propensity, and it includes all the relevant material properties and impact conditions. The fractional loss per impact 4 is considered to be some function of q. In the first instance the existence of a simple linear relationship has been explored:

where a is the proportionality constant.

To verify the theoretical predictions, ionic crystals with a cubic habit such as MgO, NaCl and KC1 have been used as model materials because their structure and properties are well-characterised. The dependence of fractional loss per impact on material properties and impact velocity was verified previously for 2 mm melt-grown crystals (Ghadiri and Zhang, 1992). A linear relationship between the fractional loss per impact and the particle size is expected, but this could not be shown unambiguously in the previous work. To examine the effect of particle size on attrition, it is necessary to measure the fractional loss per impact for several particle sizes while keeping other parameters constant. The earlier work on the size effect used commercial solution-grown PDV salt crystals in the size range of 355-500 pm. This material contains a large number of polycrystalline particles which split into individual crystals on impact, hence obscuring the mechanism of chipping under verification. To combat this problem, a special experimental procedure involving repeated impacts was developed in order to quantify an asymptotic value of fractional loss per impact. The asymptotic was considered to represent chipping, as at that stage all the polycrystalline particles had split into smaller individual crystals and been removed from the sample by sieving. This procedure has been reported in the Final Report (FRR 16-03) of the previous IFPRI project (Ghadiri and Zhang, 1992). The repeated impact technique was however considered unsatisfactory because the number of impacts could also influence the fractional loss due to work-hardening of the comers and edges of the crystals, or purely as a result of the gradual change in particle size as the test proceeds.

The choice of solution-grown crystals rather than large melt-grown crystals for the tests was at that time associated with the fact that the bore of the air-eductor used in our experimental device was too narrow to allow particles larger than 3 mm to pass through. Recently, a new attrition rig with a larger bore was constructed to overcome this shortcoming. The results of the experimental work investigating the effect of size on attrition rate for relatively perfect, large melt-grown crystals is described in this report.

In the past year some work has been carried out to verify the existing models of lateral crack formation in ionic crystals as well as other materials of interest. A new microscopic technique, the Confocal Laser Scanning Microscope has been used for this purpose, and the results are currently being analysed. This new technique is now available at the University of Surrey, and it will allow us to carry out further characterisation of sub-surface damage arising from impact or quasi-static contact.

Summary

In this research project it was intended to examine the existence of the grinding limit of fineness of product powder mainly by inliquid grinding method using media mills, such as vibration and planetary mills, and to find out the factors which determine this limit, and the laws which govern the rate of grinding to approach to this limit value.

First the results of research works on the limit fineness in dry grinding, mainly about those of the authors, were reviewed and its general tendency was conclusively summarized. And then it was confirmed using a planetary ball mill, which had shown very high grinding rate with high acceleration number, that the equilibrium particle size, or limit fineness, does exist even in &-liquid grinding, when the size is expressed by 50% average diameter, though in some cases the limit fineness has not been found. This equilibrium size reduces with decreasing ball size and is well correlated with the force excerting on a single ball by mill pot. On the other hand, the limit size obtained as specific surface area by BET gas adsorption method is found to be much smaller than the 50% diameter and also independent of the grinding conditions within most of the present experimental range.

The laws to describe the approaching process to the equilibrium state was also examined and it was found that Tanaka’s law which includes the limit specific surface area in the equation as a sort of saturation state, is not valid. More simple relation, which can be approximated to Rittinger’s law is approved as a general law until the limit fineness is attained. That means that the factors which determine the rate of grinding and the limit fineness have no or only little connection each other. This fact was also approved by the simulation calculation of the lshifting process of size distribution of ground product.

It was also found that Rosin-Rammler’s formula is extensively approved to be able to present the size distribution of ground product even in this micron and submicron order size range.

ABSTRACT

An experimental rig has been built to investigate breakage phenomena similar to that encountered in air jet milling. Particles of alumina hydragillite are accelerated by air in a nozzle and impacted on a target. Energy loss during impact is evaluated by measuring particle velocities before and after impact using two different methods: one based on observation with a high speed shutter video camera, the other based on cross-correlation of signals from an emitter-receptor optical fibre system. The particle size shape and distribution of the particle fragments from the apparatus are determined as a function of: solids loading, air flow rate, orientation and the material used for the target.

More detailed structural analysis of the debris is made by morphological characterisation based on fractal dimension. The basic hypothesis is that the breakage mechanism and kinetics depend on the initial fault network in the original particles. This is used to establish a fragmentation scheme for alumina hydragillite. Initial experiments with particles produced as a function of time in a batch operated air jet mill confirm the possibilities of the method of analysis.

Executive Summary

This report describes new size reduction work using a computer simulation of solid fracture. The details of the simulation technique were described in last year’s report. For this model, rigid elements are assembled into a simulated solid by “gluing” the elements together with compliant bouridaries. The joints fracture when the tensile strength of the glued joints is exceeded permitting a crack to propagate across the simulated solid. The great value of a computer simulation is that literally everything is known about the simulated system and is accessible to the computer experimenter. In addition, the simulation allows the independent variation of material properties such as Young’s modulus, Poisson’s ratio and ’work of fracture - flexibility, that in the laboratory, is limited by available test materials. Consequently, a computer simulation offers the ability to make investigations with a detail that is unthinkable to duplicate experimentally. As such, simulation techniques may be invaluable in areas, such as comminution and attrition, in which the actual problem is so complicated, and for which so many events happen in so short a time, that experiments are historically limited to performing post-mortems on the fragments.

The first detailed application of the simulation was to study the impact of single, disc-shaped, particles on flat plates. Where comparison is possible, the simulation appears to accurately mimic experimental results. This study shows that the size distributions are, as would be expected, most strongly dependent on the collisional energy. Of secondary importance is the ratio of the impact velocity to the sound speed within the solid material; detailed observations, permitted by the simulations, suggest a physical mechanism for this behavior. Finally, the size distributions are nearly independent of Poisson’s ratio as the tensile loads that result in breakage result from the inertia and geometry of the impact and do not depend strongly on the elastic properties.

In the past year we have also developed a 3-D version of the simulation (which to this point has been limited to two-dimensional problems.) This model has proven to be very computationally intensive. As a result, we have only applied it to a handful of impact examples. However, we have recently begun to probe deeply into the source of the problem and have discoverd ways of significantly improving the performance of the simulation. We hope to implement these shortly.

The two-dimensional model has been used to investigate Bridgewater’s IFPRI sponsored shear cell experiments on particle attrition. In one of his experiments, he notices a distinct change in the slope of his Gwyn rate curves when very low normal forces were applied to the sample. An examination of the fragments indicated that this was accompanied by a change in breakage mechanism, from corner chipping to pervasive fracture of the particles. Our simulations, duplicate the change in breakage mechanism, but indicate that the observed change in slope is due to a misinterpretation of the experimental results. In fact, the degree of breakage was observed to be directly proportional to the work performed. We have also extended the two-dimebsional model to cases in which the particle experiences large deformations. The rigid, element simulation was restricted to small deformations as the elements could not change their shapes and, thus, could not conform to any general body shape. The current extension creates a hybrid model by applying portions of the finite element technique to allow changes in the shapes of elements.

This is implemented at the element level and no global stiffness matrix is assembled; instead, the elements interact across the same compliant boundaries used in the rigid element simulation. As a result, there is no need to assemble and invert a global stiffness matrix as in the standard finite element technique. As the elements may deform in response to an elastic/plastic stress-strain law, this allows the inclusion of realistic plasticity into the model. (As discussed in last years report, only approximate plasticity could be incorporated into the rigid element model.) This model also has two unintended but beneficial side effects. The first is that, if the joints are made stiff compared to the elements, then the elastic properties of the body are determined by the elastic properties of the elements themselves; consequently, the elastic properties of the body do not change, even when it is broken into individual elements. (For the rigid element simulation, only bodies eight elements across, retained the elastic properties of the full macroscopic body.) Also, the deformation of the elements is performed in response to the stress state the element experiences. Consequently, the stresses inside the elements are computed as a natural part of this procedure permittin the internal stress state to be determined even down to the the scale of elements. For the rigid (” element model, the stress state would have to be computed by averaging the forces in the compliant contacts of many neighboring elements.) Finally, we have performed some preliminary simulations of the effects of collision geometry on particle fracture. These show that a conforming, or near conforming, impacts generate strong elastic compressive waves that, when reflected from free surfaces as a tensile wave, can generate very fine fracture. This is a completely different mechanism than that which generates breakage in non-conforming collisions such as the single particle impact studies discussed in the frrst section. This was a rather controversial result and one that will probably turn out to have no useful applications (other than increased understanding of breakage processes). However, while we have not been able to fmd experimental observations of this effect, we have been able to successfully address all objections that are raised against it. Along the way, some of the results suggested that there may be an effect of relative particle size in the breakage induced by the impact between two particles. We hope to be able to explore these possibilities in the near future.

SUMMARY

A mechanistic model of impact attrition has been developed in the previous IFPRI programme (Ghadiri and Zhang, 1992). The model was tested on ionic crystals that were known to fail under semi-brittle mode, and showed that the theoretical predictions agreed reasonably well with the experimental results. In the current programme the work is extended to glassy polymers since they represent a category of materials that is completely different from ionic crystals. Polymethylmethacrylate (PMMA) was selected as a model material because it is the most common of the glassy polymers, it is readily available, and has a wide variety of applications. The PMMA particles used in the experiments were cubic extrudates, provided by ICI, in the size range 2.36-2.80 mm.

The primary objectives of the work are as follows:

1. To identify the failure mode of PMMA particles under impact loading, by carrying out single particle impact testing, high speed photography of the process of impact, and scanning electron microscopy (SEM) of the impact damage.
2. To determine the velocities at which transitions occur from plastic deformation to chipping and from chipping to fragmentation.
3. To assess the impact attrition propensity of polymethylmethacrylate by evaluating gravimetrically the mass loss per impact as a function of impact velocity.
4. To analyse the data by comparison with the predictions of the model of impact attrition developed previously.

Single particle impact testing and high speed photography of the impact event show that PMMA fails in the semi-brittle mode under the prevailing high strain rates. However, PMMA shows a significant amount of ductility under quasi-static conditions as reflected in the relatively large size of the plastic zone when compared with the dimensions of the specimen.

The results of single particle impact tests reveal two distinct transition velocities. The first marks the onset of chipping and is about 25 m/s, while the second marks the onset of fragmentation and lies around 89 m/s. The above transition velocities apply only to the particle size used, i.e. 2.36-2.80 mm. However, the threshold velocities for other particle sizes have been estimated, based on the minimum load required for initiating various types of crack. For example, when the particle size is around 1 mm, the impact velocities for the onset of chipping and fragmentation are estimated to be approximately IS6 m/s and 556 t/s, respectively. These velocities are more relevant to comminution than attrition processes.

The SEM photographs reveal the occurrence of extensive plastic deformation beneath the impact site. The morphology of the impact site indicates that chipping is responsible for material loss.

The results of the impact attrition experiments show that attrition becomes appreciable only at very high velocities. In the range of impact velocities of interest to attrition, i.e. up to about 30 r&s, the attrition rate is very small. Within the chipping regime, the attrition rate is proportional to the impact velocity to the power of 234. If the fragmentation regime is taken into account, the power of velocity can reach up to 5.69. The attrition behaviour of PMMA does not follow the trends predicted by the model of impact attrition developed previously. The exact reasons are not clear at present, but it is considered that the lateral crack propagation in PMMA may not follow closely the model of lateral crack extension used previously. Formation of subsurface lateral cracks in PMMA requires further detailed investigations. In conclusion, PMMA appears to be an attrition-resistant material at least up to moderately high velocities.

Summary

The grinding processes in ball mills are far too complex for an exact mathematical description. The well known, simple Comminu-tion Laws and the phenomenological Population Balance Model provide equations for an estimation of the dependence between feed size, energy consumption and product size distribution. Important parameters like ball size, mill filling or mill speed, however, are not included in these equations. A more detailed quantitative description of the comminution processes seems necessary. For this purpose, the complex grinding process was split into several fundamental processes:

1. Singleparticle fragmentation: the basic process in grinding.
2. Fragmentation under packed bed conditions: packing structure determines the distribution of comminution energy flowing into individual single particles in the bed.
3. Fragmentation by an impacting ball: a packed bed with position-dependent loading intensity is formed during impact.
4. Comminution in a ball mill: if the kinetic energy distribution of the balls in the mill would be known, then the comminution process could be described in terms of a) to c).

Experimental and theoretical investigations as well as evaluation of published experimental data gave following results:

1. Based on fracture mechanics, Weibull flaw size distribution statistics and Hertz-theory of contact forces, simple equations have been derived for the probability of breakage, mass specific comminution energy and fragment size distribution of single particles. The theory contains a few parameters to be determined experimentally and was tested successfully on published experimental data.
2. The contact energy distribution of particle assemblies under packed bed conditions has been determined experimentally using packed beds of polished soft steel balls. Combining this distribution with the single particle fragmentation theory according to a), allows the prediction of the fraction of broken particles in packed bed experiments. The theoretical predictions were verified with experiments carried out on glass spheres.
3. The fragmentation of particle assemblies between a stationary and an impacting ball was studied in detail with simple equipment at the University of Karlsruhe and with the ultra fast load cell of the University of Utah. Both investigations provide evidence that the height of the particle bed is reduced to only 1 to 2 particle layers when comminution begins to be effective. This has to be taken into account in further modeling.
4. The mathematical description of ball mill grinding in terms of processes a) to c) is the final aim of the theoretical development. This, however, was beyond the scope of this research project.

Comminution is an important unit process which consumes large amounts of energy very inefficiently, only 5 to 10% of the input energy going to treat new surfaces. The development of more energy efficient processing necessarily implys a better fundamental understanding of the various mechanisms involved in the fragmentation of particules.

Part A of this report presents a literature review of the subject and situates the context in which this work was performed. Part B describes the experiments performed on the impact of 5 different types of particle on a target. Parts C and D give the experimental results and a tentative interpretation of the phenomena observed. Finally, part E presents the methods being developed for morphological analysis.

This report largely concerns new work using a computer simulation of solid fracture. The details of the simulation technique were described in previous reports. For this model, rigid elements ace assembled into a simulated solid by “gluing” the elements together with compliant boundaries. The joints fracture when the tensile strength of the glued joints is exceeded permitting a crack to propagate across the simulated solid. The great value of a computer simulation is that literally everything is known about the simulated system and is accessible to the computer experimenter. In addition, the simulation allows the independent variation of material properties such as Young’s modulus, Poisson’s ratio and work of fracture - flexibility, that in the laboratory, is limited by available test materials. Consequently, a computer simulation offers the ability to make investigations with a detail that is unthinkable to duplicate experimentally. As such simulation techniques are valuable is areas such as coxnminution and attrition, for which the actual problem is so complicated and so many events happen in so short a time, that experiments are historically limited to performing post-mortems on the fragments.

The report begins with an update on our three-dimensional extension of the model. The extension was described in last year’s report. Unfortunately though, that model had proven to be much too computationally inefficient to be put to much use. We spent quite some time this year in examining the sources of the inefficiency and discovered the culprit to be the algorithm for computing the volumetric overlap of elements undergoing collisional contacts. We realized there were many conditions for which such a general algorithm was unnecessary and a more efficient, specialized approach could be employed. This created as much as an order of magnitude speed improvement and has allowed us to make simulations of single particle fracture consisting of up to 12,000 elements. About this time, we were contacted by Albert van den Bos, a student of BrianScaclett’s who was at that time performing part of his research in Reg Davies laboratory at DuPont. His topic was the jet-milling of plastics. Some of the experimental portion of his work was being performed by Professor Guigon and he wanted to know if we would be interesting in providing a simulation component and was particularly interested in polystyrene. Given the vast IFPRI involvement in the project, we agreed to participate even though it tested the limits of the simulation technique. On the down side, the simulations we performed predicted that no large scale breakage would occur. On the up side however, that prediction was borne out by the experiments which showed that only practical cracking and chipping events were observed. Perhaps the most important result of this exercise is that it pointed out the difficulty in comparing simulations of single particle events against the product of experiments whose output is a mix of the fragments produced by many impacts from a feed stock of many particle sizes. After that experience, we searched the literature for single particle events against which we could compare the simulation results and ran a simulation of one of Arbiter’s experiments of the rapid compression of soda-lime glass spheres. We were very surprised at how accurately the simulation predicted the experiment al out come nearly exactly.

The first scientific problem we considered was the effect of impact velocity on the general pattern of particle breakage. In last year’s report, we had confirmed experimental observations that showed various changes in breakage behavior that were associated with high speed impacts - among them a change in the slope of the size distribution. A two-dimensional simulation of the impact process between round particles and a rigid plate has demonstrated that there are two modes of impact i rstoccurs between the time the particle contacts the plate and the time that the contact force brings its motion to a halt (i.e. the point of maximum compression). Such a contact induces internal tensile stresses that generate a fanlike distribution of cracks that extend radially outward from the contact point. The magnitude of the energy contained by the tensile stresses and consequently the degree of breakage ca4 be related to the degree of particle deformation. Now in last year’s report, we showed that the ratio of the impact velocity to the sound speed V,/C, could be interpreted as a characteristic particle strain. Consequently, the larger the impact velocity, the larger the particle deformation and the larger the degree of Mode I breakage. Mode II breakage occurs as the particle rebounds from the plate. The relaxation of the internal stresses both extends the fanlike cracks begun under Mode I and induces azimuthal cracks that are oriented perpendicular to the original Mode I cracks. These latter Mode II cracks were concentrated near the impact points. It turns out that the induced breakage is nearly independent of impact velocity and depends only on the impact energy. Consequently, the impact velocity determines the balance between the cracks generated by Mode I breakage and those induced by Mode II. All of the observations associated with the high speed impacts can be attributed to a tilting of the balance towards Mode I breakage. We derived a dimensionless parameter that accurately describes the degree- of Mode I breakage in terms of the impact energy, velocity and Poison’s ratio.

We also began an investigation of the effect of particle shape on the induced breakage. As a first stab at this problem, we investigated the breakage of regular polygons for which the area is held fixed as the number of sides varies. Surprisingly, we discovered that particles with odd numbers of sides break more readily than those with even numbers of sides; this effect is especially pronounced for low velocity impacts. Consequently, we began an investigation into this phenomenon and discovered a partial solution that the problem lay in the energy ~scaling. Previously, we had found that the major factor in determining the level of particle breakage was the ratio of the kinetic energy of impact to the energy required to propagate a crack across the particle, Ekin/Ecr. This investigation proved that it is critical to correctly choose the “distance” across the particle used to compute E,,. That length scale appears to be most strongly related to Mode I cracks and is thus strongly affected by the impact velocity. For very small V,/C collisions, the dominant Mode I breakage results in a single vertical crack that spans the particle. Consequently, the appropriate length scale is the vertical span of the particle. For very large VJC impacts, the Mode I cracks propagate in all directions away from the point of contact and an averaged particle dimension works quite well.

In our original proposal for the first three years of funding, we had promised to perform two-dimensional simulations of hopper flows, but had largely dropped that phase of the project and devoted most of our time to the fracture work. We decided to finish the hopper project this year and the results are given in the fourth section of this report. Originally, we were interested by experimental observations of an asymmetric unsteadiness in the flow from a hopper: i.e. the flow was observed to move alternately down either side of the hopper. We found that we could reproduce this effect for monodispersed particle sizes, but also could eliminate it by by adding a small amount of polydispersity. From this observation, we were able to understand that the unsteadiness was a result of the strong packings that monodispersed disks are prone to fall into; a small amount of polydispersity is sufficient to disrupt these packing, weaken the material and eliminate the unsteadiness. In the experiments however, we suspect that the strong packings are a result, not of monodispersity, but of that fact that the studies were performed on angular sands which tend to likewise form strong packings. This implies that this unsteadiness is a result of the non-continuum nature of granular ii breakage. The fi materials as the particle packing, shape and size distribution are non-continuum quantities. From there, we went on to examine the stress state internal to the hopper. This permitted us to evaluate common methods of mode& granular materials. In particular, we measured the internal friction coefficient and discovered that it is far from a constant. Consequently, we conclude that models based on plasticity theory that assume that the material is always at the point of imminent yield (and thus should have a constant internal friction coefficient), are questionable at best. Furthermore, we demonstrate that rapid granular flow theory, the other common modeling technique these days, is as should be expected completely inapplicable to this problem. We conclude that hopper flows (and really almost all other granular flows of industrial interest) inhabit the relatively unexplored region between these two limiting flow regimes. That is the area that granular flow research should explore in the future.

EXECUTIVE SUMMARY

A predictive model of the impact attrition of particulate solids was developed in the previous IFPRI programme (Ghadiri and Zhang, 1992):

where 5 is the fractional loss per impact, a is a proportionality constant, p is the particle density, U is the impact velocity, 1 is the particle size, H is the hardness, KC is the fracture toughness, and 4 is the constraint factor given by the ratio of the hardness to the yield stress. The model describes the chipping process and applies to materials that fail in the semi-brittle mode. The model predictions have been shown to agree reasonably well with the experimental results for ionic crystals that satisfy the semi-brittle failure conditions. In the current programme the work has been extended to glassy polymers, in order to assess the range of application of the model. The glassy polymers represent a category of materials that is completely different from ionic crystals in the material properties as well as structure. Poly-methylmethacrylate (PMMA) was selected as a model material because it is one of the most common glassy polymers and has a wide variety of applications.

Observations of the impact damage by high-speed photography showed that attrition was caused by chipping of comers and/or edges adjacent to the impact site at low impact velocities, and by fragmentation of the particle into relatively large fragments at high impact velocities. Detailed examination of the mother particles as well as debris by scanning electron microscopy and optical microscopy showed that chipping was produced by the propagation of sub-surface lateral cracks, and fragmentation by radial and median cracks. These mechanisms are associated with the semi-brittle failure mode.

A series of impact tests was carried out to quantify the extent of attrition. PMMA extrudates in the size range 2.36-2.80 mm were used. It was shown that these particles failed by chipping in the velocity range lo-30 m s-l and by fragmentation above 30 m s-l. The fractional loss per impact was measured as a function of impact velocity for 20 repeated impacts. In the chipping regime the fractional loss per impact was proportional to the impact velocity raised to the power 2.18, based on the first five impacts, and to the power 2.41, based on all the 20 impacts. The gradual increase in the power index indicates changes in the material properties with repeated impacts. However, these changes have not so far been quantified.

The effect of particle size was investigated on a theoretical basis. The limiting particle size below which fragmentation would not take place was estimated as about 300 urn for PMMA particles. Repeated impacts could reduce the limiting particle size due to fatigue effects or ductile shearing.

The conditions promoting the formation of lateral cracks were investigated by impacting rigid projectiles of various geometries on PMMA targets. It was shown that blunt projectiles were best as they induced a limited plastic deformation to initiate the cracks, and at the same time they could impart a significant amount of elastic strain energy to propagate the cracks. Further work is required to develop a mechanistic model of the fragmentation process.