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.