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.