Theoretical and experimental studies on dynamic impact on brittle solids

Abstract

Fragmentation of brittle solids under dynamic impacts is an important applied mechanics problem, which relates to a wide range of phenomena and industrial processing of powders such as pharmaceuticals, chemical, fertilizers and detergents. In civil engineering applications, impact-induced-fragmentation relates to crushing of rock mass during mining process and aggregate production. The theory of dynamic fragmentation is relatively less developed comparing to the static counterpart, and dynamic fragmentation mechanisms remain elusive. Because of the difficulty of monitoring the fragmentation sequence inside a solid under an impact of extremely short duration, our knowledge on dynamic fragmentation process is quite limited; thereby fragmentation is still mainly modeled by empirical approach. The main objective of this thesis is to provide a comprehensive approach to investigate fragmentation by using analytical, experimental and numerical analyses. First of all, the analytical solution of a solid sphere compressed dynamically between two rigid flat platens is considered. Secondly, a sequence of experiments on this setting are conducted using an impactor Dynatup 8250 at HKUST, in which both impact velocity and contact force at the impactor can be measured accurately as a function of time. Finally, a newly developed computer program at PolyU, DIFAR, is used to investigate the mechanism of dynamic fragmentation. For the analytical approach, the dynamic stress distribution within an isotropic elastic solid sphere subject to a pair of suddenly-applied patch loads (either uniform or Hertz type contact stress) along a diameter is obtained using eigen-function expansion, in which the dynamic solution is decomposed into a static solution and a series of free vibration solutions. The analytical solution is given in a double infinite series involving Legendre polynomials and spherical Bessel function. For the special case that the patch loads converges to a pair of applied point load, our solution is comparable to those obtained by Jingu and Nezu (1985); when transmission of waves through the two rigid platens is allowed the long term solutions converge to the static solution given by Hiramatsu and Oka (1966) and the solution by Chau et al. (2000) for the cases of uniform and Hertz contact loads respectively. Contour plots as a function of time reveal the effect of wave propagation, reflection and focusing. These plots provide the time evolution of dynamic stress patterns and can be used to interpret the position of fracture initiation and patterns of fragmentation. In our experiments, brittle spheres made of plaster of two different strengths and three different sizes were compressed dynamically between two rigid platens at various impact energy levels. Fractures can be broadly classified as primary and secondary, with the larger fragments resulted from primary and the finer fragments resulted from secondary fracturing. For both fragment sizes, the cumulative mass versus fragment size follows a power law or Gaudin-Schuhmann distribution, but with different size distribution modulus. As expected, the number of fragments increases with the impact energy, but surprisingly the maximum contact forces at failure remains independent of the impact energy level. The inferred specific surface energy increases with the strength but constant with the size of the spheres. In our numerical analysis, a newly developed computer program, DIFAR, is used to simulate the progressive process of dynamic fracturing and fragmentations. The computer program is based on an elastic finite element analysis of solids incorporated with a loading-rate-sensitive Mohr-Coulomb criterion with a tensile cut-off for damage checking. Both elastic modulus and strength of all elements follow a Weibull distribution spatially, thus the randomness of the initiation of fragmentation can be modeled. The simulated fragmentations of spheres subject to double impacts under various energies agree well with the general pattern of observations in experiments. The results of this thesis should provide some insight on dynamic fragmentation for spheres or non-spherical particles and a bench study for further research in the area.