A Probabilistic Damage Model of the Dynamic Fragmentation Process in Brittle Materials

Dynamic fragmentation is observed in brittle materials such as ceramics, concrete, glass, or rocks submitted to impact or blast loadings. Under such loadings, high-stress-rate tensile fields develop within the target and produce fragmentations characterized by a high density of oriented cracks. To improve industrial processes such as blast loadings in open quarry or ballistic efficiencies of armors or concrete structures against impact loadings, it is essential to understand the main properties of such damage processes (namely, characteristic time of fragmentation, characteristic density, orientation and extension of cracking, ultimate strength) as functions of the loading rate, the size of the structure (or the examination volume), and the failure properties of the brittle material concerned. In the present contribution, the concept of probability of nonobscuration is developed and extended to predict the crack density for any size, shape of the loaded volume, stress gradients, and stress rates. A closed-form solution is used to show how a brittle and random behavior under quasi-static loading becomes deterministic and stress rate dependent with increasing loading rates. Two definitions of the tensile strength of brittle materials are proposed. As shown by Monte Carlo simulations, for brittle materials, the "ultimate macroscopic strength" applies under high loading rate or in a large domain, whereas the "mean obscuration stress" applies under low stress rate or in a small domain. Next, a multiscale model is presented and used to simulate damage processes observed during edge-on impact tests performed on an ultra-high-strength concrete. Finally, the fragmentation properties predicted by modeling of six brittle materials (dense and porous SiC ceramics, a micro-concrete, an ultra-high-strength concrete, a limestone rock, and a soda-lime silicate glass) are compared.