The rapidest unloading in dynamic fragmentation events

Abstract

Solids usually break (fragmentize) into many pieces under high rate loading. Grady and co-worker have proposed one-dimensional theoretical models to estimate the average size of the fragments created in a ductile or a brittle fragmentation process. Numerical simulations have shown that the formulae, albeit identical in appearance, work well for the ductile fragmentation event, but poorly for the brittle case. In this paper we seek the physical mechanism that describes both the ductile and the brittle fragmentation processes. In analyzing the formation of multiple adiabatic shear bands, Grady and co-worker have proposed a conjecture that the bands automatically arrange their spacing so that the stress within the material is unloaded at the shortest time. In this paper, we apply Grady’s “rapidest unloading” principle to three types of solid fragmentations: the multiple adiabatic shear localizations, the ductile tensile fragmentation, and the brittle tensile fragmentation. We analyzed the simultaneous formation and growth of an array of equally-spaced defects, and the unloading wave propagations in the defect-free region. The average stress across the region was determined, from which the critical fracture time, defined as the time when the average stress drops to zero, is evaluated. It appears that for a prescribed strainrate, there always exists an optimum defect spacing corresponding to the rapidest unloading process. Assuming that in a natural fragmentation process the solids is unloaded in the fastest way, this optimum spacing provides an estimate for the average fragment size. For the three types of fragmentation events, the fragment size evaluated by using “the rapidest unloading” principle compares fairly well with the other reasonable fragment size models.