Mott waves and fragment size in viscoplastic metals

Abstract

Previous ductile fragmentation analysis involves unloading wave (Mott wave) propagations in rate-independent rigid-plastic materials. This paper studies the dynamic fragmentations of a rigid-viscoplastic material in which there exists a rigid unloading zone and a viscoplastic unloading zone. A linear diffusion equation incorporating a moving boundary condition for the viscoplastic unloading zone is presented. The fracture process of an isolated crack in the infinite medium, and that of an array of equally spaced cracks, are investigated numerically. The crack array problem reveals an optimal crack spacing upon which the material fails and unloads most rapidly. Based on the 'principle of the rapidest unloading (PRU)', this crack spacing is regarded as the measure of fragment size. Systematic numerical calculations are conducted to obtain the fragment size data as functions of strain rate and viscosity coefficient. A modified Grady-Kipp fragment size formula is proposed to accommodate the viscous effect. In the case of non-viscous material, the formula returns to the classical Grady-Kipp fragment size model. Finite element simulations using rate-dependent constitutive model show similar trend of viscous effect on the fragment sizes, and the modified fragment size formula yields better agreement with the simulated data.