Zdenek P. Bazant, Ferhun C. Caner
The modeling of high velocity impact into brittle or quasibrittle solids is hampered by the unavailability of a constitutive model capturing the effects of material comminution into very fine particles. The present objective is to develop such a model, usable in finite element programs. The comminution at very high strain rates can dissipate a large portion of the kinetic energy of an impacting missile. The spatial derivative of the energy dissipated by comminution gives a force resisting the penetration, which is superposed on the nodal forces obtained from the static constitutive model in a finite element program. The present theory is inspired partly by Grady's model for comminution due to explosion inside a hollow sphere, and partly by analogy with turbulence. In high velocity turbulent flow, the energy dissipation rate is enhanced by the formation of micro-vortices (eddies) which dissipate energy by viscous shear stress. Similarly, here it is assumed that the energy dissipation at fast deformation of a confined solid gets enhanced by the release of kinetic energy of the motion associated with a high-rate shear strain of forming particles. For simplicity, the shape of these particles in the plane of maximum shear rate is considered to be regular hexagons. The rate of release of free energy density consisting of the sum of this energy and the fracture energy of the interface between the forming particle is minimized. The particle sizes are assumed to be distributed according to Schuhmann's power law. It is concluded that the minimum particle size is inversely proportional to the (2/3)-power of the shear strain rate, that the kinetic energy release is to proportional to the (2/3)-power, and that the dynamic comminution creates an apparent material viscosity inversely proportional to the (1/3)-power of the shear strain rate.
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http://arxiv.org/abs/1306.1120
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