Statistical mechanics of fragmentation processes of ice and rock bodies

It is a well-known experimental fact that impact fragmentation, specifically of ice and rock bodies, causes a two-step (“knee”-shaped) power distribution of fragment masses with exponent values within the limits −4 and −1.5 (here and henceforth the differential distribution is borne in mind). A new theoretical approach is proposed to determine the exponent values, a minimal fracture mass, and properties of the knee. As a basis for construction of non-equilibrium statistical mechanics of condensed matter fragmentation the maximum-entropy variational principle is used. In contrast to the usual approach founded on the Boltzmann entropy the more general Tsallis entropy allowing stationary solutions not only in the exponential Boltzmann-Gibbs form but in the form of the power (fractal) law distribution as well is invoked. Relying on the analysis of a lot of published experiments a parameter β is introduced to describe an inhomogeneous distribution of the impact energy over the target. It varies from 0 (for an utterly inhomogeneous distribution of the impact energy) to 1 (for a homogeneous distribution). The lower limit of fragment masses is defined as a characteristic fragment mass for which the energy of fragment formation is minimal. This mass value depends crucially on the value of β. It is shown that for β⪡1 only small fragments can be formed, and the maximal permitted fragment (of mass m1) is the upper boundary of the first stage of the fracture process and the point where the knee takes place. The second stage may be realized after a homogeneous redistribution of the remainder of the impact energy over the remainder of the target (when β→1). Here, the formation of great fragments is permitted only and the smallest of them (of mass m2) determines a lower boundary of the second stage. Different forms of the knee can be observed depending on relations between m1 and m2.