Scaling Laws for Transport, Mechanical and Fracture Properties of Disordered Materials

Abstract

AbstractWe discuss scaling laws for scalar and vector transport properties of, and fracture processes in, disordered materials. Random resistor networks, and elastic and superelastic percolation networks are used to model the disordered material. While scalar transport properties of such systems (e.g. conductivity or diffusivity) obey universal scaling laws near the percolation threshold, vector transport properties (e.g. elastic moduli) may not follow such universal laws, and the critical exponents characterizing such scaling laws may depend on the microscopic force laws of the system. On the other hand, fracture processes in such systems appear to obey universal scaling laws. In particular, the external stress F for the fracture of the system scales with its linear size L as F ˜ Ld-1/(ln L)ψ, where d is the dimensionality of the system and ψ is a small critical exponent (ψ ≃ 0.1). Moreover, as the macroscopic fracture point of the system is approached, the ratio of various elastic moduli of the system approaches a universal fixed point, independent of the microscopic details of the system. Finally, the distribution of fracture strength in a randomly reinforced system, or in a system near its percolation threshold with a broad distribution of elastic constants, is in the form of a Weibull distribution, rather than the recently-proposed Gumbel distribution.