A mesoscopic theory of damage and fracture in heterogeneous materials

Abstract

Deformation patterns in solids are often characterized by self-similarity at the meso-level. In this paper, the framework for the mechanics of heterogeneous solids, deformable over fractal subsets, is briefly outlined. Mechanical quantities with non-integer physical dimensions are considered, i.e., the fractal stress [ σ *] and the fractal strain [ ε *]. By means of the local fractional calculus , the static and kinematic equations are obtained. The extension of the Gauss-Green Theorem to fractional operators permits to demonstrate the Principle of Virtual Work for fractal media. From the definition of the fractal elastic potential φ *, the fractal linear elastic relation is derived. Beyond the elastic limit, peculiar mechanisms of energy dissipation come into play, providing the softening behaviour characterized by the fractal fracture energy G F *. The entire process of deformation in heterogeneous bodies can thus be described by the fractal theory. In terms of the fractal quantities it is possible to define a scale-independent cohesive law which represents a true material property. It is also possible to calculate the size-dependence of the nominal quantities and, in particular, the scaling of the critical displacement w c , which explains the increasing tail of the cohesive law with specimen size, and that of the critical strain ε c , which explains the brittleness increase with specimen size.