A mesoscopic theory of damage and fracture in heterogeneous materials

Abstract

Deformation patterns in solids are often characterized by self-similarity at the mesolevel. The framework for the mechanics of heterogeneous solids, deformable over fractal subsets, is briefly outlined. Mechanical quantities with noninteger 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.