Calculation of the tensile and flexural strength of disordered materials using fractional calculus

Aim of the present paper is to show an application of the fractal integral, which was recently introduced in the framework of fractional calculus, to the mechanics of materials with disordered microstructure. Some rules of integration for simple functions performed on generalized Cantor sets have been developed. We believe that these results can be of some interest in the modeling of materials whose microstructure is fractal-like. Part of the paper is devoted, as an example, to show why concrete can be considered such a material. The fractal dimension of the stress carrying material ligament is found. Finally, by the fractal integration rules previously developed, the computation of the tensile and flexural strength for structures made by a concrete-like material is performed and consequent size effects are highlighted.