Numerical modelization of disordered media via fractional calculus

Abstract

In this paper, the framework for the mechanics of solids, deformable over fractal subsets, is outlined. Anomalous mechanical quantities with fractal dimensions are introduced, i.e., the fractal stress [σ ∗] , the fractal strain [ε ∗] and the fractal work of deformation W ∗ . By means of the local fractional operators, the static and kinematic equations are obtained, and the principle of virtual work for fractal media is demonstrated. Afterwards, from the definition of the fractal elastic potential φ ∗ , the linear elastic constitutive relation is derived. The direct formulation of the elastic problem is obtained in terms of the fractional Lamé operators and of the equivalence equations at the boundary. The variational form of the elastic problem is also obtained, through minimization of the total potential energy. Finally, discretization of the fractal medium is proposed, in the spirit of the Ritz–Galerkin approach, and a finite element formulation is obtained by means of devil's staircase interpolating splines.