The elastic problem for fractal media: basic theory and finite element formulation

In a previous paper [Comput. Methods Appl. Mech. Eng. 190 (2001) 6053], the framework for the mechanics of solids, deformable over fractal subsets, was outlined. Anomalous mechanical quantities with fractal dimensions were 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 were obtained, and the Principle of Virtual Work for fractal media was demonstrated. In this paper, the constitutive equations of fractal elasticity are put forward. From the definition of the fractal elastic potential φ∗, the linear elastic constitutive relation is derived. The physical dimensions of the second derivatives of the elastic potential depend on the fractal dimensions of both stress and strain. Thereby, the elastic constants undergo positive or negative scaling, depending on the topological character of deformation patterns and stress flux. The direct formulation of elastic equilibrium is derived 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.