Static–kinematic duality and the principle of virtual work in the mechanics of fractal media

Abstract

The framework for the mechanics of solids, deformable over fractal subsets, is outlined. While displacements and total energy maintain their canonical physical dimensions, renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions, i.e., the fractal stress [ σ *] and the fractal strain [ ε *]. A fundamental relation among the dimensions of these quantities and the Hausdorff dimension of the deformable subset is obtained. New mathematical operators are introduced to handle these quantities. In particular, classical fractional calculus fails to this purpose, whereas the recently proposed local fractional operators appear particularly suitable. The static and kinematic equations for fractal bodies are obtained, and the duality principle is shown to hold. Finally, an extension of the Gauss–Green theorem to fractional operators is proposed, which permits to demonstrate the Principle of Virtual Work for fractal media.