Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. (English) Zbl 0991.74013

Carpinteri, A.; Chiaia, B.; Cornetti, P.

Comput. Methods Appl. Mech. Eng. 191, No. 1-2, 3-19 (2001).

Summary: We outline the framework for the mechanics of solids, deformable over fractal subsets. 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 \([\sigma^*]\) and the fractal strain \([\varepsilon^*]\). We obtain a fundamental relation among the dimensions of these quantities and Hausdorff dimension of the deformable subset. 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, we propose an extension of Gauss-Green theorem to fractional operators, which permits to demonstrate the principle of virtual work for fractal media.

MSC:

74A99	Generalities, axiomatics, foundations of continuum mechanics of solids
28A80	Fractals

Keywords:

anomalous dimensions; renormalization group theory; fractal dimensions; Hausdorff dimension; local fractional operators; Gauss-Green theorem; principle of virtual work for fractal media

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