Fractals and fractional calculus in continuum mechanics. (English) Zbl 0917.73004

Carpinteri, A.; Mainardi, F.

CISM Courses and Lectures. 378. Wien: Springer. 348 p. (1997).

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This volume is one of the first books about fractals and fractional calculus in continuum mechanics. The book is divided into seven chapters. In Chapter 1, the basic concepts of scaling laws, including complete and incomplete selfsimilarity, are presented and the brittle fracture is examine by using the theory of critical phenomena. In Chapter 2, the authors describe the basic tools of fractal geometry which can be applied to extract the fractal dimensions of natural objects. The deterministic and statistical methods are explained and applied to experimentally digitized fracture patterns. Chapter 3 deals with the extension of classical mechanics to bodies with fractal boundaries and interfaces. Chapter 4 discusses flows in porous media. In Chapter 5, the authors provide an introduction to the fractional calculus and solve integral and differential equations. The attention is paid to the Laplace transform technique suitable for applied problems. Chapter 6 treats numerical aspects of the fractional calculus. Finally, in Chapter 7 the authors give a review of some applications of the fractional calculus in continuum and statistical mechanics, including the mathematical modelling of viscoelastic bodies through fractional constitutive equations and the study of unsteady motion of a particle in viscous fluid.
The results obtained in some basic problems of mechanics demonstrate that the fractional calculus becomes more popular and can provide in the future useful mathematical tools to treat complex phenomena of fractal systems.

Reviewer: L.Prášek (Plzeň)

MSC:

74-06	Proceedings, conferences, collections, etc. pertaining to mechanics of deformable solids
74Axx	Generalities, axiomatics, foundations of continuum mechanics of solids
00B15	Collections of articles of miscellaneous specific interest
28A80	Fractals
74R99	Fracture and damage

Keywords:

fractals; fractional calculus; continuum mechanics; motion of particle in viscous fluid; scaling laws; self-similarity; brittle fracture; theory of critical phenomena; fractal geometry; fractal dimensions; porous media; Laplace transform; statistical mechanics; viscoelastic bodies; fractional constitutive equations