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Zbl 0917.73004
Carpinteri, A.; Mainardi, F.
Fractals and fractional calculus in continuum mechanics. (English)
[B] CISM Courses and Lectures. 378. Wien: Springer. 348 p. (1997). ISBN 3-211-82913-X

[The articles of this volume will not be reviewed individually.] \par 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.\par 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.
[L.Prášek (Plzeň)]

MSC 2000:: *74-06 Proceedings of conferences (mechanics of deformable solids)
74Axx Generalities, etc. 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; selfsimilarity; brittle fracture; theory of critical phenomena; fractal geometry; fractal dimensions; porous media; Laplace transform; statistical mechanics; viscoelastic bodies; fractional constitutive equations

Cited in: Zbl 1035.65067

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