Experimental Determination of the Fractal Dimension of Microcrack Patterns and Fracture Surfaces
Three different investigations on the scaling properties of damage in concrete are described. In the case of uniaxial tensile tests, a laser profilometer was adopted to scan the post-mortem fracture surfaces. In a second investigation, splitting tests wer
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Series Editors: The Rectors of CISM Sandor Kaliszky - Budapest Mahir Sayir - Zurich Wilhelm Schneider - Wien The Secretary General of CISM Giovanni Bianchi - Milan Executive Editor Carlo Tasso - Udine
The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 378
FRACTALS AND FRACTIONAL CALCULUS IN CONTINUUM MECHANICS
EDITED BY
A. CARPINTERI POLYTECHNIC OF TURIN
F. MAINARDI UNIVERSITY OF BOLOGNA
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Springer-Verlag Wien GmbH
Le spese di stampa di questa volume sono in parte coperte da contributi del Consiglio Nazionale delle Ricerche.
This volume contains 152 illustrations
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 1997 by Springer-Verlag Wien Originally published by Springer-Verlag Wien New York in 1997 SPIN 10657663
In order to make this volume available as economically and as rapidly as possible the authors' typescripts have been reproduced, in their original forms. This method unfortunately I
has its typographical limitations but it is hoped that they in no way distract the reader.
ISBN 978-3-211-82913-4 DOI 10.1007/978-3-7091-2664-6
ISBN 978-3-7091-2664-6 (eBook)
PREFACE "Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line". With these words does Benoit B. Mandelbrot open his well-known essay "The Fractal Geometry of Nature". In the natural world, he observes and surveys irregularities, tortuosities and discontinuities, that cannot be described and analyzed through the Euclidean geometry or the classical infinitesimal calculus. In this context, Mandelbrot withstands Leibniz, who was used to affirm: "Natura nonfacit saltus". At the end of the present millennium, two different "Weltanschauungen" are confronted: the optimism, formerly Illuminist and then Positivist, versus the scepticism of a Critical Rationalism, that is perhaps disenchanted but even projected towards new and challenging paradigms. That nature, although with its irregularities and anomalies, could be described in mathematical terms, was anticipated by Pitagora himself. He provided a first significant example of producing complex shapes from elementary geometric elements through iterative procedures. The application of his well-known theorem, once repeated for an infinite number of
