Fractals and Fractional Calculus in Continuum Mechanics

Fractal concepts are used to study the complex shapes of fracture surfaces, as well as damage phenomena presenting statistical characteristics. While at the beginning of the loading process the microcracks can be considered as two-dimensional surfaces, as the load is increased the microcracks grow, coalesce and form an invasive fractal set with a dimension larger than two. When the evolving dimension assumes the notable value of 2.5, the microcrack set may be considered as extremely disordered, and percolation of the cracks is very likely, in particular for sufficiently large specimens. Percolation favours catastrophic and brittle behaviours and produces fracture surfaces of a Brownian character, where the local dimension is again 2.5. An additional assumption is that of considering material resisting sections at peak stress as lacunar fractals of a local dimension equal to 1.5. The above-mentioned fractalities produce the well-known scale dependence of fracture energy and tensile strength, respectively.

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 were carried out and optical microscopy was used to detect the stress-induced crack patterns. In the third investigation (compression tests), a fusible alloy was injected inside the cracked specimen under load and, afterwards, scanning electron microscopy (SEM) was applied. The fundamental concepts of fractal geometry are introduced, and the methods developed to extract the fractal dimension are described. The application of these algorithms to the experimental damage patterns shows that self-affine scaling is often provided by disordered materials, and that piecewise fractality (geometrical multifractality) comes into play due to the interaction of internal and external scale lengths. The formation and evolution of a cloud of interacting microcracks in the damage process zone (a network with fractal dimension even larger than 2.5) can be related to the characteristic softening mechanical behaviour. The catastrophic jump to the final fracture surface (with local dimension comprised between 2.0 and 2.5) identifies the rupture transition. It is shown that also the damage networks induced by compression may present a fractal dimension larger than 2.5 in the bulk, but the local dimension of each microfracture never exceeds 2.5.

The contribution to the present volume deals with the study of the influence of fractal geometry on contact problems. After a short presentation of the new mathematical tools and methods used for the correct consideration of the fractal geometry we study unilateral contact and friction problems, adhesive contact problems in interfaces of fractal geometry and finally crack problems of fractal geometry. Numerical applications illustrate the uheory. This contribution contains also an advanced mathematical section concerning the nature of the forces on a fractal boundary.

Modeling of fluid flow through porous media presents three main properties: very large range of length scales, strong heterogeneity and, in some cases, flow instability. All these properties can be studied using a fractal approach and statistical physics. The purpose of this course is to illustrate this approach for various kinds of fluid displacements.

A survey is given on some numerical methods of Riemann-Liouville fractional calculus. The topics discussed here will be: (a) approximation of fractional derivatives by generalized finite differences and their use in numerical treatment of fractional differential equations, (b) discretized fractional calculus and its use in numerical treatment of Abel type integral equations of first and second kind, (c) product integration and collocation methods for Abel integral equations, (d) the problem of ill-posedness of Abel integral equations of first kind.

We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (§1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (§2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis leads us to introduce a hydrodynamic model suitable to revisit in §3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In §4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order α with 0 < α < 2. Our analysis leads us to express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes (0 < α < 1) from intermediate processes (1 < α < 2).