O.K. Panagouli

Fractal Evaluation of Pavement Skid Resistance Variations. I: Surface Wetting

Abstract Pavement skid resistance has long been recognised as the primary factor to reduce traffic accidents. Skid resistance is affected by a large number of parameters. This is the first in a series of two papers dealing with the road surface contribution to tire-pavement skid resistance giving special emphasis to the effect of a water film development on the pavement surface. Pavement surface texture is expressed by micro⧹macrotexture depth and density. Relative regression equations giving pavement skid resistance values have been reviewed and a well fitting regression equation has been derived. The mathematical background of fractal interpolation functions is then presented. An attempt is described to approximate the pavement surface texture with fractals by taking into account its accurate geometry. Through this approximation the procedure of pavement surface wetting is evaluated, in respect with its effect to skid resistance values. This attempt yields encouraging results, although it is apparent that further research is required.

Skid resistance and fractal structure of pavement surface

Abstract Pavement skid resistance has long been recognized as the most important parameter to reduce traffic accidents. The factors affecting skid resistance are numerous and very complicated. It is a difficult task to estimate the extent and direction of their overall contribution. This paper deals with the road surface contribution to tire-pavement skid resistance. Pavement surface is expressed through micro/macrotexture depth and density. An attempt is made to approximate this pavement surface texture with fractals, in order to take into account its accurate geometry. A qualitative relationship between pavement surface texture, expressed through fractal concepts, and skid resistance is recognized. This attempt yields encouraging results, although it is apparent that further research is required.

Fractal geometry and fractal material behaviour in solids and structures

SummaryThe present paper discusses certain methods which permit us to consider the influence of the fractal geometry and the fractal material behaviour in solid and structural mechanics. The method of fractal interpolation function is introduced and the fractal quantities (boundary geometry, interface geometry and stress-strain laws) are considered as the fixed points of a given set-valued transformation. Our first aim here is to define the mechanical quantities on fractal sets using some elementary results of the theory of Besov spaces. Then we try to extend the classical finite element method for the case of fractal bodies and fractal boundaries and corresponding error estimates are derived. The fractal analysis permits the formulation and the treatment of complicated or yet unsolved problems in the theory of deformable bodies.ÜbersichtDiskutiert werden Methoden, die es erlauben, den Einfluß von fraktaler Geometrie und fraktalem Materialverhalten in der Festkörper- und Strukturmechanik zu betrachten. Die Methode der fraktalen Interpolationsfunktion wird eingeführt; die fraktalen Größen (Randgeometrie, Grenzflächengeometrie und Spannungs-Dehnungsgesetze) werden als Fixpunkte einer gegebenen mengenwertigen Transformation betrachtet. Das erste Ziel ist die Definition der mechanischen Größen auf fraktalen Mengen, wofür einige grundlegende Ergebnisse der Theorie der Besov-Räume herangezogen werden. Weiterhin wird die klassische Finite-Element-Methode auf fraktale Körper und fraktale Ränder erweitert und zugehörige Fehlerabschätzungen werden abgeleitet. Die fraktale Betrachtung gestattet die Formulierung und Behandlung komplizierter oder noch ungelöster Probleme der Theorie deformierbarer Körper.

Fractal interfaces with unilateral contact and friction conditions

Abstract Structures involving interfaces with fractal geometry are analyzed here as a sequence of classical interface subproblems. On the interface, unilateral contact and friction boundary conditions are assumed to hold. These classical subproblems result from the consideration of the fractal interface as the ‘fixed point’ (or the ‘deterministic attractor’) of a given transformation. This approximation of the fractal is combined with a two-level contact-friction algorithm based on the optimization of the potential and of the complementary energy, after some appropriate transformations relying on the singular value decomposition of the equilibrium matrix are performed. Numerical examples illustrate the theory.

On the fractal nature of problems in mechanics

Abstract This paper investigates the influence of fractal geometry and fractal material behaviour in solid and structural mechanics. For that, certain methods are proposed for the theoretical and numerical investigation of this influence based on the consideration of the fractal as the “fixed point” of a given iterated function system or as the fractal graph of a fractal interpolation function which interpolates a given set of data. First the definitions of the mechanical quantities and the mechanical laws are extended to fractal sets by using some results from the theory of Besov spaces. Then an attempt is made to extend certain calculation methods to the case of fractal interfaces and to the case of fractal nonmonotone interface laws. Finally the results of these methods are investigated.

Mechanics on fractal bodies. Data compression using fractals

Abstract The present paper deals with two interrelated subjects: the definition of a ‘correct’ mechanics on fractal bodies and the data compression method using fractals and its application to the numerical analysis of problems in engineering. The first part uses theoretical results of the theory of Bessov spaces while the second part discusses certain elements of approximation of fractals by iterated function systems (IFS).

Friction evolution as a result of roughness in fractal interfaces

In this paper, the influence of fractal interface geometry to the evolution of the friction mechanism is studied. The paper is based on fractal approaches for the modeling of the multiscale self‐affine topography of these interfaces. More specifically, these approaches are based on scale‐independent parameters such as the fractal dimension. Here, friction between rough surfaces is assumed to be the result of the gradual plastification of the fractal interface asperities. In order to study the resulting highly nonlinear problem a variational formulation is used in order to describe contact between the interfaces. The numerical method used here leads to the successive solution of quadratic optimization problems. Finally, structures with different fractal interfaces are analyzed in order to obtain results for the relation between the fractal dimension and the overall response of the structures.

Strength evalution of retrofit shear wall elements with interfaces of fractal geometry

In the present paper structures involving rough interfaces, with irregularities of all scales, are simulated with the help of fractal geometry. More specifically, the approach followed is the multilevel hierarchical approach. This approach has a multiscale self-affine structure appropriate for the simulation of rough interfaces. In the sequence the problems resulting from the consideration of a fractal interface as a profile with hierarchical structure are analyzed, with the assumption that on the interface nonmonotone adhesive contact and friction conditions hold. The method developed consists of an extension of the classical FEM to the case of fractal interfaces. The case of nonconvex energy problems is investigated, and some results from static analysis of concrete shear walls with prescribed fractal cracks are included in order to illustrate the theory.

Fractal geometry in structures. Numerical methods for convex energy problems

Abstract In the present paper certain numerical aspects of the theory of fractals in structures are presented. The fractal geometry is approximated through the I.F.S. (iterated function system) approach or through the F.I. (fractal interpolation). These approximations of the fractal are combined with the methods used in structural analysis in order to calculate stress and strain fields in fractal structures. All types of structures with convex strain energy are studied.

Fractal Geometry in Contact Mechanics and Numerical Applications

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.