Fractal Dimension and Mechanical Behaviour in Structurally Complex Formations

Filippo Catani
University of Florence, Earth Scinces Department, Via La Pira 4, - I 50121 Florence

Abstract Structurally complex formations, in which we can find rocky blocks plunged in a softer argillaceous matrix, as defined by Esu (1977), are widespread in Italy. The ones of the northern Apennines, in the area across the administrative boundary between Florence and Bologna provinces, are a very interesting example of this kind of chaotically arranged rocks, where landslide phenomena are very frequent, often generating risk factor for the human activity. This sort of complexity varies from micro to macro scale and seems to show a free-of-scale fractal pattern, at least in the frequency distribution of block dimensions. On the other hand, from an engineering and geotechnical point of view, this complexity is very difficult to understand and analyse with the classical methods of soil and rock mechanics. In order to verify the possible application of a fractal model able to improve the reliability of these approaches, we tried to test the correlation between fractal geometry and mechanical behaviour in some "chaotic complex" formations, the "argille scagliose" of the Italian authors.

In this paper, after an introduction with the problem definition, the method adopted is described as well as some first results are given. Our work starts from the consideration that the normal geotechnical testing was impossible to do in our formations, due to the extreme difficulty in sampling a representative portion of terrain and because 'in situ" tests are much too expensive. Therefore our approach was based on the comparison between fractal characteristics and simulated mechanical behaviour in numerical samples, automatically generated from a known fractal structure and analysed with a finite difference modelling software which simulates triaxial geotechnical testings. Our primary target was to understand:

  1. How fractal dimension affects the mechanical behaviour in simulated specimens having an artificial structure.
  2. How, known at least the empirical dependence stated in A, it was possible to forecast the behaviour of real specimens from simple fractal measures.

As a valid fractal model we utilised the Cantor Gasket. This object can be constructed recursively in the following way: starting from a generator unit square with the central square of side length 1/3 deleted, with the iterating process we obtain at each step N a self-similar figure in which the number of filled squares of side length l/3N is 8N. So, the fractal dimension of the Cantor Gasket is log(8)/log(3)*1.89. Besides, in order to better represent the real natural pattern of the field surveyed outcrops, the simulated specimens were generated with a similar procedure but with a random choice of the point in which the square holes are produced. By this way we built a 'Random Cantor Gasket" which looks like a good visual approximation of the real outcrop.

The first outcomes show to be in good agreement with theoretical expectation and encourage us to keep on with this work, principally in order to improve the knowledge of the empirical dependence between fractal properties and mechanical parameters.

Esu, F. 1977. "Behaviour of slopes in structurally complex formations". Proc. Int. Symp. 'The Geotechnics of Structurally Complex Formations', Capri, Italy - Vol. 2, 292-304.