applied fracture mechanics

Foundations of Measurement Fractal Theory for the Fracture Mechanics 29It is true that the physical or real fractals can be deterministic or random. In random orstatistical fractal the properties of self-similarity changes statistically from region to regionof the fractal. The dimension cannot be unique, but characterized by a mean value, similarlyto the analysis of mathematical fractals. The Figure 6b shows aspects of a statistically selfsimilarfractal whose appearance varies from branch to branch giving us the impression thateach part is similar to the whole.The mathematical fractals (or exact) and physical (or statistical), in turn, can be subdividedinto uniform and nonuniform fractal.Uniforms fractals are those that grow uniformly with a well behaved unique scale andconstant factor, , and present a unique fractal dimension throughout its extension.Non-uniform fractals are those that grow with scale factors i' s that vary from region toregion of the fractal and have different fractal dimensions along its extension.Thus, the fractal theory can be studied under three fundamental aspects of its origin:1. From the geometric patterns with self-similar features in different objects found in nature.2. From the nonlinear dynamics theory in the phase space of complex systems.3. From the geometric interpretation of the theory of critical exponents of statisticalmechanics.3. Methods for measuring length, area, volume and fractal dimensionIn this section one intends to describe the main methods for measuring the fractal dimensionof a structure, such as: the compass method, the Box-Counting method, the Sand-BoxMethod, etc.It will be described, from now, how to obtain a measure of length, area or fractal volume. Infractal analysis of an object or structure different types of fractal dimension are obtained, allrelated to the type of phenomenon that has fractality and the measurement method used inobtaining the fractal measurement. These fractal dimensions can be defined as follows.3.1. The different fractal dimensions and its definitionsA fractal dimension Dfin general is defined as being the dimension of the resultingmeasure of an object or structure, that has irregularities that are repeated in different scales(a invariance by scale transformation). Their values are usually noninteger and situatedbetween two consecutive Euclidean dimensions called projection dimension d of the objectand immersion dimension, d 1 , i.e. d D d 1 .fIn the literature there is controversy concerning the relationship between different fractaldimensions and roughness exponents. The term "fractal dimension" is used generically torefer to different fractional dimensions found in different phenomenologies, which results information of geometric patterns or energy dissipation, which are commonly called fractals[1]. Among these patterns is the growth of aggregates by diffusion (DLA - Diffusion Limited