applied fracture mechanics

32Applied Fracture Mechanicsphenomena which have self-similar properties, permitting scale transformations withoutloss of generality in the description of physical information of the phenomenon rangingfrom quantities such as volume up to energy. However, in the case described here, the Box-Counting method is performed filling the space occupied by a fractal object with boxes ofarbitrary size , and count the number N of these boxes in function its size, (Figure 9and Figure10). This number N of boxes is given as follows: ND (16)Plotting the data in a log log graph one obtains from the slope of the curve obtained, thefractal dimension of the object.In the Box-Counting method (Figure 9), a grid that recover the object is divided inton k L 0 / kboxes of equal side k and how many of these boxes that recovering the objectis counted. Then, varies the size of the boxes and the counting is retraced, and so on. Makinga logarithm graph of the number Nkof boxes that recovering the object in function of thescale for each subdivision k k / L0 , one obtains the fractal dimension from the slope ofthis plot. Note that in this case the partition maximum is reached when,N L0 / kk L0 / l0, where Lmax L0is the projected crack length l 0is thelength of the shortest practicable ruler.CFigure 9. Fragment of a crack on a testing sample showing the variation of measurement of the cracklength L with the measuring scale, k k / L0 for a partition, k variável and Lk L0(fixed), withsectioning done for counting by one-dimensional Box-Counting scaling method.