applied fracture mechanics

Therefore, the number Foundations of Measurement Fractal Theory for the Fracture Mechanics 33N k kdepending on the size, k , of these boxes is given as follows:N k k k max D(17)In the Figure 9 is illustrated the use of this method in a fractal object. Are present differentgrids, or meshes, constructed to recover the entire structure, whose fractal dimension onewants to know. The grids are drawn from an original square, involving the whole spaceoccupied by the structure. At each stage of refinement of the grid L 0 (the number of equalparts in the side of the square is divided) are counted the number of squares NL 0 whichcontain part of the structure. Repeatedly from the data found, is constructed the graph oflog L0log NL0. If the graph thus obtained is a straight line, then the fractal behavior ofthe structure has self-similarity or statistical self-affinity whose dimension D is obtained bycalculating the slope of the line. For more compact structure, it is recommended to make astatistical sampling, that is, the repeat the counting of the squares NL 0 for differentsquares constructed from the gravity center (counting center) of the in the structure. Thus,one obtains a set of values NL 0 for another set of values L 0. These data must bestatistically treated to obtain the value of fractal dimension, " D ".From the viewpoint of experimental measurement, one can consider using differentmethods of viewing the crack to obtain the fractal dimension, such as optical microscopy,electron microscopy, atomic force microscope, etc.., Which naturally have different rules kand therefore different scales of measurement k,.The fractal dimension is usually calculated using the Box-Counting shown in Figure 9, i.e.by varying the size of the measuring ruler kand counting the number of boxes, Nkthatrecover the structure. In the case of a crack the fractal dimension is obtained by thefollowing relationship:ln ND ln( l / L )(18)The description of a crack according to the Box-Counting method follows the idea shown inFigure 9, which results in:ooln 57D 1.096 . (19)ln(1 / 40)The same result can be obtained using the Box-Sand method, as shown in Figure 10.3.2.2. The sand-box counting method of the elements by static scaling of a fractal structureThe Sand-box method consists in the same way as the Box-Counting method, to count thenumber of boxes, N u , but with fixed length, u , as small as possible, extending gradually