applied fracture mechanics

Foundations of Measurement Fractal Theory for the Fracture Mechanics 434.2.3. The fractality of a crack or fracture surfaceBy observing a crack, in general, one notes that it presents similar geometrical aspects thatreproduce itself, at least within a limited range of scales. This property called invariance byscale transformation is called also self-similarity, if not privilege any direction, or selfaffinity,when it favors some direction over the other. Some authors define it as the propertythat have certain geometrical objects, in which its parts are similar to the whole in insuccessive scales transformation. In the case of fracture, this takes place from a range ofminimum cutoff scale , minuntil a maximum cutoff scale, max, contrary to the proposedby Borodich [3], which defines an infinite range of scales to maintain the mathematicaldefinition fractal. In the model proposed in this section, one used the fractal theory as a formcloser to reality to describe the fracture surface with respect to Euclidean description. Thiswas done in order to have a much better approximatation to reality of the problem and touse fractal theory as a more authentic approach.Figure 13. Self-similarity present in a pine (fractal), with different levels of scaling, k.To understand clearly the statements of the preceding paragraph, one can use the pineexample shown in Figure 13. It is known that any stick of a pine is similar in scale, the otherbranches, which in its turn are similar to the whole pine. The relationship between the scalesmentioned above, in case of pine, can be obtained considering from the size of the lowerbranch (similar to the pine whole) until the macroscopic pine size. Calling of min l 0, thesize of the lower branch and max L 0, the macroscopic size of whole pine one may bedefined cutoff scales lower and upper (minimum and maximum), subdivided, therefore, thepine in discrete levels of scales as suggested the structure, as follows:l l L o k o static case L L0 0 max 1min k max L L L dynamic case L L () to o o 0 0where an intermediate scale kmin can also be defined as follows:kmax(35)lkk .(36)Lo