applied fracture mechanics

58Applied Fracture Mechanicswhich is a self-similar relation between the projected crack length, L 0, and height of thelittle box, h 0. This relationship shows that all self-affine fractal, in the approximation of asmall scale, has a local self-similarity forming a fractal substructure, when is consideredsquare portions, L0 L0, instead of rectangular portions, L0 H0.It important to observe that L0denotes the distance between two points of the crack (theprojected crack length). The self-affine measure, L of L 0, in the fractal dimension, D , isgiven by (72). l 0is the possible minimum length of a micro-crack, which defines the scalel0 / L0under which the crack profile is scrutinized, as discussed in previous section and willbe discussed after in the section #.5.4.5. The Hurst exponent, H , is related to D by (60).In the study of a self-affine fractal there are two extremes limits to be verified. One is thelimit at which the boxes height is high in relation to its projected length, L0, i.e. ( H0 L0),which is also called local limit. The other limit is one in which the boxes height is low inrelation to its projected length, L0, i.e., ( H0 h0) which is called global limit. It will be seennow each one of this limits case contained in the expression (72).Case 1 : The self-similar or local limit of the fractalityTaking the local limit of the self-affine fractal measure as given by (72), i. e. for the casewhere, H0 L0 l0, one has:whereLL2HoH1 l oL LoL oH 1o(76)l constant(77)This equation is analogous to self-similar mathematical relationship only that the exponentis 1 Hinstead of D 1, which satisfies the relation H 2 D [3 , 40, 51 , 70].According to these results it is observed that the relation (77) has a commitment to the Hurstexponent of the profiles on the considered observation scale l0 / L0. It is observed thatthe consideration of a minimum fracture size l 01over a region, one must consider the localdimension of the fracture roughness on this scale. Similarly, if the considerations of aminimum fracture size are made in a scale that involves several regions, l 02this should takeinto account the value of the roughness global dimension on this scale, so that:H 1 1 H211 01 2 02(2 H ) l (2 H ) l constant,(78)although l 01 l 02e H1 H2.Case 2: The self-affine or global limit of fractalityTaking the global limit of the self-affine fractal measure given by (72), i.e. for the case inwhich: H0 l0 L0. Therefore the length L is independently of H and D 1 , so