applied fracture mechanics

48Applied Fracture Mechanicsknol(1 )a ~ .(40)Mishnaevsky equates with mathematical elegance, the crack propagation as the result of a"physical reaction" of interaction of a crack size, L0with a piling up of dislocations, nb ,forming a microtrinca size, a , i.e.;where a Loe nb Lo. L nb L a ,(41)oMishnaevsky proposes a fractal scaling for the fracture process since the minimum scale,given by the size a , until the maximum scale, given by the macroscopic size crack, L0.As a consequence for the existence of a minimum fracture size, recently has arosen ahypothesis that the fracture process is discrete or quantized (Passoja, 1988, Taylor et al.,2005; Wnuk, 2007). Taylor et al. (2005) conducted mathematical changes in CFM to validatethis hypothesis. Experimental results have confirmed that a minimum fractures length isgiven by:o2 K 0~ cl. 0 (42)where Kcis the fracture toughness, 0is the stress of the yielding strength before thematerial fracture.4.2.7. Fractal scaling of a self-similar rough fracture surface or profileA mathematical relationship between the extension of the self-similar contour and aextension of its projection is calculated as follows.Being A the surface extension of the fractal contour, given by a self-similar homogenousfunction with fractional degree, D , where: D .A A u(43)A0is the plane projection extension, given by a self-similar homogeneous function withinteger degree, d , in accordance with the expression:where, dd A 0A u , (44)Au is the unit area of measurement, whose values on the rugged and planesurface are the same. Thus the relationships (43) and (44) can be written in the same way asthe equations (43) and (44). Therefore, by dividing these equations, one has:A A d D0. (45)