applied fracture mechanics

54Applied Fracture MechanicsFigure 19. Self-affine fractal of Weierstrass-Mandelbrot, where k 1 / 4 and Dx 1.5 and H 0.5 ,used to represent a fracture profile (Family, Fereydoon; Vicsek, Tamas Dynamics of Fractal Surfaces,World Scientific, Singapore , 1991, p.7).LNv in vertical direction(61)l0 0v0where L0is the crack horizontal projection and vis the vertical scaling factor.Considering that the self-affine fractal extends in the horizontal direction along L 0, andoscillates in the perpendicular direction, i.e. in the vertical direction, the number of littleboxes Nh, with size, l 0in the horizontal direction, are gathered to form the projected lengthL0, while vertically the number of little boxes N v, with size h 0, overlap each other,increasing (as power law) this number in comparison to the number of little boxes gatheredhorizontally. Therefore, for the vertical direction with a projection H0, the box sidesL H , an expression for the number of boxes (or units covering) can be writen as:0 0HNh in horizontal direction .(62)h0 H h0where H is the Hurst exponent, H0is the total variation in height ( l o H 0 L o) and is the scale transformation factor in the horizontal direction.hTherefore, for the corresponding rugged crack length (real)can writes:v L , the stretch L0 H0oneL Nr(63)