applied fracture mechanics

56Applied Fracture MechanicsThe curve length in the stretch, L0 H0considering the Sand-Box method [38] whosecounting starts from the origin of the fractal, can be written as: L L,L0 L0andH H hence the equation (71) shall be given by:0 02 2H1H 0L 0L L01 ,l 0l 0 (72)whose the plot is shown in Figure 20. Note that the lengths L0and H0correspond to theprojected crack length in the horizontal and vertical directions, respectively.Applying the logaritm on the both sides of equation (72) one obtains an expression thatrelates the fractal dimension with the projected crack length: 2 2H1H 0L 0 ln ln L/ ll 0l 00 1 Df 1 (73)ln L / l 2 ln L0 00Figure 20. Graph of the rugged length L in function of the projected length L0, showing the influenceof height, H0, of the boxes in the fractal model of fracture surface: a) in the upper curves is observedthe effect of H0as it tends to unity ( H0 1.0 ), b) in the lower curves, that appearing almost overlap, isobserved the effect of H0as it tends to zero ( H0 0 ).The graph in Figure 20 shows the influence of the boxes height H0on the rugged cracklength, L , as a function of the projected crack length, L 0. Note that for boxes of lowheight ( H0 0 ), in relation to its projected length, L0, the lower curves (forH0 0.01,0.001,0.0001 ), denoted by the letter " b ", almost overlap giving rise to a linearrelation between these lengths (Figure 21). While for boxes of high height ( H0 1.0 ) inrelation to its projected length, L 0, the relation between the lengths become each moredistinct from the linear relationship for the same exponent roughness, H .