applied fracture mechanics

88Applied Fracture Mechanics6.3.1. Case – 1. Ductile self-similar limitThe local self-similar limit can be calculated applying the condition H 0 L 0 l 0inequation (79), obtainingor, with D 2 H , one hasJ2 2H l o L o Ro e pH1(97) l 0J0 2effD L 0 1D. (98)This result corresponds to the one found by Mu and Lung [26, 37] for ductile materials.Equation (98) is shown in Figure 5, where J-R curves are calculated for different values of2the fractal dimension D . 2 eff= 10.0 KJ / m is adopted and L0 l0is the crack length in l 0units. This figure shows very clearly how the surface morphology (characterized by D )determines the shape of the J-R curve at the beginning of the crack growth.Figure 5. J-R curves calculated according to the projected crack length L0 , for a fracture of unitthickness, and fractal dimensions D 1.0,1.1,1.3,1.5,1.7 and 2.0 with2 e 10 KJ / m .In Figure 6, J-R curves with fractal dimension D 1.3 are calculated according to theprojected length L 0for different measuring rulers l 0, showing how the morphology ofrugged surface cracks is best described for small values of l 0, causing the pronounced risingof J-R curve. Figure 6 and equation (98) show that the initial crack resistance is correlated tothe surface morphology characterized by dimension D , in accordance with the literature.The self-similar limit of J-R curve, given by equation (98), is valid only for regions near theonset of the crack growth in brittle materials ( H0 L0). This is due to the hardening of thematerial, which gives rise to ruggedness of the fracture surface.2