applied fracture mechanics

80Applied Fracture MechanicsThe elastic energy released rate G0can be written in terms of a fractal geometry,2m* rH 0l 0G0 L 01 (2 H)E' l 0L 02 2H2(53)5.4. The crack growth resistance R for smooth, projected and rough pathsConsidering a plane strain condition, crack growth resistance for a smooth crack is given bySubstituting equation (24) in equation (54), one findsdUlRl (54)dLlR 2(55)lObserve that if the fracture path is smooth, the specific surface energy lis a cleavagesurface energy and does not necessarily depend on the crack length. This model is onlyvalid for brittle crystalline materials where the plastic strain at the crack tip does not absorbsufficient energy to cause dependence between fracture toughness and crack length.Similarly, for a rugged crack, the fracture resistance to propagation is given bylR (56)2rThe concept of fracture growth resistance for the projected surface is given byand substituting equation (26) in equation (57), one hasdUR0 (57)dL0R 2(58)0 0Again, this model is valid for ideally brittle materials where there is almost no plastic strainat the crack tip. It basically corresponds to the model presented by Griffith, with a modifiedinterpretation introduced by Irwin with the G R curve concept.5.5. Relationship between rugged R and projected R 0fracture resistancesUsing the chain rule, and admitting Irwin´s energetic equivalence represented by equation(3), the projected fracture resistance can be written on the basis of the resistance of the realsurface,