applied fracture mechanics

74Applied Fracture Mechanicswhich corresponds to a rougher surface, this relationship is increasingly non-linear. This isreasonable since the more ruggedness, more elastic strain per unit of crack length.4.2.2. Relationship between the applied stress on the rough and projected crack lengthsComparing (8), (10) and (12), one hasm* U U U L0Ll LThen, from postulate III, i.e., the following relationship is valid only for the situation of freeloading without crack growth.2 20 r L E'0E'L 0 Using equation (13) in (19), one has the resilience as a function of the projected length L02(18)(19)2 2 0 1 H 0 l01 E0 2 E l0 L02 2H2Or, the rugged length L can be written in terms of the projected length L0, thusE'0L LE'0 r Since the elasticity modulus is independent of the crack path, one has0(20)(21) L L(22)0 0 rSubstituting equation (13) in equation (22), one has the relationship between stresses on therough and projected surfaces, rH 0l 00 1 2 l 0L 02 2H2This last result is still incomplete since it is not valid for crack propagation. For its correctionit will be considered that the elastic energy released rate G can be expressed as a function ofG0according to equation (4).1/2(23)4.2.3. The surface energy U 0 for smooth, rugged and projected cracks in accordance withfractal geometryThe surface energy of a smooth and a rugged crack are, respectively, given by