applied fracture mechanics

84Applied Fracture MechanicsKIRodL KICo.(77)dLSubstituting equation (75) and equation (76) in equation (77), one has22 Lo f o Yo o 2e pdLL f( v) ,dL w EObserve that according to the right hand side of equation (78), the ruggedness dL dL0isdetermined by the condition of the test (plane strain or stress), the shape of the sample (CT,SEBN, etc), the type of test (traction, flexion, etc) and kind of material.Considering the fracture surface as a fractal topology, one observes that the characteristics ofthe fracture surface listed above in equation (78) are all included in the ruggedness fractalexponent H. Substituting equation (60) in equation (71), one obtainso(78)JRo e p2 2H20l0H 12H l 0 L 0 2 .2 2H2H 0l 01 l 0L 0 (79)which is non-linear in the crack extension L0. It corresponds to the classical equation (70)corrected for a rugged surface with Hurst's exponent H. Experimental results [1, 2] showthat J0 and the crack resistance R0rise non-linearly and it is well known that this rising ofthe J-R curve is correlated to the ruggedness of the cracked surface [3, 4].6.2. The J0Eshelby-Rice integral for rugged and plane projected crack pathsThe J-integral concept of Eshelby-Rice is a non-linear extension of the definition given byIrwin-Orowan, for the linear elastic plastic energy released rate. In this context the potentialenergy 0is defined as0 WdV0 T . uds,V0C(80)where W the energy density integral in the in the volume V0encapsulated by the boundaryC with tractions T and displacements u , and s is the distance along the boundary C , asshown in Figure 4.Accordingly,d 0 dJ0 WdV0T.udsdL0 dL0VC (81)