applied fracture mechanics

86Applied Fracture MechanicsReturning to equation (82) and considering postulate III along with the derivative chain ruleand substituting equation (85), one hasddL dx * u dLJ0 W dy * T. ds.dL dL 0dLL dLVC 0(87)Comparing (84) with equation (87) and considering that the rugged crack is a result of atransformation in the volume of the crack, analogous to the “bakers´ transformation” of theprojected crack over the Euclidian plane, it can be concluded that( x*, y*)dx * dy*dxdy( xy , ) dx * dL dx dLdy *dydL dL dL dLoo(88)which show the equivalence between the volume elements,dV dx * dy* dxdy.(89)Therefore, the ruggedness dL / dL0of the rugged crack path does not depend on the volumeV, nor on the boundary C and nor on the infinitesimal element length ds or dy . Thus, itmust depend only on the characteristics of the rugged path described by the crack on thematerial. Finally, the integral in equation (84) can be written as dx u dLJ0W dy T.dsdLL dLVC 0(90)where the infinitesimal increment dx / dL cosiaccompanies the direction of the ruggedpath L , as show in Figure 4. Thus,uJ Wdycos i T. cos ids.xVObserve that the J-integral for the rugged crack path given by equation (91) differs from theJ-integral for the plane projected crack path given by equation (83) by a fluctuating term,cos iinside the integral. It can be observed that the energetic and geometric parts of thefracture process are separated and put in evidence the influence of the ruggedness of thematerial in the elastic plastic energy released rate,C0(91)dLJ0 J .(92)dLIt must be pointed out that this relationship is general and the introduction of the fractalapproach to describe the ruggedness is just a particular way of modeling.