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applied fracture mechanics

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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 the<strong>fracture</strong> 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.

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