applied fracture mechanics

70Applied Fracture Mechanicslower limit l 0is related to the micro-mechanics of the cracked material and the upper limitL0 maxis a function of the geometric size of the body, crack length and other factors.Figure 1. Rugged crack and its projection in the plan of energetic equivalence.III. Energy equivalence between the rugged crack surface and its projectionIrwin apud Cherepanov et al. [36] realized the mathematical complexity of describing thefracture phenomena in terms of the complex geometry of the fracture surface roughness indifferent materials. For this reason, he proposed an energy equivalence between the roughsurface path and its projection on the Euclidean plane.In the energetic equivalence between rugged and projected crack surfaces it is consideredthat changes in the elastic strain energy introduced by a crack are the same for both ruggedand projected paths,UL0 U(2)where the subscript " 0 " denotes quantities in the projected plane. Consequently, the surfaceenergy expended to form rugged fracture surfaces or projected surfaces are also equivalent,IV. Invariance of the equationsU 0L U(3)Consider a crack of length L and the quantities that describe it. Assuming the existence of ageometric operation that transforms the real crack size L to an apparent projected size L0,the length L may be described in terms of L0by a fractal scaling equation, as presented in aprevious chapter.It is claimed that the classical equations of the fracture mechanics can be applied to bothrugged and projected crack paths, i.e., they are invariant under a geometric transformationbetween the rugged and the projected paths. In the crack wrinkling operation (smooth torough) it is desired to know what will be the form of the fracture mechanics equations forthe rough path as a function of the projected length L 0, and their behavior for differentroughness degrees and observation scales.