applied fracture mechanics

Fractal Fracture Mechanics Applied to Materials Engineering 71V. Continuity of functionsIt is considered that the scalar and vector functions that define the irregular surfaces A Axy , are described by a model (as the fractal model) capable of providing analyticaland differentiable functions in the vicinity of the generic coordinate points PPxyz , , that it is possible to calculate the surface roughness. Thus, it is always possible to define anormal vector in corners.VI. Transformations from the projected to rugged path equationsAs a consequence of the previous two postulates, it can be shown using the chain rule thatthe relationship between the rates for projected and rugged paths are given by 0 0, sodf L0df L dL (4)dL dL dLThis result is used to transform the equations from the rugged to the projected path.4. Energies in linear elastic fracture mechanics for irregular mediaThe study of smooth, rough, fractal and non-fractal cracks in Fracture Mechanics requiresthe development of their respective equations of strain and surface energies.4.1. The elastic strain energy UL for smooth, rugged and fractal cracksConsider three identical plates of thickness t , with Young’s modulus E´, subjected to astress , each of them cracked at its center with a smooth, a rugged and a fractal crack asshown in Figure 2. The area of the unloaded elastic energy due to the introduction of the crackwith length Lliswhereml2lmLl lA (5) is the shape factor for the smooth crack. The accumulated elastic energy is2Ue dV(6)2 E'Thus, the elastic energy released by the introduction of a smooth crack with length Llisml2 2lLlUl mtl(7)2 E'For an elliptical crack the unloaded region can be considered almost elliptical and the shapefactor is , thusl