applied fracture mechanics

62Applied Fracture Mechanics5. Conclusionsi. It is possible, in principle, mathematically distinguish a crack in different materialsusing geometric characteristics which can be portrayed by different values of roughnessexponents in the relations (72) and (74).ii. The fractal model of the rugged crack length, L in function of the projected cracklength, L0, suggested by Alves [72 73 ,74 , 75 ] seems have a good agreement withexperimental results. This results allowed us to consolidate the model previouslypublished in the literature on fracture [72 , 73 , 75 ].iii. The rugged crack length is a response to its interaction with the microstructure. of thematerial. Therefore, mathematically is possible to portray the rugged peculiar behaviorof a crack using fractal geometryiv. The mathematical model presents a wealth (mathematical richness) that can still beexplored in terms of determining the minimum crack length, l 0for each material andthe fractal dimension as a function of test parameters and material properties.v. The mathematical model is sensitive to variations in the behavior of the crack length itis a linear or logarithmic with the projected crack length.Comparing the experimental results with the model proposed in this chapter, it isconcluded that one of the more important results obtained here are the equations (72), (74)and (82) leading to finding that the fracture surfaces of the materials analyzed are indeed(actually) self-affine fractals. Starting from this verification it becomes feasible to considerthe fractal model of rugged fracture surface and its ruggedess inside the equations of theclassical fracture mechanics, according to equation (74) and (82). As there is a closerelationship between phenomenology and structure formed by virtue of its fractalgeometry, the understanding of the formation processes of these dissipative structures, asthe cracks, should be derived from their mathematical analysis, as the close relationshipbetween the phenomenology of the formation process of dissipative structures and theirfractal geometry. Therefore, the mathematical description of fractal structures mustexceed a simple geometrical characterization, in order to correlate the pattern formed inthe process of energy dissipation with the amount of energy dissipated in the process thatgenerated it. Thus, it is possible to use the fractal geometry in order to understand othermore and more complex processes inside the fracture mechanics. Therefore, the variousmechanisms responsible by the crack deviation and by the formation of the ruggedfracture surface can then, from the fractal model, be quantified in the fractal analysis ofthis surface.The idea of obtaining a relationship between L and L 0comes the need to maintain thepresent formalism used by the CFM, showing that fractal geometry can greatly contribute tothe continued advancement of this science.On the other hand, we are interested in developing a Fractal Thermodynamic for a ruggedcrack that will be related to the CFM and the Classical Fracture Thermodynamics when thecrack ruggedness is neglected or the crack is considered smooth.