applied fracture mechanics

Foundations of Measurement Fractal Theory for the Fracture Mechanics 37 dM D ~ N ,(28)Dby analogy with equation (13), i.e. approximating to the geometric extension of the object bythe number of boxes (of the same size) necessary to recover it. But, since the definition of thebox dimension there is no optimization step, and its value is directly dependent on N (which is not the case with the Hausdorff dimension) in practice one has often the geometricextension is overestimated, particularly for large, i. e. upper limit 1and thusDB D. However, for the lower limit, i.e. 0 , the Hausdorff-Besicovitch dimensions,DHand the box dimension, D Bare equal, becoming valid the measure of geometricextension process, M at box counting algorithm.D Considering from (28) that:and thatTherefore, dividing (29) by (30) has: ~ D 1N d Dd , (29)D ~ 1N d Dd (30)maxmaxNN maxD ~ d Dd1 max (31)taking maxthe total grid extension that recover the object, one has:From as early as (31)Substituting (33) in (28) has:max 1(32)D 1N d Dd (33)MD DD ~ , (34)This equation is analogous to the fundamental Richardson relationship for a fractal length.4. Crack and rugged fracture surface modelsThe two main problematics of mathematical description of Fracture Mechanics are based onthe following aspects: the surface roughness generated in the process and the fieldstress/strain applied to the specimen. This section deals with the fractal mathematicaldescription of the first aspect, i.e., the roughness of cracks on Fracture Mechanics, usingfractal geometry to model its irregular profile. In it will be shown basic mathematical