applied fracture mechanics

Foundations of Measurement Fractal Theory for the Fracture Mechanics 41[52]. Although there are several methods to quantify the fracture area, the results aredependent on the measure ruler size used [56]. Since the last century all the existingmethods to measure a rugged surface did not contribute to its insertion into the analyticalmathematical formalism of CFM until to rise the fractal geometry. Generally, the roughnessof a fracture surface has fractal geometry. Therefore, it is possible to establish a relationshipbetween its topology and the physical quantities of fracture mechanics using fractalcharacterization techniques. Thus, with the advent of fractal theory, it became possible todescribe and quantify any structure apparently irregular in nature [1]. In fact, many theoriesbased on Euclidean geometry are being revised. It was experimentally proved that thefracture surfaces have a fractal scaling, so the Fracture Mechanics is one of the areasincluded in this scientific context.4.2.1. Importance of fracture surface modelingThe mathematical formalism of the CFM was prepared by imagining a fracture surface flat,smooth and regular. However, this is an mathematical idealization because actually themicroscopic viewpoint, and in some cases up to macroscopic a fracture surface is generally arough and irregular structure difficult to describe geometrically. This type of mathematicalsimplification above mentioned, exists in many other areas of exact sciences. However, tomake useful the mathematical formalism developed over the years, Irwin started to considerthe projected area of the fracture surface [57] as being energetically equivalent to the ruggedsurface area. This was adopted due to experimental difficulties to accurately measure thetrue area of the fracture, in addition to its highly complex mathematics. Although there aredifferent methods to quantify the actual area of the fracture [56], its equationing within thefracture mechanics was not considered, because the values resulting from experimentalmeasurements depended on the "ruler size" used by various methods. No mathematicaltheory had emerged so far, able to solve the problem until a few decades came to fractalgeometry. Thus, modern fractal geometry can circumvent the problem of complicatedmathematical description of the fracture surface, making it useful in mathematical modelingof the fracture.In particular, it was shown experimentally that cracks and fracture surfaces follow afractional scaling as expected by fractal geometry. Therefore, the fractal modeling of airregular fracture surface is necessary to obtain the correct measurement of its true area.Therefore, fracture mechanics is included in the above context and all its classical theorytakes into account only the projected surface. But with the advent of fractal geometry, is alsonecessary to revise it by modifying its equations, so that their mathematical descriptionbecomes more authentic and accurate. Thus, it is possible to relate the fractal geometriccharacterization with the physical quantities that describe the fracture, including the truearea of irregular fracture surface instead of the projected surface. Thought this idea was thatMandelbrot and Passoja [58] developed the fractal analysis by the "slit island method ".Through this method, they sought to correlate the fractal dimension with the physical wellknownquantities in fracture mechanics, only an empirical way. Following this pioneering