applied fracture mechanics

Foundations of Measurement Fractal Theory for the Fracture Mechanics 49An illustration of the relationship (43), (44) and (45) can be seen in Figure 16.Figure 16. Rugged surface formed by a homogeneous function A , with frational degee D , whoseplanar projection, A0is a homogeneous function of integer degree d , showing the unit surface areaA .uThe rugged fracture surface, may be considered to be a homogeneous function withfrational degree, D , ni.e.:Dk kA A ,(46)and its planar projection, may be considered as a homogeneous function with integer degreed 2 , i.e.:Ad0Ar r. (47)The index k was chosen to designate the irregular surface at a k -level of any magnificationor reduction. The index r has been chosen to designate the smooth (or flat) surface at a r -level, and the index, 0 , was chosen to designate the projected surface corresponding torugged surface, at the k -level.Considering that, for k r and k r, the area unit, Akandvalue and dividing relationships (46) and (47) , one has:Ar, are necessarily of equald D0 kA( ) A .(48)kThe equation (48), means that the scaling performed between a smooth and anotherd Dirregular surface, must be accompanied by a power term of type k. Thus, there is thefractal scaling, which relates the two fracture surfaces in question: a rugged or irregularsurface, which contains the true area of the fracture and regular surface, which contains theprojected area of the fracture.From now on will be obtained a relationship between the rugged and the projectedprofile of the fracture in analogous way to equation (250) for a thin flat plate (Figure 17a