applied fracture mechanics

Foundations of Measurement Fractal Theory for the Fracture Mechanics 35with maximum size, max, which inscribes the fractal object. Defining the relative scale, on the grid size, max, as being given by: (20)countting the number of boxes N( ) that have at least one point of the fractal. The boxdimension is therefore defined as:maxln N( )DB lim(21) 0ln At this point, there are two ways to obtain the actual value of the measure, or taking thelimit when 0 and allows that the dimension D fits the end value of N( ) , or it isconsidered a linear correlation in value of ln N( ) ln, which D is the slope of the line,and this defines the measure independently of the scale.In the case of numerical estimation, one can not solve the limit indicated in the equation(21). Then, DBis obtained as a slope, ln N( ) lnwhen it is small. The value N( ) isobtained by an algorithm known as Box-Counting.Self-affine fractals requiring different variations in scale length for different directions.Therefore, one can use the Box-Counting method with some care being taken, in the sensethat the box dimension DBto be obtained has a crossing region between a local and globalmeasure of the dimensions. From which follows that for each region is used the followingrelationships:l00DBgL 0000sl 0lim N L p/L L (22)for a global measurementl00DBlL 0000sl 0lim N L p/L L (23)where L0sis the threshold saturation length which the fractal dimension changes itsbehavior from local to global stage.For measurement, generally, for any self-affine fractal structure the local fractal dimension isrelated to the Hurst exponent, H , as follow,D d 1 H (24)BlAt this point, one observes that for a profile the relationship DBl 2 H commonly used,only serves for a local measurements using the box counting method. While for globalq 1