132 3 Fatigue Fracture 1.R(0)=2orR(0,0)=2; 2.R(p)=R(-p)or R(p,q)=R(-p,-q); 3.R(0)>R(p)or R(0,0)>R(p,q)which means that the autocorrelation function attains a maximum for zero shifts. With respect to the first attribute,the autocorrelation function is often normalized so that R(0)=1 or R(0,0)=1.The normalized autocorrelation function is usually denoted as r(p)or r(p,q): -1≤ro=0≤1，-1≤re,g)= R(p,q) ≤1. 2 2 Autocorrelation lengths Bp and Bo are defined as shifts p,g corresponding to a drop of the autocorrelation function to a given fraction of its initial value. The fractions and are most frequently utilized [251,253].Consequently, the surface points more distant than Bp,Ba can be assumed to be uncorre- lated.This means that the related part of the fracture surface was created by another,rather independent,process of surface generation. The character of the spectral surface can also be described in the Fourier space.The most important characteristic is the power spectral density G(p)=1F(wp)2, G(wp,wq)=|F(wp,wq)2, where wp and wg are space frequencies in the directions of coordinate axes x and y [251,254].Functions F(wp)and F(wp,wa)represent relevant Fourier transforms of the fracture surface: (3.1) F,g)= (3.2) mn Σ(+)} k=0l=0 where i is the imaginary unit [254].Equations 3.1 and 3.2 define the so- called discrete Fourier transform(DET).The conventional factors and mn might differ for various applications.Instead of the highly computationally demanding DFT the fast Fourier transform is often utilized. Fractal Parameters Fractal geometry is a mathematical discipline introduced by Mandelbrot [259] in the early 1980s.It is widely utilized as a suitable tool for the description of jagged natural objects of complicated geometrical structure.Fundamental132 3 Fatigue Fracture 1. R(0) = μ2 or R(0, 0) = μ2; 2. R(p) = R(−p) or R(p, q) = R(−p, −q); 3. R(0) ≥ |R(p)| or R(0, 0) ≥ |R(p, q)| which means that the autocorrelation function attains a maximum for zero shifts. With respect to the first attribute, the autocorrelation function is often normalized so that R(0) = 1 or R(0, 0) = 1. The normalized autocorrelation function is usually denoted as r(p) or r(p, q): −1 ≤ r(p) = R(p) μ2 ≤ 1, −1 ≤ r(p, q) = R(p, q) μ2 ≤ 1. Autocorrelation lengths βp and βq are defined as shifts p, q corresponding to a drop of the autocorrelation function to a given fraction of its initial value. The fractions 1 10 and 1 e are most frequently utilized [251,253]. Consequently, the surface points more distant than βp, βq can be assumed to be uncorre￾lated. This means that the related part of the fracture surface was created by another, rather independent, process of surface generation. The character of the spectral surface can also be described in the Fourier space. The most important characteristic is the power spectral density G (ωp) = |F (ωp)| 2 , G (ωp, ωq) = |F (ωp, ωq)| 2 , where ωp and ωq are space frequencies in the directions of coordinate axes x and y [251, 254]. Functions F (ωp) and F (ωp, ωq) represent relevant Fourier transforms of the fracture surface: F (ωp) = 1 n n −1 k=0 zk exp  −i2π kωp n , (3.1) F (ωp, ωq) = 1 mn m −1 k=0 n −1 l=0 zk,l exp  −i2π kωp m + lωq n  , (3.2) where i is the imaginary unit [254]. Equations 3.1 and 3.2 define the so￾called discrete Fourier transform (DFT). The conventional factors 1 n and 1 mn might differ for various applications. Instead of the highly computationally demanding DFT the fast Fourier transform is often utilized. Fractal Parameters Fractal geometry is a mathematical discipline introduced by Mandelbrot [259] in the early 1980s. It is widely utilized as a suitable tool for the description of jagged natural objects of complicated geometrical structure. Fundamental