134 3 Fatigue Fracture ure 3.3.By application of those methods to real objects,however,deviations from the theoretical fractal dependencies are usually observed 249,261,262]. A sigmoidal trend,obtained when calculating the parameter DH,can serve as a typical example. Figure 3.3 Some computation methods of the of fractal dimension:(a)perimeter method,(b)computation of squares,and(c)Minkowski method Calculation of the of areas fractal dimension(dr =2)is more complicated and,as usual,it is performed either by means of space versions of curve methods 263,264 or using the area-perimeter method.The latter method analyzes the fractal dimension of boundary curves of "islands"created by intersections of the horizontal plane with the object surface 264,265. Self-affinity is a more general form of self-similarity.A regular object ex- hibiting self-affinity is invariant to the transformation x→入zx,→入g,2→入22， where and:so that :y/.The ratio H=v/vy is called the Hurst exponent (the exponent of self-affinity),HE(0;1).When all contraction coefficients are equal(A=A=A2),the same transforma- tion describes the self-similarity (see Figure 3.2(e)).In the case of isotropic surfaces,Ar =Ay and H=v:.This relation also refers to the arbitrary self-affine plane curve 260.Again,the natural objects exhibit a statistical self-affinity rather than a deterministic one.Many experiments reveal that the fracture surfaces of most materials exhibit such a property [260,266,267. Indeed,the self-similarity is usually preserved in the horizontal plane r-y(the area-perimeter method is based on that assumption),whereas the self-affinity is associated with the z-coordinate. The Hurst exponent also yields information on a degree of internal ran- domness.When the object can be described by the Hurst exponent H>H*, where H the trend in the local site(e.g.low or high z-values) is most probably followed by a similar trend in every other site x+Ax (the persistence or the long-term memory).On the other hand,H<H*means an opposite tendency (antipersistence or short-term memory).The former type is typical for brittle fractures whereas the latter is typical for ductile ones 260. As a rule,the Hurst exponent is calculated by means of the so-called variable bandwidth method 267,268.First,the profile is divided into k134 3 Fatigue Fracture ure 3.3. By application of those methods to real objects, however, deviations from the theoretical fractal dependencies are usually observed [249, 261, 262]. A sigmoidal trend, obtained when calculating the parameter DH, can serve as a typical example. Figure 3.3 Some computation methods of the of fractal dimension: (a) perimeter method, (b) computation of squares, and (c) Minkowski method Calculation of the of areas fractal dimension (dT = 2) is more complicated and, as usual, it is performed either by means of space versions of curve methods [263, 264] or using the area-perimeter method. The latter method analyzes the fractal dimension of boundary curves of “islands” created by intersections of the horizontal plane with the object surface [264, 265]. Self-affinity is a more general form of self-similarity. A regular object ex￾hibiting self-affinity is invariant to the transformation x → λxx, y → λyy, z → λzz, where λy ∝ λνy x and λz ∝ λνz x so that λz ∝ λνz/νy y . The ratio H = νz/νy is called the Hurst exponent (the exponent of self-affinity), H ∈ 0; 1. When all contraction coefficients are equal (λx = λy = λz), the same transforma￾tion describes the self-similarity (see Figure 3.2(e)). In the case of isotropic surfaces, λx = λy and H = νz. This relation also refers to the arbitrary self-affine plane curve [260]. Again, the natural objects exhibit a statistical self-affinity rather than a deterministic one. Many experiments reveal that the fracture surfaces of most materials exhibit such a property [260,266,267]. Indeed, the self-similarity is usually preserved in the horizontal plane x–y (the area-perimeter method is based on that assumption), whereas the self-affinity is associated with the z-coordinate. The Hurst exponent also yields information on a degree of internal ran￾domness. When the object can be described by the Hurst exponent H>H∗, where H∗ = dT dT +1 , the trend in the local site x (e.g., low or high z-values) is most probably followed by a similar trend in every other site x + Δx (the persistence or the long-term memory). On the other hand, H<H∗ means an opposite tendency (antipersistence or short-term memory). The former type is typical for brittle fractures whereas the latter is typical for ductile ones [260]. As a rule, the Hurst exponent is calculated by means of the so-called variable bandwidth method [267, 268]. First, the profile is divided into k