《纺织复合材料》课程参考文献（Micromechanisms of Fracture and Fatigue）Chapter 3 Fatigue Fracture

130 3 Fatigue Fracture z=fx),<Z>=0 Figure 3.1 Graphical definition of basic extreme vertical parameters:the maximum hight Rp,the maximum depth Re and the vertical range R=.For comparison,the arithmetic roughness Ra is also depicted.The grey area marks the first element of the profile Length Parameters Length parameters describe the distribution of specific altitudinal levels of the surface in the horizontal plane z-y.These parameters are particularly useful for quality control in manufacturing processes.Their application in the fractography is rather rare.The most commonly used length parameters are,for example,the average spacing of profile elements on the mean line, Sm,the number of profile intersections with the mean line no or the number of peaks per the length unit mo. By evaluation of the parameter mo,for example,the peak is counted only when its horizontal distance from the previously counted peak is higher than of the vertical range R[251].Consequently,the relationR must be fulfilled,where r is the z-coordinate of the i-th peak and is that of the previously counted peak. Hybrid Parameters Hybrid parameters can be understood as a combination of altitudinal and length characteristics 250,252.In quantitative fractography,the linear roughness RL and the area roughness RA have been used for some time. These dimensionless characteristics,sometimes respectively called the rela- tive profile length and the relative surface area,are defined as L R= S RA=

130 3 Fatigue Fracture Figure 3.1 Graphical definition of basic extreme vertical parameters: the maximum hight Rp, the maximum depth Rv and the vertical range Rz. For comparison, the arithmetic roughness Ra is also depicted. The grey area marks the first element of the profile Length Parameters Length parameters describe the distribution of specific altitudinal levels of the surface in the horizontal plane x–y. These parameters are particularly useful for quality control in manufacturing processes. Their application in the fractography is rather rare. The most commonly used length parameters are, for example, the average spacing of profile elements on the mean line, Sm, the number of profile intersections with the mean line n0 or the number of peaks per the length unit m0. By evaluation of the parameter m0, for example, the peak is counted only when its horizontal distance from the previously counted peak is higher than 1 10 of the vertical range Rz [251]. Consequently, the relation xp i −xp i−1 > 1 10Rz must be fulfilled, where xp i is the x-coordinate of the i-th peak and xp i−1 is that of the previously counted peak. Hybrid Parameters Hybrid parameters can be understood as a combination of altitudinal and length characteristics [250, 252]. In quantitative fractography, the linear roughness RL and the area roughness RA have been used for some time. These dimensionless characteristics, sometimes respectively called the rela￾tive profile length and the relative surface area, are defined as RL = L L , RA = S S

3.1 Quantitative Fractography 131 where L is the fracture profile length,S is the area of the fracture surface,L' is the profile projection length and S'is the surface area projection into the macroscopic fracture plane.In the case of the profile composed of z randomly oriented linear segments RL=,whereas for the fracture surface composed of randomly oriented (nonoverlapping)facets RA =2 [214.The area roughness RA can be roughly assessed by means of the linear roughness RL as RA (月)-)+. The average slope Aa and its standard deviation Ag are defined in the following manner: n-2 2+1-2 n-1 0(z+1-) 1 It should be noted that the parameters RL and RA also provide informa- tion about the angular distribution of surface elements [258. Spectral Parameters Spectral character of the profile can be described by means of the autocorre- lation function which is a quantitative measure of similarity of the "original" surface to its laterally shifted version 251,253.Thus,the autocorrelation function expresses the level of interrelations of surface points to neighbour- ing ones.In the case of the fracture profile described by a set of n equidistant points (or m x n points obtained by sampling using constant steps in the directions of coordinate axes z,y),the autocorrelation function is defined by the following relations: n-p-1 1 R(p)= (n-p) ∑(a-(a2+p-(》 = m-p-1n-q-1 1 R(p,q)=7 (m-p)(n-q) (a,1-(2)(+pj+g-(2) i=01=0 where p and g are shifts in the directions of z and y.The autocorrelation function has the following properties:

3.1 Quantitative Fractography 131 where L is the fracture profile length, S is the area of the fracture surface, L is the profile projection length and S is the surface area projection into the macroscopic fracture plane. In the case of the profile composed of z randomly oriented linear segments RL = π 2 , whereas for the fracture surface composed of randomly oriented (nonoverlapping) facets RA = 2 [214]. The area roughness RA can be roughly assessed by means of the linear roughness RL as RA =  4 π  (RL − 1) + 1. The average slope Δa and its standard deviation Δq are defined in the following manner: Δa = 1 n − 1 n −2 i=0 |zi+1 − zi| (xi+1 − xi) , Δq =  1 n − 1 n −2 i=0  |zi+1 − zi| (xi+1 − xi) − Δa  1 2 . It should be noted that the parameters RL and RA also provide informa￾tion about the angular distribution of surface elements [258]. Spectral Parameters Spectral character of the profile can be described by means of the autocorre￾lation function which is a quantitative measure of similarity of the “original” surface to its laterally shifted version [251, 253]. Thus, the autocorrelation function expresses the level of interrelations of surface points to neighbour￾ing ones. In the case of the fracture profile described by a set of n equidistant points (or m × n points obtained by sampling using constant steps in the directions of coordinate axes x, y), the autocorrelation function is defined by the following relations: R(p) = 1 (n − p) n− p−1 i=0 (zi − z)(zi+p − z), R(p, q) = 1 (m − p)(n − q) m−p−1 i=0 n− q−1 j=0 (zi,j − z)(zi+p,j+q − z), where p and q are shifts in the directions of x and y. The autocorrelation function has the following properties:

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.Fundamental

132 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

3.1 Quantitative Fractography 133 properties of fractal objects are so-called self-similarity or self-affinity which mean an invariance with respect to scale changes.As a measure of the frac- tality the Hausdorff(fractal)dimension Dy is often used.The metrics of D can be determined by means of the Hausdorff measure T哈=lmi#∑(diam U,)4. (3.3) E-00 When calculating the Hausdorff measure,the object is covered by cells Ui.The diameter of each cell meets the following condition:diam Ui= sup {-yl:,yEUi}<s.Consequently,one searches the cell network min- imizing the sum in Equation 3.3 for an infinitely small diameter of covering cells (->0).There is only a single value of D fulfilling the conditions T=0 for each d>D and Ta=oo for each d<DH.This value is called the Hausdorff(fractal)dimension of the object.In the case of a smooth(Eu- clidean)object D=dr,where dr is the topological dimension,whereas dr D<(dr+1)holds for the fractal object.In general,DH is a rational number exceeding the topological dimension.A higher Dy-value means a higher segmentation of the object.As an example of the fractal object,Von Koch's curve is depicted in Figure 3.2 along with several first steps of its construction. (a) n=0 n=2 (e） 7=∞ 0 P (b) n=1 (d) n=3 0 Figure 3.2 Von Koch's curve:(a)fractal initiator,(b)first iteration,(c)second iteration,(d)third iteration,and (e)final fractal(D1.262) As can be seen from Figure 3.2,the length of the curve increases with increasing number of iterations and,for the final fractal,it becomes infinite. On the other hand,the area under the curve remains finite and practically unchanged.The infinite length of the fractal curve means that the marked points O and P retain the same distance during all iterations.Paradoxically, however,they coincide in the case of the final fractal (limnOP=0). Real natural objects exhibit a statistical self-similarity rather than a perfect deterministic one.This means that the self-similarity does not hold for the object itself but only for its statistical parameters (average,variation,etc.) [2601. With respect to difficulties connected with the calculation of the Haus- dorff dimension I directly according to Equation 3.3,various other simpler solutions were derived.The most widely used methods are depicted in Fig-

3.1 Quantitative Fractography 133 properties of fractal objects are so-called self-similarity or self-affinity which mean an invariance with respect to scale changes. As a measure of the frac￾tality the Hausdorff (fractal) dimension DH is often used. The metrics of DH can be determined by means of the Hausdorff measure Γd H = limε→0 inf Ui  i (diam Ui) d . (3.3) When calculating the Hausdorff measure, the object is covered by cells Ui. The diameter of each cell meets the following condition: diam Ui = sup {|x − y| : x, y ∈ Ui} ≤ ε. Consequently, one searches the cell network min￾imizing the sum in Equation 3.3 for an infinitely small diameter of covering cells (ε → 0). There is only a single value of DH fulfilling the conditions Γd H = 0 for each d>DH and Γd H = ∞ for each d<DH. This value is called the Hausdorff (fractal) dimension of the object. In the case of a smooth (Eu￾clidean) object DH = dT , where dT is the topological dimension, whereas dT < DH ≤ (dT + 1) holds for the fractal object. In general, DH is a rational number exceeding the topological dimension. A higher DH-value means a higher segmentation of the object. As an example of the fractal object, Von Koch’s curve is depicted in Figure 3.2 along with several first steps of its construction. Figure 3.2 Von Koch’s curve: (a) fractal initiator, (b) first iteration, (c) second iteration, (d) third iteration, and (e) final fractal (DH ≈ 1.262) As can be seen from Figure 3.2, the length of the curve increases with increasing number of iterations and, for the final fractal, it becomes infinite. On the other hand, the area under the curve remains finite and practically unchanged. The infinite length of the fractal curve means that the marked points O and P retain the same distance during all iterations. Paradoxically, however, they coincide in the case of the final fractal (limn→∞ |OP| = 0). Real natural objects exhibit a statistical self-similarity rather than a perfect deterministic one. This means that the self-similarity does not hold for the object itself but only for its statistical parameters (average, variation, etc.) [260]. With respect to difficulties connected with the calculation of the Haus￾dorff dimension Γd H directly according to Equation 3.3, various other simpler solutions were derived. The most widely used methods are depicted in Fig-

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 k

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 (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