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

Chapter 3 Fatigue Fracture From the practical point of view,fatigue fracture is the most important dam- age process.Available statistics show that,including corrosive assistance,fa- tigue is a leading cause of material failures registered during a long-term per- formance of engineering components and structures [246.From the microme- chanical point of view,the fatigue process can be understood as a sequence of the following stages:nucleation of cracks,stable propagation of short(small) cracks,stable propagation of long cracks and unstable fracture [247]. This chapter is divided into four sections.In Section 3.1,morphological patterns reflecting all crack growth stages on the fracture surface are briefly described.Moreover,the topological methods widely utilized in fatigue re- search and the quantitative fractography are outlined.These topics are im- portant for all subsequent sections of the chapter. The second section is devoted to the propagation of fatigue cracks under the remote opening mode(mode I).Mechanisms of nucleation and growth of short cracks are briefly reported,although these initial stages of the fatigue process were not a special subject of our research.Inclusion of these topics was,however,inevitable in order to provide a self-contained description of fatigue micromechanisms.The growth of short cracks is governed by shear stress components in favourably oriented crystallographic slip systems that are inclined at about 45 with respect to the maximal principal stress.This means that the short cracks grow in a local mixed-mode I+II and the pic- ture of dislocation emission from the crack tip as well as the related growth micromechanisms are completely different from those related to long cracks. After a certain incipient period that incorporates crystallographic and transient growths of short cracks,the fatigue cracks incline towards a di- rection perpendicular to the maximal principal stress,i.e.,nearly towards mode I loading of the crack tip.This means that the long cracks keep prop- agating so that the crack tip plasticity is produced in the opening loading mode.Thus,the second section provides the reader with a micromechanical interpretation of all important phenomena accompanying this,most frequent, type of fatigue crack growth.Knowledge of these micromechanisms is essen- 125

Chapter 3 Fatigue Fracture From the practical point of view, fatigue fracture is the most important dam￾age process. Available statistics show that, including corrosive assistance, fa￾tigue is a leading cause of material failures registered during a long-term per￾formance of engineering components and structures [246]. From the microme￾chanical point of view, the fatigue process can be understood as a sequence of the following stages: nucleation of cracks, stable propagation of short (small) cracks, stable propagation of long cracks and unstable fracture [247]. This chapter is divided into four sections. In Section 3.1, morphological patterns reflecting all crack growth stages on the fracture surface are briefly described. Moreover, the topological methods widely utilized in fatigue re￾search and the quantitative fractography are outlined. These topics are im￾portant for all subsequent sections of the chapter. The second section is devoted to the propagation of fatigue cracks under the remote opening mode (mode I). Mechanisms of nucleation and growth of short cracks are briefly reported, although these initial stages of the fatigue process were not a special subject of our research. Inclusion of these topics was, however, inevitable in order to provide a self-contained description of fatigue micromechanisms. The growth of short cracks is governed by shear stress components in favourably oriented crystallographic slip systems that are inclined at about 45◦ with respect to the maximal principal stress. This means that the short cracks grow in a local mixed-mode I+II and the pic￾ture of dislocation emission from the crack tip as well as the related growth micromechanisms are completely different from those related to long cracks. After a certain incipient period that incorporates crystallographic and transient growths of short cracks, the fatigue cracks incline towards a di￾rection perpendicular to the maximal principal stress, i.e., nearly towards mode I loading of the crack tip. This means that the long cracks keep prop￾agating so that the crack tip plasticity is produced in the opening loading mode. Thus, the second section provides the reader with a micromechanical interpretation of all important phenomena accompanying this, most frequent, type of fatigue crack growth. Knowledge of these micromechanisms is essen- 125

126 3 Fatigue Fracture tial not only for materials scientists and mechanical engineers,but also for technologists attempting to design structural materials exhibiting better re- sistance to fatigue crack propagation. When the shear components of the applied stress are dominant,both short and long cracks can grow macroscopically under shear loading modes II or III.As usual,however,pure shear-mode crack propagation persists only for a limited number of loading cycles and the cracks incline or branch to get loaded in mode I.This leads to a local mixed-mode I+II,I+III or I+II+III crack propagation.The factory roof formation under torsion loading is an instructive example of such behaviour.Therefore,Section 3.3 refers to both shear-mode and mixed-mode crack growth.The first subsections introduce theoretical models and experimental results concerning crack growth under pure-shear and torsional loading.Since a combined cyclic bending-torsion is applied to many structural components,a rather extended part of the third section reports on the results of fracture tests performed under this kind of loading. The final Section,3.4,is devoted to the application of quantitative frac- tography to failure analysis.The fracture morphology can purvey a direct link between damage micromechanisms and both initiation and propagation of cracks.This section directly outlines how the knowledge of fracture mi- cromechanisms can help to identify the reasons for failures of structural com- ponents in service.Therefore,its content can be useful for machine designers, especially for those working in the transport industry. In general,Chapter 3 attempts to convince the reader of how useful the unified nano-micro-meso-macroscopic approach can be when trying to de- scribe and interpret the behaviour of fatigue cracks. 3.1 Quantitative Fractography Micromechanisms of fatigue crack propagation can be advantageously stud- ied by means of fractographic tools.Indeed,the fracture surface morphology reflects many important stages of both stable and unstable fatigue crack propagation.Moreover,quantitative fractography is a powerful tool for fail- ure analysis.Consequently,all sections of this fatigue chapter more or less refer to morphological patterns and fractographical results.Therefore,the quantitative fractography section was placed at the very beginning of this chapter. The term "fractography"was first used by Carl A.Zapffe for the proce- dure of descriptive analysis of fracture surfaces in 1945 248.The output of this analysis is a set of numerical characteristics(number,shape,size,orien- tation,distribution)related to morphological patterns or parameters(rough- ness,fractality,texture)of the global surface topography.In both cases,the accuracy of these data is determined by knowledge of space coordinates of

126 3 Fatigue Fracture tial not only for materials scientists and mechanical engineers, but also for technologists attempting to design structural materials exhibiting better re￾sistance to fatigue crack propagation. When the shear components of the applied stress are dominant, both short and long cracks can grow macroscopically under shear loading modes II or III. As usual, however, pure shear-mode crack propagation persists only for a limited number of loading cycles and the cracks incline or branch to get loaded in mode I. This leads to a local mixed-mode I+II, I+III or I+II+III crack propagation. The factory roof formation under torsion loading is an instructive example of such behaviour. Therefore, Section 3.3 refers to both shear-mode and mixed-mode crack growth. The first subsections introduce theoretical models and experimental results concerning crack growth under pure-shear and torsional loading. Since a combined cyclic bending-torsion is applied to many structural components, a rather extended part of the third section reports on the results of fracture tests performed under this kind of loading. The final Section, 3.4, is devoted to the application of quantitative frac￾tography to failure analysis. The fracture morphology can purvey a direct link between damage micromechanisms and both initiation and propagation of cracks. This section directly outlines how the knowledge of fracture mi￾cromechanisms can help to identify the reasons for failures of structural com￾ponents in service. Therefore, its content can be useful for machine designers, especially for those working in the transport industry. In general, Chapter 3 attempts to convince the reader of how useful the unified nano- micro- meso- macroscopic approach can be when trying to de￾scribe and interpret the behaviour of fatigue cracks. 3.1 Quantitative Fractography Micromechanisms of fatigue crack propagation can be advantageously stud￾ied by means of fractographic tools. Indeed, the fracture surface morphology reflects many important stages of both stable and unstable fatigue crack propagation. Moreover, quantitative fractography is a powerful tool for fail￾ure analysis. Consequently, all sections of this fatigue chapter more or less refer to morphological patterns and fractographical results. Therefore, the quantitative fractography section was placed at the very beginning of this chapter. The term “fractography” was first used by Carl A. Zapffe for the proce￾dure of descriptive analysis of fracture surfaces in 1945 [248]. The output of this analysis is a set of numerical characteristics (number, shape, size, orien￾tation, distribution) related to morphological patterns or parameters (rough￾ness, fractality, texture) of the global surface topography. In both cases, the accuracy of these data is determined by knowledge of space coordinates of

3.1 Quantitative Fractography 127 points on the fracture surface investigated.A very intensive development of quantitative fractography is obviously directly associated with increasing accuracy of measuring methods as well as with a rapidly growing capac- ity and computing rate of computers.During the second half of the last century,two-dimensional fractography in the scanning electron microscope was widely developed.In the last 20 years,however,an extended utilization of computer-aided topography techniques has enabled enormous progress in three-dimensional methods.Similarly,a number of descriptive concepts,dis- tinguished by both the extent and the quality of utilized parameters,have been developed in the area of quantitative fractography. Nevertheless,two various approaches can be distinguished here.The first utilizes various parameters of roughness,fractality or texture of fracture sur- faces in order to find relationships between the fracture topology on one side,and the loading mode or the crack growth rate on the other side.In the first subsection,therefore,definitions of basic topological parameters are outlined.A rather different problem,usually demanding extensive research experience,constitutes the correct identification of morphological patterns as fracture facets,ridges,beach marks,tire tracks or fatigue striations.Thus, the second approach deals with the quantification of morphological patterns and their relationships to the loading parameters. 3.1.1 Topological Analysis A general description of fracture surfaces by means of topographical param- eters should be able to provide topological characteristics as well as morpho- logical patterns as special cases.This might,in principle,be achieved by an analysis of a global set and relevant subsets of topological data.However, the basic problem here is represented by a high variability of fracture surface topology measured at different resolutions as well as an extreme complexity of the related microreliefs.In spite of a rather long history,no general defini- tion of the surface topology and,consequently,no universal methodology of its quantification has been commonly accepted up to now 249-254]. In order to acquire a sufficiently wide and relevant set of topological param- eters,advanced three-dimensional topological methods are to be employed.A great majority of results presented in this book was obtained by application of two methods that are based on different physical principles.Stereophotogram- metry is a method that makes use of the stereoscopic principles in order to obtain topological data of the fracture surface under investigation.Inputs to the method are two images of the analyzed region taken from different angles of view(so-called stereoimages or the stereopair)and some additional parameters that characterize a projection used during their acquisition.Usu- ally,a scanning electron microscope(SEM)equipped with a eucentric holder is employed and the stereopair is obtained by tilting the specimen in the

3.1 Quantitative Fractography 127 points on the fracture surface investigated. A very intensive development of quantitative fractography is obviously directly associated with increasing accuracy of measuring methods as well as with a rapidly growing capac￾ity and computing rate of computers. During the second half of the last century, two-dimensional fractography in the scanning electron microscope was widely developed. In the last 20 years, however, an extended utilization of computer-aided topography techniques has enabled enormous progress in three-dimensional methods. Similarly, a number of descriptive concepts, dis￾tinguished by both the extent and the quality of utilized parameters, have been developed in the area of quantitative fractography. Nevertheless, two various approaches can be distinguished here. The first utilizes various parameters of roughness, fractality or texture of fracture sur￾faces in order to find relationships between the fracture topology on one side, and the loading mode or the crack growth rate on the other side. In the first subsection, therefore, definitions of basic topological parameters are outlined. A rather different problem, usually demanding extensive research experience, constitutes the correct identification of morphological patterns as fracture facets, ridges, beach marks, tire tracks or fatigue striations. Thus, the second approach deals with the quantification of morphological patterns and their relationships to the loading parameters. 3.1.1 Topological Analysis A general description of fracture surfaces by means of topographical param￾eters should be able to provide topological characteristics as well as morpho￾logical patterns as special cases. This might, in principle, be achieved by an analysis of a global set and relevant subsets of topological data. However, the basic problem here is represented by a high variability of fracture surface topology measured at different resolutions as well as an extreme complexity of the related microreliefs. In spite of a rather long history, no general defini￾tion of the surface topology and, consequently, no universal methodology of its quantification has been commonly accepted up to now [249–254]. In order to acquire a sufficiently wide and relevant set of topological param￾eters, advanced three-dimensional topological methods are to be employed. A great majority of results presented in this book was obtained by application of two methods that are based on different physical principles. Stereophotogram￾metry is a method that makes use of the stereoscopic principles in order to obtain topological data of the fracture surface under investigation. Inputs to the method are two images of the analyzed region taken from different angles of view (so-called stereoimages or the stereopair) and some additional parameters that characterize a projection used during their acquisition. Usu￾ally, a scanning electron microscope (SEM) equipped with a eucentric holder is employed and the stereopair is obtained by tilting the specimen in the

128 3 Fatigue Fracture SEM chamber by an angle that depends on the local roughness of the sur- face.The stereopair is processed via a matching algorithm in order to find corresponding points on both images (homologous points)and the relative z-coordinates of these points are calculated.The 3D model of the depicted surface area usually consists of ten to twenty thousand non-equidistant points and so the Delaunay triangulation must be performed [255. Optical chromatography represents another method useful for a 3D recon- stitution of the fracture surface micromorphology.The profilometer Micro- Prof FRT,Fries Research Technology GmbH,makes use of the chromatic aberration of the optical lens.Different light monochromatic components are focused at different heights from a reference plane at the output of the opti- cal fibre.The light intensity exhibits a maximum at the wavelength exactly focused on the surface and the height of the surface irregularities is deduced by using a calibration table.This optical method was usually employed only for verification of selected results obtained by stereophotogrammetry. According to their mathematical basis,recently used topological param- eters can be divided into five main categories.This classification is based on published works 250-253]and,in particular,on the work of Petropou- los et al.[256].First two categories represent vertical (altitudinal)and length roughness parameters which characterize vertical and horizontal distributions of surface points,respectively.The third group involves hybrid parameters si- multaneously describing more than one of the above-mentioned aspects.The fourth and fifth groups respectively consist of spectral and fractal character- istics of the fracture surface. 3.1.1.1 Roughness Parameters In this brief overview only vertical,length,hybrid,spectral and fractal pa- rameters are mentioned in more detail.The description of other parameters can be found elsewhere [214,249-251.For the sake of simplicity,the assump- tion of non-overlapping surface elements is accepted hereafter.This means that each pair of coordinates(ri,yi)that determine the location of a point on the reference plane perpendicular to the macroscopic fracture surface is uniquely related to one altitudinal coordinate zi.Because of a discrete data set of points utilized here to quantify surface topography,only a discrete form of definition of individual parameters is presented.Integral definition of many topological characteristics can be found,e.g.,in [251,252] Vertical Parameters Most important altitudinal parameters are characteristics associated with the probability of realization of the value zi in terms of central moments,defined as

128 3 Fatigue Fracture SEM chamber by an angle that depends on the local roughness of the sur￾face. The stereopair is processed via a matching algorithm in order to find corresponding points on both images (homologous points) and the relative z-coordinates of these points are calculated. The 3D model of the depicted surface area usually consists of ten to twenty thousand non-equidistant points and so the Delaunay triangulation must be performed [255]. Optical chromatography represents another method useful for a 3D recon￾stitution of the fracture surface micromorphology. The profilometer Micro￾Prof FRT, Fries Research & Technology GmbH, makes use of the chromatic aberration of the optical lens. Different light monochromatic components are focused at different heights from a reference plane at the output of the opti￾cal fibre. The light intensity exhibits a maximum at the wavelength exactly focused on the surface and the height of the surface irregularities is deduced by using a calibration table. This optical method was usually employed only for verification of selected results obtained by stereophotogrammetry. According to their mathematical basis, recently used topological param￾eters can be divided into five main categories. This classification is based on published works [250–253] and, in particular, on the work of Petropou￾los et al. [256]. First two categories represent vertical (altitudinal) and length roughness parameters which characterize vertical and horizontal distributions of surface points, respectively. The third group involves hybrid parameters si￾multaneously describing more than one of the above-mentioned aspects. The fourth and fifth groups respectively consist of spectral and fractal character￾istics of the fracture surface. 3.1.1.1 Roughness Parameters In this brief overview only vertical, length, hybrid, spectral and fractal pa￾rameters are mentioned in more detail. The description of other parameters can be found elsewhere [214,249–251]. For the sake of simplicity, the assump￾tion of non-overlapping surface elements is accepted hereafter. This means that each pair of coordinates (xi, yi) that determine the location of a point on the reference plane perpendicular to the macroscopic fracture surface is uniquely related to one altitudinal coordinate zi. Because of a discrete data set of points utilized here to quantify surface topography, only a discrete form of definition of individual parameters is presented. Integral definition of many topological characteristics can be found, e.g., in [251, 252]. Vertical Parameters Most important altitudinal parameters are characteristics associated with the probability of realization of the value zi in terms of central moments, defined as

3.1 Quantitative Fractography 129 n-1 Hk= (a-(), 三0 where k is the order of the central moment,n is the range of the analyzed set of data and(2)is the arithmetic average height: n-l 0 With respect to properties of (2),the first central moment is zero.The second moment(variance)yields information on the width of the distribution. This parameter is connected with the standard deviation Ro by the relation Rg=VH2. The characteristic of the third central moment,the skewness,describes a symmetry of the function p(z).It is defined as 2 A normalized form of the fourth central moment is the kurtosis: =-3=觉-8 This parameter becomes zero for the Gaussian distribution and a negative value indicates a more flat distribution.Another parameter often used in the literature is the arithmetic roughness (the centre line average): a-(z This parameter,however,is usually nearly proportional to the standard de- viation Ra [250]. The second group of vertical parameters involves characteristics based on extreme values of the set Z 257.The most widely used parameters are the highest height Rp(Rp=2max-(2))and the maximum depth Ro (R=(z)-2min)which are the maximal and minimal values of the rel- ative coordinates associated with a selected altitudinal level (2),respec- tively.Another parameter is the vertical range,R:,representing their sum R:=2max -2min (Figure 3.1)

3.1 Quantitative Fractography 129 μk = 1 n n −1 i=0 (zi − z) k, where k is the order of the central moment, n is the range of the analyzed set of data and z is the arithmetic average height: z = 1 n n −1 i=0 zi. With respect to properties of z, the first central moment is zero. The second moment (variance) yields information on the width of the distribution. This parameter is connected with the standard deviation Rq by the relation Rq = √μ2. The characteristic of the third central moment, the skewness, describes a symmetry of the function p(z). It is defined as γ1 = μ3 μ 3/2 2 = μ3 R3 q . A normalized form of the fourth central moment is the kurtosis: γ2 = μ4 μ2 2 − 3 = μ4 R4 q − 3. This parameter becomes zero for the Gaussian distribution and a negative value indicates a more flat distribution. Another parameter often used in the literature is the arithmetic roughness (the centre line average): Ra = 1 n n −1 i=0 |zi − z|. This parameter, however, is usually nearly proportional to the standard de￾viation Rq [250]. The second group of vertical parameters involves characteristics based on extreme values of the set Z [257]. The most widely used parameters are the highest height Rp (Rp = zmax − z) and the maximum depth Rv (Rv = z − zmin) which are the maximal and minimal values of the rel￾ative coordinates associated with a selected altitudinal level z, respec￾tively. Another parameter is the vertical range, Rz, representing their sum Rz = zmax − zmin (Figure 3.1)

130 3 Fatigue Fracture z=fx),=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}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