3.1 Quantitative Fractography 135 movable windows (size e)and,for each window,the local value of Ro is determined.Then the dependence of the average (global)characteristic i=0 on s is established.Finally,the Hurst exponent is calculated according to the relation W(e)as a slope of the exponential function plotted in bi-logarithmic coordinates(similarly to the fractal dimension). 3.1.2 Morphological Patterns in Fatigue The morphological features in fatigue are different when looking at the frac- ture surface under low and high magnifications.Therefore,they can simply be separated into macroscopic and microscopic patterns [149,187,269,270]. Macroscopic Patterns The fractographic investigation of fatigue fracture surfaces usually starts vi- sually or by low-magnification optical microscopy.Because of a change from a stage I slip plane that is inclined to the fracture plane macroscopically per- pendicular to the principal loading axis,one can often distinguish a boundary between stage I and stage II crack propagations.It should be noted that the stage I fracture never extends beyond about three grains around the crack initiation site at the surface.The transition from stage I to stage II is usu- ally accompanied by ridges parallel to the crack propagation direction.These ridges are formed by local shears that merge many plateaus corresponding to different fracture plains in individual grains. One of the most important features usually found on fatigue fracture sur- faces is beach marks (arrest marks),which are centred around the point of fatigue crack origin.These patterns correspond to stage II of crack propaga- tion,and occur as a result of changes in loading or frequency or by oxidation of fracture surfaces during periods of crack arrest from intermittent service of the component. The final-fracture zone can usually be identified by a fibrous morphology, which is different to that of the stage II crack growth.The size of this zone depends on the magnitude of loading,and its shape depends on the mode of external loading.Therefore,the transition boundary (line)between the stage II and final fracture zones can be related to the value of critical stress intensity factor Ke,associated with the fast (unstable)fracture.In ductile materials,shear lips(plane-stress)at approximately 45 to the fracture sur- face appear at the end of the final-fracture zone related to the free surface of
3.1 Quantitative Fractography 135 movable windows (size ε) and, for each window, the local value of Rq is determined. Then the dependence of the average (global) characteristic W = 1 k k −1 i=0 Rqi on ε is established. Finally, the Hurst exponent is calculated according to the relation W(ε) ∝ εH as a slope of the exponential function plotted in bi-logarithmic coordinates (similarly to the fractal dimension). 3.1.2 Morphological Patterns in Fatigue The morphological features in fatigue are different when looking at the fracture surface under low and high magnifications. Therefore, they can simply be separated into macroscopic and microscopic patterns [149, 187, 269, 270]. Macroscopic Patterns The fractographic investigation of fatigue fracture surfaces usually starts visually or by low-magnification optical microscopy. Because of a change from a stage I slip plane that is inclined to the fracture plane macroscopically perpendicular to the principal loading axis, one can often distinguish a boundary between stage I and stage II crack propagations. It should be noted that the stage I fracture never extends beyond about three grains around the crack initiation site at the surface. The transition from stage I to stage II is usually accompanied by ridges parallel to the crack propagation direction. These ridges are formed by local shears that merge many plateaus corresponding to different fracture plains in individual grains. One of the most important features usually found on fatigue fracture surfaces is beach marks (arrest marks), which are centred around the point of fatigue crack origin. These patterns correspond to stage II of crack propagation, and occur as a result of changes in loading or frequency or by oxidation of fracture surfaces during periods of crack arrest from intermittent service of the component. The final-fracture zone can usually be identified by a fibrous morphology, which is different to that of the stage II crack growth. The size of this zone depends on the magnitude of loading, and its shape depends on the mode of external loading. Therefore, the transition boundary (line) between the stage II and final fracture zones can be related to the value of critical stress intensity factor Kc, associated with the fast (unstable) fracture. In ductile materials, shear lips (plane-stress) at approximately 45◦ to the fracture surface appear at the end of the final-fracture zone related to the free surface of
136 3 Fatigue Fracture the component.These lips enable us to differentiate clearly the final fracture zone from that of the initial crack growth stage. Microscopic Patterns During stage II of fatigue crack propagation,striations are formed in the way shown in Section 3.2.3.According to the model relevant for the Paris-Erdogan region,the striation spacing should be equal to the local crack growth rate, which is particularly true for ductile and homogeneous materials.Fatigue stri- ations often bow out in the direction of crack propagation and generally tend to align perpendicular to the principal(macroscopic)crack growth direction. However,the crack locally propagates along multiple plateaus(facets)so that the local crack propagation directions in individual facets are usually differ- ent.The SEM picture of a striation field which documents this phenomenon is presented in Figure 3.4.Moreover,the plateaus can lie at different eleva- tions with respect to each other and join by tear ridges or walls that also contain striations. Acc.V Spot Magn Det WD Exp 20m 20.0kV4.2850x SE 12.8 48696 probe 2 Figure 3.4 The striation field in the austenitic steel Therefore,when comparing the macroscopic crack growth rate with the striation spacing for a given crack length,one has to consider an average of local crack growth rates indicated within the related striation field.This can be done by assuming the average of projections of local crack growth rates,perpendicular to the striations on individual facets,to the macroscopic propagation direction by using the formula
136 3 Fatigue Fracture the component. These lips enable us to differentiate clearly the final fracture zone from that of the initial crack growth stage. Microscopic Patterns During stage II of fatigue crack propagation, striations are formed in the way shown in Section 3.2.3. According to the model relevant for the Paris–Erdogan region, the striation spacing should be equal to the local crack growth rate, which is particularly true for ductile and homogeneous materials. Fatigue striations often bow out in the direction of crack propagation and generally tend to align perpendicular to the principal (macroscopic) crack growth direction. However, the crack locally propagates along multiple plateaus (facets) so that the local crack propagation directions in individual facets are usually different. The SEM picture of a striation field which documents this phenomenon is presented in Figure 3.4. Moreover, the plateaus can lie at different elevations with respect to each other and join by tear ridges or walls that also contain striations. Figure 3.4 The striation field in the austenitic steel Therefore, when comparing the macroscopic crack growth rate with the striation spacing for a given crack length, one has to consider an average of local crack growth rates indicated within the related striation field. This can be done by assuming the average of projections of local crack growth rates, perpendicular to the striations on individual facets, to the macroscopic propagation direction by using the formula
3.1 Quantitative Fractography 137 si cos Qi, (3.4) i=1 where n is the number of facets,si is the striation spacing on the i-th facet and oi are the related projection angles (see Figure 3.5) C. Figure 3.5 The scheme of local inclination angles of striation fields.The macroscopic crack growth direction is marked by the arrow On striation facets which are at an angle to the average plane of crack growth,so-called fissures can be formed 271,272].They appear as short (secondary)cracks related only to one surface of the local crack,are regularly spaced,and penetrate a distance below the fracture surface which is much larger than the variations in fracture surface topography.The fissures form at striations most likely due to their stress concentration effect.Because the formation of fissures causes local crack branching,the further striations created on the inclined facet start to be immediately shielded especially from the local tensile stress parallel to the fracture surface.As the crack front moves away from the fissure,the local tensile stress builds up to the point where a new fissure is formed,and the process is repeated.Therefore,the fissure spacing does not directly correspond to the local crack growth rate, similarly to the spacing of false striations created in the near-threshold region (Section 3.2). There are also other periodic patterns,sometimes observed on fatigue frac- ture surfaces generated by a shear loading,known as fibrous patterns and tire tracks.Both these markings are produced by a contact wear of crack-flank asperities.The fibrous patterns are,unlike striations,parallel to the crack growth direction and can be clearly seen in Figure 3.6.The tire tracks,re- sembling the tracks left by the tread pattern of a tire,are the result of particles or protrusions on the fracture surface being successively impressed into the mating surface part during the closing portion of the loading cycle.They are nicely depicted especially in Figure 3.52 (Section 3.3).An appearance of both fibrous patterns and tire tracks always indicates a presence of either remote or local shear mode II at the crack front.The local shear mode can also be induced in the case of a pure remote mode I by inclinations of crack front
3.1 Quantitative Fractography 137 s¯ = 1 n n i=1 si cos αi, (3.4) where n is the number of facets, si is the striation spacing on the i-th facet and αi are the related projection angles (see Figure 3.5) 1 2 3 4 5 Figure 3.5 The scheme of local inclination angles of striation fields. The macroscopic crack growth direction is marked by the arrow On striation facets which are at an angle to the average plane of crack growth, so-called fissures can be formed [271, 272]. They appear as short (secondary) cracks related only to one surface of the local crack, are regularly spaced, and penetrate a distance below the fracture surface which is much larger than the variations in fracture surface topography. The fissures form at striations most likely due to their stress concentration effect. Because the formation of fissures causes local crack branching, the further striations created on the inclined facet start to be immediately shielded especially from the local tensile stress parallel to the fracture surface. As the crack front moves away from the fissure, the local tensile stress builds up to the point where a new fissure is formed, and the process is repeated. Therefore, the fissure spacing does not directly correspond to the local crack growth rate, similarly to the spacing of false striations created in the near-threshold region (Section 3.2). There are also other periodic patterns, sometimes observed on fatigue fracture surfaces generated by a shear loading, known as fibrous patterns and tire tracks. Both these markings are produced by a contact wear of crack-flank asperities. The fibrous patterns are, unlike striations, parallel to the crack growth direction and can be clearly seen in Figure 3.6. The tire tracks, resembling the tracks left by the tread pattern of a tire, are the result of particles or protrusions on the fracture surface being successively impressed into the mating surface part during the closing portion of the loading cycle. They are nicely depicted especially in Figure 3.52 (Section 3.3). An appearance of both fibrous patterns and tire tracks always indicates a presence of either remote or local shear mode II at the crack front. The local shear mode can also be induced in the case of a pure remote mode I by inclinations of crack front
138 3 Fatigue Fracture elements from the main crack plane perpendicular to the remote loading di- rection (see also Section 3.3).The direction of the tire tracks and the change in the spacing of indentations can also indicate both the magnitude and the type of displacements that occurred during the fracture process,such as lat- eral movement from shear or torsional loading.There is,again,no simple correspondence between the spacing of tire tracks (or fibrous patterns)and the crack growth rate. 100μm Figure 3.6 Fibrous patterns and tire tracks formed under the shear(mode II)crack propagation in austenitic steel.The crack growth direction is from the bottom to the top Sometimes,periodic microstructural phases (pearlite,martensite laths, etc.)or slip traces can also be observed on fracture surfaces.Such patterns, obviously,have nothing to do with the crack growth rate.Thus,the fields of fatigue striations are the only relevant patterns that can be directly corre- lated with the rate of crack front propagation.A clear distinction between the striations and other periodical features is not trivial.Therefore,when performing a reconstitution of the fatigue process based on the fracture sur- face micromorphology,one should possess a sufficient level of fractographic experience
138 3 Fatigue Fracture elements from the main crack plane perpendicular to the remote loading direction (see also Section 3.3). The direction of the tire tracks and the change in the spacing of indentations can also indicate both the magnitude and the type of displacements that occurred during the fracture process, such as lateral movement from shear or torsional loading. There is, again, no simple correspondence between the spacing of tire tracks (or fibrous patterns) and the crack growth rate. Figure 3.6 Fibrous patterns and tire tracks formed under the shear (mode II) crack propagation in austenitic steel. The crack growth direction is from the bottom to the top Sometimes, periodic microstructural phases (pearlite, martensite laths, etc.) or slip traces can also be observed on fracture surfaces. Such patterns, obviously, have nothing to do with the crack growth rate. Thus, the fields of fatigue striations are the only relevant patterns that can be directly correlated with the rate of crack front propagation. A clear distinction between the striations and other periodical features is not trivial. Therefore, when performing a reconstitution of the fatigue process based on the fracture surface micromorphology, one should possess a sufficient level of fractographic experience
3.2 Opening Loading Mode 139 3.2 Opening Loading Mode There is common agreement within the international scientific community concerning a principal physical difference between the driving force of the fatigue(stable)crack growth and that of the brittle(unstable)fracture.While the latter is directly associated with a drop in the elastic energy (elastic strain)and the critical value of the stress intensity factor Ke,the driving force in fatigue is directly related to the range of the cyclic plastic strain at the crack tip.Because the maximum K-value during the stable crack growth lies below Ke,the crack growth can proceed only when supported by the work of external cyclic forces.Similarly,the stable growth in the case of stress corrosion cracking presumes the assistance of the chemical driving force.In 1963,Paris and Erdogan [273 proved that the diagram da/dN vs AK for so-called long cracks in the small-scale yielding range (the high-cycle fatigue) retains the advantage of LEFM,namely a satisfactory invariance in the shape and size of cracked solids.It might seem to be surprising that the linear elastic parameter also allows us to describe successfully the rate of plastic processes at the crack tip.Several years later,however,Rice 274 brought to light a theoretical reason justifying the present opinion:the small-scale cyclic plasticity (the cyclic plastic zone)at the crack tip is,indeed,controlled by the value of△K. The local value of Ak at the crack tip is determined by both external and internal stresses resulting from external forces and local plastic defor- mations(or generally from microstructural defects),respectively.The tensile and compressive elastic energies associated with internal stresses are mutu- ally compensated all over the bulk and,therefore,they do not contribute to a global tensile elastic energy of stressed solids.Consequently,they cannot be released to support the unstable fracture process.At the same time,limited amounts of elastic energy that can be released by relaxation of local tensile internal stresses in small volumes adjacent to the crack tip can significantly influence neither the onset of brittle fracture nor its crack growth rate.On the other hand,the level of internal stresses can substantially influence the stable fatigue growth rate in each step of the crack advance,since the emission of dislocations from the crack tip occurs at very low stress intensity factors.This means that even very small changes of local K-values at the crack tip can considerably modify the stable crack growth rate.Thus,unlike in the case of unstable fracture,the internal stresses created by dislocation configurations and secondary phases are to be considered as an important additional factor affecting the fatigue crack propagation rate.It is particularly this difference that elucidates a much higher complexity of shielding (or anti-shielding)ef- fects accompanying the fatigue crack growth when compared to brittle frac- ture.Unlike in brittle fracture,for example,the contact shielding in fatigue also occurs under the opening loading mode,which causes so-called crack closure phenomena.Moreover,many important phenomena associated espe- cially with the small scale yielding,e.g.,the existence of fatigue thresholds or
3.2 Opening Loading Mode 139 3.2 Opening Loading Mode There is common agreement within the international scientific community concerning a principal physical difference between the driving force of the fatigue (stable) crack growth and that of the brittle (unstable) fracture. While the latter is directly associated with a drop in the elastic energy (elastic strain) and the critical value of the stress intensity factor Kc, the driving force in fatigue is directly related to the range of the cyclic plastic strain at the crack tip. Because the maximum K-value during the stable crack growth lies below Kc, the crack growth can proceed only when supported by the work of external cyclic forces. Similarly, the stable growth in the case of stress corrosion cracking presumes the assistance of the chemical driving force. In 1963, Paris and Erdogan [273] proved that the diagram da/dN vs ΔK for so-called long cracks in the small-scale yielding range (the high-cycle fatigue) retains the advantage of LEFM, namely a satisfactory invariance in the shape and size of cracked solids. It might seem to be surprising that the linear elastic parameter also allows us to describe successfully the rate of plastic processes at the crack tip. Several years later, however, Rice [274] brought to light a theoretical reason justifying the present opinion: the small-scale cyclic plasticity (the cyclic plastic zone) at the crack tip is, indeed, controlled by the value of ΔK. The local value of ΔK at the crack tip is determined by both external and internal stresses resulting from external forces and local plastic deformations (or generally from microstructural defects), respectively. The tensile and compressive elastic energies associated with internal stresses are mutually compensated all over the bulk and, therefore, they do not contribute to a global tensile elastic energy of stressed solids. Consequently, they cannot be released to support the unstable fracture process. At the same time, limited amounts of elastic energy that can be released by relaxation of local tensile internal stresses in small volumes adjacent to the crack tip can significantly influence neither the onset of brittle fracture nor its crack growth rate. On the other hand, the level of internal stresses can substantially influence the stable fatigue growth rate in each step of the crack advance, since the emission of dislocations from the crack tip occurs at very low stress intensity factors. This means that even very small changes of local K-values at the crack tip can considerably modify the stable crack growth rate. Thus, unlike in the case of unstable fracture, the internal stresses created by dislocation configurations and secondary phases are to be considered as an important additional factor affecting the fatigue crack propagation rate. It is particularly this difference that elucidates a much higher complexity of shielding (or anti-shielding) effects accompanying the fatigue crack growth when compared to brittle fracture. Unlike in brittle fracture, for example, the contact shielding in fatigue also occurs under the opening loading mode, which causes so-called crack closure phenomena. Moreover, many important phenomena associated especially with the small scale yielding, e.g., the existence of fatigue thresholds or