150 3 Fatigue Fracture cycle,the reversed dislocation slip was suppressed approximately in one of each ten grains.This seems to be a plausible result as well. Let us finally note that similar irreversible micromechanisms also oper- ate in the cyclic plastic zone ahead of an advancing fatigue crack front (see the next subsection),and contribute to the roughness-induced shielding (see Section 3.2.4). 3.2.2 Nucleation and Growth of Short Cracks 3.2.2.1 Mechanisms of Crack Nucleation Fatigue cracks in metallic materials are nucleated by local interactions of dislocation slip bands or pile-ups with microstructural heterogeneities and defects as grain boundaries,phase boundaries,large secondary phase parti- cles or free surfaces [149,192,297.These interactions are induced by elas- tic mismatch strains evolving at boundaries of grains and microstructural phases with different stress-strain characteristics during the external loading. Although the related high elastic incompatibility stresses can be relaxed by plastic deformation (dislocation movements)in a nearly entire volume of a deformed solid,relatively high long-range stresses of dislocation arrangements in slip bands and pile-ups remain locally at microstructural boundaries.Since the cohesive strength(fracture energy)of incoherent boundaries can be very low,these stresses often lead to microcrack initiation at the weakest sites. Discrete dislocation models 298-300]revealed that the peak stresses could be higher than the cohesion stress,which depends on the surface and grain boundary energies.A high hydrostatic component of the long-range stresses intensifies a diffusion of interstitials into the stressed regions,thereby reduc- ing the cohesive strength of the boundary.Following the molecular dynamics computations of Van der Ven and Ceder [301],a fraction of 40%of oxygen (hydrogen)atoms in a {111}aluminium plane induces a relative reduction in the cohesive strength by a factor of two (three).In order to propagate further the nucleated microcrack,however,a sufficient amount of irreversible plasticity must be produced at its front.This is conditioned by a permanent fux of oxygen to avoid a repeating recovering of newly created fracture sur- faces (see hereafter).Therefore,an intergranular propagation from the bulk interior along the grain or phase boundaries by repeating decohesion mecha- nism is usually observed only in strong corrosive environments.On the other hand,a transgranular crack propagation from the internal nucleus at an in- clusion was often observed in specimens with a reinforced surface layers or in an ultra-high cycle regime (see Section 3.3.4). It should be emphasized,however,that the fatigue cracks preferentially initiate at the surface 149,192.Indeed,the free surface is a boundary that separates media with an extremely high difference in properties (lack of in-
150 3 Fatigue Fracture cycle, the reversed dislocation slip was suppressed approximately in one of each ten grains. This seems to be a plausible result as well. Let us finally note that similar irreversible micromechanisms also operate in the cyclic plastic zone ahead of an advancing fatigue crack front (see the next subsection), and contribute to the roughness-induced shielding (see Section 3.2.4). 3.2.2 Nucleation and Growth of Short Cracks 3.2.2.1 Mechanisms of Crack Nucleation Fatigue cracks in metallic materials are nucleated by local interactions of dislocation slip bands or pile-ups with microstructural heterogeneities and defects as grain boundaries, phase boundaries, large secondary phase particles or free surfaces [149, 192, 297]. These interactions are induced by elastic mismatch strains evolving at boundaries of grains and microstructural phases with different stress-strain characteristics during the external loading. Although the related high elastic incompatibility stresses can be relaxed by plastic deformation (dislocation movements) in a nearly entire volume of a deformed solid, relatively high long-range stresses of dislocation arrangements in slip bands and pile-ups remain locally at microstructural boundaries. Since the cohesive strength (fracture energy) of incoherent boundaries can be very low, these stresses often lead to microcrack initiation at the weakest sites. Discrete dislocation models [298–300] revealed that the peak stresses could be higher than the cohesion stress, which depends on the surface and grain boundary energies. A high hydrostatic component of the long-range stresses intensifies a diffusion of interstitials into the stressed regions, thereby reducing the cohesive strength of the boundary. Following the molecular dynamics computations of Van der Ven and Ceder [301], a fraction of 40% of oxygen (hydrogen) atoms in a {111} aluminium plane induces a relative reduction in the cohesive strength by a factor of two (three). In order to propagate further the nucleated microcrack, however, a sufficient amount of irreversible plasticity must be produced at its front. This is conditioned by a permanent flux of oxygen to avoid a repeating recovering of newly created fracture surfaces (see hereafter). Therefore, an intergranular propagation from the bulk interior along the grain or phase boundaries by repeating decohesion mechanism is usually observed only in strong corrosive environments. On the other hand, a transgranular crack propagation from the internal nucleus at an inclusion was often observed in specimens with a reinforced surface layers or in an ultra-high cycle regime (see Section 3.3.4). It should be emphasized, however, that the fatigue cracks preferentially initiate at the surface [149, 192]. Indeed, the free surface is a boundary that separates media with an extremely high difference in properties (lack of in-
3.2 Opening Loading Mode 151 teratomic bonds in air).In this particular case,the movement of dislocations towards the free surface is caused by attractive mirror forces (see also Sec- tion 2.3).During the first loading cycles progressive changes in the dislocation structure start to proceed.The damage processes begins at the sites of cyclic strain localization,usually called persistent slip bands(PSBs)and results in the formation of sharp surface slip patterns called persistent slip markings (PSMs).PSMs consist of extrusions and intrusions which develop on the ini- tially flat surface at emerging PSBs(see Figure 3.14).While the amplitude of reversed plastic slip in the interior of PSBs or pile-ups is highly constrained by surrounding elastic matrix,this is not the case for the parts near the PSMs on the free surface where high local plastic deformations appear.In- deed,atomic force microscopy revealed that these strains can be two orders of magnitude higher than the applied macroscopic ones(e.g.,[302]). The slip localization in surface thin bands can be explained by several mechanisms.For example,cycling of precipitation hardened alloys induces formation of thin slip bands in zones with easily shareable (coherent)pre- cipitates [303].In ductile fcc and bcc polycrystals subjected to cyclic loading the PSBs are often formed within the surface grains.The PSBs consist of hard walls (high density of edge dislocation dipoles)and soft channels (low density of screw dislocations which glide and cross-slip)creating the well- known ladder-like structure (e.g.,[304,305).This rather regular dislocation arrangement lowers the internal energy and enables high local shear defor- mations [306,307.Although the ladder-like structure is usually not observed in metals and alloys possessing a low stacking-fault energy,the PSMs occur in these materials 302]. The amount of plastic slip inside the slip bands and PSBs can be modelled by considering an elongated bulk inclusion embedded in the matrix which mimics the whole polycrystal [308].By considering analytical solutions de- rived by Eshelby for bulk inclusions [309 one can show that the primary slip in the vicinity of the free surface is higher than that in the bulk,where the deformation is constrained by the grain boundary surrounded by a hard elastic material 310.The slip within the band can be defined as the ratio between the displacement along the most active Burgers vector and the slip band thickness,h.Then the shear deformation within the surface slip bands can be expressed as =1-哈(+ 0.5∑-Tcr8 G (3.13) where L is the length of the slip band (usually the grain size),r=1.9,v is the Poisson's ratio,G is the shear modulus,is the applied stress and Ters is the critical resolved shear stress in the channel [310.The factor 0.5 corresponds to a highest possible Schmid factor of the slip band inclined 45 from the loading axis.Equation 3.13 was recently verified by finite element calculations [311].Since in large grains the values of L are tens of microns and
3.2 Opening Loading Mode 151 teratomic bonds in air). In this particular case, the movement of dislocations towards the free surface is caused by attractive mirror forces (see also Section 2.3). During the first loading cycles progressive changes in the dislocation structure start to proceed. The damage processes begins at the sites of cyclic strain localization, usually called persistent slip bands (PSBs) and results in the formation of sharp surface slip patterns called persistent slip markings (PSMs). PSMs consist of extrusions and intrusions which develop on the initially flat surface at emerging PSBs (see Figure 3.14). While the amplitude of reversed plastic slip in the interior of PSBs or pile-ups is highly constrained by surrounding elastic matrix, this is not the case for the parts near the PSMs on the free surface where high local plastic deformations appear. Indeed, atomic force microscopy revealed that these strains can be two orders of magnitude higher than the applied macroscopic ones (e.g., [302]). The slip localization in surface thin bands can be explained by several mechanisms. For example, cycling of precipitation hardened alloys induces formation of thin slip bands in zones with easily shareable (coherent) precipitates [303]. In ductile fcc and bcc polycrystals subjected to cyclic loading the PSBs are often formed within the surface grains. The PSBs consist of hard walls (high density of edge dislocation dipoles) and soft channels (low density of screw dislocations which glide and cross-slip) creating the wellknown ladder-like structure (e.g., [304, 305]). This rather regular dislocation arrangement lowers the internal energy and enables high local shear deformations [306,307]. Although the ladder-like structure is usually not observed in metals and alloys possessing a low stacking-fault energy, the PSMs occur in these materials [302]. The amount of plastic slip inside the slip bands and PSBs can be modelled by considering an elongated bulk inclusion embedded in the matrix which mimics the whole polycrystal [308]. By considering analytical solutions derived by Eshelby for bulk inclusions [309] one can show that the primary slip in the vicinity of the free surface is higher than that in the bulk, where the deformation is constrained by the grain boundary surrounded by a hard elastic material [310]. The slip within the band can be defined as the ratio between the displacement along the most active Burgers vector and the slip band thickness, h. Then the shear deformation within the surface slip bands can be expressed as γp = r(1 − ν) L h 1 + h L 2 0.5Σ − τcrs G , (3.13) where L is the length of the slip band (usually the grain size), r = 1.9, ν is the Poisson’s ratio, G is the shear modulus, Σ is the applied stress and τcrs is the critical resolved shear stress in the channel [310]. The factor 0.5 corresponds to a highest possible Schmid factor of the slip band inclined 45◦ from the loading axis. Equation 3.13 was recently verified by finite element calculations [311]. Since in large grains the values of L are tens of microns and
152 3 Fatigue Fracture the values of h are units or tens of nanometers,the ratio hL0.005<1. With regard to Equation 3.13 this means that the surface plastic deformation in the slip band is proportional to L/h,i.e.,it increases with increasing aspect ratio of the band.Clearly,the Eshelby inclusion elongated in the direction of the applied stress experiences lower back stress.Thus,the ratio of the strain EL localized in slip bands at the surface and the applied (nearly elastic)strain s can reach values as high as sL/s200.Note that the localization ratio related to the inner end of the slip band (impinged by a grain boundary)is about ten times lower 311. extrusions [101] [121] 111 dipolar walls chanels intrusion b=a/2[10i] Figure 3.14 A scheme of extrusion and intrusion patterns at the intersection of the persistent slip band with a free surface The high local reversed plasticity in slip bands and channels of PSBs produces a surface microroughness in the form of extrusions and intrusions (e.g.,[312).The movement of screw dislocations with jogs generates a sur- plus of vacancies in the channels.The counterbalancing flux of atoms inside the channels causes extrusions at the free surface,the volume of which is much higher than that of intrusions,as shown in Figure 3.14.On the other hand,volumes near the surface and close to the outer boundary of the chan- nels become depleted by atoms(or enriched by vacancies).This usually leads to formation of thin intrusions next to the extrusions [313].Such a model of combined movements of dislocations and point defects well reflects geomet- rical proportions of extrusions and intrusions 314.However,there are also many other older models of the surface relief evolution (e.g.,315,316) High stress concentrations around extrusions and,particularly,at the tip of intrusions leads to a formation of short surface cracks which start to prop- agate along the slip bands or PSBs inside the bulk.This growth is controlled by irreversible emission and absorption of dislocations at the fronts of these cracks.Owing to the localization ratio and Equation 3.13,such a damage process is most probable in the largest surface grains along slip planes with
152 3 Fatigue Fracture the values of h are units or tens of nanometers, the ratio h/L ≈ 0.005 1. With regard to Equation 3.13 this means that the surface plastic deformation in the slip band is proportional to L/h, i.e., it increases with increasing aspect ratio of the band. Clearly, the Eshelby inclusion elongated in the direction of the applied stress experiences lower back stress. Thus, the ratio of the strain εL localized in slip bands at the surface and the applied (nearly elastic) strain ε can reach values as high as εL/ε ≈ 200. Note that the localization ratio related to the inner end of the slip band (impinged by a grain boundary) is about ten times lower [311]. b extrusions [111] [121] [101] dipolar walls chanels intrusion b= /2 [101] a x y z Figure 3.14 A scheme of extrusion and intrusion patterns at the intersection of the persistent slip band with a free surface The high local reversed plasticity in slip bands and channels of PSBs produces a surface microroughness in the form of extrusions and intrusions (e.g., [312]). The movement of screw dislocations with jogs generates a surplus of vacancies in the channels. The counterbalancing flux of atoms inside the channels causes extrusions at the free surface, the volume of which is much higher than that of intrusions, as shown in Figure 3.14. On the other hand, volumes near the surface and close to the outer boundary of the channels become depleted by atoms (or enriched by vacancies). This usually leads to formation of thin intrusions next to the extrusions [313]. Such a model of combined movements of dislocations and point defects well reflects geometrical proportions of extrusions and intrusions [314]. However, there are also many other older models of the surface relief evolution (e.g., [315, 316]). High stress concentrations around extrusions and, particularly, at the tip of intrusions leads to a formation of short surface cracks which start to propagate along the slip bands or PSBs inside the bulk. This growth is controlled by irreversible emission and absorption of dislocations at the fronts of these cracks. Owing to the localization ratio and Equation 3.13, such a damage process is most probable in the largest surface grains along slip planes with
3.2 Opening Loading Mode 153 the highest Schmid factors.One should also note that the average level of cyclic plasticity is raising when transferring from a high-cycle regime to a low-cycle one.Consequently,the concentration of nucleated surface cracks follows the same trend. 3.2.2.2 Propagation of Short Cracks There is no clear notion about the moment of transition from the vacancy- assisted intrusion growth into the dislocation based propagation of the related short crack.Nevertheless,there are some plausible models explaining possible mechanisms of initial growth stages (e.g.,[149,312,317]).Since the crack nuclei are strictly aligned with the slip planes of PSBs,their further advance proceeds along these planes.This means that the cracks propagate under the shear stress coupled with the tensile normal stress (the mixed-mode I+II) that can somewhat facilitate the emission of dislocations from the crack tip by reducing the ideal shear strength(see Section 1.1).During cyclic loading in air or other corrosive environments,a passive oxide layer always forms on metallic surfaces.If the oxide layer is fractured as a consequence of deformation in the underlying polycrystal,the passivation is lost and a re-oxidation occurs. The process of repeated fracture and re-oxidation is a central principle of slip-oxidation models that are widely used to elucidate the mechanism of propagation of both short and long cracks [149,317]. The simplest model assuming the propagation along a single slip plane is depicted in Figure 3.15.In the tensile half-cycle,the edge dislocations are emitted from(or absorbed at)the crack tip which generates a new fracture surface on one of the crack flanks.The size of the new surface is equal to the number of dislocations times the Burgers vector and the length of the crack front.When assuming an immediate surface oxidization,the dislocations re- turning during the unloading cycle cannot remove the new surface.Instead of that,they will form a new fracture surface on the other crack flank,the size of which is,again,equal to the number of returning dislocations times the Burgers vector and the crack front length.Under a constant loading ampli- tude the number of returning dislocations to the crack tip is nearly equal to the number of dislocations generated during loading.Hence,the crack exten- sion per cycle is equal to the number of dislocations generated at the crack tip times the Burgers vector.In other words it is equal to the cyclic crack tip opening displacement.In this way,the crack advances during each loading cycle. In general,the crack growth rate da/dN can be assumed to be nearly pro- portional to the frequency of the oxide-layer fractures which is,again,propor- tional to the plastic strain Yp in the slip band.Consequently,the crack growth rate can be assumed to be proportional to (L/h)m,where m E(0.3,0.8)is commonly accepted [311,317].This also means that the growth rate of short cracks is higher in the long PSBs embedded in large surface grains
3.2 Opening Loading Mode 153 the highest Schmid factors. One should also note that the average level of cyclic plasticity is raising when transferring from a high-cycle regime to a low-cycle one. Consequently, the concentration of nucleated surface cracks follows the same trend. 3.2.2.2 Propagation of Short Cracks There is no clear notion about the moment of transition from the vacancyassisted intrusion growth into the dislocation based propagation of the related short crack. Nevertheless, there are some plausible models explaining possible mechanisms of initial growth stages (e.g., [149, 312, 317]). Since the crack nuclei are strictly aligned with the slip planes of PSBs, their further advance proceeds along these planes. This means that the cracks propagate under the shear stress coupled with the tensile normal stress (the mixed-mode I+II) that can somewhat facilitate the emission of dislocations from the crack tip by reducing the ideal shear strength (see Section 1.1). During cyclic loading in air or other corrosive environments, a passive oxide layer always forms on metallic surfaces. If the oxide layer is fractured as a consequence of deformation in the underlying polycrystal, the passivation is lost and a re-oxidation occurs. The process of repeated fracture and re-oxidation is a central principle of slip-oxidation models that are widely used to elucidate the mechanism of propagation of both short and long cracks [149, 317]. The simplest model assuming the propagation along a single slip plane is depicted in Figure 3.15. In the tensile half-cycle, the edge dislocations are emitted from (or absorbed at) the crack tip which generates a new fracture surface on one of the crack flanks. The size of the new surface is equal to the number of dislocations times the Burgers vector and the length of the crack front. When assuming an immediate surface oxidization, the dislocations returning during the unloading cycle cannot remove the new surface. Instead of that, they will form a new fracture surface on the other crack flank, the size of which is, again, equal to the number of returning dislocations times the Burgers vector and the crack front length. Under a constant loading amplitude the number of returning dislocations to the crack tip is nearly equal to the number of dislocations generated during loading. Hence, the crack extension per cycle is equal to the number of dislocations generated at the crack tip times the Burgers vector. In other words it is equal to the cyclic crack tip opening displacement. In this way, the crack advances during each loading cycle. In general, the crack growth rate da/dN can be assumed to be nearly proportional to the frequency of the oxide-layer fractures which is, again, proportional to the plastic strain γp in the slip band. Consequently, the crack growth rate can be assumed to be proportional to (L/h)m, where m ∈ (0.3, 0.8) is commonly accepted [311, 317]. This also means that the growth rate of short cracks is higher in the long PSBs embedded in large surface grains
154 3 Fatigue Fracture mode ll Figure 3.15 The single-slip model of short crack propagation When the growing short crack approaches a grain boundary,the emission of crack tip dislocations starts to be restricted by the back stress from a creat- ing pile-up.As a result,the growth rate rapidly decreases and,eventually,the crack could be arrested at the grain boundary.In general,the crack growth of these so-called microstructurally short cracks (MSCs)becomes retarded by various microstructural barriers of different strength.The maximal length of MSCs is determined by the distance bs of the strongest barriers such as grain or phase boundaries.As was sufficiently verified particularly by the group of K.J.Miller in Sheffield [318],the fatigue limit corresponds to a maximal stress still not high enough to propagate the longest short cracks, i.e.,to transfer it through the boundary of the surface grain to adjacent bulk grains.The distance bs also corresponds well to the maximal size of non- damaging cracks in the well known Kitagawa-Takahashi diagram [319].In specimens fractured close to the fatigue limit,therefore,many small surface cracks arrested at grain or phase boundaries can be found. Unlike in the case of long cracks,a description of MSC propagation in terms of Ak does not make too much sense.Indeed,the relative size of the plastic zone rp/a determined by the zone of emitted dislocations at the tip is not small enough to fulfil the conditions of small-scale yielding and plane strain.Moreover,there is a rapid change in the T-stress during the crack propagation from the surface towards the grain boundary.Therefore, the growth rate of MSCs is often described in the form da/dN=A△e(bs-a), where Asp is the applied plastic strain range,and A,l and k are material parameters(k<1)[247].This relationship implies a proportionality between
154 3 Fatigue Fracture T T T T mode II Kmin Kmin Kmax Figure 3.15 The single-slip model of short crack propagation When the growing short crack approaches a grain boundary, the emission of crack tip dislocations starts to be restricted by the back stress from a creating pile-up. As a result, the growth rate rapidly decreases and, eventually, the crack could be arrested at the grain boundary. In general, the crack growth of these so-called microstructurally short cracks (MSCs) becomes retarded by various microstructural barriers of different strength. The maximal length of MSCs is determined by the distance bs of the strongest barriers such as grain or phase boundaries. As was sufficiently verified particularly by the group of K. J. Miller in Sheffield [318], the fatigue limit corresponds to a maximal stress still not high enough to propagate the longest short cracks, i.e., to transfer it through the boundary of the surface grain to adjacent bulk grains. The distance bs also corresponds well to the maximal size of nondamaging cracks in the well known Kitagawa–Takahashi diagram [319]. In specimens fractured close to the fatigue limit, therefore, many small surface cracks arrested at grain or phase boundaries can be found. Unlike in the case of long cracks, a description of MSC propagation in terms of ΔK does not make too much sense. Indeed, the relative size of the plastic zone rp/a determined by the zone of emitted dislocations at the tip is not small enough to fulfil the conditions of small-scale yielding and plane strain. Moreover, there is a rapid change in the T-stress during the crack propagation from the surface towards the grain boundary. Therefore, the growth rate of MSCs is often described in the form da/dN = AΔεl p(bs − a) k, where Δεp is the applied plastic strain range, and A, l and k are material parameters (k ≤ 1) [247]. This relationship implies a proportionality between