140 3 Fatigue Fracture a delay in ratcheting,can be explained only when the behaviour of discrete dislocations is taken into account.This underlines the necessity to utilize the multiscale approaches to the fatigue crack growth phenomenon which is doc- umented in Chapter 3.Unfortunately,those fundamental diversities of stable and unstable crack growths still do not seem to be widely understood,partic- ularly among "classical"mechanical engineers dealing solely with continuum mechanics. There is,however,another important factor strongly influencing the sta- ble crack growth rate in contrast to the unstable one.This is the effect of environment,even that of the air.Simply,the stronger the chemical interac- tion between the environment and the material,the higher the fatigue crack growth rate.For a majority of metallic materials,the crack growth rates in air are two orders in magnitude higher than those in vacuum [275].Although the detailed mechanisms of chemical processes are not examined here,the micromechanical reasons for their strong influence are clearly outlined here- after.On the other hand,the growth rate of unstable(brittle)cracks remains unaffected by the environment because there is not enough time available to produce a chemical damage during the fast fracture. Let us emphasize that the mechanism of cyclic plasticity immediately elu- cidates why the fatigue cracks avoid propagating in hard (or brittle)mi- crostructural phases.In such phases,the movement of dislocations is strongly limited and,therefore,the fatigue crack is always repelled to a softer mate- rial 276,277].The same qualitative explanation holds for an increase in the fatigue limit with an increasing hardness(or ultimate strength).Indeed,the fatigue limit is related to a critical stress under which the microstructurally short cracks still remain arrested [149,247.These cracks are smaller in high- strength materials(finer microstructure)so that the stress necessary for their further growth is higher.The fundamental difference in micromechanisms of brittle and fatigue crack propagation also has a consequence in the follow- ing well-known phenomenon:whilst a continuous increase in the materials ductility along the path of propagating brittle cracks (induced,e.g,by a temperature gradient)causes their arrest,this is generally not the case of fatigue cracks. Large deformations inside the plastic zone in both the Paris-Erdogan and the near fracture regions of long-crack propagation can be described by clas- sical continuum theories (Figure 3.7).However,these regions are of less engi- neering importance than those of the short crack growth and the long-crack threshold.Here,on the other hand,the plastic zone is relatively small and the dislocation activity is confined to one or two favourable slip systems.Under such conditions,the experimentally observed phenomena can be sufficiently elucidated only when taking the discrete nature of plasticity into account. Therefore,Sections 3.2.1,3.2.2 and 3.2.3 are devoted to discrete dislocation models of cyclic plasticity and fatigue crack growth. The results presented here are essential for a sufficient grasp of both the crack closure effects and the unified model of the crack-tip shielding,as de-
140 3 Fatigue Fracture a delay in ratcheting, can be explained only when the behaviour of discrete dislocations is taken into account. This underlines the necessity to utilize the multiscale approaches to the fatigue crack growth phenomenon which is documented in Chapter 3. Unfortunately, those fundamental diversities of stable and unstable crack growths still do not seem to be widely understood, particularly among “classical” mechanical engineers dealing solely with continuum mechanics. There is, however, another important factor strongly influencing the stable crack growth rate in contrast to the unstable one. This is the effect of environment, even that of the air. Simply, the stronger the chemical interaction between the environment and the material, the higher the fatigue crack growth rate. For a majority of metallic materials, the crack growth rates in air are two orders in magnitude higher than those in vacuum [275]. Although the detailed mechanisms of chemical processes are not examined here, the micromechanical reasons for their strong influence are clearly outlined hereafter. On the other hand, the growth rate of unstable (brittle) cracks remains unaffected by the environment because there is not enough time available to produce a chemical damage during the fast fracture. Let us emphasize that the mechanism of cyclic plasticity immediately elucidates why the fatigue cracks avoid propagating in hard (or brittle) microstructural phases. In such phases, the movement of dislocations is strongly limited and, therefore, the fatigue crack is always repelled to a softer material [276, 277]. The same qualitative explanation holds for an increase in the fatigue limit with an increasing hardness (or ultimate strength). Indeed, the fatigue limit is related to a critical stress under which the microstructurally short cracks still remain arrested [149,247]. These cracks are smaller in highstrength materials (finer microstructure) so that the stress necessary for their further growth is higher. The fundamental difference in micromechanisms of brittle and fatigue crack propagation also has a consequence in the following well–known phenomenon: whilst a continuous increase in the materials ductility along the path of propagating brittle cracks (induced, e.g., by a temperature gradient) causes their arrest, this is generally not the case of fatigue cracks. Large deformations inside the plastic zone in both the Paris–Erdogan and the near fracture regions of long–crack propagation can be described by classical continuum theories (Figure 3.7). However, these regions are of less engineering importance than those of the short crack growth and the long–crack threshold. Here, on the other hand, the plastic zone is relatively small and the dislocation activity is confined to one or two favourable slip systems. Under such conditions, the experimentally observed phenomena can be sufficiently elucidated only when taking the discrete nature of plasticity into account. Therefore, Sections 3.2.1, 3.2.2 and 3.2.3 are devoted to discrete dislocation models of cyclic plasticity and fatigue crack growth. The results presented here are essential for a sufficient grasp of both the crack closure effects and the unified model of the crack-tip shielding, as de-
3.2 Opening Loading Mode 141 Paris-Erdogan 8 region near near threshold fracture region region log△K log△Kh log△Kc Figure 3.7 Scheme of the crack-growth rate us Ak dependence for long fatigue cracks scribed in Sections 3.2.4 and 3.2.5.The unified model was developed as a multiscale concept involving three basic levels;micro-crystal defects,meso -grain(phase)microstructure and macro-continuum.The discrete dislo- cation theory and the size ratio effect constitute links between these three levels.It is important to emphasize,however,that a full comprehension of the crack growth threshold phenomenon requires an insight into the atomistics of dislocation emissions from the crack tip.Section 3.2.6 presents results of an extended application of the unified model to identify the shielding com- ponents as well as the intrinsic resistance to the near-threshold fatigue crack growth in various metallic materials.Finally,Section 3.2.7 is devoted to the influence of shielding effects on the crack growth rate in the Paris-Erdogan region. 3.2.1 Discrete Dislocation Models of Mechanical Hysteresis 3.2.1.1 Hysteresis Loop The process of cyclic plastic deformation controls both the crack initiation and the rate of fatigue crack propagation.The Nabarro-Cottrell analysis [156 represents a rather simplified model of mechanical hysteresis but it is very useful for understanding the physical background of that process. This analysis was applied to provide an insight into the early stage of cyclic softening as well as to the micromechanism of ratcheting (the cyclic creep) in polycrystalline materials at room temperature [149,278,279]. A polycrystalline material at the onset of cyclic softening can be considered to be a nearly elastic aggregate containing a small number of perfectly plastic
3.2 Opening Loading Mode 141 log d /d a N log Kth log KC log K near threshold region near fracture region Paris-Erdogan region . . Figure 3.7 Scheme of the crack-growth rate vs ΔK dependence for long fatigue cracks scribed in Sections 3.2.4 and 3.2.5. The unified model was developed as a multiscale concept involving three basic levels; micro – crystal defects, meso – grain (phase) microstructure and macro – continuum. The discrete dislocation theory and the size ratio effect constitute links between these three levels. It is important to emphasize, however, that a full comprehension of the crack growth threshold phenomenon requires an insight into the atomistics of dislocation emissions from the crack tip. Section 3.2.6 presents results of an extended application of the unified model to identify the shielding components as well as the intrinsic resistance to the near-threshold fatigue crack growth in various metallic materials. Finally, Section 3.2.7 is devoted to the influence of shielding effects on the crack growth rate in the Paris–Erdogan region. 3.2.1 Discrete Dislocation Models of Mechanical Hysteresis 3.2.1.1 Hysteresis Loop The process of cyclic plastic deformation controls both the crack initiation and the rate of fatigue crack propagation. The Nabarro–Cottrell analysis [156] represents a rather simplified model of mechanical hysteresis but it is very useful for understanding the physical background of that process. This analysis was applied to provide an insight into the early stage of cyclic softening as well as to the micromechanism of ratcheting (the cyclic creep) in polycrystalline materials at room temperature [149, 278, 279]. A polycrystalline material at the onset of cyclic softening can be considered to be a nearly elastic aggregate containing a small number of perfectly plastic
142 3 Fatigue Fracture grains.The macroscopic stress-strain response of such a system of n grains corresponds to a composite of net elastic grains with Young's modulus E (a major phase)and npt plastic grains with Young's modulus E2<E (a minor phase).Denoting the relative number of plastic grains as a =npl/n (nnet),the composite modulus Ec can then be expressed as -1-10 ≈E(1-2a) in analogy to a porosity influence in ceramic materials 149,280. The aggregate strain can be expressed as E=0E for o <0o, E=Ee+Ep =a/E+2a(a-ao)/E for a>ao, (3.5) where oo is the yield stress of the composite.The stress-strain response of such aggregate is depicted in Figure 3.8 in r-y shear coordinates (E- G,0→T,E→Y) By raising the applied tensile stress T from zero,the strain remains elastic up to the point A in Figure 3.8(a),where T=To.At this moment the first dislocations are emitted from Frank-Read(F-R)sources and create pile-ups at grain boundaries.These pile-ups hinder further dislocation emissions by producing an increasing back stress to F-R sources.Note that,in engineering applications,this stress is usually declared as a residual stress that remains in the material after unloading.Thus,the dislocations are emitted from the F-R source and produce plastic strain as long as the increasing effective shear stress T-To compensates the back stress.At the point B the applied stress stops increasing (T =Tmax),and the following equilibrium equation holds: Tmax -TO-TB =0, (3.6) where TB is the back stress. Now we start to reduce the tensile stress from Tmax-Tmax -AT.Before a sufficiently high value of Ar is reached,the dislocations cannot move back and the reverse deformation proceeds in an elastic manner (the segment BC in Figure 3.8(a)).The backward motion of dislocations begins only after a stress reversal at the point C,where the sum of the applied(reversed)stress and the back stress becomes equal to the compressive yield stress: Tmax-△T-TB+T0=0. (3.7) Combining Equations 3.6 and 3.7 one obtains △T=2T0- This simple relation reveals that,for the unloading strain path,a doubled yield stress is to be taken into account.The value of the compressive yield stress Tmax-2To corresponding to the point C is lower than that of the tensile
142 3 Fatigue Fracture grains. The macroscopic stress-strain response of such a system of n grains corresponds to a composite of nel elastic grains with Young’s modulus E (a major phase) and npl plastic grains with Young’s modulus E2 E (a minor phase). Denoting the relative number of plastic grains as α = npl/n (n ≈ nel), the composite modulus Ec can then be expressed as Ec = E 1 − 1.9 α 1 + α + 0.9 α2 (1 + α)2 ≈ E(1 − 2α), in analogy to a porosity influence in ceramic materials [149, 280]. The aggregate strain can be expressed as ε = σ/E for σ<σ0, ε = εe + εp = σ/E + 2α(σ − σ0)/E for σ>σ0, (3.5) where σ0 is the yield stress of the composite. The stress-strain response of such aggregate is depicted in Figure 3.8 in τ − γ shear coordinates (E → G, σ → τ,ε → γ). By raising the applied tensile stress τ from zero, the strain remains elastic up to the point A in Figure 3.8(a), where τ = τ0. At this moment the first dislocations are emitted from Frank–Read (F-R) sources and create pile-ups at grain boundaries. These pile-ups hinder further dislocation emissions by producing an increasing back stress to F-R sources. Note that, in engineering applications, this stress is usually declared as a residual stress that remains in the material after unloading. Thus, the dislocations are emitted from the F-R source and produce plastic strain as long as the increasing effective shear stress τ − τ0 compensates the back stress. At the point B the applied stress stops increasing (τ = τmax), and the following equilibrium equation holds: τmax − τ0 − τB = 0, (3.6) where τB is the back stress. Now we start to reduce the tensile stress from τmax → τmax − Δτ. Before a sufficiently high value of Δτ is reached, the dislocations cannot move back and the reverse deformation proceeds in an elastic manner (the segment BC in Figure 3.8(a)). The backward motion of dislocations begins only after a stress reversal at the point C, where the sum of the applied (reversed) stress and the back stress becomes equal to the compressive yield stress: τmax − Δτ − τB + τ0 = 0. (3.7) Combining Equations 3.6 and 3.7 one obtains Δτ = 2τ0. This simple relation reveals that, for the unloading strain path, a doubled yield stress is to be taken into account. The value of the compressive yield stress τmax−2τ0 corresponding to the point C is lower than that of the tensile
3.2 Opening Loading Mode 143 T木 T个 B B Tmax Tmax A A To To E (a) (b) Figure 3.8 Scheme of the mechanical hysteresis behaviour in the frame of the Nabarro-Cottrell analysis:(a)the asymmetric loading cycle,and (b)the symmet- ric loading cycle yield stress.This is the well-known Bauschinger effect.Beyond the point C, the dislocations move backwards while gradually reducing the back stress up to the point D,where TB =0.If the applied compressive stress starts to be reduced now,the whole process is repeated in the frame of a closed hysteresis loop.Such a loop is also created when the reverse deformation starts before reaching the point B or proceeds beyond the point D.An example of the latter case for a symmetrical loading is shown in Figure 3.8(b).Indeed,an inverse back stress is created within the segment DE,which causes an appropriate reduction of the tensile yield stress at the point F,similarly to that of the compressive yield stress at the point C.Let us finally emphasize that such a behaviour presumes a totally reversible dislocation slip. As a rule,the elastic parts of real hysteresis loops in engineering materials are somewhat shorter,while the plastic ones are longer and curved.A main reason for that fact constitutes a statistical distribution of values of the yield stress in various grains [281].It should also be noted that all the discrete dislocation models presented in this book utilize only dislocations inevitable for simulation of cyclic plasticity phenomena,the so-called geometrically nec- essary dislocations.There are also other dislocations forming the dislocation structure in real polycrystals,the so-called statistically stored dislocations. For more details about the geometrically necessary dislocations,see [8
3.2 Opening Loading Mode 143 max 0 A B C D (a) max 0 A B C D E F (b) Figure 3.8 Scheme of the mechanical hysteresis behaviour in the frame of the Nabarro–Cottrell analysis: (a) the asymmetric loading cycle, and (b) the symmetric loading cycle yield stress. This is the well-known Bauschinger effect. Beyond the point C, the dislocations move backwards while gradually reducing the back stress up to the point D, where τB = 0. If the applied compressive stress starts to be reduced now, the whole process is repeated in the frame of a closed hysteresis loop. Such a loop is also created when the reverse deformation starts before reaching the point B or proceeds beyond the point D. An example of the latter case for a symmetrical loading is shown in Figure 3.8(b). Indeed, an inverse back stress is created within the segment DE, which causes an appropriate reduction of the tensile yield stress at the point F, similarly to that of the compressive yield stress at the point C. Let us finally emphasize that such a behaviour presumes a totally reversible dislocation slip. As a rule, the elastic parts of real hysteresis loops in engineering materials are somewhat shorter, while the plastic ones are longer and curved. A main reason for that fact constitutes a statistical distribution of values of the yield stress in various grains [281]. It should also be noted that all the discrete dislocation models presented in this book utilize only dislocations inevitable for simulation of cyclic plasticity phenomena, the so-called geometrically necessary dislocations. There are also other dislocations forming the dislocation structure in real polycrystals, the so-called statistically stored dislocations. For more details about the geometrically necessary dislocations, see [8].
144 3 Fatigue Fracture 3.2.1.2 Ratcheting The ratcheting(cyclic creep)is a process of gradual plastic elongation or con- traction of a sample during cycling loading of a constant nominal stress am- plitude.The first experimental observation of the ratcheting at room temper- ature was reported by Kennedy [282.This process can lead to a substantial reduction of fatigue life in comparison with the strain-controlled loading of the same stress range [283,284.For example,failures of helicopter airscrews can be induced by ratcheting [285].When loadings of a high asymmetry near the fatigue limit are applied,the cyclic microcreep can cause undesir- able shape changes of precisely manufactured components as spiral springs or turbine blades. In most practical cases,the prescribed load(work)level is achieved only af- ter a certain period of time (the so-called ramp-loading).During that starting period,the plastic strain is increasing and the ratcheting process can only ap- pear when the plastic strain range exceeds a certain critical value Aspe which is characteristic for a particular material.This means that,during the load- ing,no ratcheting occurs when either the highest achieved value of the plastic strain range Asp is less than Aepe or a cyclic hardening starts before reaching Aspe.In the case of cyclically softening materials such behaviour is observed even when the work load is reached just in the first loading cycle.The scheme of a typical hysteresis behaviour during the asymmetric loading of cyclically softening materials before and after the onset of ratcheting is shown in Fig- ure 3.9.However,remarkable elongations may occur even when a symmetrical loading cycle is applied (the cyclic ratio R=omin/omax =-1)[286,287]. The value of Aspe is a material characteristic and depends on the cyclic ratio (decreases with increasing R)-see e.g.,[278,287].The elongation or contraction per one loading cycle (the ratcheting rate)increases with increas- ing both the cyclic softening rate and the cycle asymmetry [149,288]. As first described in 279,the analysis based on the discrete dislocation model of the hysteresis loop can be utilized to elucidate all the experimen- tally observed phenomena.Obviously,a closed hysteresis loop means that no ratcheting can take place.According to the above-mentioned Nabarro- Cottrell analysis,the incomplete closure of the hysteresis loop must be caused by micromechanisms producing an irreversibility of the cyclic plastic strain. During initial stages of a global cyclic softening,the plastic deformation is gradually transferred from pile-ups in most favourably oriented grains into the adjacent grains 289.Simultaneously,a cyclic hardening starts to take place in some of the already plasticized grains.At the end of the global cyclic softening stage,all grains become plastic and the global response changes to the cyclic hardening.During the global cyclic softening stage the density of dislocations increases by a cooperative operation of Frank-Read sources and, in this way,primary dislocation networks and loop patches are successively created.In their vicinity,a gradual increase of internal stresses results in an activation of secondary slip systems.Schemes of both the hysteresis loop
144 3 Fatigue Fracture 3.2.1.2 Ratcheting The ratcheting (cyclic creep) is a process of gradual plastic elongation or contraction of a sample during cycling loading of a constant nominal stress amplitude. The first experimental observation of the ratcheting at room temperature was reported by Kennedy [282]. This process can lead to a substantial reduction of fatigue life in comparison with the strain-controlled loading of the same stress range [283, 284]. For example, failures of helicopter airscrews can be induced by ratcheting [285]. When loadings of a high asymmetry near the fatigue limit are applied, the cyclic microcreep can cause undesirable shape changes of precisely manufactured components as spiral springs or turbine blades. In most practical cases, the prescribed load (work) level is achieved only after a certain period of time (the so-called ramp-loading). During that starting period, the plastic strain is increasing and the ratcheting process can only appear when the plastic strain range exceeds a certain critical value Δεpc which is characteristic for a particular material. This means that, during the loading, no ratcheting occurs when either the highest achieved value of the plastic strain range Δεp is less than Δεpc or a cyclic hardening starts before reaching Δεpc. In the case of cyclically softening materials such behaviour is observed even when the work load is reached just in the first loading cycle. The scheme of a typical hysteresis behaviour during the asymmetric loading of cyclically softening materials before and after the onset of ratcheting is shown in Figure 3.9. However, remarkable elongations may occur even when a symmetrical loading cycle is applied (the cyclic ratio R = σmin/σmax = −1) [286, 287]. The value of Δεpc is a material characteristic and depends on the cyclic ratio (decreases with increasing R) – see e.g., [278, 287]. The elongation or contraction per one loading cycle (the ratcheting rate) increases with increasing both the cyclic softening rate and the cycle asymmetry [149, 288]. As first described in [279], the analysis based on the discrete dislocation model of the hysteresis loop can be utilized to elucidate all the experimentally observed phenomena. Obviously, a closed hysteresis loop means that no ratcheting can take place. According to the above-mentioned Nabarro– Cottrell analysis, the incomplete closure of the hysteresis loop must be caused by micromechanisms producing an irreversibility of the cyclic plastic strain. During initial stages of a global cyclic softening, the plastic deformation is gradually transferred from pile-ups in most favourably oriented grains into the adjacent grains [289]. Simultaneously, a cyclic hardening starts to take place in some of the already plasticized grains. At the end of the global cyclic softening stage, all grains become plastic and the global response changes to the cyclic hardening. During the global cyclic softening stage the density of dislocations increases by a cooperative operation of Frank–Read sources and, in this way, primary dislocation networks and loop patches are successively created. In their vicinity, a gradual increase of internal stresses results in an activation of secondary slip systems. Schemes of both the hysteresis loop