FATIGUE CRACK GROWTH
The fatigue life assessment of smooth bodies delivers only indirect information regarding resistance to flaws either pre-existing or forming in the course of cycling. Comprehensive understanding of fatigue properties requires evaluation of the fatigue crack growth. The attention to fatigue crack growth in NC materials is presently growing with the development of processing techniques capable of producing large enough samples from which to make the standard compact tension (CT) or center-cracked-tension (CCT) specimens. The kinetic diagram showing the crack growth rate da/dN vs. the stress intensity factor range AK (AK = YAo^fPa, where Y is the geometrical factor dependent upon the specimen and crack geometry, Ao = omax — omin is the applied stress range, and a is the crack length) for a given stress ratio R, and environment is used commonly to quantify the fatigue crack growth behavior. We should underline that the da/ dN vs. AK plot for NC materials (Fig. 15) exhibits the same stages of a stable crack propagation as those well known for conventional polycrystals, i.e., a stage of slow crack advance in a near threshold region and an intermediate stage, where the Paris relationship
(where C and m are materials properties) applies for most materials. Patlan et al.[30] found that the fatigue threshold DKth decreased after ECAP of a nonheat-treatable 5056 Al-Mg alloy (Fig. 15). Chung et al.[37] demonstrated very similar results for the UFG 6061 Al alloy and UFG ECAP low carbon steel.[39] It has been shown that the crack growth rate in the near threshold region is higher in the UFG state than in the ordinary polycrystalline state; however, the result is reversed at relatively high stress intensity factor increments AK, i.e., da/dN in UFG metals is smaller on the intermediate fatigue stage. Recently, Hanlon et al.[60] obtained similar results for electro-deposited NC Ni with grain size of 20-40 nm. The lower crack growth resistance in the near threshold regime was attributed to a less tortuous path of the intergranular crack in the UFG structure (Fig. 13a). One can see that the crack propagates nearly perfectly straight on stage I of fatigue and then, as the crack length and the related AK value increase, the tortuousity of the crack path increases with numerous deflections and attempts to switch from mode II to mode I.
Fig. 15 Crack growth rate diagram of the fine grain ECAP 5056 Al alloy in comparison with its coarse grain O-temper counterpart.
It has long been established in the fracture mechanics approach that the transition from the near-threshold slow crack growth regime to the intermediate fatigue stage is accompanied by a transition in the crack behavior from being strongly structure sensitive to structure insensitive. Such a transition is often observed when the crack tip cyclic plastic zone rcp, which is estimated as
(where l is the numerical factor of the order of 1/p and sy is the cyclic yield stress determined from the CSSC) becomes of the same order as the grain size d.[4,61] If the stress intensity factor range corresponding to the transition point is denoted as AKT, the last equation yields a transition criterion rcp(AKT)«d. However, estimations of a reverse plastic zone radius rcp at the transition point return a rcp value for the materials tested (Cu, 5056, and 6161 Al alloys), which is significantly greater than the average grain size d. For example, taking for A5056 Al-Mg alloy sy = 280 MPa and the threshold AK = AKfc= 4.3 MPa m1/2,[31] one obtains rcp=7-8 mm—which is considerably larger than d =0.3-0.4 mm. This suggests that a large number of neighboring grains may be involved into crack tip plasticity, even near threshold. Eq. 8 at the transition point can be simply rearranged as
[where d* is a characteristic scale of a materials structural unit responsible for fracture (grain size, subgrain or cell size, particle size, etc.)], which predicts AKt to increase with d; this effect is observed in low-carbon steels (for example, Refs [4,61]). Assuming that threshold AKth approximately equals AKT, the last relationship is often written in a more general, but less-transparent and less-argued form as
where A and B denote materials constants. However, observations reported in Refs. [30,37,39,60] show the reduced AK,h in UFG metals processed by SPD and NC metals, in general. If Eq. 9 were valid for these materials, a shift of AKth to lower magnitudes (1 MPa m1/2 in the 5065 Al alloy, for example) should be expected, which is far below the observed values (of 4-4.5 MPa m1/2 in the same Al alloy). These observations cast some doubt on the general applicability of a simplified plastic zone size concept to explain the fatigue transition behavior of materials. Higo et al.[62] performed a systematic investigation of AKth and da/dN dependence on the grain size and the s0.1 yield stress in Cu and Cu-Al alloy with different Al content, i.e., with different stacking fault energy ranged between 5 and 45 mJ m"2 approximately. In contrast with the linear behavior of Eq. 9, they experimentally observed that the propagation rates were slower in the fine-grained materials and da/dN varied linearly with d" 1/2 in the same manner as the s0.1 yield stress, i.e., followed the Hall-Petch relationship, which is the reverse of Eqs. 9 and 10. These effects were pronounced in alloys with higher Al content, i.e., in planar slip metals where (we have stated) a stronger dependence of fatigue life on the grain size is expected. Thus it is shown that although the fatigue crack growth behavior in NC metals reveals some similarity to that of conventional polycrystals, the quantitative explanation of the fatigue threshold reduction as well as the reduction of the crack growth rate at relatively high AK is not straightforward. More experimental data are required in the following areas:
1. crack growth rate dependence on AK at different R values for different materials (so far, data are only available for Al alloys);
2. precise investigations of fatigue crack closure; and
3. more precise experimental evaluation of the reverse plastic zone are required before the mechanism controlling the fatigue crack behavior will be identified. Despite a somewhat lower tolerance of the NC metals to small and sharp interganular cracks (at least in Al alloys), these materials demonstrate a better resistance to large cracks, perhaps, because of a smaller plastic zone size, which in turn, according to Eq. 11, appears smaller because of a higher strength and cyclic yield stress.
FATIGUE MECHANISMS IN UFG MATERIALS
The apparent similarity between the CSSCs in the coarse-grain and NC metals, which we have discussed above, suggests that the major deformation mechanisms are also alike for both kinds of materials. Thus before introducing any complication in the modeling of fatigue behavior of NC metals, one should first determine if it is possible to explain the mechanical properties of SPD metals via ordinary dislocation dynamics.
Many discussions have been put forward in the literature about the particular role of grain boundaries in the properties of UFG materials.[1,2] Obviously, the interfacial energy and higher diffusivity of grain boundaries cannot be disregarded for many phenomena (particularly in nanocrystalline materials). In fatigue of SPD metals, for instance, grain boundaries play a significant role. On one hand, the fine-grained structure usually possesses longer fatigue life—at least under stress controlled cycling—than the coarse-grain one. On the other hand, the grain boundaries appear to contribute to the relatively low stability of the UFG structure and the tendency toward recovery and grain coarsening during cycling. Furthermore, grain boundaries appear to play a role in the frequently observed shear banding as well as crack initiation and propagation. Therefore grain boundaries arguably represent the most critical structural element.
Stress-Strain Response Under Cyclic Loading
Despite the complexity of the nanocrystalline structures, the fatigue behavior of NC metals is, in many ways, more easily described than that of ordinary poly- and single crystals. The main reason for simplification is the lack of dislocation patterning in the UFG structures. Vinogradov et al.[30,35] have suggested that the shape of a stable hysteresis loop and the cyclic stress-strain curve can be described, at least semiquantitatively, by considering only the kinetics of the average dislocation density within the framework of a simplified one-parameter model first proposed by Essmann and Mughrabi[63] for dislocation annihilation. It was assumed that the mobile dislocations initiate at a grain boundary pass through the grain and disappear in the opposite grain boundary. Thus the grain boundaries act as effective sources and sinks for dislocations. Because TEM observations do not reveal any substantial difference between the initial and postfatigued structures, in some UFG metals, it is plausible to suggest that dislocations do not accumulate inside the fine grains during cycling in these cases. Following Essmann and Mughrabi[63] and Mughrabi,[64] a kinetic equation for dislocation density, p, can be written in its simplest form as
where L is the slip path of dislocations with the Burgers vector b and y is the so-called annihilation length. The first term on the right-hand side of Eq. 11 describes the rate of dislocation multiplication with the strain increase, while the second term accounts for the strain-induced decrease of dislocation density (dislocation dynamic recovery and annihilation of a general kind). At large enough strains, saturation is attained because of the equilibrium between the dislocation multiplication and annihilation, dp/de = 0.
To describe the stress-strain relationship, we assume that the shear flow stress, t, is controlled by the average total dislocation density as
where t0 is the friction stress, a is a geometrical factor of the order of 0.5, and m is the shear modulus. For numerical estimations, one can take t0 as equivalent to the cyclic shear yield stress (s0«240 MPa). Combining Eqs. 11 and 12 results in an elementary differential equation, which can be solved analytically assuming y and L do not vary with straining. While the assumption concerning the constancy of the annihilation distance seems reasonable, and has been experimentally justified (at least for Cu,[63]), the assumption regarding L is considerably less obvious. Adopting L =const condition for the sake of simplicity one can, however, propose a few arguments in a favor of this suggestion. For the cell structure, which is typical of conventional polycrystals, the slip path has been related to the cell diameter. In UFG materials, the grain size is smaller than the typical cell size, and the grain boundaries form the main barriers for dislocation motion. Therefore it is plausible to relate the dislocation mean free path to the average grain size if the grain size is small enough. Integration of Eq. 14 together with Eq. 12 and t(0) = t0 as an initial condition yields
The saturation stress ts at sufficiently high strain g ^ b/y takes the form
As an example, a typical ascending part of the stable hysteresis loop of the 5056 Al alloy is plotted in Fig. 16 in the so-called relative coordinates[5]or — epl r, where the loop is displaced in such a way that the tip of the compressive half-loop comes to the origin coordinates, and both half-loops are treated as identical. The nonlinear curve fit of experimental points by function (13) with two fitting parameters—L and y—provides a good agreement between the experimental and calculated loops with L and y of (1.2±0.6) x 10— 7 and (8.4±3.0) x 10— 9 m, respectively (Fig. 16). The ordinary relations between shear and nominal stress-strain characteristics—o = Mt and e=g/M with M« 3 as the geometrical Tailor factor— are adopted. It is worth noting that the mean free pass of dislocations L appears to be of the order of the half-grain (cell) diameter d, which is in fair agreement with the assumptions made above.
Analogously, by using Eq. 14, one can calculate peak stresses, os, at different plastic strain amplitudes examined, giving an approximation of the cyclic stress-strain curve (Fig. 8).[30,35] Because of the simplicity of this one-parameter model, the qualitative agreement between experimental results and calculations achieved is surprising. In the low strain limit, formula (14) can be rewritten as
which corresponds to Ashby’s expression of the Hall-Petch work-hardening stress t=t0 + kd—1/2 [t0 and k=k(g) are materials properties; compare with Eq. 7], if L=d/2. In other words, from Eq. 11, the ”d—1/2” dependence of the flow stress recovered theoretically agrees with the experimentally observed Hall-Petch behavior of the CSSC and the fatigue limit.
Fig. 16 Results of modeling of the hysteresis loop shape and CSSC of UFG 5056 Al-alloy within the frames of a single-parametric kinetic dislocation model.
Thus despite oversimplification, this approach is capable of qualitative explaining such experimental results as
1. saturation of the cyclic stress amplitude,
2. high saturation stress,
3. rapid hardening/softening on the early stage of cycling,
4. shape of the cyclic stress-strain curve and
5. Hall-Petch behavior of the fatigue limit.
More precise modeling would require two (or three) kinds of dislocations to be distinguished—mobile and immobile (the former are responsible the intergranular slip and the latter are attracted to the grain boundaries) as has been suggested by Estrin et al.[65] for ordinary polycrystals.
Fatigue Life Prediction
The first attempt to model the fatigue life of SPD metal was performed by Ding et al.[66] They considered the NC metal to be a two-phase composite consisting of a soft matrix representing grain interior and ”hard” reinforcement representing the heavily distorted nonequilibrium grain boundaries (or the grain boundary affected zone having a characteristic thickness of 5 nm, which they suggested was experimentally supported by TEM and X-ray structural analysis). The authors suggested the dislocation accumulation to be the main strengthening mechanism in UFG metals; however, the grain boundaries also contribute to the resultant strength. The crack growth rate da/dN was calculated from the fracture mechanics concept assuming da/dN to be proportional to the crack tip opening displacement (ACTOD). The ACTOD was related to the J-integral, and the latter was calculated in the cited work with an assumption that within the fatigue damage zone, the local cyclic stress was uniform and equal to the ultimate tensile strength of the UFG material.
The empirical Coffin-Manson and Basquin laws were finally derived as
where ai and af are the initial and final critical crack length, respectively, l is a fitting parameter having a sense of the cyclic plastic zone correction factor same as that in Eq. 8, and C and F are the so-called GB constraint and GB strengthening factors respectively, which are introduced as:
where oyUFG and oyO denote the yield stresses in the UFG and ordinary polycrystalline materials, respectively, and Ao/2 and Aoeff/2 are the average steady-state cyclic flow stress amplitude in the bulk of the material and the effective stress amplitude that contributes to local deformation, respectively. A satisfactory agreement between the calculated from Eqs. 16 fatigue life and experimental data was obtained,[66] demonstrating a potential for fatigue life prediction in frames of the proposed phenomenological approach.
EFFECT OF TEXTURE
It is now understood that crystallographic texture exerts a very strong influence on the cyclic response of polycrys-talline metals. For instance, Lukas and Kunz[13] first demonstrated that coarse-grain copper saturates at higher stresses than its small grain analog. Llanes and Laird[67] showed that this effect could be attributed to a strong (111)-(100) fiber texture formed in coarse-grain copper during annealing. The SPD processing used to produce most of the materials discussed in this report inevitably results in the formation of at least moderate tex-tures,[68-70] which can be important for fatigue because of its effect on dislocation mobility, as well as the relative sensitivity of the material to plastic instabilities. Other methods of producing NC materials (e.g., film growth or inert gas condensation) may be more or less prone to these textural issues.
EFFECT OF ENVIRONMENT
It has long been recognized that the environment plays a very important role in fatigue damage, affecting both the crack nucleation and propagation, and a survey of the experimental results concerning the cyclic response of NC materials would not be complete without a few words about environmentally assisted fatigue. Yamasaki et al.[59] investigated corrosion fatigue of ECAP copper immersed into 1 M NaNO2 aqueous solution in terms of cyclic hardening/softening behavior and surface morphology. They found that UFG copper possesses a notably better resistance to environmental attack, including corrosion fatigue, when compared to its coarse-grain counterpart. In contrast with coarse-grain Cu, which shows a trans-granular fatigue fracture, corrosion fatigue in UFG specimens occurs intergranularly. Because localized corrosion is the most deleterious, the use of NC materials (having boundaries nearly everywhere) is said to be beneficial in the case where mass loss is the same for both sample types. Further improvement of fatigue properties and corrosion fatigue resistance is possible if the instability problem of SPD materials is resolved.
EFFECT OF PROCESSING ON FATIGUE AND OPTIMIZATION OF FATIGUE PERFORMANCE
The most influential SPD processing parameter is the amount of strain imposed. Increasing number of ECA pressings results in increasing monotonic strength and fatigue limit.[32-34,36] Furthermore, the fatigue properties of UFG metals can be improved by gaining some ductility and reducing constraints for dislocation motion, i.e., by decreasing the tendency for shear banding and strain localization, which is common in many hardened metals. Thus it can be advantageous for fatigue properties to employ materials with a partially recovered structure. The positive effect of heat treatment on LCF has been already revealed in the early fatigue studies of ECAP materials.[18] It has been shown, via the acoustic emission technique and microscopic surface observations, that susceptibility to shear banding in ECAP Cu decreases dramatically after a short-term (10 min) annealing at relatively low temperature of 250°C,[56] and LCF life can be improved by a factor of 5-10 after a heat treatment that does not result in any grain growth.[ , , ] While ECAP results in considerable reduction of tensile and cyclic ductility, the same materials subjected to a post-ECAP annealing can potentially obtain a higher ductility than its conventional coarse-grain counterpart,[23,45] and shifts the Coffin-Manson line toward higher fatigue lives (Fig. 9).
Because SPD metals retain some ductility after fabrication, their tensile and high cyclic strength can be additionally improved after postprocessing conventional cold rolling with or without intermediate annealing at moderate temperature. This has been shown for several Al-Mg alloys[31] and commercial purity Ti[23,25] as discussed in the section ”High Cycle Fatigue Behavior.” The effect of precipitation in SPD NC metals is complex. On one hand, it has already been mentioned that precipitates can dramatically increase the thermal stability of SPD metals and, on the other, grain boundaries may recover during aging thereby reducing their susceptibility to strain localization and premature cracking. As an example, it has been shown that optimal aging of ECAP UFG Cu-Cr-Zr alloy results in a high-strength structure with 200 nm grain size, which remains fine after subsequent annealing at temperatures as high as 500°C.[36] Kim et al.[71] have shown that the one-pass ECA pressing of the solid-solution treated 2024 Al alloy, followed by a low-temperature aging, can impressively enhance both strength and ductility: samples aged at 100°C for 20 hr had the strength sUTS=715 MPa, and the total elongation to failure d = 16%. Chung et al.[37] have shown that the yield stress and tensile strength of 6061 Al alloy benefit from multiple ECA pressing (four passes) as compared to the one-time-pressed sample (the ECAP of solution treated billets was performed at 125°C and no subsequent heat treatment was applied.) Although the effect of the single ECA pressing of aluminum alloy 6061[37] is impressive, it is not clear whether it would be impossible to achieve the same strength and ductility in Al alloys after conventional treatment. Hence two principal competing approaches for enhancement of fatigue properties via SPD can be seen: 1) achievement of compromise between strength and ductility in a minimum number of ECA pressings—one wherever possible, i.e., a cost-effective procedure employing relatively small imposed strains; and 2) achievement of maximum possible strength and high cycle fatigue life. The results of Chung et al.[37] show a lack of correlation between the tensile strength and fatigue limit, i.e., the four-pass sample with higher sUTS has the fatigue limit lower than the one-time pressed sample. In conclusion, there are still great opportunities for the development of optimum processing scheme for desired fatigue properties of SPD materials.
CONCLUSION
The presently available experimental results concerning the fatigue behavior of NC materials have been reviewed, and the following aspects are highlighted in retrospect to the available knowledge about the grain size effect on fatigue of conventional materials:
1. The significant enhancement of high cyclic fatigue life has been demonstrated after grain refinement down to submicrocrystalline and nanoscopic size for most materials, depending on the slip mode.
2. Despite the high tensile strength and improved high cyclic fatigue properties on NC and UFG metals, the low cyclic fatigue life appears shorter than that of their coarse-grain counterparts because of some loss in ductility during SPD.
3. Fatigue damage occurs on different scale levels from the intragranular movement of individual dislocations to macroscopic strain localization in the shear bands. The nature of the shear bands on SPD metals has been discussed from the standpoint of the initial ultrafine grain structure and its evolution during cyclic deformation.
4. The susceptibility of the ECAP materials to strain localization and microcracking can be a main factor, which limits their tensile and fatigue ductility and, to a large extent, determines their fatigue performance.
5. Postprocessing annealing has proven to be capable of considerable improvement of LCF performance of SPD metals because of reduction of their susceptibility to strain localization.
6. NC and UFG materials possess lower resistance to crack propagation than their coarse-grain analogs near the threshold.
7. Both grain refinement and dislocation work hardening play an important role in the resultant properties of materials obtained by severe plastic deformation. As a consequence, most of the cyclic properties of NC materials can be rationalized, at least qualitatively, in terms of Hall-Petch and dislocation hardening within a framework of a simple approach involving one- or two-parametric dislocation generation-annihilation kinetics.
The effects of texture, processing, and environment on fatigue life, crack initiation, and propagation in NC materials have been just scarcely studied, and further investigations in this field should be of interest for both potential applications and fundamental issues of fatigue of these materials.
