Extremely-low-cycle fatigue behaviors of Cu and Cu–Al alloys: Damage mechanisms and life prediction

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ScienceDirect Acta Materialia 83 (2015) 341–356 www.elsevier.com/locate/actamat

Extremely-low-cycle fatigue behaviors of Cu and Cu–Al alloys: Damage mechanisms and life prediction ⇑

R. Liu, Z.J. Zhang, P. Zhang and Z.F. Zhang

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China Received 23 July 2014; revised 30 September 2014; accepted 2 October 2014

Abstract—The extremely-low-cycle fatigue (ELCF) behaviors of pure Cu and Cu–Al alloys are comprehensively studied following the cyclic push– pull loading tests with extremely high strain amplitudes (up to ±9.5%). Compared with the common low-cycle fatigue (LCF) region, several unique features in the ELCF regime can be noticed, including the deviations of fatigue life from the Coffin–Manson law, the non-negligible proportion occupied by the cyclic hardening stage of the whole fatigue life, special microstructures formed by cyclic loading containing deformation twins, shear bands and ultra-fine grains and the transformation of fatigue cracking modes. All these characteristics indicate the existence of special interior fatigue damage mechanisms of ELCF. To help discover the new damage mechanisms under ELCF, a model of fatigue life prediction with a hysteresis energy-based criterion is proposed. Based on the analysis of the experimental and modeling results, two intrinsic factors determining the ELCF properties were concluded: the capacity of ELCF damage, and the defusing and dispersion ability of the external mechanical work. The former can be evaluated by a parameter of the model called the intrinsic fatigue toughness W0, which is related to the microstructure evolution condition, the cyclic hardening ability, the deformation homogeneity and possibly the static toughness. The latter can be represented by the damage transition exponent b, which can be enhanced by improving the planarity, reversibility and uniformity of plastic deformation, reflecting the decline in the degree of surface damage and the dispersion of fatigue cracks. For Cu–Al alloys with increasing Al content, cooperation between an increasing damage capacity and a decreasing damage accumulation rate leads to a comprehensive improvement in the ELCF properties. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Extremely-low-cycle fatigue; Cyclic hardening; Fatigue crack; Hysteresis energy; Damage mechanism

1. Introduction Nowadays, the fatigue properties of engineering materials are commonly considered in most cases of structure design, to predict and prevent possible failure under cyclic loads [1]. Among these cases, one special circumstance has attracted attention in recent years: the cyclic loading condition with extremely large strain amplitude, such as earthquakes [2]. Under this condition, materials normally fail in fewer than 100 cycles, and perform differently from common low-cycle fatigue (LCF) behaviors. To distinguish this very-low-cycle regime from larger cycle parts of the LCF region, the fatigue process with a life of fewer than 100 cycles is termed extremely-low-cycle fatigue (ELCF) [3]. As an indispensable factor for seismic design, it is of great necessity to obtain a comprehensive and in-depth understanding of the ELCF behaviors of materials. Several notable features make ELCF a special instance that needs to be re-recognized based on the classical LCF theory [4–6], which has been well developed since the 1950s:

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(i) Microstructure evolution: the extremely large strain amplitude and relatively huge accumulated plastic strain during the ELCF tests could lead to special plastic deformation behavior followed by special microstructure evolution processes obviously different from the common LCF conditions. (ii) Cyclic hardening/softening behavior: changes occurring in plastic deformation and microstructure evolution are always related to the transformation of cyclic hardening/softening behaviors, resulting in higher hardening/softening rates, larger saturation stress amplitudes, etc. Moreover, the extremely short cyclic life changed the proportion of stabilized cycles: this shrinks with the decrease of fatigue life, even completely disappearing under some extreme conditions as specimens fail before reaching the cyclic stabilization (cyclic hardening/softening saturation) [7–10]. Hence fatigue models based on the hypothesis of cyclic stabilization are unable to make accurate predictions of lifetime in the ELCF regime [11,12]. (iii) Mode of fracture and failure: according to previous research, the failure modes between ELCF and common LCF are significantly different [7,13–15]. For instance, in several push–pull fatigue tests, the

http://dx.doi.org/10.1016/j.actamat.2014.10.002 1359-6462/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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fracture in the ELCF range often occurs in the interior of the specimen, while in the common LCF regime, the fatigue crack often starts from the surface [7]. Similar to (ii), the very short fatigue life changed the proportion of crack initiation, propagation and final fracture: due to the fast initiation of fatigue cracks in the ELCF regime, the stage of crack propagation becomes a main section of the whole fatigue life. This condition, together with the fracture mode transition [14,15] mentioned above, greatly changed the fatigue fractography [7,9], as well as the variation tendency of cyclic life [8]. These three aspects help distinguish ELCF clearly from common LCF, and make it another important category of fatigue. Based on the distinctions between common LCF and ELCF, two main research tasks of ELCF can be summarized here: one is the foundation of new theories of ELCF damage mechanisms and cracking behaviors; the other is the building of new models for ELCF life prediction. On the one hand, the special behaviors of microstructure evolution, cyclic hardening/softening [7–10] and fracture mode [7,13–15] indicate a series of new underlying fatigue damage mechanisms, which need to be considered differently to those in common LCF conditions. On the other hand, the transformation of those internal mechanisms essentially causes deviation from the Coffin–Manson law in predicting the ELCF life [2,8,9,11,12], making life prediction an unsolved problem in ELCF investigation. Due to these analyses, damage mechanism and the life prediction model are considered as two main aspects in this study. For the first aspect, pure Cu and a single-phase Cu–Al alloy system were chosen to conduct the ELCF tests and following observations, rather than the steels and other typical engineering materials mostly studied in previous ELCF research [2,7,10]. Compared with practical engineering materials, the pure Cu and Cu–Al alloys possess certain advantages for ELCF investigation as follows. Firstly, the pure Cu and Cu–Al alloys with different Al content can cover almost all the basic mechanisms of plastic deformation, making these a good choice for a comprehensive investigation of ELCF behavior, as the plastic deformation and microstructure evolution are considered the primary problem during the ELCF procedure. According to many studies [16–20], with increasing Al content, the single-phase Cu–Al alloys experience the transition of plastic deformation mechanisms from a typical wavy-slip manner (for pure Cu and alloys with low Al content) to a planar-slip manner (for alloys with more Al) and finally to the formation of stacking faults and deformation twins (for alloys with a relatively high Al content). The diversity of plastic deformation mechanisms for Cu–Al alloys is beneficial for achieving an integrated understanding of typical microstructure evolution behaviors and corresponding damage mechanisms. Secondly, a comprehensive improvement of mechanical properties is found in this alloy system with increasing Al content, including the synchronous improvement of strength and plasticity [21,22], the enhanced work-hardening ability [18,23,24] and the extended fatigue life in regimes of both HCF and common LCF [25,26]. Therefore, it is possible to extend this highly useful trend to the ELCF region, and previous mechanical research into this alloy system is of great reference value for the investigation of ELCF properties.

Moreover, both the pure metal and alloys we chose share the simplest single phase; without the interference of the second phase or complicated elements, observation and analysis are greatly simplified. This factor, together with the concise binary components, makes the Cu–Al alloy system an ideal model material for the investigations of ELCF behaviors. For the second aspect, the hysteresis energy is chosen to evaluate the ELCF damage by building a life prediction model, to replace the strain amplitude used in the Coffin–Manson law, which has been widely accepted in the study of the common LCF region. Corresponding background information will be introduced in detail at the beginning of Section 4. In this study, the ELCF properties of pure Cu and a single-phase Cu–Al alloy system are comprehensively studied by cyclic push–pull loading tests with extremely high strain amplitudes. Several ELCF damage mechanisms are then carefully investigated, including the plastic deformation mechanism, microstructure evolution process, cyclic hardening behavior, crack distributions and fracture modes. A fatigue life prediction model with a hysteresis energy-based criterion is proposed, to evaluate the ELCF damage and build a relationship between microscopic damage mechanisms and macroscopic fatigue properties. 2. Experimental procedures Pure Cu of 99.97% purity and Cu–Al alloys of three different Al contents (Cu–5 at.% Al, Cu–8 at.% Al, Cu–16 at.% Al) were investigated in this study. The pure Cu and Cu–Al alloys were cold-rolled and then annealed at 800 °C for 2 h to obtain highly homogeneous microstructures with an average grain size of 150 lm. Push–pull strain-controlled fatigue tests were then carried out on an Instron 8850–250 kN testing machine with a strain ratio of 1 and a strain rate of 1  102 s1 in ambient air at room temperature. Due to our focus on the ELCF behaviors, relative large strain amplitudes (De = 4%, 8%, 12%, 16%, 19%) were chosen, in an attempt to limit the fatigue life below 100. Correspondingly, a round bar shape with gauge dimensions of 10 mm (diameter)  12 mm was designed for the fatigue specimens, to ensure the cyclic deformation is as stable as possible. After fatigue tests, surface deformation and damage morphologies (including fractured surfaces) of specimens were observed by scanning electron microscopy (SEM) with a LEO Supra 35 field emission scanning electron microscope. The deformed microstructures were characterized by transmission electron microscopy (TEM) with an FEI Tecnai F20 microscope, operated at 200 kV. Thin foils for TEM observations were firstly cut from the fatigue specimens parallel to the loading axis by a wire cutting machine, with an original thickness of 300 lm. Then they were mechanically reduced to 50 lm thick, followed by a twin-jet polishing method in a solution of H3PO4:C2H5OH: H2O = 1:1:2 (vol.) with a voltage of 8– 10 V at 6 °C. Modeling and calculations were conducted by Matlab software. 3. Experimental results 3.1. Fatigue life The data for fatigue life for pure Cu and Cu–Al alloys at different strain amplitudes are listed in Table 1. For each metal or alloy, fatigue life decreases with increasing strain

R. Liu et al. / Acta Materialia 83 (2015) 341–356 Table 1. Results of fatigue lives in LCF (ELCF) tests of pure Cu and Cu–Al alloys. Total strain amplitude De/2 (%)

2 4 6 8 9.5

Fatigue life Nf Cu

Cu–5 at.% Al

Cu–8 at.% Al

Cu–16 at.% Al

93 40 17 10 –

172 68 45 28 –

505 128 93 52 33

1570 421 175 64 42

amplitude, and reaches below 100 in the range of ELCF. Relationships between the strain amplitude De/2 and fatigue life Nf on a log–log scale are displayed in Fig. 1b. The De/2–Nf curves show approximately linear distributions of the data for each kind of material. The general increasing tendency of both intercept and slope of the De/ 2–Nf line with increasing Al content indicates the comprehensive improvement of the ELCF properties. Similar trends were also found under the common LCF and HCF tests of Cu–Al [25,26] (see Fig. 1c and d) and analogous Cu–Zn alloy systems [27,28]. Moreover, a synchronous improvement of strength and plasticity can also be achieved by increasing Al content, which has been experimentally confirmed in Cu–Al alloys processed by severe plastic deformation (SPD) and subsequent annealing [21,22,29,30] (see Fig. 1a). All of these similarities show a comprehensive improvement in a series of mechanical properties, including strength, plasticity and resistance of fatigue destruction. Further discussions of the connections among those similar tendencies will be conducted in several other parts of this study.

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3.2. Cyclic hardening behavior Fig. 2 shows the cyclic hardening behavior of pure Cu and Cu–Al alloys during the cyclic loading tests. For the types of specimens under specific strain amplitudes in this study, the cyclic hardening process would be experienced during the first several cycles with the increasing stress amplitudes, as displayed Fig. 2a, and then saturation state would be reached (see Fig. 2d). For specimens with the same Al content but under different loading conditions, the saturation stress amplitudes increased with increasing strain amplitudes (see Fig. 2b); according to the cyclic stress–strain (CSS) curves shown in Fig. 2b and e, this increasing tendency did not reach the limitation within the range of the experimental conditions. For specimens undergoing the same strain amplitudes, the shape of the saturation stress–strain hysteresis loops and saturation stress amplitudes differed with different Al contents: the more Al component the Cu–Al alloys contain, the larger the saturation stress amplitudes achieved; the shape of hysteresis loops changed from similarly rectangular to fusiform in this process (see Fig. 2c). Two brief conclusions could be drawn from the experimental curves displayed in Fig. 2. One is that the cyclic hardening ability improved with increasing Al content. The increasing saturation stress amplitudes and CSS curves with highly significant upward trends (see Fig. 2e) indicated this tendency. To make it clearer, we divided the saturation stress amplitude DrS by a corresponding stress amplitude of the first cycle Dr1 and considered the value as a reflection of the cyclic hardening ability. As shown in Fig. 2f, there is a remarkable enhancement in DrS/Dr1 with increasing Al content, which reflects the enhanced hardening ability. The other conclusion is about the cyclic hardening rate.

Fig. 1. Corresponding trend of pure Cu and Cu–Al alloys in tensile and cyclic loading properties. (a) Relationship between ultimate tensile strength and uniform elongation of Cu, Cu–8 at.% Al and Cu–16 at.% Al after high-pressure torsion and annealing [22] and Cu–5 at.% Al and Cu–8 at.% Al after equal channel angular pressing and annealing [29]. (b) Relationships between strain amplitude De/2 and fatigue life Nf of the LCF (ELCF) tests conducted in this study. (c) Relationships between strain amplitude De/2 and fatigue life Nf of the LCF tests conducted by An [27]. (d) Relationships between stress amplitude Dr/2 and fatigue life Nf of the HCF tests conducted by An [27].

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Fig. 2. Cyclic hardening behaviors of pure Cu and Cu–Al alloys. (a) Cyclic stress–strain hysteresis loops of Cu–16 at.% Al, De/2 = 2.0%. (b) Stabilized cyclic stress–strain hysteresis loops of Cu–8 at.% Al. (c) Stabilized cyclic stress–strain hysteresis loops, De/2 = 6.0%. (d) Cyclic hardening curves, De/2 = 8.0%. (e) Cyclic stress–strain (CSS) curves. (f) Cyclic hardening abilities, DrS/Dr1.

On the one hand, the cycles for specimens to achieve the cyclic saturation state vary with the Al content. For alloys with stronger cyclic hardening ability, the number would be larger, corresponding to a longer cyclic hardening procedure (see Fig. 2d). On the other hand, the increase of strain amplitude speeds up the hardening process and limits the hardening potential. As shown in Fig. 2f, besides the improving trend with increasing Al content, there is a decreasing tendency with increasing strain amplitude, indicating the restriction of cyclic hardening. In short, cyclic hardening behavior can be varied either by the alloy component or by the loading condition, thus leading to different fatigue properties. It is worth noting that the proportion of saturated cycles shrinks due to the remarkably shorter fatigue life of the ELCF process compared with the common LCF and HCF methods, corresponding to the enlargement of the cyclic hardening region (as shown in Fig. 2d). This condition not only makes the hardening process more important in the ELCF region, but also causes certain problems in models for the ELCF life prediction. In addition, the obvious differences in the shape of the hysteresis loops and saturation stress amplitudes under the same strain amplitude, as displayed in Fig. 2c, suggest the inadequacy of damage evaluation based only on strain amplitude in some previous fatigue models based on the Coffin–Manson law [5,6]. Taking these factors into consideration, the hysteresis energy could be a more appropriate parameter for the ELCF damage measurement, as it involves the influences of both the cyclic strain and the stress amplitudes. 3.3. Surface deformation and cracking behavior Surface damage morphologies for pure Cu and Cu–Al alloys are displayed in Fig. 3, observed after cyclic loading processes with the same strain amplitude. According to

Fig. 3a, the surface of pure Cu after fatigue tests displayed conspicuous intrusions and extrusions, showing a typical surface morphology formed by cross-slip with strong deformation irreversibility. Compared with pure Cu, the slip bands of Cu–5 at.% Al alloy are much thinner and shallower (Fig. 3b), which shows the inhibition of cross-slip and the remission of deformation irreversibility. The condition of the Cu–8 at.% Al alloy (Fig. 3c) can be considered as a transient state between wavy slip and planar slip, which will be analyzed in detail later. The uniformly distributed thin bands on the surfaces of Cu–16 at.% Al alloy show that it is a typical material deforming in a planar way, with good deformation reversibility (Fig. 3d). In conclusion, by increasing the content of Al, surface damage morphologies changed from a rough surface containing thick slip bands to a relatively smooth appearance with thin deformation bands, referring to a decreasing tendency of surface damage. Distribution conditions of fatigue cracks also changed by altering Al content, as shown in Fig. 4. The fatigue fractography of pure Cu (Fig. 4a) was formed by a smoothly propagated main crack without obvious small cracks nearby. With the increasing Al content, more cracks emerged around the fracture (Fig. 4b–d). For Cu–16 at.% Al shown in Fig. 4d, the cracks were dispersed over a large region of the specimen, and the main crack was formed by connecting those scattered cracks. These phenomena indicate that the ability of fatigue damage dispersion can be enhanced by increasing the Al content. In addition, the influence of Al content on fracture behavior can also be reflected in the fractographies. As displayed in Fig. 5, by increasing the Al content, the fatigue striations become more notable and dense than those in pure Cu (Fig. 5a–d) and dimples in the final fracture regions also become smaller (Fig. 5e and f); these all indicate a slower fracture propagation speed and a higher resistance to fatigue damage.

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Fig. 3. Microscopic morphologies of slip band on surface of pure Cu and Cu–Al alloys after cyclic loading process observed by SEM: (a) pure Cu, De/2 = 6.0%; (b) Cu–5 at.% Al, De/2 = 4.0%; (c) Cu–8 at.% Al, De/2 = 4.0%; (d) Cu–16 at.% Al, De/2 = 2.0%.

Fig. 4. Distribution condition of fatigue cracks near the fractographies of pure Cu and Cu–Al alloys after cyclic loading process observed by SEM, with De/2 = 4.0%: (a) pure Cu; (b) Cu–5 at.% Al; (c) Cu–8 at.% Al; (d) Cu–16 at.% Al.

The results above concentrated on the internal factors (chemical component) that influenced the ELCF cracking behaviors; in fact, some external conditions such as the cyclic strain amplitudes are also significant factors. For instance, the fracture behavior of Cu–8 at.% Al alloy varies with cyclic strain amplitude, as displayed in Fig. 6. Under relatively low strain amplitudes (De/2 = 2.0–6.0%), the

material displayed characteristics of planar deformation (Fig. 6a), such as the smoothly distributed surface deformation bands (bands formed by slipping or twinning) and the scattered small cracks (Fig. 6c). Under higher strain amplitudes (De/2 = 6.0–9.5%), features of wavy slip such as intrusions and extrusions appeared on the surfaces of Cu–8 at.% Al alloy (Fig. 6b); cracks also tend to concentrate

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Fig. 5. Fractographies of pure Cu and Cu–Al alloys after cyclic loading process observed by SEM, with De/2 = 4.0%. (a)–(d) the crack propagation regions: (a) pure Cu; (b) Cu–5 at.% Al; (c) Cu–8 at.% Al; (d) Cu–16 at.% Al; (e)–(f) the final fracture regions: (e) pure Cu; (f) Cu–16 at.% Al.

in the region near the main crack (Fig. 6d). It can be concluded that while the fatigue cracks dispersed with increasing Al content, the increasing strain amplitude can inversely cause the concentration of fatigue damage; in other words, cyclic deformation under larger strain amplitude makes damage localization much easier. 3.4. Microstructure evolution Typical microstructures of pure Cu and Cu–Al alloys after cyclic loading process are summarized in Figs. 7 and 8. For pure Cu, dislocation walls and cells are the most common structures after fatigue tests (Figs. 7a and 8a). Due to the rather short fatigue life, regions with dislocation tangles that have not developed into regular configurations can also be observed. For Cu–5 at.% Al alloy, the deformed microstructure mainly consists of dislocation cells with much thicker walls than that in pure Cu (Fig. 7b) and elongated sub-grains (Fig. 8b); in several grains, a small number of deformation twins can also be found. For Cu–8 at.% Al alloy, deformation twins become the most common microstructure (Fig. 7c), together with special refined grains: notable grain refinement occurred during cyclic deformation, leaving some ultra-fine grains (UFGs) with a typical

size of 100–200 nm distributed in large regions (Fig. 8c). For Cu–16 at.% Al alloy, uniformly distributed twin bundles (Fig. 7d) and shear bands with complicated structures become typical features of the deformed microstructure. Deformation twins in multi-directions can be commonly observed in this alloy; these help divide the original coarse grains into nano-scale sections (Fig. 8d). The observation results introduced above agreed well with previous research on plastic deformation mechanisms. According to the previous investigations [17,21], with increasing Al content, the dominant deformation mechanism of pure Cu and single-phase Cu–Al alloys transforms from wavy slip to planar slip, then to the formation of stacking faults and deformation twins. As a result, microstructures with remarkably different features were achieved after fatigue tests of pure Cu and Cu–Al alloys (Figs. 7 and 8). The above descriptions are only a brief introduction to the microstructure evolution process; more detailed information and further investigation will be reported elsewhere. Besides the processes of microstructure evolution, the influences of microstructure evolution on damage mechanisms and fatigue behaviors will be analyzed in the following paragraphs. As is well known, within the whole LCF regime, the failure mechanism is mainly governed by the plastic

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Fig. 6. Transitions of surface morphologies and cracking distribution of Cu–8 at.% Al with the variation of strain amplitudes, observed by SEM: (a) De/2 = 2.0%; (b) De/2 = 9.5%; (c) De/2 = 6.0%; (d) De/2 = 9.5%.

Fig. 7. Typical microstructure of pure Cu and Cu–Al alloys after cyclic loading process observed by TEM: (a) pure Cu, De/2 = 6.0%; (b) Cu–5 at.% Al, De/2 = 6.0%; (c) Cu–8 at.% Al, De/2 = 9.5%; (d) Cu–16 at.% Al, De/2 = 8.0%.

damage (sometimes called ductile damage), which is characterized by microstructure deterioration such as microvoid nucleation, growth and coalescence and micro-crack initiation and consequent propagation [8]. Accordingly, almost all of the ELCF behaviors, including cyclic hardening and cracking, can be traced back to the mechanisms of plastic deformation and the process of microstructure

evolution. The following paragraphs describe some important relationships. First and foremost, the cyclic hardening behavior of ELCF is related to the scale of typical structures formed during microstructure evolution. As displayed in Fig. 8, from pure Cu to Cu–16 at.% Al alloy, with increasing Al content, the size of the minimum unit of microstructure

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Fig. 8. The minimum unit of microstructure of pure Cu and Cu–Al alloys after cyclic loading process observed by TEM: (a) pure Cu, De/2 = 8.0%; (b) Cu–5 at.% Al, De/2 = 8.0%; (c) Cu–8 at.% Al, De/2 = 9.5%; (d) Cu–16 at.% Al, De/2 = 9.5%.

decreased: for pure Cu, dislocation cells, as the smallest regions, possess a scale of 600–800 nm (Fig. 8a); the elongated sub-grains in Cu–5 at.% Al alloy share a typical size of 300–500 nm in the short axis direction (Fig. 8b); the size of UFGs in Cu–8 at.% Al alloy decreases the value to 100–200 nm (Fig. 4c); the effect of twin–twin interaction and fragmentation divided the grains of Cu–16 at.% Al alloy into regions of <50 nm (Fig. 4d). The reduced microstructure scale corresponds directly to the enhanced cyclic hardening ability, resulting in higher saturation stress amplitude, as introduced in Section 3.2. As the second aspect, the diversity of plastic deformation mechanisms significantly affects the strain uniformity, which further has an impact on the fatigue damage distribution and cracking behavior. Previous investigations indicated that the combination of twinning and slipping in Cu–Al alloys is more beneficial for the deformation homogenization than the slipping only, as occurs in pure Cu [31]; in this way, Cu–Al alloys with higher Al contents usually deform in a more uniform pattern. This trend shows good agreements with our experimental results on the distribution of fatigue cracks in Section 3.3: by improving deformation uniformity, the increase of Al content effectively prevents the concentration of fatigue damage and enhances the dispersion of fatigue cracks. Last but not least, the planarity of deformation mechanisms should have positive effects on both the hardening behavior and the damage morphologies on the surface. On the one hand, with increasing Al content, the deformation planarity can be enhanced by suppressing cross-slip, and further reduces the recovery rate of defects; it makes for a longlasting hardening potential. On the other hand, the enhanced planarity can cause the improvement of deformation reversibility, which then reduces the fatigue damage by changing the morphologies on the surface, which will go a step further to postpone the fatigue cracking and final fracture.

In summary, different deformation mechanisms cause size differences of the minimum unit of microstructure, deformation reversibility and homogeneity, which further influence the ELCF properties such as cyclic hardening ability, surface morphology, fatigue cracking distribution and fatigue life. 4. Model and analysis According to the experimental results above, the ELCF behaviors of materials differ notably from the common LCF conditions, in both the macroscopic properties and micro-scale mechanisms. To make a connection between mechanisms and properties, corresponding models and theories need to be built. However, due to the obvious differences between ELCF and common LCF, some of the previous theories that function well in the LCF region fail to explain the phenomena of ELCF. For instance, problems emerged on ELCF life prediction when using the Coffin–Manson law [2,8,9,11,12], which displays good agreements with the experimental data in the common LCF regime [5,6,25,26]. Therefore, the modification of existing models or the creation of new models is necessary to solve this problem. Recently, several modified models based on the original Coffin–Manson law were proposed for the investigation of ELCF behaviors, by making improvements mainly concerning the following two issues: the transformation of dominant failure modes from fatigue fracture of the common LCF region to accumulation of ductile damage of the ELCF regime; and the instability and inhomogeneity of the strain field caused by complex geometry of structural elements or loading conditions [2]. New functions and parameters were introduced to revise the original Coffin– Manson law from the deviation [8,9,11,12,32]. However,

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problems still exist: firstly, most of the modified models concentrate only on the external unstable factors such as structural discontinuous geometry and random loading [2,8]. But as shown in the experimental results above, the instability can also be introduced by the material itself, even under a loading condition with completely uniform stress distribution, stable strain amplitude and certain strain rate; those intrinsic behaviors of materials have not been fully reflected in those models. Secondly, the modified Coffin–Manson models remain to evaluate the damage degree by the values of strain amplitudes, which is not a suitable choice to solve problems under unstable loading conditions. Moreover, the complicated forms of the modified models restricted their practical application. In view of the above-mentioned facts, we prefer to build up a new model rather than making modifications. During the analysis of experimental results, we have noticed that the hysteresis energy should be a better choice to evaluate the ELCF damage in a model, compared with the strain amplitude. In fact, models sharing similar ideas have been proposed since the 1920s [33], and gradually formed corresponding modeling systems [34–46]. Typical achievements including the well-known Smith, Watson and Topper (SWT) model proposed in 1970 [37], which first considered the influence of the average stress. Another model is one concerning the hysteretic plastic work, which was proposed by Santner and Fine [38]; in this model, a relationship between the hysteretic plastic work and the fatigue life had been built up in a form similar to the Coffin–Manson law. Notably, models with energy-based criteria have been applied successfully to unstable or complicated loading conditions such as non-regular loadings [39,40] and multiaxial fatigue tests [41–45], while dealing with those conditions is a significant problem in the ELCF life prediction. However, except for those advantages, the earlier hysteresis energy models also have certain shortcomings. For example, arguments still exist about the method of energy calculation, mainly concentrating on the disposal of the elastic region [37,43,46]; the difficulties of hysteresis energy calculation also hindered the application of the models. Consequently, in the HCF and common LCF regimes, since the criteria of stress and strain amplitude are also an issue, the energybased models have not been generally accepted. In the ELCF regime, however, things seem to be different: on the one hand, owing to the extremely high strain amplitudes, the plastic region occupied the vast majority of the total hysteresis energy, making the influence of the elastic region insignificant; on the other hand, models based on strainamplitude-based criteria seem to be no longer appropriate for the ELCF life prediction, which provides a good chance for developing energy-based models. According to the above analysis, in this section, a concise fatigue model based on the hysteresis energy criteria will be proposed, in order to give a comprehensive consideration of the ELCF damage behaviors. Formulas and corresponding parameters are introduced as follows. 4.1. Introduction of the hysteresis energy model It is well known that models for fatigue life prediction can mainly be divided into two types: the stress-amplitude-based models and the strain-amplitude-based models. As a typical case of the former type, the Basquin law [47] is widely used for life prediction in the HCF regime, which can be briefly presented as:

ra ¼ rf ð2N f Þb

349

ð1Þ

Correspondingly, as the representative model of the latter type, the Coffin–Manson law [5,6] is usually adopted in the LCF life prediction: ep ¼ ef ð2N f Þc

ð2Þ

By calculating the damage parameter D (defined by the proportion of damage in a single cycle, 1/Nf, according to Miner’s law [4]) of the above two models, as displayed in Eqs. (3) and (4), we can easily find out that despite the two models evaluating fatigue damage by stress amplitude or strain amplitude, respectively, they share a similar form:  1=b 1 ra D¼ ¼2 ð3Þ Nf rf D¼

 1=c 1 ep ¼2 Nf ef

ð4Þ

By replacing the amplitude of stress or strain with the hysteresis energy, for the ELCF life prediction, a new form of damage parameter D can be achieved as follows: Di ¼

1 ¼ Nf



Wi W0

b ð5Þ

The factor “2” in Eqs. (3) and (4) is introduced by using the number of reversals to failure (2Nf) to calculate the total amount of cyclic loadings (see Eqs. (1) and (2)), which is twice the number of cycles to failure (Nf). Thus, the damage parameter D, a reflection of the damage of a whole cycle, should exactly contain the damage of two reversals. In contrast, to simplify the calculation process of this energy-based model, the hysteresis energy itself is counted in whole cycles, so the factor “2” in Eqs. (3) and (4) is not contained in Eq. (5). In consideration of the unstable cyclic condition causing the variation of the D value for each cycle, which is also the most effective situation for the application of this model, an i is added here as subscript. Di represents the damage parameter of the ith circle; correspondingly, Wi means the hysteresis energy of the ith circle. W0, a significant material constant of this model, sharing the same unit (J m3) with Wi, represents the fatigue damage capacity of a specific kind of material, which can be defined as the “intrinsic fatigue toughness” of materials. The larger a material’s W0 is, the more resistance of fatigue damage it achieves. Another important material constant b, named the “damage transition exponent”, is defined to evaluate the ability of damage defusing (i.e., the diminishment and dispersion of fatigue damage). In principle, the value of b should be larger than 1: if the value exactly equals 1, it means that all the mechanical work will be transformed into effective damage to the materials; the increase of b represents the improved ability for damage diminishment or dispersion. By describing fatigue damage D with a hysteresis energy based criterion, in this model, fatigue is considered as a process of energy accumulation. Materials receive mechanical energy from the cyclic loading procedure, and exclude part of them by reversible plastic deformation methods; the other part accumulates in materials with different extents of uniformity. Assuming that materials failed when the summation of fatigue damage equals 1, the relationship between Di and Nf can be described as:

350 Nf X i¼1

R. Liu et al. / Acta Materialia 83 (2015) 341–356

Di ¼

Nf X

ðW i =W 0 Þb ¼ 1

ð6Þ

i¼1

Nf X i¼1

Eq. (6) represents the general form of this hysteresis energy based model. For a special stable cycling condition with a constant hysteresis energy Wa, the formula can be simplified as follows:  b 1 Wa D¼ ¼ ð7Þ Nf W0 The above equation can be transformed into the following form: 1=b

W a ¼ W 0  Nf

ð8Þ

It is easy to notice that Eq. (8) shares a similar form with the Basquin law and Coffin–Manson law (see Eqs. (1) and (2)). In fact, if we consider the hysteresis energy as a value in proportion to the product of stress amplitude and strain amplitude (calculations in the next section confirm this assumption), this formula can be transformed into exactly the form of the above two laws, by respectively dividing out strain amplitude or stress amplitude, which are correspondingly considered as constant in the HCF and LCF processes. In other words, the model can be considered as a general form of common fatigue models, including the two typical laws as special forms in specific situations. Another special form of the model corresponding to fatigue tests with variable amplitudes can be given as follows:  b  b W1 W2 DN 1 þ DN 2 þ    þ DNn ¼ N 1  þ N2  þ  W0 W0  b Wn þ Nn  ¼1 ð9Þ W0 where W1, W2, . . . , Wn represent the hysteresis energy of each stage of cyclic amplitude, N1, N2, . . . , Nn are relevant cyclic numbers at each cyclic amplitude. This suggests that the fatigue damage under different cyclic conditions can be added up directly in terms of the energy criterion. According to the introduction given above, this model has advantages in both the reflection of the materials’ inherent quality in terms of fatigue damage resistance and the compatibility to handle various complicated unstable conditions. In all, it may be regarded as a useful model in the ELCF regime, especially for the in-depth understanding of the intrinsic damage mechanisms, as shown in the following sections. 4.2. Simplifications, results and valuations of the model To avoid complicated calculations and statistical procedures, reasonable simplifications are conducted according to the experiment results in this investigation. Firstly, according to the change of strain amplitude, the whole cyclic loading process can be roughly separated into two stages: the cyclic hardening stage and the cyclic saturation stage. Supposing the hardening procedure occupies N1 cycles of the total fatigue life: we have to calculate the hysteresis energy Wi of each cycle from 1 to N1 successively; but for the remaining saturated cycles, it can be substituted by a constant value of the saturation hysteresis energy WS. The simplified model can be expressed as follows:

Di ¼

b N1  X Wi i¼1

W0

 þ ðN f  N 1 Þ 

WS W0

b ¼1

ð10Þ

Secondly, a linear relationship can be found between Wi and the value of De  Dri, as displayed in Eq. (11), to simplify the calculation of hysteresis energy. Here De represents the total strain range and Dri is the corresponding strain amplitude of the ith cycle. W i ¼ k  De  Dri þ C

ð11Þ

Specific results of linear fitting are listed in Table 2, which shows the constant C to be very small; consequently, the relationship is mainly decided by the coefficient k. As displayed in Fig. 9a, for the same materials, data obtained from cyclic loadings with different strain range De approximately follow the same linear pattern: hence the value k here is mainly decided by species of materials. The declining tendency of k with the increase of Al content shown in Fig. 9b is probably relevant to the increasing cyclic hardening potential and the decreasing recovery rate of Cu–Al alloys. Both of the simplifications help to ensure the practical value of this model. Based on the experimental data and simplifications displayed above, values of the two material constants W0 and b with reasonable bias can be calculated. As shown in Fig. 10a and Table 3, a synchronous improvement of W0 and b is achieved by increasing Al content in Cu–Al alloys, which shares the similar tendency with the values of ef and c calculated by the Coffin–Manson law (refer to Fig. 1b). It is a beneficial trend corresponding to the comprehensive improvement of the ELCF properties (Fig. 10b): on the one hand, the enhanced value of W0 indicates an increased fatigue damage capacity, thus making the synchronous improvement of Wa and Nf possible, as displayed in the Wa–N and WS–Nf relationships shown in Fig. 11a and b; on the other hand, the increased b represents an improved ability to defuse fatigue damage, reflected in the decreasing slope of the WS–Nf relationship in Fig. 11b. It should be pointed out that as the value of Wa cannot be considered as a constant, the exact Wa–Nf relationship cannot function in a similar way to the Dep/2–Nf relationship: the WS–Nf relationship displayed in Fig. 11b is only an approximate linear relation, while the slope is also not the exact value of b. To judge the equivalence of the Coffin–Manson law and the hysteresis energy model for the evaluation of fatigue properties, a comparison between the evaluation methods of the two models based on the same group of fatigue results is conducted in Fig. 12. By making a coincidence with the two lines of Cu and comparing the data of Cu– 8 at.% Al, we can easily discover a more significant improvement tendency with the increasing Al content evaluated by the hysteresis energy modeling results. Differences between the results of the two models can be explained by considering the distinction of hysteresis energy between Cu

Table 2. Linear fitting parameter values of empirical formula to calculate hysteresis loops’ area. Parameters

Cu

Cu–5 at.% Al

Cu–8 at.% Al

Cu–16 at.% Al

k C(106 J/m3)

0.91 1.26

0.89 1.52

0.80 0.88

0.68 2.13

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351

Fig. 9. (a) The linear relationships between hysteresis loops’ area Wa and the value of De  Dr, with the corresponding fitting results; (b) relationship between the value k and the content of Al.

Fig. 10. (a) Relationships between the parameter values of hysteresis energy model (W0 and b) and the content of Al. (b) The comprehensive improvement of ELCF properties due to the synchronous increasing of W0 and b in Cu and Cu–Al alloys with the increase of Al.

Table 3. Reference parameter values of hysteresis energy model. Parameters

Cu

Cu–5 at.% Al

Cu–8 at.% Al

Cu–16 at.% Al

W0(106 J/m3) b

558 1.27

928 1.42

1321 1.64

1651 1.74

and Cu–Al alloys: under the same strain amplitude, the area of hysteresis loops increased with increasing Al content. In the Coffin–Manson law, this distinction is not directly taken into consideration, resulting in the findings not being applicable to materials with higher Al content,

which in fact receive more mechanical energy in each cycle; in contrast, the hysteresis energy model considers the distinction and provides a more realistic reflection of the intrinsic abilities of fatigue damage resistance. According to the modeling processes and modeling results introduced above, a brief evaluation of this hysteresis energy model can be conducted here, summarized as follows: (i) Compared to other energy-based fatigue models, this model achieved a relatively simple form; in addiction, a simplified calculation method of hysteresis energy has been developed, ensuring the practicability of this model.

Fig. 11. Results of the hysteresis energy model. (a) The Wa–N relationship for pure Cu and Cu–Al alloys. (b) The Ws–Nf relationship for pure Cu and Cu–Al alloys.

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damage reached the capacity of this material. In this process, the two constants W0 and b demonstrate the intrinsic fatigue properties of materials, which can be considered as the “internal factors”: the capacity of damage, the proportion of accumulated energy and its distribution uniformity are all reflected by these internal factors. According to Eq. (5), the increase of either W0 or b will lead to the decrease of Di; it means both the increment of the damage capacity and the improvement of damage defusing and dispersion ability are beneficial for the enhancement of fatigue damage resistance. Conversely, the mechanical work Wi can be considered as the “external factor”, which is decided by the loading condition. With stable W0 and b, the enlargement of Wi can cause the increase of Di, which successively decreases the fatigue life Nf. In addition, the total mechanical work WT can be considered as a reflection of the comprehensive ability of fatigue damage resistance; the value of W0/WT indicates the proportion of effective damage. The amounts of statistical work on those parameters based on experimental results of pure Cu and Cu–Al alloys are listed in Table 4. To make it clearer, some significant tendencies are displayed in Figs. 13 and 14. Analysis on the internal and external factors mentioned above will be conducted separately as follows. The influence of internal factors can be typically indicated in the following two phenomena. Firstly, the capacity of fatigue damage is notably enhanced with increasing Al content, reflecting directly on the increased value of W0 (Fig. 13b), together with some indirect reflections, such as the increased saturation hysteresis energy WS (Fig. 13a) and the total accumulated energy WT (Fig. 13b). Secondly, a decreasing tendency of the damage accumulation rate (corresponding to the increase of b) emerges together with the increasing damage capacity (W0), leading to the decreasing proportion of effective fatigue damage, in comparison to the total mechanical energy. This tendency can be reflected by the decreased value of DS (Fig. 14a) and the ratio of W0/WT (Fig. 14b) with increasing Al content. As results of the synchronous improvement of W0 and b, these two trends demonstrate a comprehensive enhancement of the ELCF properties of Cu–Al alloys by increasing Al content. The effect of external factors can be similarly described through a general decreasing tendency of WT (Table 4 and Fig. 13b) and the increasing trend of W0/WT (Figs. 13b and 14b) with the increasing strain amplitude. This phenomenon can be explained by the higher shrinking rate of Nf compared to the enlarging rate of Wi. An indirect reflection is the increased value of damage degree DS (Fig. 14a), indicating that a larger proportion of mechanical work is transformed into accumulated damage with

Fig. 12. Comparison between fatigue results evaluated by the Coffin– Manson law (marked by “C–M”) and the hysteresis energy model (marked by “HE”).

(ii) By introducing hysteresis energy instead of stress or strain amplitudes to measure the ELCF damage, this model proposes a general form of criteria for fatigue damage evaluation, together with a more acceptable manner to deal with the changing stress amplitudes caused by cyclic hardening/softening; as a result, a more equitable evaluation can be achieved. (iii) Instead of putting emphasis on the external causative situations (such as unstable loading conditions) just as previous ELCF models usually do, this model focuses on the intrinsic damage behaviors, in an attempt to reflect the distinction of ELCF properties of copper alloys with various Al contents, for further understanding of the ELCF damage mechanisms in both the macro- and the microscale. A corresponding analysis on ELCF damage mechanisms will be conducted in the following sections. 4.3. Macro-scale analysis of ELCF damage mechanisms As mentioned in Section 4.1, from a macroscopic point of view, the fatigue damage process is a procedure of energy defusing and accumulation. For instance, in the ith cycle of fatigue test, mechanical work Wi is loaded to the material, part of the work is defused and the remaining part is accumulated in the material as fatigue damage Di (calculated by Eq. (5)). The damage accumulates with the increasing circles of cyclic loading, leading to a fracture when accumulated Table 4. Cyclic plastic work density statistics of pure Cu and Cu–Al alloys. Total strain amplitude De/2 (%)

Cyclic plastic work density (106 J/m3) Cu

2 4 6 8 9.5 U

Cu–5 at.% Al

Cu–8 at.% Al

Cu–16 at.% Al

WS

WT

W0/WT

WS

WT

W0/WT

WS

WT

W0/WT

WS

WT

W0/WT

16 39 66 95 – –

1467 1526 1063 871 – 79

0.38 0.37 0.52 0.64 – 7.06

18 48 78 113 – –

3086 3208 3439 3079 – 139

0.30 0.29 0.27 0.30 – 6.67

21 52 85 123 155 –

10534 6542 7805 6279 4982 172

0.13 0.20 0.17 0.21 0.27 7.68

16 42 79 128 157 –

24612 21336 13787 8080 6591 272

0.07 0.08 0.12 0.20 0.25 6.07

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Fig. 13. Statistical data of energy densities: (a) energy densities of single saturated cycle; (b) accumulated energy densities of total cycles. “U” corresponds to the value of static toughness.

Fig. 14. Typical relationships corresponding to energy densities: (a) relationship of Ds–De/2; (b) relationship of W0/WT–De/2.

the enlarged strain amplitude, which further causes the decreases of WT based on a constant W0. In addition, by checking the values of W0/U, we found that either for pure Cu or Cu–Al alloys, the values fall in the range of 6.07–7.68 (Fig. 13b). As there are only four groups of data from a single alloy system, further conclusions could hardly be drawn; however, this possible proportionality relationship between W0 and U indicates that a consistency may exist between the static toughness U and the “intrinsic fatigue toughness W0”: they are probably both the reflections of materials’ intrinsic toughness from different points of view. 4.4. Micro-scale analysis of ELCF damage mechanisms In contrast with the damage mechanisms in the macroscale directly reflected by the variation of the modeling parameters, the energy data and the mechanical properties, the micro-scale mechanisms of ELCF damage lie behind the behavior of microstructure evolution, leading to several typical performances such as cyclic hardening and fatigue cracking. To make it clear, the following analysis will be conducted respectively along clues of the two main parameters, and summarized in a group of diagrams as displayed in Fig. 15. To begin with, the enlarged damage capacity W0 originates from the increasing capacity of defects. This is caused by the cooperation of the following two mechanisms. In one aspect, transition of the fundamental deformation mechanisms from dislocation slipping to deformation twinning (Fig. 15a) leads to the transformation of the microstructure evolution mode, which is typically reflected in

the diminishing space among defects (Fig. 15b), and resulting in smaller microstructure size and more uniformly distributed defects (Fig. 15c); on the other aspect, the improved planarity of deformation mechanisms hindered the possibility of cross-slip (Fig. 15a), thus leading to a decrease of recovery rate (Fig. 15b). In this way, the potential of defect accumulation is improved; corresponding reflections are the enhanced cyclic hardening ability and deformation homogeneity (Fig. 15c). The increase of defect density observed in the microstructure agrees with the analysis, as introduced in Section 3.4. The enhanced damage defusing ability reflected by the increase of b can thus be considered as an outcome of enhanced deformation reversibility and homogeneity. On the one hand, the improved planarity of deformation mechanisms and correspondingly repressed cross-slip method (Fig. 15a) help to improve the deformation reversibility (Fig. 15b). Enhanced deformation reversibility may effectively diminish the fatigue damage during cyclic loading, leading to a decreased damage on the surface, and further delayed the cracking initiation (Fig. 15c). On the other hand, the tendency of deformation homogenization (Fig. 15b), which is caused by the cooperation of various plastic deformation mechanisms and improved hardening ability, leads to a more uniformly distributed fatigue damage (Fig. 15c), reflecting the dispersion of fatigue cracks as introduced in Section 3.3. In summary, for pure Cu and Cu–Al alloys in this study, the essential factor in the control of almost the whole microscopic damage behaviors should be the fundamental plastic deformation mechanisms. The change of deformation mechanisms influences the planarity of deformation,

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Fig. 15. Summary of the microscopic ELCF damage mechanisms: the changes from internal properties to external reflections with the change of Al content, and details corresponding to the increasing tendency of W0 and b.

the dynamic recovery rate, the deformation reversibility and homogeneity, and successively causes a variation in several behaviors, such as cyclic hardening, deformability, surface morphology and cracking mode, ending up with the different abilities of ELCF damage resistance (Fig. 15a–c). As the two significant aspects, the cooperation between the increasing ELCF damage capacity (reflected by W0) and the decreasing ELCF damage accumulation rate (reflected by b) can lead to a comprehensive improvement tendency of the ELCF properties (Fig. 15d). 4.5. Proposals for material selection and necessity of ELCF tests in seismic design According to the analysis in Sections 4.3 and 4.4, the ELCF properties can be evaluated mainly from two aspects: one is the capacity of damage; the other is the ability of damage defusing and dispersion. Thus, recommendations for seismic design and material selection can also be proposed in two corresponding aspects. The first pattern is to increase the materials’ capacity for ELCF damage (e.g. increase W0). Different from the ordinary fatigue loading conditions as HCF or common LCF, for ELCF, the extremely high strain amplitude and mechanical energy of each cycle result in a remarkable microstructure evolution process similar to the SPD procedure. Therefore, for materials designed to resist the ELCF damage, the potential for work-hardening and the deformability are more important than the initial strength or ductility. In addition, a possible relationship between the static toughness U and the “intrinsic fatigue toughness” W0 (see Fig. 13b) suggests the significance of an appropriate match of strength and plasticity. In short, besides meeting the necessary demand for strength, the design of materials for higher ELCF damage resistance should put emphasis on enlarging the cyclic hardening potential and improving the plastic deformability.

The second pattern is to improve the ability of ELCF damage defusing and dispersion (e.g. increase b). As analyzed above, the value of b is closely related to microscopic deformation: the improvement of planarity, reversibility and uniformity of plastic deformation all help to enlarge the value of b. For instance, the alloying procedure conducted in this study, which decreased the SFE of materials, can be considered as an effective method to decrease the fatigue damage degree. In addition, according to the above experimental results and corresponding analysis, the increase of strain amplitude hindered the defusing and dispersion of mechanical energy and enhanced the accumulation and localization of ELCF damage. As a consequence, the ELCF test with extremely high strain amplitudes displays higher requirements for the comprehensive properties (especially the deformability) of materials, making it an indispensable test for seismic design. 5. Conclusions Comprehensive studies focusing on the ELCF behaviors were conducted with pure Cu and single-phase Cu–Al alloys. Cyclic push–pull loading tests with extremely high strain amplitudes were carried out, following by the investigations on fatigue life variations, cyclic hardening behaviors, microstructure evolution mechanisms, crack distributions and fracture modes. A fatigue model evaluating the ELCF damage with a hysteresis energy-based criterion was then proposed, for the prediction of fatigue life and the comprehension of intrinsic damage mechanisms. Based on the experimental and modeling results, several conclusions can be drawn as follows. (1) With the increasing Al content, a synchronous improvement of cyclic saturation stress and fatigue life was achieved under constant strain amplitude.

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(2)

(3)

(4)

(5) (i)

Moreover, compared with the common LCF region, pure Cu and Cu–Al alloys displayed several unique features in the ELCF regime, including the deviations of fatigue life from the Coffin–Manson law; the nonnegligible proportion occupied by cyclic hardening stage of the whole fatigue life; special microstructures formed by cyclic loading containing deformation twins, shear bands and UFGs; and the transformation of fatigue cracking modes. All these characteristics originate from their plastic deformation behaviors, and indicate the existence of special interior fatigue damage mechanisms of ELCF. By introducing hysteresis energy instead of stress or strain amplitudes to measure the ELCF damage degree, the hysteresis energy model proposed a general form of criteria for fatigue damage evaluation, which can be transformed into the Basquin law or the Coffin–Manson law in special conditions. In addition, this model has advantages in terms of both the compatibility to handle various complicated unstable conditions such as cyclic hardening stage, and the equitable reflection of materials’ inherent quality on fatigue damage resistance: according to the results of this model, the enhancing extent of ELCF properties with increasing Al content is underestimated by the Coffin–Manson law. From a macroscopic point of view, the fatigue damage method can be considered as a procedure of energy defusing and accumulation. The ELCF behavior is a comprehensive reflection of both the internal ability of ELCF damage resistance and the external loading condition. For the intrinsic factors, both the increase of damage capacity (evaluated by W0) and the enhancement of energy defusing and dispersion ability (reflected by b) would lead to the improvement of the ELCF properties, by decreasing the value of Di. In Cu–Al alloys, the synchronous increasing trend of W0 and b with increasing Al content suggests the improvement of intrinsic resistance of ELCF damage. For the external factor, the increment of Wi caused by the increase of strain amplitude may lead to an accelerated damage accumulation method, together with an increased proportion of mechanical work transforming into effective damage (evaluated by W0/WT). The micro-scale damage mechanisms can be analyzed from the point of two internal factors W0 and b. On the one hand, with increasing Al content, transition of the fundamental deformation mechanisms leads to the smaller microstructure size, the more uniformly distributed defects and the decrease of recovery rate, which together increase the saturation defect density, thus enlarging the capacity of fatigue damage. On the other hand, with the increasing Al content, enhanced deformation reversibility and homogeneity may effectively diminish and disperse the effective fatigue damage, reflecting the decreased damage on the surface, the delay of cracking initiation and the more uniformly distributed fatigue cracks. Several suggestions for seismic design are listed as follows: As a proportional relationship between W0 and U is discovered, materials with higher static toughness are possible to obtain larger ELCF damage capacity; moreover, the potentials of work-hardening and

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deformability are probably more important factors that should be considered than the initial strength and ductility. (ii) The improvement of planarity, reversibility and uniformity of plastic deformation mechanisms can effectively help to enhance the ability of ELCF damage defusing and dispersion, which can be reflected by the increasing value of b. (iii) The ELCF test with extremely high strain amplitudes is an indispensable method for seismic design, as it displays higher requirements for the comprehensive properties (especially the deformability) of materials, in comparison to other commonly conducted mechanical tests. Acknowledgements This work was financially supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51101162, 51201165, 51331007 and the National Basic Research Program of China under Grant No. 2010CB631006.

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