On the reversibility of dislocation slip during cyclic deformation of Al alloys containing shear-resistant particles

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Acta Materialia 59 (2011) 3720–3736 www.elsevier.com/locate/actamat

On the reversibility of dislocation slip during cyclic deformation of Al alloys containing shear-resistant particles W.Z. Han a,b, A. Vinogradov c, C.R. Hutchinson a,⇑ a

ARC Centre of Excellence for Design in Light Metals, Department of Materials Engineering, Monash University, Clayton, 3800, VIC, Australia b Los Alamos National Laboratory, Los Alamos, NM 87545, USA c Department of Intelligent Materials Engineering, Osaka City University, Osaka 558-8585, Japan Received 25 January 2011; received in revised form 23 February 2011; accepted 5 March 2011

Abstract The cyclic deformation behavior of a model Al–4Cu–0.05Sn (wt.%) alloy containing a homogeneous and well-defined distribution of shear-resistant h0 (Al2Cu) precipitate plates was used to study the effect of precipitate state on the cyclic slip irreversibility. The precipitate spacing was controlled so that it was less than the self-trapping distance of dislocations. The cyclic deformation tests were conducted under constant plastic stain amplitude mode and the evolution of the cyclic stress and cyclic hardening rate with cumulative plastic strain were monitored. The deformed and undeformed microstructures were characterized using transmission electron microscopy. The cyclic deformation behavior and the corresponding dislocation structures depend on both precipitate state and imposed plastic strain amplitude. An expression for the cyclic slip irreversibility that explicitly depends on microstructural and deformation parameters was derived based on proposed mechanisms of interaction between the mobile dislocations and the precipitates. The cyclic deformation curve was calculated using the expression for the slip irreversibility and shown to describe most features of the cyclic deformation curves well, as a function of precipitate state and imposed plastic strain amplitude, as well as describing the results of plastic strain amplitude jump tests. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Cyclic deformation; Slip irreversibility; Hysteresis loop; Precipitation strengthening; Al alloys

1. Introduction The cyclic deformation of ductile metals and alloys involves the to-and-fro motion of dislocations. There is always a degree of irreversibility associated with this dislocation motion, whether they are dislocations in the bulk of a material or near the surface, and the accumulation of the irreversible component of the cyclic slip leads to the deformation structures that influence the initiation and propagation of damage known as fatigue [1,2]. A useful definition of the cyclic slip irreversibility, p, is that fraction of the plastic shear strain that is irreversible [1]; if slip is perfectly reversible, p = 0, and if ⇑ Corresponding author.

E-mail address: [email protected] (C.R. Hutchinson).

it is perfectly irreversible, p = 1. The irreversibility will be influenced by both loading conditions and the material itself, and Mughrabi [1] has highlighted a number of microstructural mechanisms that contribute to slip irreversibility: (i) cross-slip of screw dislocations, meaning reverse glide does not occur on the same slip plane; (ii) mutual annihilation of screw and/or edge dislocations, meaning there is no reverse glide of those dislocations; (iii) slip plane asymmetry in the case of bcc metals; and (iv) the random toand-fro motion of dislocations. A clear manifestation of the irreversibility of slip during cyclic deformation is the roughening of an initially smooth sample surface leading to the generation of surface intrusions and extrusions [3–14]. This has been a topic of interest for more than 100 years since the early work of Ewing and Humfrey [15]. It is now known that, in many cases, fatigue crack initiation occurs at the sample surfaces at

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.03.007

W.Z. Han et al. / Acta Materialia 59 (2011) 3720–3736

sites of intrusions and/or extrusions and, as a result, the development of these surface features has received significant attention. An understanding of the cyclic slip irreversibility (and its dependence on cyclic deformation conditions and microstructure) is central to an understanding of the development of the surface deformation structures and subsequent fatigue crack initiation, and this has been emphasized by a number of authors [1,2,16–19]. Indeed, some models for fatigue crack initiation associate this with a critical value of the accumulated irreversible cyclic strain. However, slip irreversibility is not only important for fatigue crack nucleation; it is also expected to influence fatigue crack propagation [19,20]. If dislocation motion in the plastic zone ahead of a fatigue crack were perfectly reversible, then the crack would not propagate when subjected to cyclic loading. Of course this is not true, and illustrates the central role of the slip irreversibility in fatigue. Unfortunately, the irreversibility of cyclic slip is currently understood only qualitatively, and experimentally determined quantitative estimates have been reported in only a few special instances [1]. For example, it is expected that the degree of slip irreversibility will decrease as the plastic strain amplitude of loading decreases or as the planarity of slip increases. These expectations are borne out by elegant experiments where the surface deformation structures were measured using atomic force microscopy or scanning electron microscopy (slip band spacing, slip height, etc.) and using simplified descriptions of the dislocation behavior in the material, a numerical estimate of the cyclic slip irreversibility can be obtained [16–18]. Other researchers have developed methods to measure slip irreversibility based on the shear steps on second phase particles formed during cyclic deformation [21,22]. The results of these studies have been summarized by Mughrabi [1] and Risbet and Feaugas [12]. The irreversibilities vary over many orders of magnitude. Many of the experiments were performed on single crystals or single-phase materials, and one difficulty is that the dislocation motion and interactions in these materials are very complex and not well understood. Indeed, this is an active area of research in its own right [23,24]. To interpret the experimentally measured surface deformations in terms of cyclic slip irreversibility, models for the dislocation motion must be invoked and therefore the accuracy of the estimated slip irreversibility is contingent upon the dislocation model, in some cases, for example, describing dislocation motion within a persistent slip band [3–5]. An alternative approach that alleviates this difficulty and contributes both experimentally and theoretically to our understanding of cyclic slip irreversibility in ductile metals would be most welcome. This is the objective of this contribution. Although the surface intrusions and extrusions that result from cyclic deformation are probably the most obvious consequences of cyclic slip irreversibility, irreversible to-and-fro dislocation motion occurs throughout the bulk

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of a cyclically deformed material and the overall mechanical response (e.g. the hysteresis loop shape) also contains information about the slip irreversibility. Extraction of this information also requires the use of a dislocation model, but the opportunity exists to use specially designed and well-controlled microstructures for which the dislocation behavior is comparatively well understood. In this study we make use of a model Al–Cu alloy containing a uniform distribution of shear-resistant h0 (Al2Cu) precipitates. This system has recently been used to study and model in detail the monotonic hardening response (both kinematic and isotropic contributions) as a function of precipitate state over a large range of precipitate sizes and number densities (interparticle spacings) [25,26]. The precipitate state can be controlled so that the mean interparticle spacing is less than the self-trapping distance of dislocations. The result is that the dominant interaction for dislocations is the dislocation–precipitate interaction and not dislocation–dislocation interactions in the matrix [26]. The advantage is that the length scale of the precipitate–dislocation interaction can be controlled through suitable heat treatments. This removes an important unknown in the dislocation models and simplifies the extraction of quantitative measures of the cyclic slip irreversibility. Undoubtedly, such a multi-phase alloy is more complicated than single crystal Cu or other single-phase materials that have previously been used to extract numerical estimates of the cyclic slip irreversibility. However, as will be seen, by choosing a slightly more complicated material, with a simpler deformation structure and a better understood plastic response, the problem of studying the irreversibility of slip is made easier. Precipitation-hardened Al alloys also offer an additional advantage: the response of precipitation-hardened Al alloys to cyclic deformation has previously been studied (e.g. [27–36]) and therefore a benchmark for comparison of the general qualitative cyclic hardening behavior exists. These previous studies centered primarily on attempts to understand the origin of the cyclic softening exhibited by some Al alloys and the relationship between the softening and the effects of the cyclic deformation on the precipitate state (e.g. dissolution and/or destruction of shearable precipitates [37,38]). This is a particularly important problem since the fatigue properties of Al alloys are generally quite poor compared with other ductile alloys, despite their use in many fatigue critical applications such as aerospace [39]. Engineering alloys generally exhibit a fatigue ratio (fatigue strength/ultimate tensile strength) of 0.5 for 108 cycles of deformation. Precipitation-hardened Al alloys typically exhibit fatigue ratios of 0.3. There are therefore both scientific and technological incentives for the choice of a precipitation strengthened Al alloy for this study. 2. Experimental design and procedures An alloy of composition Al–4Cu–0.05Sn (wt.%) was cast in a steel mold from high-purity elemental components

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and, after homogenization at 500 °C for several days, the ingot was cut into several plates that were then hot-rolled to 4 mm thick sheet from which fatigue samples (16 mm gauge length and 4 mm  5 mm cross section area) were spark machined. All samples were solution treated for 1 h at 520 °C in a salt bath, water quenched to room temperature and then isothermally aged in an oil bath at 200 °C for a variety of times, ranging from 10 min to 30 days, to obtain different states of precipitation. Sn is added to the alloy as a microalloying addition because of its desirable effects in catalyzing the nucleation of exclusively shearresistant h0 precipitates (Al2Cu) with a reasonably narrow size distribution [25]. For an aging temperature of 200 °C, only plate-shaped h0 (Al2Cu) precipitates are formed for times of at least 30 days [25,26]. This provides a relatively simple, homogeneous, well-defined and controllable microstructure for the study the effect of precipitate state on the cyclic slip irreversibility. In order to study the cyclic deformation behavior, four precipitate states (aging states) were selected based on their monotonic mechanical properties (Fig. 1): 30 min (under aged), 1.5 h (peak aged), 8 h (over aged) and 4 days (over aged). The 30 min and 8 h samples have similar yield strengths and ultimate tensile strengths (UTS), but different uniform elongations. The peak-aged alloy (1.5 h) and overaged alloy (8 h and 4 days) have similar uniform elongations but different yield strengths and UTS. Cyclic deformation tests were performed in constant plastic strain amplitude mode with five different plastic strain amplitudes: Depl/2 = 1  104, 5  104, 1  103, 2  103 and 3  103. The experiments were conducted on an MTS-858 table-top system at room temperature with a frequency of 0.2 Hz. Samples surfaces were polished to a 2400# grit finish before fatigue testing. The precipitation and deformation states were assessed quantitatively by transmission electron microscopy (TEM) using a JEOL-2011 microscope operating at

200 kV. Foils were prepared using standard electropolishing techniques for Al alloys. For each aging duration, the average h0 precipitate plate length (L), thickness (h) and number density (N) was measured from micrographs taken with the electron beam oriented parallel to the [0 0 1]Al zone axis of the Al matrix. Values of L and h were averages of more than 100 individual measurements. The TEM foil thickness was measured using convergent electron beam diffraction (CBED) [40] and is necessary for estimates of N. The dislocation structures in the fatigued samples were examined using TEM. 3. Results 3.1. Precipitation microstructure The precipitate state was characterized using TEM for aging times of 30 min, 1.5 h, 8 h and 4 days at 200 °C. Bright-field (BF) TEM images are shown in Fig. 2 for the four aging conditions prior to deformation. For all the conditions examined, only plate-shaped h0 precipitates are observed. Their size is relatively uniform and their distribution is homogeneous. Two {0 0 1}a variants of h0 can be clearly seen edge-on, whilst the face-on variant is less visible due to the TEM foil thickness. Qualitatively, the observed microstructure evolution with aging time is consistent with previous investigations on Al–4Cu [41,42] and Al–Cu with ternary microalloying additions [25,26,43]. Measurements of the time evolution of the mean precipitate length L, thickness h, number density N and calculations of the volume fraction Vf are listed in Table 1. The number density of particles first increases, passes through a maximum at the peak-aged state (1.5 h) and then decreases due to coarsening of the distribution. The precipitation process has been previously studied and the precipitate number densities, lengths and thicknesses are consistent with previous measurements

Fig. 1. (a) The variation in yield stress of an Al–4Cu–0.05Sn alloy with aging time at 200 °C; (b) the variation of the UTS and uniform elongation of Al– 4Cu–0.05Sn alloy with aging time. At this aging temperature, in this alloy only h0 precipitates are formed. The four aging conditions used in this study are underlined.

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Fig. 2. BF-TEM micrographs showing the microstructure of the Al–4Cu–0.05Sn alloy after aging at 200 °C for (a) 30 min (under aged), (b) 1.5 h (peak aged), (c) 8 h (over aged) and (d) 4 days (over aged). The electron beam is close to [0 0 1]Al.

Table 1 h0 precipitate parameters as a function of aging time. Aging time

N (1021 m3)

NCalc (1021 m3)

L (nm)

h (nm)

Volume fraction (%)

a  qm (1013) (m2)

30 min 1.5 h 8h 4 days

2.7 ± 0.81 4.5 ± 1.5 1.6 ± 0.48 0.99 ± 0.3

3.3 3.0 1.8 0.75

49 ± 10 56 ± 20 67 ± 20 76 ± 30

2.3 2.3 2.74 5.6

1.18 ± 1.13 2.56 ± 2.1 1.38 ± 1.58 2.51 ± 2.7

2.5 2.5 2.5 2.5

Measurements of TEM foil thicknesses using CBED are subject to a relative uncertainty of ±30%. This uncertainty strongly affects the precision with which the number densities (N) and volume fractions (Vf) can be reported. The error associated with measurements of the length and thickness of precipitate plates is much less and errors are listed based on the widths of the particle size distributions. Vf is estimated as NphL2/4 for under-aged conditions. However, for peak-aged and over-aged conditions, the particle size distribution has been taken into account. NCalc is the precipitate number density used in the model calculations and can be seen to lie within the error range of the experimental measurements. a  qm represents the values of the parameter used in the model simulations.

Fig. 3. Experimentally acquired hysteresis loops corresponding to the cyclically stable stage for the Al–4Cu–0.05Sn alloy: (a) various aging states at Depl/ 2 = 1  103 and (b) the peak-aged state (1.5 h) under various imposed plastic strain amplitudes.

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and calculations [25]. A significant range of L, h and Vf are obtained, providing a suitable experimental data set for a study of the effect of precipitate state on the cyclic deformation behavior. 3.2. Cyclic deformation behavior The cyclic deformation behavior of the precipitatestrengthened materials under various constant plastic strain amplitudes was measured. Fig. 3a shows the comparison of the hysteresis loops for the four aging conditions at the cyclically stable stage for Depl/2 = 1  103. Fig. 3b is an example of the influence of the plastic strain amplitude on the cyclic deformation for the peak-aged (1.5 h) alloy. It is clear from Fig. 3 that the shape of hysteresis loop depends strongly on both the precipitate state and the applied plastic strain amplitude. In general, the alloy with higher monotonic strength demonstrates higher cyclic stresses, and the higher the plastic strain amplitude, the larger the hysteresis loop area. The cyclic tensile peak stress evolution with cumulative plastic strain (ecum pl ¼ 4N Depl =2; where N is the number of cycles) for each aging condition under various plastic strain

amplitudes is shown in Fig. 4. It can be seen that the peak stress vs. cumulative plastic strain curves change significantly with aging condition. In general, the higher the monotonic yield stress (Fig. 1), the higher the peak stress in cyclic loading. The cyclic peak stresses of the peak-aged state (1.5 h) are much larger than the 4 day state, whereas the peak stresses of 30 min and 8 h samples are located in between. Each of the testing conditions demonstrates a relatively stable cyclic stress–strain curve, and in some cases slight cyclic softening is observed (Fig. 4). These observations are consistent with previous studies of the cyclic deformation of alloys containing h0 precipitates [27,29,31,44]. In addition, the plastic strain amplitude has a significant influence on the stress evolution with cumulative plastic strain. Generally, the larger plastic strain amplitudes correspond to higher cyclic peak stresses. The hysteresis loops shown in Fig. 3 vary significantly with precipitate state and applied plastic strain amplitude. In order to quantitatively compare the cyclic deformation behavior, we plotted the hardening rate ðdr=deÞ at the end of each cycle. This is denoted hH, and is plotted as a function of cumulative plastic strain in Fig. 5 for each of the four precipitate states considered. hH is the lower limit

Fig. 4. Evolution of the cyclic peak stress with cumulative plastic strain for the Al–4Cu–0.05Sn alloy: (a) under aged for 30 min, (b) peak aged for 1.5 h, (c) over aged for 8 h and (d) over aged for 4 days.

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of the hardening rate during each cycle of cyclic deformation. As shown in Fig. 5, hH tends to a stable value after a few cycles of deformation, which corresponds to the stable shape of the hysteresis loop. The tests with the higher plastic strain amplitudes always exhibit a lower hH. Of the four different microstructural states investigated, the peak-aged state has the largest hH, and the 4 day over-aged state has the lowest value. The magnitude of hH, which corresponds with microyielding, varies from 10 GPa to 60 GPa, and this is much higher than the maximum macroyielding hardening rates exhibited in monotonic deformation (3.5 GPa) [26]. 3.3. Fatigue deformation microstructures The microstructures of the Al–4Cu alloy after fatigue deformation were examined using TEM as a function of applied plastic strain amplitude. BF-TEM micrographs for two representative aging conditions (1.5 h and 4 days) are displayed in Fig. 6. At small plastic strain amplitudes almost no dislocations were found either on the shear-resistant h0 particles or in the matrix region between the h0

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precipitates for the 1.5 h state (Fig. 6a). At the larger plastic strain amplitudes, the only dislocations observed are Orowan loops surrounding the h0 particles. The dark, lobe-shaped contrast adjacent to the edge-on variants of h0 in Fig. 6b is due to the Orowan loops. Representative BF-TEM micrographs for the over-aged state at the same two plastic strain amplitudes are shown in Fig. 6c and d. At the smaller plastic strain amplitude some Orowan loops are observed but clearly with a lower density than in the sample subjected to higher plastic strain amplitudes (Fig. 6d). Diffraction-contrast TEM was used to confirm that the linear contrast on the precipitate plate faces is indeed due to Orowan loops and not precipitate ledges, Moire fringes, etc. The dislocation structures formed after cyclic deformation are very different to those observed after monotonic deformation. The dislocation structures formed after monotonic loading include not only Orowan loops but also plastic relaxation dislocations in the matrix surrounding the plates [26]. Such plastic relaxation dislocations are not clearly observed during the cyclic deformation performed in this study. For a given level of applied plastic strain amplitude, the precipitate state affects

Fig. 5. Evolution of the cyclic hardening rate with cumulative plastic strain for various imposed plastic strain amplitudes: (a) under aged for 30 min, (b) peak aged for 1.5 h, (c) over aged for 8 h and (d) over aged for 4 days. Solid symbols represent experimental data and open symbols represent model calculations.

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Fig. 6. (a, b) BF-TEM images of Al–4Cu–0.05Sn alloy aged for 1.5 h and cyclically loaded to failure at Depl/2 = 1  104 and Depl/2 = 1  103 respectively; (c, d) BF-TEM images of Al–4Cu–0.05Sn alloy aged for 4 days and cyclically loaded to failure at Depl/2 = 1  104 and Depl/2 = 1  103 respectively. The electron beam is close to [0 0 1]Al.

the dislocation structures formed; the over-aged alloy (4 days) appears to have more Orowan loops than the peak-aged state (1.5 h) at Depl/2 = 1  103. 4. Discussion and modeling The microstructural and mechanical characterization results outlined in Section 3 illustrate that both the precipitate state and the plastic strain amplitude have a significant effect on the cyclic deformation behavior and the dislocation structures that are formed. Two experimental features are especially interesting: (i) the hardening rates in cyclic loading (measured from the hysteresis loop 10– 60 GPa) are very high compared to the maximum values observed in monotonic deformation (3.5 GPa) [26]; and (ii) the number of dislocation loops that are formed appear fewer compared with monotonic loading, even though the cyclic deformation leads to cumulative plastic strains of >1. In the following section, we propose a quantitative model to address these questions and to describe the cyclic deformation behavior using the concept of cyclic slip irreversibility. 4.1. Mixture of elastic and plastic deformation in cyclic loading The magnitude of cyclic hardening rates for all aging states tested are high (10–60 GPa) (Fig. 5). With increasing plastic strain amplitude, the cyclic hardening rates decrease

significantly but still maintain a relatively high level (10 GPa) compared with the maximum values in monotonic loading in this system. It is clear that a new ingredient is required to extend existing models for monotonic hardening in this system [26] to the case of cyclic deformation. Consideration of the microscopic processes that occur in an alloy containing shear-resistant precipitates helps us identify a mechanism that may account quantitatively for the observed hardening rates. Fig. 7a illustrates schematically a number of plate-shaped precipitates. During the initial forward straining the dislocation traverses from the bottom of Fig. 7a to the top, leaving behind Orowan loops on the precipitates. The stress required to do this is the bypassing stress plus any contribution from the solid solution friction stress plus a backstress that arises from the elastic loading of the precipitates by the Orowan loops. This process has been modeled quantitatively for this system by Teixeira et al. [26]. Now consider the reverse motion of the same dislocation as shown in Fig. 7b. As the dislocation moves from the top of Fig. 7b to the bottom, there is a chance that it will remove some of the dislocation loops that it deposited during its forward motion. The probability that the backtracking dislocation will remove an Orowan loop depends on the mobile dislocation and Orowan loop remaining in the same slip plane and not undergoing other dislocation reactions. If a loop is removed during the reverse motion, the backtracking dislocation does not “see” the precipitate since it does not need to bow around it. From the point of view of the retreating

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Fig. 7. Schematic representation of the interaction between a dislocation gliding on the slip plane {1 1 1}a and three h0 precipitates lying on {1 0 0}a planes, resulting in the formation of Orowan loops around the precipitates and the process of the loops being removed by the retreating dislocation: (a) Orowan loops are deposited on the precipitates during forward straining; (b) the Orowan loop on precipitate B is removed during the reverse straining leading to an increase in the “effective” precipitate spacing, k, and decreased precipitate strengthening.

dislocation, it is as if that precipitate was not there. This gives an effective decrease in the number density of precipitates from the point of view of the retreating dislocation. The removal of the dislocation loop also decreases the backstress that aids reverse flow but, as will be shown, in most cases this is a minor effect compared with the effect of the decrease in “effective” precipitate number density. This “loop removal” process is a microstructural manifestation of the reversibility of slip, and has been observed in dislocation dynamics simulations [45] and qualitatively discussed previously both in studies of fatigue [44] and in considerations of the shape of the unloading stress–strain curve during Bauschinger tests of particle containing alloys [46,47]. To develop a quantitative description of the slip irreversibility, we consider the following two features that we expect to influence the probability of a retreating dislocation removing a previously deposited Orowan loop:  the probability of a retreating dislocation removing an Orowan loop should decrease with further distances that the mobile dislocation traverses after depositing the loop, since the greater distances provide more opportunities for cross-slip events of the mobile dislocation and/ or more time for dislocation reactions or climb of the Orowan loops.  the probability of a retreating dislocation removing an Orowan loop should increase with the number of loops on the plates because the probability of the mobile dislocation being on the same slip plane as an Orowan loop is greater. The dependence of the dislocation reversibility on the distance the dislocation has traversed (strain) is a contribution to the strain-hardening rate since it gives a strain dependence to the instantaneous number density of precipitates “seen” by the mobile dislocation and hence the bypassing stress.

4.2. Slip irreversibility The key feature in the dislocation–precipitate interaction outlined in Fig. 7 is the probability that a mobile dislocation can remove one or more Orowan loops during reverse straining. This is a reflection of the slip irreversibility. For the shear-resistant precipitate-strengthened Al– 4Cu alloy, the magnitude of precipitation strengthening due to Orowan bowing during cyclic deformation is a function of the slip irreversibility. Consider the schematic illustration in Fig. 8 of a dislocation interacting with an array of precipitate plates. The dislocation begins at the righthand side of Fig. 8 and during its motion from right to left leaves behind a number of Orowan loops on the plates. During this forward motion there is a certain probability that cross-slip processes will occur, or Orowan loops on plates may climb, so that all of the Orowan loops are not expected to lie on exactly the same slip plane. Indeed, because of the effect of precipitates on dynamic recovery during monotonic loading [26,48], it would be expected that the greater the number of dislocation–precipitate interactions, the greater the chance of cross-slip or climb.

Fig. 8. Schematic illustration of the expected evolution of the slip irreversibility with forward plastic strain. Slip irreversibility is a function of the ratio between slip distance of the mobile dislocation and the average precipitate spacing.

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When the direction of straining is reversed and the dislocation begins moving from its position on the left of Fig. 8 to the right, the probability it will remove an Orowan loop is greatest for the plates closest to the left since they contain the most recently deposited loops and therefore are most likely to lie on the same slip plane as the mobile dislocation. It would be expected that the further plates are to the right in Fig. 8 the lower the probability that the dislocation can remove the loops deposited during the initial motion from right to left. The expected dependence of the slip irreversibility on the dislocation slip distance is plotted schematically in Fig. 8. The magnitude of the slip irreversibility in this case should be a function of the ratio between the average slip distance of a dislocation during a plastic strain increment, dx, and the average precipitate spacing, k. If the dislocation traverses many precipitates during a plastic strain increment, we would expect that the chances of that dislocation staying on exactly the same slip plane are relatively small. If the dislocation only traverses one precipitate during a plastic strain increment, the chances of it remaining on the same slip plane are much greater. If we assume the interaction events leading to a change in slip plane have a Poisson’s distribution, then the slip irreversibility can be expressed as:   dx p ¼ 1  exp  ð1Þ k p will be equal to 1 when dx  k and equal to 0 when dx  k. The average slip distance of a dislocation can be expressed using Orowan’s equation, dx ¼ epl =qm b, where epl is the applied plastic strain, qm is the mobile dislocation density and b is the magnitude of Burgers vector. The average precipitate spacing can be expressed as 1=2 k ¼ ðNL sin hÞ [26], where h is the dihedral angle between the plate and {1 1 1}a slip plane. For {0 0 1}a precipitate plates, h = 54.74°. Substitution for dx and k in Eq. (1), we obtain:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! epl NL sin h pðepl Þ ¼ 1  exp  qm b

ð2Þ

Eq. (2) suggests that the slip irreversibility will continuously increase with plastic strain for a given microstructure in the manner shown in Fig. 9. Qualitatively, this corresponds to the decreasing probability of a retreating dislocation removing an Orowan loop with further distances that the dislocation had traversed. However, we also expect that the probability of removing a dislocation loop should increase with the number of Orowan loops on the plates. The number of Orowan loops on the plates depends on the applied plastic strain amplitude. Indeed, when comparing the experimental hysteresis loops tested at different applied plastic strain amplitudes, the stress–strain behavior has an obvious plastic strain amplitude dependence, as shown in Fig. 10. With larger plastic strain amplitudes, the strain-hardening rate at a given plastic strain is lower. This is a reflection of the effect of the plastic strain amplitude on the irreversibility of slip. The key microstructural difference between the various applied plastic strain amplitude tests shown in Fig. 10 is the average number of Orowan loops, nOL, deposited on the plates during straining. For a given precipitate state, the number of loops deposited on a plate depends on the OL plastic strain increment: dn / Lb [26,49,50]. If there are depl more loops on the plate (at larger plastic strain amplitudes), we expect that the chances of a retreating dislocation removing a loop are greater, ceteris paribus. In order to quantitatively test this idea, we performed strain amplitude jump cyclic deformation tests on both peak-aged (1.5 h) and over-aged (4 days) samples. The strain amplitude jump tests were performed jumping from both high strain amplitude to low strain amplitude and low strain amplitude to high strain amplitude. The variation of the cyclic hardening rate hH during the tests are shown in Fig. 11. The experimental results show the same features for both the 1.5 h and 4 days samples. When the strain

Fig. 9. (a) Plot of the slip irreversibility evolving with plastic strain for peak-aged (1.5 h) Al–4Cu alloy at various imposed plastic strain amplitudes according to the proposed model; (b) the influence of the precipitate state on the slip irreversibility at Depl/2 = 1  103 based on the proposed model.

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Fig. 10. The comparison of the upper part of the experimentally measured hysteresis loop at various applied plastic strain amplitudes for the peakaged Al–4Cu–0.05Sn alloy. The five upper parts of the hysteresis loops acquired from experiments have been parallel shifted to zero for comparison of the hardening rates.

amplitude jump tests are performed from low amplitude to high amplitude (Fig. 11a (1.5 h) and c (4 days)), the cyclic

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hardening rates hH change directly from a value characteristic of the low strain amplitude tests to one characteristic of the high strain amplitude tests essentially instantaneously at the strain amplitude jump. However, when the strain amplitude jump tests are performed from high amplitude to low amplitude (Fig. 11b (1.5 h) and d (4 days)), a transient in the cyclic hardening rate hH is observed at the strain amplitude jump. This is because the larger number of Orowan loops present during the initial cyclic loading at the higher strain amplitude Depl/ 2 = 3  103 influences the slip irreversibility of the initial few cycles at the lower plastic strain amplitude Depl/ 2 = 5  104. After a few cycles of deformation after the strain amplitude jump the cyclic hardening rate gradually increases and finally reaches the level characteristic of the lower strain amplitude tests with Depl/2 = 5  104 (Fig. 11b and d). These strain amplitude jump tests are a direct test of the effect of existing Orowan loops on the reversibility of slip. One way to introduce the effect of existing Orowan loops on the irreversibility of slip is through a modification to the “effective” precipitate spacing, k, since the major effect of the additional loops is to increase the chance of

Fig. 11. Plot of the evolution of the strain-hardening rate during strain amplitude jump fatigue tests designed to study the effect of existing Orowan loops on the reversibility of slip: (a, b) the strain amplitude jump test for peak-aged (1.5 h) alloy; (c, d) the strain amplitude jump test for over aged for 4 days.

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a loop being removed by a mobile dislocation and hence rendering that precipitate invisible to the retreating dislocation. In this case, the average precipitate spacing can be modified to include the plastic strain amplitude dependence in the following way: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!   dx epl NL sin h pðepl Þ ¼ 1  exp  1  exp  bðDepl Þk bðDepl Þqm b ð3Þ where b(Depl) represents the effect of existing Orowan loops on the average “effective” precipitate spacing and is a plastic-strain-amplitude-dependent parameter. The chances of taking away a dislocation loop during strain reversal should scale with the average number of newly formed Orowan loops on the plate. For simplicity, in this first attempt, we assume the scaling is linear: bðDepl Þ ¼ anOL ¼ a

Depl ML sin h b

ð4Þ

where a is a dimensionless constant greater than 1 and M is the Taylor factor. Substituting Eq. (4) into Eq. (3), we obtain for the irreversibility of slip: ! pffiffiffiffi epl N pffiffiffiffiffiffiffiffiffiffiffiffi pðepl Þ ¼ 1  exp  ð5Þ aqm MDepl LSinh The slip irreversibility expressed by Eq. (5), is a function of microstructural parameters (N and L) and deformation conditions (plastic strain and plastic strain amplitude). Fig. 9a shows the behavior of Eq. (5) as a function of forward plastic strain for the peak-aged (1.5 h) state under various plastic strain amplitudes. It can be seen that the proposed expression for the slip irreversibility exhibits the expected behavior. For lower imposed plastic strain amplitudes, the irreversibility increases much faster due to fewer Orowan loops on the precipitates. Fig. 9b displays the influence of precipitate state on the slip irreversibility. It is expected that the increase in slip irreversibility will be much lower for the over-aged condition, due to the larger precipitate size and spacing and, on average, greater number of Orowan loops deposited on each precipitate during cyclic deformation. 4.3. Cyclic deformation behavior Several researchers have attempted to develop physically based models to describe the cyclic deformation behavior of metallic materials [51–54]. Estrin et al. [51] have extended the Kocks model to describe the deformation behavior under cyclic loading. In their model they used an idea of Kelly and Gillis [52] that during each stress reversal, a fraction of the dislocation density that was immobilized during a deformation half-cycle is considered to be re-mobilized. This idea has its origins in the dislocation slip reversibility that occurs during cyclic deformation. A description of the slip reversibility during cyclic deforma-

tion, and its dependence on microstructural parameters, is the key feature required for a physically based model of the cyclic deformation processes of metals. Based on the analysis of the dislocation–precipitate interactions outlined above, the stress and strain evolution during a cycle of deformation can be described by: rðepl Þ ¼ r0 þ rss þ hriðepl Þ þ Drppt ½pðepl ; Depl Þ et ðepl Þ ¼

r0 þ rss Dr½pðepl ; Depl Þ þ epl þ E E

ð6aÞ ð6bÞ

where r0 is the intrinsic lattice strength, rss is the solid solution strengthening, Drppt is the precipitate strengthening (and is a function of plastic strain and plastic strain amplitude due to the slip irreversibility), hri is the internal stress due to the formation of Orowan loops (also a function of plastic strain) and E is the elastic modulus of the Al–4Cu alloy, 70 GPa. In our cyclic deformation tests, we employed plastic control mode and therefore know the amount of plastic strain. The isotropic hardening contribution [26] was not included in this expression because its effect can be neglected for the relatively small plastic strain amplitudes employed in this study. 4.3.1. Precipitation hardening in cyclic deformation The precipitation strengthening is calculated using the refined version of Orowan’s equation [55]:   Mlb 1 ri Dappt ¼ pffiffiffiffiffiffiffiffiffiffiffi Ln ð7Þ ro 2p 1  m kave ðep Þ where v is Poisson’s ratio, kave is the average precipitate spacing, and ri and ro are the inner and outer cut-off radii of the dislocation, respectively. Nie and Muddle [56,57] considered the statistical evaluation of kave assuming the precipitates form as circular disks and arrived at the following expression: sffiffiffiffiffiffi pffiffiffiffi pffiffiffi 3 3 h 1 pL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð8Þ kave ¼ 2 2 8 Sinh Eff Sinh  N ðe Þ  L p

In cyclic deformation, the precipitate contribution to the strengthening is influenced by the slip irreversibility, p, through the effect of the irreversibility on the “effective” number density of precipitates seen by the mobile dislocations, NEff. The “effective” number density of precipitates during cyclic deformation can be expressed as: N Eff ¼ pðepl ; Depl ÞN

ð9Þ

4.3.2. Kinematic hardening The contribution of kinematic strengthening due to the internal stress from Orowan loops can be calculated using the approach of Brown and Clarke [58]: hriðep Þ ¼ M 2 gDlV f ep

ð10Þ

where l is the matrix shear modulus, g is the accommodation factor and D is a modulus correction factor accounting

W.Z. Han et al. / Acta Materialia 59 (2011) 3720–3736 Table 2 The values of parameters entering the kinematic hardening model, the lattice friction stress and the solid solution strengthening. Accommodation factor Modulus correction factor Shear modulus of the matrix (GPa) Lattice friction stress (MPa) Solid solution strengthening (MPa)

g D l r0 rss

0.625 1.25 25.4 28 20

The solid solution strengthening should have small differences for the different precipitate states but, since this is a very small effect compared with the contribution to precipitate hardening, an average value was used for all aging conditions.

for elastic inhomogeneity [26,58]. The values of various parameters used in calculation of the internal stress are listed in Table 2 and are taken from Ref. [26]. The calculations show that the maximum internal stress is 15 MPa, due to the relatively small maximum plastic strain.

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4.4. Application of the model Using Eqs. (5), (6) and (8)–(10), the cyclic deformation behavior of each of the four precipitates states considered (under aged (30 min), peak aged (1.5 h) and over aged (8 h and 4 days)) under the five different imposed plastic strain amplitudes have been calculated. All of the parameters entering Eqs. (5), (6) and (8)–(10) have a clear and transparent physical meaning. Most are immediately fixed through experimental measurement (e.g. N, L) or previous studies (e.g. [26]), or are microstructural independent parameters (e.g. E, v). The only unknown parameters required for the calculation are a (the dimensionless constant representing the augmenting effect of an additional Orowan loop on a plate on the probability that a loop will be removed by a retreating dislocation) and qm (the mobile dislocation density). It is likely that a will be accessible by suitable discrete dislocation dynamic simulations in the

Fig. 12. Comparison of the simulated hysteresis loop for various imposed plastic strain amplitudes based on the proposed model with the experiment results for the under-aged alloy (30 min).

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Fig. 13. Comparison of the simulated hysteresis loop for various imposed plastic strain amplitudes based on the proposed model with the experiment results for the peak-aged alloy (1.5 h).

future. For the purposes of calculation, a  qm has been used as a fitting parameter bearing in mind that its value is expected to lie somewhere in the range 1010–1015, depending on the value of a. A single value of a  qm has been chosen for all precipitate states and all applied plastic strain amplitudes. The values of the microstructural parameters used for the calculations are listed in Table 1. The calculated hysteresis loops were compared with the experimental loops acquired at the first loop (i.e. monotonic loading) and at the cyclic stable stage. These are shown in Figs. 12–15 for each of the plastic strain amplitudes and precipitate states. It can be seen that the proposed model can quantitatively describe most features of the hysteresis loops acquired from experiment, including the higher initial tension yielding compared with the yielding stress during the cyclically stable stage for all precipitate states and plastic strain amplitudes. Based on the calculated hysteresis loops, the simulated cyclic hardening rate, hH, is compared with the experimental values in

Fig. 5. Quantitatively reasonable agreement is also observed. As a further test of the proposed expression for the irreversibility of slip, simulations of the plastic strain amplitude jump tests have been performed. The peak-aged precipitate state is used as an example and comparisons of the simulated and experimental curves are shown in Fig. 16. Again the quantitative agreement is very reasonable, especially for the case of jumping from the larger plastic strain amplitude to the lower strain amplitude (Fig. 16b). The experimental hysteresis loop obtained when jumping from the lower plastic strain amplitude to the larger strain amplitude (Fig. 16a) shows points of inflections in the curve that are absent from the simulations. Such inflections have previously been observed in systems containing shear-resistant particles [44,59–61] subject to large plastic strain amplitude deformation. Asaro [60] has discussed the origin of such inflections in terms of the evolution of the contribution to kinematic hardening, which in

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Fig. 14. Comparison of the simulated hysteresis loop for various imposed plastic strain amplitudes based on the proposed model with the experiment results for the over-aged alloy (8 h).

this case has its origins in the elastic loading of the h0 (Al2Cu) particles. Whilst such inflections disappear from the experimental hysteresis loops after a few cycles of deformation, the lack of inflections in the simulated curves after a strain amplitude jump (Fig. 16a) suggests that the strain dependence of the accumulation of dislocation loops during larger strain amplitude excursions following cyclic stability at lower strain amplitudes requires further investigation. The present derivation of the effect of precipitate state on the irreversibility of slip represents a first attempt and no doubt there is much scope for refinement, enhancement and modification. Nevertheless, this relatively simple approach appears to capture the dominant features influencing the reversibility of slip under the conditions experimentally examined. Other researchers have estimated the irreversibility of slip based on measures of the surface topology [16–18] or slip steps on the surfaces of shearable particles [21,22]. In all such cases the irreversibility is reported as a single number for the deformation conditions

considered. These are “averaged” irreversibilities. Direct comparison of experimental measurements of this type with the outputs of the model outlined above (or models of this type) requires appropriate integration of the irreversibility expressed by Eq. (5). It is important to emphasize the conditions under which the present approach is expected to be valid. It is applicable only to situations where the dominant dislocation interaction is with precipitates and the precipitates are able to support Orowan loops. Furthermore, the plastic strains considered must be small enough that mostly Orowan loops are stored on the precipitates and the precipitates should not approach their natural limit for the maximum number of loops that can be supported [26]. Clearly, if the precipitate spacing is so large that the mean trapping distance of dislocations is less than the precipitate spacing, then the current approach should not be expected to describe the observed behavior very well. Similarly, if the alloy contains shearable precipitates, then a modified approach would be required.

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Fig. 15. Comparison of the simulated hysteresis loop for various imposed plastic strain amplitudes based on the proposed model with the experiment results for the over-aged alloy (4 days).

Fig. 16. Comparison of the simulation based on the proposed model with the experiment results for the strain amplitude jump test for the peak-aged (1.5 h) alloy: (a) jump from Depl/2 = 5  104 to Depl/2 = 3  103 and (b) jump from Depl/2 = 3  103 to Depl/2 = 5  104.

W.Z. Han et al. / Acta Materialia 59 (2011) 3720–3736

A final point worth discussing is the applicability of the proposed model for the irreversibility of slip to the transients in the stress–strain curve observed during Bauschinger tests (e.g. [26,50]. These tests are often used to estimate the internal stress, and it is well known that with small reverse plastic strain offsets the estimated internal stress can be larger than the total strain-hardening increment. This has typically been attributed to the complications from the transients in the unloading and reverse loading curves, and, as a result, larger plastic strain offsets are chosen when quantifying the internal stress. This is a problem, since the choices of plastic strain offset are rather arbitrary but influence the numerical value of the internal stress. Indeed, Gould et al. [46] expressed qualitatively similar concepts about the removal of Orowan loops as those articulated quantitatively in this contribution, when discussing the results of Bauschinger tests performed on dispersion hardened Cu crystals. Eq. (5) could be used as a starting point in attempts to describe the transients in Bauschinger tests of shear-resistant particle containing alloys, but there are important features of such tests that make them significantly more complicated than the cyclic deformation tests performed in this study. During Bauschinger tests, the plastic strains are much larger than those considered in this study. As a result, not only are Orowan loops stored on the particles, plastic relaxation dislocations are also stored. The plastic relaxation dislocations could influence strongly the probability of a backtracking dislocation removing a previously deposited Orowan loop and this may require additional model ingredients. This is an interesting area for future research. 5. Conclusion The use of a precipitate-strengthened Al alloy containing a well-defined, homogeneous and controllable distribution of shear-resistant precipitates has allowed for a systematic study of the effect of shear-resistant precipitates on the reversibility of dislocation slip during cyclic deformation. In particular, the effect of precipitate state and plastic strain amplitude on the reversibility could be extracted and an analytical expression for the slip irreversibility has been derived. This expression explicitly accounts for the features of the precipitate distribution and the deformation conditions and, when incorporated into a simple constitutive model for the elastic and plastic response of the material, is able to quantitatively describe the cyclic deformation behavior over a wide range of deformation conditions and precipitate states. The central idea underlying the proposed expression for the slip irreversibility is that Orowan loops deposited on precipitates have a certain probability of being removed upon reverse loading by the retreating dislocation, this probability depending on the features of the microstructure and the distances traversed by the mobile dislocations. The expression for the slip irreversibility provides a starting point for further analytical modeling studies into the cyclic deformation of alloys con-

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