Cyclic deformation behavior of Cu16at.%Al single crystals Part III: Friction stress and back stress behavior

Materials Science and Engineering, A128 (1990) 155-169

155

Cyclic Deformation Behavior of Cu-16at.°AAl Single Crystals Part IIh Friction Stress and Back Stress Behavior S. I. HONG* and C. LAIRD

Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6272 (U.S.A.) (Received January 16, 1990)

Abstract

To investigate the cyclic hardening behavior of planar slip alloys, the friction stress and the back stress acting on the dislocations were evaluated from analyses of fatigue hysteresis loops. In the initial stage of deformation the friction stress (12 MPa) is larger than the back stress (7 MPa). With accumulation of cycles the back stress increases and eventually leads the friction stress. The friction stress in the early stage of deformation, which is high relative to that of pure metals, is related to the elastic interaction of dislocations and segregated solute atoms. The cyclic hardening of this alloy is mostly caused by the increase of the back stress, which seems to be caused by the accumulation of multipoles and dislocation pile-ups frequently observed in this alloy. The equality of the back stress and the friction stress observed previously for copper breaks down for the behavior of multipoles in this alloy, because the dislocation structure is modified from that of copper to form kinked multipoles or cross-gridded multipoles, types which cannot occur in copper because of the ease of cross-slip in pure metal. The motion of these multipoles is discussed in relation to the observed friction stress and back stress behavior. 1. Introduction

In 1979, Kuhlmann-Wilsdorf and Laird [1] successfully used calculated values of the back stress and friction stress extracted from measurements of hysteresis loops to test and

*Present address: Materials Science and Technology Division, Los Alamos National Laboratory, MS G730, Los Alamos, New Mexico 87545, U.S.A. 0921-5093/90/$3.50

refine their earlier model on the cyclic deformation of pure copper [2]. Largely on the basis of the friction and back stress behavior, KuhlmannWilsdorf [3, 4] further clarified dislocation behavior in the fatigue of copper single crystals. The interpretation of the back stress and the friction stress behavior in relation to the observed dislocation structure offers a powerful tool to investigate the basic dislocation behavior during cyclic deformation. Recently, several investigators [5-7] attempted to interpret the deformation behavior of certain alloy systems in terms of the friction and back stresses obtained by the Cottrell method [8]. In 1984, Horibe et al. [5] used the friction stress and back stress of fatigued AI-Cu single crystals to investigate aging behavior after test interruptions. More recently, Pak et al. [7] ascribed the high friction stress of NiaGe single crystals to the interaction between straight screw dislocations on parallel (111) slip planes and the pinning of dislocations due to thermally activated cross-slip from (111) to (010). Yan et al. [6] also observed a high friction stress in the initial stages of the cyclic deformation of Cu-16at.% Al single crystals and explained it in terms of an alloying effect. In this study an attempt is made to investigate the friction and the back stress behavior of Cu- 16at.%A1 single crystals in relation to the dislocation structure observed after fatigue deformation [6, 9-13]. 2. Experimental details

Since the present investigation is an extension of a study reported previously [9, 10], most experimental conditions were the same as those reported earlier. Briefly, Cu-15.3at.% A1 single crystals, oriented for single slip (axis [i 35]), were grown by the Bridgman method, checked for © Elsevier Sequoia/Printed in The Netherlands

156

orientation by the Lane technique, spark machined with a gauge section of 6 m m x 4 mm x 25 mm and polished before testing. Tests were conducted under total strain control or load control in this study. Since an effect of test interruption was observed on the hardening behavior when the peak stress was lower than 35 MPa, all tests under strain control were never interrupted below 35 MPa except for one test (specimen 12) in which the total number of cycles to reach 35 MPa was so long that it was not practicable to run the tests without interruption. Even in this test there were no more than two interruptions. For detailed experimental conditions see Parts I and II of this series [9, 10].

havior of the whole sample in the later stages of deformation. Thus, through this account of the slip characteristics of the material, the friction stress and the back stress extracted from the hysteresis loops can be used to refine already established knowledge or to develop more adequate theories on the cyclic deformation of alloys with low stacking fault energy. Kuhlmann-Wilsdorf and Laird [1] proposed the following equations for the evaluation of the friction stress rE and the back stress ra: I ' E + ~s

r~=--

2

17E - - i"s

rB = - -

3. The validity of the Cottrell method Before reporting the present results, the validity of the Cottrell method [8] for measuring the friction stress and the back stress of alloys with low stacking fault energy will be briefly discussed, as follows. Kuhlmann-Wilsdorf and Laird suggested that the Cottrell method can be applied provided that the hysteresis loops can be considered to be reversible. In this context "reversible" means that the dislocation behavior and the internal stress and strain distributions are substantially unchanged (1) in successive cycles at points of the same strain and (2) from forward to reverse half-cycle, except for the sign change. The above condition is easily met for Cu-16at.% AI single crystals during cyclic deformation because of the low hardening rate and the high reversibility of slip [6, 10], except at the moments of strain bursts. When a strain burst occurs under strain control, newly activated slip bands take over most plastic strain and the old slip bands become inactive after a few strain bursts [10]. However, a few cycles after a burst under strain control, both the newly activated slip bands and the old slip bands satisfy the condition given above because the further activity of a burst is effectively suppressed by the control of strain. The friction stress and the back stress measured after a burst therefore represent averaged values for the different parts of the specimen -- both the newly activated slip bands and the old slip bands. As the overall deformation becomes rather homogeneous and the contribution of the strain bursts decreases with cycling [9, 10, 12], the back stress and the friction stress behavior measured by the Cottrell method truly represent the be-

(1)

(2)

2 where r E is the peak applied stress and rs is the yield stress in each cycle. Since the basic assumption used for the development of the above equations [1] is also true for alloys with low stacking fault energy, the above equations were used to evaluate the friction stress and the back stress in this study. For an extended discussion of the physical significance of these stresses see refs. 1-4.

4. Experimental results 4.1. The back stress and friction stress under strain control The friction stress and the back stress of Cu-16at.% AI single crystals at various strain amplitudes under total strain control are plotted in Fig. 1 as a function of the number of cycles. The back stress can be seen to increase gradually with the number of cycles as shown in Fig. 1. The friction stress, however, remains constant for a while and begins to increase when the back stress reaches about 20 MPa. At the very beginning of any test the friction stress (12 MPa) is larger than the back stress (7 MPa), but after a certain number of cycles the back stress begins to lead the friction stress and retains its dominance until failure. The number of cycles for the back stress to begin to lead the friction stress increases with decrease of strain amplitude as shown in Fig. 1. In Fig. 2 the number of cycles for the back stress and the friction stress to reach 15 MPa is plotted against the mean plastic shear strain amplitudes. The number of cycles for the total stress to reach 30 MPa is also plotted in Fig. 2. The cyclic hardening of this alloy is mainly

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caused by the increase of the back stress. No saturation of the back stress or the friction stress was observed, irrespective of the strain amplitude. 4.2. Effect of. strain bursts on the back stress and the friction stress under strain control Because of repeated strain bursts, the back stress does not increase smoothly but rather increases in a zigzag fashion. The curves for the back stress reported in Fig. 1 are smoothed out to represent the trend in behavior. The effect of strain bursts becomes smaller with increase of stress and eventually becomes negligible when the stress is larger than 36 MPa. After a strain burst, re decreases somewhat and rs increases by almost the same amount, as shown in Fig. 3. The friction stress is rarely affected by a strain burst since the total of rE and rs does not change much. On the other hand, the back stress is decreased suddenly by a strain burst since the difference between rE and r s suddenly decreases. A strain burst under strain control occurs in one or two

cycles, during which a number of dislocations are generated [9]. After a burst, however, the slip mode becomes quite reversible because of the low hardening rate and low stacking fault energy, as mentioned above in discussing "the validity of the CottreU method". 4.3. The back stress and the friction stress behavior under load control In Fig. 4(a) the back stress and the friction stress of specimen A tested under load control (peak stress 21 MPa) are plotted against the number of cycles. The behavior appears to develop in three stages as indicated in Fig. 4(a). In stage A, where the plastic strain amplitude increases very rapidly owing to a huge strain burst, the friction stress is larger than the back stress. In stage B, where the plastic strain amplitude decreases very rapidly after the huge strain burst, the relative magnitudes of the stresses are reversed; namely, the back stress is larger than the friction stress. It is questionable whether or not the slip behaviors in stages A and B are reversible

158

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Fig. 2. The number of cycles for the friction stress and the back stress to reach 15 MPa and the total stress to reach 30 MPa at various strain amplitudes.

enough to represent accurate values of the back stress and the friction stress. Since the huge strain burst under load control is larger than the strain bursts under strain control, by a factor of 2-3, the reported values of the friction stress and the back stress in stages A and B should be viewed with caution. After the huge strain burst subsides, the friction stress becomes larger than the back stress (stage C). The friction stress in stage C is very close to the initial friction stress of straincontrolled tests. In Fig. 4(b) the back stress and the friction stress behavior in a step loadcontrolled test (test B) are shown. The initial behavior before the stepwise increase of load is quite similar to that of test A. It should be noted here that the increase of load simply increases the back stress. 5. Discussion

"~s = 1.51 CF = 11.76 1; B = 9.24

1;s = 3.0 1;F = 11.8 1; S = 7.8

Fig. 3. The friction stress and the back stress just before (192 cycles) and after (212 cycles) a burst.

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5.1. Summary of possible dislocation interactions in Cu-AI alloys Recent observations on the dislocation structure of fatigued Cu-16at.%Al single crystals [9, 11, 14] enable one to explore the dislocation behavior during cyclic deformation of Cu-16at.%Al single crystals. The main conclusion of a recent study [11] is that dislocation structures in fatigued Cu-16at.%Al single crystals are very similar to those observed in Cu-AI single crystals deformed in tension, except for the formation of numerous small prismatic loops. Since no ladder structures and loop patches are observed in this alloy [9, 14], the many excellent theories available on dislocation behavior in the fatigue of copper single crystals

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Fig. 4. The friction stress and the back stress behavior under load control: (a) constant load at 21 MPa; (b) step-ascending test, stress amplitudes indicated.

[1-4, 15-19] cannot be directly applied to alloys with planar slip mode. The best possible way to explain the observed behavior of the friction stress and the back stress would be to choose

159

appropriate dislocation structures which would explain the observed behavior. Since the dislocation structures resulting from the deformation could contribute to the total resisting stress for deformation either directly or indirectly, it is very difficult to select a single dislocation mechanism responsible for the observed behavior. It would be easier to discriminate against less important mechanisms out of all possible mechanisms and then choose the more probable mechanism from the finalists. Dislocation structures which could contribute to the hardening of Cu-16at.%Al single crystals include (1) grown-in dislocations, (2) dislocation jogs, (3) dislocation pile-ups or unpaired dislocation trains, (4) dislocation multipoles, (5) prismatic dislocation loops, (6) secondary dislocation walls or forest dislocations and (7) LomerCottrell locks. Many of these features have been observed in Cu-AI alloys [9, 11, 14]. For typical dislocation structures after cyclic deformation of this alloy see refs. 11 and 14. A multipole is shown at C and some unpaired dislocations are shown at B in Fig. 5. In Fig. 6 a different view of multipoles in a (111) slice is shown. Dipoles sometimes exchange partners (at A) and assume cross-grid-like structures. Dipolar dislocations sometimes cross each other to change their equilibrium positions (at B)--in this connection it should be noted that there are two equilibrium positions (see also Fig. 9). Prismatic loops are frequently observed in this alloy after fatigue deformation as can be seen at A in Fig. 7. It is well known that the Lomer-Cottrell reaction is one of the most prominent reactions in Cu-AI alloys [20-23].

Besides the above dislocation interactions, alloying can also contribute to the strength, needless to say. Possible ~illoying effects for concentrated Cu-A1 alloys include (1) short-range order or stress-induced order, (2) elastic interaction of dislocations with individual solute atoms, (3) Suzuki atmosphere and (4) elastic interaction of dislocations and segregated solute atoms. The solid-solution-hardening mechanisms of concentrated Cu-A1 alloys will be discussed in Section 5.3.

5.2. Temperature dependence of the back stress and the friction stress It seems difficult to sort out which mechanism is responsible for the friction stress and which is responsible for the back stress. Yan et al. [6] investigated the temperature dependence of the back stress and the friction stress and found that the friction stress is significantly affected by temperature whereas the back stress is much less affected by temperature as shown in Fig. 8. The ratio of the friction stress at room temperature to that at 77 K is about 0.46 while the ratio of back stresses at room temperature and 77 K is about 0.85 (Fig. 8). For thermally activated deformation the deformation rate can be expressed as the following well-known equation:

g= go exp( U°-v*rl kT ]

where g is the deformation rate, go is a pre-exponential factor, U0 is the activation energy, v* is the activation volume, r is the applied stress and k and T have their usual meanings. By rearrangement of eqn. (3), the following equation can be obtained:

0{1

r=v~

Fig. 5. Dislocation structure ([121] section) of fatigued Cu-16 at.% AI single crystal; (plastic shear strain amplitude 5.5 x 10 - 3). Courtesy of Buchinger et al. [ 14].

(3)

-U-~0In

(4)

From the above equation the thermal stress component can be obtained as follows. U0 ( = 0.5 eV ) and v* (=2.7 x 10 -21 c m 3) for Cu-14at.%Al single crystals [24] were selected and In(go~g)is chosen as 11.5 in order to match the observed friction stress at 77 K, rV= 25 MPa, as shown in Fig. 8. Therefore the predicted ratio of the thermal stresses at 295 and 77 K from eqn. (4) is 0.49, which is very close to the measured ratio (0.46) of the friction stresses at 295 and 77 K (Fig. 8). This

160

7 Fig. 6. Dislocation structure ([111] section) in fatigued C u - 1 6 at.% AI single crystal; mean plastic shear strain amplitude 1.6 x 10- 3. Taken from ref. 11.

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Fig. 7. Dislocation structure in C u - A I alloy fatigued to 41 MPa. Small prismatic loops can be seen in local regions (at A). Taken from ref. 11.

result indicates that the friction stress is mainly caused by thermal barriers. In a similar way the temperature dependence of the back stresses can be related to the temperature dependence of the shear moduli. Unfortunately, the shear moduli of Cu-16at.%A1 alloy at low temperatures are not available. Since the mechanical behaviors and the dislocation structures of Cu-16at.%Al single crystals are very similar to those of Cu-30at.%Zn single crystals [25-27], the shear moduli data of Cu-30at.%Zn alloy can be used for purposes of comparison. The shear modulus of Cu-30at.%Zn at 77 K is 38 500 and that at 295 K is 35 300 MPa. The ratio of the shear moduli at 295 and 77 K is therefore 0.92, which is in reasonably good agreement with the measured ratio of the back stresses (0.85).

The small difference between the observed and the predicted back stress ratios could be caused by a contribution from medium-range obstacles, which might be weakly affected by thermal activation, to the back stress. Obstacles by which the movement of dislocations are resisted have been classified into two groups [28, 29]; short-range obstacles and long-range obstacles. Short-range obstacles are easily overcome by thermal activation and long-range obstacles are too strong to be overcome by thermal activation. Some obstacles which might lie between these two kinds of obstacles could be termed medium-range obstacles, which could contribute to the back stress to some degree. This could explain the small discrepancy between the observed ratio of the back stresses and the ratio of the shear moduli. Since the difference is relatively small, the contribution of the mediumrange obstacles to the back stress will be very small if there is any. The reasonably good agreement between the measured ratio of the back stresses and the ratio of the shear moduli indicates that back stresses are mostly caused by long-range obstacles. 5.3. The friction stress and the back stress behavior under strain control The behavior of the friction stress is considered to be divided into two stages; in stage 1 the friction stress remains constant and in stage 2 the friction stress slowly increases. 5.3.1. The friction stress in stage 1 The friction stress measured in stage 1 is similar to that estimated from the arrangement and the spacing of dislocations in Cu-Al alloys [30]

162

by earlier investigators. They attributed the friction stress to interactions between dislocations and solute atoms. The friction stress calculated from the curvature [30] of dislocations is 17.2 MPa for edge dislocations and 7.2 MPa for screw dislocations. If it is assumed that the density of mobile screw dislocations is the same as that of mobile edge dislocations, the estimated value for the average friction stress is 12.2 MPa. Although edge dislocations are dominant in the dislocation structures observed by transmission electron microscopy (TEM), the lower friction stress for screws could allow them to be more easily lost in thinning and to participate actively in the deformation process. As stated, the friction stress in stage 1 should be explained in terms of the short-range obstacles (thermal barriers). The friction stress of Cu-16at.% AI single crystals in stage 1 (12 MPa) is about 10 times as large as that of copper single crystals (1.3 MPa) at the beginning of cyclic deformation. This difference cannot be explained in terms of the difference in the grown-in dislocation density [31, 32]. The grown-in dislocation density of the C u - Z n and Cu-AI alloys differs from that of pure metals by less than an order of magnitude [31]. Actually, Meakin and Wilsdorf [25] observed a dislocation density of (1-5) x 106 cm-2 in a-brass, which is very close to the density observed in pure copper by Young [33]. The magnitude of the activation volume (2.7x 10 -21 cm 3) reported by Koppenaal and Fine [24] also does not favor an indirect effect from alloying as a source of the friction stress. Grown-in or forest dislocation densities in excess o f 101° c m -2 should be necessary to obtain the activation volume of 160b 3, where b is Burgers' vector. It is also unlikely that dislocation jogs exist every 160b (0.04/,m) at the beginning of a test. Prismatic loops also cannot be considered as a Source of the friction stress at the beginning of a test. In conclusion, it is impossible to explain the friction stress in stage 1 in terms of thermal dislocation barriers. Among the possible alloying effects suggested in Section 5.3, the elastic interaction of a dislocation and individual solute atoms can be eliminated as an explanation for the friction stress on the basis of activation analysis [24, 34]. For Cu-16at.% AI alloys the spacing "between" individual solute atoms will be 2.5 b if they are distributed randomly. This spacing is much smaller than the 160b expected from activation analysis

[24]. Further, short-range order and the Suzuki mechanism can also be eliminated from consideration for a source of the friction stress on the basis of the temperature dependence of the friction stress. Neither of these phenomena is expected to vary much with temperature [35] whereas the temperature dependence of the friction stress is much larger than the temperature dependence of the shear modulus. Hong and Laird [36] suggested that the elastic interaction of dislocations and segregated solute atoms is the dominant mechanism for the frictional stress. Their conclusion is supported by results [37, 38] that the strengthening in concentrated copper solid solution alloys is dependent on the atomic size misfit factor and is also compatible with the trough model of solid solution hardening by Kocks [39]. Kocks suggested that dislocations collect a significant number of solute atoms from their surroundings after being stopped by forest dislocations or other obstacles, including solute obstacles. In addition, Hong and Laird [36] showed that, on the basis of the diffusivity data of Horne and Mehl [40] for Cu-28at.% Zn alloy, an immobilized dislocation could gather solute atoms from any connected source by means of pipe diffusion in, at most, a few seconds. As well as pipe diffusion along dislocation cores, stress-induced radial diffusion toward the dislocation cores could contribute to the segregation of solute atoms. If the core radius of a dislocation is assumed to be 5b and any solutes in the core could migrate easily to the very center of the core, a simple calculation shows that a dislocation line could be continuously decorated by a line of solute atoms if the solute content is larger than 5 at.%. Therefore for concentrated solid solutions there are enough solute atoms in the core which would form a continuous string of solute atoms if they migrate to the very center of the core. Therefore long-range diffusion is not necessary to lock temporarily immobilized dislocations in place, although its occurrence would increase the amount of segregation. Cottrell and Bilby [41] developed a theory of yielding for iron based on the assumption that the central part of the atmosphere consists of a line of carbon atoms parallel to the dislocation line. For concentrated solid solutions the prerequisite for the Cottrell-Bilby model is fulfilled without longrange diffusion. As soon as a dislocation is freed from its string of impurity atoms over some distance by thermal activation and is then immobi-

163

lized by any obstacles, the dislocation would again be locked by a string of solute atoms, almost instantly, since the long-range diffusion of solute atoms is not necessary. Stress-induced short-range migration of solutes toward the very center in the dislocation core is sufficient to lock the dislocation. Needless to say, both mechanisms could contribute to the segregation of solute atoms at dislocations. The movement of dislocations in concentrated solid solutions seems to be somewhat analogous to the movement of dislocations over the Peierls barrier [42, 43]. For this model the activation area is equal to the microscopic geometrical area swept out by the bulge during the activation event [39]. This activation process definitely involves the interaction between a dislocation and many solute atoms (along the dislocation fine), which is compatible with the stress equivalence of solution hardening by Basinski et al. [34]. In conclusion, the elastic interaction of a dislocation and segregated solute atoms seems to be the dominant mechanism for the friction stress [36] in region 1, whatever the mechanism for solute segregation.

5.3.2. Back stressbehavior Generally in alloys the back stress can be caused either by alloying effects or by the effects of dislocation substructure formed during deformation. Possible alloying effects which have weak temperature dependence include (1) large incoherent precipitates, (2) short-range order and (3) Suzuki atmosphere. There are no large incoherent precipitates in this alloy. Short-range order and Suzuki atmospheres are not strong enough [31, 32] for dislocations to be piled up in such a manner as to cause an appreciable back stress at room temperature. Furthermore, regions of short-range order and Suzuki atmospheres would be destroyed by the turbulence of deformation [36]. An alloying effect therefore cannot explain the increasing nature of the back stress with deformation. Typical dislocation structures which could cause the back stress include (1) dislocation pileups at Lomer-CottreU locks, other immobilized dislocations or the surface and (2) dislocation multipoles. Other dislocation structures such as forest dislocations, dislocation jogs and prismatic loops are considered to constitute thermal barriers and these have a strong temperature dependence. Soon after dislocations are generated they could pile up against obstacles such as

Lomer-Cottrell locks [25, 44] and subboundaries [25, 44] or in surface regions [45]. Pile-ups have been frequently observed in concentrated copper alloys [25, 44-51]. Also, many dislocations of opposite sign could form multipoles, which are frequently observed in concentrated copper alloys [44-51]. Both structures could explain the increase of the back stress with cycling. The initial back stress (7 MPa) can be related to the activation of a Frank-Read dislocation source. Since the initial back stress is actually part of the flow stress, it cannot be related to the back stress due to pile-ups or multipoles. It is well known that the stress needed to activate a Frank-Read source is proportional to Gb/2, where G is the shear modulus (46000 MPa) and 2 is the length of the unpinned dislocation line. The Frank-Read stress can be expressed as follows

[52]: Gb(1-v/2)

(~)

rVR-- 2~-~(1----~ In

(5)

where v is Poisson's ratio (0.33) and b is Burgers' vector (2.61 x 10 -s cm). If the grown-in dislocation density is taken as 5 x 106 c m - 2 from the observations by Meakin and Wilsdorf [25], the critical length 2 will be 4.47x 10 -4 cm. The Frank-Read stress rFR predicted from eqn. (5) is 5.2 MPa, which is similar to the initial back stress. The temperature dependence of the initial back stress can be accounted for therefore from the temperature dependence of the shear modulus by eqn. (5). The yield stress of Cu-16at.%Al single crystals is thus represented by

Gb(1-v/2) rY=rf+rFR=rf~

(2 / 2:~(1--V) In ~

(6)

where rf is the friction stress due to alloying (12 MPa for Cu-16at.%Al single crystals). The yield stress of this alloy thus seems to be determined by the stress needed to overcome the friction stress due to alloying and the line tension of dislocations. The reason why the initial back stress due to the activation of a Frank-Read source for Cu-16at.%Al alloy (7 MPa) is larger than that for copper (1 MPa) is not clear. Probably differences in the stacking fault energy and in the density of defects including dislocations could be responsible. The wide dissociation of forest dislocations and source dislocations could also increase the

164

Frank-Read stress. Also, alloying itself might act to decrease the critical length 4. A small difference in the density of defects including dislocations could be an additional factor. According to Hazzledine [53, 54], the stress rp required to decompose multipoles, once formed, is independent of the number of dipoles in the multipole and given to a good approximation by the following equation in the case of edge dislocations: Gb rp -

(7)

8:t(1 - v)d where G is the shear modulus, v is Polsson's ratio and d is the separation of the slip planes. Since the stress rp is not affected by the number of dipoles in the multipole, further consideration will be focused on the behavior of a dipole. Depending on the relative locations of dislocations in a dipole, the force between two dislocations can be either repulsive or attractive. The repulsive and attractive shear stresses have extrema at a relative angular orientation of 67.5 ° and 22.5 ° respectively. The value of these extrema can be calculated by eqn. (7). In cubic anisotropy, for ~(110), {111 } edge dislocations, the extrema as well as the equilibrium position of the dislocations are shifted from the isotropic values [3, 55, 56]. At the same time the magnitudes of the stress extrema decrease [3], more so for the inner (the repulsive) than the outer (attractive) extremum; in the case of copper the stress extrema decrease to 78% and 87% respectively of the isotropic value given by eqn. (7). If a stress is applied to force the dislocations of a dipole together, the dipole will flip when 78% of the stress predicted by eqn. (7) is reached in case of copper. If a stress is applied further to increase the width of the dipole from its equilibrium position, the dislocations will gradually increase their equilibrium separation until the magnitude of the stress reaches the value of the outer extremum, after which the dislocations of the dipole will separate to an indefinite distance [3]. It should be noted that in elastic anistropy a new equilibrium will be reached after flipping and the stress can rise by about 10% in the case of copper before the dipole disintegrates [9]. If the flipping stress is small enough it can be considered as a frictior| stress. If the stress field caused by the outer extremum is large enough to prevent the separation of a dipole, the dislocation displaced from its equilibrium position tends to slide down the stress

field upon removal of the applied stress, which in fact is opposed by the friction stress due to alloying. When the direction of the applied stress is reversed, the back stress caused by the stress field of the outer extremum will be directed to aid the externally imposed deformation. Kuhlmann-Wilsdorf [3] theoretically proved the equality between the back stress and the larger part of the friction stress during cyclic deformation of pure copper in terms of the behavior of dipole flipping in the Taylor lattice and the interaction of the gliding dislocations and the loop patch walls. According to her, the friction stress and the back stress for the deformation of loop patch material alone would be the same. For Cu-16at.%Al alloys the friction stress does not increase for a long time although the back stress increases continuously with cycling as shown in Fig. 1. This result can be explained by assuming that either the contribution of multipoles to hardening somehow is negligible to Cu-16at.%Al alloy or the equality between the back stress and the friction stress does not hold for multipoles in Cu-AI alloys; namely, the back stress is much larger than the friction stress due to the interaction of two dislocations of a dipole. Whether or not this is true is examined as follows. Dislocation multipoles have been frequently observed for Cu-AI alloys [9, 11, 14, 46, 50]. Hazzledine [54] developed a hardening theory by assuming that the flow stress is controlled by the stress required for one group of dislocations to pass another group with the same Burgers' vector but with opposite sign on a parallel slip plane. Recently, Inui et al. [11] observed a rapid decrease of the spacing of activated primary slip planes with stress during the fatigue deformation of Cu-16at.%Al single crystals. The spacing is around 130 nm at a total shear stress of 26 MPa. From eqn. (7) the stress for flipping of a multipole with the slip plane spacing of 130 nm is 6 MPa. For this calculation G is assumed to be the same as that of copper (46 000 MPa). The values of v and b in eqn. (7) were chosen to be 0.33 and 2.61x10 -8 cm respectively. The difference of the total stress (26 MPa) and the critical resolved shear stress (19 MPa) is similar to that expected from eqn. (7). It thus seems reasonable to assume that a considerable part of the difference of the total stress and the critical resolved shear stress is related to the deformation of multipoles. Now the remaining question would be why the friction stress does not increase with cycling

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although a considerable fraction of the observed back stress seems to be caused by multipoles. This question might be answerable if the geometry of dipoles in the multipole can be modeled in the following way. If part of one dislocation or parts of both dislocations of a dipole are reoriented as shown in Fig. 9, the sense of relative orientation for both dislocations is changed at the midpoint. The flipping stress would be greatly reduced for a dipole with this configuration. In this case the zipping-unzipping of a dipole is more appropriate in describing the motion of a dipole, which would be much easier than the flipping of a whole dislocation. With the application of the stress which is smaller than that needed to separate the dipole, the cross-over point will move up or down depending on the sign of the applied stress. Dipoles with this configuration will be called kinked dipoles hereafter. One can assume that the zipping-unzipping stress is not larger than 10% of the flipping stress. Actually, many kinked dipoles have been found after cyclic deformation, e.g. at A in Fig. 6. Therefore in Cu-16at.%Al alloys the friction stress for a multipole consisting of many kinked dipoles would be much smaller than the value predicted from eqn. (7). It should be noted that the back stress due to the outer extremum is not changed from the value predicted by eqn. (7). The equality between the back stress and the friction stress breaks down for the deformation of a multipole consisting of many kinked dipoles. Unfortunately, no detailed information on the configuration of dipoles in the loop patch structure of pure copper is available because the loop patch structure is too crowded with dislocations to be resolved [1, 16]. The difference of stacking fault energy and the spacing between dipolar partners [36] for pure copper and Cu-16at.% A1 alloy could be responsible for the different behavior. If the dipolar partners in a loop patch of pure copper are kinked as in Fig. 9, cross-slip would easily occur at the cross-over point because of the high stacking fault energy and the small spacing between dipolar partners. Thus the kinks of a dipole would not be stable for copper. In copper, therefore, the equality between the back stress and the friction stress holds for the deformation of the Taylor lattice. Another interesting dislocation structure which should be mentioned is the cross-grid type of multipole illustrated in Fig. 10. See also the dislocation structure indicated by B in Fig. 6. Unlike

Fig. 9. Schematic illustration of kinked dipoles. This kind of dipole can be seen in region A of Fig. 6.

Fig. 10. Schematic illustration of cross-grid-like multipoles actually observed in region B of Fig. 6.

the kinked dipoles, both the friction stress and the back stress for the deformation of the cross-grid type of multipole would be much smaller than that predicted by eqn. (7) because dislocations in this structure easily overcome both the inner and outer extrema in the same manner as a kinked dipole easily overcomes the inner extremum. Although this cross-grid-like multipole is less stable than the normal multipole, it has the advantage of easily accommodating large cyclic strain. These cross-grid-like multipoles are more frequently observed in the heavily deformed areas than in the lightly deformed areas [11, 12]. The cross-grid type of multipole in the less constrained areas could be decomposed easily with the application of a stress large enough to overcome the sum of the friction stress due to alloying and the friction stress for zipping-unzipping. Therefore the cross-grid type of multipole in the less constrained area does not contribute much to the hardening. However, if they are constrained by obstacles such as Lomer-Cottrell locks or

166 other dislocation structures, they could contribute to the hardening by means of the increasing internal stress of piled-up dislocations. It should be noted that the internal stress of the multipoles is not affected by the cross-grid modification. Since kinked multipoles, cross-grid-like multipoles and dislocation pile-ups at Lomer-Cottrell locks or other dislocation structures are all building blocks of the dislocation substructure of fatigued Cu-16at.%Al single crystals, all of them could contribute to the cyclic hardening of the alloy. It seems, however, difficult to figure out which is dominant for cyclic hardening of Cu-16at.%A1 single crystals. Since the activity of secondary dislocations increases with increase of plastic strain amplitude, pile-ups or cross-gridlike multipoles whose motions are restricted by Lomer-Cottrell locks or other dislocation structures could become important for cyclic hardening at high plastic strain amplitudes. It was noted above that the cyclic hardening exponent when the plastic shear strain amplitude is smaller than 2 × 10 -3 is 0.27-0.35, which is very close to the 0.3 predicted from Hazzledine's hardening model [54] (where the increasing passing stress for multipoles is responsible for hardening as cycles accumulate). The cyclic hardening exponent when the plastic shear strain amplitude is greater than 2 x 10 -3 is 0.43-0.48, which is very close to the 0.5 predicted from Mott's hardening model [57] (where the increasing internal stress of piledup dislocations at Lomer-Cottrell locks is responsible for hardening). The increasing contribution of secondary systems to the crack growth behavior [10, 12, 58] also supports these results. The cyclic hardening ability below the shear strain amplitude of 2 x 10-3 is lower than that above 2 x 10 -3. For example, the ultimate strengths before fracture are between 40 and 46 MPa at low strain amplitudes while they are between 55 and 67 MPa at high strain amplitudes [10, 12]. Recent T E M observations [11] obtained in our laboratory show that the spacing of activated primary slip planes decreases with increase of stress but saturates at around 35 nm, which agrees closely with the value (40 nm) predicted by Hong and Laird [12, 36]. The predicted passing stress for a multipole with the slip plane spacing of 35 nm predicted from eqn. (7) is 21 MPa. The maximum stress for multipoles would be about 40 MPa since the total stress is the sum of the passing stress and the yield stress. At high strain

amplitudes, multipoles would be decomposed at higher stresses if they were formed at lower stresses. Also, as stated above, cross-grid-like multipoles are likely to form at high strain amplitudes to accommodate large cyclic strain. In addition, at high strain amplitudes the refinement of multipoles, once they are formed, is rather difficult because long-range movement of dislocations is difficult owing to the high activity of secondary dislocations. Further, extra unpaired dislocations generated during deformation [36] could be trapped at high strain amplitudes owing to the strong activity of secondaries and could disrupt the stress field of otherwise stable multipoles. At low strain amplitudes these surplus dislocations could easily leave the multipoles [54]. At high strain amplitudes, therefore, the back stress seems to be explained by unpaired piled-up dislocations and cross-grid-like multipoles restricted at Lomer-Cottrell locks or other dislocation structures. At low strain amplitudes, on the other hand, the deformation of kinked multipoles could be more responsible for the back stress behavior. The increase of stress at low strain amplitudes is explained by the decrease of the spacing of multipoles, which would occur by disintegration and trapping [59]. Both mechanisms are compatible with the observation that the back stress increases rapidly with cycling and the increase of the friction stress lags behind the increase of the back stress. 5.3.3. The increase of the friction stress in stage 2 As explained in Section 5.3.1, the initial constancy of the friction stress (stage 1) is related to the elastic interaction of dislocations and segregated solute atoms. If the zipping-unzipping stress is about 10% of the flipping stress, the friction stress for kinked multipoles at low strain amplitudes would not be changed much until the back stress increases considerably. As the spacing of activated primary slip planes decreases with cycles, the back stress increases as predicted from eqn. (7) at low strain amplitudes, and correspondingly the friction stress increases gradually in the later stages of cyclic deformation (stage 2). At high strain amplitudes the cross-grid-like multipoles or pile-ups (unpaired dislocation trains) could contribute to the friction stress. As explained above, the friction stress due to crossgrid-like multipoles or pile-ups is much smaller than the back stress due to obstacles such as

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Lomer-Cottrell locks. The decrease of the spacing of activated primary slip planes and the increase of the primary dislocation density and the Lomer-Cottrell locks increase the back stress and correspondingly the friction stress (stage 2). Besides multipoles or pile-ups, which increase both the back stress and the friction stress, thermal barriers such as dislocation jogs, forest dislocations and prismatic dislocation loops could contribute to the increase of the friction stress. The number of dislocation jogs observed in Cu-16at.%A1 single crystals seems to be too small to account for the increase of the friction stress. Furthermore, the number of dislocation jogs could not increase indefinitely because of the mutual annihilation of opposite jogs [2]. In the case of copper the jog density reaches saturation in about 50 cycles [2]. Apparently jog dragging cannot explain the increase of the friction stress in the later stages of cyclic deformation. The recent report [11] that the number of prismatic loops increases with increase of stress suggests that such prismatic loops could contribute to the friction stress during cyclic deformation. The number of prismatic loops increases rapidly when the total stress is higher than 30 MPa [11], compatible with the present observation that the friction stress increases gradually when the total stress is larger than 25-30 MPa. It is also apparent that the density of forest dislocations increases with increase of stress, another factor which could be responsible for the increase of the friction stress in the later stages of cyclic deformation. In conclusion, the cause of the increase of the friction stress in stage 2 is explained by combined effects: the zipping-unzipping stress of kinked multipoles and/or the passing stress of cross-grid-like multipoles and unpaired dislocation pile-ups, and the cutting stress increased by prismatic loops and forest dislocations. The contribution of the cutting stress of prismatic loops and forest dislocations would be expected to increase with increase of plastic strain amplitude because the activity of cross-slip and secondary slip increases with increase of strain amplitude. For example, consider first a result at the low plastic shear strain amplitude of 4.3 x 10-4; the predicted increase of the friction stress due to zipping-unzipping of kinked multipoles at 900 kilocycles would be approximately 1.92 MPa since the back stress is 19.2 MPa (see Fig. l(a)) (assuming the friction stress due to the zipping-unzipping stress of kinked multipoles is

10% of the back stress). Since the friction stress was actually observed to increase by 1.9 MPa from the friction stress due to alloying, most of the increase of the friction stress is consistent with the zipping-unzipping stress of multipoles. At the higher plastic shear strain amplitude of 1.4 x 10 -3 the predicted friction stress due to the zipping and unzipping stress at 500 kilocycles is about 2.5 MPa since the back stress is 25 MPa (Fig. l(c)). Since the friction stress actually increased by 3.5 MPa from the initial friction stress, about 71% of the increase of the friction stress seems to be due to the zipping-unzipping stress of multipoles. The balance is attributed to the cutting stress of the prismatic loops and the forest dislocations. At strain amplitudes higher than 2 × 1 0 - 3 the same assumption cannot be applied since the back stress is not caused by the outer extremum of the kinked dipole but by the restriction of the decomposed multipoles or unpaired dislocations (pile-ups) at LomerCottrell locks or other dislocation structures. However, if the 10% friction-back stress assumption is applied, it is seen that about half of the increase of the friction stress is caused by the cutting stress of the prismatic loops and forest dislocations at the high shear strain amplitude of 5.9 × 10 -3 (see Fig. l(e)). 5. 4. The friction stress and the back stress behavior under load control Although there is good reason to believe that the back stress is larger than the friction stress as the dislocations are pushed "all the way" until they meet strong obstacles in stage B, the validity of the measured value of the back stress and the friction stress is questionable as explained in Section 4. The behavior after the initial huge burst has subsided will therefore be discussed. In stage C the friction stress keeps constant at 12 MPa, which is consistent with the behavior in stage 1 under strain control. It is interesting to note that the back stress as well as the friction stress were observed not to change (Fig. 4(a)) after saturation, although the plastic shear strain amplitude decreased with cycles. Under load control, hardening would be caused by the mutual trapping of dislocations of opposite sign [59] which were generated during the huge burst. Since the total applied stress was 21 MPa and the intrinsic friction stress due to alloying is 12 MPa, about 9 MPa were available for the interaction of dislocations. Since the number of trapped

168 dislocations (multipoles) increased with cycles, the plastic shear strain amplitude decreased. In Fig. 4(b), which deals with the loadcontrolled step test, some of the weak multipoles saturated at one stress level would dissociate with increase Of stress and/or more dislocations would be generated; these dislocations would be manifested as a strain burst [9]. With cycling, these dissociated dislocations and/or newly generated dislocations could form more numerous and stronger multipoles and the plastic shear strain amplitude would be observed to decrease. Since many multipoles are kinked multipoles, the friction stress would not be expected to change much with increase of stress, which is also consistent with the behavior in strain-controlled tests.

6. Conclusions The results and conclusions of the present study on the friction and back stress behavior are summarized as follows. (1) Under strain control the back stress increases gradually with cycles while the friction stress keeps constant for a certain number of cycles. (2) At the very beginning of the tests under strain control the friction stress (12 MPa) is larger than the back stress (7 MPa) but the back stress increases with accumulating cycles and eventually leads the friction stress. (3) The number of cycles for the back stress to begin to lead the friction stress increases with decrease of strain amplitude. (4) The cyclic hardening of this alloy is mainly caused by the increase of the back 'stress. No saturation of the back stress and the friction stress were observed irrespective of the strain amplitudes employed for cycling. (5) The ratio of the friction stresses at room temperature and 77 K is 0.46, which is very close to the value of 0.49 predicted from a thermally activated process. (6) The ratio of the back stresses at room temperature and 77 K is 0.85, which is very close to the ratio of the shear moduli (0.92). (7) The friction stress in stage 1 is related to the elastic interaction of a dislocation with segregated solute atoms. The segregation of solutes at room temperature seems to be promoted by stress-induced radial migration of solutes as well as by pipe diffusion. The prerequisite for operation of the Cottrell-Bilby model is fulfilled with-

out long-range diffusion if the solute content is larger than 5 at.%, as applied here. (8) The initial back stress which is a part of the yield stress could be related to the stress required to activate a Frank-Read source. Good agreement between the measured back stress and the predicted Frank-Read stress was obtained. (9) The increase of the back stress with accumulation of cycles seems to be caused by the multipoles and/or pile-ups. With increase of strain amplitude the contribution of pile-ups is concluded to increase as the activity of slip on the secondary system increases. (10) The equality of the back stress and the friction stress observed previously for copper breaks down for the behavior of multipoles in this alloy because the multipolar structure is modified to form kinked multipoles or cross-gfidded multipoles, types which cannot occur in copper because of the ease of cross-slip in pure metal. (11) The friction stress is concluded to be much smaller than the back stress for the deformation of the modified multipoles or pile-ups. (12) The increase of the friction stress in stage 2 under strain control is partly caused by the zipping-unzipping stress of accumulating kinked multipoles or the passing stress of cross-grid-like multipoles and unpaired dislocation pile-ups and partly caused by the cutting stress of an increasing density of prismatic loops and forest dislocations. (13) Under load control, hardening is concluded to be caused by the mutual trapping of dislocations of opposite sign generated during the initial huge burst which normally occurs in the load type of test. Increase of the total applied stress results only in an increase of the back stress, which is consistent with the behavior observed under strain control when the total stress is lower than 30 MPa. (14) A n approach to explain the yield stress of alloys in terms of a single mechanism is considered questionable. As a result of the present study the yield stress in Cu-A1 alloy can be related to the stress needed to overcome the friction stress due to alloying and the dislocation line tension.

Acknowledgments The present research was supported by the Laboratory for Research on the Structure of Matter under Grant No. DMR 88-19885 from the National Science Foundation. We are grateful

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for this support and also to our colleagues D. Ricketts and A. L. Radin for their excellent support in experimentation.

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