Dislocation behavior in fatigue VI: Variation in the localization of strain in persistent slip bands

Materials Science and Engineering, 50 (1981) 127 - 136

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D i s l o c a t i o n B e h a v i o r in F a t i g u e V I : V a r i a t i o n in t h e L o c a l i z a t i o n o f S t r a i n in P e r s i s t e n t Slip B a n d s

CAMPBELL LAI R D

University of Pennsylvania, Philadelphia, PA 19104 (U.S.A.) J. M. FINNEY

Structures Division. Aeronautical Research Laboratories, Melbourne (Australia) DORIS KUHLMANN-WILSDORF

University of Virginia, Charlottesville, VA 22901 (U.S.A.) (Received March 27, 1981)

SUMMARY

It is n o w widely accepted that the strain localized in the persistent slip bands (PSBs) o f fatigued pure metals is about 0.01, so much so that the average nature o f this strain appears to have been forgotten. Accordingly, we offer interferometric observations on the details o f slip offsets within PSBs. These were obtained by repolishing the gauge surfaces o f specimens cycled into saturation and restraining to the strain amplitude; they thus represent PSB behavior in transitu. Strain concentrations up to 1000 are s h o w n to exist at the surface well into saturation. The implications o f these results for understanding dislocation behavior in fatigue are discussed: the slip step behavior can be explained by the glide o f groups o f like-handed screw dislocations in the PSBs and is n o t consistent with a model in which any significant part o f the strain is due to edge dislocations bowing o u t o f the PSB walls.

1. INTRODUCTION

By observing the fraction of the gauge surface of a fatigued single crystal which was occupied by persistent slip bands (PSBs) Winter concluded that the strain carried by a PSB is about 0.01 [1, 2]. Winter observed the PSBs from the start of cycling and thus the measurement of strain localization was based on the integrated effect of all the cycling. 0025-5416/81/0000-0000/$02.50

Finney and Laird [3], however, who used interferometry on specular specimens periodically repolished during cycling, measured the localized strain at a given p o i n t in life. They confirmed that the localized strain was about 0.01 (actually shear strain amplitude equals 0.006 25), but this was an average value. Since then, the value of 0.01 has acquired the inflexibility expected of the velocity of light, and its average nature appears to have been overlooked [4]. Since attempts to understand cyclic deformation require a realistic values (or values) of the localized strain [5 - 9] and since fatigue cracks are initiated in PSBs, almost certainly as a result of the notch peak topography due to strain irreversibility, quantitative values of the variation in strain localization are also needed. The purpose of the present paper is to demonstrate the degree of localization observed in copper single crystals oriented for single slip, to give some idea of the numbers of dislocations involved in the volumes of concentrated plastic strain and to explore dislocation behavior with reference to interferometric observations.

2. EXPERIMENTAL DETAILS

The results to be reported here were obtained in the same investigation as that reported by Finney and Laird [3] but were not reported then because of the need for concise publication. Thus the exper© Elsevier Sequoia/Printed in The Netherlands

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imental details, including the specimen serial numbers, are exactly the same as previously reported [3]. Briefly, copper single crystals of rectangular section (about 6.4 mm × 4.8 mm), of gauge length 12.7 mm and oriented for single slip were cycled at strains consonant with the "plateau" of the cyclic stress-strain curve. Thus, in saturation, PSBs were the dominant site for deformation. The gauge surfaces of the rectangular-sectioned specimens were polished until specular at the start of cycling and subsequently during test interruptions for numbers of cycles up to 30 000. The PSB structures were studied by interferometry both immediately after cycling and again after repolishing, r e e n t r y of the test specimen and after 1 cycle or after several cycles. By this technique the deformation behavior of a PSB at any desired point in life could be established. Moreover, the variation in strain localization was observed by photographing and measuring the slip offsets for whole PSBs and for individual offsets within bands. Further experimental details can be obtained from the earlier paper [3]. Since it is possible that, in the unstressed state before, during and after repolishing, the dislocations in fatigued copper could be anchored by Cottrell atmospheres of point defects produced by the cycling [7], it is appropriate to question the validity of the interferometric approach used here. The duration of the test interruptions to repolish the specimens and to record the interferograms were frequently lengthy enough to permit the point defects to diffuse to the dislocations. Did this so interfere with the PSBs that the subsequent behavior (during the imposition of ¼ cycle) became more akin to monotonic deformation than to cyclic deformation ? Whether or not the point defects did so diffuse, we have unequivocal evidence to indicate that the approach is valid. Let us consider, for example, ref. 3, Fig. 10. This figure shows interferograms taken, after res cycle, 1~1 cycles and polishing, at ~ cycle, ~ 103 cycles. In all these interferograms the localized strain within any one band was almost the same after each treatment and the variations in strain between bands were approximately the same. Moreover, before the test was stopped to record the 103 cycle, the specimen had just received 9 regular fatigue cycles. Any "monotonic" type of behavior

could not have survived these 9 cycles. Furthermore, any effect of locking would have shown up in the measurements of the hysteresis loops, but no such effect was observed.

3. RESULTS AND DISCUSSION

3.1. The magnitudes o f localized strain Strain concentration is a readily understood concept, but to quantify it requires the specification of a gauge length for any welldefined zone of deformation. Obviously a discrete step on a surface implies nearly infinite strain concentration and the concept is therefore useful only when the gauge length is large compared with lattice dimensions. For a given deformation band a plastic strain concentration factor can be defined as the measured step height across the band divided by that expected from the applied plastic strain distributed uniformly over the whole gauge length. To apply this definition, it is necessary to identify the bands in the specimens. These are shown schematically in Fig. 1. It will be noted that C26, cycled at a 4 4 0 0 CYCLES

50 0 0 0 CYCLES

(a) 2 7 5 0 CYCLES

(b) I 4 2 0 CYCLES

12 0 0 0 CYCLES

(c) Fig. 1. Surface slip traces associated with PSBs resulting from 1 cycle of tensile plastic strain after repolishing at the-numbers o f cycles indicated: (a) specimen C26, Ag'p ffi 0.0025; (b) specimen C14, A~/p ffi 0,005; (c) specimen C29, ATp ffi 0.01. (From ref. 3.)

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plastic shear strain amplitude of 0.001 25, contained one large macroband composed of smaller bands, sometimes very narrow and appearing as a single step of coarse slip b u t mostly in packets of easily observed width, with no slip between these smaller bands. Typical examples of packets of slip for this specimen produced by 1 cycle after repolishning are shown in Fig. 2. At present there is no terminology to distinguish the macroband's from the microbands; both are loosely referred to as PSBs. Accordingly we here introduce the terms "macro-PSBs" and "micro-PSBs". The plastic strain concentration for the macro-PSB in C26 after 30 000 cycles is 1.9, i.e. the shear strain in it is 0.24%, whereas the plastic strain concentration factor for micro-PSBs within this macroband shown in Fig. 2 are as follows (designating the bands 1 - 5 in Fig. 2 from left to right): band 1, 11; band 2, 14; band 3, 13; band 4, 13; band 5, 17; this means that the plastic shear strain amplitudes are 11 X 0.001 25% = 1.4%, 14 X 0.001 25% = 1.8%, 13 × 0.001 25% = 1.6%, 13 × 0.001 25% = 1.6% and 17 × 0.001 25% = 2.1% respectively. Within these micro-PSBs then, the variation in

Fig. 2. A reasonably uniform concentration of strain within micro-PSBs (of which five are visible in the field of view) in specimen C26 tested at ATp = 0.0025 for 30 000 cycles, repolished and then given a plastic tensile strain increment equal to its strain amplitude. In order to gauge the magnification in this and subsequent interferograms, it should be noted that the fiducial marks are 100 pm apart. The strong strain concentration in the lower half of the middle band, as well as the remnant of negative strain in the left band, should also be noted.

strain concentration is less than a factor of 2 and the individual micro-PSBs carry o f the order of 5 - 10 times as much strain as the macro-PSBs do because the strain in the macro-PSBs is the sum of the strains in the micro-PSBs, if the matrix strain [8] is disregarded.

Throughout this paper the strain accomm o d a t e d by the loop patches between the PSBs, and the attendant AG effect, is neglected. The specimens were tested at constant strain at zero load between successive half-cycles. Consequently, considering the whole sample, the strain a c c o m m o d a t e d b y the dislocations moving in the PSBs and giving rise to the macro-PSBs and micro-PSBs is the applied strain minus that due to loop "flipping" or reorientation outside the PSBs. The corresponding correction is less important at larger than at smaller plastic strains b u t can easily amount to a few hundredths per cent, even in the plateau region, as shown in ref. 8. However, since the fringes were calibrated with reference to the whole gauge section of the samples, the above values for the strain in the bands are not affected. Returning to the discussion of the micrographs it should be noted that C14 cycled at 7p = 0.0025 contained five macro-PSBs (Fig. 1). Within each, a large area was taken up with a seemingly continuous band of slip in which discrete steps, including indications of back slip, were frequently observed. A typical example of this type of macro-PSB is shown in Fig. 3. However, discrete steps and narrow packets of slip isolated between areas of no slip, similar to those observed in C26, were also observed, e.g. Fig. 4. The plastic strain concentration factors for three of the macro-PSBs (Fig. 1) were measured: C, 0.8; D, 1.7; E, 3.0 (with corresponding shear strain amplitudes of 0.2%, 0.43% and 0.75% respectively); a variation of a factor of 4 is evident. Reference to Fig. 1 shows that C29 (which has a shear strain amplitude of 0.005 and this

Fig. 3. Slip steps across a PSB (Fig. l(b), specimen C14, face 3, PSB E) (2750 cycles). Again the segment with negative strain on the levelof the fiducial marks at the right edge of the wide band should be noted.

(From ref. 3.)

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Fig. 4. Discrete slip steps and narrow micro-PSBs

between regions of non-slip, or perhaps up to a few hundredths per cent of dispersed slip due to loop reorientations, in specimen C14 (face 3) tested at ATp = 0.005 for 2750 cycles, repolished and given-~cycle in tension.

coincides with the uppe~ end of the plateau in the cyclic stress-strain curve) contained one continuous macro-PSB, a "super-PSB", which almost filled the whole gauge section. Similar to those of C14, this macro-PSB consisted mainly of a continuous deformation within which large steps were observed, together with some isolated narrow microPSBs and seemingly discrete steps. An example of this structure is provided in Fig. 5, again obtained from ~ c y c l e after repolishing. Comparison with Fig. 2 shows that the localized strain concentration, as judged by the slope of the interference fringes (the magnification is the same in these figures), is comparable for both specimens. A note of caution in making such a comparison is appro-

Fig. 5. Typical discrete steps, narrow micro-PSBs and wider micro-PSBs (showing continuous deformation) in specimen C29 (face 1) tested at ATp = 0.01 for 1420 cycles, repolished and then given 1 cycle in tenslon.

priate here: Fig. 2 was obtained with monochromatic light while white light was used for the fringes in Fig. 5. Moreover, in any interferogmm the separation of the fringes depends on the setting of the interferometer (compare Figs. 2 and 3), so that, in general, it might be quite misleading to compare the slopes of fringes of any pair of figures. It just happens that the comparison is approximately valid for Figs. 2 and 5. The above observations relate to single excursions of strain amplitude. Both under these circumstances and in excursions involving several cycles, many instances were observed where quite large steps were formed in very narrow bands. A typical example of these large strain concentrations is shown in Fig. 6. The specimen shown in this figure had been cycled 27633 cycles, repolished and then subjected to 11 cycles, thus being returned to the state of zero stress and zero strain. Since the repolishing zero for the slip steps was at the compressive strain limit, the slip steps observed at zero stress and zero strain for the specimen correspond to a single excursion of tensile strain amplitude.

Fig. 6. An intense concentration of plastic strain at forward as well as unreversed slip steps in specimen C14 tested at ATp = 0.005 for 2 7 6 3 3 cycles, repolished and then 11 cycles of plastic*strain, i.e. the end point was zero stress and zero strain.

It should be noted that the steps do not all have the same sign, and thus some of the steps were formed in the previous compression stroke; therefore they were thus wholly or partially unreversed. If only one atomistic slip plane was involved in forming each of these steps, which is unlikely but possible, then their plastic strain concentration factors would be of the order of 105 . We feel that a value of about 1000 is more realistic, assuming that the widths of the steps were comparable with those of the finer details of PSBs observed in repolished and restrained spec-

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imens by scanning electron microscopy and in as-fatigued single crystals by replication electron microscopy [10]. Another example taken from this specimen, for an identical stressing and polishing history, is shown in ref. 3, Fig. 11, but that example showed generally lower levels of strain concentration. Examination of the micrographs discussed above, and especially Fig. 6, shows that the shear strain is not constant along the individual slip lines. Clearly, unlike the PSBs as a whole, individual slip lines in PSBs do not pass right through the specimen. Indeed, individual slip lines within the PSBs are estimated to be only about 0.2 m m long; there is much variation in this value, and it is not possible to tell how long the lines are in the complicated middle regions of some micro-PSBs. It is important to emphasize the difference in the experimental approach used here from Winter's technique [2] for measuring PSB behavior. Since he did not repolish his specimens he could not discriminate between macro-PSBs and micro-PSBs, and his technique obscures the presence of regions between micro-PSBs in which no localized slip occurs during saturation, because such regions are covered with transient slip.

with increase in strain amplitude because the areas of fairly continuous deformation increased. The results, although rather subjective, are reported in Table 1. From an examination of the films and from these figures we conclude that (a) the number of micro-PSBs, like the volume fraction of macro-PSBs [2, 3], increases with applied strain amplitude and (b) the n u m b e r of microPSBs probably does not increase with cycling during saturation. We feel that the number of PSBs alone is not a sufficient measure of behavior and it is of at least equal importance to establish the total width of the micro-PSBs. However, to measure widths with any degree of accuracy would be nearly impossible for many of the bands owing to insufficient surface plane resolution in the interferograms.

3.2. Does the localized strain change during saturation ? In order to understand the mechanisms both of cyclic deformation and of fatigue failure, it is important to know whether or not the PSBs settle down to a much more uniform behavior after many cycles of saturation. Do the discrete steps die out, and does such behavior depend on the strain amplitude ? It will be recalled that three film strips of interferograms were recorded on both polished faces of each specimen [3]. The faces were chosen as those on which edge dislocations with the primary Burgers vector "emerged", i.e. those on which screw dislocations " k n i f e " along, perpendicular to the line of intersection between the surface and the slip planes. Using some of these records, we counted the number of micro-PSBs along the gauge sections of the specimens of interest. In this, a micro-PSB was taken to be any band that produced a quite noticeable fringe offset. The count became progressively more difficult

TABLE 1 The densities of micro-PSBs during saturation

Specimen Plastic shear identification strain amplitude

Number of Number of cycles micro-PSBs in a 12.7mm gauge length

C26 C26

0.00125 0.00125

4400 30000

C14

0.0025

2750

98

C29 C29

0.005 0.005

1420 12000

186 182

69 86

The dependence of plastic strain concentration factors on cycling during saturation can be determined by examining the slopes of slip offsets across specific micro-PSBs and dividing these values by the average step height gradient along the gauge length, namely 2.11, 4.81 and 9.91 white light fringe spacings per millimeter for C26, C14 and C29 respectively. In summary, on the basis of many examinations we believe that (a) for micro-PSBs of similar width the concentration factors did not alter during saturation (this can be seen by comparing the slopes of the fringe spacings in Fig. 7, taken after 4400 cycles, with those in Fig. 2, taken after 30 000 cycles) and {b) again for micro-PSBs of similar width the concentration factors were about the same for all specimens, independent of the applied strain amplitude or

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Fig. 7. Well-behaved micro-PSBs together with one sharp slip step in specimen C26 (face 1) early in saturation after it had been subjected to 4400 cycles, repolished and strained I cycle in tension. It should be noted that the fringes deviate in an opposite sense from those in the other figures. This does not mean that the steps are compressive, only that the interferometer was set up in the opposite mode.

depth of saturation. Let us compare Figs. 2, 4 and 7, allowing for the average step height gradient and the caution mentioned above in comparing Figs. 2 and 5. It is encouraging to have obtained these results, because the observed uniformity of PSB dislocation walls is consistent with them and so are other aspects of PSB theory [5]. It should be noted that the above comparison was limited to micro-PSBs of similar width, i.e. the well-behaved micro-PSBs in which the plastic strain does not vary from the accepted value of 0.01 by much more than a factor of 2. However, as noted in Section 3.1, the variation in this strain is enormous and we find the variation to be unchanged during the course of the saturation that we studied. As examples let us compare the interferograms in Fig. 2 and Fig. 7, which show well-behaved micro-PSBs, with those o f Fig. 6 and Fig. 8(a). In Fig. 8(a) there are very intense slip steps, together with regions in which the fringe slopes are quite shallow. Figure 8(a) corresponds to a very early stage of saturation while Fig. 8(b) is quite a late stage (the life of the specimen is about 55 000 cycles [3] ). We also believe that the variation in strain localization is independent of strain amplitude. For example, let us compare the interferogram shown in Fig. 9 with those of Fig. 4 and Fig. 8. These figures show typical results for the three specimens studied in detail. As an example of the magnitude of this variation,

(a)

(b) Fig. 8. PSBs containing well-behaved micro-PSBs and discrete steps in specimen C29, showing that the variation in localized strain does not change during saturation: (a) after 1420 cycles, repolished and strained 1 cycle in tension, face 3; (b) after 12 000 cycles, reapolished and strained 1 cycle in tension, face 1.

Fig. 9. Micro-PSBs of varying width and strain localization in specimen C26 (face 1) cycled at ATop= 0.0025 for 30 000 cycles, repolished and strained 4 cycle in tension.

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it should be noted that the micro-PSBs shown in Fig. 2 had strain concentration factors varying by about 1.5 (except the lower half of the line in the middle, for which the factor would be very large); of those shown in Fig. 9 for the same specimen and the same number of cycles, eight micro-PSBs were examined, and the concentration factors were found to vary by a factor of 9.4; for some of the bands in both figures we even found an inversion in the direction of the strain.

3.3. Dislocation behavior in persistent slip bands To produce the step heights shown in the various interferograms, large numbers of dislocations must intersect the surface. The number of dislocations involved and their average spacing can be calculated easily. Typical numbers for the bands shown in Fig. 2 are listed in Table 2. Thus for the five bands shown in this figure, on the average, one dislocation must have intersected the surface at a spacing of approximately 70 slip planes to produce the step height changes observed. In instances where there is a greater concentration of plastic strain, such as in Fig. 6, the spacing of the active atomistic slip planes to produce the observed step is much smaller. These figures are to be compared with the requirement of one dislocation sweeping over about every 1000 slip planes (for specimen C26) to achieve the applied plastic strain uniformly along the specimen. TABLE 2 Numbers of dislocations required to produce slip offsets

Band designation (Fig. 2, from left to right)

Number of Average number dislocations to of slip planes per produce step dislocation heights

1 2 3 4 5

2590 820 520 780 1180

83 66 69 70 55

The relevance of these findings to recently reported results will now be discussed. One point which emerges clearly is the difficulty of obtaining transmission electron microscopy (TEM) samples which are representative of re-

gions of extremely high strain concentration within PSBs, if we assume for the moment that these exist throughout the specimen and do not represent strain concentrations local to the surface. For example, if it was desired to cut sections parallel to the primary slip plane in order to study the arrangements of the screw dislocations which interlink the walls of the PSBs (and this could be carried out "under load" and after neutron irradiation to pin the dislocations in place) then only a limited number of specimens could be cut. With luck, one or two could be taken from C26 and perhaps five from C14. In the subsequent thinning of these specimens the TEM viewing plane would occur somewhere inside the micro-PSBs, but it has a low probability of coinciding with a region of the highest strain concentration within a micro-PSB even in high voltage TEM. Sections taken at an angle to the primary plane so as to include the primary Burgers vector would have a higher probability of locating a region of high strain concentration. Unfortunately such sections are of lesser value in identifying the signs of the screw dislocations. However, they have already served to show that the walls of the PSBs, at least in the centers of specimens, are reasonably uniform in thickness [2 - 4, 11 13]. The time-honored question of the representativeness of TEM observations, often raised in ignorance of the power of TEM, is thus seen to have limited validity in the context of studying PSBs. Although there appears to be agreement about the dislocation structure of PSB walls and the fact that screw dislocations are the agents for transferring edge dislocations from one wall to its neighbor [4, 5, 11], there is disagreement about how the screw dislocations so act. Kuhlmann-Wilsdorf and Laird [5] have described a mechanism in which screw dislocations, in groups of the same sign, move in coordination between PSB walls, so as to lay down simultaneously edge dislocations of opposite sign in dipolar configuration at the walls. They support this mechanism by arguments concerned with energetics [5], and also with early TEM observations by Mughrabi [14]. Mughrabi [4, 11], however, has championed an older mechanism [3, 5] in which the edge dislocations have repeatedly to be created by the bowing-out of the PSB walls and traversal of the channels between

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the walls. This mechanism predicts that the screw dislocations would equally be represented in opposing signs and thus would encounter each other by cross-slip and annihilate. Mughrabi's later TEM studies [4, 11] have tended to support this mechanism, but of course the observations would be subject to the criticisms noted above. We believe that both mechanisms occur but to different degrees and that the evidence presented here strongly indicates the coordinated motion of the screw dislocations. The most relevant evidence in this connection is that the slip line length in the PSBs is about 200 p m while the channel width of the PSBs is 1.3 ~m. Such long slip line lengths, and their slow variation in shear strain along the length, is fully consistent with a model of groups of screw dislocations gliding in coordination but cannot be explained by the bowing-out of edge dislocation loops from walls, since that would presuppose the prior existence of edge dislocation pile-ups in the wails. Furthermore, the slip lines do not traverse the whole specimen, i.e. opposing groups of screw dislocations do evidently meet on sufficiently closely spaced atomic planes that they can mutually annihilate by cross-slip. However, it seems clear that some edge dislocation bowing must also occur, i.e. screw dislocations are annihilated where they meet others of opposite sign and where they emerge at free surfaces. Therefore, although perhaps many screw dislocations move to and fro for several cycles, the majority are liable to be annihilated within any half-cycle. These must be regenerated. Some of them may come from free surfaces. Again, slip lines do not traverse completely through a sample and thus the majority of the screw dislocations must be regenerated in the interior. The only mechanism by which this can occur appears to be the bowing-out of edge dislocations. Because of the ordering of the edge dislocations into tilt wall-like arrangements in the PSBs [5 - 9], the screws re-form into their peculiar grouping. Now, let us consider the strain. If edge and screw dislocations are of comparable total line lengths (namely one or a few microns commensurate with the channel width per dislocation, and two screw dislocations formed by any one bowing edge loop), t h e y are bound to contribute glide in proportion to their mean

free paths, i.e. as 1.3/~m to I × 200/~m, i.e. as 1:100. Correspondingly, the bowing edge dislocations provide screw dislocations but contribute very little to the strain. Also, the evidence most strongly suggests that the stress of bowing does not control the flow stress at any point. Next it is necessary to examine dislocation behavior in the light of the other evidence obtained by interferometry. Even after thousands of cycles in saturation, regions of high strain concentration occur within PSBs, as we saw. Clearly, this is further strong evidence of coordinated dislocation motion which, on the basis of the preceding conclusions, must be that of screw dislocations. The question to be answered, then, is why the observed strong strain concentrations arise. The high strain concentrations are associated with the discrete slip lines. Two possibilities about these present themselves directly. Firstly, t h e y may be a surface effect, i.e. they do not penetrate into the interior of the sample b e y o n d the first one or two PSB walls, with the implication that the PSBs in the interior operate in accordance with the theory developed previously [5 - 9]. (It should be noted that, if the dislocations behaved at the surface as precisely as described in ref. 5 for the bulk, then the defor mation would be mechanically reversible and failure would not occur.) Such a concentration of glide at the surface from a distributed condition in the interior could readily arise via cross-slip into closely spaced slip planes. This cross-slip could be triggered by stress enhancement at slip steps, and because deep steps possess a lower surface energy than m a n y atomic-sized steps, on account of the excess surface energy of ledges. Cross-slip due to these two causes would be confined to the outermost few microns, b e y o n d which the dipolar structure and uniformity of the PSB walls would remain intact and enforce the coordinated behavior already described. This is the interpretation which we strongly favor at this point. At any rate, if the deep slip lines originated at (or were confined to) the surface, they could not be due to edge bowing because the propensity of the screw dislocations to annihilate would demand a large local supply of bowing links, and this not only would locally thicken the PSB walls but would cause large pile-ups of edge dislocations

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at the PSB walls. However, significant local wall thickness variations are rarely observed in the inner volumes of the material, and no pile-ups are seen. Furthermore, the stresses associated with pile-ups attendant on the formation of discrete steps b y gliding edge dislocations would be prohibitively large and would destroy the loop patch and PSB wall structure. The second possibility is that the deep slip lines are not a surface effect b u t represent what in essence would be ordinary unidirectional glide cutting through large numbers of dipolar walls. We believe this to be less likely b u t not impossible. This effect could be compatible with almost any theory of dislocation behavior ever proposed b u t would tend to disrupt the PSB wall structure near

(a)

the surface, for which there is no evidence. In order to make an informed judgment of this problem, statistics of the frequency of deep slip lines and of the distribution of strain concentration factors on t w o different sets of surfaces should be made on the same sample. Thus, if cross-slip of screw dislocations due to stress perturbations at surfaces were the essential factor, slip lines should be much more uniform where the surface makes a large angle with the Burgers vector. So far, except for the above brief discussion, theory has not y e t addressed [4, 5] the large variations in localized strain observed here. These variations pose no problem to the considerations in refs. 5 - 9 as long as the micro-PSBs are well behaved, i.e. as long as their strain concentrations vary by less than an order of magnitude with respect to the accepted value of 0.01 in a region in which the slopes of the interference fringes are gentle and continuous. These would simply require local thickening of the PSB walls. Luk~s and his coworkers [15, 1 6 ] , who have concentrated on observing near-surface volumes with TEM, have indeed observed variations in the thickness of PSB walls. A typical micrograph taken from their studies is shown in Fig. 10, together with a micrograph obtained by Woods [ 1 3 ] , taken from deeper within a specimen. Such variations in local PSB wall thickness would be consistent with variations in plastic strain of a factor of a b o u t 3-5.

4. CONCLUSIONS

(b) Fig. 10. Dislocation structures o f PSBs, showing variations in wall thickness in copper single crystals cycled at long life in tension-compression: (a) nearsurface foil of ~121)orientation (it was specifically identified as being taken about 50 pm below the surface); (b) deeper within a specimen. ((a) By courtesy of Luk~s e t al. [15] ; (b) by courtesy of Woods [13].)

A re-examination was made of detailed interferometric observations on which ref. 3 was based b u t which were published only in part, and the following conclusions are offered. (1) It is confirmed that the average strain localized in the PSBs of fatigued pure f.c.c. metals is a b o u t 0.01. (2) However, PSBs consist of macro-PSBs encompassing clumps of narrower microPSBs. The localized strain of the micro-PSBs can vary from a b o u t the accepted value of 0.01 to several times this value. (3) Micro-PSBs are either well behaved, i.e. the localized strain within them is uniform and within an order of magnitude of the

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accepted value, or can exist as discrete steps showing surface strain concentration factors of 1000 or more. (4) The interferometric observations favor a model of dislocation behavior in PSBs in which like-handed screw dislocations traverse the PSB channels in coordinated fashion during cycling. Edge dislocations bowing o u t of the PSB wails will replenish the populations of screw dislocations, most of which mutually annihilate every cycle. However, it is considered that the bowing-out stress is not controlling and bowing edge dislocations contribute only a b o u t 1% of the plastic strain. (5) Existing PSB theory [5] is consistent with the behavior of well-behaved microPSBs. The variations in strain observed are consistent with local variations observed by TEM in the thickness of PSB walls. (6) It is proposed that at the surface there exist marked differences in deformation behavior (e.g. extremely high strain concentrations) with respect to that of the bulk. This behavior is explained as a surface effect, in which groups of like-handed screw dislocations cross-slip into closely spaced slip planes because of stress enhancement b y slip steps. Consequently, large slip offsets are observed. The surface topography resulting from such differences in behavior is the primary cause of crack nucleation. (7) Although both macro-PSBs and microPSBs can, and c o m m o n l y do, pass right through the specimen, the micro-PSBs are shown to contain slip lines only a b o u t 0.2 mm in length. The variation in the strain along the lines is considerable.

ACKNOWLEDGMENTS The experimental work was originally supported b y the Army Research Office under Contract DAHC-04-74-G0026 and by the National Science Foundation under Contract NSF-5-25-880 under the auspices of the Materials Failure Thrust Area, Laboratory

for Research on the Structure of Matter, and latterly by the National Science Foundation (C.L.) under Grant DMR77-13934. J.M.F. acknowledges the award of an Australian Public Service Board Postgraduate Scholarship which provided the opportunity for this work. D.K.W. gratefully acknowledges the financial support of the Materials Sciences Division, Office of Naval Research, Arlington, VA.

REFERENCES

1 J. D. Atkinson, L. M. Brown, R. Kwadjo, W. M. Stobbs, A. T. Winter and P. J. Woods, Proc. 3rd Int. Conf. on the Strength o f Metals and Alloys, Cambridge, Cambridgeshire, August 20 - 23, 1973, Iron and Steel Institute, Institute of Metals,

London, 1974. 2 A. T. Winter, Philos. Mag., 30 (1974) 719. 3 J. M. Finney and C. Laird, Philos. Mag., 31 (1975) 339. 4 H. Mughrabi, in P. Haasen, V. Gerold and G. Kostorz (eds.), Proc. 5th Int. Conf. on the Strength o f Metals and Alloys, Aachen, August 1979, Pergamon, Oxford, 1979. 5 D. Kuhlmann-Wilsdorf and C. Laird, Mater. Sci. Eng., 27 (1977) 137. 6 D. Kuhlmann-Wilsdorf and C. Laird, Mater. Sci. Eng., 37 (1979) 111. 7 D. Kuhlmann-Wilsdorf, Mater. Sci. Eng., 39

(1979) 127. 8 D. Kuhlmann-Wilsdorf, Mater. Sci. Eng., 39 (1979) 231. 9 D. Kuhlmann-Wilsdorf and C. Laird, Mater. Sci. Eng., 46 (1980) 209. 10 J. M. Finney, Ph.D. Thesis, University of Pennsylvania, 1974. 11 H. Mughrabi, F. Ackermann and K. Herz, A S T M Spec. Tech. Publ. 675, 1979, p. 69. 12 H. Mughrabi, Mater. Sci. Eng., 33 (1978) 207. 13 P. J. Woods, Philos. Mag., 28 (1973) 155. 14 H. Mughrabi, J. Microsc. Spectrosc. Electron., 1 (1976) 571. 15 P. Luke, M. Klesnil and J. Krej~i, Phys. Status Solidi, 27 (1968) 545. 16 M. L u k ~ and M. Klesnil, in O. F. Devereux, A. J. MeEvily and R. W. Staehle (eds.), Corrosion Fatigue -- Chemistry, Mechanics and Microstructure, National Association of Corrosion Engineers, Houston, TX, 1971, pp. 118 - 132.