Cyclic stress—strain response of polycrystalline copper under fatigue conditions producing enhanced strain localization

MateriaLsScience and Engineering, tO0 ( 1988157 68

57

Cyclic Stress-Strain Response of Polycrystalline Copper Under Fatigue Conditions Producing Enhanced Strain Localization ZHIRUI WANG and CAMPBELL LAIRD l)epartment of Materials Science and Engineering. UniversiO'oJ'Pennsylvania. Philadelphia. PA 19104-(~202(U.S.A.) (Received May 26, 1987: in revised form July 21, 1987)

Abstract

It is" now es'tablis'hed that the cyclic stress-strain curve of polycrystalline wavy slip metal, e.g. copper, does not contain a plateau if the tests" are carried out in conventional strain control. Since fracture research indicates that such tests homogenize the strain, and tests initiated by ramp loading appear to give higher degrees of strain localization, an investigation of cyclic response has been made in polycrystalline copper using the ramp-start method. After specimens were ramped to a desired stress" level the)' were then tested in strain control. The results" showed a plateau to occur at 98 MPa, which corresponds" to the plateau shear stress for monocrystals corrected by the Taylor factor. Observations of persistent slip band behavior on the gage surfaces, and of the dislocation structures by transmission electron microscopy (7"k.'M) support the exis'tence of the plateau and demonstrate that it is associated with higher degrees of strain localization induced by starting the tests by ramp loading. 1. Introduction

In the last fifteen years there have been numerous studies of cyclic deformation using single crystals because they have been found especially helpful in understanding fatigue behavior. In order to put this information to practical use, there have also been attempts to relate the deformation behavior of single crystals to that of polycrystals. In this connection, Laird [1] showed that a strain fatigue limit inferred from monocrystalline behavior applied to polycrystals as well. One of the most marked features of monocrystalline cyclic response is the occurrence of a plateau in the cyclic stress-strain curve. Bhat and Laird [2] suggested, on the basis of early (and, as it later turned out, incomplete) data by Lukas and Klesnil [3], that polycrystalline material should also show a plateau. This suggestion was controversial, because (1(725-5416/88/$3.50

persistent slip bands (PSBs) had not been observed in the central regions of polycrystalline copper. However, systematic searches for PSB structures (including regular ladder structures) in polycrystals then revealed their existence in the bulk of the material [4-8]. However, measurements of the cyclic stress-strain curve by several workers yielded mixed results: ( 1 ) all interested workers now seem to agree that the curve, as measured by tests in constant strain control, does not contain a plateau [4, 6, 8, 9] in copper but the shape of the curve reflects PSB behavior; (2) Figueroa et al. [6] observed a plateau at about 70 MPa for tests carried out in load control, and they also showed by means of descending step tests that the shape of the curve is history dependent; (3) plateaus occur in the curves for polycrystalline AI-Cu alloy containing shearable precipitates, in which the localized strain is extremely high [10]; also, the shapes of the curves are dependent on grain size; (4) for a while it appeared that large-grained polycrystalline copper might show a plateau [11], but extensive testing by Lukas and Kunz [9] revealed this to be a multicrystal effect rather than a plateau for a true polycrystal. The question of plateau behavior in the cyclic stress-strain curves of polycrystalline copper is explored further here because of two recent findings: (1) Liang [12] has shown that intergranular crack nucleation in copper is promoted by homogeneous deformation and this mode of cracking occurs at low strains from cycling in strain control; this suggests in turn that a plateau does not occur for strain-controlled tests because the slip is not sufficiently localized; (2) Yan and coworkers [13, 14] have found that the cyclic deformation tends to be more strongly localized than in tests at constant amplitude if the tests are initiated by the ramp-loading method used by Neumann and his coworkers for many years. Accordingly, the ramp-loading method has been used here to control the strain © Elsevier Sequoia/Printed in The Netherlands

58 localization behavior, and the influence of history effects on cyclic response is reported.

2. Theoreticalconsiderations In cyclic deformation, as in monotonic deformation, it is possible to relate the flow stress of the single crystal to that of the polycrystal by at least three models, those of Taylor, Sachs and the maximum Schmid factor. All three can be formulated as: o = Mr where o is the applied normal stress in the polycrystal, r is the resolved shear stress for the single crystal and M is the orientation factor, 3.06, 2.24 and 2 for the Taylor, Sachs and Schmid factor models respectively, for f.c.c, material. The Taylor model, as is well known, assumes that the plastic strain is uniform through the material and the same as the applied strain. Thus the model considers compatibility, and allows up to five independent slip systems to act in a single grain. However, Kocks has pointed out [15] that there are problems about this model and the interested reader is directed to his paper for further details. The Sachs model treats the polycrystalline aggregate as an assembly of parallel but free single crystals. This arrangement will fulfill the conditions of yield and equilibrium, but will violate the compatibility conditions. The Sachs factor has been popular for applying to cyclic deformation because the plastic strain amplitude is small, compatibility can be accommodated elastically, and it is possible for only one slip system to be active in a grain, and usual for one to be dominant by a large margin. In applying these models here, all were considered in the experimental design. The ramp-loading method of initiating a test requires that the highest stress in the ramp be defined, that the load be increased uniformly and that a large number of cycles ( >i 20 000) be used in the course of the ramp. When the ramp is completed, PSBs may or may not have formed, depending on the maximum stress in the ramp, but the grains should be expected to be hardened uniformly by formation of dislocation loop patches [13]. In order to measure the cyclic stress-strain response for the ramp-loaded condition, it is possible to change the control mode of the test from load to strain and then to conduct an incremental step test in strain control. This procedure was followed here for three choices of maximum ramp stress given by the maximum Schmid factor, the Sachs factor, and the Taylor factor. The findings of Mughrabi and Wang [4] that the first PSB is formed at the stress given by the maximum Schmid factor, for tests in constant amplitude, lead

to the belief that a plateau can be expected at this stress amplitude, provided that the strain in cyclic deformation is sufficiently localized. The resolved shear stress (monocrystal) at which the loop patches can be expected to become unstable was taken as 32 MPa rather than the regular plateau stress of 28 MPa, because 32 MPa is the stress observed to nucleate PSBs for the ramp-loading method of initiating a test [13] under load control. Thus, the highest normal stress values in ramp loading for the polycrystalline material were 64, 72 and 98 MPa for the maximum Schmid factor, the Sachs factor and the Taylor factor respectively.

3. Experimentalprocedure An oxygen free high conductivity (OFHC) copper rod was machined into cylindrical threaded specimens with gage sections 12.7 mm in length and 6.25 mm in diameter. Before cycling, all specimens were annealed in a vacuum at 650°C for 3-3.5 h, and the resulting grain size was between 0.07 and 0.08 ram. In assessing the grain size, twin boundaries were counted as grain boundaries, since the grain boundaries themselves were rather fiat and straight after this annealing treatment. All specimens were electropolished before testing to allow the strain localization to be assessed by surface observation. Testing was carried out in a conventional MTS servohydraulic machine, except that, to obtain a ramp-loading start for the tests, two function generators were employed. A special device was designed to combine the signal from one generator with that from another so as to obtain a ramp-loading mode, shown in Fig. 1. Both the number of cycles and the maximum stress in the ramp could be adjusted to meet the requirements of the experiments. Three specimens were first ramp loaded under load control from zero MPa each to a different degree, namely to 64, 72 and 98 MPa. Following

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59 this, as mentioned above, the specimens were then tested under the strain-control mode on the same machine in order to measure the cyclic stress-strain response by the incremental step method. About 1500-2500 cycles were carried out at each strain step and this was found ample to describe the saturated condition. A fourth sample was directly tested by the straincontrol method to obtain the cyclic stress-strain curve for virgin material. This specimen provided the control measurement. For this specimen and all the others, the plastic strain amplitude was defined as half of the maximum width of the hysteresis loop. The frequency used for all tests was 1 Hz. This is much lower than the 34 Hz routinely used by Neumann and his coworkers for their tests. However, Yah and Laird have found that a low frequency does not affect the phenomena associated with the ramp-loading method [14] except in minor ways. The low frequency was preferred here to improve control and data recording. In order to observe the dislocation structures, another group of specimens was tested in the same way as that described above to provide a source of thin foils for observation by TEM. A conventional TEM sample preparation method was employed and thin foils were made from specimens tested to different stages with different testing modes. Mainly, two types of samples were made. In the first type, samples were cut from specimens which had just reached the maximum stress in the ramp. in the second type, samples were cut from those tested after different histories during the incremental step tests following ramp loading. Both longitudinal and transverse samples were examined. 4. Results and discussion

4.1. The polycrystalline cyclic response Cyclic hardening measurements for ramp-loaded polycrystals are rare. Since the plastic strain was monitored during the cyclic load-controlled ramp, it is possible to plot the relationship between the stress and the strain amplitudes during the ramp. This plot is shown in Fig. 2 for three typical specimens, the first of which was ramped to 98 MPa and the second to 120 MPa. As for the third one, the specimen was ramped to 98 MPa; the test was then not interrupted as was usual at the top of the ramp but continued under a constant stress of 98 MPa for a further 5000 cycles. During these cycles, the plastic strain amplitude gradually increased, but after 5000 cycles, the rate of increase had begun to lessen considerably. This is normal behavior if the

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maximum stress in the ramp is sufficient to create PSBs, Except for this behavior, all of the three curves in Fig. 2 show good agreement. Neumann and his coworkers have reported ramp hardening results for single crystals for many years and have consistently shown the occurrence of strain bursts. They have not reported bursts for polycrystals, however. We also did not observe bursts in the present polycrystalline specimens. Observing bursts is best done using a continual monitoring and recording device. In these tests, the strain behavior during ramping was observed by means of an oscilloscope. Within the limitations of such a tool used in this application, no bursts were observed. After ramping, as in regular virgin specimens, cyclic response was measured by step tests in strain control, allowing ample opportunity for saturation to be attained at each step. These experiments permitted measurement of the cyclic stress-strain response. The cyclic stress-strain curve (CSSC) of the specimen which was conventionally tested by the incremental step method and not ramp loaded is shown in Fig. 3(a). No plateau appears in the curve, but a bulge is found on it which extends from 60 to 100 MPa in stress amplitude and from 7 x 10 5 to 5 x 10 -4 in plastic strain amplitudes. This result is very similar to that reported by Lukas and Kunz [9j for tests in constant strain amplitude, and in reasonable agreement with the curve measured by Kuokkala using an incremental test [16]. As for the ramp-loaded specimens, the CSS curves were found to be quite different. Figures 3(b) to 3(d) show the CSS curves for the three different values for the maximum ramp stress. It is markedly noticeable that the strain-hardening rates of the three ramped specimens are much lower than that shown in Fig.

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Fig. 3. Cyclic stress-strain curves for polycrystalline O F H C copper having different histories: (a) annealed, incremental step-test; (b), (c) and (d) ramp-loaded to 64, 72 and 98 MPa respectively in 20000 cycles, and then subjected to incremental step tests in strain control. T h e stresses and strains corresponding to the peaks of the ramps are indicated. For comparison, the curve measured by Lukas and Kunz [9] for constant amplitude tests and that measured by Kuokkala [16] using an incremental test are provided in (a).

3(a). Furthermore, the CSSCs for the ramped specimens do show plateaus. These were particularly noticeable during incremental testing because, in the plateau, the stress was observed not to change when the strain amplitude was adjusted upwards to the next step. We therefore claim the existence of these plateaus very positively, although their extents are not as great as that of the monocrystalline plateau. The plateaus on the two curves shown in Figs. 3(c) and 3(d) are most pronounced and lie at 9 7 - 9 9 MPa in stress and 7 × 10 4 to 1 × 10 -3 in strain. The curve, in Fig. 3(b), for a ramp stress given by the maximum Schmid factor appears to have a weak plateau at about 72 MPa and shows other jogs in the curve at about 90 and 98 MPa. Although the plateaus are not well defined for this specimen, there are marked reductions in slope which reflect PSB behavior. Also, we believe that the final plateau at 98 MPa in this specimen would have been even more marked if its fracture had not terminated the test while it was in the plateau. In this specimen, 2500 cycles were applied at each

step, whereas only 1500 were used on the other two ramp-loaded specimens and they were able to sustain steps to still higher amplitudes. Comparison of the four CSSCs shows that ramp loading using at least 20 000 cycles in the ramp and an appropriate stress as the ramp peak, does condition polycrystalline material to behave, to some extent, like a single crystal. The plateau stress in the CSSC of the ramp-loaded specimens is defined as about 98 MPa, which fits the result of the Taylor model. Figure 4 shows the CSSC of all the histories studied here; the stresses and the strains have been transposed from normal values to the shear components using the Taylor factor. Also, the wellaccepted CSSC of monocrystalline copper is given in the same plot for comparison. It can be noted from Fig. 4 that the three curves of the ramp-loaded specimens are parallel to each other, implying that they have a similar hardening rate. The curve for the maximum Schmid factor is lowest and that for the Taylor factor is highest with the Sachs curve lying in between. This behavior

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must be the result of the initial ramp loading, which would produce dislocation loop patches with different sizes and densities, and, therefore, would harden the polycrystals to different levels, depending upon the peak ramp stress. T h e curve obtained without r a m p loading has a higher strain hardening rate than that of all the other three curves. Since the specimen had no previous hardening history, its flow stress at the lowest strain is the lowest because cyclic hardening under constant small amplitudes is not as effective as in ramp loading. T h e reduction in the hardening rate at higher amplitudes in the ramploaded specimens is attributed to strain localization and persistent slip behavior.

4.2. Surjace observations In order to check the slip band appearance, and to gain information on slip localization, observations of surface behavior were carried out. It was found that ramp loading alone, to the stress level of 64 MPa and to 2 0 0 0 0 cycles, does not cause slip bands to show in every grain (Fig. 5(a)). However, maintaining the sample at the peak stress of the ramp, for an additional 1 0 0 0 0 cycles would give rise to the appearance of slip in most of the grains (Fig. 5(b)). This behavior is consistent with the observations of N e u m a n n and coworkers who find that, in single crystals, extended cycling at a constant shear stress of 32 MPa will eventually cause PSBs to form [17]. T h e rate at which they form is much slower in single crystals than in polycrystals because of stress concentrations between neighboring grains. From Fig. 5(b), it can be seen that most of the grains which show slip bands have only one slip system operating. This is consistent with the Sachs theory. T h o s e grains which do not show any slip bands are usually smaller grains (Fig. 5(c)). This is

Fig. 5. Surface appearance of OFHC copper after different cycling histories: (a) after ramping in load control to 64 MPa in 20000 cycles; (b) maintained in load control after ramping for an additional 10000 cycles at 64 MPa; (c) small grains are less likely to show slip bands than large grains. Stress direction vertical.

ascribed to the result of elastic compatibility between adjacent grains where smaller grains have stronger constraints and, therefore, have a more complex stress state with the same m a x i m u m applied stress. After the specimens were step tested to higher stresses, as well as strains, the surfaces showed different features. Figure 6(a) illustrates a typical example from a specimen which was first ramp-

62

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Fig. 6. Slip-band morphology in OFHC copper first ramp loaded to 98 MPa and then incrementally step tested in strain control -- interruption stress 98 MPa and strain 1.1 × 10-3: (a) grain with several persistent slip bands; (b) PSBs cross a twin boundary; the PSB protrusions may be seen. Stress direction vertical.

loaded to 98 MPa and then incrementally step tested to the end of the plateau in the mode of strain control. The final strain amplitude in this specimen was 1.1 x 10 -3. The large grain in the central area, in Fig. 6(a), shows several families of slip bands operating. Slip band family P-1 is the strongest. It crosses a twin boundary (Fig. 6(b)) and finally extends to, but does not cross over, the adjacent grain boundary. Slip band family P-2, seen in Fig. 6(a), indicates that twin boundaries can cause PSBs to nucleate, since from these high magnification photographs the slip bands are easily recognized as PSBs. Figure 7 shows another example in which PSBs cross twin boundaries and cause the development of an interesting serrated notch. It is important to notice that the operation of slip band P - I ' (Fig. 6(a)) is obviously due to the incompatibility stress from the adjacent grains, this being high since the plastic strain amplitude is high. Figure 8(a) shows a

Fig. 7. PSBs cross a twin boundary and produce a damaging notch. Stress direction vertical. Same stress history as specimen in Fig. 5. (a) and (b) at different magnifications.

strongly operating PSB in a grain and a PSB caused by incompatibility stress which is located in the top right of the same grain near a triple point. Figure 8(b) gives an example of how a strongly operating family of PSBs impinges on a grain boundary (G.B.) and is completely blocked by the boundary. Microcracks have formed along the prominent PSBs and are clearly visible. In Fig. 9, two PSBs can be seen to operate on different systems, to meet and to cause microvoid formation at the point of intersection.

4.3. Dislocation structures

T E M observations made to analyze the dislocation structures associated with the different testing modes and stages, can be divided into two categories: (a) dislocation structures produced by ramp loading to different peak stresses, namely 64 MPa and 98 MPa; and (b) dislocation structures produced during subsequent measurement of the cyclic

63

Fig. 9. Interacting PSBs cause microvoid formation. Specimen ramp loaded to 98 MPa and then step tested to the end of the plateau. Stress direction horizontal. The fine horizontal lines are polishing marks.

Fig. 8. Slip-band morphology in OFHC copper first ramp loaded to 72 MPa and then step tested to the end of the plateau: (a) large-grain morphology; (b) PSBs blocked by a grain boundary. Stress direction horizontal.

response. The results are described in the two categories as follows: 4.3.1. Dislocation structures after ramp loading Figure 10(a) shows the dislocation structures after ramp loading a typical specimen to 64 MPa in 20 000 cycles. Dislocation loop patches with relatively low dislocation density appear everywhere with extraordinary uniformity. Depending upon the orientations of the grains with respect to the incident electron beam (and thus to the stress axis), loop patches may show different appearances in different grains. Figure 10(b) gives the details of the loop patches under higher magnification. If the peak

stress of the ramp were higher, one would expect that the loop patclaes would become more dense and vein structure would form. Figure 11 shows the results from a specimen ramp loaded to 98 MPa and obviously the loop patches have been densified to the stage of vein structure. The structures of these loop patches are interesting in that the exterior shells of the patches, which normally are denser than the interior, are unusually dense. Figure 12(a) shows the dislocation loop patches in three adjacent grains. The dislocation densities of loop patches and veins both in Fig. 11 and in Fig. 12 are clearly higher than that in the 64 MPa specimen. Figure 12(b) shows the same area as in Fig. 12(a) but with different orientation, the better to show the structures in the left lower grain. The structure in this grain contains "wall"-like elements (arrowed), implying that, at the end of ramp loading to 98 MPa, some grains have begun to form PSBs. Also, the loop patches show channels at a large angle to one another, a common feature when the density of secondary dislocations is high. Dislocation-free channels adjacent and parallel to certain grain boundaries are also visible. Such a feature is commonly observed in polycrystals [4-8]. The structures reported here are similar in essentials to those observed in monocrystals but are different from those previously reported for polycrystals in showing the uniformity of structures expected of monocrystals. 4.3.2. Dislocation structures during step tests in strain control As described in Section 4.1, CSSCs were measured after the specimens were ramp loaded to dif-

64

(b) Fig. 10. Dislocation loop patches uniformly distributed in a grain: (a) and (b) shown at different magnifications. Specimen ramp loaded to 64 MPA.

Fig. 11. Dense loop patches in a specimen ramped to 98 MPa.

ferent stresses. A specimen which had been ramp loaded to 98 MPa and then step tested in strain control to the plateau of 98 MPa was examined by T E M in both transverse and longitudinal sections. Both of these types of sections showed structures

typical of the upper part of the plateau, namely, loop patches, dipolar walls, PSBs and labyrinth structure. Since regular ladder-like PSBs have been reported previously for polycrystals [4, 5], we show here only two interesting special cases involving

65

Fig. 12. Loop patches in three adjacent grains, same specimen as in Fig. 1 I.

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Fig. 13. Typical ladder-like PSBs in the bulk of polycrystalline copper: (a) straddling a twin boundary; (b) contained entirely within a narrow twin; specimen rampqoaded to 98 MPa, and then step-tested in strain control to the midpoint of the plateau at 98 MPa.

twin b o u n d a r i e s (Fig. 13). In Fig. 13(a), a regular PSB is seen parallel to, and straddling, a twin b o u n dary, so that the PSB plane is the same as the twin plane. Such an example would c o r r e s p o n d to the case where strain localization occurs at a twin b o u n d a r y and crack nucleation occurs there. This

p h e n o m e n o n has been widely r e p o r t e d experimentally and the stress distribution necessary to cause it was recently calculated with F E M calculations [18]. A n o t h e r PSB parallel to a twin b o u n d a r y is shown in Fig. 13(b), but in this case the twin is very narrow, having the width of a typical PSB, and

66 the PSB is confined within, and defined by, the whole twin. If a crack were to nucleate in such a PSB, and did so at the most usual location, the interface between the PSB and its matrix, the crack would be positioned precisely on the twin boundary, although the localized deformation would be "entirely" beside the twin boundary. For a specimen cycled to the end of the plateau, well-developed cell structures appear in most of the grains (Fig. 14). The cells seen in Fig. 14 are either equiaxed or elongated to the point of being considered the dipolar walls of PSBs. However, the size of the cells and the width of the elongated cells is the same, approximately 0.5-0.6 /~m. It can also be seen in Fig. 14 that cell structures are formed uniformly in the grains and the cells can abut the grain boundary directly. Figure 15 provides another example of cells which, in addition to showing the general character of the structures, also shows that the orientation difference between groups of wellaligned adjacent cells is only one degree or so (compare (a) and (b) in Fig. 15). Such misorientations are commonly observed in single crystals cycled at the upper end of the plateau [19, 20]. Since the stress state may differ at different locations of a grain owing to the interaction stress between grains, the dislocation structure in the same grain was found to vary across its width. Figure 16 illustrates an example of this behavior. On the left side of the micrograph, dense dipolar walls from one slip system are evident; towards the right side, one sees more and more evidence of secondary dislocation operation and finally typical labyrinth structures are seen interacting with the primary slip system. This means, we believe, that

the slip in two different systems in this area is roughly equal. This is taken from a specimen also tested to the end of the plateau. Again, all these structures are typical of plateau behavior in monocrystals at the upper end of the plateau.

Fig. 14. Uniaxial and elongated dislocation cell structures, specimen ramp loaded to 98 MPa and incrementally step tested to the end of plateau.

Fig. 15. Dislocation cell structures, same specimen as in Fig. 14: (b) is approximately 1 degree different in orientation from (a).

5. Conclusions

From all the above observations and discussion, the following conclusions can be drawn: (1) A polycrystalline specimen tested in the ordinary mode of strain control does not show a plateau in its cyclic stress-strain curve. However, specimens which were initially ramp loaded all showed some type of plateau behavior and as long as the peak ramp stress for a specimen is higher than a certain value and the number of cycles in the ramp is large enough, in the present case, 64 MPa and 20 000 cycles respectively, it can be expected to show a marked plateau in its CSSC.

67

Fig. 16. Variation in dislocation structure from dipolar walls on the left to labyrinth structure on the right caused by a gradient in the secondary dislocation density. Ramp-loaded specimen, step tested to the end of the plateau at 98 MPa.

(2) The plateau extends from l x l 0 ) to 3 x 10 -~ in strain amplitude and the plateau stress o equals o = M r = 98 MPa (Fig. 4), where M = 3.(16 is the Taylor factor and r = 32 MPa is the plateau stress in the CSSC of monocrystalline copper measured by the ramp-loading method of initiating the test. (3) Ramp loading decreases the hardening rate as well as reducing the flow stress in the CSSC subsequently measured by techniques of strain control. (4) Within the plateau range of the CSSC of polycrystalline copper, compatibility slip bands appear frequently and many of them can become compatibility-induced PSBs. (5) The dislocation structures of ramp-loaded polycrystals are quite similar to those observed in monocrystals and much more uniform than those previously observed in polycrystals tested by conventional methods. In order to explain these results and conclusions, it is necessary to emphasize the connection between the observed plateaus and the other observations. Plateau behavior is observed when the following conditions are met: (1) PSBs are observed in the great majority of the grains; (2) PSBs are distributed uniformly in the gauge length; (3) incompatibility between different grains is accommodated by secondary slip. Compatibility slip, i.e. secondary slip bands, may well give rise to secondary PSBs and help to homogenize the PSB distribution. It is not apparently important whether or not the PSBs are formed by the externally applied stress or by compatibility-induced stresses.

The above conditions are met when the strain is high, i.e. corresponding to the upper end of the plateau observed for single crystals. Under these circumstances the continuity of localized slip through the polycrystal is sufficient to give rise to plateau behavior. When the strain is low, i.e. corresponding to the low strain end of the monocrystalline plateau, then the observed polycrystalline plateaus are either weak or non-existent. Under these circumstances, PSBs are indeed formed, but only in association with single slip and a minority of the most favorably-oriented grains. Increase in the applied strain cannot be carried by the existing PSBs because of increased back stresses from adjacent grains. Thus, secondary slip is excited in the already active grains, or as is more likely, less favorably oriented grains nucleate PSBs. Whatever the mechanism, the stress is required to increase and the plateau is suppressed. Also, there is no continuous path for localized slip as occurs in a single crystal after the formation of a single PSB. It is an interesting question why the Taylor stress (98 MPa) plateau does not usually appear in a conventional fatigue test and why ramp loading is necessary for polycrystalline material to show a plateau of the single-crystal type. This is explained as follows: in a conventional test, as the applied stress increases at the end of rapid hardening, or the applied stress is increased by a step test, the number of grains in which PSBs form also increases in proportion. The PSBs which form in the preferentially oriented grains are able to "absorb" more strain by

68 exerting a secondary stress on the adjacent grains. These two phenomena develop in synchrony. Thus, even at high strain, an increase in applied strain still requires a higher stress to activate more slip systems to overcome the incompatibility between grains. Moreover, since rapid hardening occurs especially rapidly in polycrystals subject to conventional tests, secondary dislocations are created earlier in the testing and act to produce large variations in the stress to nucleate PSBs. Thus the PSBs which form show a wide range of flow stresses. This militates against plateau behavior. However, if the specimen is ramp loaded to a chosen stress level in a large number of cycles, the loop patches accumulate more uniformly, the dislocation structure from grain to grain is more uniform, and the mix of primary and secondary dislocations is more uniform. Therefore the dislocation structure is somewhat more stabilized and the flow stresses of the PSBs are more homogeneous. When secondary slip occurs, it then also tends to form PSBs. Figure 17 illustrates the consequences of this argument schematically. Here, PA and PB are two primary slip lines in grains A and B respectively, and these lines may or may not have developed into PSBs during the ramp loading, depending on the peak value of the ramp. As the applied stress is increased during ramping, the increase in stress on a secondary system, resulting from the grain interaction in addition to the resolved component of the applied stress, will trigger SA and SB to operate. SA and SB are termed (grain boundary) compatibility slip lines. During the strain cycling which follows arrival at the peak of the ramp loading, the precursors of PSBs or actual PSBs, including both PA, PB and SA, SB, would serve as the sources of "new" PSBs giving rise to the formation of a plateau because their flow stresses would be very uniform. Furthermore, this interpretation forces the conclusion that the plateau stress

B

level must be consistent with the Taylor model, because compatibility slip is essential. The importance of compatibility slip then eliminates the Sachs model as the controlling mechanism. Choice of the Sachs stress for the peak ramp stress failed to produce an associated plateau.

Acknowledgments This work was supported by the National Science Foundation under Grant No. DMR 8513259. We are grateful for this support and also to the Laboratory for Research on the Structure of Matter which provided testing facilities under Grant No. D M R 85-19059. The active help of A. Radin and S. Macri in the development of instrumentation is greatly appreciated, as are the discussion of the personnel of the fatigue group at the University of Pennsylvania.

References 1 2 3 4

5 6 7 8

9 10 11

12 13 14 15 16 17 18

Fig. 17. Schematic illustration of the operations of secondary slip in two adjacent grains.

19 20

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