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Frequency effects on cyclic plastic strain of polycrystalline copper under variable loading
MATERIALS SCIENCE & ENGINEERING ELSEVIER
Materials Science and Engineering A194 (1995) 137-145
A
Frequency effects on cyclic plastic strain of polycrystalline copper under variable loading Herwig MayerCampbell Laird University of Pennsylvania, Department of Materials Science &Engineering, Philadelphia, PA 19104-6272, USA Received 21 April 1994; in revised form 20 July 1994
Abstract Studies of material frequency effects in cumulative damage are rare. Since new work has shown that the frequency of cycling affects persistent slip band (PSB) formation, with large effects on cyclic plastic strain in load control tests, step-descending tests have been performed at various frequencies. The results show that in regimes of stress-strain where loop patches and PSBs are the prevailing structures, cyclic deformation following a load-descending step is significantly affected by frequency. For example, a frequency of 0.5 Hz at the low level can increase the current plastic strain in the specimen by as much as nearly two times over the same step test performed at 8 Hz. If the prevailing structure from which the step test is initiated consists of cell structure, the effect of frequency differences is found to be small. These results, interpreted in terms of structural changes and secondary dislocation behavior, show that frequency effects could be significant in the accumulation of fatigue damage, especially in variable loading.
Keywords: Plasticity; Strain; Crystals; Copper; Fatigue, Frequency
1. Introduction
In a previous part of the present investigation [1], the role of frequency was explored for fatigue tests run under load control. Please see [1] for a review of the literature relating to frequency effects in cyclic deformation. It was found that frequency has a significant effect on the degree of localized strain. Low frequencies encourage the formation of persistent slip bands (PSBs). Thus, under load control, tests run at low frequencies show higher plastic strains than at higher frequencies. Only the deformation aspects of this behavior were studied; the consequences for fatigue fracture were not. However, it can be expected that a frequency effect on life to failure should follow such behavior. It was also found that the frequency effect, through its involvement with PSBs, depends on stress-strain
iOn leave from: Institute for Meteorology and Physics, Universitfit fiir Bodenkultur, Tiirkenschanzstrasse 18, A-1180 Vienna, Austria. 0921-5093/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSDI 0921-5093(94)09670-8
amplitude, because PSBs occur only in certain regimes of cyclic stress and strain. It seems reasonable to suggest, therefore, t h a t there should also be a frequency effect in tests conducted under variable loading. Reports of investigations into frequency effects in cumulative damage are scarce. Since the effect might be important in damage assessment if cycle-counting methods based on cyclic deformation or the rain flow approach are used, and could influence life behavior as well, we have preliminarily explored the cyclic deformation aspects of the frequency effect in variableloading fatigue and offer the following report.
2. Experimental details
The experiments were run on exactly the same batch of polycrystalline copper as used in the previous report [1], and details of the material, specimen and testing methods can be found there. Briefly, the copper of 99.99% purity was tested in the annealed condition, had a grain size of 65 ktm (not including twins, and 50 ktm if twins are included), and the following mono-
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tonic mechanical properties: 0.2% flow stress, 12 MPa; UTS, 205 MPa; RA, - 8 6 . An electropolished specimen of cylindrical geometry was tested under load control in an electrohydraulic machine using Wood's metal grips, and an extensometer was attached to the gauge length in order to monitor the strain. The present investigation involves making simultaneous changes in load amplitude and frequency during testing. We believe that such changes induce changes in dislocation structure that depend on load as well as frequency, and that these changes can be perceived indirectly by measuring the stress-strain hysteresis loop. However, a possible ambiguity could arise in the evaluation, if the hysteresis loop could be affected not only by the frequency-induced changes of dislocation structure but also by the rate sensitivity of these structures. Therefore, for evaluating the hysteresis loop, the same loading frequency of 0.5 Hz was always used. Fatigue tests were conducted to investigate the influence of cyclic frequency on the plastic strain behavior when the stress level was reduced. After cycling and making measurements at the reduced level, we would raise the stress to the former level and loading frequency, re-equilibriate at this level, and subsequently reduce the stress to another lower level/frequency, and so on. Thus, the same specimen was used in repeated tests, and results are also available for up-loading changes. The measurement focus, however, was on the effect of stress reduction. The specimens were initially ramp-loaded with linearly increasing stress amplitude, until the chosen test amplitude was reached, after which cycles at constant load amplitude were applied. Ramp loading was used to initiate the test for two reasons: (1) The test amplitudes were much greater than the flow stress of the annealed copper. Application of such amplitudes from the first cycle would have caused unacceptable deformations. (2) The rate of increase in the ramp can be used to control the dislocation structure, and especially a slow rate tends to produce homogeneous dislocation structures of loop patches, ideally adapted to forming PSBs. On completion of the ramp, the specimens were cycled at constant stress amplitude for about 20-30% of their expected lives to failure. The details of the various pre-treatments which were used are as follows: (1) The specimen was ramp-loaded with a linearly increasing cyclic stress amplitude of 0.018 MPa/ cycle ("long ramp"). After the stress amplitude reached 87 MPa, the specimen was loaded for another 55 000 cycles with this amplitude. For the ramp as well as for further cycling at constant stress amplitude, a frequency of 2 Hz was used. At a stress of 87 MPa, the saturated dislocation struc-
ture would consist of a mixture of loop patches and PSBs. (2) First a long ramp was applied and after the stress reached 94 MPa, the specimen was subjected to cyclic loading at this constant stress amplitude for 35 000 cycles. The loading frequency for the ramp as well as for the subsequent constant stress amplitude cycling was 2 Hz. At 94 MPa, the dislocation structure would contain not only loop patches and PSBs, but some grains would contain cells. (3) The increase of the ramp was 1 MPa/cycle up to a stress of 94 MPa (i.e. a much faster increase than in starts (1) and (2)), and a frequency of 0.5 Hz was used during the ramp-loading ("short ramp"). Subsequently, the specimen was loaded with a constant stress amplitude of 94 MPa, for 35 000 cycles, during the first 20 cycles using a cycling frequency of 0.5 Hz, and then 2 Hz. Because of the short ramp, there would be strong multiple slip during the test start, and the dislocation structures would be more cellular than for the long ramp. (4) A short ramp was applied first followed by constant stress amplitude loading at 110 MPa for 18000 cycles. The loading frequency during the ramp and for the first 20 cycles at constant stress amplitude was 0.5 Hz and afterwards 2 Hz. This stress amplitude is high enough to produce a wholly cellular dislocation structure. After the initial saturation produced by the abovedescribed treatments, the cyclic stress amplitude was reduced to a certain lower value. At this level the specimen was cycled either until the cyclic plastic strain amplitude showed no further change for increasing numbers of cycles or, if saturation could not be defined, for 25 000 cycles. The cyclic frequency for the lower level was either 0.5 Hz, 2 Hz or 8 Hz chosen for similarity to the frequencies used in the first part of the investigation [ 1]. After three experiments with the different frequencies at the same low stress level, some of the specimens were tested in a similar way using another low level. In detail, the specimen initially loaded at 87 MPa was tested at lower levels of 68 MPa, 74 MPa and 80 MPa. The specimen initially subjected to the long ramp and constant stress amplitude loading with 94 MPa was investigated at lower levels of 80 MPa and 87 MPa. For the stress level 94 MPa with initial short ramp, the lower load level was 80 MPa, and for the initial treatment at the high stress level of 110 MPa, the lower level was 94 MPa. The number of cycles at the initial stress level between two successive load reduction tests was at least 10 000 cycles in the case of the specimen tested at the high stress level 87 MPa, 8000 cycles or more for
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both specimens tested at the high stress level 94 MPa, and at least 3000 cycles when the high stress level was 110 MPa. These numbers of cycles were chosen to reproduce about the same cyclic plastic strain amplitude at the high stress level prior to all load-reduction tests of the same specimen.
3. Results
As detailed above, all specimens were initially ramploaded and then cycled at constant stress amplitude prior to the load reduction tests. The plastic strain amplitudes vs. numbers of cycles at constant stress amplitude for the different load levels and pre-treatments are shown in Fig. 1. For the stress level 87 MPa, the plastic strain amplitude subsequent to the long ramp decreased first and showed a minimum of 0.20 x 10 -3 after 5000 cycles. For higher numbers of cycles, the specimen softened and the cyclic plastic strain amplitude increased to 0.25 x 10 -3 after 55 000 cycles. According to the literature [2,3] this value for the plastic strain indicates ill-defined veins and PSBs as the dominant dislocation structure of this specimen. The cyclic plastic strain response for a stress level 94 MPa and a long ramp as pre-treatment is also shown in Fig. 1. A slight hardening in the first 2000 cycles leads to a minimum plastic strain amplitude of 0.30 x 10 -3. For higher numbers of cycles the specimen softened and after 35 000 cycles the plastic strain
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amplitude was 0.41 × 10 -3. Investigations of dislocation structures of similarly loaded specimen by Llanes and Laird [3] showed that, in some parts of the specimen, a dislocation structure of ill-defined veins and PSBs was dominant, whereas in other areas a cell and labyrinth structure was formed. The saturation behavior of a specimen first ramploaded with a short ramp and subsequently cycled at a constant stress amplitude of 94 MPa again is shown in Fig. 1. The plastic strain amplitude decreased after the ramp and showed a minimum of 0.45 × 10 -3 in the regime of 1500-1800 cycles. For higher numbers of cycles, the specimen softened slightly and the plastic strain amplitude increased to 0.48 × 10 -3 after 35 000 cycles. This plastic strain amplitude indicated that both a structure of ill-defined veins and PSBs as well as a cell structure will be present in some parts of the specimen. Nevertheless, due to the higher initial plastic strain after the short ramp, the volume fraction of cell structure will be larger than that in the specimen cycled with the initial long ramp [3]. The results of plastic strain amplitude vs. numbers of cycles for a specimen loaded with a short ramp and subsequently cycled with a constant stress amplitude of 110 MPa (Fig. 1) show that the specimen hardened first and a plastic strain amplitude of 0.90× 10 -3 developed after 250 cycles. In the range from 250 cycles to 5000 cycles, the specimen softened slightly and for larger numbers of cycles it hardened again. After 18000 cycles a plastic strain amplitude of 0.9× 10 -3 was measured. This cyclic plastic strain amplitude as well as the relatively high initial plastic strains subsequent to the ramp pre-treatment indicate that the dislocation structure of the specimen consisted of cells [2,3]. After loading the specimens with a ramp and constant stress amplitude, load reduction experiments were initiated. Fig. 2 shows the plastic strain response of a specimen first cycled at 87 MPa and 2 Hz (high level) and then at 68 MPa and 0.5 Hz (low level). In the regime from 112 000 cycles to 117 000 cycles the plastic strain amplitude for the constant stress amplitude of 87 MPa was quite constant at 0.254 x 10 -3. After the load and frequency reduction, the plastic strain amplitude at the low level shows that the specimen hardened and then softened. After 25 000 cycles at the low level, the stress was increased to 87 MPa again, and the frequency increased to 2 Hz. As indicated, from 142000 cycles to 156000 cycles the plastic strain amplitude at the high level showed hardening first and subsequent softening to a value for the plastic strain of 0.253 x 10 -3. This plastic strain amplitude is about the same as determined in the range from 112 000 cycles to 117 000 cycles previous to the load reduction experiment, indicating that history
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Materials Science a n d E n g i n e e r i n g A 1 9 4 (1995) 1 3 7 - 1 4 5
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independence applies in the cyclic response when the amplitude is increased. To simplify the comparison of the cyclic plastic strain response for different lower levels and different frequencies, the cyclic response at the lower levels for all tests starting with the same high level are summarized in one figure. For example the plastic strain for all load reduction tests with a high stress level of 87 MPa are shown in Fig. 3. T h e abscissa indicates the numbers of cycles at the low level after load reduction. T h e cyclic strain amplitudes after load reduction to lower load levels of 68 MPa, 74 MPa and 80 MPa are conveniently separated in the y-direction. T h e maximum value shown on the y-axis of 0.252 x 10 3 indicates the mean plastic strain amplitude for the high level 87 MPa previous to the load reduction experiments. T h e same specimen was used to make all the load reduction tests from 87 MPa and it is necessary to report that the plastic strain amplitude returned faithfully to about 0.25 x 10 -3 at 87 MPa after each experiment. In detail, the strain lay between 0.250 x 10 -3 and 0.254 x 10 3 At the low level of 68 MPa, the specimen hardened during the first 100 cycles for all three frequencies examined. (Measurements of the plastic strain amplitude at the low level for frequencies larger than 0.5 Hz started after 50 cycles (2 Hz) or 100 cycles (8 Hz), because the measurement p r o c e d u r e made it necessary to reduce frequency to 0.5 H z for about 3 cycles. Since this might have had an influence on the saturation behavior at the low stress level for low numbers of cycles, the evaluation was started later.) After about 100 cycles, hardening stopped in the case of a loading frequency of 0.5 Hz and the plastic strain amplitude was about constant for the next 5000 cycles. For higher numbers of cycles, the specimen showed p r o n o u n c e d
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softening, which led to a plastic strain amplitude of 0.063 × 10 -3 in the range between 1 5 0 0 0 cycles and 25 000 cycles. For the loading frequency of 2 Hz, the specimen hardened for numbers of cycles less than 1000 and then the plastic strain amplitude of 0.043 × 10 -3 became constant. For a frequency of 8 H z at the low level the initial hardening continued up to 5000 cycles, before the plastic strain amplitude became constant at 0.035 x 10-3. At the lower stress levels of 74 and 80 MPa, the initial hardening after load reduction was less pronounced (almost absent at 80 MPa) compared to behavior at the low level of 68 MPa. T h e hardening was greater and more profound for higher numbers of cycles, the greater the frequency. Softening subsequently developed as cycling was continued for every condition, but was always greatest for the lowest frequency. Softening was particularly p r o n o u n c e d at 74 MPa and 0.5 H z (Fig. 3). Table 1 shows the details of the plastic strain amplitudes for the various frequencies. T h e cyclic plastic strain response after rapid load reduction for a specimen initially loaded with a long ramp and constant cyclic stress amplitude of 94 MPa is shown in Fig. 4(a). T h e maximum plastic strain value indicated at the y-axis is 0.417 x 10 -3, which is the mean value for the plastic strain amplitude at the high level prior to the load reduction experiments. (To
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Materials Science and Engineering A194 (1995) 137-145 0.417
Table 1 Plastic strain amplitudes measured at lower stress levels after stress reduction
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simplify c o m p a r i s o n of the results o f load r e d u c t i o n tests for different high a n d low levels, all results (Figs. 3 and 4(a)-(c)) s h o w the s a m e logarithmic division for the x- as well as the y-axis.) T h e range of the saturation values m e a s u r e d during cycling at 94 M P a a n d 2 H z was (0.412 to 0 . 4 2 2 ) × 10 -3. T h e cyclic plastic strain r e s p o n s e after load r e d u c tion to 80 M P a for a s p e c i m e n initially l o a d e d with a short r a m p and c o n s t a n t stress a m p l i t u d e of 94 M P a is p r e s e n t e d in Fig. 4(b). T h e m a x i m u m value indicated o n the y-axis is 0.461 × 10 -3, w h i c h is the average cyclic plastic strain a m p l i t u d e for 94 MPa, p r i o r to the load r e d u c t i o n experiments. Similar results after load r e d u c t i o n to 94 M P a for a s p e c i m e n initially l o a d e d with a short r a m p and c o n s t a n t stress a m p l i t u d e of 110 M P a are given in Fig. 4(c). T h e m a x i m u m value indic a t e d on the y-axis is 0 . 8 4 3 × 10 -3, a v e r a g e d as usual. Similar to the results s h o w n in Fig. 3, t h o s e for the load r e d u c t i o n tests p r e s e n t e d in Fig. 4(a)-4(c) s h o w an
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influence of frequency at the low level on the course of the plastic strain amplitude. This influence is generally more pronounced the greater the load reduction, as can be seen in Fig. 3 and Fig. 4(a). After a load reduction has been made, some hardening occurs immediately but the cyclic response then gives way to softening. The number of cycles over which hardening occurs is greater the higher the frequency and the subsequent softening is more marked at lower frequencies. Comparison of the cyclic response after a load reduction of order 15% from 87 MPa, 94 MPa (long and short ramp) and 110 MPa (lower levels: 74 MPa, 80 MPa and 94 MPa) shows that the influence of frequency on saturation is more pronounced for lower initial load levels. Generally softening at the low amplitude is more pronounced the greater the initial stress. Comparison of results between Fig. 4(a) and 4(b) for the same constant amplitude but different ramps shows that a short ramp leads to slightly more pronounced softening at the low amplitude for all frequencies. Nevertheless, the frequency influence on softening shows no significant difference for both initiating ramps. Table 1 tabulates the details of the measured plastic strain amplitudes for the various testing conditions.
4. Discussion
The influence of frequency on cyclic plastic strain response after a load reduction, the results show, is more pronounced the lower the upper level from which the reduction is started and the larger the reducing step (Figs. 3 and 4). In the light of the dislocation structure this means that a structure of ill-defined veins and PSBs (Fig. 3) shows a pronounced influence of frequency after load reduction whereas, in a cell structure, the loading frequency at the low level is of minor influence (Fig. 4(c)). When the cyclic response after load reduction from 87 MPa (Fig. 3) is analysed in more detail, it is seen that the specimen tends to initial hardening, which is extended for higher frequencies, and softening follows, more pronouced for lower frequencies. The mechanism of softening or hardening after load reduction for a matrix-PSB dislocation structure was studied by Ma and Laird [4,5] with load reduction experiments using copper single crystals. Investigations of PSB activity showed that a decrease of stress amplitude from 28 MPa (plateau stress) to values lower than 25 MPa caused passivation of most of the PSBs and a significant reduction of the plastic deformation in still active PSBs. This activity declined for higher numbers of cycles. Die-out of PSB activity is an indication of
hardening and is combined with changes in the dislocation structure of the PSBs, namely an increase of wall thickness and formation of debris in PSB channels [4]. When the lower level is at least 25 MPa [5], PSBs are potentially active, and reactivating of PSBs, starting from areas of currently active PSBs, causes softening of the specimen. Nevertheless, the plastic strain of the PSBs after load reduction stays below the average localized strain of 0.01, typical for constant-amplitude loading. A process similar to that of single crystals may be expected in polycrystals but the process will certainly be influenced by grain boundaries and grain orientation. Therefore, it may be expected that at the low level 80 MPa most of the PSBs will be potentially active and therefore can be reactivated in sufficiently high numbers of cycles. In contrast, at the lower level 68 MPa, the resolved shear stress in the slip direction of the PSBs in many grains will remain under the threshold stress for reactivation and PSB die-out will have a pronounced influence. Only PSBs contained in grains with favorable slip orientations and relatively high Schmid factors will be reactivated. The influence of cyclic frequency at the low level is superimposed on this general behavior. For all low levels, sufficiently high numbers of cycles lead to softening when the cycling frequency is 0.5 Hz, whereas softening is less pronounced and starts at higher numbers of cycles, if at all, in the case of 8 Hz. Softening is caused by the spreading of active PSBs in suitably oriented grains, and reactivation of other PSBs, processes which are obviously more probable the lower the loading frequency. The mechanism for reactivation is not clearly understood, but changing of the matrix structure between currently active PSBs and potentially active PSBs could be responsible for increasing the volume fraction of active PSBs. Change in the matrix structure is expected to be caused by emission of secondary dislocations from currently active PSBs. In any case, the spread of active PSBs should be strongly affected by the frequency of cycling. This hypothesis is supported by two arguments: (1) It was previously shown that increasing the frequency from 0.5 Hz to 2 Hz in a well-developed matrix-PSB structure leads to an average reduction of plastic strain amplitude of 4%, and a further increase from 2 Hz to 8 Hz reduces the plastic deformation by another 5.5-7% [1]. After load reduction, besides the decrease of plastic deformation due to the negative step, higher frequencies reduce the plastic deformation of the PSBs additionally. This makes it more probable that PSB dieout will occur for higher frequencies than for lower ones. A hint for the validity of this explanation is provided by the fact that the saturation strain after load reduction from 87 MPa to 80 MPa is 6%
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lower for 2 Hz than for 0.5 Hz, whereas a comparison between 2 Hz and 8 Hz shows a decrease of 15%. This reflects the more pronounced influence on plastic strain amplitude when frequency is changed from 2 Hz to 8 Hz compared with changing from 0.5 Hz to 2 Hz. (2) The process of emission of secondary dislocations at the boundary between currently active PSBs and matrix involves thermally activated processes. Since these kinds of processes are more likely for lower frequencies, the emission is more probable for lower than for higher frequencies. Therefore, a reactivation of potentially active PSBs is more probable for lower than for higher frequencies. For a cell structure, developed at the stress level 110 MPa (Fig. 4(c)), the frequency influence after load reduction is far less pronounced than for the structure containing PSBs discussed above. Studies of saturation behavior after load reduction for a cell structure were performed by Lukfis and Klesnil [6] and Figueroa and Laird [7]. Both sets of workers agree that a cell structure formed initially at a high level is stable after a load reduction to a stress level corresponding to that for which a virgin specimen would develop matrix-PSB structure. For some conditions of cycling and microstructure, especially in the interiors of larger grains, the cell morphology can change from an equiaxed form at the higher level to a more elongated form at the lower level. Nevertheless, stress levels necessary to cause this change of cell shape lie below the stresses studied in the present work and therefore this effect is considered to be of minor importance here. Plastic deformation in a cell structure is presently understood by deformation in the cell interior first, and as the stress increases plastic deformation of the hardened cell walls subsequently develops [8]. After load reduction, softening of the cell structure can be hardly affected by frequency of loading. To explain why dislocation glide is nevertheless somewhat easier at lower frequencies, we point out that the activation of such a process may be affected by internal stresses as well as by thermal activation, which is more probable for lower frequencies. When load reduction from the high level 94 MPa is analysed, a more complex dislocation structure, containing matrix-PSB areas as well as volumes with cell structures, must be taken into account. When it is considered that the plastic deformation of a specimen containing different dislocation structures derives from the plastic deformation of respective parts [1], the results discussed above can be taken to explain the frequency influences. On this basis, an influence of frequency on saturation behavior after load reduction can be expected due to the volumes containing
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matrix-PSB structure, but the influence will be less pronounced than in the case of the high level 87 MPa, because softening of the volumes containing cells will be only slightly affected by frequency. This general description fits quite well with the results shown in Fig. 4(a). A given reduction of the stress amplitude ( ~ 15%) from 94 MPa leads to a less pronounced influence of frequency than the same percentage of load reduction starting from 87 MPa, and the influence is more pronounced than after a similar percentage of load reduction starting from 110 MPa. The results for the specimen reported in Fig. 4(b) can be taken to support the validity of this simple model. To increase the volume fraction of cells, this specimen was initially loaded with a short ramp and subsequently cycled at the constant amplitude of 94 MPa. After load reduction of -~ 15% to 80 MPa, this specimen showed a more pronounced influence of frequency than in the case of the high stress 110 MPa (Fig. 4(c)), and the frequency influence after load reduction is less pronounced compared with the negative step loading from 87 MPa to 74 MPa (Fig. 3). Nevertheless, the difference from the specimen initially loaded with a long ramp (Fig. 4(a)) is relatively small, except that the softening was somehow more pronounced, which may have been caused by the increased volume fraction of cells. An interesting point can be noted from comparing the numbers of cycles necessary to stimulate softening after stress reductions from 87 MPa and 94 MPa to a low level 80 MPa. For high level 94 MPa (Figs. 4(a) and (b)) softening starts after about 4 0 - 7 0 cycles for 0.5 Hz, between 100 and 200 cycles for 2 Hz, and 4 0 0 - 4 5 0 cycles for 8 Hz. Similar numbers of cycles are necessary to cause softening in the case of stress reduction from 87 MPa to 80 MPa. This comparison indicates that the onset of the spreading of the active PSBs is more dependent on the current load and frequency than on the percentage of load reduction. For all negative step tests the scope of the cyclic response after load reduction shows initial hardening, which is more pronounced for greater negative steps. One reason for the initial hardening is related to the Bauschinger effect. The load level was routinely changed from the upper level to the lower level when the actual stress was zero, subsequent to compression, which caused in the first cycles at the low level a larger plastic deformation during tension than during compression. This behavior increased the overall plastic deformation in the first cycles after load reduction. Nevertheless, after 10 to 20 cycles the stress-strain loop adjusted to zero mean plastic strain and the influence diminished. In the case of load reduction from 87 MPa to 68 MPa and loading frequency 8 Hz, initial hardening continued up to more than 1000 cycles. Hardening for
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such high numbers of cycles cannot be explained by the Bauschinger effect but die-out processes of PSB activity cause the decrease of the plastic strain amplitude. An influence of frequency on PSB activity after load reduction, shown by Ma and Laird [5], is that the number of temporarily active PSBs increases with high frequencies. This may serve to explain why the unfavorable influence of high frequencies for PSB activity is less pronounced initially after load reduction from 87 MPa than for continual cycling. Nevertheless, for higher numbers of cycles temporarily active PSBs die out, and therefore this effect does not contribute to frequency-induced differences in the saturation strain. It would be interesting to explore whether or not the results obtained by Ma and Laird [5] on single crystals of copper can be applied to polycrystals at still lower levels than 68 MPa. For example, since the stress required to activate PSBs in polycrystalline copper is twice the monocrystalline plateau stress (for the maximum Schmid factor), the stress necessary to prevent total die-out in PSBs produced at a higher level can be expected to be twice 25 MPa. Predicting life behavior in cumulative fatigue damage under variable loading is difficult because many factors are involved, and test results are subject to heavy scatter. The influence of frequency has usually been neglected in the evaluation of cumulative damage, except in the broadest of probabilistic models [9], and the influence of frequency, in the present context, appears to have been ignored even for accelerated testing [9]. A typical procedure used in accelerated testing is to increase frequency; the present results indicate that such a procedure could be non-conservative, even in the absence of such mechanical effects as temperature increase at higher frequencies due to internal friction, and environmental effects caused by frequency-environment interaction. The present results also show that, under load control, frequency can significantly influence the degree of localized strain, and thus failure mechanisms. Moreover, frequency effects in conjunction with load variations can influence the current plastic strain levels. This behavior indicates that making use of the cyclic stress-strain curve for cycle counting and damage assessment, without reference to frequency, is bound to introduce errors, and explains in part the wide prevalence of scatter in cumulative damage test results.
5. Conclusions Measurements of cyclic response obtained in step descending and ascending tests performed at various
frequencies lead to the following conclusions: (1) A dislocation structure of ill-defined veins and PSBs shows a pronounced influence of frequency after load reduction, whereas the plastic strain response after a descending step in a cell structure is relatively insensitive to frequency. (2) When descending steps occur in the presence of PSBs/veins, the activity of a fraction of the formerly active PSBs tends to die out initially. With extended cycling, the PSBs in more voluminous parts of favorably oriented grains gradually become more active; spreading of active PSBs is preferred at low frequencies because secondary slip is facilitated. (3) The relative insensitivity of the frequency effect in structures containing cells is caused by the greater resistance of cell structures to change when the load is reduced. (4) Improved methods of cycle counting and damage assessment in long-life, variable-amplitude fatigue will require consideration of the frequency effect on the cyclic plastic strain, as a significant material property.
Acknowledgments We are grateful to the National Science Foundation for supporting this work under grant No. D M R 9014381. Facility support was provided by the Laboratory for Research on the Structure of Matter under NSF Grant No. D M R 91-20668. Generous grants to one of us (H.M.) from several sources--the Vorarlberger Landesregierung, the Exportakademie der Bundeswirtschaftkammer and the Fulbright Scholar Program--encouraged the collaboration which made the work possible. We also thank our colleagues in the fatigue group at the University of Pennsylvania for helpful discussions and machine time. References [1] H. Mayer and C. Laird, Influence of cyclic frequency on strain localization and cyclic deformation in fatigue, Mater. Sci. Eng. A, 187(1994) 23-35. [2] J.C. Figueroa, S.P. Bhat, R. de la Veaux, S. Murzenski and C. Laird, The cyclic stress-strain response of copper at low strains--I. Constant amplitude testing, Acta Met., 29 (1981) 1667-1678. [3] L. Llanes and C. Laird, Effect of loading mode on the cyclic response and the associated substructure of polycrystalline copper in the high-cycle regime, Fatigue Fract. Eng. Mater. Struct., 16 (1993) 165-179. [4] B. Ma and C. Laird, Dislocation structures of copper single crystals for fatigue tests under variable amplitude, Mater. ScL Eng., 102 (1988) 247-258.
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[51 B. Ma and C. Laird, Overview of fatigue behavior in copper single crystals--IV. Strain and load interaction effects for tests under variable amplitude loading, Acta Met., 37 (1989) 357-368. [6] P. Lukfis and M. Klesnil, Cyclic stress-strain response and fatigue life of metals in low amplitude region, Mat. Sci. Eng., 11 (1973) 345-356.
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[7] J.C. Figueroa and C. Laird, The cyclic stress-strain response of copper at low strains--II. Variable amplitude loading, Aeta Met., 29 (1981) 1679-1684. [8] H. Mughrabi, Dislocation wall and cell structures and long range internal stresses in deformed metal crystals, Acta Met., 31 (1983) 1367-1379. [9] J.L. Bogdanoff and F. Kozin, Probabilistic Models of Cumulative Damage, J. Wiley & Sons, New York, 1985.
Materials Science and Engineering A194 (1995) 137-145
A
Frequency effects on cyclic plastic strain of polycrystalline copper under variable loading Herwig MayerCampbell Laird University of Pennsylvania, Department of Materials Science &Engineering, Philadelphia, PA 19104-6272, USA Received 21 April 1994; in revised form 20 July 1994
Abstract Studies of material frequency effects in cumulative damage are rare. Since new work has shown that the frequency of cycling affects persistent slip band (PSB) formation, with large effects on cyclic plastic strain in load control tests, step-descending tests have been performed at various frequencies. The results show that in regimes of stress-strain where loop patches and PSBs are the prevailing structures, cyclic deformation following a load-descending step is significantly affected by frequency. For example, a frequency of 0.5 Hz at the low level can increase the current plastic strain in the specimen by as much as nearly two times over the same step test performed at 8 Hz. If the prevailing structure from which the step test is initiated consists of cell structure, the effect of frequency differences is found to be small. These results, interpreted in terms of structural changes and secondary dislocation behavior, show that frequency effects could be significant in the accumulation of fatigue damage, especially in variable loading.
Keywords: Plasticity; Strain; Crystals; Copper; Fatigue, Frequency
1. Introduction
In a previous part of the present investigation [1], the role of frequency was explored for fatigue tests run under load control. Please see [1] for a review of the literature relating to frequency effects in cyclic deformation. It was found that frequency has a significant effect on the degree of localized strain. Low frequencies encourage the formation of persistent slip bands (PSBs). Thus, under load control, tests run at low frequencies show higher plastic strains than at higher frequencies. Only the deformation aspects of this behavior were studied; the consequences for fatigue fracture were not. However, it can be expected that a frequency effect on life to failure should follow such behavior. It was also found that the frequency effect, through its involvement with PSBs, depends on stress-strain
iOn leave from: Institute for Meteorology and Physics, Universitfit fiir Bodenkultur, Tiirkenschanzstrasse 18, A-1180 Vienna, Austria. 0921-5093/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSDI 0921-5093(94)09670-8
amplitude, because PSBs occur only in certain regimes of cyclic stress and strain. It seems reasonable to suggest, therefore, t h a t there should also be a frequency effect in tests conducted under variable loading. Reports of investigations into frequency effects in cumulative damage are scarce. Since the effect might be important in damage assessment if cycle-counting methods based on cyclic deformation or the rain flow approach are used, and could influence life behavior as well, we have preliminarily explored the cyclic deformation aspects of the frequency effect in variableloading fatigue and offer the following report.
2. Experimental details
The experiments were run on exactly the same batch of polycrystalline copper as used in the previous report [1], and details of the material, specimen and testing methods can be found there. Briefly, the copper of 99.99% purity was tested in the annealed condition, had a grain size of 65 ktm (not including twins, and 50 ktm if twins are included), and the following mono-
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tonic mechanical properties: 0.2% flow stress, 12 MPa; UTS, 205 MPa; RA, - 8 6 . An electropolished specimen of cylindrical geometry was tested under load control in an electrohydraulic machine using Wood's metal grips, and an extensometer was attached to the gauge length in order to monitor the strain. The present investigation involves making simultaneous changes in load amplitude and frequency during testing. We believe that such changes induce changes in dislocation structure that depend on load as well as frequency, and that these changes can be perceived indirectly by measuring the stress-strain hysteresis loop. However, a possible ambiguity could arise in the evaluation, if the hysteresis loop could be affected not only by the frequency-induced changes of dislocation structure but also by the rate sensitivity of these structures. Therefore, for evaluating the hysteresis loop, the same loading frequency of 0.5 Hz was always used. Fatigue tests were conducted to investigate the influence of cyclic frequency on the plastic strain behavior when the stress level was reduced. After cycling and making measurements at the reduced level, we would raise the stress to the former level and loading frequency, re-equilibriate at this level, and subsequently reduce the stress to another lower level/frequency, and so on. Thus, the same specimen was used in repeated tests, and results are also available for up-loading changes. The measurement focus, however, was on the effect of stress reduction. The specimens were initially ramp-loaded with linearly increasing stress amplitude, until the chosen test amplitude was reached, after which cycles at constant load amplitude were applied. Ramp loading was used to initiate the test for two reasons: (1) The test amplitudes were much greater than the flow stress of the annealed copper. Application of such amplitudes from the first cycle would have caused unacceptable deformations. (2) The rate of increase in the ramp can be used to control the dislocation structure, and especially a slow rate tends to produce homogeneous dislocation structures of loop patches, ideally adapted to forming PSBs. On completion of the ramp, the specimens were cycled at constant stress amplitude for about 20-30% of their expected lives to failure. The details of the various pre-treatments which were used are as follows: (1) The specimen was ramp-loaded with a linearly increasing cyclic stress amplitude of 0.018 MPa/ cycle ("long ramp"). After the stress amplitude reached 87 MPa, the specimen was loaded for another 55 000 cycles with this amplitude. For the ramp as well as for further cycling at constant stress amplitude, a frequency of 2 Hz was used. At a stress of 87 MPa, the saturated dislocation struc-
ture would consist of a mixture of loop patches and PSBs. (2) First a long ramp was applied and after the stress reached 94 MPa, the specimen was subjected to cyclic loading at this constant stress amplitude for 35 000 cycles. The loading frequency for the ramp as well as for the subsequent constant stress amplitude cycling was 2 Hz. At 94 MPa, the dislocation structure would contain not only loop patches and PSBs, but some grains would contain cells. (3) The increase of the ramp was 1 MPa/cycle up to a stress of 94 MPa (i.e. a much faster increase than in starts (1) and (2)), and a frequency of 0.5 Hz was used during the ramp-loading ("short ramp"). Subsequently, the specimen was loaded with a constant stress amplitude of 94 MPa, for 35 000 cycles, during the first 20 cycles using a cycling frequency of 0.5 Hz, and then 2 Hz. Because of the short ramp, there would be strong multiple slip during the test start, and the dislocation structures would be more cellular than for the long ramp. (4) A short ramp was applied first followed by constant stress amplitude loading at 110 MPa for 18000 cycles. The loading frequency during the ramp and for the first 20 cycles at constant stress amplitude was 0.5 Hz and afterwards 2 Hz. This stress amplitude is high enough to produce a wholly cellular dislocation structure. After the initial saturation produced by the abovedescribed treatments, the cyclic stress amplitude was reduced to a certain lower value. At this level the specimen was cycled either until the cyclic plastic strain amplitude showed no further change for increasing numbers of cycles or, if saturation could not be defined, for 25 000 cycles. The cyclic frequency for the lower level was either 0.5 Hz, 2 Hz or 8 Hz chosen for similarity to the frequencies used in the first part of the investigation [ 1]. After three experiments with the different frequencies at the same low stress level, some of the specimens were tested in a similar way using another low level. In detail, the specimen initially loaded at 87 MPa was tested at lower levels of 68 MPa, 74 MPa and 80 MPa. The specimen initially subjected to the long ramp and constant stress amplitude loading with 94 MPa was investigated at lower levels of 80 MPa and 87 MPa. For the stress level 94 MPa with initial short ramp, the lower load level was 80 MPa, and for the initial treatment at the high stress level of 110 MPa, the lower level was 94 MPa. The number of cycles at the initial stress level between two successive load reduction tests was at least 10 000 cycles in the case of the specimen tested at the high stress level 87 MPa, 8000 cycles or more for
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both specimens tested at the high stress level 94 MPa, and at least 3000 cycles when the high stress level was 110 MPa. These numbers of cycles were chosen to reproduce about the same cyclic plastic strain amplitude at the high stress level prior to all load-reduction tests of the same specimen.
3. Results
As detailed above, all specimens were initially ramploaded and then cycled at constant stress amplitude prior to the load reduction tests. The plastic strain amplitudes vs. numbers of cycles at constant stress amplitude for the different load levels and pre-treatments are shown in Fig. 1. For the stress level 87 MPa, the plastic strain amplitude subsequent to the long ramp decreased first and showed a minimum of 0.20 x 10 -3 after 5000 cycles. For higher numbers of cycles, the specimen softened and the cyclic plastic strain amplitude increased to 0.25 x 10 -3 after 55 000 cycles. According to the literature [2,3] this value for the plastic strain indicates ill-defined veins and PSBs as the dominant dislocation structure of this specimen. The cyclic plastic strain response for a stress level 94 MPa and a long ramp as pre-treatment is also shown in Fig. 1. A slight hardening in the first 2000 cycles leads to a minimum plastic strain amplitude of 0.30 x 10 -3. For higher numbers of cycles the specimen softened and after 35 000 cycles the plastic strain
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139
amplitude was 0.41 × 10 -3. Investigations of dislocation structures of similarly loaded specimen by Llanes and Laird [3] showed that, in some parts of the specimen, a dislocation structure of ill-defined veins and PSBs was dominant, whereas in other areas a cell and labyrinth structure was formed. The saturation behavior of a specimen first ramploaded with a short ramp and subsequently cycled at a constant stress amplitude of 94 MPa again is shown in Fig. 1. The plastic strain amplitude decreased after the ramp and showed a minimum of 0.45 × 10 -3 in the regime of 1500-1800 cycles. For higher numbers of cycles, the specimen softened slightly and the plastic strain amplitude increased to 0.48 × 10 -3 after 35 000 cycles. This plastic strain amplitude indicated that both a structure of ill-defined veins and PSBs as well as a cell structure will be present in some parts of the specimen. Nevertheless, due to the higher initial plastic strain after the short ramp, the volume fraction of cell structure will be larger than that in the specimen cycled with the initial long ramp [3]. The results of plastic strain amplitude vs. numbers of cycles for a specimen loaded with a short ramp and subsequently cycled with a constant stress amplitude of 110 MPa (Fig. 1) show that the specimen hardened first and a plastic strain amplitude of 0.90× 10 -3 developed after 250 cycles. In the range from 250 cycles to 5000 cycles, the specimen softened slightly and for larger numbers of cycles it hardened again. After 18000 cycles a plastic strain amplitude of 0.9× 10 -3 was measured. This cyclic plastic strain amplitude as well as the relatively high initial plastic strains subsequent to the ramp pre-treatment indicate that the dislocation structure of the specimen consisted of cells [2,3]. After loading the specimens with a ramp and constant stress amplitude, load reduction experiments were initiated. Fig. 2 shows the plastic strain response of a specimen first cycled at 87 MPa and 2 Hz (high level) and then at 68 MPa and 0.5 Hz (low level). In the regime from 112 000 cycles to 117 000 cycles the plastic strain amplitude for the constant stress amplitude of 87 MPa was quite constant at 0.254 x 10 -3. After the load and frequency reduction, the plastic strain amplitude at the low level shows that the specimen hardened and then softened. After 25 000 cycles at the low level, the stress was increased to 87 MPa again, and the frequency increased to 2 Hz. As indicated, from 142000 cycles to 156000 cycles the plastic strain amplitude at the high level showed hardening first and subsequent softening to a value for the plastic strain of 0.253 x 10 -3. This plastic strain amplitude is about the same as determined in the range from 112 000 cycles to 117 000 cycles previous to the load reduction experiment, indicating that history
H. Mayer, C. L a i r d
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Materials Science a n d E n g i n e e r i n g A 1 9 4 (1995) 1 3 7 - 1 4 5
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independence applies in the cyclic response when the amplitude is increased. To simplify the comparison of the cyclic plastic strain response for different lower levels and different frequencies, the cyclic response at the lower levels for all tests starting with the same high level are summarized in one figure. For example the plastic strain for all load reduction tests with a high stress level of 87 MPa are shown in Fig. 3. T h e abscissa indicates the numbers of cycles at the low level after load reduction. T h e cyclic strain amplitudes after load reduction to lower load levels of 68 MPa, 74 MPa and 80 MPa are conveniently separated in the y-direction. T h e maximum value shown on the y-axis of 0.252 x 10 3 indicates the mean plastic strain amplitude for the high level 87 MPa previous to the load reduction experiments. T h e same specimen was used to make all the load reduction tests from 87 MPa and it is necessary to report that the plastic strain amplitude returned faithfully to about 0.25 x 10 -3 at 87 MPa after each experiment. In detail, the strain lay between 0.250 x 10 -3 and 0.254 x 10 3 At the low level of 68 MPa, the specimen hardened during the first 100 cycles for all three frequencies examined. (Measurements of the plastic strain amplitude at the low level for frequencies larger than 0.5 Hz started after 50 cycles (2 Hz) or 100 cycles (8 Hz), because the measurement p r o c e d u r e made it necessary to reduce frequency to 0.5 H z for about 3 cycles. Since this might have had an influence on the saturation behavior at the low stress level for low numbers of cycles, the evaluation was started later.) After about 100 cycles, hardening stopped in the case of a loading frequency of 0.5 Hz and the plastic strain amplitude was about constant for the next 5000 cycles. For higher numbers of cycles, the specimen showed p r o n o u n c e d
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softening, which led to a plastic strain amplitude of 0.063 × 10 -3 in the range between 1 5 0 0 0 cycles and 25 000 cycles. For the loading frequency of 2 Hz, the specimen hardened for numbers of cycles less than 1000 and then the plastic strain amplitude of 0.043 × 10 -3 became constant. For a frequency of 8 H z at the low level the initial hardening continued up to 5000 cycles, before the plastic strain amplitude became constant at 0.035 x 10-3. At the lower stress levels of 74 and 80 MPa, the initial hardening after load reduction was less pronounced (almost absent at 80 MPa) compared to behavior at the low level of 68 MPa. T h e hardening was greater and more profound for higher numbers of cycles, the greater the frequency. Softening subsequently developed as cycling was continued for every condition, but was always greatest for the lowest frequency. Softening was particularly p r o n o u n c e d at 74 MPa and 0.5 H z (Fig. 3). Table 1 shows the details of the plastic strain amplitudes for the various frequencies. T h e cyclic plastic strain response after rapid load reduction for a specimen initially loaded with a long ramp and constant cyclic stress amplitude of 94 MPa is shown in Fig. 4(a). T h e maximum plastic strain value indicated at the y-axis is 0.417 x 10 -3, which is the mean value for the plastic strain amplitude at the high level prior to the load reduction experiments. (To
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Materials Science and Engineering A194 (1995) 137-145 0.417
Table 1 Plastic strain amplitudes measured at lower stress levels after stress reduction
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Lower stress level (MPa)
Frequency (Hz)
Cycles to reach min. plastic strain
Minimum plastic strain (10 -3 )
"Saturation" plastic strain (10 -3 )
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0.049 0.043 0.035 0.092 0.086 0.080 0.160 0.154 0.150
0.063 0.043 0.035 0.130 0.106 0.087 0.200 0.188 0.164
From 94 MPa; long ramp 80 0.5 50 80 2 200 80 8 300 87 0.5 N/A a 87 2 N/A 87 8 N/A
0.142 0.132 0.126 (0.270) b (0.260) (0.258)
0.226 0.193 0.161 0.330 0.322 0.307
From 94 MPa; short ramp 80 0.54 40 80 2 120 80 8 300
0.150 0.142 0.138
0.248 0.216 0.180
From 110 MPa; short ramp 94 0.5 40 94 2 N/A 94 8 N/A
141
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simplify c o m p a r i s o n of the results o f load r e d u c t i o n tests for different high a n d low levels, all results (Figs. 3 and 4(a)-(c)) s h o w the s a m e logarithmic division for the x- as well as the y-axis.) T h e range of the saturation values m e a s u r e d during cycling at 94 M P a a n d 2 H z was (0.412 to 0 . 4 2 2 ) × 10 -3. T h e cyclic plastic strain r e s p o n s e after load r e d u c tion to 80 M P a for a s p e c i m e n initially l o a d e d with a short r a m p and c o n s t a n t stress a m p l i t u d e of 94 M P a is p r e s e n t e d in Fig. 4(b). T h e m a x i m u m value indicated o n the y-axis is 0.461 × 10 -3, w h i c h is the average cyclic plastic strain a m p l i t u d e for 94 MPa, p r i o r to the load r e d u c t i o n experiments. Similar results after load r e d u c t i o n to 94 M P a for a s p e c i m e n initially l o a d e d with a short r a m p and c o n s t a n t stress a m p l i t u d e of 110 M P a are given in Fig. 4(c). T h e m a x i m u m value indic a t e d on the y-axis is 0 . 8 4 3 × 10 -3, a v e r a g e d as usual. Similar to the results s h o w n in Fig. 3, t h o s e for the load r e d u c t i o n tests p r e s e n t e d in Fig. 4(a)-4(c) s h o w an
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Fig. 4. Plastic strain amplitudes vs. numbers of cycles after load reduction to the indicated stress levels, for the following initial loadings: (a) long ramp and constant stress amplitude 94 MPa; (b) short ramp and constant stress amplitude 94 MPa; (c) short ramp and constant stress amplitude 110 MPa. The symbols for thc different frequencies are the same as those used in Fig. 3.
142
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Materials Science and Engineering A194 (1995) 137-145
influence of frequency at the low level on the course of the plastic strain amplitude. This influence is generally more pronounced the greater the load reduction, as can be seen in Fig. 3 and Fig. 4(a). After a load reduction has been made, some hardening occurs immediately but the cyclic response then gives way to softening. The number of cycles over which hardening occurs is greater the higher the frequency and the subsequent softening is more marked at lower frequencies. Comparison of the cyclic response after a load reduction of order 15% from 87 MPa, 94 MPa (long and short ramp) and 110 MPa (lower levels: 74 MPa, 80 MPa and 94 MPa) shows that the influence of frequency on saturation is more pronounced for lower initial load levels. Generally softening at the low amplitude is more pronounced the greater the initial stress. Comparison of results between Fig. 4(a) and 4(b) for the same constant amplitude but different ramps shows that a short ramp leads to slightly more pronounced softening at the low amplitude for all frequencies. Nevertheless, the frequency influence on softening shows no significant difference for both initiating ramps. Table 1 tabulates the details of the measured plastic strain amplitudes for the various testing conditions.
4. Discussion
The influence of frequency on cyclic plastic strain response after a load reduction, the results show, is more pronounced the lower the upper level from which the reduction is started and the larger the reducing step (Figs. 3 and 4). In the light of the dislocation structure this means that a structure of ill-defined veins and PSBs (Fig. 3) shows a pronounced influence of frequency after load reduction whereas, in a cell structure, the loading frequency at the low level is of minor influence (Fig. 4(c)). When the cyclic response after load reduction from 87 MPa (Fig. 3) is analysed in more detail, it is seen that the specimen tends to initial hardening, which is extended for higher frequencies, and softening follows, more pronouced for lower frequencies. The mechanism of softening or hardening after load reduction for a matrix-PSB dislocation structure was studied by Ma and Laird [4,5] with load reduction experiments using copper single crystals. Investigations of PSB activity showed that a decrease of stress amplitude from 28 MPa (plateau stress) to values lower than 25 MPa caused passivation of most of the PSBs and a significant reduction of the plastic deformation in still active PSBs. This activity declined for higher numbers of cycles. Die-out of PSB activity is an indication of
hardening and is combined with changes in the dislocation structure of the PSBs, namely an increase of wall thickness and formation of debris in PSB channels [4]. When the lower level is at least 25 MPa [5], PSBs are potentially active, and reactivating of PSBs, starting from areas of currently active PSBs, causes softening of the specimen. Nevertheless, the plastic strain of the PSBs after load reduction stays below the average localized strain of 0.01, typical for constant-amplitude loading. A process similar to that of single crystals may be expected in polycrystals but the process will certainly be influenced by grain boundaries and grain orientation. Therefore, it may be expected that at the low level 80 MPa most of the PSBs will be potentially active and therefore can be reactivated in sufficiently high numbers of cycles. In contrast, at the lower level 68 MPa, the resolved shear stress in the slip direction of the PSBs in many grains will remain under the threshold stress for reactivation and PSB die-out will have a pronounced influence. Only PSBs contained in grains with favorable slip orientations and relatively high Schmid factors will be reactivated. The influence of cyclic frequency at the low level is superimposed on this general behavior. For all low levels, sufficiently high numbers of cycles lead to softening when the cycling frequency is 0.5 Hz, whereas softening is less pronounced and starts at higher numbers of cycles, if at all, in the case of 8 Hz. Softening is caused by the spreading of active PSBs in suitably oriented grains, and reactivation of other PSBs, processes which are obviously more probable the lower the loading frequency. The mechanism for reactivation is not clearly understood, but changing of the matrix structure between currently active PSBs and potentially active PSBs could be responsible for increasing the volume fraction of active PSBs. Change in the matrix structure is expected to be caused by emission of secondary dislocations from currently active PSBs. In any case, the spread of active PSBs should be strongly affected by the frequency of cycling. This hypothesis is supported by two arguments: (1) It was previously shown that increasing the frequency from 0.5 Hz to 2 Hz in a well-developed matrix-PSB structure leads to an average reduction of plastic strain amplitude of 4%, and a further increase from 2 Hz to 8 Hz reduces the plastic deformation by another 5.5-7% [1]. After load reduction, besides the decrease of plastic deformation due to the negative step, higher frequencies reduce the plastic deformation of the PSBs additionally. This makes it more probable that PSB dieout will occur for higher frequencies than for lower ones. A hint for the validity of this explanation is provided by the fact that the saturation strain after load reduction from 87 MPa to 80 MPa is 6%
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lower for 2 Hz than for 0.5 Hz, whereas a comparison between 2 Hz and 8 Hz shows a decrease of 15%. This reflects the more pronounced influence on plastic strain amplitude when frequency is changed from 2 Hz to 8 Hz compared with changing from 0.5 Hz to 2 Hz. (2) The process of emission of secondary dislocations at the boundary between currently active PSBs and matrix involves thermally activated processes. Since these kinds of processes are more likely for lower frequencies, the emission is more probable for lower than for higher frequencies. Therefore, a reactivation of potentially active PSBs is more probable for lower than for higher frequencies. For a cell structure, developed at the stress level 110 MPa (Fig. 4(c)), the frequency influence after load reduction is far less pronounced than for the structure containing PSBs discussed above. Studies of saturation behavior after load reduction for a cell structure were performed by Lukfis and Klesnil [6] and Figueroa and Laird [7]. Both sets of workers agree that a cell structure formed initially at a high level is stable after a load reduction to a stress level corresponding to that for which a virgin specimen would develop matrix-PSB structure. For some conditions of cycling and microstructure, especially in the interiors of larger grains, the cell morphology can change from an equiaxed form at the higher level to a more elongated form at the lower level. Nevertheless, stress levels necessary to cause this change of cell shape lie below the stresses studied in the present work and therefore this effect is considered to be of minor importance here. Plastic deformation in a cell structure is presently understood by deformation in the cell interior first, and as the stress increases plastic deformation of the hardened cell walls subsequently develops [8]. After load reduction, softening of the cell structure can be hardly affected by frequency of loading. To explain why dislocation glide is nevertheless somewhat easier at lower frequencies, we point out that the activation of such a process may be affected by internal stresses as well as by thermal activation, which is more probable for lower frequencies. When load reduction from the high level 94 MPa is analysed, a more complex dislocation structure, containing matrix-PSB areas as well as volumes with cell structures, must be taken into account. When it is considered that the plastic deformation of a specimen containing different dislocation structures derives from the plastic deformation of respective parts [1], the results discussed above can be taken to explain the frequency influences. On this basis, an influence of frequency on saturation behavior after load reduction can be expected due to the volumes containing
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matrix-PSB structure, but the influence will be less pronounced than in the case of the high level 87 MPa, because softening of the volumes containing cells will be only slightly affected by frequency. This general description fits quite well with the results shown in Fig. 4(a). A given reduction of the stress amplitude ( ~ 15%) from 94 MPa leads to a less pronounced influence of frequency than the same percentage of load reduction starting from 87 MPa, and the influence is more pronounced than after a similar percentage of load reduction starting from 110 MPa. The results for the specimen reported in Fig. 4(b) can be taken to support the validity of this simple model. To increase the volume fraction of cells, this specimen was initially loaded with a short ramp and subsequently cycled at the constant amplitude of 94 MPa. After load reduction of -~ 15% to 80 MPa, this specimen showed a more pronounced influence of frequency than in the case of the high stress 110 MPa (Fig. 4(c)), and the frequency influence after load reduction is less pronounced compared with the negative step loading from 87 MPa to 74 MPa (Fig. 3). Nevertheless, the difference from the specimen initially loaded with a long ramp (Fig. 4(a)) is relatively small, except that the softening was somehow more pronounced, which may have been caused by the increased volume fraction of cells. An interesting point can be noted from comparing the numbers of cycles necessary to stimulate softening after stress reductions from 87 MPa and 94 MPa to a low level 80 MPa. For high level 94 MPa (Figs. 4(a) and (b)) softening starts after about 4 0 - 7 0 cycles for 0.5 Hz, between 100 and 200 cycles for 2 Hz, and 4 0 0 - 4 5 0 cycles for 8 Hz. Similar numbers of cycles are necessary to cause softening in the case of stress reduction from 87 MPa to 80 MPa. This comparison indicates that the onset of the spreading of the active PSBs is more dependent on the current load and frequency than on the percentage of load reduction. For all negative step tests the scope of the cyclic response after load reduction shows initial hardening, which is more pronounced for greater negative steps. One reason for the initial hardening is related to the Bauschinger effect. The load level was routinely changed from the upper level to the lower level when the actual stress was zero, subsequent to compression, which caused in the first cycles at the low level a larger plastic deformation during tension than during compression. This behavior increased the overall plastic deformation in the first cycles after load reduction. Nevertheless, after 10 to 20 cycles the stress-strain loop adjusted to zero mean plastic strain and the influence diminished. In the case of load reduction from 87 MPa to 68 MPa and loading frequency 8 Hz, initial hardening continued up to more than 1000 cycles. Hardening for
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such high numbers of cycles cannot be explained by the Bauschinger effect but die-out processes of PSB activity cause the decrease of the plastic strain amplitude. An influence of frequency on PSB activity after load reduction, shown by Ma and Laird [5], is that the number of temporarily active PSBs increases with high frequencies. This may serve to explain why the unfavorable influence of high frequencies for PSB activity is less pronounced initially after load reduction from 87 MPa than for continual cycling. Nevertheless, for higher numbers of cycles temporarily active PSBs die out, and therefore this effect does not contribute to frequency-induced differences in the saturation strain. It would be interesting to explore whether or not the results obtained by Ma and Laird [5] on single crystals of copper can be applied to polycrystals at still lower levels than 68 MPa. For example, since the stress required to activate PSBs in polycrystalline copper is twice the monocrystalline plateau stress (for the maximum Schmid factor), the stress necessary to prevent total die-out in PSBs produced at a higher level can be expected to be twice 25 MPa. Predicting life behavior in cumulative fatigue damage under variable loading is difficult because many factors are involved, and test results are subject to heavy scatter. The influence of frequency has usually been neglected in the evaluation of cumulative damage, except in the broadest of probabilistic models [9], and the influence of frequency, in the present context, appears to have been ignored even for accelerated testing [9]. A typical procedure used in accelerated testing is to increase frequency; the present results indicate that such a procedure could be non-conservative, even in the absence of such mechanical effects as temperature increase at higher frequencies due to internal friction, and environmental effects caused by frequency-environment interaction. The present results also show that, under load control, frequency can significantly influence the degree of localized strain, and thus failure mechanisms. Moreover, frequency effects in conjunction with load variations can influence the current plastic strain levels. This behavior indicates that making use of the cyclic stress-strain curve for cycle counting and damage assessment, without reference to frequency, is bound to introduce errors, and explains in part the wide prevalence of scatter in cumulative damage test results.
5. Conclusions Measurements of cyclic response obtained in step descending and ascending tests performed at various
frequencies lead to the following conclusions: (1) A dislocation structure of ill-defined veins and PSBs shows a pronounced influence of frequency after load reduction, whereas the plastic strain response after a descending step in a cell structure is relatively insensitive to frequency. (2) When descending steps occur in the presence of PSBs/veins, the activity of a fraction of the formerly active PSBs tends to die out initially. With extended cycling, the PSBs in more voluminous parts of favorably oriented grains gradually become more active; spreading of active PSBs is preferred at low frequencies because secondary slip is facilitated. (3) The relative insensitivity of the frequency effect in structures containing cells is caused by the greater resistance of cell structures to change when the load is reduced. (4) Improved methods of cycle counting and damage assessment in long-life, variable-amplitude fatigue will require consideration of the frequency effect on the cyclic plastic strain, as a significant material property.
Acknowledgments We are grateful to the National Science Foundation for supporting this work under grant No. D M R 9014381. Facility support was provided by the Laboratory for Research on the Structure of Matter under NSF Grant No. D M R 91-20668. Generous grants to one of us (H.M.) from several sources--the Vorarlberger Landesregierung, the Exportakademie der Bundeswirtschaftkammer and the Fulbright Scholar Program--encouraged the collaboration which made the work possible. We also thank our colleagues in the fatigue group at the University of Pennsylvania for helpful discussions and machine time. References [1] H. Mayer and C. Laird, Influence of cyclic frequency on strain localization and cyclic deformation in fatigue, Mater. Sci. Eng. A, 187(1994) 23-35. [2] J.C. Figueroa, S.P. Bhat, R. de la Veaux, S. Murzenski and C. Laird, The cyclic stress-strain response of copper at low strains--I. Constant amplitude testing, Acta Met., 29 (1981) 1667-1678. [3] L. Llanes and C. Laird, Effect of loading mode on the cyclic response and the associated substructure of polycrystalline copper in the high-cycle regime, Fatigue Fract. Eng. Mater. Struct., 16 (1993) 165-179. [4] B. Ma and C. Laird, Dislocation structures of copper single crystals for fatigue tests under variable amplitude, Mater. ScL Eng., 102 (1988) 247-258.
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[51 B. Ma and C. Laird, Overview of fatigue behavior in copper single crystals--IV. Strain and load interaction effects for tests under variable amplitude loading, Acta Met., 37 (1989) 357-368. [6] P. Lukfis and M. Klesnil, Cyclic stress-strain response and fatigue life of metals in low amplitude region, Mat. Sci. Eng., 11 (1973) 345-356.
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[7] J.C. Figueroa and C. Laird, The cyclic stress-strain response of copper at low strains--II. Variable amplitude loading, Aeta Met., 29 (1981) 1679-1684. [8] H. Mughrabi, Dislocation wall and cell structures and long range internal stresses in deformed metal crystals, Acta Met., 31 (1983) 1367-1379. [9] J.L. Bogdanoff and F. Kozin, Probabilistic Models of Cumulative Damage, J. Wiley & Sons, New York, 1985.