Materials Science and Engineering A297 (2001) 48 – 53 www.elsevier.com/locate/msea
Masing behavior in copper single crystals fatigued under load control Mohammad Ahsan Jameel *, Pedro Peralta 1, Campbell Laird Department of Materials Science and Engineering, Uni6ersity of Pennsyl6ania, Philadelphia, PA 19104, USA Received 7 February 2000; received in revised form 20 June 2000
Abstract In a previous paper it was demonstrated that copper single crystals cycled under load control do not saturate in plastic strain once PSBs have nucleated. Thus, a powerful tool for describing and analyzing information from the stress – strain hysteresis loop is lost. This paper approaches the details of the cyclic stress – strain response using Masing behavior as a tool for understanding the microstructural changes that prevent saturation from occurring. It is shown that under constant amplitude load control copper single crystals do not strictly exhibit Masing behavior. However, Masing behavior is observed in crystals that do saturate because the cycling conditions are not such as to lead to the nucleation of PSBs. This implies that the microstructure is constantly evolving meaning, that the number of PSBs is increasing throughout life. The change in the microstructure is gradual and Masing behavior can be said to apply to the first degree. Under variable amplitude load control, however, copper single crystals exhibit Masing behavior if the current load is lower than the previous load. This implies that the previously established microstructures are capable of supporting the deformation specified at the current load amplitude. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Copper; Single; Crystals; Masing; Load-control
1. Introduction The cyclic stress – strain behavior of metals is usually reported in the form of a cyclic stress – strain curve (CSSC) which is a curve of the stress at saturation as a function of applied strain. The CSSC for copper single crystals and for other materials can be simply obtained as the locus of the tips of the hysteresis loops at saturation. Thus, the understanding of the changes in the hysteresis loops as damage develops in the crystal is crucial. The development of the hysteresis loop is dependent on the microstructure and the shape of the CSSC is a direct consequence of this dislocation structure. For example the three major regions in the CSSC for copper single crystals cycled under strain control correspond to three different microstructures prevalent in the crystal. * Corresponding author. Present address: Honeywell Engines and Systems, 1130 W. Warner Road, Tempe, AZ 85284, USA; Tel.: +1-480-5922564. E-mail address: [email protected]
(M.A. Jameel). 1 Present address: Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA.
In the case of copper single crystals cycled under stress control, the definition of such a curve is not possible as the crystals do not saturate after developing PSB’s [1,2]. Thus, a powerful tool for describing and analyzing information from the stress–strain hysteresis loop is lost. Another method that is frequently employed is the model proposed by Masing  for materials with two or more phases. A material is said to exhibit Masing behavior if there is a unique relationship that describes its cyclic behavior. In practical terms Masing behavior is exhibited if the ascending parts of the hysteresis loops obtained at different strain amplitudes are the same. This makes the definition of the CSSC for materials that exhibit Masing behavior relatively simple as the loop shape and the shape of the CSSC are the same. The classification of materials as Masing or nonMasing has practical implications as the description of the crack kinetics and the prediction of life in cyclic deformation by the J-integral approach is predicated on the presence of Masing behavior . Abdel Raouf et al.  has discovered that some materials exhibit Masing behavior whereas others do not. Other authors
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[4,6,7] have found that in wavy slip materials the occurrence of Masing behavior depends upon the loading history. In constant strain amplitude tests conducted on copper polycrystals by Christ and Mughrabi  and single crystals by Li and Laird  Masing behavior was found not to be present. Copper polycrystals tested under conventional stress control did not show Masing behavior either. The explanation for the absence of this phenomenon can be found by a consideration of the microstructure. A crystal tested under constant strain control with an amplitude in the plateau region will form a certain number of PSB’s. The microstructure will essentially be two phase with the soft PSB’s carrying most of the strain and the hard matrix essentially strain free. Another crystal tested at another strain amplitude in the plateau will have a different volume fraction of PSB’s. There will be a difference in microstructure which will lead to the non Masing behavior. Thus, Masing behavior applies when there is no change in the microstructure and this assumption is implicit in Masing’s treatment. Christ and Mughrabi  tested copper polycrystals using a special incremental step test that leads to constancy in the microstructure and found that Masing behavior applied. Wang and Laird  found Masing behavior to be present in polycrystalline copper tested under ramp loading because this type of test leads to a uniform two phase microstructure of PSB’s and matrix structure. Li and Laird  quantified the necessary condition for the observance of the Masing behavior by analysis of the hysteresis loop with the Cottrell method using the ideas of friction and back stresses. They specify that Masing behavior will be observed if the short and long range obstacles to dislocation movement are the same at the same strain level. In terms of PSB dislocation structures, it means that the PSB ladder spacings have to remain unchanged for Masing behavior to occur.
Fig. 1. The resolved plastic shear strain amplitude as a function of cycles for copper single crystal c18 tested at a resolved shear stress amplitude of 26 MPa.
The deformation is carried by standing screw dislocations in the channels and by dislocations bowing out of the PSB channel walls and shuttling back and forth as screw dislocations. The requirement for Masing behavior would specify that their number and behavior also remain unchanged. The present paper details whether Masing behavior applies to copper single crystals cycled under both constant and variable amplitude load control.
2. Experimental procedures The details of the material used, specimen specifications, and mechanical test parameters are specified in another paper  along with techniques of hysteresis loop shape analysis. However, for convenience, pertinent details are provided as follows: Pure copper single crystals oriented for single slip [1,4,8], were cycled in load control at ambient temperature and 1 Hz, in an electrohydraulic machine; tests were run in load control at stresses around and below the plateau stress (28 MPa) of the CSSC under conditions which minimized cyclic creep. Specific crystals are identified by numbers. These crystals may have been given complex test histories, which are detailed elsewhere [1,2].
3. Results: analysis of hysteresis loops
3.1. Constant amplitude load control The analysis of the loops with respect to Masing behavior is illustrated here for a crystal (crystal c18) which was tested at 26 MPa until final fracture around 250 000 cycles. The cyclic stress–strain response of copper single crystals under constant amplitude load control is detailed elsewhere [1,2]. Fig. 1 shows the behavior of plastic strain as a function of cycles for this specimen. The cyclic stress–strain hysteresis loops elaborating the cyclic response of crystal c18 are shown in Fig. 2. The hysteresis loops have been shifted up and to the left so that the lower left tips of the loops coincide, making a Masing comparison possible. The stresses are normal stresses, not shear stresses resolved on the primary slip plane. The startup for this crystal lasted 350 cycles, i.e. the steady state stress value was reached in 350 cycles. Fig. 2a shows the loops at 350, 400, 450, 500, 550, 600 and 1000 cycles. The widest loop corresponds to 350 cycles and the narrowest to 1000 cycles with the other loops falling in between in series. Notice that the loops do not show Masing behavior. The difference is minuscule for the cycles from 350 to 600.Fig. 2b shows the loops taken at 1000, 2000, 3000, 6000, 7000, 8000 and 10 000
M.A. Jameel et al. / Materials Science and Engineering A297 (2001) 48–53
Fig. 2. Cyclic stress – strain hysteresis loops for copper single single crystal c18 tested at a resolved shear stress amplitude of 26 MPa. (a) For cycles 350, 400, 450, 500, 550, 600 and 1000. The widest loop corresponds to 350 cycles and the narrowest to 1000 cycles with the rest of the loops falling in between in series. (b) For cycles 1000, 2000, 3000, 6000, 7000, 8000 and 10 000 cycles. The widest loop corresponds to 1000 cycles and the narrowest to 10 000 cycles with the rest of the loops falling in between in series. (c) For cycles 20 000, 40 000, 100 000, 120 000, 160 000, and 180 000. The narrowest loop corresponds to the lowest cycle number and the widest to the highest. (d) For cycles 180 000, 200 000, 220 000, 240 000, and 245 000 cycles at which point fracture was imminent. The narrowest loop corresponds to the lowest cycle number and the widest to the highest.
cycles. Again the widest loop was taken at 1000 cycles while the narrowest corresponds to 10 000 cycles. Masing behavior is conspicuously absent this time. However, notice that loops taken for cycles between 6000 and 10 000 are almost identical while loops for 2000 and 3000 cycles are very different from the rest. Fig. 2c shows the loops between 20 000 and 180 000 cycles. In these loops the narrowest loop corresponds to the lowest cycle number and the widest to the highest as cyclic softening has developed. It appears that Masing behavior does apply here but a more careful observation shows that there are indeed small but insignificant differences in the loops. Fig. 2d shows the loops near the end of life and it is evident that Masing behavior is not present. The narrowest loop is for the lowest cycle which is 180 000 and the widest for 245 000 cycles which was extremely close to failure.
cycled at 28 MPa. The stress was then lowered to 26 MPa and then increased again to 28 MPa. The cyclic stress–strain response of this crystal (c 35) is shown in Fig. 3 and the history is given in Table 1. The variable amplitude cyclic stress–strain response of copper single crystals is detailed elsewhere [1,2].
3.2. Variable amplitude load control The applicability of Masing behavior to copper single crystals cycled under variable amplitude load control will be illustrated here through a crystal that was first
Fig. 3. The resolved plastic shear strain as a function of the number of cycles in copper single crystal c35 tested under variable amplitude load control.
M.A. Jameel et al. / Materials Science and Engineering A297 (2001) 48–53 Table 1 The stress history of crystal c 35 cycled under variable amplitude load control
28 28 28 26 26 26 28
20 000 20 000 10 000 8000 22 000 20 000 16 000
Cumulative cycles 20 000 40 000 50 000 58 000 80 000 100 000 116 000
than the previously applied stress means that the cyclic softening being observed is due to increased activity in the existing PSB’s and not nucleation of new ones. However, when the stress is raised back to the higher value, the nucleation of PSB’s starts again and Masing behavior is found to be absent. There are some subtleties involved in the change in microstructure which are revealed through a careful examination of the evolution of the hysteresis loops throughout life. At the start of life the plastic strain reaches a value that is higher than at any other time during life except when the crystal is very close to
The absence of Masing behavior is observed in crystal c 35 till the amplitude is decreased to 26 MPa. Fig. 4a shows the cyclic stress – strain hysteresis loops at 20 000 and 40 000 cycles. Masing behavior is clearly absent here. However, when the stress is lowered to 26 MPa Masing behavior is observed (Fig. 4b). This figure shows cyclic stress – strain hysteresis loops at 60 000 and 70 000 cycles. When the stress is increased back to 28 MPa Masing behavior is again found to be absent (Fig. 4c). In Fig. 4c the two loops are at 105 000 and 116 000 cycles.
4. Discussion The results on Masing behavior presented above show that Masing behavior indeed does not apply to copper single crystals cycled under constant amplitude stress control. It was mentioned previously that Masing behavior is observed when the underlying microstructure does not change implying that the microstructure in copper single crystals cycled under stress control is constantly in flux. This is an extremely important observation. The behavior of the plastic strain has been detailed in another paper . It is seen that the plastic strain increases constantly, with a rate of increase that is essentially constant for the majority of life. This increase in the plastic strain could be due to two reasons. Either the population of PSB’s remains unchanged but the existing PSB’s localize increasing amounts of strain in which case Masing behavior should be observed. Or the number of PSB’s increases throughout life where this constantly changing microstructure would result in the absence of Masing behavior. Since Masing behavior is absent in copper single crystals cycled under constant amplitude stress control, it is concluded that the population of PSB’s increases throughout life. The observation that Masing behavior applies in a variable amplitude test when the applied stress is lower
Fig. 4. Cyclic stress – strain hysteresis loops for copper single crystal c35 tested at a resolved shear stress amplitude of 28 MPa followed by cycling at a resolved shear stress amplitude of 26 MPa. (a) For cycles 20 000 and 40 000 at 28 MPa. (b) For cycles 60 000 and 70 000 at 26 MPa with a previous excursion at 28 MPa. (c) For cycles 105 000 and 116 000 at 28 MPa with previous excursions at 28 and 26 MPa.
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Fig. 5. TEM micrograph showing the boundary between a PSB and the matrix structure in a copper single crystal fatigued under strain control .
fracture. Previous investigators  have shown that Masing behavior applies in a descending step test under strain control in which the steps are small enough such that saturation is not reached. The behavior of the plastic strain at the start of this test is very similar to a descending step test conducted under strain control. Thus, we would expect some parallels in behavior. However, we see an opposite effect in that Masing behavior is seen not to apply (Figs. 2 and 4). The reason Li and Laird  see Masing behavior was first that the microstructure was PSB’s in all the steps and second that since the microstructure never reached saturation, the number of the PSB’s did not change from step to step. Thus, the volume fraction of PSB’s developed at the start was enough to handle the strain imposed later. This is easy to see in a descending step test. However, Li and Laird  observed Masing behavior in an ascending test also in which the steps were small. The significant observation here is that these step tests were conducted after the crystals had already been saturated at an amplitude that was not lower than the amplitude at any subsequent time in these step tests. Thus, a population of PSB’s was established that was not changed at any time in the step tests because the crystals were not cycled for a long enough time. The interpretation of our results and the demonstration of their consistency with previously published results requires the understanding of the dislocation processes that lead to the formation of PSB’s [8–13]. We believe that the Masing behavior experienced in crystals cycled under stress control provides compelling experimental evidence supporting the theory of PSB formation offered by Kuhlmann-Wilsdorf and Laird
. Instead of viewing the breakdown of the loop patches leading to the formation of the PSB’s as a catastrophic event  these authors take the point of view that this process happens gradually and continuously. A TEM picture published by Peralta et al.  is presented as supporting evidence (Fig. 5). This figure shows a PSB forming out of, and coincident with, the matrix structure consisting of loop patches and channels. The instability involved in the formation of PSB’s does not seem to be sudden and is on a small scale. The process of conversion of loop patches to PSB’s has been suggested to involve dislocation glide on secondary planes which have a component normal to the primary plane. This glide acts to reposition the positive and negative edge dislocations arranged in the loop patches, which are essentially (and idealistically) in a structure similar to a Taylor’s lattice  so that dislocations of opposite sign line up against each other and annihilate by glide. The stress, therefore, has to be high enough locally to initiate slip on these secondary planes. This can be achieved by either a high applied stress or more probably by a combination of the applied stress and by the forces exerted by dislocations on each other. These forces will become high when the dislocation density is high enough because of the interactive stresses of the statistically stored dislocations. Notice that this analysis suggests that if the dislocation density is high enough locally, then this slip can initiate even at low applied stresses, leading to PSB formation. Conditions which would lead to such an inhomogeneous dislocation distribution would undoubtedly result in PSB formation at stresses lower than 28 MPa in copper single crystals. In the experiments conducted for the present investigation the virgin crystal was cyclically deformed in a startup that leads to an inhomogeneous distribution of dislocations. These dislocations are initially arranged in loop patches. The density of dislocations in some loop patches is postulated to be so high that secondary slip of the type required for annihilation of loop patches is initiated locally, even at 26 MPa. This process essentially starts very soon after the final stress amplitude is reached. Thus, when the loops between cycles 350 and 1000 are compared (Fig. 2a) it is seen that Masing behavior does not apply even though the loops for the low number of cycles are very similar. The process of transition from loop patches to PSB’s has thus started but not been completed yet. The dislocation structure is constantly evolving, leading to differences in loop shapes. The cyclic regime when a PSB population is completely nucleated lies between 1000 and 2000 cycles and thus there is a clear difference between the shapes of these loops and that at a lower number of cycles. However, after the nucleation of the first PSB the primary mechanism of deformation becomes localized
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in the PSB and Masing behavior becomes dependent on the behavior of the PSB’s. The loops after 2000 cycles (Fig. 2b and c) thus show only minute differences because the deformation mechanism does not change. The only change is in the number of PSB’s. Close to the end of life, the cracks are big enough to significantly influence the microstructure locally and on a large scale. This leads to the formation of cell structure and the changes in the loops are dramatic (Fig. 2d). Thus we see evidence of a transition from loop patches to PSB’s that is not catastrophic and instead takes place over hundreds of cycles. After the formation of PSB’s, we see an increase in the number of PSB’s throughout life in which the PSB’s provide the primary deformation mechanism. At the end of life a transition to cell structures is observed.
5. Conclusions Masing behavior is not strictly observed for crystals cycled under constant amplitude stress control conditions. However, since the dislocation microstructure changes gradually through life, Masing behavior is observed to a first approximation. The microstructure for copper single crystals that are cycled under stress control conditions that develop PSB’s undergoes a transition from loop patches to PSB’s to cell structures near the end of life. Masing behavior is observed in variable amplitude load control tests in which the applied amplitude is lower than the previously applied amplitude.
Acknowledgements We are grateful to the Laboratory for Research on the Structure of Matter at the University of Pennsylvania which provided support for this investigation from National Science Foundation funding, as well as to the University itself. The authors are especially grateful for the support provided by Alex Radin, Rollin Lakis, and Bill Romanow.
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