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The Bauschinger effect in cold drawn patented wire
Acta metail. Vol. 33, No. 5, pp. 771-783, Printed
in Great
Britain.
1985 All rights reserved
Copyright
OOOI-6160/85 %3.00 + 0.00 Q 1985 Pergamon Press Ltd
THE BAUSCHINGER EFFECT IN COLD DRAWN PATENTED WIRE Department
W. M. STOBBS and S. PAETKEt of Metallurgy and Materials Science, University of Cambridge, Pembroke Street, Cambridge, England (Received 29 August 1984; in revised form 30 October 1984)
Abstract-The magnitude of the Bauschinger effect in cold drawn patented wire is used to assess the importance of elastic stresses, built into it during the drawing process, in determining its flow behaviour. Results are described for both room temperature and liquid nitrogen temperature tests on wire given a variety of post drawing heat treatments as well as stress relaxation fatiguing cycles. The work hardening behaviour of fine pearlite is also discussed in relation to the data obtained. RCsum&Nous utilisons la gandeur de I’effet Bauschinger dans un fil patent6 i.tiri. g froid pour prkciser I’importance des contraintes elastiques produites au tours de 1’Ctirage pour la coractbrisation de son Ccoulement plastique. Nous dicrivons les rtsultats d’essais B la tempkrature ambiante et B la temptrature de l’azote liquide sur un fil qui a Bte soumis B divers traitements thermiques aprbs ttirage ainsi qu’8 des cycles de fatigue par relaxation de contrainte. Nous discutons tgalement le durcissement de la perlite fine en relation avec les rksultats obtenus. Zusammenfassung-Die GriiDe des Bauschinger-Effektes an kaltgezogenen bleigehlrteten Drlhten wird benutzt, urn die Bedeutung der wlhrend des Ziehens erzeugten inneren elastischen Spannungen fiir das FlieDverhalten abzuschitzen. Hierzu wurden Drghte, die nach dem Ziehen zur Spannungsreiaxation unterschiedlichen WIrmebehandlungen und Ermiidungszyklen unterworfen worden waren, bei Raumtemperatur und 78 K untersucht. Die Ergebnisse werden dargestellt. AuDerdem wird das Verfestigungsverhalten des feinen Perlits im Hinblick auf diese Ergebnisse diskutiert.
INTRODUCTION
The high flow stress and surprising ductility of cold drawn patented wire have attracted considerable interest in the relationships between its strength and microstructure. Langford [l] has reviewed recent work on the deformation of pearlite and the observed flow stress is generally taken to be governed by the reciprocal of the square root of the average slip distance (and thus, in patented wire, the interlamellar spacing) in terms of a pile-up model. Sevillano [2] has however demonstrated the plausibility of a linear relationship of Orowan form between the yield stress and the reciprocal of the interlamellar spacing given a further contribution to the strength due to internal stresses associated with the cementite. Such an approach would be supported both by the ductility of the material and by recent results on its recovery behaviour as described by Paetke [3]. It is thus the evaluation of this contribution with which we are interested here given, in particular, the well known ductile flow behaviour of cementite during the wire drawing process [e.g. 11. The measurement of the magnitude of the Bauschinger effect has been used for several years to provide a means of evaluating the inhomogeneity of tNow at Shell Research Ltd, Thornton Research Centre, P.O. Box 1, Chester CHl 3SH, England. 777
the internal stress state of a material containing a hard dispersion as a function of its deformation history. Once the contribution to the hardening from this source is known, the total work hardening behaviour can often be unambiguously modelled in terms of the “mean matrix stress”, (a),, and other terms due, for example, to forest hardening. A system for which the approach has been markedly successful is that of a ductile copper matrix containing a dispersion of hard silica particles [e.g. 4, 51. In this case the plastic deformation of the matrix results in the hard particles being elastically strained to a degree, prior to any plastic load relaxation, which provides a “perfect memory” of the applied plastic strain. While the locally varying stress fields around the particles provide a “source shortening” term increasing the magnitude of the Orowan stress for flow in either the forward or reverse direction, the uniform elastic shear of the particles results, for a finite matrix, in a superposed uniform elastic stress in the matrix, (o),+,. This stress is of a sense to oppose forward and to aid reverse flow. Prior to local stress relaxation (o)~ is proportional both to the volume fraction of the hard particles and to the applied plastic strain. The magnitude of (o)~ can be determined for any system of ellipsoidally shaped particles by applying the approach developed by Eshelby [6-81, and Brown and Clarke [9] have provided prescriptions for composite materials containing rib-
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bons, fibres, discs or spheres. In the copper-silica system (o)~ can be measured experimentally by comparing the reverse flow stress, as plotted in the forward sense as a function of the cumulative strain, with the extrapolated forward flow stress. The reverse flow curve as plotted in this manner becomes parallel with the forward flow curve and thus exhibits a “permanent softening” Aa, which is directly related to (a),+,, at the strain at which flow is reversed, in such a way that Ag, = 2(o),+,. This has been most convincingly demonstrated for iron based systems by the X-ray work of Wilson [lo] and Wilson and Konnan [1 11. Given that a system exhibits a permanent softening, (u)~ can be evaluated even after local stress relaxation has started at the hard particles and, again in copper-silica, it has been shown that (Q),,, is then built up proportionately to the square root of the applied plastic strain after relaxation starts, the stress relaxation becoming increasingly difficult because of the local forest hardening associated with the process [12]. The approach of measuring the Bauschinger effect has in fact not only allowed the hardening behaviour of copper-silica to be fully modelled [13] but has also been shown to have broad application to a variety of different types of composite systems as recently reviewed by Brown [14]. The results of the application of this technique are used here to assess the extent to which inhomogeneous stresses associated with the cementite plates, contained in the ferritic matrix, determine both the initial flow behaviour of cold drawn patented wire and its subsequent work hardening behaviour in a tensile test. 2. EXPERIMENTAL The measurement of the Bauschinger effect for a system such as cold drawn patented wire is experimentally difficult not only because of its high yield strength (N 1600 MPa) but also because of its typically small specimen cross section which necessitates tensionshort the use of correspondingly compression specimens. The material tested was obtained from wire of highly uniform composition and structure to BS 1408M C 0.82%
Si 0.23%
S 0.025%
P 0.015%
Mn 0.86%.
After lead patenting, the wire was acid pickled and then bond coated in borax both to protect and lubricate it during the drawing process. A final diameter of 4.25 mm and a total reduction of approximately 70% were achieved using an eight capstan drawbench. The tungsten carbide dies used had rake angles ranging from 12” to 8” giving successively smaller individual reductions down to approximately 12%. The lubricant used was a petroleum based grease and the bulk wire temperature during drawing
was maintained below 130°C. No back stress was applied. A closed-loop servo hydraulic testing machine was developed to give high-accuracy strain-control, the strains being measured across the specimen grips using LVDT’s. The tests required the construction of hardened push-pull grips which could be accurately aligned and which were free from backlash for the high loads involved. Alignment was maintained during testing by means of pre-loaded roller-bearing guides so that any misalignment during a test could be identified easily from the stress-strain data and subsequently confirmed by the appearance of the specimen. A guage length was centreless ground onto the wire with a 30” taper at each end to facilitate its gripping in split, hardened inserts which screwed into the main grips. These tapers were preloaded during specimen insertion sufficiently to ensure that the specimens, when tested, were at all times securely forced into the inserts and main grips. This was accomplished by grinding the ends of the specimens accurately square and back loading them by means of a hardened rod. This was done under load control to avoid pre-loading the gauge length. The test specimens, of 6 mm gauge length and 3 mm2 area, were electro polished, after grinding, to remove any stresses introduced by this process though the final lubricated grinding passes were designed to give reductions of less than 10 pm per pass and would have introduced little damage. It should be noted however that the reduction in cross sectional area of the test specimens would itself have the necessary effect of altering the inhomogeneous stress distribution introduced into the wire during the original drawing process. The grip design included an effective thermal barrier to allow either uniform heating or cooling of the specimen and adjacent grips to + 1°C over the period of a test. 3. RESULTS Since we were interested in both demonstrating the presence of internal stresses associated with the wire drawing process and in examining how these might be modified during subsequent deformation, Bauschinger tests were carried out for a range of initial tensile and compressive strains. The room temperature true stress/true strain flow behaviours of the as received (“AR”) wire in both compression, “CF”, and tension, “TF”, are shown in Fig. 1 together with similar curves for wire given a variety of pre-test heat treatments. These included a low temperature, 200°C 30min anneal which had been previously demonstrated to improve the tensile flow stress [3] and two further 30min treatments, at 550” and 69O”C, to promote spheroidisation. Bauschinger data as obtained for room temperature (RT) tests are given in Fig. (2a, b) and (c) for the wire as received (AR) and after the 200” and 550°C heat treatments respectively. In each case the reverse flow behaviour after a variety
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Fig. 2. Room temperature true stress, true strain curves for: (a) AR material (b) 200°C anneal (c) 550°C anneal. The forward curve is (-), the subsequent reverse flow is plotted in the forward sense (---), In each case the Bauschinger effect behaviour is shown both after an initial tensile stroke (TF) (on the left) and initial compressive stroke (CF) (on the right). For each reverse flow curve the zero stress axis is marked at the point at which a linearly extrapolated elastic unloading line would have met this axis. Note the general tendencies, irrespective of heat treatment to: earlier flow, greater initial work hardening and lower Bauschinger effect for the (TF) relative to the (CF) specimens. A.M
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of forward strains is plotted in the foreward sense. Because of the differences in initial tensile and compressive flow stress, as discussed below, two sets of curves are given as a function of the sense of initially applied deformation as either tensile (TF) or compressive (CF). It may be seen that in all states plastic flow starts considerably earlier in tension than in compression and that the differences in compressive and tensile flow stresses once the curves become parallel are relatively independent of heat treatment and of the order of 100-150 MPa (some 3-5 times the variability of the flow behaviour between tests). The light heat treatment (200°C) results in an increase in both the tensile and compressive flow stresses above approximately 2% plastic strain but also an increase in the difference between the tensile and compressive flow behaviour at lower strains. While it would appear that the light heat treatment was sufficient to modify any retained internal stresses affecting the
Fig. 1. Room temperature true stress, true strain curves for patented wire 1,2 are as received (AR), curve 1 being tensile (TF) curve 2 compressive (CF) 3, 4 are after a light 200°C anneal, curve 3 being (TF) and curve 4 (CF) 5, 6 are after a 550°C anneal, curve 5 being (TF) and curve 6 (CF) 7, 8 are after a 690°C anneal, curve 7 being (TF) and curve 8 (CF) 9 is for both (TF) and (CF) on specimens 550°C heat treated and then fatigued as described in text.
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EFFECT IN PATENTED WIRE
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Fig. 3. True stress, true strain, roam temperature tested Bauschinger curves for AR specimens after a pre-test fatigue treatment. The curves are plotted as in Figure 2 and the spread in the behaviour is indicated by the differences in the behaviour of specimen 1 and ---- and specimen 2
+ + + f, -l-l-l-. flow behaviour, it is clear that if this is the origin for the difference in flow behaviour in tension and compression annealing, and even particle spheroidisation, does not remove the effect. At first sight the differences in tensile and compressive flow behaviour might be ascribed to the progressive fracturing of the cementite lamellae during the tensile test, though the maintenance of the difference in behaviour even after considerable spheroidisation, as in the 690°C heat treatment, would make such an explanation unlikely. A method which has often been used to relax internal stresses, is to give a light fatiguing treatment at low loads [15]. This was carried out for both the AR and 550°C heat treated material, each being given 1250 cycles at f 660 MPa. In both cases this resulted in the flow behaviour becoming indistinguishable in tension and compression. For the 550°C material, the resultant work hardening became intermediate between that of the compressive and tensile curves of the unfatigued material at low strains while tending to the tensile curve for the unfatigued material above approximately 2% plastic strain. The AR fatigued material curve simply tended to that of the tensile curve of the unfatigued AR specimens. Flow curves for two AR prefatigued specimens are shown in Fig. 3 together with the reverse flow behaviour after different forward strains (whether compressive or tensile) as plotted in the forward sense. Similar curves for 550” heat treated and fatigued specimens are shown in Fig. 4. The flow behaviour as exhibited after fatigue, discounts the above potential explanation for the yield stress asymmetry being due to cementite lamellae failure in tension. It also demonstrates that the effect is not due to any change in modulus because of the high elastic strains necessary for flow, in agreement with order of magnitude calculations of such a second order elastic modulus effect [16]. However, it makes the greater ease of flow in tension rather than compression after the wire drawing process all the more surprising since if this were naively modelled as simply a plastic elongation, the reverse would be expected.
EFFECT IN PATENTED WIRE
That the origin of the asymmetry effect is unlikely to lie in the undoubtedly strong texture of the wire drawn material is itself demonstrated by the similarity of the forward and reverse curves after fatigue. The anisotropy of slip on (112) planes was first discovered by G. I. Taylor for b-brass [17] as noted by Hirsch [18] who further interpreted the results of Argon and Maloof [19] on tungsten in terms of the anisotropy of slip of a dislocation dissociated in b.c.c. on {112) and/or {110) planes. The effect is general for b.c.c., if variable from material to material, and it is interesting to note from the more recent work of Mughrabi and Wilthrich [20] on cc-iron that such asymmetry would be expected to persist after fatigue. That is does not for the strongly textured patented wire is in itself surprising but possibly indicates a high population of potentially active dislocations on all slip systems after the wire drawing process. In any case it should be noted that the flow asymmetry observed in the wire prior to its fatigue is of opposite sense to that which has been observed in other b.c.c. materials when tested at the orientation of the wire drawing texture. It would seem to be inescapable that at least once the drawn wire is reduced in cross section, as described, the cementite lamellae are left in elastic longitudinal compression. The magnitude of this compression can be qualitatively assessed by examination of the comparative development of a Bauschinger effect as a function of applied strain on the fatigued material. Such a comparison of, for example, Fig. 1 and Fig. 3 suggests that the observed asymmetry in flow behaviour of the wire would be explained if the wire underwent reverse plastic flow, from an imagined reference point of a state of zero internal stress, by between f and 1% on leaving the wire drawing dies. This estimate neglects any asymmetry of opposite sense which might naturally be present due to the wire’s texture. If it were, then the reverse compressive flow undergone by the wire would have to be correspondingly larger to explain its subsequent tensile and compressive behaviour. Such a reverse plastic
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Fig. 4. True stress, true strain room temperature tested Bauschinger curves for 550°C annealed specimens also given a pre-test fatigue treatment. The curves are plotted as in Fig. 2 and the spread in the behaviour is indicated by the behaviour of specimen 1 and --- and specimen 2+ + + and -l-l-.
STOBBS and PAETKE:
BAUSCHINGER EFFECT IN PATENTED WIRE
flow on unloading is not in itself surprising: it may be seen that after only 4% forward plastic flow the tensile curve shown in Fig. 2 exhibits N l/3% reverse plastic flow on unloading. This reverse flow is driven in this case by the relief of the elastic stresses built into the cementite lamellae during the previous forward deformation. However, it should be noted that the associated relief of the stresses in the cementite on unloading is, in such a tensile test, only partial and subsequent deformation in compression is easier than in tension. The initial behaviour of the unfatigued central portion of the wire after its fabrication in the wire drawing process is very different and indicates that the reverse plastic flow on leaving the die must be substantially larger than the above figure since prior to this flow it is inconceivable that the internal stress in the ferrite matrix, as associated with those in the cementite lamellae, was not of a sense to aid reverse flow. It is however important to realise both that the wire drawing process involves substantial dilatational compression of the material in the dies, allowing the observed plastic deformation of the cementite, and that the outer portions of the wire are generally considerably more heavily deformed than the central region. It is in this latter point that we find a possible explanation for the flow behaviour of the central portion of the wire as tested. The greater flow in the outer regions of the wire than at its centre would result, during the drawing process, in the retention of larger elastic tensile strains in the cementite lamellae around the periphery of the wire than in those at its centre. It is then necessary to infer that the mean matrix stress, as the wire leaves the die, is sufficiently high to cause bulk reverse flow of sufficient magnitude that the elastic strains retained in the lamellae at the centre of the wire are reversed in sense. It would still of course be expected that if a Bauschinger test could be carried out on the whole wire rather than on only its central portion, as necessitated by the fabrication of a gauge length, then the wire would exhibit a greater flow stress in tension than in compression. While the above explanation for the observed asymmetry in the flow behaviour of the drawn wire is self consistent it remains possible that it could either arise from, or be modified by, a difference in the expansion coefficients of cementite and ferrite, even though the wire was drawn under conditions such that its temperature did not exceed 13o”C, provided that the expansion coefficient of ferrite is markedly larger than that of cementite. While the sense of this difference might be expected we are unable to find reliable figures for either the expansion coefficient or the various compliances of cementite and this has hampered a fully quantitative assessment of our results. The above hypothesis can however be tested by seeing whether or not the flow stress asymmetry shows any substantial increase on testing at a reduced temperature. Tensile and compressive
781
flow curves for both the AR and 550°C heat treated material are presented as obtained at liquid N, temperature in Fig. 5. Bauschinger data are also included for a full comparison with the results shown in Fig. 2. It may be seen that while all the initial flow stresses are markedly increased the asymmetry of the flow behaviour is virtually unchanged. This indicates that any stresses caused by differences in expansion coefficients have little affect on the yield stress and cannot be responsible for the asymmetry in this value in the as drawn material as tested at room temperature. Turning to the work hardening behaviour of the wire it will be remembered that this was found to be similar irrespective of heat treatments which both substantially lowered the initial yield stress and undoubtedly initiated a marked degree of cementite spheroidisation (see Fig. 1). Given both the asymmetry of the tensile and compressive flow stresses and the ill defined initial yield stress the work hardening behaviour of the material is best examined in the “stress-relieved” fatigue state. Examining Figs 3 and 4 it is apparent that after the pre-test fatigueing treatments the material exhibits a potentially readily characterisable Bauschinger effect: both the transient and permanent softening increase in a well defined way as a function of the applied forward strain. The main difficulty in determining the relationship between the work hardening and, for example, the permanent softening lies in delineating the initial yield stress. If the mean matrix stress is assessed by measuring Aa,, a further difficulty is however caused by the large reverse strains required before the reverse and extrapolated forward flow curves become parallel. Whereas for particle-hardened systems the internal stresses are relaxed by a reverse plastic strain of approximately l/3 the forward plastic strain [12], in a fibre-hardened system complete relaxation would not be expected before the reverse strain is equal to the applied strain. It is however clear from Figs 3 and 4 that, at least qualitatively, the permanent softening is of the order of twice the work hardening increment suggesting that virtually the entire work hardening increment has its origin in the way the mean matrix stress is built up with the applied strain. It remains interesting to examine the way (r~)~ builds up with the applied strain. If for example plastic relaxation occurs around the lamellae with local forest hardening, (o)~ might be expected to be proportional to the root of the applied plastic strain, whereas if no such relaxation occurs (a), would be proportional to the applied strain [5]. Given the difficulties in measuring Ao,, as mentioned above, the alternative semiquantitative approach was taken of assessing the transient softening. There are a variety of ways of doing this and we have applied the simple method of measuring the reverse plastic strain as a function of the forward strain at a given reverse load. The reverse strains observed at 660 MPa are plotted in Fig. 6(a) as a function of e,, and in Fig. 6(b) as a function of
STOBBS and PAETKE:
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Fig. 5. Liquid Nitrogen temperature tested Bau~hinger curves for: in (a) AR and in (b) 550°C annealed specimens. The curves are plotted as in Fig. 2 and (TF) specimens are on the left, (CF) specimens on the right.
e,)‘lz for both the AR and 550°C heat treated material as prefatigued. It is interesting and surprising that as plotted in this manner for a given reverse load the data for the differently heat treated specimens fall on a single line and do not do so if the reverse strains are taken at for example, a given fraction of the forward flow stress prior to stress reversal. Given the spread in the reverse Row behaviour observed, as indicated by the differences in the curves shown in Fig. 4, this is likely to be fortuitous and the data in Fig. 2 if plotted in a similar manner show less spread if plotted for a constant fraction of the yield stress, Equally while these latter data further demonstrate the greater Bauschinger effects observed for compression specimens if tested without a prefatigue treatment (CF and TF data being approximately parallel but relatively displaced), they do not assist in differentiating between the two ways in which the mean matrix stress might be being built up as a function of the applied strain. Both functional behaviours appear to fit the data shown in Fig. 6 equally well. Reconsidering however the Bauschinger data obtained at liquid nitrogen temperature it may be noted that the Bauschinger effect associated with deformation introduced at this temperature is considerably larger than at room temperature and this suggests that the build up of internal stresses at the lamellae at room temperature is in fact associated with partial local plastic relaxation. Equally however, that the Bauschinger effect introduced at liquid nitro-
gen temperature, in particular for the 550°C heat treated material, is so large and asymmetric as a function of the sense of the deformation first applied is difficult to understand. At face value it would appear that for this lower strength material, when deformation is applied in the tensile sense the Bauschinger effect, and thus the internal stress aiding reverse flow, develops much more slowly than when the initial defo~ation is applied in the reverse sense. While this is of course consistent with the interpretation given above, that the material as tested and prior to deformation contains a mean matrix stress aiding tensile flow, the magnitude of the asymmetry seen here is inconsistent both with the relatively small differences in initial flow stress in tension and compression and with the smaller asymmetries seen for the unheat-treated allow. Since the 550°C heat treated wire must have undergone at least partial
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6. (a) The value of the reverse transient strain e, at the constant compressive stress of 660 MPa as a function of the applied forward plastic strains, e,,, for prefatigued AR and 550°C specimens (from Figs 3 and 4). (b) e, as defined for (a) as a function of the root of the applied plastic strain.
STOBBS and PAETKE:
BAUSCHINGER EFFECT IN PATENTED WIRE
relaxation of the stresses introduced in the drawing process it would appear that we must look for an explanation for its asymmetric low temperature Bauschinger effect which does not have its origin entirely in the form of the inhomogeneous stresses built up in the material during forward flow. The essential difference between the AR and 550°C material will lie in the partial spheroidisation of the carbides in the latter specimens and this would allow cross slip mechanisms to become much more important in relieving internal stresses than in the material containing essentially continuous cementite lamellae. This in turn suggests that, as a function of the texture of the wire, there might be a strong asymmetry in the tem~rature dependence of the orientation dependent change in the most active slip system on reversing the flow. The form of the effect equally implies that new deformation sources are required for flow otherwise the asymmetry would have been as strong for the initial flow, as a function of its sense, given the retained inhomogeneous stresses in the wire as a result of the drawing process. 4. CONCLUSIONS It has been demonstrated that internal stresses built into patented wire during the drawing process play a major role in dete~ining the yield stress and that these stresses are not easily relieved by annealing, but are on fatiguing the wire at stresses well below the macroscopic yield stress. It has also been argued that the form of the asymmetry of the initial flow behaviour of the central portion of such wire indicates that the wire undergoes substantial reverse plastic flow on leaving the drawing dies to the extent that, at least in the central region of the wire, the cementite lamellae are in longitudinal elastic compression in a matrix in which plastic flow is thus made easier in tension than in compression. The effects observed could not be explained either on the basis of the well known asymmetries of flow stress as a function of texture or as being due to stresses built up as a function of differences in expansion coefficient of the cementite and ferrite on cooling from the drawing temperature. It would be interesting to examine the changes in initial flow stress asymmetry as a function
783
of the diameter of the gauge length relative to that of the drawn wire. This would allow the internal stresses to be mapped across the specimen and indirectly indicate the degree of inhomogeneity of flow in the wire drawing process. It is also concluded that the form of the development of the Bauschinger effect as a function of further applied deformation indicates that the work hardening of drawn wire of this type is almost exclusively due to the build up of a mean matrix stress in the ferrite even though, at room temperature, plastic relaxation would appear to be partially relaxing the stresses built up in the cementite lamellae as deformation proceeds. are grateful to Professor R. W. K. Honeycombe F.R.S. for the provision of laboratory facilities. ~c~~o~~e~ge~e~~s-we
REFERENCES 1. G. Langford, Metalf. Trans. SA, 861 (1977). 2. J. G. Sevillano, 5th Int. ConJ Strength Metals and ABoys, Aachen (edited by P. Haasen, V. Gerhold and G. Kostor), p. 819. Pergamon Press, Oxford (1981). 3. S. Paetke. Ph.D. thesis. Cambridge Univ. (1981). 4. L. M. Brbwn and W. &I. StobbsrPhil. A4&. 2j, 1185 (1971). 5. L. M. Brown and W. M. Stobbs, Phil. Mag. 23, 1201 (1971). 6. J. D. Eshelby, Proc. R. Sot. A241, 376 (1957). 7. J. D. Eshelby, Proc. R. Sot. A252, 561 (1959). 8. J. D. Eshelby, Prog. Solid Me&. 2, 89 (1961). 9. L. M. Brown and D. R. Clarke, Acta metall. 23, 821 (1975). 10. D. V. Wilson, Acta metall. 13, 807 (1965). 11. D. V. Wilson and Y. A. Konnan, Acta metall. 12, 617 (1964). 12. J. D. Atkinson, L. M. Brown and W. M. Stobbs, Phi% Mag. 30, 1247 (1974). 13. L. M. Brown and W. M. Stobbs, Phil. Mug. 34, 351 (1976). 14. L. M. Brown, 5th Int. Co@ Strength Metals and Alloys, Aachen (edited by P. Haasen, V. Gerhold and G. Kostor), p. 1551. Pergamon Press, Oxford (I 98 I). 15. Z. S. Basinski, private communi~tion (1983). 16. L. M. Brown, private communication (1983). 17, G. I. Taylor, Proc. R. Sot. All& 1 (1928). 18. P. B. Hirsch, Proc. Int. Cons. on Strength of Metals and Alloys, Suppl. Trans. Jap. Inst. Metals 9, XXX (1968). 19. A. S. Argon and S. R. Maloof, Acta metall. 14, 1449 (1966). 20. H. Mughrabi and A. Wiithrich, Phil. Mag. 33, 963 (1976).
in Great
Britain.
1985 All rights reserved
Copyright
OOOI-6160/85 %3.00 + 0.00 Q 1985 Pergamon Press Ltd
THE BAUSCHINGER EFFECT IN COLD DRAWN PATENTED WIRE Department
W. M. STOBBS and S. PAETKEt of Metallurgy and Materials Science, University of Cambridge, Pembroke Street, Cambridge, England (Received 29 August 1984; in revised form 30 October 1984)
Abstract-The magnitude of the Bauschinger effect in cold drawn patented wire is used to assess the importance of elastic stresses, built into it during the drawing process, in determining its flow behaviour. Results are described for both room temperature and liquid nitrogen temperature tests on wire given a variety of post drawing heat treatments as well as stress relaxation fatiguing cycles. The work hardening behaviour of fine pearlite is also discussed in relation to the data obtained. RCsum&Nous utilisons la gandeur de I’effet Bauschinger dans un fil patent6 i.tiri. g froid pour prkciser I’importance des contraintes elastiques produites au tours de 1’Ctirage pour la coractbrisation de son Ccoulement plastique. Nous dicrivons les rtsultats d’essais B la tempkrature ambiante et B la temptrature de l’azote liquide sur un fil qui a Bte soumis B divers traitements thermiques aprbs ttirage ainsi qu’8 des cycles de fatigue par relaxation de contrainte. Nous discutons tgalement le durcissement de la perlite fine en relation avec les rksultats obtenus. Zusammenfassung-Die GriiDe des Bauschinger-Effektes an kaltgezogenen bleigehlrteten Drlhten wird benutzt, urn die Bedeutung der wlhrend des Ziehens erzeugten inneren elastischen Spannungen fiir das FlieDverhalten abzuschitzen. Hierzu wurden Drghte, die nach dem Ziehen zur Spannungsreiaxation unterschiedlichen WIrmebehandlungen und Ermiidungszyklen unterworfen worden waren, bei Raumtemperatur und 78 K untersucht. Die Ergebnisse werden dargestellt. AuDerdem wird das Verfestigungsverhalten des feinen Perlits im Hinblick auf diese Ergebnisse diskutiert.
INTRODUCTION
The high flow stress and surprising ductility of cold drawn patented wire have attracted considerable interest in the relationships between its strength and microstructure. Langford [l] has reviewed recent work on the deformation of pearlite and the observed flow stress is generally taken to be governed by the reciprocal of the square root of the average slip distance (and thus, in patented wire, the interlamellar spacing) in terms of a pile-up model. Sevillano [2] has however demonstrated the plausibility of a linear relationship of Orowan form between the yield stress and the reciprocal of the interlamellar spacing given a further contribution to the strength due to internal stresses associated with the cementite. Such an approach would be supported both by the ductility of the material and by recent results on its recovery behaviour as described by Paetke [3]. It is thus the evaluation of this contribution with which we are interested here given, in particular, the well known ductile flow behaviour of cementite during the wire drawing process [e.g. 11. The measurement of the magnitude of the Bauschinger effect has been used for several years to provide a means of evaluating the inhomogeneity of tNow at Shell Research Ltd, Thornton Research Centre, P.O. Box 1, Chester CHl 3SH, England. 777
the internal stress state of a material containing a hard dispersion as a function of its deformation history. Once the contribution to the hardening from this source is known, the total work hardening behaviour can often be unambiguously modelled in terms of the “mean matrix stress”, (a),, and other terms due, for example, to forest hardening. A system for which the approach has been markedly successful is that of a ductile copper matrix containing a dispersion of hard silica particles [e.g. 4, 51. In this case the plastic deformation of the matrix results in the hard particles being elastically strained to a degree, prior to any plastic load relaxation, which provides a “perfect memory” of the applied plastic strain. While the locally varying stress fields around the particles provide a “source shortening” term increasing the magnitude of the Orowan stress for flow in either the forward or reverse direction, the uniform elastic shear of the particles results, for a finite matrix, in a superposed uniform elastic stress in the matrix, (o),+,. This stress is of a sense to oppose forward and to aid reverse flow. Prior to local stress relaxation (o)~ is proportional both to the volume fraction of the hard particles and to the applied plastic strain. The magnitude of (o)~ can be determined for any system of ellipsoidally shaped particles by applying the approach developed by Eshelby [6-81, and Brown and Clarke [9] have provided prescriptions for composite materials containing rib-
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bons, fibres, discs or spheres. In the copper-silica system (o)~ can be measured experimentally by comparing the reverse flow stress, as plotted in the forward sense as a function of the cumulative strain, with the extrapolated forward flow stress. The reverse flow curve as plotted in this manner becomes parallel with the forward flow curve and thus exhibits a “permanent softening” Aa, which is directly related to (a),+,, at the strain at which flow is reversed, in such a way that Ag, = 2(o),+,. This has been most convincingly demonstrated for iron based systems by the X-ray work of Wilson [lo] and Wilson and Konnan [1 11. Given that a system exhibits a permanent softening, (u)~ can be evaluated even after local stress relaxation has started at the hard particles and, again in copper-silica, it has been shown that (Q),,, is then built up proportionately to the square root of the applied plastic strain after relaxation starts, the stress relaxation becoming increasingly difficult because of the local forest hardening associated with the process [12]. The approach of measuring the Bauschinger effect has in fact not only allowed the hardening behaviour of copper-silica to be fully modelled [13] but has also been shown to have broad application to a variety of different types of composite systems as recently reviewed by Brown [14]. The results of the application of this technique are used here to assess the extent to which inhomogeneous stresses associated with the cementite plates, contained in the ferritic matrix, determine both the initial flow behaviour of cold drawn patented wire and its subsequent work hardening behaviour in a tensile test. 2. EXPERIMENTAL The measurement of the Bauschinger effect for a system such as cold drawn patented wire is experimentally difficult not only because of its high yield strength (N 1600 MPa) but also because of its typically small specimen cross section which necessitates tensionshort the use of correspondingly compression specimens. The material tested was obtained from wire of highly uniform composition and structure to BS 1408M C 0.82%
Si 0.23%
S 0.025%
P 0.015%
Mn 0.86%.
After lead patenting, the wire was acid pickled and then bond coated in borax both to protect and lubricate it during the drawing process. A final diameter of 4.25 mm and a total reduction of approximately 70% were achieved using an eight capstan drawbench. The tungsten carbide dies used had rake angles ranging from 12” to 8” giving successively smaller individual reductions down to approximately 12%. The lubricant used was a petroleum based grease and the bulk wire temperature during drawing
was maintained below 130°C. No back stress was applied. A closed-loop servo hydraulic testing machine was developed to give high-accuracy strain-control, the strains being measured across the specimen grips using LVDT’s. The tests required the construction of hardened push-pull grips which could be accurately aligned and which were free from backlash for the high loads involved. Alignment was maintained during testing by means of pre-loaded roller-bearing guides so that any misalignment during a test could be identified easily from the stress-strain data and subsequently confirmed by the appearance of the specimen. A guage length was centreless ground onto the wire with a 30” taper at each end to facilitate its gripping in split, hardened inserts which screwed into the main grips. These tapers were preloaded during specimen insertion sufficiently to ensure that the specimens, when tested, were at all times securely forced into the inserts and main grips. This was accomplished by grinding the ends of the specimens accurately square and back loading them by means of a hardened rod. This was done under load control to avoid pre-loading the gauge length. The test specimens, of 6 mm gauge length and 3 mm2 area, were electro polished, after grinding, to remove any stresses introduced by this process though the final lubricated grinding passes were designed to give reductions of less than 10 pm per pass and would have introduced little damage. It should be noted however that the reduction in cross sectional area of the test specimens would itself have the necessary effect of altering the inhomogeneous stress distribution introduced into the wire during the original drawing process. The grip design included an effective thermal barrier to allow either uniform heating or cooling of the specimen and adjacent grips to + 1°C over the period of a test. 3. RESULTS Since we were interested in both demonstrating the presence of internal stresses associated with the wire drawing process and in examining how these might be modified during subsequent deformation, Bauschinger tests were carried out for a range of initial tensile and compressive strains. The room temperature true stress/true strain flow behaviours of the as received (“AR”) wire in both compression, “CF”, and tension, “TF”, are shown in Fig. 1 together with similar curves for wire given a variety of pre-test heat treatments. These included a low temperature, 200°C 30min anneal which had been previously demonstrated to improve the tensile flow stress [3] and two further 30min treatments, at 550” and 69O”C, to promote spheroidisation. Bauschinger data as obtained for room temperature (RT) tests are given in Fig. (2a, b) and (c) for the wire as received (AR) and after the 200” and 550°C heat treatments respectively. In each case the reverse flow behaviour after a variety
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Fig. 2. Room temperature true stress, true strain curves for: (a) AR material (b) 200°C anneal (c) 550°C anneal. The forward curve is (-), the subsequent reverse flow is plotted in the forward sense (---), In each case the Bauschinger effect behaviour is shown both after an initial tensile stroke (TF) (on the left) and initial compressive stroke (CF) (on the right). For each reverse flow curve the zero stress axis is marked at the point at which a linearly extrapolated elastic unloading line would have met this axis. Note the general tendencies, irrespective of heat treatment to: earlier flow, greater initial work hardening and lower Bauschinger effect for the (TF) relative to the (CF) specimens. A.M
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of forward strains is plotted in the foreward sense. Because of the differences in initial tensile and compressive flow stress, as discussed below, two sets of curves are given as a function of the sense of initially applied deformation as either tensile (TF) or compressive (CF). It may be seen that in all states plastic flow starts considerably earlier in tension than in compression and that the differences in compressive and tensile flow stresses once the curves become parallel are relatively independent of heat treatment and of the order of 100-150 MPa (some 3-5 times the variability of the flow behaviour between tests). The light heat treatment (200°C) results in an increase in both the tensile and compressive flow stresses above approximately 2% plastic strain but also an increase in the difference between the tensile and compressive flow behaviour at lower strains. While it would appear that the light heat treatment was sufficient to modify any retained internal stresses affecting the
Fig. 1. Room temperature true stress, true strain curves for patented wire 1,2 are as received (AR), curve 1 being tensile (TF) curve 2 compressive (CF) 3, 4 are after a light 200°C anneal, curve 3 being (TF) and curve 4 (CF) 5, 6 are after a 550°C anneal, curve 5 being (TF) and curve 6 (CF) 7, 8 are after a 690°C anneal, curve 7 being (TF) and curve 8 (CF) 9 is for both (TF) and (CF) on specimens 550°C heat treated and then fatigued as described in text.
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EFFECT IN PATENTED WIRE
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Fig. 3. True stress, true strain, roam temperature tested Bauschinger curves for AR specimens after a pre-test fatigue treatment. The curves are plotted as in Figure 2 and the spread in the behaviour is indicated by the differences in the behaviour of specimen 1 and ---- and specimen 2
+ + + f, -l-l-l-. flow behaviour, it is clear that if this is the origin for the difference in flow behaviour in tension and compression annealing, and even particle spheroidisation, does not remove the effect. At first sight the differences in tensile and compressive flow behaviour might be ascribed to the progressive fracturing of the cementite lamellae during the tensile test, though the maintenance of the difference in behaviour even after considerable spheroidisation, as in the 690°C heat treatment, would make such an explanation unlikely. A method which has often been used to relax internal stresses, is to give a light fatiguing treatment at low loads [15]. This was carried out for both the AR and 550°C heat treated material, each being given 1250 cycles at f 660 MPa. In both cases this resulted in the flow behaviour becoming indistinguishable in tension and compression. For the 550°C material, the resultant work hardening became intermediate between that of the compressive and tensile curves of the unfatigued material at low strains while tending to the tensile curve for the unfatigued material above approximately 2% plastic strain. The AR fatigued material curve simply tended to that of the tensile curve of the unfatigued AR specimens. Flow curves for two AR prefatigued specimens are shown in Fig. 3 together with the reverse flow behaviour after different forward strains (whether compressive or tensile) as plotted in the forward sense. Similar curves for 550” heat treated and fatigued specimens are shown in Fig. 4. The flow behaviour as exhibited after fatigue, discounts the above potential explanation for the yield stress asymmetry being due to cementite lamellae failure in tension. It also demonstrates that the effect is not due to any change in modulus because of the high elastic strains necessary for flow, in agreement with order of magnitude calculations of such a second order elastic modulus effect [16]. However, it makes the greater ease of flow in tension rather than compression after the wire drawing process all the more surprising since if this were naively modelled as simply a plastic elongation, the reverse would be expected.
EFFECT IN PATENTED WIRE
That the origin of the asymmetry effect is unlikely to lie in the undoubtedly strong texture of the wire drawn material is itself demonstrated by the similarity of the forward and reverse curves after fatigue. The anisotropy of slip on (112) planes was first discovered by G. I. Taylor for b-brass [17] as noted by Hirsch [18] who further interpreted the results of Argon and Maloof [19] on tungsten in terms of the anisotropy of slip of a dislocation dissociated in b.c.c. on {112) and/or {110) planes. The effect is general for b.c.c., if variable from material to material, and it is interesting to note from the more recent work of Mughrabi and Wilthrich [20] on cc-iron that such asymmetry would be expected to persist after fatigue. That is does not for the strongly textured patented wire is in itself surprising but possibly indicates a high population of potentially active dislocations on all slip systems after the wire drawing process. In any case it should be noted that the flow asymmetry observed in the wire prior to its fatigue is of opposite sense to that which has been observed in other b.c.c. materials when tested at the orientation of the wire drawing texture. It would seem to be inescapable that at least once the drawn wire is reduced in cross section, as described, the cementite lamellae are left in elastic longitudinal compression. The magnitude of this compression can be qualitatively assessed by examination of the comparative development of a Bauschinger effect as a function of applied strain on the fatigued material. Such a comparison of, for example, Fig. 1 and Fig. 3 suggests that the observed asymmetry in flow behaviour of the wire would be explained if the wire underwent reverse plastic flow, from an imagined reference point of a state of zero internal stress, by between f and 1% on leaving the wire drawing dies. This estimate neglects any asymmetry of opposite sense which might naturally be present due to the wire’s texture. If it were, then the reverse compressive flow undergone by the wire would have to be correspondingly larger to explain its subsequent tensile and compressive behaviour. Such a reverse plastic
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flow on unloading is not in itself surprising: it may be seen that after only 4% forward plastic flow the tensile curve shown in Fig. 2 exhibits N l/3% reverse plastic flow on unloading. This reverse flow is driven in this case by the relief of the elastic stresses built into the cementite lamellae during the previous forward deformation. However, it should be noted that the associated relief of the stresses in the cementite on unloading is, in such a tensile test, only partial and subsequent deformation in compression is easier than in tension. The initial behaviour of the unfatigued central portion of the wire after its fabrication in the wire drawing process is very different and indicates that the reverse plastic flow on leaving the die must be substantially larger than the above figure since prior to this flow it is inconceivable that the internal stress in the ferrite matrix, as associated with those in the cementite lamellae, was not of a sense to aid reverse flow. It is however important to realise both that the wire drawing process involves substantial dilatational compression of the material in the dies, allowing the observed plastic deformation of the cementite, and that the outer portions of the wire are generally considerably more heavily deformed than the central region. It is in this latter point that we find a possible explanation for the flow behaviour of the central portion of the wire as tested. The greater flow in the outer regions of the wire than at its centre would result, during the drawing process, in the retention of larger elastic tensile strains in the cementite lamellae around the periphery of the wire than in those at its centre. It is then necessary to infer that the mean matrix stress, as the wire leaves the die, is sufficiently high to cause bulk reverse flow of sufficient magnitude that the elastic strains retained in the lamellae at the centre of the wire are reversed in sense. It would still of course be expected that if a Bauschinger test could be carried out on the whole wire rather than on only its central portion, as necessitated by the fabrication of a gauge length, then the wire would exhibit a greater flow stress in tension than in compression. While the above explanation for the observed asymmetry in the flow behaviour of the drawn wire is self consistent it remains possible that it could either arise from, or be modified by, a difference in the expansion coefficients of cementite and ferrite, even though the wire was drawn under conditions such that its temperature did not exceed 13o”C, provided that the expansion coefficient of ferrite is markedly larger than that of cementite. While the sense of this difference might be expected we are unable to find reliable figures for either the expansion coefficient or the various compliances of cementite and this has hampered a fully quantitative assessment of our results. The above hypothesis can however be tested by seeing whether or not the flow stress asymmetry shows any substantial increase on testing at a reduced temperature. Tensile and compressive
781
flow curves for both the AR and 550°C heat treated material are presented as obtained at liquid N, temperature in Fig. 5. Bauschinger data are also included for a full comparison with the results shown in Fig. 2. It may be seen that while all the initial flow stresses are markedly increased the asymmetry of the flow behaviour is virtually unchanged. This indicates that any stresses caused by differences in expansion coefficients have little affect on the yield stress and cannot be responsible for the asymmetry in this value in the as drawn material as tested at room temperature. Turning to the work hardening behaviour of the wire it will be remembered that this was found to be similar irrespective of heat treatments which both substantially lowered the initial yield stress and undoubtedly initiated a marked degree of cementite spheroidisation (see Fig. 1). Given both the asymmetry of the tensile and compressive flow stresses and the ill defined initial yield stress the work hardening behaviour of the material is best examined in the “stress-relieved” fatigue state. Examining Figs 3 and 4 it is apparent that after the pre-test fatigueing treatments the material exhibits a potentially readily characterisable Bauschinger effect: both the transient and permanent softening increase in a well defined way as a function of the applied forward strain. The main difficulty in determining the relationship between the work hardening and, for example, the permanent softening lies in delineating the initial yield stress. If the mean matrix stress is assessed by measuring Aa,, a further difficulty is however caused by the large reverse strains required before the reverse and extrapolated forward flow curves become parallel. Whereas for particle-hardened systems the internal stresses are relaxed by a reverse plastic strain of approximately l/3 the forward plastic strain [12], in a fibre-hardened system complete relaxation would not be expected before the reverse strain is equal to the applied strain. It is however clear from Figs 3 and 4 that, at least qualitatively, the permanent softening is of the order of twice the work hardening increment suggesting that virtually the entire work hardening increment has its origin in the way the mean matrix stress is built up with the applied strain. It remains interesting to examine the way (r~)~ builds up with the applied strain. If for example plastic relaxation occurs around the lamellae with local forest hardening, (o)~ might be expected to be proportional to the root of the applied plastic strain, whereas if no such relaxation occurs (a), would be proportional to the applied strain [5]. Given the difficulties in measuring Ao,, as mentioned above, the alternative semiquantitative approach was taken of assessing the transient softening. There are a variety of ways of doing this and we have applied the simple method of measuring the reverse plastic strain as a function of the forward strain at a given reverse load. The reverse strains observed at 660 MPa are plotted in Fig. 6(a) as a function of e,, and in Fig. 6(b) as a function of
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e,)‘lz for both the AR and 550°C heat treated material as prefatigued. It is interesting and surprising that as plotted in this manner for a given reverse load the data for the differently heat treated specimens fall on a single line and do not do so if the reverse strains are taken at for example, a given fraction of the forward flow stress prior to stress reversal. Given the spread in the reverse Row behaviour observed, as indicated by the differences in the curves shown in Fig. 4, this is likely to be fortuitous and the data in Fig. 2 if plotted in a similar manner show less spread if plotted for a constant fraction of the yield stress, Equally while these latter data further demonstrate the greater Bauschinger effects observed for compression specimens if tested without a prefatigue treatment (CF and TF data being approximately parallel but relatively displaced), they do not assist in differentiating between the two ways in which the mean matrix stress might be being built up as a function of the applied strain. Both functional behaviours appear to fit the data shown in Fig. 6 equally well. Reconsidering however the Bauschinger data obtained at liquid nitrogen temperature it may be noted that the Bauschinger effect associated with deformation introduced at this temperature is considerably larger than at room temperature and this suggests that the build up of internal stresses at the lamellae at room temperature is in fact associated with partial local plastic relaxation. Equally however, that the Bauschinger effect introduced at liquid nitro-
gen temperature, in particular for the 550°C heat treated material, is so large and asymmetric as a function of the sense of the deformation first applied is difficult to understand. At face value it would appear that for this lower strength material, when deformation is applied in the tensile sense the Bauschinger effect, and thus the internal stress aiding reverse flow, develops much more slowly than when the initial defo~ation is applied in the reverse sense. While this is of course consistent with the interpretation given above, that the material as tested and prior to deformation contains a mean matrix stress aiding tensile flow, the magnitude of the asymmetry seen here is inconsistent both with the relatively small differences in initial flow stress in tension and compression and with the smaller asymmetries seen for the unheat-treated allow. Since the 550°C heat treated wire must have undergone at least partial
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6. (a) The value of the reverse transient strain e, at the constant compressive stress of 660 MPa as a function of the applied forward plastic strains, e,,, for prefatigued AR and 550°C specimens (from Figs 3 and 4). (b) e, as defined for (a) as a function of the root of the applied plastic strain.
STOBBS and PAETKE:
BAUSCHINGER EFFECT IN PATENTED WIRE
relaxation of the stresses introduced in the drawing process it would appear that we must look for an explanation for its asymmetric low temperature Bauschinger effect which does not have its origin entirely in the form of the inhomogeneous stresses built up in the material during forward flow. The essential difference between the AR and 550°C material will lie in the partial spheroidisation of the carbides in the latter specimens and this would allow cross slip mechanisms to become much more important in relieving internal stresses than in the material containing essentially continuous cementite lamellae. This in turn suggests that, as a function of the texture of the wire, there might be a strong asymmetry in the tem~rature dependence of the orientation dependent change in the most active slip system on reversing the flow. The form of the effect equally implies that new deformation sources are required for flow otherwise the asymmetry would have been as strong for the initial flow, as a function of its sense, given the retained inhomogeneous stresses in the wire as a result of the drawing process. 4. CONCLUSIONS It has been demonstrated that internal stresses built into patented wire during the drawing process play a major role in dete~ining the yield stress and that these stresses are not easily relieved by annealing, but are on fatiguing the wire at stresses well below the macroscopic yield stress. It has also been argued that the form of the asymmetry of the initial flow behaviour of the central portion of such wire indicates that the wire undergoes substantial reverse plastic flow on leaving the drawing dies to the extent that, at least in the central region of the wire, the cementite lamellae are in longitudinal elastic compression in a matrix in which plastic flow is thus made easier in tension than in compression. The effects observed could not be explained either on the basis of the well known asymmetries of flow stress as a function of texture or as being due to stresses built up as a function of differences in expansion coefficient of the cementite and ferrite on cooling from the drawing temperature. It would be interesting to examine the changes in initial flow stress asymmetry as a function
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of the diameter of the gauge length relative to that of the drawn wire. This would allow the internal stresses to be mapped across the specimen and indirectly indicate the degree of inhomogeneity of flow in the wire drawing process. It is also concluded that the form of the development of the Bauschinger effect as a function of further applied deformation indicates that the work hardening of drawn wire of this type is almost exclusively due to the build up of a mean matrix stress in the ferrite even though, at room temperature, plastic relaxation would appear to be partially relaxing the stresses built up in the cementite lamellae as deformation proceeds. are grateful to Professor R. W. K. Honeycombe F.R.S. for the provision of laboratory facilities. ~c~~o~~e~ge~e~~s-we
REFERENCES 1. G. Langford, Metalf. Trans. SA, 861 (1977). 2. J. G. Sevillano, 5th Int. ConJ Strength Metals and ABoys, Aachen (edited by P. Haasen, V. Gerhold and G. Kostor), p. 819. Pergamon Press, Oxford (1981). 3. S. Paetke. Ph.D. thesis. Cambridge Univ. (1981). 4. L. M. Brbwn and W. &I. StobbsrPhil. A4&. 2j, 1185 (1971). 5. L. M. Brown and W. M. Stobbs, Phil. Mag. 23, 1201 (1971). 6. J. D. Eshelby, Proc. R. Sot. A241, 376 (1957). 7. J. D. Eshelby, Proc. R. Sot. A252, 561 (1959). 8. J. D. Eshelby, Prog. Solid Me&. 2, 89 (1961). 9. L. M. Brown and D. R. Clarke, Acta metall. 23, 821 (1975). 10. D. V. Wilson, Acta metall. 13, 807 (1965). 11. D. V. Wilson and Y. A. Konnan, Acta metall. 12, 617 (1964). 12. J. D. Atkinson, L. M. Brown and W. M. Stobbs, Phi% Mag. 30, 1247 (1974). 13. L. M. Brown and W. M. Stobbs, Phil. Mug. 34, 351 (1976). 14. L. M. Brown, 5th Int. Co@ Strength Metals and Alloys, Aachen (edited by P. Haasen, V. Gerhold and G. Kostor), p. 1551. Pergamon Press, Oxford (I 98 I). 15. Z. S. Basinski, private communi~tion (1983). 16. L. M. Brown, private communication (1983). 17, G. I. Taylor, Proc. R. Sot. All& 1 (1928). 18. P. B. Hirsch, Proc. Int. Cons. on Strength of Metals and Alloys, Suppl. Trans. Jap. Inst. Metals 9, XXX (1968). 19. A. S. Argon and S. R. Maloof, Acta metall. 14, 1449 (1966). 20. H. Mughrabi and A. Wiithrich, Phil. Mag. 33, 963 (1976).