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The shear band deformation process in microcrystalline Pd80Si20
Ad. m&a. Vol. 31. pp. I IO 8. 1983 Printed in Great Britain. All rights reserved
THE SHEAR BAND DEFORMATION PROCESS MICROCRYSTALLINE PdsoSizo P. EL DONOVAN
IN
and W. M. STOBBS
Department of Metallurgy and Materials Science. University of Cambridge. Cambridge CB2 3QZ. England (Received 22 December 1981; in revised&m
f8 August
1982)
Abstract-We have observed the mode of deformation of initially amorphous PdseSiss which had been heat-treated to transform it to a microcrystalline state. In this condition the material undergoes localised deformation in apparently the same manner as the glass, in that transmission electron microscope images of the shear bands show the same kind of structural changes as we have previously obaewed in deformed amorphous metals. We.have also studied the surface slip-steps on microcrystalline specimens deformed by bending at 78 K and at room temperature by scanning electron microscopy. We found that the morphology of the shear bands formed under a compressive load depends on the deformation temperature; becoming, at 78 K, more like that seen at all temperatures in amorphous PdseSis,. The similarities and di&rences in the deformation behaviour of amorphous and microcrystalline PdseSile allow us to discuss a realistic model for the shear band deformation process. R&sum&Nous avons observe le mode de deformation dun PdseSir, initialement amorphc, ayant subi un traitement thermique afin de passer B un &at microcristallin. Dans ces conditions, le mattriau prtsente une ~fo~tion localis& ap~mment semblable a celle dun verre, en ce que les micrographis &ctroniques en transmission des bandes de cisaillement prCsentent le mime type de changements structuraux que ceux que nous avons antineurement observis darts des mttaux amorphes deform&s. Nous avons tgalcment ttudiC par microscopic Clectronique fi balayage les marches de &sement superficielks sur des Cchantillons microcristaliins deform& en flexion a 78 K et B la temptrature. ambiante. La morphologic des bandes de cisaillement form&es en compression dependait de la tern*ture de deformation et ellc devenait B 78 K semblable B celle qu l’on observe I toutes les temptratures dans PdssSise amorphe. Les similitudes et les differences dans la deformation de PdseSise amorphe et microcristallin nous permettent de discuter un modHe rCaliste pour le mtcanisme de d&formation par bandes de cisaillement. Zuaammeafatmmg-Die Verformung von mikrokristallinem Pds&e wurde untersucht Das ursprtlnglich amorphe Material wur& hierxu mit einer Wirmebehandlung in den rnikro~s~ti~n Zustand tibergefiihrt. Das Material verformt aich tokal, scheinbar wie im Glasxustand. Die ekktronenrmkro+opische Dur~~~~g der S&erb%nder im ~~k~s~li~n Material zeigt diiben strukturelIen Anderungen wie im verformten amorphen Metall. Die durch Biegung des ~~ok~s~l~en Materials bei 78 K und Raumtemperatur entstandenen Gleitatufen wur&n im Rast~elek~o~nmi~k~ UntersuchL Die Struktur der U&K Drucklast entstandenen ScherbSinder hiingt van der Verfo~u~~-~~ sb. +i 78 K iihnelt sie der, die bei silmtlichen Temperaturen am amorphen PdseSise beobachtet wird. Ahnlichkeiten und Unterschiede im Verformungsverhalten zwischen amorphen und mikrokristaflinen PdssSise erm@lichen uns, ein realistisches Mode11Wr den Scherbandverformungsprozefi zu diskutieren.
vicinity of a free surface. Where a free surface was present during plastic flow small voids, less than 1 nm in diameter, are observed in the shear bands to a depth of the order of the shear band width, which is between 10 and 2Onm in all the metallic glasses we have examined [4,5]. The determination of the width of shear bands was an important result of our previous work, but we were surprised to find that operating shear bands are as wide as this because, firstly, it is difficult to visualise such a large region undorgoing the necessary expansion for shear without becoming unstable and, secondly, most models for deformation in metallic glasses are based on much more localised atomic rearrangements [ 1,263. In order to obtain further information about the deformation process in amorphous metals we have
1. INTBODUCIION
At low temperatures and high strain-rates metallic glasses undergo ~~y-l~ii~ ‘inhomogeneous’ defor~tion [ 11, during which plastic flow is confined to a few narrow regions of the matfzriak known as ‘shear bands’. It has been suggested that this type of deformation may produa structural changes in the shear bands, such as a decrease in density [2] or disruption of the local chemical o&ring 133. We have previously investigated the structure of shear bands in the unloaded state by transmission electron microscopy (TFM) [“I]. Our observations indicate that inhomogeneous deformation involves an increase in the volume of the deforming material, which is recovered when shear ceases except in the A.M. 31/i--*
1
2
DONOVAN
and STOBBS:
SHEAR BAND DEFORMATION
investigated the behaviour of shear bands in two classes of crystallised glass. In the initial stages of crystallisation most metallic glasses form a dispersion of relatively large crystals in the amorphous matrix. The behaviour of this type of material is discussed elsewhere [SJ. Here we report our observations on Pd,,SizO, which transforms initially to a microcrystalline state [7]. 2. EXPERIMENTAL The amorphous PdeoSilo ribbon used in this investigation was prepared at the Cavendish Laboratory by a single-roller quenching method and supplied by Dr P. H. Gaskell. Specimens were prepared for TEM examination by electropolishing in a solution of 10% perchloric acid in glacial acetic acid at 283 K. As described previously [4], the specimens were deformed by bending and then given a coating of lacquer on one side before being polished to perforation from the opposite side. In this way the original surface of the deformed glass was retained and slip steps marked the positions of the shear bands. The surfaces of specimens which had been deformed at room temperature or at 78 K were examined by scanning electron microscopy (SEM). These specimens were deformed by bending them across the edge of a razor blade to give approximately the same total strain in each case. 3. ORSERVATIONS 3.1 Strucrure of the annealed Pd8&, After the glass had been annealed for 63 h at 563 K it was found to be brittle, so that on bending the specimens shattered. On examining this material in the TEM it was found to have transformed almost completely to a state comparable to that described by Masumoto and Maddin [7j as MS-II, with an average grain size of about 0.5 pm. After annealing for 46 h at 563 K the ribbon was still quite ductile. TFM and X-ray investigation of the ribbon in this state demonstrated that it was fully microcrystalline, with an average grain size of ap proximately 5 nm. The X-ray diffraction patterns showed only distinct lines, broad because of the small grain-size of the material but with no trace of the diffuse rings characteristic of amorphous materials, and this limits the volume fraction of any amorphous material in the specimens to less than about 5%. The d-spacings of the microcrystalline phase are listed in Table 1. They do not correspond to those of the f.c.c. microcrystalline phase reported by Masumoto and Maddin 17-J. Some of the d-spacings correspond to those of palladium, but the remaining spacings do not match any of the stable phases in the Pd-Si phase t Even a random structure will give rise to bright speckle in dark-field images, but the size of this speckle is determined by the Airy-disc size of the objective aperture. Larger bright specks can be interpreted as images of crystallites.
Table I. X-ray diffraction results for the metastable crystalline
Measured lattice spacing d(A) 2.77 2.71 2.52 2.36 2.30 2.24 2.16 2.10 2.03 1.95 1.89 1.79 1.66 1.63 1.57 1.54 1.49 1.47 1.39 1.37 1.34 1.32 1.31 1.29 1.28 1.26 1.23 1.185 1.171 1.157 1.121
Observed intensity I
micro-
phase
Pd lattice spacing (PDF)
Relative intensity IIIi (PDF)
2.246
100
1.945
42
1.376
25
1.1730
24
0.9723
3
0.8924 0.8697
13 11
VW W VW
m vw S
m m W m VW VW VW VW VW W mw W W W W mw W W VW W VW mw VW VW W
0.942 0.911
VW
0.869 0.841 0.828 0.807 0.7%
VW
VW
VW VW VW VW
diagram. It seems that the crystallisation process in Pds&, glass is very sensitive to the precise. initial composition of the material and the annealing conditions [S, 91. The crystal structure of the metastable microcrystalline state was not determined in this work, since for our investigation of shear band deformation only a knowledge that the material was fully microcrystalline and of the relative sizes of the crystal grains and the shear bands was required. The morphology of the aged material was investigated by high-resolution bright-field and dark-field TRM. A high resolution axial bright-field image is shown in Fig. la, and a dark-field image in Fig. lb. The degree of crystallinity of the material may be estimated by observing the area covered by lattice fringes in the bright-field image and by non-aperturelimited bright speckle in the dark-field image.t For the imaging conditions used for the bright-field image,
DONOVAN and STOBBS:
SHEAR BAND DEFORMATION
Fig. 1. TEM images of a PdlaSizo specimen after it had been annealed for 48 h at 563 K. (a) Highresolution bright-field image, (b) dark-field image. The edge of the specimen is indicated by arrows. Moire fringes from overlapping cryatallites can be seen in the relatively thick parts of the specimen.
a fully crystalline specimen with random gram orientations and a thickness of one crystal diameter would be expected to show lattice fringes over about 15% of its area, although this is obviousIy dependent on the crystal size (e.g. 10). Quite clearly the fringe coverage in the region shown is much greater than thih hich is mainly because the specimen thickness was In ore than one crystal diameter (5 nm). However, the pimen was not more than about 10nm thick in the
region shown, as can be seen from the simple nature of the Moire fringes formed between di~rent crystalIites in the dark-field image, Fig. lb. While there are problems in detecting isolated crystal&s in thick amorphous specimens by bright-field imaging [ 1I], and images of amorphous material obtained with non-axial illumination can appear to be crystalline under certain circumstances [12], the high degree of fringe coverage in the image shown, which was obtained using axial illumination, strongly suggests that the material contains no amorphous regions. This is confirmed by the dark-field image, Fig. lb, which has 30% coverage of non-a~rture-lined speckle. Again, this figure is higher than would be expected (< 10%) for a completely crystalline specimen one crystal diameter in thickness, 3.2 TEM observations of shear bands in microcrystalline Pd80Si,0
Fig. 2. TEM image of shear bands formed on the ‘tension’ side of a bent Pd&lo specimen after it had ban annealed for 48 h at 563 K.
Figure 2 is a low-magnification TEM image of a group of shear bands formed on the ‘tension’ side of a bent microcrystalline Pdt,,,Si10 specimen. At this magnification the shear bands are indistinguishable from those formed in amorphous Pds&,, as shown in Fig. 3. A series of TEN images of a shear band formed at room temperature on the ‘tension’ side of a micr~rys~lline specimen is shown in Fig 4. The feature marked A in Fig. 4a is a small crack with its
4
I~ONOVAN
side of a bent amorphous Pda,&,
wd
SI‘OBBS:
SHEAR
ribbon.
associated plastic zone (EC) formed in the thin specimen after electropolishing. In the bright-field image, t We have discussed elsewhere [43 the general problem of ~~erentiating between isoIated voids and more diffuse low density regions by dark field techniques and here the term ‘void’ is used for bath types of irregularity.
HAN11
Illif-ORMATION
Fig. 4a, it can be seen that the edges of the crack are bumpy, suggesting that the crack has propagated through the grain-bottndi~ry Iaycrs between the crystaflites. A similar bumpiness can be seen on the edge of the shear step (DE), suggesting that during bulk shear plastic flow also occurred in the grain-boundary layers. There is no evidence to suggest that the crystals themselves have been plastically deformed. The bright speckle contrast from the plastic zone around the crack in the dark-field images (4&f) shows that a high concentration of voids has been formed there, just as they are formed when thin films of amorphous metals undergo plastic deformationt. Our previous investigation showed that a much lower density of voids is produced when shear bands are formed in a bulk amorphous specimen than when they are formed in a thin film. Unfortunately, it is not possible to determine from the images shown in Fig. 4 whether a concentration of voids comparable to that which we have observed in shear bands in amorphous material is present in the bulk shear band,. DE, because of the relatively intense low-angle scattering from the crystallites. Small specks of bright contrast
Fig. 4. TEM images of a shear band. DE, and a crack, AC, in micr~r~tailine Pd&,,. (a) Bright-field image, (b-f) dark-field images with the di~ra~tion conditions shown inset. The contrast of both the shear band and the plastic zone around the crack is the same as that which we have previously observed in amorphous
Fe,oNi40B,0.
DONOVAN
and STOBBS:
SHEAR BAND DEF~)RMAT[ON
Fig. 5. SEM images of surfaa slipstcps on PdsOSizOribbons formed on the compression side of a bend: (a) amorphous, deformed at room temperature, (b) amorphous deformed at 78 K, (c) ~~~~iin~ deformed at room temperature, (d) micr~ys~llin~ deformed at 78 K. &ale marker represents 1 pm, (e) microcrystalline deformed at room temperature, scak marker represents 0.1~~5, (1) microcrystalline deformed at 78 K, scale marker represents 10 @n.
5
phous metal the small differences between the stip steps formed on the different surfaces and at the different temperatures are probably not significant, since the deformation was not strictly controlled. On the ‘tension’ side of a microcrystalline specimen there is substantial ‘faceted’ cracking, whether the material is deformed at room temperature or at 78 K. Indeed in these images (Fig. 6c and d) no shear bands are apparent which have not opened out into cracks. However we know from Fig. 3 that slip steps are in fact formed without cracking in rni~~ys~~me material at lower applied pIastic strains. This implies that shear bands in micr~rys~lline Pds&, can carry less plastic strain than those in amorphous PdBoSi20 when the net stress is tensile. The images of the ‘compression’ side of bent microcrystalline material are remarkable. At room temperature (Fig. 5~) the slip lines are wavy and fine crosslinking slip-steps have been formed. Furthermore the slip is now often associated with the extrusion of material in rounded slivers. However, after deformation at 78 K (Fig. 5d) well defined surface steps are observed, indistinguishable from those on a bent ~o~hous specimen. The bottom surface of a rapidly quenched metallic ribbon, which is in contact with the wheel during quenching, is usually rough. Typically there are pits where gas bubbles have been trapped berween the molten metal and the wheel. Pits of this type were present on the .Pd&&, ribbon used in this investigation, but on both the amorphous and ‘the microcrystalline specimens the nature of the deformation did not, in general, appear to be affected by these
due to the crystallites can be seen all over the thin part of the specimen in Fig. 4b for example, and it is this contrast which masks the presence of any small voids. We cannot, therefore, determine whether the structure of a shear band in microcrystalline PdsoSilo is more, or less, disordered after deformation than that of a shear band in amorphous Pds&,.
The appearance of the slip steps associated with shear bands in microcrystalline material was found to depend on whether the bands were formed on the ‘tension’ or ‘compression’ side of the bend, and also on the deformation temperature. In some cases deformed microcrystalline Pds&& looked quite different from the deformed glass. SEM images of bent amorphous and microcrystalline PdsoSizo are shown in Fig. 5 and 6. In the amor-
Fig. 6. SEM images of surface slip-steps on Pdl,Si,, ribbons formed on the tension side of a bend: (a) amorphous deformed at room temperature, (b) amorphou; dofor&d at 78 K, (c) microcrystalline deformed at room temperature, (d) microcrystalline deformed at 78 K.
6
DONOVAN and STOBBS: SHEAR BAND DEFORMATION
irregularities. However, Fig. Sf shows a pit on the compression side of a microcrystalline specimen deformed at 78 K where the deformation appears to have been altered. A crack has formed in the pit across the primary shear bands, and at the head of the pit (labelled A) additional shear bands have formed, some of which have cracked open. This might be due to the presence of a ‘quenched-in’ crystal in the pit, since a gas-bubble would have reduced the local cooling-rate during solidification [6]. Alternatively the ‘secondary’ shear observed could have been caused by the stress-raising effect of the surface irregularity. This is the only example of this type of behaviour we have observed. 4. THE FORMATION OF SHEAR BANDS IN MICROCRYSI’ALLINE MATERIAL That shear bands can form in microcrystalline material is, at least initially, surprising, and as far as we are aware this behaviour has not been reported before. However, the TEN images shown in Fig 4 clearly indicate that the deformation of microcrystalline PdaoSizo gives rise to the same type of structural changes, at least after deformation has stopped and the load has been removed, as does the deformation of an amorphous metal [4]. We have previously suggested that the structural changes observed in unloaded shear bands formed in a glass are due to partial relaxation of the stresses associated with the dilatation of the shear bands during plastic flow. This dilatation is apparently necessary before shear can occur. Hence we can reasonably infer that an operating shear band in mi~~rys~line Pds&e is also dilated. Both the similarities and the differences in behaviour between the amorphous and the microcrystaIline material, as characterised by SEM, then provide further insights into how deformation in shear bands occurs. How then can apparently completely rnicrocrystalline material deform by the same shear band mechanism as a metallic glass? The basic similarity of the deformation mechanism in amorphous and microcan be understood if the very crystalline Pd,&e small size of the crystallites is considered. The bumpy nature of the shear bands, on a scale less than their width and comparable to that of the crystal&es, suggests that in microcrystalline material the shear deformation is accomplished in the grain-boundaires. It is reasonable to assume that these regions can be dilated more easily than the crystals. Furthermore, for current models of grain-boundaries between randomly oriented crystals [13,14] and for crystals of the size observed, the ‘volume fraction’ of disordered material forming the grain-boundaries is similar to that of the crystals. We can thus consider the grain-boundary regions to be effectively a continuous quasi-amorphous layer with a shear modulus perhaps 40% lower than that of the crystals it surrounds [IS], Local&d deformation shoufd then proceed by the dilatation of
this grain-boundary layer, so that the shear process may be visual&d as involving the movement of the crystallites in a ‘sea’ of dilated grain-boundary. Plastic shear of the crystallites themselves seems unlikely: while the Orowan stress for the motion of an existing dislocation within a typical crystallite is comparable with the theoretical yield strength of the glass, the stress required to form the dislocations is higher than this. 5. DISCUSSION OF TWE OPERATION OF A SHEAR BAND IN MICROCRYSTAUIME AND AMORPHOUS MATERL4L The relative motion of the crystallites within a shear band in microcrystalline material would lead to them impinging on one another, which would give rise to large, locally inhomogeneous elastic stresses. These stresses might be partially relaxed by non-uniform expansion and contraction of different grain boundaries across the operating shear band, but even for crystals as small as Snm in diameter this effect would be insu~ent to allow them to move past one another during shear. Hence there may be a tendency for voids to form at the interfaces between the crystallites during plastic flow. Although it can be seen from Fig. 4 that there arc no voids of a size comparable to the crystallites ln the relaxed shear bands. smaller voids (- 1 nm) could be present and larger voids could be formed in an operating and dilated band and yet be healed in the bulk of the material when shear ceases. We will thus consider a simple model for their fo~tion. The release of the energy associated with an elastically strained spherical crystal by the forrnation of a hemispher~~l capping void, will be energetically possible when
where p is the shear modulus of the crystal, E is the difference between elastic shear strain in the crystal and in its surroundings, r is the crystal radius and y is the surface energy. If we take r = 5 mn, y z 0.5 JmT2 and fi z 3.10” Nmw2, E must be about 10% for void formation. Since there is considerable uncertainty in the appropriate value of y, which could easily be l/3 of the value we have used, and since some stressrelaxation would occur if a smaller void were to be formed, which is probable given Goods and Brown’s development of the simple model above [16], it seems likely that voids are indeed nucleated in an operating shear band in microcrystalline material. In fact differences between the dilatation of the crystals and of their ‘amorphous surroundings* would probably be as important as the differences in relative elastic shears in promoting voiding, so we have not developed the above argument. Under a tensile stress voids would tend to grow, leading naturally to crack-formation, as was observed. Void-formation is less likely at smaller grain sizes, so it is also necessary to consider whether the in-
DONOVAN
and STOBBS:
SHEAR BAND DEFORMATtON
homogeneous shear stresses within the shear bands could be diffusionally relaxed. The strain-rate within a shear band is of the order of 10-l se’ [I73 and it therefore seems unlikely that significant diffusional relaxation could occur even at room temperature [18], although the activation energy for diffusion in the dilated grain-~undary layer must be inherently low. However, the SEM observations suggest that some local diffusion does occur within the shear bands during plastic flow at room temperature. This is demonstrated by the change in the appearance of the shear steps produced during compressive deformation of microcrystalline material as the deformation temperature is changed. At room temperature large shear strains are apparently carried by each band, with the formation of large rounded extrusions. The wavy appearance of the slip-steps and the presence of fine angled slip-lines suggests that ‘secondary’ slip is also occurring. At 78 K, on the other hand, there is no sign of ‘secondary’ slip, the shear steps are relatively straight and sharp-edged and the shear displacements are smaller. The large ‘primary’ shear strains obsenrMf at room temperature in the microcrysta&te material suggest that the nucleation of shear bands is more difficult than in the glass, but that once formed they can carry a large shear, presumably with some difFusional relaxation of local stress inhomogeneities. Hence a given plastic strain is carried by fewer bands than are required at lower temperatures (or in amorphous materhf~ However, the large plastic strains associated with these ‘primary bands’ produce iarge inhomogeneous stress-fields within the surrounding matrix and since these are not relaxed by diffusion this leads to ‘secondary’ slip= The extrusions, which are thicker than the width of a single shear band, indicate that slip is in fact occurring in closely associated bands. At low temperatures the primary shear bands are more evenly spaced and there is apparently no ‘secondary’ slip. This suggests that the relative lack of diffusional accomodation of the motion of the crystallites at 78 K is limiting the amount of plastic strain that can occur in any one shear band so that more are nucleated. Given that the slip is thus finer, and despite the fact that diffusional relaxation is clearly more difficult at 78 K than at room tem~rature, the inhomogeneity in the elastic stress fields is insu~cient to nucleate ‘secondary’ slip. The apparent difficulty in nucleating a shear band in microcrystalline Pd,&, is consistent with an increase of the tensile fracture stress [17], and it indicates that a greater dilatation is necessary in microcrystalline material than in a glass before cooperative plastic flow can occur.
an operating shear band in an amorphous
metal may be non-uniform, as it is in microcrystalline Pds&,. This suggests a shear process in amorphous material involving inhomogeneous and fluctuating density changes across the band, and furthermore it lends some insight into the physically surprising large width of shear bands in general. A non-uniform dilatation within a shear band will, ind&d, be energetically favourabfe. The energy increment involved in increasing the volume of the forming shear band by a small amount is proportional to the eiastic modulus, E, but at the large values of elastic strain which occur in a shear band the forces between the atoms will decrease as the strain increases and the atoms are moved further apart. An overall volume increase of 10% will, therefore, require less energy if the dilatation is nonuniform, and it is possible that in parts of the operating shear band the locai increase in volume may approach 20%. Secondly, our observations indicate that the models of inhomogen~us deformation in metallic glasses which have been developed previously are inadequate. These treatments [I, 2,6] have considered the motion of only small groups of atoms under an applied stress, as is more appropriate when anelastic deformation is considered [ 19,20]. However, such an approach does not provide a good description of shear band deformation, given these observations. Acknowledgements-We would like to thank Dr P. H. Gaskclf for providing the Pds,$i,, glass, S. E. Fielding for assistance with the scanning electron microscopy,
Professor R. W. K. Honeycombe FRS for provision of laboratory facilities and the SERC for financial support.
RE!?ERENCES 1. F. Spaepen, Acta merait.U, 407 (1977). 2. A. S. Argon and H. Y. Kuo, Mater. Sci. Engng 39, 101 (1979). 3. H. S. Chen, J. Non-Cryst. Solids 22, 135 (1976). 4. P. E. Donovan and W. M. Stobbs, Acta metal!. 29, I419 (1981). 5. P. E. Donovan and W. M. Stobbs. J. Nun-Cryst Solids. To be published. 6. A. S. Argon, Acta me&. 27,47 (1979). 7. T. Masumoto and R. Maddin, Acta metalk 19. 72s (1971). 8. P. Duhaj, D. Barancok and A. Ondryka, 3. Non-Cryst, Solids 21, 41 I (1976).
9. 10. 11. 12. 13.
CONCLUSIONS These observations have several interesting implications. Firstly, they imply that the local dilatation in
7
14. 15.
Giessen, in Rapidty Quenched Metab III (edited by 3. Cantor)? Vol. I, p. 438. Metals Society, London (1978). P. G. Self, H. K. R. H. Bhadeshia and W. M. Stobbs, Ultramicroscopy6, 29 (1981). 0. L. Krivanik, Election kicroscopy 1976 (edited by Brandon), Vol. I, p. 275. Tal Int. Publ. Co. S. C. McFarlane and W. Cochrane, J. Phys. CX, 1311 (1975). &%.F. Ashby, F. Spaepen and S. Williams, Acta met& 26, 1647 (1978). A. P. Sutton and V. Vitek, in ~ist~~~to~ Modetling of Physicai Systems (edited by Ashby, Butlough, Hartiey and HirthX p. 549. Pergamon Press, Oxford (~980). J. J. Gilman, J. a&. Phys. 46, 1625 (1975). 5. C.
8
DONOVAN
and STUBBS:
SHEAR BAND DEFORMATION
16. S. M. Goods and L. M. Brown. Acru twtail. 27, f i 1979). 17. M. Neuhauser. Scriptn melull. 12,471 (19783. 18. W. M. Stobbs, Phil. May. 27, 1073 (1973).
19. J. Logan and M. F. Ashby, Acta net&. 22, 1047 (1974). 20. A. S. Argon, in Disiocatiorr iMllcre&ng tJPi~ysicu1 Syswnrs (edited by Ashby, Buflough, Hartley and Mirth), p. 393. Pcrgamon Press, Oxford (1980).
THE SHEAR BAND DEFORMATION PROCESS MICROCRYSTALLINE PdsoSizo P. EL DONOVAN
IN
and W. M. STOBBS
Department of Metallurgy and Materials Science. University of Cambridge. Cambridge CB2 3QZ. England (Received 22 December 1981; in revised&m
f8 August
1982)
Abstract-We have observed the mode of deformation of initially amorphous PdseSiss which had been heat-treated to transform it to a microcrystalline state. In this condition the material undergoes localised deformation in apparently the same manner as the glass, in that transmission electron microscope images of the shear bands show the same kind of structural changes as we have previously obaewed in deformed amorphous metals. We.have also studied the surface slip-steps on microcrystalline specimens deformed by bending at 78 K and at room temperature by scanning electron microscopy. We found that the morphology of the shear bands formed under a compressive load depends on the deformation temperature; becoming, at 78 K, more like that seen at all temperatures in amorphous PdseSis,. The similarities and di&rences in the deformation behaviour of amorphous and microcrystalline PdseSile allow us to discuss a realistic model for the shear band deformation process. R&sum&Nous avons observe le mode de deformation dun PdseSir, initialement amorphc, ayant subi un traitement thermique afin de passer B un &at microcristallin. Dans ces conditions, le mattriau prtsente une ~fo~tion localis& ap~mment semblable a celle dun verre, en ce que les micrographis &ctroniques en transmission des bandes de cisaillement prCsentent le mime type de changements structuraux que ceux que nous avons antineurement observis darts des mttaux amorphes deform&s. Nous avons tgalcment ttudiC par microscopic Clectronique fi balayage les marches de &sement superficielks sur des Cchantillons microcristaliins deform& en flexion a 78 K et B la temptrature. ambiante. La morphologic des bandes de cisaillement form&es en compression dependait de la tern*ture de deformation et ellc devenait B 78 K semblable B celle qu l’on observe I toutes les temptratures dans PdssSise amorphe. Les similitudes et les differences dans la deformation de PdseSise amorphe et microcristallin nous permettent de discuter un modHe rCaliste pour le mtcanisme de d&formation par bandes de cisaillement. Zuaammeafatmmg-Die Verformung von mikrokristallinem Pds&e wurde untersucht Das ursprtlnglich amorphe Material wur& hierxu mit einer Wirmebehandlung in den rnikro~s~ti~n Zustand tibergefiihrt. Das Material verformt aich tokal, scheinbar wie im Glasxustand. Die ekktronenrmkro+opische Dur~~~~g der S&erb%nder im ~~k~s~li~n Material zeigt diiben strukturelIen Anderungen wie im verformten amorphen Metall. Die durch Biegung des ~~ok~s~l~en Materials bei 78 K und Raumtemperatur entstandenen Gleitatufen wur&n im Rast~elek~o~nmi~k~ UntersuchL Die Struktur der U&K Drucklast entstandenen ScherbSinder hiingt van der Verfo~u~~-~~ sb. +i 78 K iihnelt sie der, die bei silmtlichen Temperaturen am amorphen PdseSise beobachtet wird. Ahnlichkeiten und Unterschiede im Verformungsverhalten zwischen amorphen und mikrokristaflinen PdssSise erm@lichen uns, ein realistisches Mode11Wr den Scherbandverformungsprozefi zu diskutieren.
vicinity of a free surface. Where a free surface was present during plastic flow small voids, less than 1 nm in diameter, are observed in the shear bands to a depth of the order of the shear band width, which is between 10 and 2Onm in all the metallic glasses we have examined [4,5]. The determination of the width of shear bands was an important result of our previous work, but we were surprised to find that operating shear bands are as wide as this because, firstly, it is difficult to visualise such a large region undorgoing the necessary expansion for shear without becoming unstable and, secondly, most models for deformation in metallic glasses are based on much more localised atomic rearrangements [ 1,263. In order to obtain further information about the deformation process in amorphous metals we have
1. INTBODUCIION
At low temperatures and high strain-rates metallic glasses undergo ~~y-l~ii~ ‘inhomogeneous’ defor~tion [ 11, during which plastic flow is confined to a few narrow regions of the matfzriak known as ‘shear bands’. It has been suggested that this type of deformation may produa structural changes in the shear bands, such as a decrease in density [2] or disruption of the local chemical o&ring 133. We have previously investigated the structure of shear bands in the unloaded state by transmission electron microscopy (TFM) [“I]. Our observations indicate that inhomogeneous deformation involves an increase in the volume of the deforming material, which is recovered when shear ceases except in the A.M. 31/i--*
1
2
DONOVAN
and STOBBS:
SHEAR BAND DEFORMATION
investigated the behaviour of shear bands in two classes of crystallised glass. In the initial stages of crystallisation most metallic glasses form a dispersion of relatively large crystals in the amorphous matrix. The behaviour of this type of material is discussed elsewhere [SJ. Here we report our observations on Pd,,SizO, which transforms initially to a microcrystalline state [7]. 2. EXPERIMENTAL The amorphous PdeoSilo ribbon used in this investigation was prepared at the Cavendish Laboratory by a single-roller quenching method and supplied by Dr P. H. Gaskell. Specimens were prepared for TEM examination by electropolishing in a solution of 10% perchloric acid in glacial acetic acid at 283 K. As described previously [4], the specimens were deformed by bending and then given a coating of lacquer on one side before being polished to perforation from the opposite side. In this way the original surface of the deformed glass was retained and slip steps marked the positions of the shear bands. The surfaces of specimens which had been deformed at room temperature or at 78 K were examined by scanning electron microscopy (SEM). These specimens were deformed by bending them across the edge of a razor blade to give approximately the same total strain in each case. 3. ORSERVATIONS 3.1 Strucrure of the annealed Pd8&, After the glass had been annealed for 63 h at 563 K it was found to be brittle, so that on bending the specimens shattered. On examining this material in the TEM it was found to have transformed almost completely to a state comparable to that described by Masumoto and Maddin [7j as MS-II, with an average grain size of about 0.5 pm. After annealing for 46 h at 563 K the ribbon was still quite ductile. TFM and X-ray investigation of the ribbon in this state demonstrated that it was fully microcrystalline, with an average grain size of ap proximately 5 nm. The X-ray diffraction patterns showed only distinct lines, broad because of the small grain-size of the material but with no trace of the diffuse rings characteristic of amorphous materials, and this limits the volume fraction of any amorphous material in the specimens to less than about 5%. The d-spacings of the microcrystalline phase are listed in Table 1. They do not correspond to those of the f.c.c. microcrystalline phase reported by Masumoto and Maddin 17-J. Some of the d-spacings correspond to those of palladium, but the remaining spacings do not match any of the stable phases in the Pd-Si phase t Even a random structure will give rise to bright speckle in dark-field images, but the size of this speckle is determined by the Airy-disc size of the objective aperture. Larger bright specks can be interpreted as images of crystallites.
Table I. X-ray diffraction results for the metastable crystalline
Measured lattice spacing d(A) 2.77 2.71 2.52 2.36 2.30 2.24 2.16 2.10 2.03 1.95 1.89 1.79 1.66 1.63 1.57 1.54 1.49 1.47 1.39 1.37 1.34 1.32 1.31 1.29 1.28 1.26 1.23 1.185 1.171 1.157 1.121
Observed intensity I
micro-
phase
Pd lattice spacing (PDF)
Relative intensity IIIi (PDF)
2.246
100
1.945
42
1.376
25
1.1730
24
0.9723
3
0.8924 0.8697
13 11
VW W VW
m vw S
m m W m VW VW VW VW VW W mw W W W W mw W W VW W VW mw VW VW W
0.942 0.911
VW
0.869 0.841 0.828 0.807 0.7%
VW
VW
VW VW VW VW
diagram. It seems that the crystallisation process in Pds&, glass is very sensitive to the precise. initial composition of the material and the annealing conditions [S, 91. The crystal structure of the metastable microcrystalline state was not determined in this work, since for our investigation of shear band deformation only a knowledge that the material was fully microcrystalline and of the relative sizes of the crystal grains and the shear bands was required. The morphology of the aged material was investigated by high-resolution bright-field and dark-field TRM. A high resolution axial bright-field image is shown in Fig. la, and a dark-field image in Fig. lb. The degree of crystallinity of the material may be estimated by observing the area covered by lattice fringes in the bright-field image and by non-aperturelimited bright speckle in the dark-field image.t For the imaging conditions used for the bright-field image,
DONOVAN and STOBBS:
SHEAR BAND DEFORMATION
Fig. 1. TEM images of a PdlaSizo specimen after it had been annealed for 48 h at 563 K. (a) Highresolution bright-field image, (b) dark-field image. The edge of the specimen is indicated by arrows. Moire fringes from overlapping cryatallites can be seen in the relatively thick parts of the specimen.
a fully crystalline specimen with random gram orientations and a thickness of one crystal diameter would be expected to show lattice fringes over about 15% of its area, although this is obviousIy dependent on the crystal size (e.g. 10). Quite clearly the fringe coverage in the region shown is much greater than thih hich is mainly because the specimen thickness was In ore than one crystal diameter (5 nm). However, the pimen was not more than about 10nm thick in the
region shown, as can be seen from the simple nature of the Moire fringes formed between di~rent crystalIites in the dark-field image, Fig. lb. While there are problems in detecting isolated crystal&s in thick amorphous specimens by bright-field imaging [ 1I], and images of amorphous material obtained with non-axial illumination can appear to be crystalline under certain circumstances [12], the high degree of fringe coverage in the image shown, which was obtained using axial illumination, strongly suggests that the material contains no amorphous regions. This is confirmed by the dark-field image, Fig. lb, which has 30% coverage of non-a~rture-lined speckle. Again, this figure is higher than would be expected (< 10%) for a completely crystalline specimen one crystal diameter in thickness, 3.2 TEM observations of shear bands in microcrystalline Pd80Si,0
Fig. 2. TEM image of shear bands formed on the ‘tension’ side of a bent Pd&lo specimen after it had ban annealed for 48 h at 563 K.
Figure 2 is a low-magnification TEM image of a group of shear bands formed on the ‘tension’ side of a bent microcrystalline Pdt,,,Si10 specimen. At this magnification the shear bands are indistinguishable from those formed in amorphous Pds&,, as shown in Fig. 3. A series of TEN images of a shear band formed at room temperature on the ‘tension’ side of a micr~rys~lline specimen is shown in Fig 4. The feature marked A in Fig. 4a is a small crack with its
4
I~ONOVAN
side of a bent amorphous Pda,&,
wd
SI‘OBBS:
SHEAR
ribbon.
associated plastic zone (EC) formed in the thin specimen after electropolishing. In the bright-field image, t We have discussed elsewhere [43 the general problem of ~~erentiating between isoIated voids and more diffuse low density regions by dark field techniques and here the term ‘void’ is used for bath types of irregularity.
HAN11
Illif-ORMATION
Fig. 4a, it can be seen that the edges of the crack are bumpy, suggesting that the crack has propagated through the grain-bottndi~ry Iaycrs between the crystaflites. A similar bumpiness can be seen on the edge of the shear step (DE), suggesting that during bulk shear plastic flow also occurred in the grain-boundary layers. There is no evidence to suggest that the crystals themselves have been plastically deformed. The bright speckle contrast from the plastic zone around the crack in the dark-field images (4&f) shows that a high concentration of voids has been formed there, just as they are formed when thin films of amorphous metals undergo plastic deformationt. Our previous investigation showed that a much lower density of voids is produced when shear bands are formed in a bulk amorphous specimen than when they are formed in a thin film. Unfortunately, it is not possible to determine from the images shown in Fig. 4 whether a concentration of voids comparable to that which we have observed in shear bands in amorphous material is present in the bulk shear band,. DE, because of the relatively intense low-angle scattering from the crystallites. Small specks of bright contrast
Fig. 4. TEM images of a shear band. DE, and a crack, AC, in micr~r~tailine Pd&,,. (a) Bright-field image, (b-f) dark-field images with the di~ra~tion conditions shown inset. The contrast of both the shear band and the plastic zone around the crack is the same as that which we have previously observed in amorphous
Fe,oNi40B,0.
DONOVAN
and STOBBS:
SHEAR BAND DEF~)RMAT[ON
Fig. 5. SEM images of surfaa slipstcps on PdsOSizOribbons formed on the compression side of a bend: (a) amorphous, deformed at room temperature, (b) amorphous deformed at 78 K, (c) ~~~~iin~ deformed at room temperature, (d) micr~ys~llin~ deformed at 78 K. &ale marker represents 1 pm, (e) microcrystalline deformed at room temperature, scak marker represents 0.1~~5, (1) microcrystalline deformed at 78 K, scale marker represents 10 @n.
5
phous metal the small differences between the stip steps formed on the different surfaces and at the different temperatures are probably not significant, since the deformation was not strictly controlled. On the ‘tension’ side of a microcrystalline specimen there is substantial ‘faceted’ cracking, whether the material is deformed at room temperature or at 78 K. Indeed in these images (Fig. 6c and d) no shear bands are apparent which have not opened out into cracks. However we know from Fig. 3 that slip steps are in fact formed without cracking in rni~~ys~~me material at lower applied pIastic strains. This implies that shear bands in micr~rys~lline Pds&, can carry less plastic strain than those in amorphous PdBoSi20 when the net stress is tensile. The images of the ‘compression’ side of bent microcrystalline material are remarkable. At room temperature (Fig. 5~) the slip lines are wavy and fine crosslinking slip-steps have been formed. Furthermore the slip is now often associated with the extrusion of material in rounded slivers. However, after deformation at 78 K (Fig. 5d) well defined surface steps are observed, indistinguishable from those on a bent ~o~hous specimen. The bottom surface of a rapidly quenched metallic ribbon, which is in contact with the wheel during quenching, is usually rough. Typically there are pits where gas bubbles have been trapped berween the molten metal and the wheel. Pits of this type were present on the .Pd&&, ribbon used in this investigation, but on both the amorphous and ‘the microcrystalline specimens the nature of the deformation did not, in general, appear to be affected by these
due to the crystallites can be seen all over the thin part of the specimen in Fig. 4b for example, and it is this contrast which masks the presence of any small voids. We cannot, therefore, determine whether the structure of a shear band in microcrystalline PdsoSilo is more, or less, disordered after deformation than that of a shear band in amorphous Pds&,.
The appearance of the slip steps associated with shear bands in microcrystalline material was found to depend on whether the bands were formed on the ‘tension’ or ‘compression’ side of the bend, and also on the deformation temperature. In some cases deformed microcrystalline Pds&& looked quite different from the deformed glass. SEM images of bent amorphous and microcrystalline PdsoSizo are shown in Fig. 5 and 6. In the amor-
Fig. 6. SEM images of surface slip-steps on Pdl,Si,, ribbons formed on the tension side of a bend: (a) amorphous deformed at room temperature, (b) amorphou; dofor&d at 78 K, (c) microcrystalline deformed at room temperature, (d) microcrystalline deformed at 78 K.
6
DONOVAN and STOBBS: SHEAR BAND DEFORMATION
irregularities. However, Fig. Sf shows a pit on the compression side of a microcrystalline specimen deformed at 78 K where the deformation appears to have been altered. A crack has formed in the pit across the primary shear bands, and at the head of the pit (labelled A) additional shear bands have formed, some of which have cracked open. This might be due to the presence of a ‘quenched-in’ crystal in the pit, since a gas-bubble would have reduced the local cooling-rate during solidification [6]. Alternatively the ‘secondary’ shear observed could have been caused by the stress-raising effect of the surface irregularity. This is the only example of this type of behaviour we have observed. 4. THE FORMATION OF SHEAR BANDS IN MICROCRYSI’ALLINE MATERIAL That shear bands can form in microcrystalline material is, at least initially, surprising, and as far as we are aware this behaviour has not been reported before. However, the TEN images shown in Fig 4 clearly indicate that the deformation of microcrystalline PdaoSizo gives rise to the same type of structural changes, at least after deformation has stopped and the load has been removed, as does the deformation of an amorphous metal [4]. We have previously suggested that the structural changes observed in unloaded shear bands formed in a glass are due to partial relaxation of the stresses associated with the dilatation of the shear bands during plastic flow. This dilatation is apparently necessary before shear can occur. Hence we can reasonably infer that an operating shear band in mi~~rys~line Pds&e is also dilated. Both the similarities and the differences in behaviour between the amorphous and the microcrystaIline material, as characterised by SEM, then provide further insights into how deformation in shear bands occurs. How then can apparently completely rnicrocrystalline material deform by the same shear band mechanism as a metallic glass? The basic similarity of the deformation mechanism in amorphous and microcan be understood if the very crystalline Pd,&e small size of the crystallites is considered. The bumpy nature of the shear bands, on a scale less than their width and comparable to that of the crystal&es, suggests that in microcrystalline material the shear deformation is accomplished in the grain-boundaires. It is reasonable to assume that these regions can be dilated more easily than the crystals. Furthermore, for current models of grain-boundaries between randomly oriented crystals [13,14] and for crystals of the size observed, the ‘volume fraction’ of disordered material forming the grain-boundaries is similar to that of the crystals. We can thus consider the grain-boundary regions to be effectively a continuous quasi-amorphous layer with a shear modulus perhaps 40% lower than that of the crystals it surrounds [IS], Local&d deformation shoufd then proceed by the dilatation of
this grain-boundary layer, so that the shear process may be visual&d as involving the movement of the crystallites in a ‘sea’ of dilated grain-boundary. Plastic shear of the crystallites themselves seems unlikely: while the Orowan stress for the motion of an existing dislocation within a typical crystallite is comparable with the theoretical yield strength of the glass, the stress required to form the dislocations is higher than this. 5. DISCUSSION OF TWE OPERATION OF A SHEAR BAND IN MICROCRYSTAUIME AND AMORPHOUS MATERL4L The relative motion of the crystallites within a shear band in microcrystalline material would lead to them impinging on one another, which would give rise to large, locally inhomogeneous elastic stresses. These stresses might be partially relaxed by non-uniform expansion and contraction of different grain boundaries across the operating shear band, but even for crystals as small as Snm in diameter this effect would be insu~ent to allow them to move past one another during shear. Hence there may be a tendency for voids to form at the interfaces between the crystallites during plastic flow. Although it can be seen from Fig. 4 that there arc no voids of a size comparable to the crystallites ln the relaxed shear bands. smaller voids (- 1 nm) could be present and larger voids could be formed in an operating and dilated band and yet be healed in the bulk of the material when shear ceases. We will thus consider a simple model for their fo~tion. The release of the energy associated with an elastically strained spherical crystal by the forrnation of a hemispher~~l capping void, will be energetically possible when
where p is the shear modulus of the crystal, E is the difference between elastic shear strain in the crystal and in its surroundings, r is the crystal radius and y is the surface energy. If we take r = 5 mn, y z 0.5 JmT2 and fi z 3.10” Nmw2, E must be about 10% for void formation. Since there is considerable uncertainty in the appropriate value of y, which could easily be l/3 of the value we have used, and since some stressrelaxation would occur if a smaller void were to be formed, which is probable given Goods and Brown’s development of the simple model above [16], it seems likely that voids are indeed nucleated in an operating shear band in microcrystalline material. In fact differences between the dilatation of the crystals and of their ‘amorphous surroundings* would probably be as important as the differences in relative elastic shears in promoting voiding, so we have not developed the above argument. Under a tensile stress voids would tend to grow, leading naturally to crack-formation, as was observed. Void-formation is less likely at smaller grain sizes, so it is also necessary to consider whether the in-
DONOVAN
and STOBBS:
SHEAR BAND DEFORMATtON
homogeneous shear stresses within the shear bands could be diffusionally relaxed. The strain-rate within a shear band is of the order of 10-l se’ [I73 and it therefore seems unlikely that significant diffusional relaxation could occur even at room temperature [18], although the activation energy for diffusion in the dilated grain-~undary layer must be inherently low. However, the SEM observations suggest that some local diffusion does occur within the shear bands during plastic flow at room temperature. This is demonstrated by the change in the appearance of the shear steps produced during compressive deformation of microcrystalline material as the deformation temperature is changed. At room temperature large shear strains are apparently carried by each band, with the formation of large rounded extrusions. The wavy appearance of the slip-steps and the presence of fine angled slip-lines suggests that ‘secondary’ slip is also occurring. At 78 K, on the other hand, there is no sign of ‘secondary’ slip, the shear steps are relatively straight and sharp-edged and the shear displacements are smaller. The large ‘primary’ shear strains obsenrMf at room temperature in the microcrysta&te material suggest that the nucleation of shear bands is more difficult than in the glass, but that once formed they can carry a large shear, presumably with some difFusional relaxation of local stress inhomogeneities. Hence a given plastic strain is carried by fewer bands than are required at lower temperatures (or in amorphous materhf~ However, the large plastic strains associated with these ‘primary bands’ produce iarge inhomogeneous stress-fields within the surrounding matrix and since these are not relaxed by diffusion this leads to ‘secondary’ slip= The extrusions, which are thicker than the width of a single shear band, indicate that slip is in fact occurring in closely associated bands. At low temperatures the primary shear bands are more evenly spaced and there is apparently no ‘secondary’ slip. This suggests that the relative lack of diffusional accomodation of the motion of the crystallites at 78 K is limiting the amount of plastic strain that can occur in any one shear band so that more are nucleated. Given that the slip is thus finer, and despite the fact that diffusional relaxation is clearly more difficult at 78 K than at room tem~rature, the inhomogeneity in the elastic stress fields is insu~cient to nucleate ‘secondary’ slip. The apparent difficulty in nucleating a shear band in microcrystalline Pd,&, is consistent with an increase of the tensile fracture stress [17], and it indicates that a greater dilatation is necessary in microcrystalline material than in a glass before cooperative plastic flow can occur.
an operating shear band in an amorphous
metal may be non-uniform, as it is in microcrystalline Pds&,. This suggests a shear process in amorphous material involving inhomogeneous and fluctuating density changes across the band, and furthermore it lends some insight into the physically surprising large width of shear bands in general. A non-uniform dilatation within a shear band will, ind&d, be energetically favourabfe. The energy increment involved in increasing the volume of the forming shear band by a small amount is proportional to the eiastic modulus, E, but at the large values of elastic strain which occur in a shear band the forces between the atoms will decrease as the strain increases and the atoms are moved further apart. An overall volume increase of 10% will, therefore, require less energy if the dilatation is nonuniform, and it is possible that in parts of the operating shear band the locai increase in volume may approach 20%. Secondly, our observations indicate that the models of inhomogen~us deformation in metallic glasses which have been developed previously are inadequate. These treatments [I, 2,6] have considered the motion of only small groups of atoms under an applied stress, as is more appropriate when anelastic deformation is considered [ 19,20]. However, such an approach does not provide a good description of shear band deformation, given these observations. Acknowledgements-We would like to thank Dr P. H. Gaskclf for providing the Pds,$i,, glass, S. E. Fielding for assistance with the scanning electron microscopy,
Professor R. W. K. Honeycombe FRS for provision of laboratory facilities and the SERC for financial support.
RE!?ERENCES 1. F. Spaepen, Acta merait.U, 407 (1977). 2. A. S. Argon and H. Y. Kuo, Mater. Sci. Engng 39, 101 (1979). 3. H. S. Chen, J. Non-Cryst. Solids 22, 135 (1976). 4. P. E. Donovan and W. M. Stobbs, Acta metal!. 29, I419 (1981). 5. P. E. Donovan and W. M. Stobbs. J. Nun-Cryst Solids. To be published. 6. A. S. Argon, Acta me&. 27,47 (1979). 7. T. Masumoto and R. Maddin, Acta metalk 19. 72s (1971). 8. P. Duhaj, D. Barancok and A. Ondryka, 3. Non-Cryst, Solids 21, 41 I (1976).
9. 10. 11. 12. 13.
CONCLUSIONS These observations have several interesting implications. Firstly, they imply that the local dilatation in
7
14. 15.
Giessen, in Rapidty Quenched Metab III (edited by 3. Cantor)? Vol. I, p. 438. Metals Society, London (1978). P. G. Self, H. K. R. H. Bhadeshia and W. M. Stobbs, Ultramicroscopy6, 29 (1981). 0. L. Krivanik, Election kicroscopy 1976 (edited by Brandon), Vol. I, p. 275. Tal Int. Publ. Co. S. C. McFarlane and W. Cochrane, J. Phys. CX, 1311 (1975). &%.F. Ashby, F. Spaepen and S. Williams, Acta met& 26, 1647 (1978). A. P. Sutton and V. Vitek, in ~ist~~~to~ Modetling of Physicai Systems (edited by Ashby, Butlough, Hartiey and HirthX p. 549. Pergamon Press, Oxford (~980). J. J. Gilman, J. a&. Phys. 46, 1625 (1975). 5. C.
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DONOVAN
and STUBBS:
SHEAR BAND DEFORMATION
16. S. M. Goods and L. M. Brown. Acru twtail. 27, f i 1979). 17. M. Neuhauser. Scriptn melull. 12,471 (19783. 18. W. M. Stobbs, Phil. May. 27, 1073 (1973).
19. J. Logan and M. F. Ashby, Acta net&. 22, 1047 (1974). 20. A. S. Argon, in Disiocatiorr iMllcre&ng tJPi~ysicu1 Syswnrs (edited by Ashby, Buflough, Hartley and Mirth), p. 393. Pcrgamon Press, Oxford (1980).