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Strain concentration at grain boundaries in AlMgSi alloys
Materials Science and Engineering, A157 (1992) 1-8
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Strain concentration at grain boundaries in A1-Mg-Si alloys P. Singh* a n d J. T. E v a n s Department of Mechanical, Materials and Manufacturing Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7R U (UK) N. J. H. H o l r o y d Alcan International, Banbury, Oxon. OX16 7SP (UK) (Received July 8, 1991; in revised form December 17, 1991 )
Abstract Grain boundary sliding has been observed in an A1-Mg-Si alloy deformed at room temperature. The extent of sliding was primarily a function of the macroscopic strain whether this was produced by creep or in slow-strain-rate tensile tests but was more intense at low strain rates than at high strain rates. Grain boundary microcracks were observed to form after a critical amount of sliding at low strain rates. At high strain rates, microvoids nucleated at larger second-phase particles rather than in the grain boundaries. A consideration of the mechanisms which could operate to produce localized deformation indicates that the phenomenon is due initially to slip on the grain boundary plane itself but is quickly intensified by plastic flow within the precipitate-free zone after microvoids have nucleated. Plastic flow without the prior nucleation of holes cannot be sufficiently concentrated to account for the observed strain localization.
1. Introduction High purity A1-Mg-Si alloys can exhibit low ductility in the peak precipitation-hardened state because of the occurrence of intergranular fracture [1-4]. Detailed observation of the fracture surfaces reveals the presence of fine-scale dimples (spacing, 1-4/~m) which suggests that concentrated deformation localized near the grain boundary is responsible for the fracture. The addition Of small amounts of manganese or chromium greatly improves the properties of A1-Mg-Si alloys by suppressing intergranular fracture, giving instead transgranular fracture with coarse dimples. Dowling and Martin [5] noted that small non-coherent or semicoherent precipitates from manganese or chromium are effective in dispersing coarse transgranular slip in aluminium alloys and they proposed that the beneficial effect of manganese or chromium arises from the homogenization of slip and suppression of the heavy stress concentrations which occur when coarse slip bands impinge on grain boundaries.
*Present address: Department of Materials Science and Engineering, Case Western Reserve University, Cleveland, OH 44106, USA. 0921-5093/92/$5.00
Manganese additions are effective in suppressing intergranular fracture in A1-Mg-Si strained at normal rates. However, there is evidence that intergranular fracture reappears in commercial alloys containing manganese [6, 7] when plastic straining is very slow or when specimens are subjected to sustained loading. Lewandowski et al. [7] observed dimpled intergranular fracture after slow crack growth in AI-Mg-Si alloys containing manganese, similar to the intergranular fracture surfaces produced in the more rapid fracture of pure AI-Mg-Si. Lewandowski et al. also observed a second mode of intergranular fracture at low crack velocities (less than 10 -3 /~m s -1) at stress intensities just above the threshold for slow crack growth; here, fracture surfaces were smooth and undimpled and associated with the presence of small quantities of lead which can be present as an impurity in commercial alloys. An implication of the slow-crack-growth observations is that the strain concentrations at grain boundaries increase when the rate Of strain is very low or under sustained loading conditions. This was confirmed by Lewandowski et al. [7] and Kim et al. [8] in supplementary tests on unnotched specimens. The object of the present work was to measure the deformation at grain boundaries as a function of strain rate © 1992--Elsevier Sequoia. All rights reserved
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Strain concentration at grain boundaries in AI-Mg-Si
and to relate the strain concentrations to the development of ductile intergranular fracture.
2. Experimental method
A feature of the deformation of a number of aluminium alloys is the development of prominent grain boundary offsets. In the present authors' experience, the development of these offsets after small deformations at room temperature is particularly pronounced in A1-Mg-Si alloys [9], although it can also occur in high purity aluminium [10] and in A I - Z n - M g alloys [11-13]. This is easily observed in specimens which have been electropolished and then strained a few per cent. An example is shown in Fig. 1. At first sight, the offsets may be mistaken for microcracks but, although microcracks do eventually form at these features, careful examination reveals little evidence that the boundaries are cracked when the offsets first become visible. In order to study the grain boundary deformation in a semiquantitative way the following experiments were performed to measure the in-plane component of the relative displacement of the neighbouring grains. Tests were made on the alloy with the composition shown in Table 1. The impurity lead level was 20 ppm. Plates were supplied in the peak aged condition (a yield stress of 320 MPa and a tensile strength of 355 MPa at a strain rate of 1.3 x 10 -3 S-l) from experimental casts which had been homogenized, hot rolled, solution treated, quenched and aged. The microstructure con-
sisted of elongated grains with the longer axis parallel to the rolling direction (average grain diameter in this direction, 100 /~m). Creep tests and slow-constantstrain-rate tests were performed on small tensile specimens (12.5 mm gauge length x 5 mm x 1 mm) machined with the tensile axis parallel to the short direction of the plate. One surface of each specimen was polished to a high quality optical finish. The penultimate stage in polishing consisted in lapping on a cloth impregnated with colloidal silica. In the final stage, fine scratches were deliberately introduced by polishing on a cloth impregnated with 0.25/zm diamond paste to enable the grain boundary offsets to be measured [10]. The following procedure was adopted. The polished specimens were deformed at room temperature in creep or'with a constant applied strain rate (7 x 10 -7, 7 x 10- 5 or 7 x 10- 3 s- l) to a predetermined strain. The specimen was then removed from the testing machine and examined in the scanning electron microscope. At this stage, areas showing prominent grain boundary offsets were identified and photographed and their position was recorded. The specimen was then remounted in the testing machine, straining was continued and the above process was repeated a number of times. In this way the development of grain boundary sliding (GBS) and the nucleation of microcracks at specific locations were recorded. A smaller number of specimens were examined in a similar way after creep deformation under constant load.
3. Results
TABLE 1. Alloy composition
Grain boundaries were observed to slide and eventually to crack at a rate determined by the accumulation of macroscopic strain. In order to characterize the properties of the alloy, constant-load creep tests were performed on small tensile specimens at room temperature. Little deformation was observed at stresses smaller than about 85% of the nominal yield stress (measured with an applied strain rate of 1 . 3 x 1 0 -3 s-l). At larger stresses, creep curves showing the classical form were obtained, with decelerating creep in stage I, a constant creep rate in stage II and accelerating creep leading to fracture in stage HI. At lower stresses, only stages I and II were monitored because of the large times required to reach stage HI. The results are summarized in Fig. 2 where the minimum creep rate dminis shown as a function of a/o0 on a log-log plot, where o is the applied stress and o0 is the nominal yield stress. The curve in Fig. 2 suggests a relation of the form
Element Mg Si Mn Fe Cu Zr A1 Amount(wt.%) 0.62 1.00 0.55 0.24 0.08 0.02 Balance
e=m=eo 7o
Fig. 1. Offsets produced by GBS after a macroscopic strain of 2% (peak aged alloy with the composition shown in Table 1). In this case the slip vector is out of the plane of the surface.
(o)n
(1)
P. Singh et aL 10-2
I
r
l
[
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Strain concentration at grain boundaries in AI-Mg-Si
3
I
O
o
I
-z,
10 "7 .c
10-6
10-8
I 1°, I i 0.8
1.0 o/00
1-2
Fig. 2. Logarithmic plot of the minimum creep rate as a function of applied stress for specimens loaded at room temperature.
where ~0 and n are constants. This is similar to power law creep [14] which is often observed at higher temperatures but in the present case the exponent n is much larger (greater than 100) than normally associated with power law creep. The difference can be interpreted as indicating the presence of a large internal stress [15], somewhat smaller than the short-term yield stress. The ductility of the material in the creep tests at the faster rates r~lnged from 12 down to 10% elongation to fracture compared with 15% for a short-term tensile test. Examination of the final fracture surfaces showed mixed transgranular and intergranular fracture with an increasing proportion of intergranular fracture produced at low strain rates. Thus the numbers of grain boundaries showing sliding and cracking increased as the duration of the test increased. However, the extent to which intergranular cracking produces embrittlement in the material was not studied in detail in this work since studies of slow crack growth in notched specimens [7] indicate that triaxial tension of the sort which appears at the tip of a sharp crack may be required to develop embrittlement to its full extent. Observations of the surface of polished specimens deformed in creep revealed the nucleation and growth of grain boundary microcracks. These were always formed at grain boundaries which were highlighted by the formation of sliding offsets prior to the nucleation of the cracks. The grain boundaries showing prominent offsets were often those where the trace of the boundary plane on the surface was oriented at about 45 ° to the tensile axis, although offsets at boundary traces with a small angle were also observed (e.g. Fig. 1). In these cases the slip vector had a large out-of-plane component at the surface. An example showing a sequence of offset formation and microcrack nucleation is shown in Figs. 3(a) and 3(b). The microcracks widen as they lengthen and then the cracked bound-
Fig. 3. Sequence showing crack nucleation at grain boundary offsets in specimens loaded at a constant stress (300 MPa): (a) after 1 day; (b), (c) after 5 days; (c) the dimpled fracture surface of the grain boundary microcrack in (b).
aries were seen to be dimpled on a fine scale (Fig. 3(c)). It appears that localized shear produces microvoids in the boundary which eventually link to give the microcracks. We were unable to identify the particles which forni the nuclei for the dimples because of their very small diameters (less than 0.25/~m), but circumstantial evidence suggests that they are probably Mg2Si precipitates. A fine dispersion of MnA16is also expected
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Strain concentration at grain boundaries in AI-Mg-Si
to be present at grain boundaries but, since fine dimpled intergranular fracture occurs profusely in manganese-free A1-Mg-Si alloys [5], it seems likely that Mg2Si precipitates at the grain boundaries are responsible in both cases. Slow-strain-rate tests were performed on polished a n d scratched specimens to measure the in-plane grain boundary sliding. Specimens were examined at intervals in the course of a test, first to locate those boundaries where grain offsets occurred and then to follow the increase in the localized deformation as a function of strain. An example is shown in Figs. 4(a)-4(d). Grain boundary sliding was first detected from the misregistry of the scratch marks. After a small critical strain the formation of microcracks was detected as is also indicated by the opening up of gaps between the ends of the scratch marks across the boundary. The effect of the applied strain rate on the offsets parallel to the boundary plane (maximum recorded values) as a function of the applied strain is shown in Fig. 5. The offset increases approximately as the square root of the strain, the largest offsets being produced at the lowest strain rate. Microcrack nucleation was detected after an offset of about 0.6/~m at about 2% strain when the applied strain rate was 7 x 10 -5 s-1 or less, but no microcrack nucleation was detected at the faster strain rate of 7 x 10- 3 s- 1. Instead, transgranular cracking occurred at higher strains (7%) nucleated by microcracking of coarse ironcontaining second-phase particles in the grain interiors (Fig. 4(e)). These particles are present in relatively small numbers as a consequence of the solidification process rather than as a result of subsequent heat treatment. Evidently, the GBS is not sufficiently developed by fast straining (Fig. 5) to produce visible microcracking in the boundaries.
4. Discussion
Time-dependent intergranular cracking occurs during creep, and our results show that its nucleation is due to the accumulation of strain since similar amounts of cracking were obtained in creep and slow-strain-rate tests. After nucleation the microcracks grew somewhat discontinuously at rates between 0.1 and 0.01/~m s-1. The observations indicate a link between the development of localized strain at the grain boundaries and the subsequent formation of microcracks in the grain boundaries. However, the effect is most pronounced at low strain rates. At high strain rates the concentration of deformation at the grain boundaries is smaller and transgranular microcracks at second-phase particles are formed in preference to the grain boundary cracks. In contrast, grain boundary ductile fracture is observed to occur at high strain rates in high purity AI-Mg-Si
alloys [5]. The grain boundary deformation in the present alloy is sufficiently localized at low strain rates to give the appearance of grain boundary sliding, so that the grains slide past one another by slip either on the boundary plane or in a very narrow zone adjacent to the grain boundary, possibly corresponding to the precipitate-free zone (PFZ). The possibility of the latter is discussed below. The phenomenon is similar to the GBS observed in high temperature creep, but the mechanism must be different because of the virtual absence of diffusional flow at room temperature. In their model of GBS, Raj and Ashby [16] treated the grain boundary itself as if it were an atomically thin region which can support a tensile stress acting normal to the boundary but which has no resistance to shear tangential to the boundary. Given that a grain boundary is not generally a planar interface and possesses local curvature and undulations, the rate of inelastic sliding must be controlled by the accommodation of strain in the neighbouring grains. Raj and Ashby treated some idealized geometries with accommodation by diffusional flow. However, in the present experiments the observed rate of sliding is, at best, about five orders of magnitude larger than could be explained by diffusional flow accommodation at room temperature. It follows that any accommodation of the strains produced by GBS must be accomplished by dislocation glide and plastic flow in the grains adjacent to the boundary, a possibility recognized but not treated by Raj and Ashby [16]. The fact that the extent of GBS is mainly determined by the applied strain (Fig. 5) supports this conclusion, assuming that local accommodation can be accomplished within the latitude provided by general yielding. However, the extent of GBS increases at slow applied strain rates, which suggests thermal activation of the process of sliding on the boundary plane. Unfortunately little is known about this aspect of localization of strain at grain boundaries. Dahmen and Hornbogen [10] investigated GBS in pure aluminium and found two values of activation energy: one characteristic of sliding and migration at temperatures above 200 °C and another lower value at temperatures below 200 °C. The lower value was believed to be stress dependent. Dahmen and Hornbogen also found evidence from the transmission electron microscopy (TEM) examination of a high dislocation density either at the grain boundary or in a localized region within 0.2/~m of the boundary in deformed specimens. In the present AI-Mg-Si alloy there is also the possibility of an additional effect associated with PFZs at the grain boundary of the sort shown in Fig. 6. The relative softness of the PFZ is often envisaged to be responsible for the localization of strain at the grain boundaries, but in the present authors' opinion material continuity pre-
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Strain concentration at grain boundaries in A l-Mg-Si
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Fig. 4. (a)-(d) GBS observed by the misregistry of scratch marks across a boundary after deformation at various strains at 7 x 10-5 s- 1, showing the nucleation of microvoids: (a) 0.6%; (b) 1.2%; (c) 2.4%; (d) 3.2%. (e) At fast strain rates (7 x 10 -3 S-l), microvoids are nucleated at 7% strain at large second-phase particles rather than in the grain boundary. cludes this possibility unless some special conditions apply. This is now discussed with the aid of a simple model. Consider a P F Z (Fig. 7(a)) with a local shear strength r i and width w. T h e material adjacent to the P F Z has a
larger strength ry. In approximate terms, one can identify ry with o0/31/2, where tr0 is the macroscopic yield stress (ry ~- 185 MPa) whereas, since ri refers to the Mg2Si-depleted PFZ, it can be identified with the shear yield strength of a grossly overaged specimen of
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Strain concentration at grain boundaries in AI-Mg-Si
the alloy, Le. r~ ~- 60 MPa. In fact, the precise values of ry and ri have little effect on the outcome as shown in the following argument. The softness of the zone can show up to the full extent if the slip plane happens to be parallel to the grain boundary plane (Fig. 7(b)). In the general case, however, the slip planes are inclined steeply to the boundary plane and a rough estimate shows that only a very small fraction of boundaries will be favourably oriented for single-plane slip parallel to the boundary. If the width of the PFZ is w and the length of a slip line contained within the PFZ is/, then, on the assumption of a random distribution of boundary orientation relative to a {111} slip plane, the fraction of grains with PFZ slip lines greater or equal to l is given by 43-(w/l)2. If the ratio l/w > 1000 (small angle of inclination to the boundary plane), which it has to be to account for the localized deformation, the fraction of grains of a suitable orientation is less than 10 -6. The fraction of grain boundaries showing sliding offsets is
2
I
,
I
i
~
I
E
much larger than this. In order to have a general shear produced by slip on planes not parallel to the boundary, the von Mises condition must be met [17], with cooperative slip on five independent crystal systems ({111}(110)) (Fig. 7(c)). In this case the strain can only be confined within the PFZ if dislocation pile-ups are formed at the interface between the PFZ and the grain interior. Clearly this is not tenable since there is no physical barrier at this interface. The only difference in the two regions is the difference in the local yield strengths ri and ry. This point can be demonstrated using the Bilby, Cottrell and Swinden (BCS) model [18] of a slip band. As illustrated in Fig. 7(d), we consider just one of the slip lines with a dislocation source located at the boundary. Slip in the PFZ operates with an effective stress r - ri (between 0 and w), whereas slip in the grain interior is driven by the effective stress r - r y , where r is the local component of the applied shear stress. For the moment we assume that ri < r < ry.
I
oe
s~
.1_, t-
7x10-s
.
~0 t-i ttl
o
~5
:%.
® ~ ' - - ,, 2
0
, "--'-'~', 4 Stroin */,
7×10-31 6
8
Fig. 5. Curves showing the effect of strain rate on the GBS as a function of applied strain. Microvoid nucleation was first detected at strains indicated by the arrows. No grain boundary microvoid nucleation was detected at fast strain rates (7 X 10-3 s-l).
Fig. 6. Precipitate-free zone at a grain boundary in the AI-Mg-Si alloy (TEM observation).
"v
Q
"gy
-, '- : - : ' i - . - - , "gi
(a)
(b)
-I
(cl
(d)
(e)
Fig. 7. Schematic illustration of mechanisms of GBS: (a) PFZ; (b) slip in the PFZ when a slip plane is parallel to the boundary plane; (c) deformation in the PFZ by slip on planes not parallel to the boundary; (d) a slip line extending from within the PFZ into the neighbouring grains; the slip line extends from 0 to w within the PFZ and from w to a in the rest of the grain; (e) localization of strain within the PFZ after nucleation of microvoids.
P. Singh et al.
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Strain concentration at grain boundaries in AI-Mg-Si
The slip band is assumed to extend a distance a from the grain boundary as shown in Fig. 7(d). The ratio w/a is given by [18] W=cos{at ~--ri /
a
(2)
~2 Vy-ril
A measure of the concentration of slip in the PFZ is given by the relative displacement of the material on each side of the slip line at the PFZ boundary u(w) in comparison with the displacement at its centre u(0). When the concentration index r=u(w)/u(O)=O the strain is sharply concentrated within the PFZ. When r = 1 there is no concentration. The ratio is easily calculated from the BCS model (see for example ref. 19) as U(W)
r = u ( 0 ) = 2 In
(a)(l+[l+(w/a~]l/21-1+[l~] In
(3)
The relation between r and a/w from (3) is shown in Fig. 8. To estimate the strain concentration we need to estimate a/w. Since, by definition, r ~- ty, it is clear using (2) that a/w must be large. On the assumption that a can be identified with a grain diameter and taking w ~ 0.1 pm, the magnitude of a/w must be of the order of 200. From Fig. 8 we see that r > 0.8, which is much too large to produce the observed strain concentration. Thus the presence of a PFZ cannot of itself produce sharply localized grain boundary deformation. This conclusion is in line with that of Vasudevan and Doherty [20] who deduced from experimental observations that the presence of large grain boundary precipitates, rather than that of a PFZ, is responsible for ductile grain boundary fracture. If we exclude plastic flow by dislocation glide within the PFZ as a cause of strain localization there are two remaining possibilities: (i) that microvoids are nucleated in the grain boundary at a very early stage in plastic deformation and cause strain localization, before the voids are detected on the strained surface; (ii) that GBS causes void nucleation and occurs by slip
1.0
~
0.5
1
10
10 2
10 3
10 ~
alW
Fig. 8. Curve showing the relation between strain concentration within the PFZ as a function of the length of the slip band.
7
on the boundary plane itself. These possibilities are now discussed. (i) If microvoids nucleate in sufficient densities, the presence of free surfaces created by the voids allows th e sharp localization of strain to occur as illustrated in Fig. 7(e), with slip lines linking the voids. GBS by this mechanism is contrary to appearances since microvoids appear to be nucleated as a consequence of sliding rather than the reverse. Attempts to locate subsurface early void nucleation in the vicinity of slipped boundaries by examination of lightly strained and polished specimens gave negative results. Despite this, it is possible that the voids are nucleated below the surface layers and remain undetected because they are so small. Such microvoids would have to nucleate at low strains contrary to expectations, considering that holes do not nucleate at larger particles in the grain interior until larger strains (7%) have been accumulated (Fig. 4(e)). In contrast the effect of low strain rates in intensifying GBS might be explained on the basis of different activation energies for flow in the PFZ and in the precipitation hardened interiors. (ii) Very little is known about slip on the grain boundary plane except that it occurs with relative ease at high temperatures. It is of course the accommodation process which is rate controlling so that sliding alone is difficult to study directly as was pointed out by Raj and Ashby [16]. Sliding by movement of extrinsic grain boundary dislocations has been observed in some TEM studies [21, 22] and, if such movement were thermally activated, might account for the effect of slow strain rate in intensifying sliding. On balance it seems most likely that slip on the grain boundary plane is the event responsible for the nucleation of grain boundary microvoids because there is no obvious mechanism for nucleating holes at very small precipitates without extensive strain localization. Once nucleated, the presence of microvoids can further intensify the localization of strain at the boundary. In addition, the nucleation of voids will produce dilatation of the material in the PFZ which should assist local accommodation of strain where the grain boundary deviates from planarity by a small amount. The observations made in the present work are consistent with the observation of dimpled intergranular fracture in studies of crack growth in precracked specimens of AI-Mg-Si alloys [7, 8]. As observed here, slow strain rates intensify localization of strain at grain boundaries corresponding to the conditions likely to be met at the tip of a slowly growing crack [23]. When AI-Mg-Si-Mn pre-cracked specimens are strained rapidly, fracture is transgranular. The observations in the present work confirm that this is because there is much less localization of strain at grain boundaries at fast strain rates. At low strain rates,
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Strain concentration at grain boundaries in AI-Mg-Si
strain is localized at the grain boundary leading to ductile intergranular microfracture.
4 M. Abe, K. Asano and A. Fujiwara, Metall. Tram., 4 (1973) 1499. 5 J. M. Dowling and J. W. Martin, The influence of manganese addition on the deformation and fracture behaviour of an AI-Mg-Si alloy, Proc. 3rd Int. Conf. on the Strength of Metals and Alloys, Cambridge, Cambs., August 20-25, 1973, Vol. 1,
5. Conclusions (1) GBS has been observed at room temperature in an A I - M g - S i alloy containing manganese, the intensity of which is primarily a function of the applied strain. (2) T h e intensity of the strain localization produced by GBS is increased when specimens are deformed at a slow rate. (3) It is believed that the localized grain boundary strain is produced initially by sliding on the grain boundary plane rather than by plastic deformation within the PFZ. However, a small amount of sliding leads to microvoid formation at grain boundary precipitates which further intensifies strain localization, eventually producing grain boundary cracks. (4) T h e present observations are consistent with previous observations and support the idea that embrittlement of pre-cracked specimens under sustained loading is due in part to the localization of strain at grain boundaries.
References
6 7 8
9 10 11 12 13 14 15 16 17 18 19 20 21
1 J. D. Evensen, N. Ryum and D. J. Embury, Mater. Sci. Eng., 18 (1975) 221. 2 E. Ness and D. J. Embury, Z. Metallkd., 64 (1973) 805. 3 N. Ryum, Acta Metall., 16 (1968) 327.
22 23
Iron and Steel Institute, Metals Society, London, 1974, pp. 170-174. M. Guttmann, B. Quantain and Ph. Dumolin, Met. Sci. J., 16 (1983) 123. J. Lewandowski, V. Kohler and N. J. H. Holroyd, Mater. Sci. Eng., 96 (1987) 185. Y. S. Kim, N. J. H. Holroyd and J. J. Lewandowski, Pbinduced solid-metal embrittlement of A1-Mg-Si alloys at ambient temperature, Proc. Conf. on Environment-Induced Cracking of Metals, National Association of Corrosion Engineers, Houston, TX, 1989, pp. 371-377. K. Matsuda, Y. Uetani, S. Tada and S. Ikeno, J. Jpn. Inst. Light Met., 40 (1990) 580 (in Japanese). U. Dahmen and E. Hornbogen, Z. Metallkd., 66 (1975) 255. P.N.T. Unwin and C. G. Smith, J. Inst. Met., 97(1969) 229. T. Kawabata and O. Izumi, Acta Metall., 24 (1976) 817. M. Gr~ifand E. Ho rnbogen, A cta Meta ll., 25 ( 1977) 883. H.J. Frost and M. F. Ashby, Deformation Mechanism Maps, Pergamon, Oxford, 1982. R. S. W. Shewfelt and L. M. Brown, Philos. Mag., 35 (1977) 945. R. Raj and M. F. Ashby, Metall. Trans., 2 (1971) 1113. G.I. Taylor, J. Inst. Met., 67(1938) 307. B.A. Bilby, A. H. Cottrell and K. H. Swinden, Proc. R. Soc. London, Ser. A, 272 (1963) 304. J.T. Evans, Eng. Fract. Mech., 22 (1985) 347. A.K. Vasudevan and R. D. Doherty, Acta Metall., 35 (1987) 1193. G. B~iro, H. Gleiter and E. Hornbogen, Mater. Sci. Eng., 3 (1968)92. R.G. Gates, Acta Metall., 21 (1973) 855. N. Behnood, H. Cai, J. T. Evans and N. J. H. Holroyd, Mater Sci. Eng. A, 119 (1989) 23.
1
Strain concentration at grain boundaries in A1-Mg-Si alloys P. Singh* a n d J. T. E v a n s Department of Mechanical, Materials and Manufacturing Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7R U (UK) N. J. H. H o l r o y d Alcan International, Banbury, Oxon. OX16 7SP (UK) (Received July 8, 1991; in revised form December 17, 1991 )
Abstract Grain boundary sliding has been observed in an A1-Mg-Si alloy deformed at room temperature. The extent of sliding was primarily a function of the macroscopic strain whether this was produced by creep or in slow-strain-rate tensile tests but was more intense at low strain rates than at high strain rates. Grain boundary microcracks were observed to form after a critical amount of sliding at low strain rates. At high strain rates, microvoids nucleated at larger second-phase particles rather than in the grain boundaries. A consideration of the mechanisms which could operate to produce localized deformation indicates that the phenomenon is due initially to slip on the grain boundary plane itself but is quickly intensified by plastic flow within the precipitate-free zone after microvoids have nucleated. Plastic flow without the prior nucleation of holes cannot be sufficiently concentrated to account for the observed strain localization.
1. Introduction High purity A1-Mg-Si alloys can exhibit low ductility in the peak precipitation-hardened state because of the occurrence of intergranular fracture [1-4]. Detailed observation of the fracture surfaces reveals the presence of fine-scale dimples (spacing, 1-4/~m) which suggests that concentrated deformation localized near the grain boundary is responsible for the fracture. The addition Of small amounts of manganese or chromium greatly improves the properties of A1-Mg-Si alloys by suppressing intergranular fracture, giving instead transgranular fracture with coarse dimples. Dowling and Martin [5] noted that small non-coherent or semicoherent precipitates from manganese or chromium are effective in dispersing coarse transgranular slip in aluminium alloys and they proposed that the beneficial effect of manganese or chromium arises from the homogenization of slip and suppression of the heavy stress concentrations which occur when coarse slip bands impinge on grain boundaries.
*Present address: Department of Materials Science and Engineering, Case Western Reserve University, Cleveland, OH 44106, USA. 0921-5093/92/$5.00
Manganese additions are effective in suppressing intergranular fracture in A1-Mg-Si strained at normal rates. However, there is evidence that intergranular fracture reappears in commercial alloys containing manganese [6, 7] when plastic straining is very slow or when specimens are subjected to sustained loading. Lewandowski et al. [7] observed dimpled intergranular fracture after slow crack growth in AI-Mg-Si alloys containing manganese, similar to the intergranular fracture surfaces produced in the more rapid fracture of pure AI-Mg-Si. Lewandowski et al. also observed a second mode of intergranular fracture at low crack velocities (less than 10 -3 /~m s -1) at stress intensities just above the threshold for slow crack growth; here, fracture surfaces were smooth and undimpled and associated with the presence of small quantities of lead which can be present as an impurity in commercial alloys. An implication of the slow-crack-growth observations is that the strain concentrations at grain boundaries increase when the rate Of strain is very low or under sustained loading conditions. This was confirmed by Lewandowski et al. [7] and Kim et al. [8] in supplementary tests on unnotched specimens. The object of the present work was to measure the deformation at grain boundaries as a function of strain rate © 1992--Elsevier Sequoia. All rights reserved
2
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Strain concentration at grain boundaries in AI-Mg-Si
and to relate the strain concentrations to the development of ductile intergranular fracture.
2. Experimental method
A feature of the deformation of a number of aluminium alloys is the development of prominent grain boundary offsets. In the present authors' experience, the development of these offsets after small deformations at room temperature is particularly pronounced in A1-Mg-Si alloys [9], although it can also occur in high purity aluminium [10] and in A I - Z n - M g alloys [11-13]. This is easily observed in specimens which have been electropolished and then strained a few per cent. An example is shown in Fig. 1. At first sight, the offsets may be mistaken for microcracks but, although microcracks do eventually form at these features, careful examination reveals little evidence that the boundaries are cracked when the offsets first become visible. In order to study the grain boundary deformation in a semiquantitative way the following experiments were performed to measure the in-plane component of the relative displacement of the neighbouring grains. Tests were made on the alloy with the composition shown in Table 1. The impurity lead level was 20 ppm. Plates were supplied in the peak aged condition (a yield stress of 320 MPa and a tensile strength of 355 MPa at a strain rate of 1.3 x 10 -3 S-l) from experimental casts which had been homogenized, hot rolled, solution treated, quenched and aged. The microstructure con-
sisted of elongated grains with the longer axis parallel to the rolling direction (average grain diameter in this direction, 100 /~m). Creep tests and slow-constantstrain-rate tests were performed on small tensile specimens (12.5 mm gauge length x 5 mm x 1 mm) machined with the tensile axis parallel to the short direction of the plate. One surface of each specimen was polished to a high quality optical finish. The penultimate stage in polishing consisted in lapping on a cloth impregnated with colloidal silica. In the final stage, fine scratches were deliberately introduced by polishing on a cloth impregnated with 0.25/zm diamond paste to enable the grain boundary offsets to be measured [10]. The following procedure was adopted. The polished specimens were deformed at room temperature in creep or'with a constant applied strain rate (7 x 10 -7, 7 x 10- 5 or 7 x 10- 3 s- l) to a predetermined strain. The specimen was then removed from the testing machine and examined in the scanning electron microscope. At this stage, areas showing prominent grain boundary offsets were identified and photographed and their position was recorded. The specimen was then remounted in the testing machine, straining was continued and the above process was repeated a number of times. In this way the development of grain boundary sliding (GBS) and the nucleation of microcracks at specific locations were recorded. A smaller number of specimens were examined in a similar way after creep deformation under constant load.
3. Results
TABLE 1. Alloy composition
Grain boundaries were observed to slide and eventually to crack at a rate determined by the accumulation of macroscopic strain. In order to characterize the properties of the alloy, constant-load creep tests were performed on small tensile specimens at room temperature. Little deformation was observed at stresses smaller than about 85% of the nominal yield stress (measured with an applied strain rate of 1 . 3 x 1 0 -3 s-l). At larger stresses, creep curves showing the classical form were obtained, with decelerating creep in stage I, a constant creep rate in stage II and accelerating creep leading to fracture in stage HI. At lower stresses, only stages I and II were monitored because of the large times required to reach stage HI. The results are summarized in Fig. 2 where the minimum creep rate dminis shown as a function of a/o0 on a log-log plot, where o is the applied stress and o0 is the nominal yield stress. The curve in Fig. 2 suggests a relation of the form
Element Mg Si Mn Fe Cu Zr A1 Amount(wt.%) 0.62 1.00 0.55 0.24 0.08 0.02 Balance
e=m=eo 7o
Fig. 1. Offsets produced by GBS after a macroscopic strain of 2% (peak aged alloy with the composition shown in Table 1). In this case the slip vector is out of the plane of the surface.
(o)n
(1)
P. Singh et aL 10-2
I
r
l
[
/
Strain concentration at grain boundaries in AI-Mg-Si
3
I
O
o
I
-z,
10 "7 .c
10-6
10-8
I 1°, I i 0.8
1.0 o/00
1-2
Fig. 2. Logarithmic plot of the minimum creep rate as a function of applied stress for specimens loaded at room temperature.
where ~0 and n are constants. This is similar to power law creep [14] which is often observed at higher temperatures but in the present case the exponent n is much larger (greater than 100) than normally associated with power law creep. The difference can be interpreted as indicating the presence of a large internal stress [15], somewhat smaller than the short-term yield stress. The ductility of the material in the creep tests at the faster rates r~lnged from 12 down to 10% elongation to fracture compared with 15% for a short-term tensile test. Examination of the final fracture surfaces showed mixed transgranular and intergranular fracture with an increasing proportion of intergranular fracture produced at low strain rates. Thus the numbers of grain boundaries showing sliding and cracking increased as the duration of the test increased. However, the extent to which intergranular cracking produces embrittlement in the material was not studied in detail in this work since studies of slow crack growth in notched specimens [7] indicate that triaxial tension of the sort which appears at the tip of a sharp crack may be required to develop embrittlement to its full extent. Observations of the surface of polished specimens deformed in creep revealed the nucleation and growth of grain boundary microcracks. These were always formed at grain boundaries which were highlighted by the formation of sliding offsets prior to the nucleation of the cracks. The grain boundaries showing prominent offsets were often those where the trace of the boundary plane on the surface was oriented at about 45 ° to the tensile axis, although offsets at boundary traces with a small angle were also observed (e.g. Fig. 1). In these cases the slip vector had a large out-of-plane component at the surface. An example showing a sequence of offset formation and microcrack nucleation is shown in Figs. 3(a) and 3(b). The microcracks widen as they lengthen and then the cracked bound-
Fig. 3. Sequence showing crack nucleation at grain boundary offsets in specimens loaded at a constant stress (300 MPa): (a) after 1 day; (b), (c) after 5 days; (c) the dimpled fracture surface of the grain boundary microcrack in (b).
aries were seen to be dimpled on a fine scale (Fig. 3(c)). It appears that localized shear produces microvoids in the boundary which eventually link to give the microcracks. We were unable to identify the particles which forni the nuclei for the dimples because of their very small diameters (less than 0.25/~m), but circumstantial evidence suggests that they are probably Mg2Si precipitates. A fine dispersion of MnA16is also expected
4
P. Singh et al.
/
Strain concentration at grain boundaries in AI-Mg-Si
to be present at grain boundaries but, since fine dimpled intergranular fracture occurs profusely in manganese-free A1-Mg-Si alloys [5], it seems likely that Mg2Si precipitates at the grain boundaries are responsible in both cases. Slow-strain-rate tests were performed on polished a n d scratched specimens to measure the in-plane grain boundary sliding. Specimens were examined at intervals in the course of a test, first to locate those boundaries where grain offsets occurred and then to follow the increase in the localized deformation as a function of strain. An example is shown in Figs. 4(a)-4(d). Grain boundary sliding was first detected from the misregistry of the scratch marks. After a small critical strain the formation of microcracks was detected as is also indicated by the opening up of gaps between the ends of the scratch marks across the boundary. The effect of the applied strain rate on the offsets parallel to the boundary plane (maximum recorded values) as a function of the applied strain is shown in Fig. 5. The offset increases approximately as the square root of the strain, the largest offsets being produced at the lowest strain rate. Microcrack nucleation was detected after an offset of about 0.6/~m at about 2% strain when the applied strain rate was 7 x 10 -5 s-1 or less, but no microcrack nucleation was detected at the faster strain rate of 7 x 10- 3 s- 1. Instead, transgranular cracking occurred at higher strains (7%) nucleated by microcracking of coarse ironcontaining second-phase particles in the grain interiors (Fig. 4(e)). These particles are present in relatively small numbers as a consequence of the solidification process rather than as a result of subsequent heat treatment. Evidently, the GBS is not sufficiently developed by fast straining (Fig. 5) to produce visible microcracking in the boundaries.
4. Discussion
Time-dependent intergranular cracking occurs during creep, and our results show that its nucleation is due to the accumulation of strain since similar amounts of cracking were obtained in creep and slow-strain-rate tests. After nucleation the microcracks grew somewhat discontinuously at rates between 0.1 and 0.01/~m s-1. The observations indicate a link between the development of localized strain at the grain boundaries and the subsequent formation of microcracks in the grain boundaries. However, the effect is most pronounced at low strain rates. At high strain rates the concentration of deformation at the grain boundaries is smaller and transgranular microcracks at second-phase particles are formed in preference to the grain boundary cracks. In contrast, grain boundary ductile fracture is observed to occur at high strain rates in high purity AI-Mg-Si
alloys [5]. The grain boundary deformation in the present alloy is sufficiently localized at low strain rates to give the appearance of grain boundary sliding, so that the grains slide past one another by slip either on the boundary plane or in a very narrow zone adjacent to the grain boundary, possibly corresponding to the precipitate-free zone (PFZ). The possibility of the latter is discussed below. The phenomenon is similar to the GBS observed in high temperature creep, but the mechanism must be different because of the virtual absence of diffusional flow at room temperature. In their model of GBS, Raj and Ashby [16] treated the grain boundary itself as if it were an atomically thin region which can support a tensile stress acting normal to the boundary but which has no resistance to shear tangential to the boundary. Given that a grain boundary is not generally a planar interface and possesses local curvature and undulations, the rate of inelastic sliding must be controlled by the accommodation of strain in the neighbouring grains. Raj and Ashby treated some idealized geometries with accommodation by diffusional flow. However, in the present experiments the observed rate of sliding is, at best, about five orders of magnitude larger than could be explained by diffusional flow accommodation at room temperature. It follows that any accommodation of the strains produced by GBS must be accomplished by dislocation glide and plastic flow in the grains adjacent to the boundary, a possibility recognized but not treated by Raj and Ashby [16]. The fact that the extent of GBS is mainly determined by the applied strain (Fig. 5) supports this conclusion, assuming that local accommodation can be accomplished within the latitude provided by general yielding. However, the extent of GBS increases at slow applied strain rates, which suggests thermal activation of the process of sliding on the boundary plane. Unfortunately little is known about this aspect of localization of strain at grain boundaries. Dahmen and Hornbogen [10] investigated GBS in pure aluminium and found two values of activation energy: one characteristic of sliding and migration at temperatures above 200 °C and another lower value at temperatures below 200 °C. The lower value was believed to be stress dependent. Dahmen and Hornbogen also found evidence from the transmission electron microscopy (TEM) examination of a high dislocation density either at the grain boundary or in a localized region within 0.2/~m of the boundary in deformed specimens. In the present AI-Mg-Si alloy there is also the possibility of an additional effect associated with PFZs at the grain boundary of the sort shown in Fig. 6. The relative softness of the PFZ is often envisaged to be responsible for the localization of strain at the grain boundaries, but in the present authors' opinion material continuity pre-
P. Singh et al.
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Strain concentration at grain boundaries in A l-Mg-Si
5
Fig. 4. (a)-(d) GBS observed by the misregistry of scratch marks across a boundary after deformation at various strains at 7 x 10-5 s- 1, showing the nucleation of microvoids: (a) 0.6%; (b) 1.2%; (c) 2.4%; (d) 3.2%. (e) At fast strain rates (7 x 10 -3 S-l), microvoids are nucleated at 7% strain at large second-phase particles rather than in the grain boundary. cludes this possibility unless some special conditions apply. This is now discussed with the aid of a simple model. Consider a P F Z (Fig. 7(a)) with a local shear strength r i and width w. T h e material adjacent to the P F Z has a
larger strength ry. In approximate terms, one can identify ry with o0/31/2, where tr0 is the macroscopic yield stress (ry ~- 185 MPa) whereas, since ri refers to the Mg2Si-depleted PFZ, it can be identified with the shear yield strength of a grossly overaged specimen of
6
P. Singh et al.
/
Strain concentration at grain boundaries in AI-Mg-Si
the alloy, Le. r~ ~- 60 MPa. In fact, the precise values of ry and ri have little effect on the outcome as shown in the following argument. The softness of the zone can show up to the full extent if the slip plane happens to be parallel to the grain boundary plane (Fig. 7(b)). In the general case, however, the slip planes are inclined steeply to the boundary plane and a rough estimate shows that only a very small fraction of boundaries will be favourably oriented for single-plane slip parallel to the boundary. If the width of the PFZ is w and the length of a slip line contained within the PFZ is/, then, on the assumption of a random distribution of boundary orientation relative to a {111} slip plane, the fraction of grains with PFZ slip lines greater or equal to l is given by 43-(w/l)2. If the ratio l/w > 1000 (small angle of inclination to the boundary plane), which it has to be to account for the localized deformation, the fraction of grains of a suitable orientation is less than 10 -6. The fraction of grain boundaries showing sliding offsets is
2
I
,
I
i
~
I
E
much larger than this. In order to have a general shear produced by slip on planes not parallel to the boundary, the von Mises condition must be met [17], with cooperative slip on five independent crystal systems ({111}(110)) (Fig. 7(c)). In this case the strain can only be confined within the PFZ if dislocation pile-ups are formed at the interface between the PFZ and the grain interior. Clearly this is not tenable since there is no physical barrier at this interface. The only difference in the two regions is the difference in the local yield strengths ri and ry. This point can be demonstrated using the Bilby, Cottrell and Swinden (BCS) model [18] of a slip band. As illustrated in Fig. 7(d), we consider just one of the slip lines with a dislocation source located at the boundary. Slip in the PFZ operates with an effective stress r - ri (between 0 and w), whereas slip in the grain interior is driven by the effective stress r - r y , where r is the local component of the applied shear stress. For the moment we assume that ri < r < ry.
I
oe
s~
.1_, t-
7x10-s
.
~0 t-i ttl
o
~5
:%.
® ~ ' - - ,, 2
0
, "--'-'~', 4 Stroin */,
7×10-31 6
8
Fig. 5. Curves showing the effect of strain rate on the GBS as a function of applied strain. Microvoid nucleation was first detected at strains indicated by the arrows. No grain boundary microvoid nucleation was detected at fast strain rates (7 X 10-3 s-l).
Fig. 6. Precipitate-free zone at a grain boundary in the AI-Mg-Si alloy (TEM observation).
"v
Q
"gy
-, '- : - : ' i - . - - , "gi
(a)
(b)
-I
(cl
(d)
(e)
Fig. 7. Schematic illustration of mechanisms of GBS: (a) PFZ; (b) slip in the PFZ when a slip plane is parallel to the boundary plane; (c) deformation in the PFZ by slip on planes not parallel to the boundary; (d) a slip line extending from within the PFZ into the neighbouring grains; the slip line extends from 0 to w within the PFZ and from w to a in the rest of the grain; (e) localization of strain within the PFZ after nucleation of microvoids.
P. Singh et al.
/
Strain concentration at grain boundaries in AI-Mg-Si
The slip band is assumed to extend a distance a from the grain boundary as shown in Fig. 7(d). The ratio w/a is given by [18] W=cos{at ~--ri /
a
(2)
~2 Vy-ril
A measure of the concentration of slip in the PFZ is given by the relative displacement of the material on each side of the slip line at the PFZ boundary u(w) in comparison with the displacement at its centre u(0). When the concentration index r=u(w)/u(O)=O the strain is sharply concentrated within the PFZ. When r = 1 there is no concentration. The ratio is easily calculated from the BCS model (see for example ref. 19) as U(W)
r = u ( 0 ) = 2 In
(a)(l+[l+(w/a~]l/21-1+[l~] In
(3)
The relation between r and a/w from (3) is shown in Fig. 8. To estimate the strain concentration we need to estimate a/w. Since, by definition, r ~- ty, it is clear using (2) that a/w must be large. On the assumption that a can be identified with a grain diameter and taking w ~ 0.1 pm, the magnitude of a/w must be of the order of 200. From Fig. 8 we see that r > 0.8, which is much too large to produce the observed strain concentration. Thus the presence of a PFZ cannot of itself produce sharply localized grain boundary deformation. This conclusion is in line with that of Vasudevan and Doherty [20] who deduced from experimental observations that the presence of large grain boundary precipitates, rather than that of a PFZ, is responsible for ductile grain boundary fracture. If we exclude plastic flow by dislocation glide within the PFZ as a cause of strain localization there are two remaining possibilities: (i) that microvoids are nucleated in the grain boundary at a very early stage in plastic deformation and cause strain localization, before the voids are detected on the strained surface; (ii) that GBS causes void nucleation and occurs by slip
1.0
~
0.5
1
10
10 2
10 3
10 ~
alW
Fig. 8. Curve showing the relation between strain concentration within the PFZ as a function of the length of the slip band.
7
on the boundary plane itself. These possibilities are now discussed. (i) If microvoids nucleate in sufficient densities, the presence of free surfaces created by the voids allows th e sharp localization of strain to occur as illustrated in Fig. 7(e), with slip lines linking the voids. GBS by this mechanism is contrary to appearances since microvoids appear to be nucleated as a consequence of sliding rather than the reverse. Attempts to locate subsurface early void nucleation in the vicinity of slipped boundaries by examination of lightly strained and polished specimens gave negative results. Despite this, it is possible that the voids are nucleated below the surface layers and remain undetected because they are so small. Such microvoids would have to nucleate at low strains contrary to expectations, considering that holes do not nucleate at larger particles in the grain interior until larger strains (7%) have been accumulated (Fig. 4(e)). In contrast the effect of low strain rates in intensifying GBS might be explained on the basis of different activation energies for flow in the PFZ and in the precipitation hardened interiors. (ii) Very little is known about slip on the grain boundary plane except that it occurs with relative ease at high temperatures. It is of course the accommodation process which is rate controlling so that sliding alone is difficult to study directly as was pointed out by Raj and Ashby [16]. Sliding by movement of extrinsic grain boundary dislocations has been observed in some TEM studies [21, 22] and, if such movement were thermally activated, might account for the effect of slow strain rate in intensifying sliding. On balance it seems most likely that slip on the grain boundary plane is the event responsible for the nucleation of grain boundary microvoids because there is no obvious mechanism for nucleating holes at very small precipitates without extensive strain localization. Once nucleated, the presence of microvoids can further intensify the localization of strain at the boundary. In addition, the nucleation of voids will produce dilatation of the material in the PFZ which should assist local accommodation of strain where the grain boundary deviates from planarity by a small amount. The observations made in the present work are consistent with the observation of dimpled intergranular fracture in studies of crack growth in precracked specimens of AI-Mg-Si alloys [7, 8]. As observed here, slow strain rates intensify localization of strain at grain boundaries corresponding to the conditions likely to be met at the tip of a slowly growing crack [23]. When AI-Mg-Si-Mn pre-cracked specimens are strained rapidly, fracture is transgranular. The observations in the present work confirm that this is because there is much less localization of strain at grain boundaries at fast strain rates. At low strain rates,
8
P. Singh et al.
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Strain concentration at grain boundaries in AI-Mg-Si
strain is localized at the grain boundary leading to ductile intergranular microfracture.
4 M. Abe, K. Asano and A. Fujiwara, Metall. Tram., 4 (1973) 1499. 5 J. M. Dowling and J. W. Martin, The influence of manganese addition on the deformation and fracture behaviour of an AI-Mg-Si alloy, Proc. 3rd Int. Conf. on the Strength of Metals and Alloys, Cambridge, Cambs., August 20-25, 1973, Vol. 1,
5. Conclusions (1) GBS has been observed at room temperature in an A I - M g - S i alloy containing manganese, the intensity of which is primarily a function of the applied strain. (2) T h e intensity of the strain localization produced by GBS is increased when specimens are deformed at a slow rate. (3) It is believed that the localized grain boundary strain is produced initially by sliding on the grain boundary plane rather than by plastic deformation within the PFZ. However, a small amount of sliding leads to microvoid formation at grain boundary precipitates which further intensifies strain localization, eventually producing grain boundary cracks. (4) T h e present observations are consistent with previous observations and support the idea that embrittlement of pre-cracked specimens under sustained loading is due in part to the localization of strain at grain boundaries.
References
6 7 8
9 10 11 12 13 14 15 16 17 18 19 20 21
1 J. D. Evensen, N. Ryum and D. J. Embury, Mater. Sci. Eng., 18 (1975) 221. 2 E. Ness and D. J. Embury, Z. Metallkd., 64 (1973) 805. 3 N. Ryum, Acta Metall., 16 (1968) 327.
22 23
Iron and Steel Institute, Metals Society, London, 1974, pp. 170-174. M. Guttmann, B. Quantain and Ph. Dumolin, Met. Sci. J., 16 (1983) 123. J. Lewandowski, V. Kohler and N. J. H. Holroyd, Mater. Sci. Eng., 96 (1987) 185. Y. S. Kim, N. J. H. Holroyd and J. J. Lewandowski, Pbinduced solid-metal embrittlement of A1-Mg-Si alloys at ambient temperature, Proc. Conf. on Environment-Induced Cracking of Metals, National Association of Corrosion Engineers, Houston, TX, 1989, pp. 371-377. K. Matsuda, Y. Uetani, S. Tada and S. Ikeno, J. Jpn. Inst. Light Met., 40 (1990) 580 (in Japanese). U. Dahmen and E. Hornbogen, Z. Metallkd., 66 (1975) 255. P.N.T. Unwin and C. G. Smith, J. Inst. Met., 97(1969) 229. T. Kawabata and O. Izumi, Acta Metall., 24 (1976) 817. M. Gr~ifand E. Ho rnbogen, A cta Meta ll., 25 ( 1977) 883. H.J. Frost and M. F. Ashby, Deformation Mechanism Maps, Pergamon, Oxford, 1982. R. S. W. Shewfelt and L. M. Brown, Philos. Mag., 35 (1977) 945. R. Raj and M. F. Ashby, Metall. Trans., 2 (1971) 1113. G.I. Taylor, J. Inst. Met., 67(1938) 307. B.A. Bilby, A. H. Cottrell and K. H. Swinden, Proc. R. Soc. London, Ser. A, 272 (1963) 304. J.T. Evans, Eng. Fract. Mech., 22 (1985) 347. A.K. Vasudevan and R. D. Doherty, Acta Metall., 35 (1987) 1193. G. B~iro, H. Gleiter and E. Hornbogen, Mater. Sci. Eng., 3 (1968)92. R.G. Gates, Acta Metall., 21 (1973) 855. N. Behnood, H. Cai, J. T. Evans and N. J. H. Holroyd, Mater Sci. Eng. A, 119 (1989) 23.