Macroscopic aspects of Lüders band deformation in mild steel

Acta metoll, mater. Vol. 39. No. 12, pp. 3153-3160, 1991

(D56-7151/91 $3.00+ 0.00 Copyright ~ 1991 Pergamon Press plc

Printed in Great Britain. All rights reserved

MACROSCOPIC ASPECTS OF LI~DERS BAND DEFORMATION IN MILD STEEL V. S. A N A N T H A N t and E. O. HALL Department of Mechanical Engineering, The University of Newcastle, N.S.W. 2308, Australia (Received 26 January 1990; in revised form 18 March 1991)

Almtract--Details are given of experiments on the propagation of single Lfiders bands in mild steel. The results cover the effect of varying grain size on the appearance of the bands in specimens of differing cross-section, and concentrate on the macroscopic shape changes observed following the passage of the band. The results are generally consistent with a shear-flow concept [E. O. Hall, N. J. Carter and G. Vitullo, ICSMA 6, p. 393. Pergamon Press, Oxford (1983)], whereby the Lfiders strain arises from a shear which creates both the macroscopic shape changes and sufficient free dislocations for the flow to continue at the lower yield stress. The shear component is strongly dependent on grain size, and at large grain sizes the Liiders band front becomes diffuse. The flow component is more nearly constant. However, the direction of shear is not in the line of greatest slope in the Lfiders front, and this points to some other component, such as grip constraints or the bending moment of the specimen, as factors which may influence the shear direction. The same factors may be responsible for influencing the orientation of the Liiders band front. R~anr---g)n drcrit en drtail des expdriences sur la propagation de bandes de Lfiders isolres darts un acier doux. Les rrsultats sont relatifs d reffet de la variation de la taille du grain sur l'apparition des bandes dans des 6chantillons de differentes sections; de plus, on 6tudie particulirrement los variations macroscopiques de forme observres pendant le passage de la bande. Les rrsultats sont grnrralement on accord avec le concept de fluage-cisaillement [E. O. Hall, N. J. Carter and G. Vitullo, ICSMA 6, p. 393. Pergamon Press, Oxford (1983)], scion lequel la drformation de Lfiders se produit ~ partir d'un eisaillement qui crre la fois les variations macroscopiques de forme et suffisamment de dislocations fibres pour que le fluage continue fi la limite 61astique infrrienre. La composante de cisaillement drpend fortement de la taille du grain, et pour de grandes tailles de grain le front de la bande de Lfiders devient diffus. La c o m t e de fluage est presque constante. Cependant, la direction du eisaillement n'est pas sur la ligne de plus grands pente du front de L/iders, ce qui indique l'existence d'une autre composante, telle que les contraintes de serrage ou le moment de flexion de l'rchantillon, qui seraient un faetenr pouvant influencer la direction du eisaillement. Les m~mes facteurs peuvent 6tre responsables de I'orientation du front de la bande de Lfiders. Zaummmfamaag~Experimente zur Ausbreitung einzelner Liidersb/inder in Fluflstalfl werden ausf'fihrlich dargestellt, lnsbesondere betreffen die Ergebniss¢ den Einflul3 der Korngr6~ auf das Auftreten der B~nder in proben mit unterschiedlichem Querschnitt und die makroskopischen Form~nderungen naeh der Passage eines Bandes. Die Ergebnisse sind im allgemeinen vertr~glich mit einem Scherl~eS-Modell [E. O. Hall, N. J. Carter and G. Vitullo, ICSMA 6, p. 393. Pergamon press, Oxford (1983)], bei dem die Lfidersdehnung yon einer Scherung herrfihrt, die sowold makroskopische Formfmdta'ungtm als auch genfigend freie Versetzungen erzeugt, damit das Flieflcn beim unteren FlieBpunkt andauert. Die Scherkomponente h/ingt stark yon der Korngr6Be ab; bei gro~n K6mern wird die Front des Lfdersbandes diffus. Die FlieBkomponente ist n~herungsweise konstant. Allerdings liegt die Scherrichtung nicht in der Richtung der gr613ten Steigung in der Lfidersfront; dieser Umstand weist auf andere Komponenten, wie etwa Einfl/isse der Fassung oder Biegemomente in der Probe, als Faktoren lain, die die Scherrichtung beeinflussen krnnen. Dieselben Faktoren k6nnen verantwortlich flit die Be¢influssung der Orientierung der Front des Lfidersbandes sein.

INTRODUCTION Mild steel deforms non-uniformly at the onset of yielding, and this type of non-homogeneous deformation is characterized by the formation of a Lfiders band (after Liiders [1], 1860). Liiders bands are usually initiated from the regions of the specimen held in the grips, or from the shoulders of the test piece, and propagate from one end of the specimen

$Prescnt address: Materials Department, Riso National Laboratory, Postboks 49, DK-4000 Roskilde, Denmark.

to the other at the Liiders strain (£L)" The band front itself may be viewed as a small localized region o f plastic deformation, spreading the deformation along the elastic region of the specimen at the constant lower yield stress (aL). The yield point phenomena in mild steel was studied extensively [2] and a number of theories have emerged as a consequence [3-5]. The yield stress was found to be dependent on grain size (d-t"~) [6, 7], and this relation has been validated by later studies [8]. Considerable interest was also shown over the years concerning the nature and formation of the Liiders band in different materials [9-12], but

3153

3154

ANANTHAN and HALL: LODERS BAND DEFORMATION IN MILD STEEL

EZ

~B

EC EB

B face

.,\\ \ \

\\ "x \x ~ .

/ I / o}

/ ~ ~

/~i "Sheor&flow I L~Shear only I Unstrained

/I/

flow 1 [ l~Sheor only \ I Unstrained

b)

Fig. 1. An exaggerated representation of transformation of a circular cross-section into an ellipse with the passage of Liiders band in the case of (a) hexagonal specimens, (b) square specimens. a study of the literature reveals that two important aspects which could influence the morphology of these bands, that is, the specimen shape and the grain size, have not been given enough attention. In early work [6, 9] these two factors were thought to affect the nature of the bands, and hence prominence was given to these factors in this study. The published investigations have so far concentrated on one or more of the following aspects: (i) nature and propagation of the Lfiders band [13-17], (ii) velocity of the Liiders front [18-21], (iii) Lfiders strain [5, 11, 12, 21-23], (iv) orientation of the Lfiders front [9, 24-26], (v) kink angle and shear at the Lfiders front [6, 15, 27], and (vi) kinetics of the Liiders front propagation [28-30]. While all these aspects combine to provide a macroscopic picture of the bands, the basic nature of their formation and propagation is still debatable. The shear characteristics of the band were demonstrated by Sylwestrowicz and Hall [18], when they observed specimens of circular cross-section become elliptical with the passage of the Liiders band. The concept of the presence of macroscopic shear at the band front was supported by other workers [9, 17,24,31], while some of them [9,31] clearly saw that shear alone will not be able to explain the reduction in cross-sectional area. Recently Hall et al. [22] proposed a theory based on a "shear-flow" concept, where shear was considered to nucleate free dislocations in the material for the flow to continue under a constant applied stress (OL). The present work enabled the authors to verify such a theory based on experimental results from specimens of varying shapes and grain sizes. STRAIN ANALYSIS The model used for the calculation of the Lfiders strain is based on the "shear-flow" theory of Hall

et al. [22]. This theory basically underlines the fact that a circular cross-section transforms into an elliptical one with the Lfiders deformation. Figure l(a,h) show an exaggerated depiction of the deformation of this nature in the case of specimens with hexagonal and square cross-sections, where the strains ~^, %, ~c represent the transverse strains that could be measured on different faces of the specimen. The strains q], E, refer to the principal strains along the major and minor axis of the ellipse, and E~2 is associated with the shear component. Considering a simple diagram with only one face (say side A) (Fig. 2) the strain E^ can be resolved into components of the principal strains and shear component as [32]

~A= ~H COS2OZ+ ~Z2sin2 ef + 2~12sin,~ cos -

(1)

and the strain perpendicular to e, as ~ = q2(cos' • - sin' ~) + (E, - ql) sin oz cos oz. (2)

1~2L

"

E12

L~22

Fig. 2. Resolving components of strain in a simple cue with one face.

ANANTHAN and HALL: LODERS BAND DEFORMATION IN MILD STEEL

5.Smm

21mm

59rnm

Z..3mrn

3155

15.Srnrn

Fig. 3. The different cross-sections used in this study.

Similar equations follow for sides B, C, etc, from which the principal strains and the shear component can be calculated as

e,2/=/cos

sin2~ LCOS2V sin 2"~ /

e22Ji

sin2 sin 2

es . ec

(3)

In the above equation ell, en and eu are not independent of each other. This implies that at least three measurements of transverse strains are needed to evaluate the principal and shear components of the strain. Specimens with six or more sides readily enable one to measure three or more transverse strains. In square specimens the measurement of the third strain becomes difficult because of its geometry, and the problem has to be treated slightly differently. Consider the deformation of a square specimen as shown in Fig. l(b). Equation (1) applies to CAand ee. It can be shown that the shear component in terms of angular change is ~.,,s = - ~

= w/~

= ,~/2

(4)

where ~ is the twist angle of the specimen and w, d are referred to in Fig. l(b). As can be seen from the diagram, a transformation of the cross-section to an ellipse should also induce a twist in the specimen, in addition to the kink that has been observed earlier [6, 15]. Hence, by measuring the twist angle a third independent equation can be written to calculate the principal strains. Equation (2) can now be used with (eAs,~) or @Bs,P) and either will give the same equation. With the calculation of the principal strains, the Ltiders strain as components of shear and flow can be worked as shown by Hall et al. [22]. The shear component of the strain was calculated assuming that the m a x i m u m shear direction lies along the line of greatest slope in the shear surface, i.e. the Lfiders band front, from the modified Schrnid and Boas equations for single crystals [33].The Lfiders strain is the summation of flow and shear component of the extension; i.e.the Lfiders strain e L == eLS "3L eLF

(5)

where ELS refers to the shear component and eLF refers to the flow component of the Lfiders strain. To summarize, this model involves the calculation of the flow and shear components of the Lfiders strain from the measurements of the macroscopic parameters of the band front. AM 39/12--N

EXPERIMENTAL Polygonal specimens were found to form a stable L~3ders band [24] that led to the choice of differing cross-sectional specimens, so that a perspective of the band formation could be obtained when the crosssection changed from one extreme (circular) to the other (rectangular); accordingly, octagonal, hexagonal and square cross-sections were examined to provide the transition (Fig. 3). In addition to these specimens a few square samples of side length I0 m m were also tested. The chemical compositions of the specimens are given in Table I. All the specimens, 200 m m in length, were prepared by machining the drawn rods to give the different cross-sections shown in Fig. 3 and then vacuum annealed at different temperatures, in the range 750-1300°C. This annealing treatment gave a range of grain sizes varying from 0.03 to 0.33 mm. The initial texture of the annealed specimens measured for circular specimens (by using the neutron diffraction technique at the Rise National Laboratory, Denmark) was rather weak (maximum orientation density in the O D F was 2.7 × random orientation). The specimens wcrc then polished by mechanical and electrolytic methods to glvc a smooth surface, and tensile tested in a Shimadzu OSS-5000 tensile machine with self-aligning grips at a test speed of 0.1 mm/min, to initiate a single Lfiders band. The Liiders strain (~c) was measured in all the specimens by making indentation markings 10 m m apart on all sides of the specimens and measuring their extension after the passage of the L~ders band. The transverse strains (EA, ~e, ~c) at the band front were measured using a micrometer which had an accuracy of 0.01 ram. The specimens were mounted on a specially made bench fitted with the micrometer, and by rotating and translating the specimen, the crosssection widths on both sides of the Liiders band were measured. Three to five measurements were taken on each side of all the specimens, and the average transverse strains were calculated. Table I. The chemical compositions of the steels used in this work Hexagonal circular % % % % % %

carbon nitrogen phosphorus manganese silicon sulphur

0.09-0.10 0.006 0.029-0.031 0.39--0.41 0.034-0.036 0.019-0.200

Rectangular

Square octagonal

0.10 0.009 0.040 0.20-0.50 -0.050

0.12 0.005 0.027 0.37 0.030 0.020

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ANANTHAN and HALL: LODERS BAND DEFORMATION IN MILD STEEL

The other parameters of the band front, the kink angle and orientation angle were measured as described in Ref. [27] to accuracies of +0.05 ° and _+0.5° respectively. The twist angles of square and rectangular specimens were also measured in a similar manner to the kink angles, but by transversing the profiler of the Taly-contour machine along the transverse direction to the specimen axis. The average twist angles were calculated to an accuracy of 4-0.05 ° from three measurements made across the front from deformed and undeformed regions. The yield stresses of the specimens were calculated directly from the load-extension curves. RESULTS

(i) Nature o f the bands The tensile testing of most of the specimens was carried out at a crosshead speed of 0.1 mm/min. This crosshead speed was chosen to minimize the test duration, and to minimize the generation of multiple Liiders bands. Hexagonal specimens of a limited range of grain sizes were tested at 0.05 mm/min, where the test duration was doubled. The 10ram square specimens produced multiple/complex bands even when tested at a test speed of 0.001 ram/rain. Sharp, single bands were produced in finer and medium grain sized specimens. Two factors, length/thickness (l/t) ratio of the specimens and the test speed (in turn the strain rate), had profound effects on the formation of single Lfiders bands. The effect of (l/t) was earlier noted by Lomer [9], but the present study showed that formation of single bands also depends on the test speed. In Table 2 is listed the l/t ratio and the nature of the bands produced in specimens tested in this work. Single bands were

:. ~ . : ~ :. -, . : : .2.- : • :

•

. !

.

Table 2. Nature of the Lfiders band observed for different I/t (length/thickness)ratio for specimensof differentcross-sections Test speed Cross-section I/t (ram/rain) Natureof the bands Circular 36 0.I Singlebands Octagonal 46 0.1 Singlebands Hexagonal 33 0.1 Singlebands 33 0.05 Singlebands Square 46 O.1 Singlebands 20 0.1 Complex bands 0.05 Complex bands 0.01 Complex bands 0.001 Complex bands RectangularJ 12:74 0.1 Two singlebands from either end of the grips, especiallyin specimens with coarser grain size "The l/t ratio is consideredas ratio of (length/flatside thickness): (length/thinside thickness). generally produced in all the specimens when tested at a crosshead speed of 0.1 mm/min for a l/t ratio > 30. This value appears higher than the lit ratio of 12 obtained by Lomer [9], but the test speed was not mentioned by Lomer. It was observed in this work that by increasing the test speed complex bands were observed even in slender specimens. There may also be some effect of the grain size on the formation of single bands. It was observed that two single bands, one from each of the grips were produced in rectangular, square and hexagonal specimens beyond the grain sizes of 0.065, 0.09 and 0.17 mm respectively. Multiple bands were obtained in rectangular specimens above the grain sizes of 0.11 mm. Circular and octagonal cross-sectional specimens produced single bands up to a range of 0.25 mm; however, the bands were diffuse in all the specimens with grain sizes > 0.15 mm. Figure 4 shows examples of different Lfiders bands observed in this study.

1

-.

.:

_..

1

(a)

(b)

(c)

(d)

Fig. 4. Nature of different types of bands observed. (a) Single, sharp band in a square specimen (grain size 0.034ram). (b) Complex band in a 10mmsquare specimen (grain size 0.04 mm). (c) Diffuse band in a square specimen (grain size 0.227 mm). (d) Single curved band in a rectangular specimen (grain size 0.046 nun).

ANANTHAN

and HALL:

LODERS

BAND

DEFORMATION 500

Table 3. Values of a 0 and ky obtained for specimens of different cross-sections Cross-section Hexagonal Square Circular Octagonal Rectangular

o 0 (MN/m 2) 97.8 + 80.1 + 71.6 ± 130.4 + 150 +

10.8 (SD) 13.2 (SD) 16.5 (SD) 28. I (SD) 9.5 (SD)

i

SD refers to the standard deviation.

(ii) Hall-Petch relation The lower yield stress was plotted against d -1/2, where d is the mean grain diameter, in accordance with the HalI-Petch equation (Fig. 5). In Table 3 the values of the slope, k,, and the friction stress, a0, for all the specimens tested are given. The HalI-Petch relation was found to be valid for all the specimens, but the rectangular specimens showed a low value of ky which could be associated with the change in chemical composition (refer Table 1). The other differences in Fig. 5 are small, especially when one takes into account of the different batches of samples used in this work. Hence a common regression line is drawn for all the samples in Fig. 5, excluding the values for rectangular samples, to show the general trend of the variation of the yield stress with grain size, and the regression line equation is given along with the figure.

(iii) Liiders front angles The pole angles of the Liiders front were calculated from the measurements of the angle between the tensile direction and the band front on all sides of the different cross-sectional specimens. The measured angles were projected in a stereogram and the band front angles ~, #, y, etc. of the different sides 1, 2, 3, etc. were determined as described by Hall et al. [22]. The angle between the Liiders front and the specimen axis varied from 45 to 55 ° . The angle deviated from 45: to a higher value in larger grain sized specimens, approaching a value as high as 70 ° in very coarse grained samples. In some cases, it was noted that the band front was straight as it nucleated from the grip and tended to become curved as it traversed along the specimen. This was particularly evident in rectangular specimens on the flat faces [Fig. 4(d)]. Measurements of the angles of such curved shaped band fronts were difficult.

(it,) Liiders strains The Liiders strains of the specimens were calculated in terms of the components of the shear and flow strains. The measured transverse strains ~^, ~B, ~c, etc. and the calculated Lflders front angles n, B, ~/ etc. were used in the calculation of the principal strains ~]~, Ez2 and E~2. In square and rectangular specimens the twist angles were used in the calculation of the principal strains. In square specimens, the Liiders strains were calculated independently using the average twist angle ~ of the sides 1 and 3, and the average twist angle ~e of the sides 2 and 4

i

- - ~,00

Hexagonal o CircuLar o Rectangular

z

• Squoni o Octagonal

ky (MN/m 3n) 0.74 + 0.08 (SD) 0.92 :L 0.10(SD) 0.97 __.0.13 (SD) 0.69 + 0.19 (SD) 0.22 + 0.07 (SD)

IN MILD

STEEL

3157

i

i

i

z 3

I /.

I 5

-~ 3o0 Ln t,1 UJ {:E I--

200

9 w >

100

0

[ 1

0

I 2

d'"2 (mm"~)

Fig. 5. Dependence of lower yield stress on grain size in specimens of different cross-sections. The regression line, ¢y = 7 9 . 6 + 0.9 d -I:~ (Mn/m2). (sides I, 2 are opposite to sides 3, 4 respectively). The strains calculated independently with 0, and 02 agreed very closely. The principal strain q, is related to the flow strain, and the shear strains of the specimens were calculated using the kink angles (k(av) values, Ref. [27]), the Lfiders front angles, and the principal strains. The experimental values of Lfiders strain compared well with the calculated values, considering the high percentage error inherent in the calculated Lfiders strain arising in the kink and twist angle measurements, the orientation angles and the transverse strains. The shear and flow components of the Lfiders strain were plotted against d - m and shown in Figs 6 i

r

i

i

• % Shear s t r a i n ~7 % Flow stra,n (3 % Luders s t r a i n , c a l c u l a t e d o % tQders stra,n,experimental l.

u-~

~n 2

c

o

• b

1

~

0

I 1

~

;

~

~7 ~7 v ~7

~7 0

I 2

I 3

f z.

I 5

d'1~ {mm'l~)

Fig. 6. Variation of the flow, shear calculated and experimental Lfiders strains with grain size in case of the hexagonal specimens. The flow, shear and experimental Lfiders strains may deviate by approximately 5%, and the calculated Lt]ders strain may deviate by approximately 10% from the plotted values. The regression lines a, b, c and d refer to the flow, shear, calculated and experimentul Lfiders strains respectively, and the corresponding regression line equations are: % flow strain = 0.72 + 0.035 d- '/';% shear strain -- 0.263 + 0.252 d - Ja; % Lfiders strain (cai)--0.983 +0.288d -ta, and % Lfiders strain (exp) = 0.256 + 0.436 d-i/2.

3158

ANANTHAN and HALL: LI]'DERS BAND DEFORMATION IN MILD STEEL

and 7, for hexagonal and square specimens. The linear regression fits are drawn as unbroken lines to show the linearity of such a variation and the regression equations are given along with the figures. Similar plots drawn [34] for the circular and octagonal specimens were of the nature of the hexagonal specimens, while the rectangular specimens were similar to the square specimens. A reasonably good fit of the shear strain with d -1/2 was observed for all the specimens of different cross-sections. The shear component of the Liiders strain decreased with increasing grain size for all the specimens of different cross-sections. The scatter of the flow strains with grain size was larger than that of the shear component of the Liiders strain, and the regression fit of the flow stains was not found to be very good. However, an approximately constant flow strain was observed for various grain sizes for all the cross-

6 5

=

~Shear strain~ ~ % Flow stram o % L~ders strain, calculated o % LOders straln.expenmenta'.

~ o "~] / ~ |

3

d

2

0

i

i

i

i

I

1

2

3 d-~J2 (ram-l/2)

4

5

6

Fig. 7. Variation of the flow, shear the calculated (using the twist angle Ol) and experimental Liiders strains with the sectional specimens, except the square specimens. grain size in square specimens. The flow, shear and experIn both the rectangular and square specimens (where imental Lfiders strains may deviate by approximately 5%, the twist angles were used for calculatingthe strains), and the calculated Lfiders strain may deviate by approxia high percentage of flow strain contributed to the mately 10% from the plotted values. The regression lines a, calculated Liiders strain. Figures 6 and 7 also show b, c and d refer to the flow, shear, calculated and experimental Liiders strains respectively, and the corresponding the oomparison for hexagonal and square specimens regression line equations are: % flow strain ffi0.196+ when the calculated and experimental Lfiders strains 0.43d-lfz; % shear strain f0.08+0.30d-I/2; % Lfiders were plotted vs d-,/~.Both the calculatedand exper- strain (cal) ffi 0.28 + 0.73 d -in, and % Lfider strain (exp) = -0.04 + 0.55 d-i/2. imental Liiders strains seem to follow a Hall-Perch

type equation in agreement with H u [12]and Hall [2]. The variationin the Lfldcrsstrainwith grain sizecan be attributed to its shear component, since the flow component was approximately constant in the range of grain sizes studied. DISCUSSION The present study confirms the suggestion [18] that Liiders deformation is not just the result of a pure shear at the boundary separating deformed/ undeformed regions. Lomer [9] arrived at the same conclusion but his experimental data were limited to justifying the two possible mechanisms (shear + flow) operating at the Lfiders front. The model used in this work is based on the "shear-flow" theory of Hall et al. [22], and the experimental results suggests that the model, in fact, comes closer to reality; however, to achieve an ideal model to describe fully a nonuniform deformation of this nature one has to address the various aspects of the Lfiders band. In doing so, we were able to identify two factors, grip and bending moment effects, contributing in an effective way to the deviations of the macroscopic parameters of the L~ders band, and these aspects are discussed in the next section.

(i) Grip and bending moments effects The angle of the L~ders front with respect to the specimen axis obtained by stereographic projection varied from 45 to 55 ° (with a mean value orS1 + 1°) for various cross-sectional specimens. These values compare with those in the earlier results of Lomer [9]

48.5 + 1.5°, Jaoul [31] 50 °, Iriciber et al. [15] 55 °, and Hall et al. [22] 51 °. However, there were notable exceptions observed in this study, both in rectangular and circular specimens, where in a few coarser grained specimens the angle deviated to a large extent from the plane of maximum shear, being virtually perpendicular to the stress axis. Although such a large deviation from the plane of maximum shear in coarser grained specimens was not present in other cross-sectional specimens, a tendency for the angle to deviate away from the plane of maximum shear was seen in coarser grained specimens of hexagonal and octagonal specimens, while square specimens showed an approximately constant angle. The present observations do not yield a consistent picture either in relation to Lormer [9] who predicted a uniform deviation from 45 ° based on the shear associated with the shear kink in coarse grained specimens, or to the work of Delwichie and Moon [24] who expected the front angle to 45 ° to be independent of specimen geometry. It was noted in our earlier paper [27] that the direction of maximum kink did not lie in the direction of the greatest slope and it was suggested that grip effects and bending moments at the front would have caused such a deviation. The irregular variations of the Lt3ders front angles in this work and the earlier work of others suggest that grip effects and bending moments associated with the band front also influence the stability and orientation of the Liiders band front. The bending moment associated with the shear kink of the band front, together with the tensile load

ANANTHAN and HALL: LODERS BAND DEFORMATION IN MILD STEEL acting in the specimen should balance to stabilize the orientation of the Lfiders band front in the plane of maximum shear stress. It was noted by Lomer [9] that the kink is straightened rapidly behind the front by the applied load. The large l/t ratio of the specimens (Table 2) would certainly have induced the bending moment effects. Self-aligning grips as used in this work should in theory contribute little to misalignments in the specimen, but this may not be true in actual practice. The effect of specimen grips on the orientation of the Lfiders band front was not considered in the earlier works, but we believe that this significantly influences the band orientation. It is experimentally difficult to determine if grip and bending moment effects produce a deviation of the angles either in a positive or a negative sense and the magnitude of such deviations, but it is possible that for a particular specimen shape and lit ratio such effects would be nearly constant.

(ii) "Shear-flow" model The LiJders strains calculated based on the model derived in the earlier section agree reasonably with the experimental values for all the cross-sectional specimens, and the experimental data gathered in this work demonstrate the need for two separate mechanisms operating at the Lfiders front. However, it should be pointed out that the accuracy of the experimental data needs to be improved (especially in the measurements of kink and twist angles), and the model derived here is too simple. We have used the Schmid and Boas equation for single crystals [33] which may be an oversimplification, as the direction of maximum kink did not lie in the direction of greatest slope [27]. However, the basic principle used in this model, i.e. the transformation of a circular cross-section into an elliptical one, is valid for two reasons: (i) if conceptually the kink is caused by a shear process, a decrease in kink angle should be reflected with a similar decrease in shear strains with increasing grain sizes and this was found true in all the cross-sectional specimens (refer for example Figs 6 and 7); and (ii) the initial texture of the samples was weak. The shear component of strain only accounts to about 60-70% of the total Lfiders strain (experimental) (Figs 6 and 7) in specimens with small grain-size and even smaller fraction in coarse grained specimens. While shear explains the elliptical transformation of cross-section, a reduction in cross-sectional area at the band front can only be accounted for with a flow deformation. Our calculations of flow strains show that this is indeed the case in all the specimens except the square (Fig. 7) and rectangular specimens. In both square and rectangular specimens we have to include the twist angle at the band front in the calculations of strain [equation (4)]. This angle is small (fraction of a degree) and the difficult nature of its measurement makes its accuracy questionable.

3159

This might have resulted in an overestimation of the flow strains and also cansed larger scatter of the flow strain values in these specimens. However, the general trend that emerges from the experiments is clear; L f a t m deformation can not be accounted for only by a shear at the band front. Two separate mechanisms as postulated earlier by Hall [6] contribute together to the total L~ders deformation, thereby supporting the hypothesis of a model of the Lfiders band based on "shear-flow" mechanism. The deviations observed in the calculated strains using the model outlined earlier (Fig. 7, for example) result mainly because of the inherent constraints of the experiments and the limitations of the experimental accuracies we were able to achieve. CONCLUSIONS Summarizing the effect of the contribution of the specimen shape and grain size to the various parameters of the macroscopic aspects of the Lfiders band, the present work reveals the following conclusions: I. The nature of the band front will depend on the grain size and the geometry of the cross-section. The generation of the single bands depends on the l/t ratio of the specimens and test speed, whereas the diffuse nature of the Lfders bands mainly depends on the grain size. Specimens with flatter cross-section (rectangular, square, etc.) tended to yield multiple and/or complex bands compared with the specimens of circular cross-sections, and specimens of coarser grain size (>0.15 ram) generally produced diffuse bands in this study. 2. The grip effects and the bending moment associated with the shear kink influence the deviation of the orientation angle of the band front and the direction of maximum kink angle. The direction of maximum kink angle did not lie in the direction of the greatest slope in the shear plane. The angle of the band front with the specimen axis (average value of 51 ± l ° for all the specimens) observed in this study is in agreement with the results reported earlier. However, this angle deviated away from 45 ° in coarser grained material in all the specimens except the square ones. 3. The Liiders strains calculated based on a model of "shear-flow" compares well with the experimental values of the strain. Further modifications to this model should include factors such as grip effects, specimen geometry, bending moment associated with the kink, test speed, etc. 4. The Hali-Petch relation was found valid for all the specimen shapes. The Lfiders strain may also be related in a similar fashion within the range of grain sizes studied. The Liiders deformation will mainly consist of shear in finer grained specimens while, as the bands become diffuse in coarser grained specimens, the shear contribution to the total deformation is lowered. Hence, the dependency of the Lfiders

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strain on grain size is because of its shear component dependency on grain size, whereas the grain size may have little effect on its flow component. Acknowledgements--The authors thank Mr J. B. Marsh for preparing the specimens and Mr J. A. Grahame for his assistance with the experiments. One of the authors (V.S.A.) acknowledges the University of Newcastle, N.S.W., for the award of a post-graduate scholarship. We also thank Dr D. Juul Jmascn for texture measurements and Dr T. Leffers for his comments on the manuscript.

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