A criterion for the prediction of shear band angles in F.C.C. metals

A criterion for the prediction of shear band angles in F.C.C. metals

Acta metall, mater. Vol. 39, No. 3, pp. 411-417, 1991 0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie Printed in Great Britain. All ri...

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Acta metall, mater. Vol. 39, No. 3, pp. 411-417, 1991

0956-7151/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press pie

Printed in Great Britain. All rights reserved

A CRITERION FOR THE PREDICTION OF SHEAR BAND ANGLES IN F.C.C. METALS W. B. LEE and K. C. CHAN Department of Manufacturing Engineering, Hong Kong Polytechnic, Hung Hom, Hong Kong

(Received 17 January 1990; receivedfor publication 19 September 1990) Abstraet--The formation of shear bands has been followed in the rolling of an age-hardened A1-4%Cu aluminium alloy. While some grains were found to shear readily, some grains were not prone to localized shear. Current criteria of shear band angle prediction are ambiguous. Analysis in this paper shows that the most likely shear band propagation angle is the one which has the greatest texture softening among the ones with the same minimum number of slip systems and shear strength. Whether a shear band will form or not will then depend on the balance between the strain hardening rate of the slip planes and the texture softening factor. The prediction agrees well with the frequent occurrence of ~ 19° and ~ 35 ° shear bands found in the aluminium alloy and is successful in accounting for shear band angles in other systems. R r s u m r ~ O n a suivi la formation des bandes de cisaillement pendant le laminage d'un alliage d'aluminium 4% de cuivre, durci par vieillissement. Alors que certains grains se cisaillent tout de suite, d'autres n'ont pas tendance fi un cisaillement localisr. Les critrres courants utilisrs pour la prrvision de l'angle des bandes de cisaillement sont ambigus. L'analyse prrsentre dans cet article montre que l'angle de propagation le plus probable pour une bande de cisaillement est celui qui prrsente l'adoucissement par texture le plus 61ev6 parmi ceux qui ont le m~me nombre minimal de systrmes de glissement at la mrme rrsistance au cisaillement. I1 y aura ou non formation d'une bande de cisaillement suivant l'rquilibre entre le taux de consolidation des plans de glissement et le facteur d'adoucissement par texture. Ces prrvisions sont en bon accord avec le fait que l'on observe frrquemment des bandes de cisaillement ~ ~ 19° et ~35 ° dans l'alliage d'aluminium &udir. Elles rendent compte aussi avec succrs des angles de bandes de cisaillement observrs dans d'autres systrmes.

Zusammenfassung--Die Bildung von Scherb/indern wird w~ihrend des Walzens einer gealterten Legierung A1-4%Cu verfolgt. Einige KSrner werden sofort geschert, einige waren hierfiir nicht anf'fillig. Gegenwfirtige Kriterien der Voraussage der Scherbandwinkel sind vieldeutig. Die Analyse in dieser Arbeit zeigt, dab der am wahrscheinlichsten auftretende Scherbandwinkel derjenige ist, der zur grSBten Texturentfestigung fiihrt unter allen mit derselben geringsten Zahl an Gleitsystemen und derselben Scherfestigkeit. Ob sich ein Scherband bildet oder nicht, h/ingt dann von dem Gleichgewicht zwischen der Verformuugsverfestigung der Gleitebenen und dem Faktor der Texturentfestigung ab. Die Voraussagen stimmen mit dem h~iufigen Auftreten der Scherb/inder mit ~ 19° und ~ 35 ° in der Aluminiumlegierung iiberein und erkl~iren die Scherbandwinkel in anderen Systemen.

INTRODUCTION

years as the successful modelling o f the shear b a n d angles give hints to the physical m e c h a n i s m o f shear b a n d formation. In this paper, the occurrence of shear b a n d s in the rolling of a n A I - 4 % C u a l u m i n i u m alloy are reported a n d the observed shear b a n d angles c o m p a r e d with o u r current u n d e r s t a n d i n g o f shear b a n d f o r m a t i o n based o n the crystallographic models p u t forward by Dillamore et al. [5], V a n H o u t t e [6], C a n o v a [7], a n d the m o r e recent " c o m p o s i t e m o d e l " of Y e u n g a n d D u g g a n [8].

Shear b a n d i n g is a highly localized form o f plastic d e f o r m a t i o n which is frequently observed in metalw o r k i n g processes such as rolling, extrusion a n d forging. T h e f o r m a t i o n o f shear b a n d s c a n limit the ductility available for the required deformation, These shear b a n d s a p p e a r as d a r k - e t c h i n g b a n d s a n d m a y occur o n a grain-scale or sample-scale of the d e f o r m e d materials. Flow localization in the f o r m o f shear b a n d s is a n i m p o r t a n t p a r t o f the ductile fracture m e c h a n i s m as fracture eventually occurs a l o n g these bands. Exhaustive shears o n these b a n d s have been associated with failure d u r i n g rolling o f some a l u m i n i u m a n d c o p p e r alloys [1]. T h e continu u m analysis [2-4] a l t h o u g h successful in describing some essential features o f the localized shear do n o t take adequately the effect o f material a n i s o t r o p y of the materials into accont. T h e prediction o f shear b a n d angles in textured polycrystals has been a subject o f m u c h intensive investigation over the past

EXPERIMENTAL A commercial A 1 - 4 % C u h o t rolled plate 7 m m thick a n d 3 0 m m wide was used as the starting material. H a l f o f the plates were a n n e a l e d at 410°C for 2.5 h, a n d the o t h e r half solution treated at 493°C for 2 h a n d then aged at r o o m t e m p e r a t u r e for 5 days. T h e initial h a r d n e s s o f the a n n e a l e d plate a n d the h a r d e n e d one were 59 V H N a n d 133 V H N 411

412

LEE and CHAN: SHEAR BAND ANGLES IN F.C.C. METALS

respectively. They were rolled on a 2 high laboratory rolling mill without end to end reversal. The rolls were lubricated with paraffin oil and the rolled specimens were quenched into water between each pass to remove any heat accumulation. Samples were removed after various cold reductions to examine the strutural changes. SEM specimens were prepared from the longitudinal sections of the rolled plate. The structural changes of both the annealed and age-hardened alloys were followed after various cold reductions. Hardness was higher in the age-hardened sample when compared with the annealed one for the same rolling reduction. Slip bands were observed at low rolling strains of both alloys but at a higher reduction of about 40%, grain-scale shear bands were found in material that had been age-hardened. Single sets of shear bands (shear bands with the same sense of orientation) and two sets of shear bands are observed. At higher rolling strain, ~ 35 ° shear bands then dominate. Grains of different orientations behaved differently in terms of shear band formation. Not all grains formed shear bands and it seems some grains were not prone to localized shear. The set of shear bands observed at the reduction of 80% was not symmetrical as shown in Fig. 1. One set formed at ~ 19° and the other set at ~ 35 °. While the ~ 35 ° shear bands are frequently reported in low stacking fault energy f.c.c, materials such as ~t-brass [8], Cu-AI [5] and stainless steels [9], the ~ 19° shear band has received relatively less attention. These low angle shear bands do not occur only in the age-hardened aluminium alloy investigated in this paper but also in a coarse-grained 70/30 brass (Fig. 2).

~

. . . . . . . . . . . . . . Fig. 2. Optical microstructure of coarse-grained 70/30 brass cold rolled ~84% showing both the ~ 19° and ~35 ° shear bands [10]. where ~ is the equivalent stress or the plastic work per unit volume strain and ~ is the critical resolved shear stress on the active slip systems. M is the Taylor factor which relates the effective strain d(w and the total dislocation shear strain d F M = dF/d~w.

(2)

The Taylor-factor M can be calculated readily from the maximum work principle of Bishop and Hill [11] and is related to the orientation of the crystals with respect to the principal stress axes. When more than one grain is considered in the sample volume, M will be averaged over all grain orientations being considered. A large value of M indicates a large shear strength of the grain. The symmetric strain tensor in the band referred to the sheet geometry (Fig. 3) is given by dy ~ - s i n 2 f l cos2fl ] ~w = 2 - L - cosEfl sin2fl]

CURRENT CRYSTALLOGRAPHIC SHEAR BAND THEORIES The propagation of shear bands in textured polycrystals requires a constitutive model which accounts for the material anisotropy. For simplicity, the material constitutive equation for an individual grain can be written as = Mz~

.......... .......... . . . . . . . . . . . . .

(1)

(3)

where fl is the shear angle. The macroscopic imposed strain tensor will then be transformed from the work piece coordinate system (x~ - y ~ ) to the crystallographic axes of the crystal ( x 2 - Y2) i.e. ~ = PEwP-~

(4)

where ~ is the strain tensor referred to the cube axes of the crystals and P is the transformation matrix. For an isotropic material the orientation of the shear band should coincide with the direction of maximum shear stress and makes an angle 45 ° with the rolling direction. If the shear angle deviates from 45 ° by an angle ~t, the shear strain y in the band will be increased by a factor of 1/cos2~ in order to produce the same

S EET S R ACE

Fig. 1. Scanning electron micrograph showing both the ,-~19° and ~35Q shear bands in age-hardened Al-4%Cu aluminium alloy after ~80% cold rolling.

~

~ Fig. 3. Geometry of shear bands.

.

LEE and CHAN: SHEAR BAND ANGLES IN F.C.C. METALS amount of macroscopic strain. Since the resolved shear stress on the inclined plane decreases as 1/cos2~ from the 45 ° plane, an effective Taylor factor M ' is defined by M/cos2c~. The shear is considered by Dillamore et al. [5] to occur at an angle such that the plastic work in deforming the material will be mini• mum, i.e. at which the value of M/cos2~t is minimized. Very often the variation of M/cos2a with shear band angle is associated with a plateau and a range of shear angle is then possible based on the principle of minimum work alone. For example in {112}(111) orientation, the permissible positive shear band angles based on minimum M ' has a range from 19.47 ° to 54.74 ° (Fig. 4).

1.o o.s ~ 0.6 r~ 0.4 ~ .~ o.z ~ 0.0 ~ -0.z ~ -04

An°ther criteri°n used f°r the determinati°n ° f shear band in the literature [5, 6] is based on the texture softening parameter. A shear band will form when

1° ~ 0.6 ~ o.s ~ 0.4

413

,,~,~. ~

5

/

~

o

lo

6o ~ ~

,,~i,~ ~

~- -0.6 -0.8 -1.o

1.0 o.a 0.6 0.4 o.z ~ o.o ";o

/

ao

;

t~°

-o.4-°z , -0.6 -0.8

(a) positve shear band

-1.o '~'~ ~ o ~ 6 o

i

t 1"° 0.a 0.6 0.4

.~

0.2

o.z -

1 d~

~<0.

-

(5)

6 dgw Differentiating equation (I), we have

1 d6

1 dM

M dz¢

6 dgw = -M- dgw + -~¢~-f ~0.

(6)

~

lo

o.o ~

20

ao

4o

v

o -0.z

I

~ -0.4

,~. u

~-0.8 -o.a

To'

.--

6] ~ 5~ . | "~ 4

Illz~ oriented gr~m \

~' ~ ¢ z ~ o

o

i

lo

J

2o

i

p

i

i

i

ao 40 50 so 7o so ~o She~r Angle t~o Fig. 4. Variation of effective Taylor factor M' for positive shear band {112}(111) oriented grains,

-0.z ~i-0.4 i-°'s

(b) negative shear band

-1.0

The term (l/M)(dM/d(w) represents the texture softening or hardening factor S, and (M/z~)(dz~/dF) represents the slip plane hardening contribution. At the early stages of deformation, (M/%)(d%/dF) is usually positive and a shear band will only develop when (I/M)(dM/d~w) is most negative. The texture softening factor calculated for single orientations sometimes contains several minima such that no unique solution can be found. An example of this behaviour is shown for the {111}(112) oriented grains (Fig. 5). The global minimum S is - 0 . 6 3 at 54.75 ° for the positive shear band and - 0 . 2 8 at 70.54 ° for the negative shear band. Based on the consideration of texture softening factor alone, the {112}(111) oriented grain would be expected to shear at 54.75 ° and 70.54 ° (Fig. 5) instead of the observed ~ 19° or ~ 3 5 °. An interesting point about Fig. 5 is that although there is a sharp change in S

o

oo

I -o.a -1.0

Fig. 5. Variation of texture softening factor for shear banding in {112}(111) oriented grains. near ~ 54 ° or ~ 70 °, numerical expression for calculating the S values shows that the curve is still continuous. In Yeung and Duggan's "composite model" of shear band formation [8], the texture softening factor is not used but the effective Taylor factor is modified by introducing a factor # which is the ratio of the in-plane shear stress component between the boundaries of the twin/matrix elements to the plane strain flow strength of the twin/matrix. The effective Taylor factor used by Yeung and Duggan (M") is then given by M" =

M cos2~ + 2/~[1 + sin2a - c o s 2 ~ ] '

(7)

Any deviation in the local stress state and hence g would cause a deviation of the shear band angle from 35 °. With a proper choice of the g value, Yeung and Duggan is then able to show the uniqueness of the 35 ° shear band in the twinned volume of or-brass. This ratio p cannot be predicted from microstructural parameters and is difficult to be determined from experiments. This led Yeung [12] later to predict the shear band angle based on the minimum value of M ' and the minimum number of slip system only. However, based on Yeung's method, no unique shear angle can be found for orientations such as {l lO} (O01). In the {l lO} (OO1) orientation there are tWO possible shear angles with the same minimum M ' and the same minimum number of slip systems. The difficulties of the current crystallographic models in

414

LEE and CHAN: SHEAR BAND ANGLES IN F.C.C. METALS

the prediction of grain-scale shear band angle are thus either:

lattice rotation and the associated change in the Taylor factor, i.e.

(i) the variation of S with shear band angle shows several minima and the observed shear band angle does not occur at the global minimum of S, or (ii) the variation of M ' with the shear band angle is associated with a plateau, or (iii) modification of M ' to give an unique minimum requires an arbitrary # factor which cannot be determined experimen-

dM dgw

dr dM d(w dr)"

(8)

The criterion for selecting the slip systems is based on Bishop and Hill's principle of maximum work [10, 13] for full constraints deformation. Equal hardening of slip systems is assumed. The ambiguity in identifying the set of five slip systems is resolved in this paper by solving the set which minimizes the second order plastic work as proposed by Renouard and Winterberger [14]. The effective Taylor factor as a function of the shear band angle M'(fl) and the texture softening factor S(fl) were computed for the rolling texture components commonly found in aluminium. For comparison purpose, the {111} (112) orientation is also included. The shear angles at which the minimum value of M ' or S occur are listed in Table 1, except for highly symmetrical grain orientations such as {100} (001) where no unique minimum of M ' or S existed. F o r such symmetrical orientation, no unique shear angle is found based on the minimum effective Taylor factor or the minimum texture softening factor alone as S can have several local minimum values. When the number of operating slip systems for each shear band configuration is taken into consideration, a minimum number of two slip systems is always found at the lower and upper limit of the permissible range of shear band angles based on M ' , while more than two slip systems are required for other shear bands. Thus, based on the effective

tally, In summary, no current criteria can unambiguously predict the shear band angles, THE PROPOSED CRITERION The criterion proposed in this paper is based on the combination one of the effective Taylor factor M ' , the number of slip systems and the texture softening factor in sequence until a unique solution is found, To predict the most likely shear band angles, the minimum M ' is calculated first. If a range of shear angles all possess the same minimum M ' then the shear angle with both the minimum M ' and the smallest number of slip system N will be selected. If still no unique shear band angle is found, then the one with both minimum M ' , minimum N and with the smaller S of the minimum N will be the most likely shear band angle. The texture softening factor is computed numerically for both the positive and negative shear bands by determining the rate of

Table 1. (a) Prediction of positive shear band angles (IV) Permissible angles based on min. M '

(I) Orientation

(II) M

(III) Min. M'

{110}(001)

2.45

1.22

35.26°-54.74 °

{110}(112)

3.27

1.63

30°~0 °

{112}(111)

3.67

1.84

19.47 ° 54.74 °

{111}012 )

3.67

1.84

35.26°-70.53 °

{100}(001)

2.45

1.22

45 °

(V) Angles of (IV) with min. no. of slip systems (1) (2) (1) (2) (l) (2) (1) (2)

35.26 " 54.74 ° 30" 60 ° 19.47 ° 54.74 ° 35.26 ° 70.53 ° 45 °

(VI) Slip systems* corresponding to (V) (1) (2) (1) (2) (1) (2) (1) (2)

al,a 2 bl,b 2 al,c I b2,e 2 a,,a 2 c 3, d 3 c3,d3 bl,b 2 a2,d 2

(VII) S factor corresponding to (V) (1) (2) (1) (2) (1) (2) (1) (2)

0 0.6284 0 0.866 0 0.6284 -0.6284 0 0

(VIII) Shear band angle based on a smaller S in (VII)

(IX) Min. deviation of (VIII) from slip plane

35.26 °



30 °

19.47 °

19.47 °



35.26 °

35.26 °

45 °

35.26 °

(VIII) Shear band angle based on a smaller S in (VII)

(IX) Min." deviation of (VIII) from slip plane

35.26 °



30 °

19.47 °

35.26 °

54.73 °

19.47 °

19.47 °

45 °

35.26 °

*In Bishop and Hill's nomenclature 113]. Table 1. (b) Prediction of negative shear band angles (IV) Permissible angles based on rain. M '

(I) Orientation

(II) M

(III) Min. M'

{110}(001)

2.45

1.22

35.26°-54.74 °

{110}(112)

3.27

1.63

30°-60 °

{112}(111)

3.67

1.84

35.26o-70.53 °

{111}(112)

3.67

1.84

19.47°-54.74 °

{100} ( 0 0 1 )

2.45

1.22

45 °

*In Bishop and Hill's nomenclature [13].

(V) Angles of (IV) with rain. no. of slip systems (1) (2) (1) (2) (1) (2) (1) (2)

35.26 ° 54.74 ° 30 ° 60 ° 35.26 ° 70.53 ° 19.47 ° 54.74 ° 45 °

(VI) Slip systems* corresponding to (V) (1) (2) (1) (2) (1) (2) (1) (2)

b~,b 2 a~,a2 b2,c2 a I, cI c3,d 3 al,a 2 b~,b 2 c3,d3 b2,c 2

(VII) S factor corresponding to (V) (1) (2) (1) (2) (1) (2) (1) (2)

0 0.6284 0 0.866 0 0.9778 -0.9778 0 0

LEE and CHAN: SHEAR BAND ANGLES IN F.C.C. METALS Taylor factor and slip localization on a limited number of slip systems as suggested by Yeung [12], there is no unique solution. F o r example, according to Table l, shear in the { l l l } ( l l 2 ) orientation can occur at either ~ 35 ° or ~ 70 ° to the rolling direction as they are both associated with the same minimum of M ' and same minimum number of slip systems, However, if the texture softening factor S corresponding to these two shear angles with both minimum M ' and minimum number of slip systems is further determined, the shear angle will always be the one with the lower S value. The shear band angle determined in this manner for the 5 texture cornponents are shown in column (VIII) of Table l(a) for positive shear bands and Table l(b) for negative shear bands. From the above analysis, the major rolling texture components of aluminium would exhibit shear band angles of 19.47 °, 30 °, 35.26 ° for the positive shear bands and 30 ° and 35.26 ° for the negative shear bands. Excluding the { 110} (112) orientation which is known to be stable and to give rise to no microstructural changes such as twins or shear bands [10, 15], the predicted shear angles 19.47 ° and 35.26 ° agrees well with those found in the A1-4%Cu aluminium alloy. These are also common shear band angles reported in other aluminium alloys [l, 5]. Although the shear bands are basically noncrystalographic, one of the reasons for their frequent occurrence may be that the shear bands in the { 112} (111 ) oriented grains coincide with the active slip plane. Figure 6 is a schematic diagram showing how the shear band angle is determined from the set of minimum values of M ' , minimum number of slip systems N and the minimum value of S. The set of shear angles that possess minimum S (i.e. both local or global minimum) may or may not satisfy the set of minimum M ' or N and is therefore drawn to intersect with other sets only. The most likely shear band angle fl is the one which has the smallest S among the ones with the minimum number of slip systems which are also associated with the minimum

I

M ' . Although the procedure is complicated, the physical meaning is clear. If a unique minimum shear strength occurs in some directions, shear banding will occur accordingly. If not, some balancing of the texture softening factor and the slip plane hardening rate will need to be sorted out as indicated by equation (6). The requirement of a reduction in the number of slip systems is a logical prerequisite as a decrease in the slip plane hardening rate will favor shear banding. Although a range of shear band angles has been reported in the rolling of f.c.c, metals by some authors [17, 18], it is surprising to note that the frequently occurring angles concentrate on two i.e. ~ 35 ° and ~ 19° shear band are the most common. Table 2 shows the shear band angles that would occur in the { l l 0 } ( 0 0 1 ) and {112} (111) oriented grains if they are displaced about their TD (transverse direction) axis by _+5° and + 10°. The results show that grains within 10° of the spread of the two orientations about their TD axis give a unique shear band angle based on the minimum value of M ' which is also associated with the minimum number of slip systems. Although a spread in the grain orientations from their ideal positions in the rolling texture is expected, the shear band angles corresponding to these minimum values are not commonly found in rolled aluminium alloys.

DISCUSSION A criterion based on the consideration of effective Taylor factor, the number of slip systems and the texture softening factor for determining the most likely shear band angle on a grain-scale has been suggested. The calculation of shear band angles in { 112} (111) and (110} (001) orientations agrees well with those frequently observed shear band angles in the aluminium alloys and ~-brass. The criterion used in this paper is different from the one used by Dillamore [5]. Dillamore calculated the texture softening factor directly for a group of texture

Mmin

/ gminSetof

M'mtn = ~fl: ~ w i t h t h e rain. M"]

s

Nmtn=

~: /8 with the nlin. no. of slip systems~

Smin = ~fl:fl with the min. S[ S s

~ rain

~-~ r:~

= I~: fl with a smaller S in N rain}

DiUamo~'s c,-i~e~n [5] Du.gg=n & reu~g's crater,,on [ 8 ] dzt/2rk & I~vies' criterio~ [16] c'ri,terio~ "used £~ tA'~s I~xzpe~,

Fig. 6. Criteria used in the prediction of shear band angles. AM 39/3---K

415

416

LEE and CHAN: SHEAR BAND ANGLES IN F.C.C. METALS Table 2. (a) Theoretical shear band angles for the { 112} (11 ! ) oriented grain displaced about the T D axis Positive shear band Rotation angle 0° 5° --5 ° 10 ° -10 °

M

Min. M'

Shear band angle

Slip systems

3.67 3.62 3.62 3.45 3.45

1.84 1.76 1.53 1.73 1.35

19,47 ° 49,74 ° 24.47 ° 44.74 ° 29.47 °

al, a2 c3, d 3 al, a2 c3,d 3 al,a 2

Negative shear band S

Min. M'

shear band angle

Slip systems

S

0 -0.6869 0 -0.6,139 0

1.84 1.76 1.53 1.73 1.35

35.26 ° 40.26 ° 65.53 ° 45.26 ° 60.53 °

a 1, a 2 c3 , d 3 al, a2 c3,d 3 al,a 2

0 0 --0.4013 0 -0.5179

Table 2. (b) Theoretical shear band angles for the { I 10} ( 0 0 1 ) oriented grain displaced a b o u t the T D axis Positive shear band Rotation angle 0° 5° -5 ° 10 ° - 10 °

M

Min. M'

Shear band angle

Slip systems

2.45 2.41 2.41 2.34 2.34

1.23 1.17 1.17 1.15 1.15

35.26 ° 49.74 ° 40.26 ° 44.74 ° 45.26 °

a I, a2 hi, b 2 al,a 2 bl,b 2 a I, a2

components and a unique minimum S can often be found. The difficulty of applying Dillamore's approach to grain-scale shear band has been discussed by Yeung and Duggan [8] in the analyzing of the ~35 ° shear bands in ~-brass. The texture softening factors calculated for shear banding in single oftentation contains several local minima. The single texture softening factor criterion as used by Dillamore is suitable for a group of orientations only because of the averaging effect of different grains in the calculation of the S value. This led Yeung and Duggan [8] to give up the texture softening factor and introduce a in-plane shear in the calculation of M ' such that a unique minimum M ' could be found. In this paper, the ratio p can be ignored if the texture softening and the number of slip system are both considered in the selection of the shear band angle, The general applicability of this approach can be illustrated in other alloy system such as Pb--Ca-Sn in which the shear band is reported to form at 30° to the rolling direction [19]. The minimum effective Taylor factor was found to occur at 28034 ' for the positive shear band and 61°26 ' for the negative shear band. Further inspection will reveal that the shape change for both shear bands can be accommodated by one slip system only with the same dislocation activity, However, when the S factor is further determined, it is found that S is smaller at 28034 ' and should be formed according to equation (6). The {112}(111)and {110}(001)orientations are stable in plane strain rolling and they have higher M-values compared with grains which are oriented a few degrees from them. Chang et al. [20] are of the opinion that the Goss {110} (001) oriented grains in the rolled f.c.c, metals are easier to deform (i.e. they have lower M-value) and the deformation is easier to be loealised than the others. Their simple view is not supported here. According to the analysis performed in this paper and the experimental results reported by Lee and Duggan [21] in the rolling of ct-brass, it is always the harder grains (i.e. grains with high M

Negative shear band S

Min. M'

shear band angle

Slip systems

S

0 - 1.6961 0 -1.124 0

1.23 1.17 1.17 1.15 1.15

35.26 ° 40.26 ° 49.74 ° 45.26 ° 44.74 °

al, a 2 b I , b2 al,a 2 bl,b 2 a I, a 2

0 0 -1.6961 0 - 1.124

factor and have higher dislocation activities) which form shear bands and not vice versa. It has also been shown by Duggan and Lee [22] in the cross rolling of ~-brass that shear bands were found in grains with a high M-value. The analytical results as shown in Table 2 is of particular interest. According to the orientation distribution function (ODF) analysis of the rolling texture, there should be an isotropic scattering region of the ideal rolling texture cornponents [23]. If grains misoriented __+5° and _ 10° about the TD axis of {112}(111) and {110}(001) orientations are considered, the predicted shear band angles seldom occur. The implication of this finding is that shear bands will not occur in other grain orientations unless they have converged to the ideal orientations of {112}(111) or {110}(001) in the rolling texture. In addition, the {112} (111) and {110}(001) oriented grains have the highest M-values (i.e. they are harder grains) compared with grains which are misoriented by a few degrees from them in Eulerian space. It must be emphasized that the criterion used (Fig. 5) for the prediction does not tell whether the grain will form localised shear or not. What the criterion suggests is t h a t / f shear bands do occur in that orientation, the shear angle will be governed by conditions listed in the criterion. A truly predictive theory for shear band angle has to take into account the micromechanics expressed through a constitutive equation such as expressed by equation (1). The instability criterion of equation (5) implies that the attainment of a negative or zero work-hardening rate is crucial for the occurrence of shear bands. For grain orientations listed in Table 1 and with the minimum S factor equal 0, whether shear bands form or not depends strongly on the strain hardening term dzc/dF of the slip planes. When S is 0, shear bands will occur at a strain E only if dxc/dF becomes negative. This dependency on the slip plane hardening rate explains why it is easier to find shear bands in age-hardened aluminium alloy but not the annealed one. In the low

LEE and CHAN:

SHEAR BAND ANGLES IN F.C.C. METALS

stacking fault energy metal such as ~t-brass, mechanical twins of {111 } (112) orientations have been associated with profuse shear band formation [24]. The S-factor in column VII of Table 1 for {111}(112) is negative and this means that a small but positive work hardening rate can still give rise to shear banding in this orientation. This may partly explain the abundant shear bands found in this orientation. The dependency of shear band formation on the grain orientation is through the development of a lamellar structure such as mechanical twins in ~t-brass [8] or a layered dislocation arrangement in aluminium as reported by Morii [25]. The work-hardening rate is low in these types of geometrical structures. All these observations reflect the importance of the residual hardening capacity for the occurrence of shear bands. In the heat-treatable aluminium alloy, the presence of G.P. zones lower the mobility of slip dislocations which leads to a lower strain-hardening capacity. The importance of slip plane hardening rate on the onset of shear localization has been demonstrated by Chang and Asaro [26] in a A1-2.8%Cu single crystal. They pointed out that a critically low value of the ratio of slip plane hardening rate to the current tensile stress is needed if shear deformation is to be localized. CONCLUSION The propagation of shear bands has been studied in the rolling of an A1-4%Cu alloy. No shear bands were found in the alloy that was rolled in the annealed state. Harder grains (grains with a higher M-factor) tend to form shear bands more readily, while softer grains with a low M factor are not prone to shear band formation. A criterion has been found for the explanation of the shear direction on a grain-scale based on both the minimum Taylor factot, the minimum number of slip system and the minimum texture softening factor. This is capable of explaining the dominance of the ~19 ° and ~35 ° shear bands commonly found in aluminium alloy and g-brass. Whether a shear band will occur or not in a

417

particular grain will then depend on the balance between the strain hardening state of the slip planes and the texture softening factor. Acknowledgement--The authors wish to express their thanks to Dr B. J. Duggan for helpful discussion and his careful reading of the manuscript.

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