A dislocation avalanche theory of shear banding

Acre metali. ma&~. Vol. 42, No. 3, pp. x57-860, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved

Pergamon

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A DISLOCATION AVALANCHE SHEAR BANDING

THEORY

OF

C. S. LEE’-’ and B. J. DUGGAN’ ‘Department

of Mechanical ZPresent Address:

Engineering, University of Hong Kong, Pokfulam Road, Hong School of Metallurgy and Materials, University of Birmingham, Birmingham B15 2TT, England

(Received 16 November 1992; in revised form

Kong

and

16 July 1993)

Abstract-While there are extensive studies of shear band formation based on Dillamore’s equation, most of the effort has concentrated on the geometrical softening factor. Recently, a yield type explanation has been proposed in some specific situations such as 2-stage cross rolling in which geometrical softening can have only minor effects. This yield theory involving dislocation pile-ups against moderately strong barriers is further developed here to explain shear band formation in certain materials which form fine twin lamellae during conventional rolling and it is possible that a similar mechanism applies in some precipitate hardened alloys.

1. INTRODUCTION

Shear bands are a form of inhomogeneous deformation which occurs in a wide range of metals and alloys including Cu-20% Ni [l], aluminium [2, 31, 70/30 brass [4, 51, copper [6,7] and austenitic stainless steel [8]. In these materials they are sheet-like regions cutting through a layered microstructure where highly localised shearing has occurred and often appear as dark etched lines. In rolled materials they usually form at 20-50” to the rolling direction, with 35” as the most widely reported angle for f.c.c. materials. Stacking fault energy (SFE) is an important factor and shear bands are found most frequently in low SFE materials and particularly in the finely twinned regions where normal slip is restricted [5]. Based on the instability criteria do/dt < C, Dillamore et al. (9) have derived the general instability condition 1 da -.__=; D dc

n

m

+T.dt+

di

l+n+m A4

dM ‘dt

m dN --.-GO N dc

(1) where

N = mobile

dislocation

density,

m = strain

and A4 = Taylor factor. They suggested that the term in N is important for Liiders type phenomena, while shear bands formed in rolling should be related to (l/M) (dM/dc). This term corresponds to geometrical softening such that instability is favoured if it causes a lattice rotation into a geometrically softer orientation. Following this, extensive research concentrated on the geometrical softening term. However, Lee et al., [lo] in a novel experiment found that t( brass after cold rolling to 80% reduction, when re-rolled a small amount in the transverse direction, suddenly pro-

rate

sensitivity,

n = work

hardening

coefficient

duced extensive shear banding and this phenomenon could not be explained by geometrical softening as the additional rolling produced no discernible texture change. They therefore considered the term in N, and by using a three layer model of total width w containing a dislocation pile-up, were able to obtain a relationship between the density of mobile dislocations freed by cutting through the central lamina and the resultant shear strain. From this highly simplified model they obtained m dN _._= N dt

mM sin b Nwb

(2)

where, fl, w and b are respectively the shear band angle, the total thickness of the laminae and the Burgers vector. Clearly the pile-up stress has to reach a critical value for this process to occur, but there is no need to attempt to evaluate this as the critical stress will be reached when the rolling pressure, and the number of dislocations in the pile-ups are sufficient to overcome the barriers and any back stresses. The issue of interest is whether homogeneous deformation or localised deformation in the form of shear banding occurred, and this was considered to depend on the instability condition given by equation (1). The fact that the shear bands were macroscopic rather than grain scale, as is usually the case at rolling reductions of less than 90% in c( brass, was explained by the notion of the “alien dislocation” in the new rolling geometry, in the specific sense meant by Jackson and Basinski [l l] for this case of cross rolling. The insights gained from the cross rolling experiment and the simplified model devised to explain the results provide the starting point for a more general theory shear band formation in conventional rolling.

858

LEE and DUGGAN: DISLOCATION AVALANCHE THEORY OF SHEAR BANDING (a)

2. THEORY

The material is assumed to be a low SFE metal or alloy rolled to a strain where extensive twinning has occurred followed by coupled rotation of the twinmatrix bundles until they are approximately parallel to the rolling plane [5]. Twin/matrix lamellae are thin, typically 0.1-0,5 # m in width, and of the same order of length as the rolled-out grain size. Slip is restricted to the plane parallel to the twin boundaries, operation of other slip systems result in dislocation pileups against these moderately strong barriers. The theory presented here suggests that shear bands form in a catastrophic yield-like manner. The last term in the Dillamore equation is now treated in detail. Consider a finely twinned grain with a potential shear band at an angle fl as shown in Fig. 1. Due to the high density of twin boundaries, those slip systems which intercept the twinning planes cannot operate easily and their dislocations must pile up against the twin boundaries. As the shear band operates these piled up dislocations are freed. The density of dislocations freed is given by d N = np. sin fl tw

(3)

where np = number of dislocations in the pile up, t = shear band thickness and the strain is given by dE =-~ I~,1 M

(4)

with ?~ = shear strain of the ith slip system and the summation is done over the active slip systems in the shear band. Now ~ is given by npib Y; = t sin ~i"

(5)

sin fl

(1 1 1)[-1-1 2] 1.4--

-5

1.2 -~ 1.0 ..~ o.8 ~ 0.6

2 ~

•~ 0.4 ~ 0.2 I

I

10

20

I

I

I

I

I

30 40 50 60 70 Shear band angle

0

80 90

(b) (1 1 1)[1 1-2] 1.4

-5

1.2

A

M'

4

1.0 ~ 0.8 ¢¢ 0.6 e~ o

2 o .>

0.4 0.2

~5

0 0 10 20 30 40 50 60 70 80 900 Shear band angle

Fig. 2. Dislocation avalanche factor (A) and effective Taylor factor (M') for a positive shear band in (a) (111)[I-1-2], (b) (111)[11"2] crystals. In this equation, npi is the number of pile-up dislocations of the ith slip system, ~i is the angle between the ith slip system and the rolling direction and b is the Burgers vector, sin ¢i can be obtained from the dot product of unit vectors of the rolling direction, _R and normal of the ith slip system, ri~ as IR "till. Putting this into equation (5) gives npi =

?,.

(6)

The total number of dislocations in the pile-up is given by summing the number of dislocations in each slip system

~Potential

a)

np = ~ t/p~

Shear Band

t

the ith slip system

- b sin//

X l h ' ti, r,[.

(7)

Putting this into equation (3) gives Twin

i.e.

1

d N = ~w X l R " h,Y,I

(8)

hence the last term in equation (1) is

b)

\ Potential Shear Band

Fig. 1. (a) Geometry of a finely twinned grain with potential shear band. (b) Dislocation pile up on the ith system of a twin boundary.

m dN m M Y.lR'tl iTi[ N dE = N w b EI?il

(9)

If we define A as the dislocation avalanche factor A= M

~:1-~ ",i,y,I El~'il

(10)

LEE and DUGGAN: DISLOCATION AVALANCHE THEORY OF SHEAR BANDING

859

Table

then m dN

m

- -

N dE

Nwb

1. Angle o f m i n i m u m A for different orientations, where ~ is the angle o f rotation f r o m {111}(112) about {111} and fl is the shear band angle at which A is m a x i m u m

{ 111}(uvw)

A.

(ll)

RD

As this term is naturally negative, the larger A, the more likely is instability to arise according to equation (1). 3. NUMERICAL RESULTS A is computed for the twin related orientations (111)[-1-i"2] and (111)[11~] for positive shear bands in Fig. 2, and also plotted are the effective Taylor factors [9]. It is important to note that A has relatively large values over the range of angles allowed for which shearing requires minimum energy i.e. minimum M ' . The orientation {111}(110) is also of importance in the brass texture and A and M ' for both ( I l l ) l I T 0 ] and (111)[110] are shown in Fig. 3. Only one curve is necessary due to the symmetry of this orientation, and because the curve in A is rather flat over the angular range where M ' is minimum, i.e. from ~ 30 ° to 60 °, instability can occur over a wide angular range. While A indicates the tendency for instability, it is not considered as determining the angle of shearing in this formulation. In other words, it is a factor related to shear band initiation but not to propagation. It is believed that the shear band angle is determined by some minimum energy principle such as the minimum effective Taylor factor and reduced slip condition, (12) but the value of A along with 1/M din/de must be such as to trigger instability, i.e. the instability criterion, equation (1), must be met. In this sense A and the propagation angles are related. Clearly A is near to maximum at 35 ° in (111)['~'2], M ' is minimum, and only two slip systems are needed for shear at ~ 35 °, Fig. 2(a), and conversely these conditions are not met for a positive shear band making ~ 60 ° to the rolling direction in its twin, Fig. 2(b). The angles for maximum A for orientations along the 7-fibre (i.e. {111}(uvw)) are also calculated and shown in Table 1. The range and average angle

{1 1 1]-(-1 I O)

1.2 "G

5

A

M'

1.0

4

"~ 0.8

~

3

0.6

2 .-,

•~

0.4

.,

0

0 10

20

30

40

50

Shear band

60

70

80

90

angle

Fig. 3. Dislocation avalanching factor (A) and effective Taylor factor (M') for a positive shear band in {111}(110) crystals.

0 3 6 9 12 15 18 21 24 27 30

--1 -5 --9 --4 --6 -7 -7 -1 -4 -1 0

-- 1 --6 --13 --7 --13 --19 -25 -5 -31 -16 -- 1

2 ll 22 11 19 26 32 6 35 17 1

32 ° 32 ° 32 '~ 32 ° 33 ° 40 '~ 40 :~ 42 ~ 42 ° 43" 45"

Average: 37.5 °

are similar to experimentally observed shear band angles [4, 5]. Although A can be calculated from equation (11), the magnitude of the term m/N dN/dE is still unknown. This term can only be calculated with knowledge of the mobile dislocation density, N, which is obviously difficult to estimate. However, it is possible to turn the problem around and by assuming reasonable values for other parameters, and by using equation (1) the critical mobile dislocation density can be found and compared with expected values. Assuming that for a material with n = 0.2, m --0.1 and the material becoming unstable when E = 0.7, a value for which there is some experimental support (5), for Dillamore's average texture, (9), (1 + n + m)/M dM/dE = 0.65 and n/E = 0.29, thus mA/Nwb ~ 1 when equation (l) has a value of zero. Hence N ~ 3 × l0 u m -2. This figure has to be compared with the total dislocation density for annealed and heavily cold worked materials, commonly given a s 1012 and 1016m-2 respectively. This value for the density of mobile dislocations when E = 0.7 is considered reasonable. 4. APPLICATION TO PARTICLE HARDENED SYSTEMS Although the model has been developed for laminar structure, in principle any system capable of producing pile-ups should be covered. It is well known that precipitate hardened aluminium produces well developed shear bands on rolling, while commercially pure materials do not, and it is necessary to associate shear banding with the presence of particles. Plane strain conditions favour precipitate cutting when the inter-particle spacing is small rather than looping or climbing, hence the term in dN/de determines the onset of instability. The importance of this dislocation avalanche term suggests that the conclusion drawn by Dillamore et al., that shear bands can only occur in a well developed texture is not necessary. Engler and Lficke [3] show shear bands in weakly textured particle hardened materials, with a rather larger spread of angles than found in single phase low S.F.E. materials.

860

LEE and DUGGAN:

DISLOCATION AVALANCHE THEORY OF SHEAR BANDING

5. CONCLUSIONS A yield theory of shear band formation in brass is proposed. It is pointed out that the dislocation avalanche term coined here and used in Dillamore's instability criterion is important and has the right order of magnitude to cause instability. Application of this model is not restricted to the twin/matrix lamellar structures in or-brass, but can be applied to any situation which involves dislocation pile-ups against a barrier. One example of this is precipitate hardened aluminium alloys in which cutting is the significant deformation mode.

Acknowledgements--C. S. Lee gladly acknowledges the

Croucher Foundation for financial support in the form of a Studentship and subsequently a Fellowship. The foundation is also thanked for its provision of electron microscope facilities at the University of Hong Kong.

REFERENCES

I. F. Adcock, J. Inst. Metals 27, 73 (1922). 2. A. Korbel, F. Dobrazanski and M. Richert, Acta metalL 31, 293 (1983). 3. K. L/icke and O, Engler, Mater Sci. Technol. 6, 1113 (1990). 4. B. Fargette and D. Whitwham, M~m. Scient. Revue M~tall. 76, 197 (1976). 5. B. J. Duggan, M. Hatherly, W. B. Hutchinson and P. T. Wakefield, Metals Sci. 12, 343 (1978). 6. J. Grrwen, T. Noda and D. Sauer, Z. Metallk. 68, 260 (1977). 7. M. Hatherly and A. S. Malin, Metals Technol. 6, 308 (1979). 8. M. Blicharski and S. Gorczyca, Metals Sci. 12, 303 (1978). 9. I. L. Dillamore, G. J. Roberts and A. C. Bush, Metals Sci. 13, 73 (1979). 10. C. S. Lee, W. T. Hui and B. J. Duggan, Scripta metall. Mater. 24, 757 (1990). 11. P. J. Jackson and Z. S. Basinski, Can. J. Phys. 45, 7D7 (1976). 12. W. Y. Yeung and B. J. Duggan, Acta metall. 35, 541 (1987).