Effect of rate sensitivity on the stability of torsion textures

000-6160 xx $3.00 f0.00

Acra m<,!r///.Vol. 36, No. 12. pp. 3077-3091. 1988 Printed in Great Britain. All rights reserved

EFFECT

OF RATE

STABILITY L. S. TOTH,t Department

Copyright

of Metallurgical

(Received

SENSITIVITY

OF TORSION P. GILORMINIS

c 1988 Pergamon Press plc

ON THE

TEXTURES

and J. J. JONAS

Engineering. McGill University Montreal. Canada H3A 2A7

29 July 1987; in revisedform

3450 University

24 March

Street,

1988)

Abstract-The yield stress potentials pertaining to the rate sensitive deformation of cubic crystals are described; they are shown to be strictly convex, with shapes that depend on the rate sensitivity exponent. M. A four-variable maximization procedure is presented, which permits the stress state associated with full constraint conditions to be found rapidly. The rigid body, lattice and glide rotation rates are clearly distinguished and specified; the lattermost rate decreases as m is increased, going to zero when m = I. It is demonstrated that none of the torsion ideal orientations can be permanently stable under rate sensitive conditions of flow. An orientation stability parameter is introduced, which is used for the construction of orientation stability maps (OSM’s) in Euler space. The OSM’s are employed to characterize the tubes associated with fixed end testing. Finally, a “cloud” model of texture development is proposed; it is employed to account for the initial anti-shear rotations of experimental textures and for their subsequent migration parallel to the rigid body rotation, i.e. to the applied shear. With the aid of an axial stress map in Euler space, the cloud model is also used to account for the development of compressive, followed by tensile, axial stresses. R&urn&On dkcrit les potentiels de limite klastique correspondant i la sensibilitk de la diformation i la vitesse pour les cristaux cubiques. On montre qu’ils sont strictement convexes, avec des formes qui dkpendent de l’exposant de la sensibilitk B la vitesse rn. Une prockdure de maximisation ii quatre variables est prksentke, elle permet de trouver rapidement I’etat de contrainte associe g des conditions d’emboitagc. les vitesses de rotation du corps rigide, du r&au et du glissement sont clairement distingukes et sp&cifit-es. la dernitre de ces vitesses dkcroit si m croit, atteignant 0 pour nl = I. On dtmontre qu’aucune des orientations idtales de torsion ne peut ttre stable de maniire permanente dans des conditions de sensibilitk de I’tcoulement $ la vitesse. Un paramktre de stabilitt: en orientation est introduit, il est utilise pour la construction de cartes de stabiliti en orientation dans I’espace de Euler (CSO). Les cartes CSO sont utilistes pour caracttriser les tubes associ&s g des essais $ extrtmites fix&es. Finalement. un modile en “nuage” du dtveloppement des textures est propoi. 11est utilisk pour rendre compte des rotations initiales en sens inverse du cisaillement dans les textures expkrimentales et de leur migration ult&rieure parallele i la rotation du corps rigide, c8d g la contrainte appliquCe. G&e ri une carte de contrainte axiale dans I’espace d’Euler, le modtle en “nuage” est kgalement utilis6 pour rendre compte du dtveloppement de contraintes axiales d’abord en compression, puis en traction. Zusammenfassung-Die Fliefispannungspotentiale, die zur ratenempfindlichen Verformung kubischer Kristalle gehiiren, werden beschrieben; es wird gezeigt, daB diese strikt konvex sind, wobei die Form von dem Exponenten m der Ratenempfindlichkeit abhJngt. Es wird eine Maximierungsprozedur mit ricr Variablen vorgelegt, mit der der slmtlichen Einschrlnkungen entsprechende Spannungszustand rasch aufgefunden werden kann. Die Festkorper-, Gitter- und Gleitrotationsraten werden klar unterschieden und spezifiziert; die letztere nimmt mit zunehmendem tn zu und geht fiir m = 0 durch Null. Es wird gezeigt, dal3 keine der torsionsidealen Orientierungen unter den Bedingungen des ratenempfindlichen FlieDens dauernd stabil bleiben kann. Ein Parameter der Orientierungsstabilitlt wird eingefiihrt, mit dem Karten der Orientierungsstabilitat im Euler-Raum konstruiert werden. Diese Karten werden benutzt, urn die mit den Verformungsversuchen bei festgehaltenen Enden zusammenhangenden Riihren zu charakterisieren. Zum SchluR wird ein “Wolkenmodell” der Texturentwicklung vorgeschlagen. Mit diesem werden die anfgnglichen Anti-Scher-Rotationen der experimentell untersuchten Texturen und deren anschliellende Bewegung parallel zur Festk(irperrotation, d.h. zur %uBeren Scherspannung. erklart. Mit Hilfe einer Karte der achsialen Spannungen im Euler-Raum wird dieses Wolkenmodell such benutzt, die Entwicklung von Druck- und danach von Zugspannungen zu erklaren.

1. INTRODUCTION

The torsion testing of metals has fascinated scientists and engineers for a considerable period and for a

number of reasons. One is that, in contrast to uniaxial tension, uniaxial compression, and plane strain tension or compression, torsion involves appreciable rigid body rotation. For the former strain paths, the end textures

ton

leave from Institute for General Physics, E6tv6s University, 1445 Budapest, P.O.B. 323, Hungary. SLaboratoire de Micanique et Technologie, Ecole Normale

tion

SupCrieure. 61 Av. Prksident Wilson, 94230 Cachan, France. AU IhIZ B

rate drops

are simply to zero.

reached In torsion,

when

the glide

on the other

rotahand,

because of the high rigid body rotation rates, texture stability requires the countervailing presence of high glide rotation rates, as will be seen in more detail

3077

3078

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STABILITY OF TORSION TEXTURES

below. Thus torsion is an example of a strain path for which the consequences of rigid body and glide rotation must be clearly recognized and taken into account. A further reason for the investigation of torsion as a deformation process involves the textures that are developed and the anisotropic effects that accompany them. Of primary interest are the axial effects: compression or tension under fixed end conditions of testing [l-.5], and lengthening or shortening under free end conditions [ 1,6,7]. Although compression and lengthening have been accounted for by rate insensitive theories of texture evolution [8], tension [9] and shortening [IO, 1I] cannot be modelled without recourse to a rate sensitive analysis. Further topics of study have been the rates of texture development 141, which increase with temperature, and the frequently observed “tilts” of the textures away from their ideal orientations, i.e. from their positions of perfect symmetry. The latter are generally rotated in the sense opposite to the shear at small strains and parallel to the shear at large strains [4, 12, 131. The present study was undertaken to clarify some of the above questions. In order to do so, the main elements of a rate sensitive analysis will first be described. By this means it is shown that the rate of grain rotation increases with rate sensitivity (and therefore with temperature). A clear distinction is drawn between the rigid body and lattice rotation rates, and an Euler map of the latter is introduced, from which the relative stabilities of the various texture components can be readily visualized. This form of representation also leads to a simple explanation for the ‘tilts’ or asymmetries of the experimental ideal orientations, and for the way in which they evolve with strain. An Euler map of axial stress is also prepared, from which an explanation of the axial stress variations commonly observed is derived. Finally, one of the applications of the present analysis is that it permits the shortening of the free end samples deformed at high temperatures to be readily simulated. The lattermost topic, however, will not be described here, but will he addressed in separate publications [ 10, 111.

Peirce et al. [ 171, Asaro and Needleman [ 181, NematNasser and Obata [19] and Canova et al. [20]. In the present paper m, r. and ii0are assumed to be constant and the same for ail the systems of a given crystal and for all the constituent crystals when a polycrystal is being considered. The resolved shear stress tS is related to the deviatoric part Q of the stress tensor applied to the crystal through the following relation 7* =

m$ai,

where rni = b;nj is defined by the components of the unit normal ns to the slip plane and the unit vector bs parallel to the slip direction of slip system s. The usual summation convention over repeated subscripts is used here, and exceptions to this rule are indicated. Finally, the (100) crystal axes are employed as the reference axes. The sign term in equation (1) requires the shear rate to be of the same sign as the resolved shear stress, which permits the use of a single index s to designate two opposite slip systems with the same ns and opposite vectors b”. If m is the same for all the systems and all the grains in a polycrystal, it is readily shown that pn is also the macroscopic rate sensitivity of the polycrystal as a whole. Let Z and B be an external deviatoric stress applied to the polycrystal and the associated macroscopic strain rate, respectively. _!?is the average of the IocaI strain rates i induced in the grains by the shear rates which are associated with the local stress tensor (r induced by 6. If 0 is multiplied by the same positive scalar t( everywhere in the polycrystal, the resulting local stress field is still in equilibrium and its average is 0rZ. Since ail the resolved shear stresses are multiplied by 01,the shear rates are multiplied by ~1”“’on all the systems in all the crystals, and consequently i is multiplied by CIlirn and so is the average strain rate. Thus clC is associated with tl”‘“& and this means that m is the rate sensitivity of the polycrystal. This paper concerns the rate sensitivity of f.c.c. crystals, which means that there are I2 possible slip systems (plus their opposites) in each crystal, defined by the {111) planes and (110) directions (Table 1). The results presented in this section are also applicable to b.c.c. crystals with {l lO}(l 11) slip, since changing t71: into rnj! only reverses the sign of the 2. RATE SENS~IVE ANALYSIS OF glide rotation rate (see below). In general, CJinduces POLYSLIP DEFORMATION a non zero resolved shear stress on all the slip systems, and consequently 12 systems are active The deformation of rate sensitive crystals is usually modelled by a power law relationship between the simultaneously. The cases where less than 12 systems shear rate y” and the resolved shear stress r” which is are active can be investigated systematically by studying the set of 12 equations in oii defined by equation applied to a slip system identified by the index s (2), and the main results are shown in Table 2. Note that a stress state applied to an f.c.c. rate sensitive crystal cannot activate only 1, 2, 3 or 5 systems. In equation (1) m is positive and defines the system Because the common stable ideal orientations rate sensitivity, to is a reference shear stress and Ij, a observed in torsion are highly symmetric and therereference shear rate. The rate sensitivity was repre- fore only involve the operation of 1 or 2 systems in sented as n = l/m by Hut~hinson 1141 and Canova the Bishop and Hill analysis, it is not clear from and Kocks [151,but m was used by Pan and Rice [ 161, Table 2 how these orientations remain stable within

TOTH er al.:

STABILITY

OF TORSION

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Table 1. F.c.c. slip systems Slip system index

siip alane

Slip direction

the context of a rate sensitive analysis. This is a point to which we will return below. 2. I. The stress potential and extremum principles The components of the strain rate tensor associated with a given stress state are readily obtained from equations (1) and (2) iu=Ctcma+m;)~‘=~Zi(~~+~~,) s 5 xm”,,o,,ImRa,,I”im)-’

(3)

where F6’= a,i, is the rate of plastic work. The equipotentiais cover all stress space, since any nonzero stress state leads to plastic yielding of a rate sensitive crystal. The shapes of all the equipotentials related to a given value of m are the same, sinceJis a homogeneous function of degree 1 + I/m in the c~, It should also be noted that the direction of i is fixed for a given direction of a; only the magnitude oft’ is modified when the magnitude of 0 is changed. The Hessian tensor deduced from the stress potential

(no sum over i andj). It should be observed that the strain rate can also be deduced from a stress potential (e.g. Hutchinson, Ref. [ 141): .

6

I,

=-.

af(0) 1

withf(a)

Odi/

shows that any of the equipotential convex, since

= .-!_-!Zx m + I 7$m

~lf(m;+m~,)oijl”~“‘+’ = 3

e(e)

‘i

x(mSk,+msk)IZS(6)I(‘im)-i

surfaces is strictly

(4)

Table 2. Cases where less than 12 systems are active in a rate sensitive f.c.c. crystal and relation with the Bishop and Hill polyhedron for a rate insensitive crystal. The number of active systems is equal to the number of non zero resolved shear stresses Number of active svstems

Number of possible stress states

I

0

2 3

0 0

4

6

5

0

6

16

Remarks (n = degree of freedom for the stress states)

n = 1,directions of the middle of the 4A edges, include three zero z’s on two slip planes, the non zero T’Sare all equal

n = 1, directions of the B and D six-system vertices, include three zero t’s on one slip plane, the non zero T’S are all equal

I

12

n = 2, include three zero T’Son one slip plane

8

33

n = 1 for 6 cases, directions of the E eight-system vertices, the non zero T’Sare all equal; n = 2 for 3 cases, directions containing a pair of 4A edges; n = 2 for 12 cases, directions containing a 413 edge; n = 2 for 12 cases, directions containing a 4E edge, include three zero T’Son one slip plane

9

40

n = 2 for 32 cases, including 24 cases containing a 4H edge; n = 3 for 8 ewes, including 4 cases with three zero t’s on one slip plane, and 4 cases with one zero t on three slip planes

10

42

n=3

11

12

JZ=4

3080

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et al.:

STABILITY

for any non zero symmetric x and any CT.Here x is any stress state in the vicinity of CT.The strict convexity of any equipotential implies that (a - o*):i(a)

3 0

(5)

for any u * located on the same equipotential as cr, i.e. f (cr*) = f (CT),and equality applies only when o * = cr. Double dots indicate contracted products of tensors. We now introduce the following function @(ci, (T)=f(&) -f(cr)

+ (o - &):P(cT)

where 8 and crare two stress states. There is a positive 1 such that ci = Aa*, where (T* is on the same equipotential as cr and therefore @ can be written as @(8,a)=I”/“‘f’

f(a*) >

-f(a)

m+l

a:i(o)

where equation (5) was used, and where the equality applies only for d * = g. The function of d on the right hand side of the above inequality is positive, and is zero only for A = 1, i.e. for u = a *. As a consequence, Cp is positive, and @ is zero only for 6 = CT.This demonstrates the following extremum prin~~pie, given by Hutchinson [14] without detailed proof: > ci:i(a) -f(S)

where equality applies only if there is a a such that <* is proportiona to i. It follows that

and
(i) there is a unique stress state c associated with a given i, and (ii) B is proportional to e* if C: is proportional to 6*, equality in equation (7) applies only if a*(t:*) is proportional to o(C). Equation (7) can thus be used to determine a cr* in the direction of u associated with a given i with a four-variable maximization procedure, and the precision of the result can always be tested by comparing the left hand term of equation (7) to unity. The four parameters can be the four @angles introduced by Gilormini et al. [21] in the five-dimensional stress space defined by Lequeu et al. 1221,where a stress state is defined by the vector

There is an immediate geometrical interpretation of equation (7) in this five dimensional space where a vector can also be associated with a strain rate tensor by using the above equations: the scalar product Z .c’* is equal to ]&I lL*l when i* is parallel to i, i.e. when (r* is in the direction of 6. When a set of @ angles has been obtained which fulfills equation (7), a unit vector a* is found in the direction u and the problem is finally solved by using the relation

(6)

for any 8, where equality applies only for ci = cr. The application of this principle to a and 6 in turn leads immediately to the uniqueness of the stress state associated with a given strain rate. [The uniqueness of the strain rate associated with a given stress state is evident from equation (3).] This principle is usually used to find the stress state associated with a given strain rate, since it is not possible to solve equation (3) directly when m # 1. The maximization procedure must be carried out numerically and involvesfiue variables which are the independent components of a. A simpler method is proposed here, which involves just four variables, since it is only necessary to determine the direction of a, which depends on the direction of the prescribed C, and its magnitude can then be deduced easily from the magnitude of i. From the following relation

2i:i* -3 P

TEXTURES

+ (ET- la*):i(a)

ml”‘“‘+’ - (m + 1)a + 1

o:C(a) -f(o)

OF TORSION

(7)

where C = (2i :i/3) ‘I2 denotes the equivalent strain rate, and the equality again applies only if i* is proportional to i. Since it has been shown above that

(T=-

i*(u*)

m rJ*

(8) i > ( which results from equation (3). The method outlined above was employed in the uniform strain rate (Taylor) calculations described below. In the case of “relaxed constraints”, when part of the strain rate and the complementary part of the stress applied to the crystal are specified, the above mentioned angles cannot be used because some of the ui are zero in the sample system. So in the RC case, the ui themselves can be used as variables in the maximization procedure. 2.2. The shapes of the equipotentials and closed subspaces It was mentioned above that the shape of the equipotential is determined only by the m value, and Fig. l(a) to (f) show six two-dimensional sections in the present stress space for m = 1, 0.2, 0.1, and 0.0% When the crystal symmetries are taken into consideration, these six sections are sufficient to describe the 10 sections in the subspaces generated by any pair of the coordinate axes. One equipotential is shown for each value of m, namely for f(u) = z,-jom/(m + I), which means that they all correspond to the same value of the rate of plastic work, T,,&, .The relation to the rate insensitive case (shown as broken lines) is clearly observed; the curves in Fig. 1 all tend to the Bishop and Hill [23] locus for a critical resolved shear stress equal to r0 when m tends to 0. This can be proved as follows for any equipotential, any stress direction and any value of Ij,. Let a stress direction

T6TH

et al.:

STABILITY

be chosen and CJbe the stress state corresponding to a value fof the stress potential associated with a rate sensitivity equal to m. There is a positive I such that a = I.cr*, where CJ*is located on the Bishop and Hi11 polyhedron, i.e. Irnf,o:) < to and equality applies only for a maximum of eight values of s which correspond to the activated systems. The definition of the stress potential, equation (4), leads to the following value for i -pm + ,) Tm + 1 f

OF TORSION TEXTURES

308 1

It can be verified that 1 tends to 1 when m tends to 0 for any value off, and this can be done for any stress direction. As a consequence, D tends to CT *, and all the equipotentials tend to the Bishop and Hill polyhedron. For m = I, the equipotentials are spheroidal and given by

in the present five-dimensional stress space. The above considerations suggest that the type of function corresponding to an equipotential can be used to

d)’

3082

Tt)TH et al.:

STABILITY OF TORSION TEXTURES

Fig. 1. Sections of the equipotentialf(u) = (m/m + I)r,j,, in (a) the n-plane (u, , uZ),(b) the (a,, UJ shear stress plane [(u), us) and (Us, us) sections identical], (c) the (u, , us) plane, (d) the (ur, u5) plane (e) the (a,, ug) plane ((u, , u4) section identical), and (f) the (a,, a,) plane [(ur, ~~4)section identical]. The curves associated with m = 1, 0.2, 0.1, 0.05 are located at increasing distances from the center of each figure. Broken lines indicate the respective sections of the Bishop and Hill polyhedron. The values of 7, and j0 are taken equal to 1.

fit the yield surface of a rate insensitive polycrystal in a manner similar to that employed by Lequeu et al. [24]. It should be noted that the spheroidal equipotential for m = 1 has the same shape as the fitting surface used by Montheillet et al. [S] in the special case a = /I (in their notation). Not all the sections shown in Fig. 1 are closed. The

concept of closed subspaces in stress space was introduced by Canova et al. [8] and means that the strain rate (or at least one of the possible strain rates if not unique) associated with any admissible stress state in a given section is also located in this section. The closed subspaces which can be defined by sets of coordinate axes in the five-dimensional stress space introduced above can be readily determined by using equation (3). They are the following for m # 1: (i) the 3 four-dimensional subspaces defined by the n-plane (a,, +) and two shear components; (ii) the 4 three-dimensional subspaces defined either by the n-plane and one shear component or by the three shear components; (iii) the 6 two-dimensional subspaces defined by the rc-plane, by u5 and either u, or ur, or by two shear components; and (iv) finally, 5 one-dimensional subspaces consisting of the coordinate axes. By contrast, because of the spheroidal shape of the equipotential, any subspace defined by a set of coordinate axes is closed for m = 1. As a consequence, the sections in Fig. l(a) to (d) are closed, and the sections in Fig. l(e) and (f) are closed for m = 1 only. It is also of interest that Fig. l(f) is the only non closed section in Fig. 1 for the rate independent Bishop and Hill yield locus.

2.3. Relation between the material, glide and lattice rotation rates It will now be useful to define three rotation rates which are involved in the analysis of texture evolution. (1) The rotation rate fl with respect to the laboratory of material lines which are parallel to the principal axes of i. This is given by fi=d--i

(10)

where h is the gradient of the velocity of the material with respect to the laboratory. In what follows, fi is referred to as the material or rigid body rotation rate. (2) The rotation rate of the same material lines with respect to the crystal lattice. This is called the plastic spin or glide rotation rate, 6, and is defined by cj;,=g-_i

(11)

where g is the gradient of the velocity of the material with respect to the lattice. It is induced by the crystallographic slips, as expressed by

(3) The rotation rate of the lattice with respect to the laboratory h. Known as the lattice rotation rate, it is the rotation rate of principal interest in the study of texture evolution, and it can be deduced and (2) above because fl=ti+n

from (1)

(12)

TdTH

et

al.: STABILITY

i.e. the material/laborato~ spin is the sum of the material/lattice and lattice/laboratory spins. The elements of the h tensor can be calculated from relations (I I) and (12), which can be combined to give

It should be noted that not all the components of & are imposed or known beforehand (especially not in the RC deformation mode). However, if the orientations of at least one specific direction and one plane normal are kept constant in the laboratory system during a test, then & is compietely defined because of its antisymmetric nature. For example, in the case of simple shear, when the orientations of shear plane 2 and shear direction 1 are kept constant, ri has three nul prescribed elements: &, = d,, = & = 0 (in laboratory coordinates) which are sufficient for the calculation of fi. During a jinite time increment &, the lattice rotation SR cannot be deduced from h&, which is not orthogonal. The method used in the present work to solve this problem is similar to that described in Ref. 1251. 2.4. Absence of glide rotation when m = I In the case of { 1I I)( 1IO} slip, the Schmid factors are such that the following summation has a nul value for 63 sets of ijkf (in the crystal system)

In the remaining relations apply:

18 cases, the following

C m;m;, = 413. 1 m;rnZ = -213 \ b

if

simple

i #j,

and xm;tmF, = -213

if

i #j

(no sum over i and j). When introduced in equation (3), and for m = 1, the above relations lead to

t3F TORSION

TEXTURES

3083

(no sum over i and j), for i #j and for any stress state. (This result was obtained previously, but in a more complex way, in Ref. [26].) As a consequence, when m = 1, the Iattice rotation rate 0 is equal to the material rotation rate fi for any crystal orientation. When a Taylor model is used, with the hypothesis of a uniform velocity gradient throughout the polycrystal, there is therefore no evolution of the crystallographic texture. In the cases of rolling, axisymmetric tension and compression (fi = 0), the pole figures remained fixed as long as G;,= 0, as indicated above. By contrast, in the case of torsion, the material rotation rate j has two non-zero components & = -& = $12, where y > 0 is the shear rate applied in the z plane and along the Q direction. Thus, given again that LJ = 0, each grain is simply rotated around the r axis in the sense of the shear. Similar conclusions have been drawn for the m = 1 case by Canova et al. [20] and Harren er at. [9]. Thus one complete rotation (2~) of any given grain requires the application of a torsional shear strain y = 47-c.Moreover, the invariance of the torsion test with respect to a rotation of the specimen by n about the r-axis implies that the macroscopic shear stress and axial stress (if any) vary cyclically with a period equal to y = 211(see for instance Ref. [26]). 3. ORIENTATION STABILITY The results described above show that there are no stable orientations when m = 1. By contrast, when 0 < m < 1, as observed experimentally, apparently stable orientations are found which are generally labelled Al A*, B, C [4, .5] (Table 3). In this section, the primary factors affecting orientation stability under rate sensitive conditions are introduced and discussed. The following convention is employed for the reference system: directions 1, 2 and 3 are the shear direction, shear plane normal and radial direction, respectively. This system was used by Van Houtte and coworkers 128,291 in their ODF analysis. In this paper, however, for reasons of clarity, these directions will also be referred to as 0, z and r, respectively. 3.1. Effect of’ rate sensitivity on lattice rotation

and

“o=II!(;-6,)c, YO

(14)

(no sum over i and j). Another consequence of the above relations is that, for m = 1, the glide rotation rate is zero, as given by

In this section, the effect of rate sensitivity on the glide and lattice rotation rates corresponding to the torsion ideal orientations is considered. The results pertaining to fixed-end (full constraint) conditions are given in Table 4. It can be seen that, for each ideal orientation, the lattice rotation rate increases, while the glide rotation rate decreases with rate sensitivity. It is also evident from the slip distributions that these differ more and more from the rate insensitive (Bishop and Hill) case as nz is increased. Under rate sensitive conditions, all 12 slip systems operate, although some of them only produce very small shears (for low values of m). As m is increased further, the slip distribution becomes more and more even among

Ti)TH et al.:

3084

STABILITY

OF TORSION

TEXTURES

Table 3. Experimentally observed torsion ideal orientations for positive torsion. Slip system indices refer to Table 1 (negative sign means negative shear in the slip system). Bishop and Hill stress states are given according to the notation of Kocks et al. [Z’?]

Orientation

Shear plane

Shear direction

A

cl!0

proi

R AT A: 3 B

llf:;

t:::;

i;::; (iiz) (100)

C

&

Active slip system indices for m = 0

Slip ratios in case of two systems for M =o

BishopHill stress state, no. of active vertices

3 2,3 3

I:1 -

facet ( 16) facet (16) 2A edge (8) 2A edge (8) 2B edge (6) 28 edge (6) 28 edee (6)

35.26 180

4 35.26 45 35.26

42 45 45 0

Qil] t:::;

144.74 0

45 54.74

0 45

-8, -9 3, -12

I:1 3:l

lOTI

180 90

54.14 45

45 0

3, -12 -4. -10

3:l I:1

0

the slip systems. These observations are also discussed in Ref. [ZO], with some examples. The effect of slip distribution on the lattice rotation rate can now be explained in terms of its influence on the glide rotation rate. The rigid body rotation rate fi is independent of m and also independent of grain orientation (under full constraint conditions). It follows that the lattice rotation rate, which is the difference between the rigid body and glide rotation rates [equation 13(b)], is controlled by the glide rotation rate, which is rate-dependent; an increase in m leads to a decrease in the glide rotation rate (see Table 4). The relation between the glide rotation rate and the slip distribution can be made more evident by employing the rotation rate vectors corresponding to individual slip systems. For example, when only one slip system operates (which is possible only for m = 0), all the crystallographic shears are concentrated into a single slip system, producing a given amount of glide rotation around the b x n axis (this is also the axis of the rotation rate vector). As m is increased, shears on the other slip systems appear and increase and the shear on the original (m = 0) slip system decreases. Each of the slip systems of interest produces a rotation rate component, the resultant of which is the overall glide rotation rate &. Because the b x n vectors are all differently oriented, it is apparent that more even slip distributions lead to smaller glide rotations; furthermore, given that fl is constant, an increase in m leads to an increase in the lattice rotation rate, ti. Finally, we can conclude that, as long as m #O, none of the torsion ideal orientations listed in Table 4 is permanently stable. However, below a certain limit, which is about m = 0.05, the lattice rotation rate is so small that individual grains can remain near particular ideal orientations over appreciable strains, and the

predictions obtained from the rate insensitive analysis (Bishop and Hiti) are at least approximately valid. 3.2. Orientation stability in Euler space under rate sensitive conditions In the previous section, the question of orientation stability was examined at specific points in Euler space. Although all the ex~rimental ideal orientations were studied, it is at least conceivable that other stable orientations could appear in calculations pertaining to rate sensitive conditions. In part for this reason, the examination was extended to the complete Euler space. In order to represent the results in a readily assessible manner, it is useful to introduce an orientation stability parameter S, which is defined as follows

Here &, Qt,& are the Euler angles introduced by Bunge, &is the von Mises equivalent strain rate, and Ih(~,,#,b12,~)I=(h:,+n:,+ht*)“*

(16)

is the absolute value of the lattice rotation rate. It is clear from this definition that S is high when the corresponding orientation is particularly stable. The only instance when S is undefined is when ]h] = 0. We will see below, however, that this does not occur in simple shear under the present rate sensitive conditions. The smallest possible section of Euler space required for the description of torsion testing in the case of cubic metals corresponds to [30] #, = 0” + 180”,

Cp= 0” + #,,

rjz = 0” -+ 90

with #, = arc tan(l/cos 4,) if & G 45”, and 4, =arc tan(l/sin &) if (p2> 45”. In the ODF in-

Table 4. Glide and lattice rotation rates for the torsion ideal orientations with different rate sensitivities for the full constraint deformation mode. The rotation rates are normalized by the van Mises equivalent strain rates. The rigid body rotation rate is 0.866 Glide rotation rate

Lattice rotation rate Orientation

m = 0.05

m =0.125

nl =0.2

A,A

2.348-6 2.068-6 2.83E-6 3.3 I E-6

0.0097 0.0090 0.0116 0.0134

0.079 0.071 0.088 0.102

A:,A: 8, B c

m=l 0.866 0.866 0.866 0.866

m = 0.05

m =O.lZj’

m = 0.2

0.866 0.866 0.866 0.866

0.856 0.857 0.853 0.853

0.778 0.795 0.778 0.762

iZ=l 0 0 8

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Levels = 0.2 0.4 0.6 0.8 1.0. 1.2 1.4 1.6 1.8 2.0 2.2 2.4

3085

@l

Fig. 2. Orientation stability map (OSM) in Euler space for fixed end torsion. The tl tube extends from C (top right), through B, A:, A, AT and B to C (bottom left). The centro-symmetric a’ tube extends from C (top left), through L?,A :, A, A _? and B to C (top right).

vestigations carried out by Van Houtte and coworkers [28,29], the range of 4, was 180”. In our orientation stability investigations, however, the range of 4, was extended to 360” because the A, 2 and B, B orientations, taken singly, are not centrosymmetric, i.e. not identical [see Figs 2 and 3(a)]. Taken together, they are centro-symmetric, although the individual orientations were implicitly treated as

centro-symmetric in Ref. [29]. Because of symmetries, the extension of the range of 4, is not necessary for the description of ODF’s; it is nevertheless useful for the visualization of the tubes and for a full understanding of texture transitions in the & = 45- plane (see section 3.5 below). In this section, the orientation stability results pertaining to fixed-end torsion are presented. The

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(a)

(b)

Fig. 3. (111) pole figure representation of ideal orientations and OSM tubes. (a) The nomenclature of single ideal orientations and fibres; (b) appearance of the a OSM tube along the skeleton line in Euler space for fixed end torsion. The numbers 1, 2, 3, 4 indicate the starting positions of the four (111) plane normals (a C orientation); the same numbers with primes denote the terminal points.

observations that apply to other cases will be published elsewhere [lo, 111. In order to represent earlier experimental results on Cu at high temperatures [4] and to emphasize the effect of rate sensitivity (as well as to lessen the acuity of the textures), the m value chosen was 0.125. The stability parameter S was calculated by holding & constant in 5” increments (only every third & section is presented here for simplicity) at points separated by 2” along the 4, and 4 directions (95,760 points in the full representation). The iso-S value map obtained in this way is illustrated in Fig. 2. This will be referred to here as “orientation stability map” (OSM). In this OSM, the isovalue curves form “tubes” in Euler space. These tubes contain the ideal orientations determined above, which are also given in the form of a (111) pole figure [Fig. 3(a)]. Here, the two well-known torsion texture fibres; {11 l}(uuw) (the “A”-fibre) and {hk[}( 110) (the “B”-fibre) are also plotted (broken lines). Figure 2 clearly shows the regions where the orientation stability is high, i.e. the spaces where grains can remain for appreciable times. It is therefore expected that these regions will be the ones most populated by grains. It then follows that this map will have some similarity or relation to experimental or predicted ODF’s. If the OSM presented in Fig. 2 is compared with the experimental ODF published by Van Houtte and coworkers [28,29], the similarity is obvious. There are important differences, however, between ODF’s and OSM’s. For a given rate sensitivity, the latter isfixed and independent of strain (if there are no latent hardening or grain shape effects). By contrast, in an ODF, the intensities of the idealized texture components and fibres evolve along a given strain path. This question will be studied in more detail in section 3.4. and 3.5. The strain dependence of texture formation in

Euler space can also be examined by analysing the rotation field. This method was employed by Clement and co-workers [31-331. They studied the formation of rolling textures on the basis of the connection between the rotation field and the continuity equation of orientation flow. With respect to the rate sensitive analysis of torsion testing, we have also studied the rotation field, which can be obtained by evaluating the lattice rotation rate vector at any point g = g(d, , #J,&) in Euler space. The rotation field Ag was calculated at 966 points in two series of sections of Euler space and for an increment of shear and Ay = 0.05. These are the & = constant & = constant sections, where the $r~ and 4,-$ components of Ag are plotted, respectively (see Fig. 4). It can be seen from this figure that the arrows are oriented mainly in the direction of decreasing 4, (right hand side) or parallel or opposite to the direction of the & coordinate axis (left hand side). It follows from the definition of the coordinate system that, if there is no change in the 4 coordinate during a rotation increment and the vectors are exactly parallel to the 4, axis in Fig. 4, then the orientation change takes place solely around the r axis of the specimen. Moreover, if an arrow is oriented opposite to the 4I axis, then the rotation rate is parallel to the rigid body rotation rate. In the cases where the rotation vectors are located outside the tubes and their 4,-$ components are small (especially above the tubes), the & component (which can be seen in the left hand side of Fig. 4) is always large and the arrows are oriented in the direction of the decreasing 4 2. 3.3. Description of theJibres In Fig. 2, two kinds of continuous fibres (here entitled the CIand a’ tubes) can be distinguished. In

TOTH et al.:

Levels = 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

42 ~0

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STABILITY OF TORSION TEXTURES

+1 30 ,...-..___a illlIt -

60

90

0 1-

30

60

* I --... t

90 . I ---

120

150

t

.__

180

Fig. 4. The rotation field in the case of fixed end torsion. The scale of the rotation vectors is three times higher than that for the df’s in order to make the arrows more visible.

order to visualize these tubes as they appear on (111) pole figures, the traces of their four (111) plane normals produced by motion along the skeleton lines of the OSM tubes are presented in Fig. 3(b). The c1 tube starts at the C orientation [the four reflections are identified as 1, 2, 3, 4 in Fig. 3(b)] at the Euler point (270”, 45”, 0”); it then goes to the B orientation along the B fibre. Here, it leaves the B fibre and merges into the A fibre passing first through the A f orientation, After leaving A:, it follows the A fibre towards 2, but does not puss t~~~~g~ it, tracing instead a path 1” from A in the direction of B. It is important to note here that there is a strong maximum in the OSM (see Fig. 2) which extends from A to B (and also from A to B, by symmetry), in which the maximum value of S is located about I” from the A (and A) orientations. After leaving the neighbourhood of the 2 orientation, the c(tube continues along the A fibre through

the A f orientation; finally, it passes from the A to the B fibre, going through B, and terminating at the C orientation. [The latter four reflections are identified as l’, 2’, 3’, 4’ in Fig. 3(b).] The second tube, the 01’,is simply the dttube rotated around the r axis of the (I 11) pole figure by 180”. It also starts at a C orientation, but at the Euler point (90’, 45”, 0”); it then passes through the ideal orientations I?, A :, A, A: and B in turn, ending up at C. The CI’tube is not plotted in (I 11) pole figure form in Fig. 3(b) as it can be obtained from Fig. 3(b) by a rotation of 180’, with the 2 and B orientations replaced by A and g, respectively. Fig. 3(b), taken in conjunction with Fig. 2, clearly shows that the A and B fibres belong to a single orientation tube. The transition from the B to the A fibre takes place between B (or i?) on the B fibre and A: (or A:) on the A fibre; note that the A orientation is completely bypassed in this process [see Fig. 3(b)]. Another

T6TH

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sense of shear

Levels 1,2,4,7, lo

Levels1.2.4.7,10

Inlax=10.5

lmax =

A

15.5 6

Fig. 5. Simulated fixed end torsion textures for (a) low (y = 1) and (b) high (y = 12) strains (m = 0.125).

important observation is that the B fibre is not continuous between the A and A orientations [see also Ref. [25], Fig. 9(b)]. The OSM of Fig. 4 indicates that the most stable part of Euler space is fairly extensive, and is located between the A and B ideal orientations. This result is in good agreement with observed textures in Al and Cu heavily deformed at high temperatures. These materials displayed macroscopic rate sensitivities of about 0.125 and, at large strains, the pole figure components became stabilized between these two orientations [4]. It is also of interest that the grain orientations with the highest S-values (A /A and B/B) correspond to cases where most of the slip is concentrated on a single system. From the OSM investigations, we can conclude that the introduction of rate sensitivity does not lead to additional ideal orientations beyond those predicted on the basis of a rate independent analysis. Another important conclusion is that OSM’s, as constructed by the present method, can provide much useful information about fibres and texture development in addition to that obtained from experimental or theoretical ODF’s. 3.4. Examples

of simulated

textures

The present rate sensitive model was applied to the simulation of the textures developed under fixed end conditions. Eight-hundred grains were considered, the positions of which were selected by a random number generator at zero strain. Uniform strain rates were applied to the grains, the non-zero shear rate component was i,, = iZO= j /2. The applied shear rate was i = 0.05, and the constants z,, and v,, were taken equal to 1. Rate sensitivities in the range 0.054.2 were investigated. However, the results shown below correspond to m = 0.125, as this was the value employed to derive the OSM; it is also approximately equal to the value observed [4] in the experiments that

we wanted to simulate. The calculations were carried out for positive torsion up to y = 12 shear strain. The texture predictions corresponding to low (y = 1) and high (y = 12) strains are presented in (111) pole figure form in Fig. 5. It can be seen from Fig. 5(a) that all the ideal texture components are present at a strain of y = 1, but with different intensities. It is of interest that the A: ideal component is considerably stronger than the AT. It is also clear from Fig. 5(a) that every component appears to be rotated in a sense opposite to the shear by about 5 degrees. This rotation is also observed experimentally [4, 12, 131 as well as in earlier simulations [9, 341 and its origin will be discussed in section 3.5. At large strains [y = 12, see Fig. 5(b)], the strengthening of the C and AT texture components is observed. The A; is also present, but at a much lower intensity than the A T . This indicates that the A : and A: intensity levels are interchanged with respect to the small strain case. Another important observation is that at large strains all the observed texture components are now rotated in the same sense as the shear by approximately 2-5”. These observations are again in accord with experiment [4], and an explanation of this behaviour will be advanced in the next section. 3.5. Discussion of texture formation In the case of a randomly textured polycrystal, the Euler space is fairly evenly populated by grains (the population density is exactly uniform when the orientation increments are taken as d4,, d cos 4, d&). When texture evolution begins, most of the grains are located outside the two OSM tubes, tl and a’, described above (Fig. 2). When deformation begins, grains will rotate quickly to one of the tubes, although not necessarily to the nearest one, because a common rotation direction is favoured in simple shear: i.e. rotation around the r axis of the specimen

TOTH

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STABILITY

in the sense of the applied shear, as shown in Fig. 4. Grains which are external to tubes are not at recognized ideal (or symmetrical) orientations; accordingly. multiple slip conditions generally apply and the glide rotation rate is therefore generally quite small. Under these conditions, the lattice rotation rate is governed mainly by the rigid body rotation rate 8. Now [I is constant, and corresponds to a rotation around the r axis of the specimen in the sense of the applied shear. This means that an initially homogeneous “cloud” in Euler space formed by individual grains starts moving in the /i direction, which is the direction of decreasing 4,. Not every grain follows this motion exactly, of course. Nevertheless, it can be shown that the average lattice rotation rate, which is made up of all the individual rotation rate vectors, is exactly parallel to the Y axis of the specimen.? This average has the highest value when the texture is random and decreases as the grains approach the OSM tubes i.e. as the texture develops. The initial 5 degree rotation of the predicted (11 I) pole figure [Fig. S(a)] in the sense opposite to the shear can now be explained in terms of the “cloud” model of texture formation introduced above. As plastic deformation begins, the initially homogeneous cloud starts to move to the left (decreasing 4,) in Euler space (Fig. 4). The grains to the left of each skeleton line move into regions of increasing h and are therefore accelerated; by contrast, those to the right move into regions of decreasing h and are decelerated. This process leads to an unsymmetrical population around the skeleton line of the OSM, and is the reason why initially the right sides of the tubes will be more densely populated. Such a process of grain concentration on the OSM can be converted into the equivalent process on a pole figure by noting that a displacement to the right in Euler space (increasing 4,) corresponds to a rotation in the sense opposite to the shear on a pole figure. In Fig. 5(a), (y = I), this process takes place around the r axis, and leads to the observed counter-clockwise “tilts” of the ideal orientations. This type of model also explains the intensity differences pertaining to the A F and A: components in Fig. 5(a). It is clear from Fig. 4 that the AT catchment area (i.e. the space to the right of the A: skeleton line in Fig. 4 which extends to the A: skeleton line) is greater than the A: catchment area tlf a G grain is located at the point (4,) 4, &) in Euler space then, because of the centro-symmetry of simple shear. there must bc another grain, say G’, located at the point (4, f 180 (4, &). We now apply the same strain rate to Ihe two grains; this means that the absolute values Ifl(‘I, /0” 1 of the lattice rotation rates will be the same. Because of the centro-symmetry, the directions of the twc~ vectors R” and R” must be symmetrical with respect to the r axis of the specimen. It therefore follows that the resultant lattice rotation rate vector must be parallel to the r axis. By extending this conclusion to all pairs of grams, we can see that the average lattice rotation of a polycrystal takes place around the r axis of a specimen.

OF TORSION

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TEXTURES

(i.e. the space to the right of the A: skeleton line which extents to the C skeleton line). It then follows that more grains will be captured by the A: orientation than by the A T (when the orientation distribution is initially uniform). Furthermore, the differences in shape between the right sides of the AT and A f tubes will accentuate this effect. At strains beyond 7 = I, the rate of texture development slows down, but does not stop. The asymmetrically located “clouds” continue their motion in the sense of the shear, but very slowly. (The speed depends on m as discussed in section 3. I .). At a strain of about y = 3, the clouds arrive at their positions of symmetry. This value for ;I agrees with the one observed in experimental textures by Montheillet et al. [4]. At strains 7’ > 3. the clouds slowly leave their ideal orientations, rotate in the sense of the shear, and develop the frequently observed opposite tilts [4], see Fig. 5(b). Because of this process a part of a tube can even disappear at large strains. This is the case for the At component. As a result of the lattice rotation around r, the grains from the A: area move towards the C position. This process strengthens the C component, which itself loses grains for the same reason to the AT component. (In a similar way, A f grains, of which there are few, move towards the A* position.) The net result is a weakening of AT, and the strengthening of C and A f This type of texture transition was also observed in the rate sensitive simulations carried out by Lowe and co-workers [9,34,35]. For the present conditions, the C component seems to be the most stable because the tube is at its widest at the C orientation. 3.6. Axial stresses during jixed

end torsion

It is commonly observed experimentally (see Refs [l-5]) and also in all simulated textures [8,9,34-361 that a compressive axial stress first develops when torsion textures begin to form. In experiments, the compressive stress passes through a maximum at a shear strain of about 1.5. Subsequently, the stress decreases to a minimum at a shear strain of about 5 [4], beyond which the axial stress usually exhibits another compressive peak. If not preceded by fracture, it becomes permanently tensile at large strains and high temperatures [4]. The first qualitative explanation of this phenomenon was based on the continuum mechanics of textured polycrystals (CMTP) [5]. In that study, using a smoothed yield surface, the effects of individual ideal orientations were examined and the axial stresses were related to the small tilts of these orientations, anti-shear when compressive and shear parallel when tensile, about their positions of perfect symmetry. In order to clarify the relation between axial effects and texture evolution, an Euler-map of the normal stress component (a,_) was made (Fig. 6), using the present rate sensitive model. In this figure, the range of the & axis has been shifted by 45 degrees from O-90 to 45--135 The positions of the idcal orien-

3090

TdTW et al.: STABILITY OF TORSION TEXTURES

lines are located very close to the isovalue lines of zero axial stress. At the initiation of straining, the axial stress is zero (if the texture is random) because the tensile and compressive areas in Fig. 6 are equally densely pop ulated. When the texture starts to develop, a strong A: fibre is formed initially, together with weaker A; and Bf components (see section 3.4). However, because of the initial anti-shear tilts of all fibres (as accounted for by the “cloud’y model in section 3.4.), the axial stress “tubes” are initially more heavily populated on the right sides. It is clear from Fig. 6 that the axial stress is compressive on the right sides of both the At and A; fibres, and tensile on the right of the B, fibre. The net result is compressive stress because the B, fibre-cloud will usually be overwhelmed by the A: and A, fibre-clouds. As the deformation proceeds, the clouds will tilt to the opposite sides of their tubes, reversing the signs of the associated axial stresses (tensile for A: and A; and compressive for Bf). At large strains, the net result will be tensile because the At and A; contributions will generally overwhelm that attributable to B,. The above schematic picture of axial stress evolution indicates that the initial anti-shear tilt of the texture is responsible for the compressive nature of axial stress at small strains. By contrast, in the studies of Lowe and co-workers [9,34,35], the initial compressive stress was attributed to the axial effects of grains which are not close to the tubes. The further development of the axial stress was then accounted for by the different evolutions of the weights of the ideul At, A; and Bf iibres. Our map, taken in conjunction with the cloud model and the effect of rigid body rotation, indicates instead that the axial effect is closely linked to the evolution of the pole figure tilts, from inclinations which are anti-shear initially, to shear parallel at large strains. Finally, it should be emphasized that the detailed Fig. 6. Map of axial stress (crzz)in Euler space under fixed shape of the axial stress map depends on the rate end conditions. White and shaded areas correspond to sensitivity, m. At low values of m, it will be similar compressive and tensile stresses. The stress values were to the Bishop and Hill (rate insensitive) map. When calculated for a strain rate 1’= 0.05. The values of’~0and $0 are taken equal to 1, the rate sensitivity parameter is m = 1 (linear case) on the other hand, the map will lead to predictions similar to those of the CMTP m = 0.125. approach. tations and skeleton lines of the OSM tubes of Fig. 4. CONCLUSIONS 2 are also indicated. For the ided orientations, the axial stress is compressive only for the A : comThe purpose of this work was to study the stability ponent, tensile only for the A;, and zero for the C, of torsion textures under rate sensitive conditions. A/A and B/B orientations. Concerning the tubes, it The full constraint torsion case was considered and rate sensitivities applicable to high temperature deis clear from Fig. 6. that the axial stress is just tensile along the A/B-A f-B skeleton line. Similarly, the formation were employed. The textures simulated in figure indicates that the stress is zero along the this way were compared with experimental data as B-C-B skeleton line and just compressive along the well as with the results of other theoretical predic&A f-A /B skeleton line. In the following, we will use tions. These calculations have led to the following the notation of Lowe and co-workers [9,34,35] for general conclusions. 1. Under rate sensitive conditions, the number n of the identification of these fibres: their A: fibre is equivalent to the A/B-A:-B; their 8, fibre to the slip systems which are simultaneously active is n = 4, or 6 < n < 12; the cases ~~esponding to 1 < n 6 3 B-C-B; and their A; to the B-AT-A/B. However, or n = 5 are not permitted. The associated yield stress it should be noted that both the At and A ; skeleton

TbTH

ef al.:

STABILITY

potentials are strictly convex, which provides the basis for a four-variable maximization procedure to find the stress state when the strain rate is fully prescribed. 2. When m # 0, not one of the experimentally observed ideal orientations is permanently stable. As the former is increased, the crystallographic slips become more and more evenly distributed by comparison with the rate insensitive (Bishop and Hill) case. This leads to a decrease in the glide rotation rate, and therefore an increase in the lattice rotation rate with nz. 3. The lattice rotation rates were examined in the complete Euler space and an orientation stability parameter was introduced to represent these results in terms of an orientation stability map (OSM). The OSM is similar in form to the ODF of experimental torsion textures [28,29]. The rotation field was also presented in Euler space. It is apparent that a common orientation flow direction is favoured, which is that of the applied shear, i.e. of the rigid body rotation. 4. Textures were simulated up to large strains (;I = 12) with a rate sensitivity m = 0.125 and compared to the experimental results of Montheillet et ul. [4]. ‘The calculated textures clearly show the initial anti-shear and high strain shear-sense rotations, as well as the variations in texture component intensity observed experimentally. 5. Using the rotation field map and the OSM, a “cloud” model of texture development was presented. The latter is particularly useful in explaining the anti-shear and shear-sense rotations observed in the experimental textures determined under fixed end conditions of testing. 6. A map was presented of the axial stress distribution in Euler space under fixed end testing conditions. It is evident that the axial stresses attributable to the A:, A, and B, fibres are highly sensitive to orientation assymmetries associated with tilts opposite or parallel to the sense of the applied shear. The origin of the initial compressive and large strain tensile stresses commonly observed experimentally can be readily interpreted using such axial stress maps and the “cloud” model of texture development. ArknoM,led~~mmr.s~The authors are indebted to Drs R. J. Asaro and T. C. Lowe for numerous stimulating discussions. They are thankful to the Canadian Steel Industry Research Association and the Ministry of Education of Quebec (FCAR program) for financial support. LST acknowledges with gratitude the International Exchange Award granted by the Natural Sciences and Engineering Research Council of Canada as well as the period of sabbatical leave accorded by the Eiitviis University in Hungary. Finally, he is thankful to Dr Brigitte Bacroix for initiating him into the mysteries of the computer simulation of torsion textures. REFERENCES I. D. Hardwick RC'TW

Metal/.

and W. J. McG. 58, 869 (1961).

Tegart, M&I.

scim.

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2. D. E. R. Hughes, J. Iron Steel Insr. 170, 214 (1952). 3. P. A. Portevin, R&we M&all. 67, 761 (1970). 4. F. Montheillet, M. Cohen and J. J. Jonas, Acta metall. 32, 2077 (1984). 5. F. Montheille;, P. Gilormini and J. J. Jonas, Acta me&l. 33, 705 (1985). 6. H. W. Swift, Engineering 163, 253 (1974). 7. F. Morozumi, Nippon Kokan Techn. Rep. No. 4, 67 (1965). 8. G. R. Canova, U. F. Kocks, C. N. Tom&and J. J. Jonas, J. Mech. Phys. Solids 33, 371 (1985). 9. S. Harren, T. C. Lowe, R. J. Asaro and A. Needleman, Phil. Trans. R. Sot. To be published. 10. L. S. T6th, J. J. Jonas, P. Gilormini and B. Bacroix, Int. J. P&icily. In press. II. L. S. T6th and J. J. Jonas, Proc. 8th Int. Conf. Textures ofMaterials, Santa Fe, New Mexico (1987). 12. G. J. Richardson and H. P. Stiiwe, Iron Steel Inst. Rep. No. 108, Discussion C, 131, (1968). 13. J. Gil Sevillano, P. Van Houtte and E. Aernoudt, Z. Metallk. 66, 367 (1975). 14. J. W. Hutchinson, Proc. R. Sot. A348, 101 (1976). 15. G. R. Canova and U. F. Kocks, Proc. ICOTOM7 (edited by C. M. Brakman et al.), p. 573. Noordwiikerhaut. Holland (1984). 16. J. Pan and J. R. Rice, Xt. j. Solids Slruct. 19, 973 (1983). 17. D. Peirce, R. J. Asaro and A. Needleman, Acta metall. 31, 1951 (1983). 18. R. J. Asaro and A. Needleman, Acre me&l. 33, 923 (1985). 19. S. Nemat-Nasser and M. Obata, Proc. R. Sot. A407, 343 (1986). A. Molinari and U. F. 20. G. R. Canova, C. Fressengeas, Kocks, Acta metall. 36, 1961 (1988). 21. P. Gilormini, B. Bacroix and J. J. Jonas, Acta metall. 36. 231 (1988). 22. Ph. Lequeu, P. Gilormini, F. Montheillet, B. Bacroix and J. J. Jonas, Acta mefall. 35, 439 (1987). 23. J. F. W. Bishop and R. Hill, Phil. Mug. 42, 1298 (1951). 24. Ph. Lequeu, P. Gilormini, F. Montheillet, B. Bacroix and J. J. Jonas. Acta mefall. 35. 1159 11987). 25. G. R. Canova, U. F. Kocks and j. J. Jonas, Acta met&l. 32, 211 (1984). 26. G. R. Canova, A. Molinari and C. Fressengeas, in Rheology of Anisotropic Malerials, Proc. 19th GFR Colloquium (edited by C. Huet, D. Bourgoin and S. Richemond), p. 327, CEPADUES, Toulo&e (1986). 27. U. F. Kocks, G. R. Canova and J. J. Jonas, Acta metal/. 31, 1243 (1983). 28. J. Gil Sevillano, P. Van Houtte and E. Aernoudt, Prog. Mater. Sci. 25. 69 (1980). 29. P. Van Houtte, E. ‘Aerioudt and K. Sekine, in Proc. ICOTOM6 (edited by S. Nagashima), p. 337. Tokyo, Japan, ISIJ (1981). 30. P. Van Houtte and E. Aernoudt, Muter. Sci. Engng 23, 11 (1976). 31. A. Cltment and P. Coulomb, Scripta metall. 13, 899 (1979). 32 K. Wierzbanowski and A. ClOment, Crystal Res. Tech. 19, 201 (1984). 33. K. Wierzbanowski, A. Hihi, M. Berveiller and A. Climent, in Proc. ICOTOM7 (see Ref. [14]), p. 179 (1984). 34. T. C. Lowe and R. J. Asaro, Proc. IUTAMjICM Symp., Yielding, Damage qfAnisorropic Solids (1987). In press. 35. T. C. Lowe and R. J. Asaro, Proc. 8th Int. Conf. Textures ofMaterials, Santa Fe, New Mexico (1987). In press. 36. B. Bacroix, Ph.D. thesis, McGill University, Montreal (1986).