The yield surface of textured polycrystals†

OO22Z5096/85 $3.00+O.OO Pergamon Press Ltd.

J. Mech.Phys. SolidsVol. 33,No. 4, pp. 371-397,1985 Printed in Great Britain.

THE YIELD SURFACE G.

OF TEXTURED

POLYCRYSTALS?

R. CANOVA,~ U. F. KOCKS, C. N. TOM@

Center for Materials Science, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.

and J. J. JONAS Department of Metallurgical Engineering, McGill University, 3450 University Street, Montreal, Canada H3A 2A7 (Received 24 July 1984)

ASTRACT THE PLASTICanisotropy of a material is characterized in part by its yield surface. It is shown that conventional

descriptions, based on extensions of the von Mises hypothesis for isotropic materials, are experimentally and theoretically inadequate in many instances. Symmetry arguments are used to derive the dimensionality and extent of the space necessary for representing the yield surface under various conditions of anisotropy. A useful concept is introduced : “closed” subspaces, in which sections and projections of the yield surface are identical and in which, therefore, normality is complete. Yield surfaces of heavily rolled or sheared sheets are derived from a computer simulation of polycrystal plasticity. It is found that even mild textures give rise to significant departures from “oval” yield surfaces: they develop sharp ridges and extensive flats. The anisotropy coefficients for in-plane tension of rolled sheets have been calculated. For torsion testing under fixed and free end conditions, respectively, the axial force and the length change have been calculated, as well as the change in the ratio of wall thickness to diameter.

1. INTRODUCTION THE ANISOTROPY of the mechanical properties after large plastic deformations is usually severe and a problem of considerable technological importance. Its description has attracted a fair amount of attention, primarily from a phenomenological point of view. Even though it is widely recognized that the anisotropy arises from the non-random polycrystalline nature of the materials, only a few papers have actually taken the crystallographic aspect of the problem into account. Among these are : the evaluation of average elastic and yield properties of rolled materials based on experimentally determined orientation distribution functions (ODF) (PARNIBRE and SAUZAY, 1976; VIANA, KALLEND and DAVIES, 1979), the yield characterization of metals with

t Work supported by the U.S. Department of Energy. $ Permanent address : L.P.M.M., Faculte des Sciences, Ile du Saulcy, 57045 Metz, France. §On leave from IFIR-CONICET, Universidad National de Rosario, Pellegrini 250, 2000 Rosario, Argentina. 371

312

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transversely isotropic properties (HILL, 1950; BASSANI, 1977 ; LOGAN AND HOSPORD, 1980), and the interpretation ofmeasured yield loci in tubular specimens on the basis of ideal texture components (ALTHOFF and WINCIERZ, 1972). In the present work, we have used modern texture predictions, based on polycrystal plasticity theory, to derive complete yield surfaces after arbitrary strains. These calculations were carried out as a part of a comprehensive program (KOCKS, CANOVA and JONAS, 1983 ; CANOVA, KOCKS and JONAS, 1984a; JONAS, CANOVA, TOME and KOOKS, 1983 ; TOME, CANOVA, KOCKS, CHRISTODOULOUand JONAS, 1984 ; CANOVA, KOOKS and STOUT, 1984b; CANOVA AND KOCKS, 1984), with the aid of a computer code originally developed at McGill University. Large prestrains of various kinds (tension, compression, rolling, and torsion) were imposed, and subsequently the yielding behavior of the textured polycrystal probed in all directions. Torsion is the most challenging of these cases, inasmuch as the symmetry of the test is here not orthotropic. It is a fundamental question how the test symmetries influence the texture symmetries, and how these in turn determine the symmetry elements of the yield surface. A substantial part of the present paper addresses this symmetry question. While a five-dimensional representation is in general required for a complete description of the yield surface, symmetry properties sometimes allow a representation of all the needed information in one or more suhspuces of the five-dimensional yield surface. Some of these subspaces are “closed” in the sense that the normality of the strain-increment to the yield surface holds within this subspace, and no strain increments outside this subspace occur when there are no stresses outside it. We will derive general rules for a number of common symmetries. Another part of the symmetry problem is the extent of each subspace needed. For example, when there is inversion symmetry of the flow stress, only half of each subspace contains nonredundant information. Another familiar example is isotropy (with inversion symmetry), where a 30” sector in the deviatoric normal-stress plane (the “z-plane”) suffices. The shape of anisotropic yield surfaces has generally been assumed to be ellipsoidal (HILL, 1948, 1950) or a higher-order oval (HOSFORD, 1972; BASSANI, 1977; HILL. 1979), with mirror symmetry across all coordinate planes of stress space. Some calculations of length changes during torsion (HILL, 1948) relied on the orthotropy hypothesis. AS is demonstrated in what follows, actual polycrystal calculations, even with relatively mild textures, exhibit much less regular shapes and often vertices that are not on coordinate axes. It will be here assumed that the anisotropy of the yield surface is entirely due to crystallographic texture. In addition, we will allow, as a separate case, sign-dependence (i.e. a lack of inversion symmetry) of the yield surface due to a Bauschinger effect or active twinning systems. The operational definition of “yield” for which these assumptions are presumed realistic is that of the back-extrapolated flow stress, or a large offset (3 0.2:;). While the symmetry arguments themselves would, in principle, hold for any definition, the “micro-yield-stress” (e.g. offset of 5 x 10 ‘) is more strongly influenced by details of the recent history, as reflected, for example, in mobiledislocation distributions, than by texture (STOUT, MARTIN, HELLING and CANOVA, 1984). As an application we will give a few examples of yield surfaces after rolling and after

Yield surface of textured

polycrystals

313

torsion. For rolling, we show how they can be used for predicting the strain ratio R at various angles in the plane of the rolled sheet, For torsion, length changes, axial forces, and the shrinking of thin walled tubes are derived.

2.

THE YIELD SURFACE

The concept of a yield surface, first stated by VON MISES (1928) assumes criterion of yielding under combined stresses can be given by f(oij)

= const,

that a

(I)

wherefis a function of the components of the stress tensor which defines a hypersurface in stress space. If, according to flow theory,fis proportional to the plastic potential, the relation between the yield stress and the strain-rate is given by the “flow rule” as

where ;Lis a positive factor of proportionality. Throughout this work we will assume that the yield locus is equal to the plastic potential such as to guarantee normality. However, for materials in which the yield surface and plastic potential are not proportional to each other, all our derivations still hold for the plastic potential. Since only deviatoric stresses and strain rates are relevant, it is simpler sometimes to write stress and strain tensors as five-dimensional vectors of the form 0 = (~ir~2;BLJ,cJ4,(TJ, i = (&,E,;&,&,d,).

(3) (4)

The last three components are customarily defined as being proportional to the offdiagonal tensor components (KOCKS, 1970; KOCKS et ul., 1983; TOMI? and KOCKS, 1985), and any arbitrary convention for contracting the three diagonal tensor components into the first two vector components (which are separated off with a semicolon) may be used, so long as stress and strain-rate are work-conjugated : Qij

&ij

=

bkEk.

(5)

It is demonstrated in Appendix 1 that normality holds in this vector space if it holds in the tensor space. Both the vectorial and the tensorial notations are going to be used throughout this work: while the former is more suitable for describing geometrical representations of the yield surface, the latter is more appropriate for discussing its symmetry properties. One of the subspaces we will continually refer to is the so-called “rc-plane” : the projection of the normal stresses on the deviatoric plane, such that the unit vectors for the diagonal components of the stress deviator tensor form a coplanar set at 120” to each other. It is in this space that the two orthogonal first components of the stress vector defined in (3) are represented. Sometimes the term “(7-r)-space” will also be used to designate the three deviatoric diagonal components ; but note that we do not imply principul coordinates when using this term.

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G. R. CANOVA et ~1.

FIG. 1. Schematic three-dimensional representation of a yield surface showing the difference section g3 = 0 and an orthogonal projection de, = 0, both in the (cr. u,} subspace.

between

a

In the following, we will discuss the definition of sections and of projections of a given yield surface. The concepts are sketched in Fig. 1 where an arbitrary surface is represented in the three-dimensional subspace (a,, 02, rr3), together with both its section by and its projection on the {a,, (TV}space. All projections are meant to be orthgonal to the subspace considered. Note that in this particular example the normals to the yield surface at the locus of the section have a component outside the plane {oi, c2). Conversely, while the normals to the yield surface are completely represented in the projection, only two of the three non-zero stress components are contained in it. In general, a sedan (KCEKS, 1970) of the yield surface in the p-dimensional subspace of stresses {CJJ (where k has p values between 1 and 5) is defined as the locus of points obtained by intersecting the yield surface successively by the hyperplanes cj = 0 (j = 1,s # k). For example, a (a,, 02) section is performed by crossing the whole surface by cj = 0, o4 = 0 and cr5 = 0. An orthogonal pr-ejection (BISHOP, 1953) of the yield surface onto the p-dimensional subspace of stresses {ok} along the complementary subspace {oj) G = I,5 # k) is obtained by representing in the subspace {gJ all points for which di = 0 (j = 1,5 + k). For example, in a {pi, (TV)subspace, the projection consists of the oi, ~~ components of stress such that equation (2) makes C, = & = & = 0. In general, the complementary components aj of a projection along (oj,j # k} arc nonzero, just as the ij components associated with a (ck} section. When both section and projection are coincident, however, then all the non-zero components of the strain-rate vector are given by the normal to the yield surface sectioned by that subspace (“yield subsurface”).? We shall call such a subspace “closed”. As we shall see, for a given subspace to be closed, the yield surface (and therefore the texture) has to possess certain symmetry elements. The relation between the symmetry of the mechanical properties and the symmetry of the t More generally, when the yield surface exhibits sharp edges or corners so that the normal is not unique (as in single crystals), the subspace is considered to be closed if it contains at least one of all the possible normal vectors.

Yield surfaceof texturedpolycrystals

yield surface was explicitly taken into three mutually orthogonal planes symmetric positive-definite bilinear plastic potential. This can be written

37s

account by HILL (1948) who, for the case in which of symmetry exist in the material, proposed a form (an ellipsoid in stress space) to describe the in matrix form as

f(a) = Qijoioj = 1,

(6)

where cr is the vectorial representation of the stress (3) and Q is a 5 x 5 positive-definite symmetric matrix. If the intersections of the three orthogonal symmetry planes are chosen as reference axes, Q reduces to the following form

Q=

AC00

0

CBOO

0

OOLO

0

?

(7)

OOOMO 0000

N

where A, B, C, I,, M, N are independent constants. (A, B, C are linearly related to the parameters F, G, H used by HILL (1948) and the relation depends on the coordinate system used in equation (3)) The flow rule (2) can now be expressed as d, = LQijaj.

(8)

It can be easily seen from (7) and (8) that if 6 belongs to the subspace {ol, c2) or {cr3} or {cJ~} or (a,}, the corresponding i will have no component outside the respective subspace, meaning that each of these sets is closed. As will be shown in the following section, this result is only a consequence of the assumed orthotropic symmetry and the implicit inversion symmetry of the yield surface and does not require the plastic potential to be represented by an ellipsoid or any other specific function.

3.

TEXTURE SYMMETRYAND YIELD SURFACE SYMMETRY

When an initially random polycrystal is deformed plastically it builds up a texture that is characteristic of the test applied and invariant against any operation that leaves the displacement gradient tensor unchanged. This property is important in texture calculations because it allows a reduction of the extent of the Euler space in which the initial orientations have to be given (VAN HOUTTE and AERNOUDT, 1976). However, when the material exhibits an initial texture, the final texture symmetry may not reflect the test symmetry only, and in order to treat the texture development in such cases the totality of the Euler space has to be accounted for. The symmetry analysis that we address in this section is a different one, namely: starting with a material whose texture exhibits certain symmetries, we derive the symmetries of the associated yield surface, the closed subspaces in the sense defined in the previous section, and the minimum extent or irreducible section of the stress space that must be considered to obtain a complete description of the yield surface. In what follows it is important to distinguish between the symmetry of the texture

G. R.

376

CANOVAet al

and the symmetry of the yield surface: while the former is related to the crystallographic orientation of the grains and as such refers to a three-dimensional space, the latter relates to the five-dimensional space of the independent stress components. The symmetry elements considered in this work in association with the texture are mirror planes and rotation axes. Inversion symmetry is always present in the texture because only the translation lattice of the constituent grains is relevant, which is by definition centro-symmetric. In the jjield surface, inversion symmetry presupposes that the opposite strain rate is produced when the stress is reversed. To avoid confusion with the inversion symmetry of the texture we will refer to the above property as sign independence of the yield surface. This is to be expected if the yield surface of the single crystal is itself signindependent. Clearly, this is not the case when twinning systems or a Bauschinger effect are present. In‘what follows, results are derived for both sign-independent and signdependent types of yield surface. When sign-independence is present, it reduces by half the extent of the space necessary to describe the yield surface. For example, in the full five-dimensional description, the strain rate associated with a stress vector having cI < 0 will be the opposite of the strain rate associated with the opposite stress state (having o1 > 0) and, as a consequence, the description of the yield surface for values of g1 3 0 contains all the necessary information. A concept that constitutes the core of all the symmetry arguments that follow refers to the transformation properties of stress strain relations under the operations that leave the texture invariant. It may be emphasized that there is an important difference between, on the one hand, expressing the same tensor quantity in two different reference systems and. on the other, describing two tensors related by some transformation operator in the sarlze reference system. In the first case, if a stress state c,, promotes a strain rate 8,, and the rejbence system s1 is rotated by the operation R then in the new cystem .s?, g,, = R’o,,R will be associated with dS2= R’.$,R. In the second case, the promote a strain rate transform of the stress 0:: = Ra,,R’ will not necessarily +$, == R.G,,R’, unless R is a symmetry operator so that it also leaves the texture invariant. F‘or example, the strain rate that is produced by a stress 0

CT

0 / cannot

be derived, for an arbitrary

0 0

-0

0

0

0

texture, from the strain rate produced 0

ff

0’

0

0

0,

0

0

0

by

even though the stresses are related through a rotation of $4 around x3. Another example is that of an isotropic texture which is, by definition, invariant against any rotation, and for which, therefore, the mechanical properties can be calculated simply by tensor rotations ; however, this does not necessarily give rise to a

311

Yield surface of textured polycrystals

spherical (von Mises) yield surface: rotations in stress space are not the same as rotations in real space and the only requirement is that the yield function can be expressed in terms of the invariants of the stress tensor. The basic postulate of the following study is : any geometric operator R that leaties the texture invariant is so that, if 0 promotes C (in a given set of axes), RaR’ will promote a strain rate RiR’ (along the same axes). In what follows, different textures are considered, in order of increasing symmetry. Their symmetry elements are also evident in the pole figures, if the representation axes have been properly chosen. When necessary, a property of the closed subspaces that is demonstrated in Appendix 2 will be used, namely that the intersection of two closed subspaces is itself closed.

3.1. One mirror plane This type of texture develops during tests on initially isotropic materials for which any diagonal and one off-diagonal component of strain may differ from zero (e.g. torsion with fixed or free ends, see VAN HOUTTE and AERNOUDT, 1976). The scheme employed here for analyzing the symmetry properties of the yield surface will be described in some detail for this case. Assume that the distribution of grain orientations (and so the texture) is invariant under a mirror operation. The mirror plane will be considered to be perpendicular to the axis x2, so that the matrix representation of the symmetry operator is 0

1 M,=

0

-1

of an arbitrary

=

(9)

1

stress tensor 0 by this operation -012

011 d

0. 0

0 The transform

0

M,aM;

=

622

is

013

(10)

-g23 033

Similarly

for the strain rate tensor --El2

El1

d’ = M,iM;

=

E22

it3 -&23

/ .

(11)

E33

Now observe that, according

to (1 l), the transform El1 ,$=

0

43

822

0

of a special tensor of the form

E33

gives the same tensor. Uniqueness requires that the stress state must also be identical, which imposes the condition on the related stress (lo), that o12 = 623 = 0. As a result the subspace (71,o13} is closed when there is a mirror plane perpendicular to x2.

378

G.

R. CANOVA

et al.

If now a strain rate that belongs to the complementary namely I0 i=

&

0

0

d,, 3

subspace is considered,

0 the mirror operation (11) transforms it into its opposite, . 0 ’ 0 --&I2 d’= 0 -&, = -i, 0 while 0’ adopts the general form of (10). No further simplification results unless the yield surface is sign-independent, in which case for (--a) to be associated with (-ci), err, @2?s@33r and c1 3 must be equal to zero and, as a consequence, {gl 1, cz3> is also closed.? In what concerns the irreducible section of the space necessary for a complete description of the yield surface, according to (10) and (ll), stress states that are mirror related have cr12and gz3 components of the opposite sign, and their associated strain rates are the mirror transforms of each other. As a consequence, the stress states with, say, crz2 2 0 contain all the necessary info~ation about the yield surface, which reduces by half the extent of {rrr2, (rz3). The remaining space is further reduced if sign-independence is also present. Note that if the mirror symmetry of (9) is superimposed on the inversion symmetry -1 I=

-1 --_I i

(which is always present in the texture), a two-fold rotation axis results : -1

c, = IM2

=

f -1

colinear with .x2 and perpendicular to M,. M, and C2 are then equivalent; one will be evident in the pole figures, depending on the axes chosen.

either

This case is characteristic of (mid-plane) rolling textures ; or, in general, of any test where only the diagonal strain components may be different from zero (VANHOUTTE and AERNOUJBT, 1976). Note that the assumption of two mirror planes implies the third, 7 Note that the “closedness” of the ~mpjementary spaces does not imply that a general stress state can be decomposed into two components, each one being analyzed separately in each subspace. It just guarantees that the analysis can be restricted to a particular subspace when the tensor components in the complementary space are zero.

Yield surface of textured

using inversion

symmetry.

The matrix

representation

for the operators

; M,=

1

is

1

1

-1 M, =

379

polycrystals

and

-1

1

M, =IM,M,

=

1

(12) -1

1

Extending the results derived in the previous section for one mirror plane it can be concluded that now the, subspaces (n, cz3}, (71,c13} and (n,c~,,} are closed. Furthermore, using the property of the intersection of closed subspaces demonstrated in Appendix 2, (z} itself is also closed. This is the case of orthotropic symmetry usually considered. Concerning the extent of the irreducible space, it can be proved, using the same arguments as before, that only l/4 of (oz3, c13, fl12} is necessary for having a complete description of the yield surface in that subspace: any stress state in this subspace can be transformed, by means of a mirror operation, into one having all positive or all negative stress components. Under the additional assumption of sign-independence of the yield surface, (012, oIS>, {~12,~~31 and (g 13, cz3) are also closed, and when these subspaces are intersected, it follows that {c12}, {cJ~~} and (cz3) are closed separately. Furthermore, any subspace formed by the union of one or more of these three and the {z} subspace is also closed. Nevertheless, a 5-dimensional space is necessary for a full description of the yield surface, but now only non-negative values of the three non-diagonal components is added to the mirror are required (l/8 of {a,,, g13, a,,}), when sign-independence operations. When the {oij, i # j} are separate l-dimensional closed spaces, then the normal to the yield surface is colinear with those axes at the intersection points. However, the fact that the z-plane is also closed does not necessarily support a similar conclusion for the intersection with the three diagonal stress axes. For example, the 71plane representation of HILL’S (1948) ellipsoid expressed in the “principal axes of anisotropy”, equations (6) and (7), is an ellipse for which the lengths of the main axes and their inclination are functions of the parameters A, B and C.

3.3. Cubic symmetry Cubic symmetry is a particular case of the problem discussed above and as such the same conclusions apply with regard to the symmetry of the yield surface. It is not a realistic case except for the infinitely sharp (i.e. single crystal) texture of a cubic material. In addition to the previous symmetry elements, it also exhibits a three-fold rotation axis C, in the (111) direction and a mirror plane MI, at 45” with respect to M, and M,. Both are represented in Fig. 2a and their matrix operators are of the form 0 c,=

1

0

0

0

1

1

0

0

and

MI,=

0

1

0

1

0

0 .

0

0

1

(13)

It is clear that the operation of C, permutes the three coordinate axes cyclically while M,, permutes the axes x1 and x2. The combined operation of C, and M,, provides a

380

G. R.

CANOVA

et al.

\

SW\ andependent\ \ \ \

FIG. 2. (a) Symmetry elements associated with cubic crystals (n-fold rotation axes C, and mirror planes M). (b,c) Reduced regions in the {n) and {cz3,0 ,,,u12} subspaces, respectively, which result from the transformation properties of the corresponding stress components under the cubic symmetry operations.

“fan” of three mirror according to :

planes

that intersect

at C,. They transform

tensor

quantities

(14)

and 3 0’

=

M,,aM;,

=

611

I For the following

013

.

(15)

033

analysis, two subspaces are considered, namely {rc} E {a,,, (To*,oJ3} and {cz3, g31r cl2 } ; according to the above results, the former is closed while the latter requires sign independence in order to be closed (in which case {Cam}, {rr3r}, (ark} are

Yield surface of textured

381

poiycrystals

individually closed) but that assumption is not necessary here. Also, as was deduced above, the irreducible section of each subspace is l/2 for {z) if sign-independence is assumed ; and l/8 or l/4 for (aij, i # j> depending on whether sign-independence is assumed or not respectively. Observe that, according to (14) and (15), the corresponding tensor components in each space transform as the components of a three-dimensional vector under the cubic symmetry operations C, and M12. As a consequence, the subsurfaces should exhibit cubic symmetry when represented in those subspaces and the extent of their irreducible sections should reduce, accordingly, to l/6 of the original extent. For {crij,i # j> this means that a fraction l/S x l/6 = f j48, or 1/4x l/6 = l/24 of the subspace is irreducible (like one of the triangles in a stereographic projection of direction space), while for {?-clthe irreducible section is either l/Z x 1/6 = l/12 or l/6, depending on whether sign-independence is assumed or not. The two irreducible sections are sketched in Fig. 2b, c. It is the existence of these symmetry properties which allows the 56 pentaslip vectors (vertices) of the yield surface of cubic crystals to be classified in five groups and to be generated from the knowledge of a representative member in each group (BISHOP,1963 ; KOCKSet ul., 1983). The minimum set of vertices from which the rest can be derived via symmetry operations has been called the “irreducible set of vertices” by TOMBand KOCKS(1985).

This is another case of interest only for the extreme case of single crystals with hexagonal structure. The basic elements associated with hexagonal symmetry are a sixfold rotation axis C,, contained in a mirror plane M, and a mirror plane M, that coincides with the basal plane (see Fig. 3). As was demonstrated in section 3.2, the presence of two orthogonal mirror planes implies the existence of a third plane (M, in this case) perpendicular to both of these.? The repeated operation of Cs produces a “fan” of six mirror planes out of M, and M,. The presence of the orthogonal mirror planes guarantees that the subspaces {R) and (71,aij} are closed as well as the individual (bij, i # j > subspaces if sign-independence is assumed. The effect of the mirror and rotation operations is discussed in general in what follows since the conclusions are valid for any symmetry exhibiting an n-fold axis in the x,-direction and a mirror plane containing xj. The form of the symmetry operators is 1cos #,

C, =

-sin&

0

sin 4”

cos 4,

0

0

0

1

’ cos 2@

and

M=

sin2@ 0

sin 2d) -cos2@ 0

0 0 ,

(36)

1

where (6, = 27c/nand @ is the angle between the mirror plane and the axis x1 (see Appendix 3). $ The existence M,.

of M, also results from the combined

operation

of C, and

M, without

having to assume

382

G.

FIG. 3. Symmetry

The components

R.

=

cJ11+a22 ~-

+

+ozz

with hexagonal

2 011-c22

- ~--

cos4@+o12sin4~,

cos 4@ - o12 sin 4@,

G.;2

=

43

=

0;2

=

a;3

=

g13

cos

43

=

D13

sin 20 - fs23 cos 2@

and the components

2

=

2

(17)

033,

c11--(722

.

sm 4Q, - CT1 2 cos 4@,,

2

2Q + 02x sin 2@,,

of r~’ = C,aCi cfll

crystals.

tensor CT’= MaM’ are of the form 011-022

2 Cl1

et al.

elements associated

of the transformed Oil

CANOVA

arc

~ll+~zz -~

c711-@22

2 +

+

g;2

Cl =-------

0;3

=

g33:

a;2

=

Cl1 __--_

0;3

=

(713 cos

a;,

=

(213

2

CT22

2 011

- ---

-c22

2

cos 24-a,,

sin

24,

cos 24 + cr, 2 sin 24,

(18) 022

9

sin 245+ o12 cos 24,

+c23

sin 4,

sin f$ -t a2 3 cos 4.

Note that neither M nor C, mix the components ofthe {n, G,,) with the components of the {o13, oz3} space. N evertheless this result does not imply that the latter is closed

383

Yield surfaceof textured polycrystals

(unless sign-independence is assumed) because no condition can be imposed over the components of the strain-rate. From (17) and (18) it can be seen that (~rs and (bus transform in the same way as the components of a vector under the corresponding symmetry operations. As a result, the subspace {Ur3, cZ3} exhibits the same “basal” symmetry as the texture. For the particular case of hexagonal symmetry this means that only a section of 30 in the {a,,, oZ3} subspace (l/12) is necessary for describing the yield surface (see Fig. 3). On the other hand, the form of (17) and (18) suggests that the five independent components should be defined such that 01

=

~11+022 2

Cl1 ,

O2=

in which case the transformation

-D22 2

>

(T3=023,

by M adopts

04=g13,

OS=O12

(18’)

the form

0; = 01, 0; = (TVcos 40 + (TVsin 4@,

(19)

f~; = fs2 sin 4Q - us cos 4Q and for C, a; = fJr, 0; = cr2 cos 24 - fr5 sin 24,

(20)

0; = o2 sin 24 + rr5 cos 24. What can be inferred from (19) and (20) is that (iI remains invariant under the action of M and C,, while c2 and c5 transform as the basal components of a vector would do under both a mirror operation with respect to a plane at an angle of 2@ to xl and a rotation of 24 around the vertical axis, respectively. As a consequence one might expect an n/2-fold symmetry around g1 in the subspace (ol, 02, 05). For hexagonal symmetry it implies an irreducible region of 60” in the plane {(a, 1- g1 J/2, or 2}.

3.5. Fiber texture The symmetry associated with a fiber texture (such as would be observed after wire drawing or, in general, after any axisymmetric test on an initially isotropic material, see VAN HOUTTE and AERNOUDT, 1976) may be regarded as one of a mirror plane and an cofold rotation axis C,. The angles Q and Q,in (17) and (18) may adopt any value since n is arbitrarily large and as a consequence the n and n/2-fold symmetries of subspaces {a,, rr4} and {02, c5} increase ad injinitum. Accordingly, the irreducible section in each of these subspaces narrows until it becomes a line, meaning that only one of the two dimensions suffices for describing the yield surface, which now appears as a circle when represented in those subspaces. From the point of view of the stress components this means that it is always possible to find values for the reflection and rotation angles CD and 4 such as to make oi2 and 0; 3 simultaneously zero in (17) and (18). This result is important in that it allows a complete description of the five-dimensional yield surface if its three-dimensional description is known in the space (71,g23}.

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G. R.

CANOVA et al

3.6. lsotro~y In this case the properties of the material are invariant against any symmetry operation and the outcome is already well known. Now, any of the coordinate axes can be taken as an m-fold axis and the results of the above section guarantee that the representation of the yield surface is a circle in any subspace made up of two shear components, like for example the subspaces {(cl1 ---a&/2, cr,,J, (a,,, or,}, {or3, cr,J, etc., resulting in a sphere in the subspace {g2, as, oh, (~~1 if the definition (18’) is used. Note that none of the symmetry considerations involve the normal component CF~, which means that two dimensions are still necessary to represent the yield surface. Since any spatial rotation is allowed, the stresses can always be transformed to a diagonal form in which case the two-dimensional representation of the closed n-plane contains all the information about the yield surface. The symmetries derived for the cubic case are also valid here and the description can be reduced to a section of 30” or 60”, depending on whether sign-independence is assumed or not. 4.

APPLICATIONS

The concepts defined above will now be used to derive the yield surface of polycrystals after large deformations in rolling or torsion. This will be done on the basis of a specific theory of polycrystal deformation; however, the qualitative features were also found using a simple BISH~PHILL (1951) approach. The “McGill-A” code was used to simulate the defo~ation of an aggregate consisting of 800 initially randomly oriented grains. The procedure, which has been thoroughly described by TOMB et ~2. (1984), CANOVA et al. (1984b) and CANOVA and KOCKS (1984), is briefly the following. The grains are assumed to undergo the same strain as the polycrystal (TAYLOR, 1938) so long as they are approximately equiaxed (i.e. their shape does not have a preferred orientation): this is the fully constrained (FC) model. As the grain shape becomes more and more distorted, some of the strain constraints are gradually relaxed (RC approach); namely, those that, over a significant fraction of the volume of the grain, are not constrained by compatibility with a neighboring grain. The calculation is done on the basis of the single crystal yield surface (BISHOP and HILL, 1951), adjusted to treat pth order plastic states (p = 5,4,3), as required by the RC model (KOCKS et al. 1983). Orientation changes are calculated on the basis of certain fixed material planes and lines, characte~stic of the test geometry (KO~ZKSand CHANDRA, 1982). The only innovation contained in this work with respect to the procedure described by TOMB et al. (1984), refers to the method used to resolve the ambiguity in the selection of active slip systems. While previously, we used an averaging approach, here a rate sensitivity principle is employed, which consists in slightly rounding off the vertices and edges of the single crystal yield surface (CANOVA and KOCKS, 1984). Under the normality rule the stress vector is shifted from the vertex (or edge) to the point where the imposed strain is normal to the surface. As a result, the resolved shear stress is not the same for all the systems associated with the active quasi-vertex (or quasi-edge). The with rate sensitivity criterion for choosing the active systems is, in agreement principles, to pick the p independent ones where the strain-rate is highest.

385

Yield surface of textured polycrystals

The increment of plastic work per unit volume that takes place in an aggregate when a strain increment d& is imposed and all the grains are assumed to undergo the same deformation is oij dEij = qj deij, (21) where CJis the macroscopic stress tensor, cg is the stress in each grain that maximizes the plastic work and the bar denotes grain average. The inner envelope of hyperplanes defined by (21) characterizes the yield surface of the polycrystal (strictly speaking an upper limit to it, BISHOPand HILL,1951) and constitutes the basis of the procedure used here for the numerical derivation of the yield surfaces. It can be proved that the average of the crystal stresses aTj for each d&belongs to the yield surface defined by (21). According to the principle of maximum work demonstrated by BISHOP and HILL (195 l), if a crystal is caused to deform plastically through an increment of strain ds, the work done by the required stress by is not less than that done by any other stress bg* not violating the yield condition (oyj - r$) deij > 0. Taking

averages

over all grains, the following (ofj -

relation

(22) is obtained

for the polycrystal,

o$ ) dEij 3 0,

(23)

which is the condition that the vectors of the yield surface have to obey if convexity and normality are to be fulfilled. If the latter is assumed, then gg and 0 must be the same and the locus defined by d coincides with the yield surface of (21) (HILL, 1967). The principle of the present calculations is to apply a strain increment dsij = E: de in the direction co to the polycrystal, which is defined by a collection of grains with characteristic shape and orientation. .Y’has to be normalized for comparison purposes and here this is done with respect to axisymmetric tension, i.e. &;E;

=

312.

(24)

The strain increment is expressed in the principal axes of each grain and is imposed on it, except for the strain components which, according to the RC model, have to be relaxed. As a result, the stress compatible with the strain increment cg = z PM is obtained. Here z is the critical resolved shear stress, p refers to the number of imposed strain components and *M is a dimensionless orientation dependent tensor also called the @h-order plastic state (KOCKS et al., 1983). The density of plastic work (equation (21)) may now be written oij&; = TM E C&f,

(25)

M = PMij~;

(26)

where

is the Taylor factor, defined as the projection direction, and z”==

of the stress onto the normalized

strain

TM

M

(27)

386

G. R. CANOVA et al.

is an average critical shear stress weighted by the Taylor factor. As demonstrated by TOMBet al. (1984), while fis a function of the average state of hardening of the material, it is relatively insensitive to the Taylor factor distribution.+ This quantity
_FThis is trivially true if the critical stress T is assumed to have reached saturation in every grain after large plastic deformation. 1 Sharp corners appear in the polycrystal yield surface even if the vertices in the single-crystal yield surface have been smoothed out due to rate sensitivity. In real materials, one would expect some rounding.

Yield surface of textured

387

polycrystals

(a)

(b) FIG. 4. (a) Reference plane representation

Cc)

system for rolling used in this work and relative orientation of the tensile axis xi. (b) zof the yield surface after a rolling reduction of E33 = -2. (c) Corresponding projection onto the subspace {(u,~ -os3)/2, o,,}.

vector providing softening and therefore flow localization. At this level of thickness strain, grains being very flat, the shear strain c13 is not necessarily continuous throughout the thickness of the sample and grains may choose more “economical” paths, which leads to the appearance of shear bands. A complete analysis of shear banding in rolling, carried out by CANOVA et al. (1984b) using these concepts, has been able to explain the shear patterns and orientations that are characteristic of this type of deformation. Note that, while the appearances of ridges on the polycrystal yield surface is expected for any crystallographic theory, the possibility of local shear band formation is present only in a relaxed-constraints model. Another interesting feature of the subspace representation of the yield surface concerns the evaluation of the Lankford coefficient for the “rolled” polycrystal. The latter is defined as the ratio of lateral strain increments R = d.$Z/dE;3 that take place in

Gw

orientation)

Cm

M

A4 C,//@, 0,1)

M,, M,, M G,//(O,@I)

Ml*

G/AL A 1)

M3

isotropic

fiber {tension and compression : axisym. strain

hexagonal (hcp monocrystals)

cubic (fee and bee monocrystals)

M,, M,, M,

M2,

orthotropic (rolling, plane strain)

MI,

(torsion)

Type of texture

M, (or C,)

Symmetry elements

yes

no

as above

as above

no

yes

as above

yes

___.-

as above

_-~

as above

as above

Closed subspaces

no

yes

no

yes

no

yes

no

Signindep.

TABLE 1.

2D ljh of {rrj 1112 of {rc}

Irreducible space

.---

_____-

sphere for

circle for (a,,, c3i}

hexag. for {a,,, IJ~I )

Special shapes

% R

5 5

P

P

?

Yield surface of textured polycrystals

389

4 012

FIG. 5. Schematic representation

of the yield surfacein the closed subspace {x, o1*}showing the locus defined by the tensile tests done on the plane of the sheet at dierent angles 4.

a bar cut from the rolled sheet at an angle 4 with respect to the rolling direction and subjected to a tensile test in the direction xi. The primed set of axes is indicated in Fig. 4a and it is evident that a tensile stress state in that system will exhibit a nonzero o12 component when expressed in the main rolling axes. As a consequence, the n-plane representation of Fig. 4b is insufficient for a description of this test (unless 4 = 0 or 4 = n/2, in which case the tensile and coordinate axes coincide) and the (12,o12} closedspace representation has to be used. As 4 takes values in the interval 0 < 4 < n/2 the stress states associated with the corresponding tensile tests describe a locus on this subsurface similar to the one shown schematically in Fig. 5. Here we adopt a two- rather than a three-dimensional approach, which consists in evaluating the n-plane projection of the yield surface for different sets of axes xi, xi, xj rotated by A4 = 15” from each. other.? The ratio R is then geometrically derived from the normal to the yield surface at the point where it intersects the a;, axis. This procedure amounts to prescribing FI ai may differ from zero. ‘12. -- 0 and so the corresponding Some of the n-plane representations used to deduce the R values are plotted in Fig. 6 for different values of 4. Those corresponding to 4 = 0 and 4 = 7~12can be inferred from Fig. 4b. The resulting strain ratios R are plotted in Fig. 7 as a function of 4 together with experimental values measured by HIRSCH, MUSICK and LUCKE (1978) in high purity copper cold-rolled to 95% reduction (E~~ = - 3). It can be seen that both the predicted and the experimental values exhibit the same trend, namely : R < 1 for C$= 0, R > 1 for 4 = 7112and a peak at around 4 = 7114.Quantitative differences could, at least in part, be due to the difference in thickness reduction associated with the two sets of points. The usual continuum mechanics approach to this problem assumes in-plane isotropy and is based on a particular form of Hill’s ellipsoid, characterized by a constant R value. Empirical variations on this form having up to seven adjustable parameters were proposed by HILL (1979) and by BASSANI(1977) in order to take into

t The range for 4 is (0, n/4) ; the range (n/4, n/2) can be obtained by interchanging u’,1and &.

G. R. CANOVA et al. * fl,,

FIG. 6. n-plane representations of the yield surface corresponding to a rolling reduction of cjIl = - 2 for the primed systems xi, xi, xi rotated by 4 = 15”, 30” and 45” with respect to the system x1, x2, xj. The Lankford coefficient R is the ratio of the components of the normal to the figure at the intersection with the axis o’, , (or with & for angles n/2 - 4).

account the “anomalous behavior” and in-plane anisotropy, evident in the curves of Fig. 7. On the other hand, anisotropy and “anomalous behavior” are accounted for naturally when a polycrystalline approach (not necessarily the one presented here) is used.

0”

15”

30”

45”

60

75”

90

FIG. 7. Calculated values for the strain ratio R (0) corresponding to a rolling reduction of sj3 = - 2 as a function of the tensile angle 4. Superimposed are experimental values (A) for high-purity copper cold-rolled to ea3 = - 3 (HIRSCH et al., 1978).

Yield surface of textured polycrystals

391

4.2. Results for torsion

This case falls into the category of Section 3.1. The pole figure associated with a torsion test (of an initially isotropic material) exhibits a two-fold axis in the radial direction r, when r is plotted in the center, or a mirror plane perpendicular to r, when r is

placed in the periphery of the pole figure, The reference system used here (shown in Fig. 8a) has the l-, 2-, and 3-axes in the tangential, radial and axial directions, respectively. According to Table 1, (n, cri3j is a closed subspace and, since the yield surface is signindependent, also (cJ~~,~~~~ is closed. A complete discussion of the torsion problem clearly requires a three-dimensional representation of the yield surface in the first subspace, which contains both the prescribed shear direction and the n-plane. In addition to the prescribed shear component e13, the boundary conditions on the specimen imposed by the kinematics (CANOVA, 1982; CANOVA et al., 1984a) are El1 = &zz= -E&2.

(28)

Since no radial dependence is allowed for the strain components in our calculations, they must be regarded as describing the deformation of a thin ring-layer imbedded in a solid bar. In what follows, two different cases will be considered concerning the axial constraints : (i) for a jxed-ends torsion test &33= 0 has to be imposed and in general axial forces ((733# 0) are developed ; (ii) if instead the sample ends are leftfree (flT33= 0) then length changes are observed. Represented in Fig. 8 are three orthogonal projections of the yield surface associated with a fixed-ends torsion test carried out up to &13= 0.5 (y z 1). Figure 8b is a n-plane projection : it resembles a Tresca yield surface and shows a slight tilting with respect to the perpendicular axes (rs3 and (cl1 --cJ~~). Figure 8c is a {os3, CJ~~)projection and displays the axial effects : if the condition de,, = 0 has to be obeyed, the corresponding stress state must include a compressive component cr33 < 0. This compressive axial stress was evaluated as a function of the deformation and is reported in Fig. 9, normalized by the corresponding shear component u,~. The simulation results for a ring layer should overpredict experimental results for solid rods ; this was in fact found in a comparison with the results of MONTHEILLET, COHEN and JONAS (1984) for torsion of Gu and Al rods at room temperature. Nevertheless, the qualitative behavior is very similar. In a free-ends test, the condition g‘33= 0 would give rise to a positive increment de, 3 and thus to an axial elongation of the sample, as can be inferred from Fig. SC. Figure 8d is a projection on the {(a, 1 -0,,)/2, a13} subspace and allows some of the deformation tendencies of a thin walled tube to be predicted, if this is assumed to have undergone the same deformation path and texture development as the cylindrical layer in the solid bar. The boundary condition (28), now has to be replaced by the condition over the tangential and radial stress components 011 =a,,=o.

(29)

Equation (29), together with the condition of fixed ends (de,, = 0), leads to a yield vector along cl3 in Fig. 8d, which promotes a strain increment such that (de, 1 -de,,) < 0 and, since in this case de1 1+de,, = 0, dEll = -de,,

< 0

(30)

392

G.

R.

CANOVA

et a[.

AD

x,=RD

dE

dC

FIG. 8. (a) Reference system for torsion used in this work. (b) z-plane projection of the yield surface corr~ponding to a shear strain of sE3 = 0.5. (c) Projection on the subspace {cz~~,sT~~~,which indicates the dc,,-component when o,~ = 0, and the cr,,-component wh-n ds 33 = 0. (d) Projection on the subspace o,~), which indicates the ratio of diameter decrease to wall-thickness increase when the i@, * -o&/2, hoop and radial stresses are zero.

393

Yield surface of textured polycrystals

the wall thickness of the sample increases whereas its diameter shrinks, a behavior that has been confirmed by experiment (STOUT, 1984). If now the ends of the tube are left free and Fig. SCis used in combination with Fig. 8d, the strain increments must be such that

meaning

that

d& =- 0,

(31)

dEs3 > 0 and, as a result, dq, = -d&Z2 -dej3 < 0 meaning that the diametral contraction is further enhanced by the axial elongation. In order to derive the length changes of a thin ring layer imbedded in a solid bar during torsion it is sufficient to work in the {cs3, 0 13} subspace: the reasons are that {x, c1 3}is a closed subspace and the kinematic condition, (28), projects the yield surface on the {cJ~~,cT~~}space. The “McGill” code was modified to calculate, at each deformation step, the value of dEs3 that would provide a zero average value of crs3 over the constituent grains. The simulation shows a monotonically increasing lengthening, which is observed experimentally during low-temperature torsion of cubic polycrystals (SWIFT, 1947). The results are shown in Fig. 9, superimposed on the axial stress associated with the fixed-ends test. The reason why the two curves exhibit dissimilar

0.25

t t

0

1

2

3

4

5

FIG. 9. Predicted variation with the shear strain s,s of the axial stress osl in a fixed-ends torsion test, and of the axial strain .sss in a free-ends

torsion

test.

394

G. R.

CAN~VA et al.

behavior is as follows: while ds,, is related to the slope of the yield surface at the intersection with the or3 axis (Fig. 8c), the axial stress crs3 is given by the separation of the left summit from the or3 axis. One is related to the tilting of the yield surface, the other to its flattening. Thus it is possible for es3 to exhibit a monotonic variation while this is not the case for 033. The textures developed under fixed and free ends conditions are quite similar but not identical : the components of the texture are the same, the only difference is a greater tilting of the whole texture about the radial axis in the free-ends case.

5.

CONCLUSIONS

A relationship has been established between the texture symmetry and the yield surface symmetry and general properties have been derived for both sign-independent and sign-dependent types of yield surfaces. Emphasis has been put on the notion of “closed” subspaces associated with a yield surface (any stress state in that subspace will produce a strain rate tensor belonging to the same subspace) and on the minimum extent of stress space necessary for a complete representation of the yield surface. It has been proved in particular that whenever the texture exhibits a mirror plane Mi, the subspace {rc,~jjk} withj, k # i (and also {aij, cik} if sign-independence is assumed) is closed. For the orthotropic symmetry associated with rolling textures this result implies that the subspaces {x} and {n, cjk) (also (nij> and (oij, oik) when the yield surface is sign-independent) are closed, independently of the particular shape that the yield surface may adopt. A summary of these relationships is given in Table 1 where, for completeness, also the symmetries of cubic and hexagonal single crystal yield surfaces are described. As an application, the construction of a few n-plane representations enabled us to calculate the Lankford coefficients after a simulated rolling process using the Relaxed Constraints theory for polycrystal deformation, combined with a kind of rate sensitive approach to eliminate the problem of ambiguity in slip system selection. Similarly, for the case of torsion, it is demonstrated how mechanical problems, such as the development of axial forces, axial length changes, or the diametral shrinking of thin walled tubes, are linked to texture problems. In conclusion, yield surfaces calculated on the basis of polycrystalline models differ, sometimes substantially, from the ones that have been postulated empirically; this was shown for one particular theory, but holds generally. These calculated yield surfaces have proved to be well suited for describing the anisotropic behavior of textured materials.

ACKNOWLEDGEMENTS The authors are very grateful to S. S. Hecker, S. Shrivastava and M. G. Stout for stimulating discussions. This work was supported by the U.S. Department of Energy and, in part, at McGill University under sponsorship of the Ministry of Education of Quebec and the Natural Sciences and Engineering Research Council of Canada. One of us (C.T.) was supported by an External Fellowship from the Consejo National de Investigaciones Cientificas y Tbcnicas de la Republica Argentina.

Yield surfaceof texturedpolycrystals

395

REFERENCES ALTHOFF,J. and WINCIERZ, P. BASSANI,J. L. BISHOP,J. F. W. BISHOP,J. F. W. and HILL, R. CANOVA, G. R. CANOVA, G. R. and KOCKS, U. F. CANOVA, G. R., KOCKS, U. F. and JONAS,J. J. CANOVA, G. R., KOCKS, U. F. and STOUT, M. G. HILL, R. HILL, R.

1972

Z. ~e~a~~k. 63,523.

1977 1953 1951

Int. J. Mech. Sci. 19,651. Phil. Mag. 44, 51. Phil. May. 42,414, 1298.

1982 1984

Doctoral Thesis, McGill University, Montreal. fnt, Co& of Textures of ~at~~al~ ICOTOM-~.

1984a

Acta MetalI. 32,211.

1984b

Scripta Met. 18,437.

1948 1950

HILL, R. HILL, R. HIRSCH,J., MUSICK, R. AND LOCKE, K. HOSFORD,W. F. HOUTTE, P. VAN, and AERNOUDT,E. JONAS,J. J., CANOVA, G. R., TOMB, C!. and Kocxs, U. F. KOCKS, U. F. KOCKS, U. F. and CHANDRA, H. KOCKS, U. F., CANOVA, G. R. and JONAS,J. J. LOGAN, R. W. and HOSFORD,W. F. MISES, R. VON MONTHEILI.ET,F., COHEN, M. and JONAS,J. J. PARN&RE,P. and SAUZAY, C. STOUT, M. G.

1967 1979 1978

Proc. R. Sot. A193,281. The mathematical Theory of PZastic~~y,Clarendon Press, Oxford. J. Mech. Phys. Solids l&79. Math. Proc. Camb. Phil. Sot. 85, 179. Textures of Materials (edited by G. GOTTSTEINand

1972 1976

J. appl. Mech., Trans., ASME E39,607. Mater. Sci. ~ngng 23, 11.

1983

J. Metall. d’ilutomne de la SFM.

1970 1982

Met. Trans. 1, 1121. Acta Metatf. 30,695.

1983

Acta Metaff. 31, 1243.

1980

Int. J. Mech. Sci. 22,419.

1928 1984

Z. angew. Math. Mech. 8, 161.

1976

Mater. Sci. Engng 22,271.

1984

Los

STOUT, M. G., MARTIN, P. L., HELLING,D. E. and CANOVA, G. R. SWIFT, H. W. TAYLOR, G. I. TOME, C., CANOVA, G. R.,

1984

K. LijcKE), Vol. II, p. 437. Springer, Berlin.

Acta Metall. 32,2077.

Alamos National Laboratory, private communication. International Symposium en Current Theories of Plasticity and Applications, in press.

1947 1938 1984

Engineering 163,253. J. Inst. Metals 62, 307. Acta Metall. 32, 1637.

1985 1979

Acta Metall. 33,603. Int. 1. Mech. Sci. 21,355.

KOCKS,U. F., CHRISTODOULOU, N. and JONAS,3. J. TOMB, C. and KOCKS, U. F. VIANA, C. S. da C., KALLBND, J. S. and DAVIES, G. J.

396

R.

G.

CANOVA

et al.

APPENDIX 1 The flow rule, as stated in equations (2) of this work, is expressed in terms of the tensor components of stress and strain-rate as

(A-1) wheref(crij) is the yield function (assumed to be proportional to the plastic potential). Since the discussion of the symmetry properties of the yield surface is referred here to the vector space defined by equations (3) and (4), it is necessary to prove that the normality rule also holds for the five-dimensional representation of the yield surface. Assume that the tensor components of stress and strain rate are linearly related to the corresponding vector components through a general equation of the form bij = iii

=

aijmom,

64-2)

(A-3)

Pii”&.

As a result of the invariance condition on the work rate, (5), the linear coefficients in (A-2) and (A-3) must obey the condition %jmBijn

When the plastic potentialf(aij) f*(a,J such that

=

(A-4)

6nm.

is expressed in terms of the vector components it adopts a form

ft”ij)

=

f(aij(gki))

(A-5)

=f*(ak).

Using (A-l) through (A-5) it can be proved that laf*=nafij=B aa,aa, 80,

(‘4-6)

k'

which is the expression of the flow rule for the five-dimensional surface.

representation

of the yield

APPENDIX 2 In this appendix we demonstrate a general property of closed subspaces, namely, that the subspace which results from the intersection of two closed subspaces is itself closed. For simplicity the demonstration is restricted to a particular case, but the arguments and the result are completely general and applicable to any arbitrary case. Assume that the subspaces {a, (Tag}and {x, ei3} are closed. This implies that the stress and strain-rate states are related by 011

b=

u12

0

022

0

.

El1 )

d =

.

El2

0

1,,

0

033

(A-7)

E33

OI 011 6=

0

613

422

0 633

El1 * )

i=

0

43

ez2

0 E33

(‘4-8)

391

Yield surface of textured polycrystals

respectively. Assume now that a particular stress state is imposed on the material, having nonzero components only in the intersection of the two previous subspaces : Cl1 U=

0

0

022

0

(A-9)

.

433

This represents a particular case of both equations (A-7) and (A-S), which predict that the related strain-rate components i1 3 and 6, 2 have to be zero respectively. As a consequence, the associated strain rate state adopts the form . El1 i=

0

0

* E22

0.

(A-10)

%3

From (A-9) and (A-10) it is evident that {rr} is also a closed subspace. Note that the subspace formed by the union of two closed subspaces is not closed unless some further conditions (such as the yield surface being sign independent and the texture exhibiting certain mirror planes) are obeyed.

APPENDIX3 Here we derive the general form of the matrix representation for the operator associated with a mirror symmetry plane characterized by the direction cosines nr, n2 and n3 of its normal ri. An arbitrary vector X is related to its mirror-reflected vector X’by (X+X’)*ri = 0, (A-11) (X-X’) = Iri or equivalently : n&x, + xi) + n&c2 + XL)+ n3(x3 x1-x;

x2-x;

-=-=-9 nl

+ x;)

=

0,

x3-x;

n2

n3

which leads to the solution X’ = Mx, where M =

(l-2n:)

-2n,n,

-2n,n,

-2n,n,

(l-2n2,)

-2n,n,

-2n,n,

-2n,n,

(l-2n:)

(A-12)

represents the mirror symmetry operator. The inverse of this operator is the operator itself. For example, if a plane containing xg and at an angle @ with respect to x1 is a mirror plane, then its normal vector is given by ri = (-sin CD,cos @‘,0) and the above operator adopts the form M =

cos 2@

sin 2@

sin 2@

-cos2Qp

0

0

0 0 . 1

(A-13)