Optimal bounds on texture coe cients 1 Introduction

Optimal bounds on texture coecients Roberto Paroni Mathematical Institute, University of Oxford 24-29 St. Giles', Oxford OX1 3LB, UK Let w be the orientation distribution function of a polycrystalline aggregate of crystallites with symmetry Gcr and with group of texture symmetry Gtex . In this paper we obtain a \recipe" on how to derive optimal bounds on the texture coecients Wlmn associated with w. In particular, we nd explicit bounds in the case in which Gtex is a group with orthorhombic symmetry and Gcr is either a group with cubic symmetry or a group with hexagonal symmetry.

Abstract.

1 Introduction A polycrystalline material is an aggregate of numerous crystallites or grains, each having its own shape and particular orientation of its crystallographic axes. Manufacturing processes such as rolling, forging, extrusion, annealing and aging usually leave the grains in certain preferred orientations so that the metal acquires a slight anisotropy. Unless the orientation of the crystallites is completely random, we say that the polycrystalline aggregate has preferred crystallographic orientations and is textured. The texture of a polycrystalline aggregate can be mathematically described by a probability (Borel) measure } de ned on the group of rotations SO(3), which, under appropriate assumptions, generates an orientation distribution function w (ODF). It is usual practice to write the ODF w in a series of generalised spherical functions (cf. Bunge [1], Roe [2])

w( ; ; ) =

1 X l X l X l=0 m=?l n=?l

Wlmn Zlmn (cos )e?im e?in ;

where Zlmn are the normalized Jacobi polynomials and ( ; ; ) are the Euler angles, and we use here the convention adopted by Roe p [2]. From the normalization condition }(SO(3)) = 1 we always have W000 = 1=(4 22). We call those expansion coecients Wlmn with l  1 the texture coecients. When the crystallites comprising the aggregate and the aggregate itself have some symmetries, many of the texture 1

coecients Wlmn are no longer independent and some of them are identically equal to zero. Moreover in many applications in elasticity, as we will see below, it suces to consider only the texture coecients Wlmn with l  6. For instance, in the case of an orthorhombic aggregate of cubic crystallites there are, under an appropriate choice of the reference, only 7 independent texture coecients which have relevance in the theories discussed below: W400; W420; W440; W600; W620; W640 and W660. Hence in this case the symmetries force W401, for instance, to be equal to zero. It is then natural to ask the following question: do the symmetries imply any bounds on the seven independent coecients? And, in general: do a generic crystal symmetry and a generic texture symmetry imply any bounds on the independent texture coecients? The main aim of this paper is to answer this question. A partial answer to the question above was given by Walpole [3]; in fact he deduced, for a orthorhombic aggregate of cubic crystallites (see Eqs. (37) and (38) of Walpole [3]) that p p 2 7 7 ? 482  W400  3222 ; p p 5 7 jW420j  482 ; jW440j  3235 2 : Walpole derived these bounds via an algebraic method, essentially using dierent properties of fourth order tensors that he de ned. It seems, at least to the present author, not obvious how to use Walpole's method to derive bounds on the W6mn texture coecients. The approach followed in this paper is completely dierent. We rst write the problem variationally; then we show that the problem admits a solution (in a particular space) and nally we evaluate the bounds. In this way we shall not only derive optimal bounds on the texture coecients, but we shall also nd out how the crystallites should be arranged to achieve the bounds. This last piece of information might be useful in problems of optimal design. This paper is organized in the following manner. In Section 2 we introduce some preliminaries and we also establish our notation. The main result of Section 3 is Theorem 3.1. Roughly speaking, Theorem 3.1 gives the recipe on how to derive optimal bounds on the texture coecients whatever the crystallite and texture symmetries are. In particular it also shows that the bounds are achieved for ideal textures (i.e., all the crystallites have the same orientation). In Sections 4 and 5 the recipe given by Theorem 3.1 is applied to the case of orthorhombic aggregates of cubic and hexagonal crystallites, respectively. In Section 4 we shall obtain the same bounds on W400 and W440 obtained by Walpole, while we shall improve the bound on W420. The bounds on the W6mn coecients of Section 4 and the bounds found in Section 5 seem to be new. We now brie y motivate why we restrict our attention to the texture coecients Wlmn with l  6. To this end, we cite the theories which we hope the bounds derived in this paper may nd applications. The elastic properties of a polycrystalline 2

material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged. The elastic constants of a textured polycrystalline material could be estimated by averaging the elastic constants of the single crystallite in a generic orientation over all orientations, where the orientational average is weighted by the orientation distribution function. Man and Paroni [4, 5] and Paroni and Man [6] developed simple micromechanical models for prestressed and textured polycrystals which consist of crystallites of the same chemical composition. According to these models, given the ODF of the polycrystal and the second- and third-order elastic constants of the crystallites comprising the aggregate, the elasticity tensor C and the acoustoelastic tensor D of the polycrystalline aggregate can be evaluated by orientational averaging through appropriate integrations over the rotation group. More recently, Man [7] and Paroni and Man [8] (see also Paroni [9]) obtained, under a particular \physical hypothesis", representations of constitutive equations depending on the ODF. In particular Man [7] showed, again under the same hypothesis, that the elasticity tensor C of a weakly-textured orthorhombic aggregate of cubic crystallites with orientation distribution function w can be written as: C (w)[e] = (tr e)I + 2e + c W (W400; W420; W440)[e];

(1)

here e is the in nitesimal strain, a symmetric second-order tensor; I is the identity tensor; ;  and c are material constants; W is an explicitly determined orthorhombic fourth-order tensor. Paroni and Man [8] showed that, for a weakly-textured orthorhombic aggregate of cubic crystallites, the \physical hypothesis" dictates that the acoustoelastic tensor D (w) must have the following form D (w) = D iso +

5 X

p=1

pP(p)(w);

where D iso is a constant sixth-order isotropic tensor (it is what D would be if the crystallites have no preferred orientations), the p's are material constants, and the P(p)(w)'s (for p = 1; :::; 5) are explicitly determined sixth-order orthorhombic tensors that depend only on Wlmn for l  6.

2 Preliminaries A polycrystalline material is an aggregate of many crystallites, each of which has its own orientation in space. Let us consider a single crystallite, which has an unstressed natural con guration , and a polycrystalline material in a con guration . We describe the texture at the con guration (X ) of the aggregate point X (i.e., the totality of orientations of crystallites at X ) by a probability (Borel) measure 3

} de ned on the rotation group SO(3). We call } the orientation measure of the aggregate point X . For a Borel subset A of SO(3), }(A) gives the probability that we nd a speci c crystallite, in the con guration (X ), having an orientation, with respect to the con guration , represented by an element in A. Sometimes, in what follows, we will write }; in place of } to emphasize that the chosen con gurations for the single crystallite and the polycrystalline material are  and , respectively. We now characterize the relationship between } and the symmetries of the single crystallite and of the aggregate. To do so we nd it convenient to change the reference con gurations  and . Let  be a reference con guration which can be obtained by a rotation Q of the reference ; we write  = Q. Let  } ; be the probability measure associated with the references  and . Then, if R and R are the orientations of a crystallite in the con guration  with respect to the con gurations  and , respectively, we have that R = RQT . Thus if A  SO(3) we have R 2 A , R 2 AQ, and from this equivalence it follows that }(A) = } ; (AQ). We write this last result in the following more suggestive way: }(A) = };(A) = }Q;(AQ); (2) for every Q 2 SO(3). Similarly, by changing the reference  and by keeping the reference  xed, we obtain }(A) = };(A) = };Q(QA); (3) for every Q 2 SO(3). Let us denote by Gcr the symmetry group of the crystallite in the reference con guration . Then, by de nition we have  = Q = ; for every Q 2 Gcr . From Eq. (2) and the previous equation, we conclude that the measure } has to satisfy the following identity }(A) = }(AQ); (4) for every Q 2 Gcr and every Borel A  SO(3). Analogously, we de ne Gtex to be the set of Q 2 SO(3) for which }; = };Q; the group Gtex measures the symmetries of the reference con guration  in a probabilistic sense. From Eq. (3) we nd that }(A) = }(QA); (5) for every Q 2 Gtex and every Borel A  SO(3). Following Man & Paroni [4], we call Gtex the group of texture symmetry. Hereafter we suppose the orientation measure } to be absolutely continuous with respect to the unique probability Haar measure }H on SO(3), }  }H (cf. Rudin 4

[10, 11]). We can then de ne the orientation distribution function w in terms of } and }H by d} ; w = 812 d} (6) H

where d}=d}H denotes the Radon-Nikodym derivative of } with respect to }H . Henceforth we assume that w is square-integrable on SO(3) with respect to the Haar measure }H . Let us choose a Cartesian coordinate system in space. After parametrising the rotation group SO(3) with the Euler angles ( ; ; ), the de nition of which we follow the convention adopted by Roe [12], we may expand w as an in nite series of the generalized spherical functions:

w( ; ; ) =

1 X l X l X l=0 m=?l n=?l

Wlmn Zlmn (cos )e?im e?in ;

(7)

where Zlmn are the normalized Jacobi polynomials. We call the expansion coecients Wlmn , for l  1, the texture coecients. The orientation distribution function can also be rewritten as

w( ; ; ) =

l 1 X l X X l=0 m=?l n=?l

l ( ; ; ); clmnDmn

(8)

l are the Wigner D-functions (cf. Man [7], Varshalovich et. al. [13]), and where Dmn

r

clmn = (?1)m?n 2l 2+ 1 Wlmn : (9) For a given orientation distribution function w the coecients clmn , and hence the texture coecients Wlmn , can be determined by using the orthogonality condition satis ed by the Wigner D-functions (cf. Varshalovich et. al. [13]) Z l (Q) d} (Q) = 1    ; (10) Dqrp (Q)Dms H 2l + 1 pl qm rs SO(3) where  denotes the complex conjugate; in fact it follows that Z p l (Q) d} (Q): m ? n 2(2l + 1) w(Q)Dmn (11) Wlmn = (?1) H SO(3)

5

3 A general theorem The goal of this section is to determine the possible values of Wlmn for a given texture symmetry group Gtex and for a given crystallite symmetry group Gcr . Let M be the set of probability measures on SO(3). We de ne

Ms = f} 2 M : }(QA) = }(A) 8Q 2 Gtex; }(AQ) = }(A) 8Q 2 Gcr ; (12) and for everyBorel set Ag; 

and

 d} 2 M2 = } 2 M : }  }H ; d} 2 L (SO(3); B; }H ) ; H Ma = Ms \ M2;



(13)



d} : W = w 2 L (SO(3); B; }H ) : 9} 2 Ma; such that w = 81 2 d} (14) H The set Ms is the subset of probability measures which satisfy the required symmetries, while M2 is the set of measures which generate a square integrable orientation distribution function. The set W is the set of square integrable orientation distribution functions with the required symmetries. We can now state, in view of Eq. (11), our problem as follows: Find 2

inf J l [w]; w2W mn where

l [w]; and sup Jmn w2W

l [w] = (?1)m?n p2(2l + 1) Jmn

Z SO(3)

l (Q) d} (Q): w(Q)Dmn H

(15)

l [w]" we really mean two separate problems, namely nding By \Find inf w2W Jmn l [w]) and inf w2W Im(J l [w]), where Re(z ) and Im(z ) denote the real inf w2W Re(Jmn mn and imaginary parts of z. The same method of solution however, applies to both of these problems. Hence we simply suppress the Re(
) and Im(
) notations here, which should be restored in the appropriate context. A similar remark applies for the sup problem. Let us analyse the inf problem rst. Let fwk g  W be a sequence such that l l [w]. Now we would like, as suggested by the direct method of Jmn[wk ] ! inf w2W Jmn the calculus of variations, to nd a topology on W which makes W itself sequentially l lower semicontinuous. Usually in such problems compact and the functional Jmn some weak topology will work. Since W  L2(SO(3); B; }H ) we may try to use the weak topology of L2(SO(3); B; }H ). The problem, though, is that the sequence fwk g is not bounded (as it will be clear later) in L2, and hence the direct method does

6

not apply. On the other hand the sequence fwk g is bounded in L1(SO(3); B; }H ), in fact kwk kL1 = 1; the space L1, however, is not re exive, so again the direct method does not apply. This, however, leads us to imbed the space of functions W into the space of probability measures Ma. Let w 2 W . Then there exists a measure d} , and hence by using the Radon-Nikodym theorem } 2 Ma such that w = 81 2 d} H we have Z Z 1  l l (Q) d}(Q): w(Q)Dmn (Q) d}H (Q) = 82 Dmn (16) SO(3)

SO(3)

By using Eqs. (15), (16) and the one-to-one correspondence between W and Ma, we can restate our problem as follows: Find inf I l [}]; }2Ma mn where

l [}]; and sup Imn }2Ma

p

2(2l + 1) Z Dl (Q) d}(Q): (17) mn 82 SO(3) The same remarks about the inf and sup made immediately after Eq. (15) apply here as well. Henceforth we say that a sequence of probability measures f}ng converges weakly to the probability measure }, written }n * } in M, if l [}] = (?1)m?n Imn

Z

SO(3)

f (Q) d}n (Q) !

Z

SO(3)

f (Q) d}(Q)

(18)

l is a continuous function on for every continuous function f on SO(3). Since Dmn l is weakly continuous. We also have, from SO(3) we observe that the functional Imn Prohorov's theorem (cf. Billingsley [14]), that any family of probability measures on a compact metric space is sequentially weakly compact. Hence, there are two measures }? and }+ in M such that l [}? ] = inf I l [}]; and I l [}+ ] = sup I l [}]: Imn mn mn }2Ma mn }2Ma

(19)

We now show that the space M2 is not weakly closed. Let Q be any element of SO(3) and let fSk g  SO(3) be a sequence of Borel sets which shrink to Q nicely. Then the probability measures }k de ned by Z 1 }k (E ) = } (S ) Sk (R) d}H (R) H k E belongs to M2, and converges weakly to the Dirac measure Q 2= M2 concentrated at Q. In this way, we have also proved the following lemma. 7

Lemma 3.1 For every Q 2 SO(3) the Dirac measure Q belongs to the weak closure of M2 . We now instead prove that Ms is weakly closed. Lemma 3.2 The space of measures Ms is weakly closed. Proof. Let fng  Ms be a sequence of probability measures such that n *  in M for some  2 M. Let us choose and x Q 2 Gtex and Q 2 Gcr. To each probability  Q ) for measure  we associate the probability measure ` de ned by ` (B ) =  (QB every Borel set B , and to each continuous function g on SO(3) we associate the continuous function `g(Q) = g(Q T QQ T ). Let f be any continuous function on SO(3). Then by changing variables we nd

Z

SO(3)

f (Q) d`n (Q) =

!

Z

Z

SO(3)

SO(3)

`f (Q) dn (Q) `f (Q) d(Q) =

Z SO(3)

f (Q) d`(Q);

from which we conclude that

`n * `; in M: But `n = n for every n, and thence, from the uniqueness of the weak limits, we conclude that ` = , which is equivalent to  2 Ms.  We are now ready to state the main result of this section. Theorem 3.1 Let Gtex = fQ 1; : : :; Q Ntex g and Gcr = fQ 1; : : :; Q Ncr g. Then for every l; m; n there are two elements R? ; R+ 2 SO(3) such that Ncr Ncr N N tex X tex X X X 1 1 + ? } =N N Q R?Qj ; Q R+Qj ; (20) } =N N tex cr i=1 j =1 i tex cr i=1 j =1 i where }? and }+ are de ned in Eq. (19). Moreover, l [}] = max Al (Q); l (Q); I A sup inf I l [}] = Q2min mn mn }2M mn Q2SO(3) mn SO(3) }2Ma

a

where and

Almn(Q) = (?1)m?n

p

l X l 2(2l + 1) X l l T l 82 k=?l p=?l Cnk Dkp (Q )Tpm

Ncr X l (Q j) Dnk Cnkl = N1

N tex X l = 1 l (Q  i) Tpm Dpm N

cr j =1

tex i=1

8

(21) (22) (23)

Proof. We prove the part of the theorem concerning the in mum, the other part is proved in a similar way. Since }? is the weak limit of measures in Ma = Ms \M2 we have, by Lemma 3.2, that }? 2 Ms; hence for every Borel set A the following identity holds

}? (A) = }?(Q iAQ j ); for every i = 1; : : : ; Ntex and j = 1; : : :; Ncr . With a change of variables we nd p Z 2(2 l + 1) l m ? n l (Q) d}? (Q) inf I [}] = (?1) D (24) mn 2 }2Ma mn 8 SO(3) p Z 2(2 l + 1) l (Q m ? n  iQQ j ) d}? (Q) D = (?1) mn 2 8 SO(3) for every i = 1; : : : ; Ntex and j = 1; : : :; Ncr . Now summing over i and j we obtain p Z NX Ncr tex X 2(2 l + 1) 1 l m ? n l (Q  iQQ j ) d}? (Q) I [ } ] = ( ? 1) inf D mn mn 2 }2Ma 8 NtexNcr SO(3) i=1 j=1

and hence, using the fact that }? is a probability measure we obtain p Ncr tex X 2(2l + 1) 1 NX l (Q l [}]: m ? n  iQQ j )  inf Imn D min ( ? 1) mn }2Ma Q2SO(3) 82 NtexNcr i=1 j=1 Let Q 2 SO(3) and let

(25)

Ncr N tex X X  = N 1N Q QQ : tex cr i=1 j =1 i j Clearly  is a probability measure. We now show that in fact it belongs to Ms. Let B be a Borel set of SO(3). Then Ncr N tex X X 1    (QpB Qq) = N N Q QQ (Q pB Q q ) tex cr i=1 j =1 i j Ncr N tex X X 1 = N N QT Q QQ QT (B ) =  (B ); tex cr i=1 j =1 p i j q for every p = 1; : : : ; Ntex and q = 1; : : : ; Ncr . From Lemma 3.1 we deduce also that  belongs to the weak closure of M2. Thence, p Z 2(2 l + 1) l (R) d (R) l m ? n D I [ } ]  ( ? 1) inf mn mn 2 }2Ma 8 SO(3) p Ncr N tex X X 2(2 l + 1) 1 l  (Q m ? n  iQQ j ): D = (?1) mn 2 8 NtexNcr i=1 j=1

9

From Eq. (25) and the previous equation we get (?1)m?n inf I l [}] = Q2min }2Ma mn SO(3)

p

Ncr tex X 2(2l + 1) 1 NX l (Q  iQQ j ) D mn 2 8 Ntex Ncr i=1 j=1

(26)

and, denoting by R? the element in which the minimum of the right hand side of the previous equation is achieved, we also have

}? =

Ncr tex X 1 NX NtexNcr i=1 j=1 QiR?Qj :

We nish the proof by showing the equivalence between Eq. (21) and Eq. (26); to do this we will use the following two identities (cf. Varshalovich et. al. [13]) l (Q) = Dl (QT ); Dms sm

l (QR) = Dms

l X k=?l

l (Q)Dl (R); Dmk ks

(27)

which hold for every Q; R 2 SO(3). By using Eqs. (27) we nd Ncr Ncr N tex X tex X X 1 NX 1    l l  T T T Ntex Ncr i=1 j=1 Dmn (QiQQj ) = NtexNcr i=1 j=1 Dmn (Qj Q Qi ) l X l Ncr X N tex X X 1 l (Q l (QT )Dl (Q T  Tj )Dkp =N N Dnk pm  i ); tex cr i=1 j =1 k=?l p=?l

and therefore the theorem is proved.  Remark. The rotations R? and R+ determined in Theorem 3.1 are not unique. Indeed, using the notation of the theorem, we have that }(A) = }(Q iAQ j ) for i = 1; : : : ; Ntex, j = 1; : : :; Ncr and for every A  SO(3). This identity implies w(R) = }(Q iRQ j ) for every R 2 SO(3); and hence the ODF w assumes the same value at at least Ncr  Ntex rotations.

10

4 Orthorhombic aggregate of cubic crystallites In this section we apply Theorem (3.1) to the case of an aggregate of cubic crystallites with orthorhombic texture. For this aggregate it is well known that Ncr = 24 and that Ntex = 4; moreover, under a Cartesian coordinate system where the coordinate axes agree with the axes of orthorhombic symmetry of the texture and with the three 4-fold axes of cubic symmetry of the reference crystallite, it is well known (cf. Roe [12, 2]) that all the Wlmn are real, Wlmn = 0 for 1  l  3 and for l = 5, and there are only three independent W4mn and only four independent W6mn, which we can choose to be W400; W420; W440; W600; W620; W640 and W660. We apply Theorem (3.1) to the case described above in order to nd the numerical bounds on Wlmn , for l = 1; : : : ; 6. l de ned in Eq. The rst thing to do is to evaluate the constants Cnkl and Tpm (23). To determine Cnkl , it suces to evaluate the Euler angles corresponding to the rotations Q j , j = 1; : : : ; 24, belonging to the cubic symmetry group, and evaluate l (Q  j ) by means of the formula Dnk

r

2 Z (cos())e?im e?in; l ( ; ; ) = (?1)m?n Dmn (28) 2l + 1 lmn which can be derived from Eqs. (7){(9). The normalized Jacobi polynomials in question can be derived by using a recursive formula (see for instance Eqs. (8) and (9) of Man [7]), or they can be found tabulated in Morris and Heckler [15]. The values that we found are: l = Cl = Cl ; Cnkl = Cnk kn nk

Cnkl = 0 for l = 1; 2; 3; 5 or for n = 1; 2; 3; 5; 6; C000 = 1; p 7 5; 4 4 C00 = 12 ; C40 = 2470 ; C444 = 24 p 1 6 6 C00 = 8 ; C40 = ? 1614 ; C446 = 167 : l coecients. We obtained: In a similar way we computed the Tpm l = 0 if jpj 6= jmj; Tpm 8 1 if p = 0 and l is even, > > < and l is odd, Tppl = Tppl = Tppl = Tppl = > 00 ifif jppj=is0 odd, > : 1=2 if p 6= 0 and jpj is even.

11

Now the functions Almn, de ned in Eq. (22), are easily established. These functions, though, are highly oscillating and therefore the evaluation of the maximum and the minimum is not completely trivial. We show in some detail how the bounds were derived for two cases. Since we are claiming that the bounds on W420 found by Walpole [3] are not sharp, we look at this case rst. Afterwards we derive the bounds for W620, since, as we shall see, they look \suspicious". The bounds for the other coecients can be derived in a similar way, except those for W400 and W600; the functions A400 and A600 do not depend on , and hence it is easy to derive the bounds on W400 and W600 by direct inspection. l reported above we nd, from Eq. (22), By using the values of Cnkl and Tpm

p

A (Q) = 12822 f14(D024 (QT ) + D024 (QT )) p 4 T 4 (Q ) + D42 + 70(D42 (QT ) + D424 (QT ) + D424 (QT ))g: Now, let ( ; ; ) be the Euler angles corresponding to the rotation QT , and let us de ne x = cos : Then by using the identities p 3 Z402(x) = 8 5 (?7x4 + 8x2 ? 1); p 3 Z442(x) = 1614 (?x4 ? 2x3 + 2x + 1); Zlmn (x) = (?1)m+n Zlmn(x) = Zlmn (?x) 4 20

together with Eq. (28), we obtain

p

7 5 (1?x2)f(7x2 ?1) cos(2)+(1+x2) cos(2) cos(4 )?2x sin(2) sin(4 )g: A = 128 2 We can now proceed to nd the bounds. Obviously 4 20

p

7 5 (1 ? x2)fj7x2 ? 1j + (1 + x2)j cos(2) cos(4 )j + 2jxjj sin(2) sin(4 )jg; jA j  128 2 and by using the elementary inequality 2ab  a2 + b2 we deduce 4 20

p

7 5 (1 ? x2)fj7x2 ? 1j + (1 + x2)(j cos(2) cos(4 )j + j sin(2) sin(4 )j)g jA j  128 p2 7 5 (1 ? x2)fj7x2 ? 1j  128 2 12 4 20

+ 1 +2 x (cos2(2) + cos2(4 ) + sin2(2) + sin2(4 ))g; p p 5 7 7 = 1282 (1 ? x2)(j7x2 ? 1j + 1 + x2)  6452 : p These bounds are sharp, for the maximum is attained at ( = 0 ; x = 2=2;  = 0) p while the minimum is achieved at ( = 0; x = 2=2;  = =2). In a similar way we deduce 2

A

6 20

p

2730 (1 ? x2)f(33x4 ? 18x2 + 1) cos(2) = 2048 2 ? (33x4 ? 10x2 + 1) cos(2) cos(4 ) + (44x3 ? 20x) sin(2) sin(4 )g:

p

To nd the bounds of A620, we rst look at the case jxj  1=3 + 2 22=33. In this region we have that 33x4 ? 10x2 + 1  j44x3 ? 20xj and hence with a computation similar to that done for A420 we nd

p

2730 (1 ? x2)(j33x4 ? 18x2 + 1j + 33x4 ? 10x2 + 1); jA j  2048 2 and hence p p 2730(1729 + 181 181) : jA620j  2 1881792p  We now look at the case in which jxj  1=3 + 2 22=33. In this case we can bound 6 20

A620 as follows:

p

2730 (1 ? x2)fj33x4 ? 18x2 + 1jj cos(2)j jA j  2048 2 + (33x4 ? 10x2 + 1)j cos(2)j + j44x3 ? 20xjj sin(2)jg: 6 20

Since the right hand side of the the previous inequality is a function of only two variables we can easily nd the max either by inspection or by graphing. If we do so we nd p p p p p 390 1320 ? 55 301(83 7 + 56 43) ; 6 jA20j  19360002 but this bound is tighter then the one we found in the previous case, hence we conclude that p p 2730(1729 + 181 181) : 6 jA20j  2 1881792 Again, these p p bounds are sharp; in fact the maximum is attained at ( = =4; x = p517 + 22p181=33;  = 0), while the minimum is achieved at ( = =4; x = 517 + 22 181=33;  = =2). 13

We conclude the section by reporting the bounds we found, namely:

p

p

7 2  W  7 2; ? 48 400 32p2 p2 7 5  W  7 5; ? 64 420 64 p 2 p 2 35 ;  W ? 3235 440  2 32 p 2 p 26  W  26 ; ? 13 600 362 p5122 p p p 2730(1729 + 181 181) 2730(1729 + 181 181) ;  W ? 620  2 2 1881792 p p 1881792 91 ; ? 6491  W 640  2 642 p  p 6006  W  6006 : ? 864 660 2 8642 In the next table we report the values ( ; x = cos ; ) where the extrema are attained. (Recall that ( ; ; ) are the Euler angles corresponding to the inverse of the rotation which takes the orientation of the reference crystallite to that of the crystallite in question.)

W400 W420 W440 W600 W620 W640 W660

pMin (=4p ; 3=3; any)

Max (0;p0; any) (0; 2=2; =2) (0; 2=2; 0) (0p; 0; =4) (0p; 0; 0) (=4; 2=2p ; any) p (0; 2=2p; any) p (=4; 517 + 22 181=33; =2) (=4; 517 + 22 181=33; 0) (0p; 0; 0) (0; 0p; =4) (=4; 3=3; =6) (=4; 3=3; 0)

The rotations that achieve the max and min of Wlmn are not unique; see the remark at the end of Section 3. Indeed every max and min is attained by at least 24  4 = 96 rotations.

5 Orthorhombic aggregate of hexagonal crystallites In this section we treat the case of an aggregate of hexagonal crystallites with orthorhombic texture. For this aggregate it is well known that Ncr = 12 and that 14

Ntex = 4; moreover, under a Cartesian coordinate system where the coordinate axes agree with the axes of orthorhombic symmetry of the texture and the 3-axis is parallel to the 6-fold axis of hexagonal symmetry of the reference crystallite, it is well known (cf. Morris and Heckler [16]) that all the Wlmn are real, Wlmn = 0 for l = 1; 3; 5, and there are only two independent W2mn , three independent W4mn and eight independent W6mn, which we can choose to be W200; W220; W400; W420; W440, W600, W620, W640, W660, W606, W626, W646 and W666. We apply Theorem (3.1) to the case described above to nd the numerical bounds on Wlmn , for l = 1; : : :; 6. The procedure is identical to that of Section 4; hence, for brevity, here we report only the values we found. l are: The nontrivial Cnk C00l = 1 for l = 0; 2; 4; 6; C666 = C666 = C666 = C66l = 12 ; l are equal to those found in Section 4 (as in the previous while the coecients Tpm section we are here dealing with orthorhombic texture symmetry). We conclude this section by reporting the bounds deduced:

p

 ? 1610 p 2  ? 1615 2  p 9 ? 5622  p 27 ? 22452  p 3 ? 64352  p p 26(3 + 7 5 ? 29042 15)  p p 2730(2 + 5 ? 21782 5)  p 125 ? 4356912  p 6006  ? 256 p 2 6006  ? 256 2

p

W200  8102 ; p W220  1615 ; 2  p 3 W400  822 ; p 27 W420  22452 ; p 3 W440  64352 ; p W600  8262 ; p p 5) ; 2730(2 + 5 W620  2 p2178 91 ; W640  125 p43562 6006 ; W660  256 p 2 6006 ; W606  256 2 15

p

p

3 1430 ; ? 35121430  W  626 p5122 p p p 2 ? 14) ; ? 429(5216102 ? 14)  W646  429(521610 2  p p 26 :  W ? 1626 666  2  162 As in the previous section we now report the values ( ; x; ) where the extrema are achieved. (Recall that ( ; ; ) are the Euler angles corresponding to the inverse of the rotation which takes the orientation of the reference crystallite to that of the crystallite in question.)

W200 W220 W400 W420 W440 W600 W620 W640 W660 W606 W626 W646 W666

Min Max (any, 0, any) (any, 1, any) (any, 0 ; = 2) (any, 0; 0) p (any, p21=7, any) (any, p 1, any) (any, 2 7=7; =2) (any, 2 7=7; 0) =4) (any, 0; 0) p (any, 0; p (any, p495 + 66 p15=33, any) p(any, 1, any) p (any, 561p+ 132 5=33; =2) (any, 561p+ 132 5=33; 0) (any, 429=33; =4) (any, 429=33; 0) (any, 0; =6) (any, 0; 0) (=6; 0, any) (0; 0, any) p(=p6; 0; 0) p p(0; 0; 0) (=6; 6 10 ? 15=3; 0) (0; 6 10 ? 15=3; 0) (=6; 1; 0) (0; 1; 0)

Acknowledgements: The present paper has its origin in Section 2.5 of the au-

thor's doctoral dissertation, which was accepted by the University of Kentucky in May 1998. It is a pleasure to thank Prof. C.-S. Man for many valuable discussions concerning polycrystalline aggregates. The author wishes also to acknowledge the support of the Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University, as well as the support of the National Science Foundation under Grant CMS-9522829.

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