On the bending of microstretch elastic plates

International Journal of Engineering Science 37 (1999) 1309±1318

On the bending of microstretch elastic plates Michele Ciarletta

*

Department of Information Engineering and Applied Mathematics, University of Salerno, I-84084 Fisciano (Sa), Italy Received 12 February 1998; accepted 9 September 1998

Abstract This paper is concerned with the linear theory of microstretch elastic solids introduced by Eringen (A.C. Eringen, in: Prof. Dr. Mustafa Inan Anisina, Ari Kitabevi Matbaasi, Istanbul, 1971, pp. 1±18; A.C. Eringen, Int. J. Eng. Sci. 28 (1990) 1290±1301). First, a theory of bending of homogeneous and isotropic plates is studied. Then, a uniqueness theorem, with no de®niteness assumptions on the elastic coecients, is presented. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Bending; Microstretch; Plates

1. Introduction The theory of microstretch elastic solids has been introduced by Eringen in Refs. [1,2]. The material points of these bodies can stretch and contract independently of their translations and rotations. A microstretch body can model composite materials reinforced with chopped elastic ®bers and various porous solids. The theory of microstretch elastic bodies is a generalization of the micropolar theory [3]. In Ref. [4], Eringen presented a theory of micropolar elastic plates. The ®eld equations and the boundary conditions are derived for both the plane problem and the bending problem. Some basic theorems on plate energy and on the uniqueness of solutions have been also established. In this paper, by using the results presented by Eringen [4], we derive a theory of bending of microstretch elastic plates. The basic equations of the linear theory of microstretch elastic bodies are contained in Section 2. Section 3 is devoted to the bending theory of elastic plates. In Section 4 we establish a uniqueness result. Following Ref. [5], we derive a reciprocal relation which leads to a uniqueness theorem with no de®niteness assumptions on the elastic coecients.

*

Address: Dip. di Ingegneria dell'Informazione e Matematica App., Via Ponte Don Melillo, 84084 Fisciano, Salerno, Italy. Tel.: 00 39 089 964251; fax: 00 39 089 964 191; e-mail: [email protected]. 0020-7225/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 8 ) 0 0 1 2 3 - 2

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2. Basic equations We refer the motion of the continuum to a ®xed system of rectangular Cartesian axes Oxk …k ˆ 1; 2; 3†. We shall employ the usual summation and di€erentiation conventions: Latin subscripts (unless otherwise speci®ed) are understood to range over the integers …1; 2; 3†, whereas Greek subscripts are con®ned to the range (1; 2); summation over repeated subscripts is implied and subscripts preceded by a comma denote partial di€erentiation with respect to the corresponding Cartesian coordinates. Let B be a regular region of Euclidean three±dimensional space occupied by a homogeneous  We call oB the boundary of B,  and isotropic microstretch elastic solid. Let B be the interior of B. and designate by ni the components of the outward unit normal to oB. In this section we present the ®eld equations of the linear theory of microstretch elastic solids established by Eringen in Refs. [1,2]. We denote by ui the components of the displacement vector and by ui the components of the microrotation vector. Let w be the microstretch function. We denote by U the seven-dimensional vector ®eld on B de®ned by U ˆ …ui ; ui ; w†. The strain measures associated with U are de®ned by eij ˆ uj;i ‡ jir ur ;

jij ˆ uj;i ;

ci ˆ w;i

…1†

where irs is the alternating symbol. The constitutive equations for the linear theory of isotropic microstretch elastic solids are tij ˆ kerr dij ‡ …l ‡ j†eij ‡ leji ‡ awdij ; mij ˆ ajrr dij ‡ bjji ‡ cjij ; ki ˆ rci ;

…2†

s ˆ aerr ‡ bw; where tij is the stress tensor, mij the couple stress tensor, ki =3 the microstress vector, s=3 the microstress function, and k; l; j; a; ab; c; r and b are constitutive constants. The equation of motion are ui ; tji;j ‡ fi ˆ q mji;j ‡ irs trs ‡ gi ˆ j ui ; …3†  kj;j ÿ s ‡ H ˆ J w; on B  T ; where fi , gi and H are the body loads, q is the reference mass density, j…> 0† a coecient of inertia, T ˆ …0; 1†, J ˆ 3j=2 and a superposed dot denotes the partial derivative with respect to the time t. We assume that q and j are given constants. The surface tractions at regular points of oB are given by …4† ti ˆ tji nj ; mi ˆ mji nj ; p ˆ ki ni ; where ti is the stress vector, mi the couple stress vector and p the surface microstress function. To the ®eld equations (1±3) we must adjoin boundary conditions and initial conditions. We note that the strain energy density is given [2] by 1 W ˆ kerr ess ‡ …l ‡ j†eij eij ‡ leij eji ‡ ajrr jss ‡ bjij jji ‡ cjij jij ‡ 2awerr ‡ rci ci ‡ bw2 : …5† 2

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Let M and N be non-negative integers. We say that u is of class C M;N on B  T if u is continuous on B  T and the functions  n  om ou ; m 2 f0; 1; 2; :; Mg; n 2 f0; 1; 2; :; Mg; m ‡ n 6 maxfM; N g; oxi oxj . . . oxs otn exist and are continuous on B  T . We write C M for C M;M . We assume that fi , gi and H are continuous on B  T . Let U ˆ …ui ; ui ; w†. We say that U is an admissible process on B  T provided ui ; ui and w are of class C 2 on B  T and of class C 1 on B  ‰0; 1†: 3. Bending of the plates We stipulate that the region B from here on refers to the interior of a right cylinder of length 2h with the bounded cross-section R and the smooth lateral boundary P. The rectangular Cartesian coordinate frame is supposed to be chosen in such a way that the plane x1 Ox2 is middle plane. Thus, the plane faces of the plate are situated in the planes x3 ˆ h. We assume that the generic cross-section R is a simply connected region. Let L be the boundary of R. Thus, B ˆ fx : …x1 ; x2 † 2 R; ÿh < x3 < hg;

P ˆ fx : …x1 ; x2 † 2 L; ÿh < x3 < hg;

An admissible process U ˆ …ui ; ui ; w† is a state of bending on B  T provided [4,6] ua …x1 ; x2 ; x3 ; t† ˆ ÿua …x1 ; x2 ; ÿx3 ; t†; u3 …x1 ; x2 ; x3 ; t† ˆ u3 …x1 ; x2 ; ÿx3 ; t†; ua …x1 ; x2 ; x3 ; t† ˆ ua …x1 ; x2 ; ÿx3 ; t†;

…6†

u3 …x1 ; x2 ; x3 ; t† ˆ ÿu3 …x1 ; x2 ; ÿx3 ; t†; w…x1 ; x2 ; x3 ; t† ˆ ÿw…x1 ; x2 ; ÿx3 ; t†;

…x1 ; x2 ; x3 † 2 B; t 2 T :

It follows from Eqs. (1), (2) and (6) that the functions tij , mij , ki and s have the symmetries tab …x1 ; x2 ; x3 ; t† ˆ ÿtab …x1 ; x2 ; ÿx3 ; t†; t33 …x1 ; x2 ; x3 ; t† ˆ ÿt33 …x1 ; x2 ; ÿx3 ; t†; ta3 …x1 ; x2 ; x3 ; t† ˆ ta3 …x1 ; x2 ; ÿx3 ; t†; t3a …x1 ; x2 ; x3 ; t† ˆ t3a …x1 ; x2 ; ÿx3 ; t†; mab …x1 ; x2 ; x3 ; t† ˆ mab …x1 ; x2 ; ÿx3 ; t†; m33 …x1 ; x2 ; x3 ; t† ˆ m33 …x1 ; x2 ; ÿx3 ; t†; ma3 …x1 ; x2 ; x3 ; t† ˆ ÿma3 …x1 ; x2 ; ÿx3 ; t†; m3a …x1 ; x2 ; x3 ; t† ˆ ÿm3a …x1 ; x2 ; ÿx3 ; t†; s…x1 ; x2 ; x3 ; t† ˆ ÿs…x1 ; x2 ; ÿx3 ; t†; ka …x1 ; x2 ; x3 ; t† ˆ ÿka …x1 ; x2 ; ÿx3 ; t†; k3 …x1 ; x2 ; x3 ; t† ˆ k3 …x1 ; x2 ; ÿx3 ; t†: Clearly, we assume that the body loads have the following properties of symmetry

…7†

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fa …x1 ; x2 ; x3 ; t† ˆ ÿfa …x1 ; x2 ; ÿx3 ; t†; f3 …x1 ; x2 ; x3 ; t† ˆ f3 …x1 ; x2 ; ÿx3 ; t†; ga …x1 ; x2 ; x3 ; t† ˆ ga …x1 ; x2 ; ÿx3 ; t†;

…8†

g3 …x1 ; x2 ; x3 ; t† ˆ ÿg3 …x1 ; x2 ; ÿx3 ; t†; H …x1 ; x2 ; x3 ; t† ˆ ÿH…x1 ; x2 ; ÿx3 ; t†: We derive a theory of thin plates of uniform thickness assuming that the ®elds ui , ui and w do not vary violently with respect to x3 . We introduce the notations 1 sij ˆ 2h s ˆ

1 2h

H ˆ

Zh tij dx3 ; ÿh

1 lij ˆ 2h

Zh s dx3 ; ÿh

1 2h

Fi ˆ

1 2h

Zh mij dx3 ; ÿh

1 pi ˆ 2h

Zh fi dx3 ;

Gi ˆ

ÿh

1 2h

Zh ki dx3 ; ÿh

Zh gi dx3 ;

…9†

ÿh

Zh H dx3 : ÿh

It follows from Eqs. (7) and (8) that sab ˆ 0; s33 ˆ 0; la3 ˆ 0; pa ˆ 0; s ˆ 0; Fa ˆ 0; G3 ˆ 0; H  ˆ 0:

…10†

We assume that the functions ti , mi and p are prescribed on the surface x3 ˆ h. If we integrate the equations of motion (3) with respect to x3 between the limits ÿh and h, then with the aid of Eqs. (6) and (10) we obtain the following three partial di€erential equations  sa3;a ‡ F ˆ qw;

 ; lba;b ‡ qa3 …s3q ÿ sq3 † ‡ Ka ˆ jw a

on R  T ;

…11†

where 1 wˆ 2h

Zh u3 dx3 ; ÿh

1 wa ˆ 2h

Zh ua dx3 ;

…12†

ÿh

1 q ˆ t33 …x1 ; x2 ; h; t†; h …13† 1 Ka ˆ Ga ‡ ga ; ga ˆ m3a …x1 ; x2 ; h; t†: h Now multiply by x3 the equations …3†1 and …3†3 and integrate from x3 ˆ ÿh to x3 ˆ h. In view of Eqs. (6)±(9), we obtain the following equations F ˆ F3 ‡ q;

Mba;b ÿ 2hs3a ‡ Ha ˆ qI va ; Qa;a ÿ 2hp3 ÿ S ‡ P ˆ f u; on R  T ;

…14†

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where Zh Mab ˆ

Zh x3 tab dx3 ;

Qa ˆ

ÿh

x3 ka dx3 ;

Sˆ

ÿh

Zh Iva ˆ

Zh ÿh

Zh x3 ua dx3 ;

Iu ˆ

ÿh

x3 s dx3 ;

2 I ˆ h3 ; 3

x3 w dx3 ; ÿh

…15† f ˆ JI;

Zh Ha ˆ qa ‡ La ;

qa ˆ 2ht3a …x1 ; x2 ; h; t†;

La ˆ

x3 fa dx3 ; ÿh

…16†

Zh P ˆ v ‡ R;

v ˆ 2hk3 …x1 ; x2 ; h; t†;

Rˆ

x3 H dx3 : ÿh

Eqs. (11) and (14) constitute the balance equations in the theory of bending of microstretch plates. Clearly, the functions F , Ka , Ha and P are prescribed. In what follows we restrict our attention to the state of bending characterized by ua ˆ x3 va …x1 ; x2 ; t†; u3 ˆ w…x1 ; x2 ; t†; ua ˆ wa …x1 ; x2 ; t†; u3 ˆ 0; w ˆ x3 u…x1 ; x2 ; t†;

on B  T :

…17†

We note that, in the context of micropolar theory, the representation (17) has been introduced by Eringen [4]. From Eqs. (1) and (17) we obtain eab ˆ x3 ab ; ea3 ˆ a3 ; e3a ˆ 3a ; e33 ˆ 0; jab ˆ gab ; ja3 ˆ j3a ˆ 0; j33 ˆ 0; ca ˆ x3 na ;

c3 ˆ u;

…18†

where ab ˆ vb;a ; gab ˆ wb;a ;

a3 ˆ wa ‡ 3ab wb ;

3a ˆ va ÿ 3ab wb ;

na ˆ u;a :

…19†

It follows from Eqs. (2) and (18) that tab ˆ x3 ‰kqq dab ‡ …l ‡ j†dab ‡ lba ‡ audab Š; ta3 ˆ …l ‡ j†a3 ‡ l3a ; t3a ˆ …l ‡ j†3a ‡ la3 ; t33 ˆ x3 …kqq ‡ au†; mjm ˆ agqq djm ‡ bgmj ‡ cgjm ; ma3 ˆ m3a ˆ 0;

m33 ˆ agqq ;

ka ˆ rx3 na ; k3 ˆ ru; s ˆ x3 …aqq ‡ bu†:

…20†

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In view of Eqs. (9), (15) and (20) we arrive at sa3 ˆ …l ‡ j†a3 ‡ l3a ; s3a ˆ …l ‡ j†3a ‡ la3 ; ljm ˆ agqq djm ‡ bgmj ‡ cgjm ; Mab ˆ I‰kqq dab ‡ …l ‡ j†ab ‡ lba ‡ audab Š; Qa ˆ rIna ; p3 ˆ ru; S ˆ I…aqq ‡ bu†:

…21†

Thus, the basic equations of the theory of bending of microstretch elastic plates consist of the equations of motion (11) and (14), the constitutive equations (21) and the geometrical equations (19). To these equations we adjoin the initial conditions va …x1 ; x2 ; 0† ˆ v0a …x1 ; x2 †;

w…x1 ; x2 ; 0† ˆ w0 …x1 ; x2 †;

wa …x1 ; x2 ; 0† ˆ w0a …x1 ; x2 †; u…x1 ; x2 ; 0† ˆ u0 …x1 ; x2 †; _ 1 ; x2 ; 0† ˆ x…x1 ; x2 †; v_ a …x1 ; x2 ; 0† ˆ ma …x1 ; x2 †; w…x  _ 1 ; x2 ; 0† ˆ v…x1 ; x2 †; on R; w_ a …x1 ; x2 ; 0† ˆ va …x1 ; x2 †; u…x

…22†

and the boundary conditions ~ a ; sa3 na ˆ ~s; Mba nb ˆ M ~ on L  T : Qa na ˆ Q;

~a ; lba nb ˆ l

…23†

The functions appearing on the right-hand sides of the conditions (22) and (23) are prescribed. By an elastic process on R  T , corresponding to the loads fF , Ka , Ha , P g, we mean a suciently smooth array q ˆ fva , w, wa , u, ab , a3 , 3a , gab , na , Mab , Qa , S, sa3 , s3a , lab , p3 g that satis®es Eqs. (11), (14), (19) and (21) on R  T . The problem of bending consists in ®nding an elastic process on R  T corresponding to the loads fF , Ka , Ha , P g that satis®es the initial conditions (22) and the boundary conditions (23). The internal energy of the plate due to bending is de®ned by 

Zh

W ˆ

W dx3 :

…24†

ÿh

It follows from Eqs. (5), (17) and (24) that W ˆ

1 I‰kqq mm ‡ …l ‡ j†ab ab ‡ lab ba ‡ 2auqq ‡ rna na ‡ bu2 Š ‡ 2h‰…l ‡ j†…3a 3a 2 ‡ a3 a3 † ‡ 2l3a a3 ‡ agqq gmm ‡ bgab gba ‡ cgab gab ‡ ru2 Š :

We note that 2W  ˆ Mab ab ‡ Qa na ‡ Su ‡ 2h…sa3 a3 ‡ s3a 3a ‡ p3 u ‡ lab gab †:

…25†

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Clearly, if we assume that the internal energy of the plate is a positive de®nite quadratic form, then the system (11), (14), (19) and (21) is hyperbolic (we have already assumed that q, j and I are strictly positive). The rather large number of papers devoted to re-examination, rederivations and extensions of the equations of the linear theory of thin shells and plates, in itself may be a sucient indication that the foundations of the linear theory, as derived from the classical theory of elasticity, is not as yet ®rmly established. The model of a Cosserat surface provides a fruitful means for characterization and direct development of a general theory for shells and plates. Green and Naghdi [7] have shown that the equations of the linear theory of an elastic Cosserat plate can be obtained as a ®rst approximation to an asymptotic expansion of an exact linearized three-dimensional theory of a generalized continuum which admits a director. For an extensive review of the literature on the theory of shells and plates the reader is referred to Refs. [6,8]. In Ref. [4], Eringen derived a theory of elastic plates from the three-dimensional equations of micropolar elasticity. The theory presented in this section is an extension of the theory of bending of micropolar elastic plates established by Eringen [4]. The microstretch continuum is a generalization of the micropolar continuum and it is easy to see that under simple assumptions the ®eld equations (11), (14), (19) and (21) reduce to those presented in Ref. [4]. 4. Uniqueness In the ®rst part of this section we establish a reciprocal theorem. Following Ref. [5] we show that the reciprocity relation implies a uniqueness theorem with no de®niteness assumption on the constitutive coecients. ~ …q† 0…q† ~…q† Consider two external data systems D…q† ˆ fF …q† ; Ka…q† ; Ha…q† ; P …q† ; Ma…q† ; ~s…q† , l a ; Q ; va ; …q† …q† …q† …q† …q† …q† …q† …q† …q† …q† …q† …q† w0…q† ; . . . ; v…q† g; …q ˆ 1; 2†. Let A…q† ˆ fv…q† a ; w ; wa ; u ; ai ; 3a ; gab , na ; sai ; s3a ; lab ; Mab ; Qa ; …q† …q† …q† p3 ; S g be a solution corresponding to D . We introduce the notations …q†

Ma…q† ˆ Mba nb ;

…q†

…q†

s…q† ˆ sa3 na ;

l…q† a ˆ lba nb ;

Q…q† ˆ Q…q† a na :

…26†

Theorem 4.1. Let Z n …j† …m† …j† …m† Ejm …r; s† ˆ Ma…j† …x; r†v…m† a …x; s† ‡ Q …x; r†u …x; s† ‡ 2h‰s …x; r†w …x; s† L

‡

o

…m† l…j† a …x; r†wa …x; s†Š

dx ‡

Z n

…j† Ha…j† …x; r†v…m† …x; r†u…m† …x; s† a …x; s† ‡ P

R

Z n o …j† …m† …m† …x; s† ‡ F …x; r†w …x; s†Š da ÿ qI v…j† ‡ 2h‰Ka…j† …x; r†w…m† a …x; r†va …x; s† a R

‡ f u…j† …x; r†u…m† …x; s† ‡ for all r; s 2 T . Then

 …j† …x; r†w…m† …x; s† 2h‰jw a a

o  …j† …x; r†w…m† …x; s†Š dax ; ‡ qw

…27†

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Eab …r; s† ˆ Eba …s; r†

…28†

for all r; s 2 T . Proof. Let …q†

…m†

…m† …q† …m† Iqm …r; s† ˆ Mab …r†ab …s† ‡ Q…q† a …r†na …s† ‡ S …r†u …s† …q†

…m†

…q†

…m†

…q†

…m†

…q†

‡ 2h‰s3a …r†3a …s† ‡ sa3 …r†a3 …s† ‡ lab …r†gab …s† ‡ p3 …r†u…m† …s†Š;

…29†

where, for convenience, we have suppressed the argument x. It follows from Eqs. (21) and (29) that Iqm …r; s† ˆ Imq …s; r†: On the other hand, in view of Eqs. (11), (14) and (19), we have h i h i …j† …m† …j† …j† …m† …m† ‡ 2h sa3 …r†w…m† …s† ‡ lab …r†wb …s† Ijm …r; s† ˆ Mab …r†vb …s† ‡ Q…j† a …r†u …s† ;a ;a h i …j† …j† ‡ Ha…j† …r†v…m† …r†u…m† …s† ‡ 2h Ka…j† …r†w…m† …r†w…m† …s† a …s† ‡ P a …s† ‡ F h …j† i …j† …j† …m† …m† …m†  …r†w…m† …s† ‡ qw  …r†v …s† ÿ f u …r†u …s† ÿ 2h j w …r†w …s† : ÿ qI v…j† a a a a

…30†

…31†

If we integrate this relation over R, then we obtain, with the aid of Eq. (30) and the divergence theorem, the desired result.  Theorem 4.2. Let Z G…r; s† ˆ fHa …r†va …s† ‡ P …r†u…s† ‡ 2h‰Ka …r†wa …s† ‡ F …r†w…s†Šg da R

Z

‡

fMa …r†va …s† ‡ Q…r†u…s† ‡ 2h‰s…r†w…s† ‡ la …r†wa …s†Šg dx;

…32†

L

for all r; s 2 T . Then Z d  qIva va ‡ fu2 ‡ 2h‰jwa wa ‡ qw2 Š da dt R

Zt ˆ

‰G…t ÿ s; t ‡ s† ÿ G…t ‡ s; t ÿ s†Šds 0

‡

Z n

_ _ qI‰v_ a …2t†va …0† ‡ v_a …0†va …2t†Š ‡ f‰u…2t†u…0† ‡ u…0†u…2t†Š

R

o _ _ ‡ w…0†w…2t†Š da: ‡ 2hj‰w_ a …2t†wa …0† ‡ w_ a …0†wa …2t†Š ‡ 2hq‰w…2t†w…0†

…33†

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Proof. By Theorem 4.1, Zt

Zt E11 …t ‡ s; t ÿ s† ds ˆ

0

E11 …t ÿ s; t ‡ s† ds:

…34†

0

Now we apply the relation (34) to the process A ˆ fva ; w; wa ; u; ai , s3a ; lab ; Mab ; Qa ; p3 ; Sg. It follows from Eqs. (27) and (32) that Zt

Zt E11 …t ‡ s; t ÿ s† ds ˆ

0

G…t ‡ s; t ÿ s† ds ÿ

Z n

qI va …t ‡ s†va …t ÿ s† ‡ f u…t ‡ s†u…t ÿ s†

R

0

3a ; gab ; na ; sai ,

 …t ‡ s†w …t ÿ s† ‡ 2hqw…t  ‡ s†w…t ÿ s† ‡ 2hjw a a

o

da:

…35†

Similarly Zt

Zt E11 …t ÿ s; t ‡ s† ds ˆ 0

G…t ÿ s; t ‡ s† ds ÿ

Z n

0

qI va …t ÿ s†va …t ‡ s† ‡ f u…t ÿ s†u…t ‡ s†

R

 …t ÿ s†w …t ‡ s† ‡ 2hqw…t  ÿ s†w…t ‡ s† ‡ 2hjw a a

o

da:

…36†

If we use the relations Zt

f…t ‡ s†g…t ÿ s† ds ˆ f_…2t†g…0† ÿ f_…t†g…t† ‡

0

Zt

_ ÿ s† ds; f_…t ‡ s†g…t

0

Zt

Zt _ _ g…t ÿ s†f …t ‡ s† ds ˆ g…t†f …t† ÿ g…0†f …2t† ‡

0

…37† _ ÿ s†f_…t ‡ s† ds; g…t

0

then Eqs. (34)±(36) imply Eq. (33).  We now use Theorem 4.2 to establish the following uniqueness result. Theorem 4.3. Assume that q and j are strictly positive. Then, the boundary-initial-value problem of the bending of microstretch elastic plates has at most one solution. Proof. Suppose that there are two solutions. Then their di€erence fva ; w ; wa , u ; ai ; 3a ,  ; Qa ; p3 ; S  g corresponds to null data. Thus, we conclude from Eq. (33) that gab ; na ; sai ; s3a , lab ; Mab Z n o qIva va ‡ f…u †2 ‡ 2hjwa wa ‡ 2hq…w †2 Š da ˆ 0; R

which implies va ˆ 0; w ˆ 0; wa ˆ 0; u ˆ 0. 

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Uniqueness results with no de®niteness assumptions on the elastic coecients are important in the case of prestressed bodies. Further remarks on this point are made in the books by Knops and Payne [9] and Iesan [10]. Acknowledgements I express my gratitude to the referees for their helpful suggestions. This work was performed under the auspices of G. N. F. M. of Italian Research Council (C. N. R.) and with the grant 40% M. U. R. S. T. References [1] A.C. Eringen, Micropolar elastic solids with stretch, in: Prof. Dr. Mustafa Inan Anisina, Ari Kitabevi Matbaasi, Istanbul, 1971, pp. 1±18. [2] A.C. Eringen, Theory of thermo-microstretch elastic solids, Int. J. Eng. Sci. 28 (1990) 1291±1301. [3] A.C. Eringen, Linear theory of micropolar elasticity, J. Math. Mech. 15 (1966) 909±923. [4] A.C. Eringen, Theory of micropolar plates, ZAMP 18 (1967) 12±30. [5] D. Iesßan, Reciprocity, uniqueness and minimum principles in elastodynamics, An. St. Univ. ``Al. I. Cuza'' Iasßi, s. Matematica 36 (1990) 175±183. [6] P.M. Naghdi, The theory of shells and plates, in: C. Truesdell (Ed.), Handbuch der Physik, vol. VI a/2, Springer, Berlin, 1972. [7] A.E. Green, P.M. Naghdi, The linear theory of an elastic Cosserat plate, Proc. Cambridge Philos. Soc. 63 (1967) 537±550. [8] J.E. Lagnese, J.L. Lions, Modelling, Analysis and Control of Thin Plates, Collection RMA, vol. 6, Masson, Paris, 1988. [9] R.J. Knops, L.E. Payne, Uniqueness Theorems in Linear Elasticity, Springer Tracts in Natural Philosophy, vol. 19, Springer, Berlin, 1971. [10] D. Iesßan, Prestressed Bodies, Pitman Research Notes in Mathematics Series, Longman Scienti®c and Technical, Essex, 1988.