Natacha Buannic

Patrice Cartraud

Higher-order eective modeling of periodic heterogeneous beams. I. Asymptotic expansion method

Keywords: Asymptotic analysis, Beams, Constitutive model, Eective property, Homogenization methods, Timoshenko

This paper is concerned with the elastostatic behavior of heterogeneous beams with a cross-section and elastic moduli varying periodically along the beam axis. By using the two-scale asymptotic expansion method, the interior solution is formally derived up to an arbitrary desired order. In particular, this method is shown to constitute a systematic way of improving Bernoulli's theory by including higher-order terms, without any assumption, in contrast to Timoshenko's theory or other re®ned beam models. Moreover, the incompatibility between the interior asymptotic expansions and the real boundary conditions is emphasized, and the necessity of a speci®c treatment of edge eects is thus underlined.

Introduction

Heterogeneous structures consisting of elements arranged periodically are widely used in civil engineering and industry. Using standard numerical methods (such as ®nite elements) to predict the overall behavior of these structures leads to heavy computations. However, when the size of the heterogeneity is small compared with the macroscopic dimension of the structure, the latter can be regarded as a homogeneous continuous medium. Thus, the method of homogenization can be applied.

The study presented here concerns the homogenization of structures having one large global dimension in comparison with the others, and a periodic heterogeneity only in this direction. One can for example think about repetitive lattice structures or any other periodic structure displaying overall beamlike behavior. Such structures possess two small parameters: e, which measures the ratio of the width of the crosssection to the total length L of the structure, and e, which is the ratio of the length of the heterogeneity to the length L.

The method of homogenization consists in letting these two small parameters tend to zero starting from the three-dimensional (3D) elasticity problem. Hence several methods exist, depending on the order in which one realizes these two limits. The commutativity of the limiting processes has been studied from a theoretical point of view by [START_REF] Geymonat | Sur la commutativit e des passages a la limite en th eorie asymptotique des poutres composites[END_REF]. The authors established that for a beam with a variable cross-section (transversal heterogeneity), dierent one-dimensional (1D) homogenized models are obtained when letting e 3 0 then e 3 0, or the inverse. More precisely, the use of convergence theorems leads in both cases to a limit behavior corresponding to Bernoulli's model, but associated with dierent eective stinesses. The method consisting in letting ®rst e tend to zero and then e amounts to study a Bernoulli's beam with rapidly varying properties [START_REF] Cioranescu | Homogenization of Reticulated Structures[END_REF][START_REF] Miller | The eigenvalue problem for a class of long, thin, elastic structures with periodic geometry[END_REF]. Following the other method, we take the limit with respect to e ®rst (which corresponds to averaging the eect of cross-section and material variations) and afterwards the limit with respect to e (which consists in applying Bernoulli's theory to the resulting beam).

Another way of homogenization is to assume that the two small parameters simultaneously become vanishingly small. It leads to apply the method of the asymptotic expansion with only one small parameter. This approach has been initiated in [START_REF] Caillerie | Thin elastic and periodic plates[END_REF] for periodic plates, and extended to the case of periodic beams in [START_REF] Kolpakov | Calculation of the characteristics of thin elastic rods with a periodic structure[END_REF] and [START_REF] Kalamkarov | Analysis, Design and Optimization of Composite Structures[END_REF]. At the ®rst order, this method leads to a generalization of Euler Bernoulli Navier's model. The way of obtaining this limit behavior is widely explained in [START_REF] Kolpakov | Calculation of the characteristics of thin elastic rods with a periodic structure[END_REF], where convergence results are also established.

Therefore, three methods are available to homogenize the structure, and the question of de®ning their respective range of applicability naturally arises. When the limit processes are carried out successively, the method is a priori valid only if the parameter tending ®rst to zero is much smaller than the other one. On the other hand, the method consisting in letting both parameters simultaneously tend to zero is a priori appropriate if e and e are of the same order of magnitude, i.e. if the basic cell is neither very long and thin, nor very short and fat. However, the application of the latter method to dierent examples shows that its domain of validity can be enlarged [START_REF] Buannic | Etude comparative de m ethodes d'homog en eisation pour des structures p eriodiques elanc ees[END_REF]. In that reference, a periodic lattice structure is studied. The equivalent characteristics are identi®ed from a classical study of a beam made of a large number of basic cells, and compared to those obtained from the two homogenization methods: method 1 (e 3 0 then e 3 0) and method 2 (e 9 e 3 0). It turns out that the method 2 gives very accurate results whatever the value of the ratio e=e, while the method 1 is valid if e ( e. Similar results have been obtained in the case of honeycomb plates [START_REF] Bourgeois | Mod elisation num erique des panneaux structuraux l egers[END_REF]. This is in the same line as the conclusion drawn in Lewi nski (1991b), in which the author claims that the only restrictions to the method 2 are e ( 1 and e ( 1. The latter method will therefore be applied here.

The present paper aims at deriving the successive terms of the interior asymptotic expansions for periodic heterogeneous beams. As already mentioned, the ®rst order terms correspond to Bernoulli's model. Consequently, the latter give a good approximation of the 3D behavior only if e ( 1 or if the applied loading does not involve any transverse shearing force within the structure. But in practice, e is never in-®nitely small, and it may be necessary to characterize the higher-order terms of the expansions, which is the purpose of this paper. The expression of these terms is well known in the case of homogeneous isotropic elastic beams from Cimeti ere et al. (1988) in the nonlinear case, [START_REF] Rigolot | Sur une th eorie asymptotique des poutres droites[END_REF], [START_REF] Fan | Re®ned engineering beam theory based on the asymptotic expansion approach[END_REF] or [START_REF] Trabucho | Mathematical modelling of rods[END_REF] in the linear case for an arbitrary cross-section, and [START_REF] Duva | The usefulness of elementary theory for the linear vibrations of layered, orthotropic elastic beams and corrections due to two-dimensional end eects[END_REF] for a narrow rectangular cross-section treated in plane stress analysis. The case of transversely nonhomogeneous isotropic rods is also treated in [START_REF] Trabucho | Mathematical modelling of rods[END_REF]. We extend here these works to a periodic heterogeneous beam, with arbitrary variable cross-section, and within the framework of anisotropic elasticity.

Section 2 contains the formulation of the initial 3D elasticity problem and the de®nition of the notations. In Section 3, the asymptotic expansion method will be presented. It leads to a sequence of microscopic cellular problems (Section 4) as well as successive macroscopic 1D models (Section 5).

In this part, most of the attention is focused on the outer expansion of the beam equations. The treatment of end eects and the derivation of the boundary conditions will be given in Part II of this paper.

Throughout this paper, Latin indices take values in the set f1; 2; 3g while Greek indices in f1; 2g. We also use the Einstein summation convention on repeated indices. Moreover, the partial derivatives o=oz 3 , o 2 =oz 2 3 and o 3 =oz 3 3 will be denoted o 3 , o 33 and o 333 .

The initial three-dimensional problem

The 3D slender structure X e considered herein is formed by periodic repetition of the periodicity cell Y e over the e $ 3 direction (see Fig. 1). Any kind of heterogeneity, geometrical or material, can be studied, and the structure is not assumed to present any particular symmetry (material or geometrical) with respect to the middle axis x 1 x 2 0.

The periodicity cell Y e is de®ned by (see Fig. 2):

Y e x $ x i =l e 1À x 2 ; x 3 < x 1 < l e 1 x 2 ; x 3 ; l e 2À x 1 ; x 3 < x 2 < l e 2 x 1 ; x 3 ; À l e 3 2 < x 3 < l e 3 2 1
where the functions l e aAE are assumed to be periodic in x 3 with period l e 3 . Let Y Ãe be the solid part of the cell with boundary oY Ãe (see Fig. 2) such that oY Ãe oY e a oY e b oY e c with oY e a the plane surfaces perpendicular to the e $ 3 direction, oY e b the lateral outer boundary of the cell and oY e c its inner boundary (cell holes). The elastic moduli of the beam, a e ijkl x $ , are periodic in x 3 with period l e 3 , and satisfy the following classical relations: The beam is considered to be under body forces f $ e and tractions g $ e on the outer boundary C e b (see Fig. 2). The holes boundary C e c is supposed to be free of traction. The left end S e 0 is clamped and stress data r e 3i x 1 ; x 2 are prescribed on the right end S e L . The static problem P e of linear elasticity consists in ®nding the ®elds r e , e e and u $ e , such that:

i a e ijkl x $ a e jikl x $ a e klij x $ ; x $ P X e ii W m > 0 such that Vs=s ij s ji ; ms ij s ij 6 a e ijkl x $ s ij s kl iii W M such that M sup a e ijkl x $ ; x $ P X e 2
div $ x r e Àf $
e r e a e x $ : e e u $ e e e u $ e grad s x u $ e

r e Á n $ g $ e on C e b r e Á n $ 0 $ on C e c r e Á e $ 3 r e 3i x 1 ; x 2 Á e $ i on S e L u $ e 0 $ on S e 0 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : 3
where grad s x and div $ x correspond respectively to the symmetric strain and divergence operators, with respect to the spatial coordinate x $ . The vectors n $ in Eq. (3) (third and fourth equations) and e $ 3 in Eq. ( 3) (®fth equation) denote the outer normal of the corresponding boundary. The superscript e in the formulation of P e indicates that the solutions depend on the values of the two small parameters of the structure, e and e, which are assumed to be equal, as explained in Section 1.

A unique solution r e , e e , u $ e exists for the problem (3) under conditions (2) and assuming that the functions f $ e , g $ e and r e i3 x 1 ; x 2 are suciently smooth, and the boundary oX e regular.

The asymptotic expansion method

The ®rst step of the method consists in de®ning a problem equivalent to the problem (3), but now posed on a ®xed domain which does not depend on the small parameter e.

To this end, we apply the technique of [START_REF] Caillerie | Thin elastic and periodic plates[END_REF] and [START_REF] Kolpakov | Calculation of the characteristics of thin elastic rods with a periodic structure[END_REF], and so introduce the following changes of variables, to take into account successively the slenderness of the beam cross-section and the smallness of the beam heterogeneity:

z 1 ; z 2 ; z 3 x 1 e ;
x 2 e ; x 3 ; y 1 ; y 2 ; y 3 z 1 ; z 2 ; z 3 e 1 e x 1 ; x 2 ; x 3 since e e 4

Consequently, z 3 represents the slow or large scale or macroscopic variable of the problem and y

$

x $ =e the fast or small scale or microscopic one.

According to this change of variable, we associate the new strain and divergence operators in the following manner: grad s

x : grad s z 3 : 1 e grad s y : div $ x : div $ z 3 : 1 e div $ y :

( 5 where grad s z 3 and div $ z 3 correspond to partial dierentiations with respect to the only variable z 3 , while grad s y and div $ y are the dierential operators with regard to the three microscopic variables y i . As a second step, it is necessary to presuppose the order of magnitude of the loadings which are applied to the structure. Especially, we set: f e 3 x 1 ; x 2 ; x 3 e 1 Á f 3 z 3 ; y 1 ; y 2 g e 3 x 1 ; x 2 ; x 3 e 2 Á g 3 z 3 ; y 1 ; y 2 f e a x 1 ; x 2 ; x 3 e 2 Á f a z 3 ; y 1 ; y 2 g e a x 1 ; x 2 ; x 3 e 3 Á g a z 3 ; y 1 ; y 2 r e 33 x 1 ; x 2 e 1 Á r 33 y 1 ; y 2 r e a3 x 1 ; x 2 e 2 Á r a3 y 1 ; y 2 6 Furthermore, the elasticity moduli a e ijkl are assumed to be independent of e, so we have: a e x $ ay $ 7

Remark 3.1. The homogenized limit 1D model depend on the orders of magnitude of the applied loadings with respect to e, for example [START_REF] Karwowski | Asymptotic models for a long, elastic cylinder[END_REF] for the theory of asymptotic modeling of rods, or [START_REF] Caillerie | The eect of a thin inclusion of high rigidity in an elastic body[END_REF] and [START_REF] Millet | Contribution a l'analyse asymptotique en th eorie des plaques et des coques[END_REF] for the case of plates. The assumptions ( 6) and ( 7) are usually made in order that the limit behavior (e 3 0) of the 3D slender structure is that of a beam [START_REF] Trabucho | Mathematical modelling of rods[END_REF][START_REF] Cimeti Ere | Asymptotic theory and analysis for displacements and stress distribution in nonlinear lastic straight slender rods[END_REF].

Third, following a standard technique, the solution u $ e of the P e problem is sought in the form used in [START_REF] Kolpakov | Calculation of the characteristics of thin elastic rods with a periodic structure[END_REF]:

u $ e x $ û0 a z 3 e a $ eu $ 1 z 3 ; y $ e 2 u $ 2 z 3 ; y $ Á Á Á 8
where every function u $ k z 3 ; y i is periodic in the variable y 3 with period l 3 l 3 l e 3 =e, which will be denoted y 3 -periodic in the following.

Consequently, using Eq. ( 5) (®rst equation) and the constitutive relations, the strains and stresses expansions are given by: e e x $ e 0 z 3 ; y

$ ee 1 z 3 ; y $ e 2 e 2 z 3 ; y $ Á Á Á r e x $ r 0 z 3 ; y $ er 1 z 3 ; y $ e 2 r 2 z 3 ; y $ Á Á Á 9
Remark 3.2. The form of the ®rst term of the expansion (8), which is composed of only the de¯ections û0 a z 3 , is not an assumption, in the sense that applying the asymptotic method with the relations (6) leads to that expression of u $ e [START_REF] Trabucho | Mathematical modelling of rods[END_REF]. A similar result has been established in the case of periodic plates [START_REF] Caillerie | Thin elastic and periodic plates[END_REF], where the ®rst term of the expansion is found to be reduced to the de-¯ection of the middle plane (i.e. u 0 3 z a e $ 3 ) if appropriate magnitude order assumptions are made on the applied loadings.

Because the beam asymptotic model obtained under assumptions (6) satis®es the relation (8), it is usual to scale the displacement components [START_REF] Trabucho | Mathematical modelling of rods[END_REF][START_REF] Cioranescu | Homogenization of Reticulated Structures[END_REF]. For example, in [START_REF] Trabucho | Mathematical modelling of rods[END_REF], the authors associate with the displacement ®eld u $ e x $ the scaled functions u $ z 3 ; y $ e through the following scalings: and these scaled functions are then assumed to have the asymptotic expansion:

u i z 3 ; y $ e u 0 i z 3 ; y $ eu 1 i z 3 ; y $ e 2 u 2 i z 3 ; y $ Á Á Á
The justi®cation of the scalings (10) is purely mathematical and is related to convergence results of the asymptotic method as e 3 0. However, this convergence aspect will not be treated at all in the present paper. Thus, such scalings will not be used here, since all the results presented here remain formal, i.e. without any convergence study.

When introducing the relations (4) (8) into the P e problem (3) and equating the terms of a same order with respect to e, we replace the problem P e by a family of problems. The ®elds involved in the latter are functions of the two kinds of variables y i and z 3 , but no longer depend on the small parameter e. So, when treating the z 3 -and y i -coordinates as independent, and considering the ®elds function of the only variable z 3 as given data, we can regard each of these problems as a microscopic problem, which is posed on the scaled period Y. In that sense, these successive problems are commonly named the cellular periodic (or basic cell) problems, and will be denoted herein P k cell , where the superscript k stands for the order of the current problem with respect to e. As it will be seen in Section 4, the solution of these cellular problems enables us to determine the periodic parts of the expansions (8) and ( 9). Then, expressing the existence conditions of solutions for the P k cell problems, we obtain the formulation of homogenized 1D-macroscopic problems, denoted by P k hom , the solution of which gives the macroscopic (i.e. nonperiodic) parts of the ®elds ( 8) and ( 9). The formulation and solution of the P k hom problems are treated in Section 5.

The set of cellular problems P k cell

From the change of variable (4) and of operators (5), and inserting the asymptotic expansion (8) for the displacement ®eld u $ e into the initial P e problem, one can derive an in®nite set of cellular P k cell problems, with k starting from À1.

General formulation of the kth cellular problem P k cell

For an arbitrary power k of the small parameter e, the P k cell problem is posed on the ®xed period Y de®ned as:

Y y $ y i =l 1À y 2 ; y 3 < y 1 < l 1 y 2 ; y 3 ; l 2À y 1 ; y 3 < y 2 < l 2 y 1 ; y 3 ; À l 3 2 < y 3 < l 3 2
with l aAE y b ; y 3 l e aAE ey b ; ey 3 =e and l 3 l e 3 =e;

and with boundary oY Ã such that oY Ã oY a oY b oY c . oY a , oY b , oY c denote the scaled lateral boundary surface obtained from oY e a , oY e b , oY e c respectively. The P k cell problem consists in ®nding the ®elds r k1 , e k1 and u $ k2 satisfying the following equations:

div $ y r k1 Àf $ k À div $ z 3 r k r k1 ay $ : e k1 e k1 grad s y u $ k2 grad s z 3 u $ k1 r k1 Á n $ g $ k1 on oY b r k1 Á n $ 0 $ on oY c r k1 i3 and u $ k2 y 3 -periodic 8 > > > > > > > > > > > > < > > > > > > > > > > > > : 12
with k P À1 and where the negative powers of r k and e k vanish. f

$ k and g $ k correspond respectively to the body and surface densities of forces which occur at the order e k . We recall that these forces are assumed to obey the relations (6), so that f k 3 g k1

3 0 if k T 1 and that f k a g k1 a 0 if k T 2.
Finally, the periodic boundary conditions (12) (sixth equation) result from the structure periodicity, see Eq. ( 8), and due to the opposite values of r k1 Á n $ on opposite sides of oY a where n $ AEe 3 $ .

Remark 4.1. When solving the cellular problem P k cell at order k, we consider that the preceding P kÀ1 cell problem has already been solved and thus that the ®elds r k and u $ k1 have been determined. Consequently, the parameters div $ z 3 r k and grad s z 3 u $ k1 constitute macroscopic given ®elds for the current problem P k cell : the ®rst one can be regarded as a ®ctive volume force and the second as an initial strain state in the period Y.

Let us introduce

W Y fw $ P H 1 Y 3 ; y 3 -periodicg. The P k cell problem is equivalent to: ®nd the dis- placement ®eld u $ k2 belonging to W Y such that Vw $ P W Y ; Z Y Ã r k1 : grad s y w $ dY Z Y Ã div $ z 3 r k f $ k Á w $ dY Z oY b g $ k1 Á w $ dC 13
where the stress ®eld r k1 is related to the displacement ®eld u $ k2 following Eq. ( 12) (second equation) and where dY dy 1 dy 2 dy 3 . According to the variational form Eq. ( 13), it is easy to show that the P k cell problem possesses a solution provided that the data div $ z 3 r k , f $ k , g $ k1 verify the following relation:

Vv $ P R; Z Y Ã div $ z 3 r k f $ k Á v $ dY Z oY b g $ k1 Á v $ dC 0 14
where R corresponds to the set of the y 3 -periodic rigid body motions for the period Y, and is given by:

R fv $ z 3 ; y $ =v $ vi z 3 Á e i $ uz 3 y 1 Á e 2 $ À y 2 Á e 1 $ g 15
Under the necessary condition ( 14), the solutions r k1 , e k1 and u $ k2 (determined up to an element of R) exist and can be linearly expressed with respect to these data. The compatibility condition (14) will enable us to formulate the macroscopic problems, as we shall see in Section 5.

In the next sections, we give the solution of the cellular problems which leads to the determination of the microscopic parts of the displacement ®eld u $ e and consequently to a formal expression of the latter.

Solution of the cellular problem P À1 cell

The ®rst cellular problem occurs for k À1. Since we have assumed that no force f $ À1 or g $ 0 is applied at this order, it can be written as follows:

div $ y r 0 0 $ r 0 ay $ : e 0 e 0 grad s y u $ 1 grad s z 3 u $ 0 r 0 Á n $ 0 $ on oY b oY c r 0 i3 and u $ 1 y 3 -periodic 8 > > > > > > > > < > > > > > > > > : 16
The only data of the problem are thus contained in the tensor grad s z 3 u $ 0 and, according to the form of the ®eld u $ 0 , we have:

grad s z 3 u $ 0 0 0 1 2 o 3 û0 1 z 3 0 1 2 o 3 û0 2 z 3 sym 0 2 6 6 4 3 7 7 5 17
where o 3 û0 a z 3 are the two macroscopic data of the P À1 cell problem and where sym stands for the symmetric part of the matrix.

The compatibility condition ( 14) is satis®ed identically for the problem ( 16), ensuring thus the existence of the solution. Furthermore, one can easily establish that this problem possesses a direct solution which is:

u $ 1 part Ày a Á o 3 û0 a z 3 Á e 3 $
and r 0 e 0 0 18

In that sense, the two data o 3 û0 a z 3 do not constitute eective data, since the associated solution corresponds to a zero deformation state [START_REF] Sanchez-Hubert | Introduction aux M ethodes Asymptotiques et a l'Homog en eisation Masson[END_REF].

The displacement ®eld given in Eq. ( 18) is obtained up to an element of R, so the complete solution of the P À1 cell problem has to be written:

u $ 1 û1 i z 3 Á e i $ u 1 z 3 y 1 Á e 2 $ À y 2 Á e 1 $ À y a Á o 3 û0 a z 3 Á e 3 $ u $ 1 z 3 ; y $ 19
4.3. Solution of the zeroth order cellular problem P 0 cell Since r 0 0, the P 0 cell problem consists in ®nding the ®elds r 1 , e 1 and u $ 2 which satisfy:

div $ y r 1 0 $ r 1 ay $ : e 1 e 1 grad s y u $ 2 grad s z 3 u $ 1 r 1 Á n $ 0 $ on oY b oY c r 1 i3 and u $ 2 y 3 -periodic 8 > > > > > > > > < > > > > > > > > : 20
As with the preceding problem, the compatibility condition ( 14) is satis®ed identically for the problem (20).

According to the expression (19) of u $ 1 obtained at the preceding order, the data of the zeroth order cellular problem can be written as follows:

grad s z 3 u $ 1 0 0 1 2 o 3 û1 1 z 3 À y 2 o 3 u 1 z 3 0 1 2 o 3 û1 2 z 3 y 1 o 3 u 1 z 3 sym o 3 û1 3 z 3 À y a o 33 û0 a z 3 2 6 6 6 4 3 7 7 7 5 21
The two data o 3 û1 a z 3 will provide a direct solution u $ 2 part similar to expression (18). The four other data which are contained in grad s

z 3 u $ 1 , namely o 3 û1 3 z 3 , o 33 û0 a z 3 , o 3 u 1 z 3
, correspond respectively to a macroscopic extension, two macroscopic curvatures and a macroscopic torsion rotation. Due to the linearity of the problem (20), the displacement ®eld u $ 2 can be expressed as a linear function of these four eective data. Adding the direct solution u $ 2 part provided by the two other data o 3 û1 a z 3 as well as the rigid motion, the complete displacement ®eld at the second order assumes the following form:

u $ 2 u $ 2 v $ 1E y $ Á o 3 û1 3 z 3 v $ 1Ca y $ Á o 33 û0 a z 3 v $ 1T y $ Á o 3 u 1 z 3 22 where u $ 2 z 3 ; y $ û2 i z 3 Á e i $ 2 z 3 y 1 Á e 2 $ À y 2 Á e 1 $ À y a Á o 3 û1 a z 3 Á e 3 $ .
For later consistency of notations, we introduce the four-components vector e $ 1 z 3 and the 3 Â 4 matrix v 1 y $ so that we have:

u $ 2 u $ 2 z 3 ; y $ v 1 y $ Á e $ 1 z 3 23 with e $ 1 z 3 t fo 3 û1 3 z 3 ; o 33 û0 1 z 3 ; o 33 û0 2 z 3 ; o 3 u 1 z 3 g 24 v 1 y $ v $ 1E y $ ; v $ 1C 1 y $ ; v $ 1C 2 y $ ; v $ 1T y $ 25
In Eq. ( 24), the four eective data have been grouped in the vector e $ 1 z 3 , with the result that the latter represents the ®rst order macroscopic strain vector.

Remark 4.2. The problem (20) does not have an analytical form solution for the unknown v 1 y $ in general, except in the case of homogeneous rods, see for example [START_REF] Trabucho | Mathematical modelling of rods[END_REF] for an isotropic material.

In the same manner as the displacement ®eld, the stress ®eld r 1 solution of P 0 cell has a linear expression with regard to the data:

r 1 s 1E y $ Á o 3 û1 3 z 3 s 1Ca y $ Á o 33 û0 a z 3 s 1T y $ Á o 3 u 1 z 3 26 with s 1E ij a ij33 a ijkh o y h v 1E k s 1Ca ij Ày a a ij33 a ijkh o y h v 1Ca k s 1T ij Ày 2 a ij13 y 1 a ij23 a ijkh o y h v 1T k 8 > > < > > : 27
which will be formally denoted as:

r 1 s 1 y $ Á e $ 1 z 3 28
where s 1 y $ corresponds to the regrouping of the four elementary stress tensors s 1E , s 1Ca , s 1T so that:

r 1 ij s 1 ij1 y $ Á o 3 û1 3 z 3 s 1 ij2 y $ Á o 33 û0 1 z 3 s 1 ij3 y $ Á o 33 û0 2 z 3 s 1 ij4 y $ Á o 3 u 1 z 3 s 1 ijm e 1 m ; m P 1; 4 29 with s 1 ij1 s 1E ij ; s 1 ij2 s 1C 1 ij ; s 1 ij3 s 1C 2 ij ; s 1 ij4 s 1T ij 30
4.4. Solution of the ®rst order cellular problem P 1 cell It follows from Eqs. ( 6), ( 22) and ( 26) that the P 1 cell problem comprises the following relations:

div $ y r 2 Àdiv $ z 3 r 1 À f 3 Á e 3 $ r 2 ay $ : e 2 e 2 grad s y u $ 3 grad s z 3 u $ 2 r 2 Á n $ g 3 e 3 $ on oY b r 2 Á n $ 0 $ on oY c r 2 i3 and u $ 3 y 3 -periodic 8 > > > > > > > > > > > > < > > > > > > > > > > > > : 31 with N uannic, Cartraud nternational ournal o olids and tructures 1 1 1 1 div $ z 3 r 1 s 1E i3 y $ Á o 33 û1 3 z 3 s 1Ca i3 y $ Á o 333 û0 a z 3 s 1T i3 y $ Á o 33 u 1 z 3 Á e i $ 32
and

grad s z 3 u $ 2 0 0 1 2 o 3 û2 1 z 3 À y 2 o 3 u 2 z 3 0 1 2 o 3 û2 2 z 3 y 1 o 3 u 2 z 3 sym o 3 û2 3 z 3 À y a o 33 û1 a z 3 2 6 6 6 4 3 7 7 7 5 0 0 1 2 v 1E 1 y $ o 33 û1 3 z 3 v 1C a 1 y $ o 333 û0 a z 3 v 1T 1 y $ o 33 u 1 z 3 0 1 2 v 1E 2 y $ o 33 û1 3 z 3 v 1Ca 2 y $ o 333 û0 a z 3 v 1T 2 y $ o 33 u 1 z 3 sym v 1E 3 y $ o 33 û1 3 z 3 v 1Ca 3 y $ o 333 û0 a z 3 v 1T 3 y $ o 33 u 1 z 3 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 33
This problem admits a solution up to an element of R if and only if the data div $ z 3 r 1 , 0; 0; f 3 and 0; 0; g 3 satisfy the relation ( 14). From Eq. ( 32), div $ z 3 r 1 can be expressed as a function of o 3 e $ 1 , which is the ®rst gradient of the strains e $ 1 z 3 . Thus, the compatibility conditions ( 14) lead to a relation between o 3 e $ 1 and f 3 and g 3 , which enables us to express the ®ctive volume force div $ z 3 r 1 in the form: 14), which can be linearly expressed with respect to the data, in the same manner as at the preceding orders.

div $ z 3 r 1 h 1 $ o 3 e $ 1 h 2 $ f 3 h 3 $ g 3 34 where h 1 $ ; h 2 $ ; h 3 $ are
Firstly, the solution of the well posed elementary problem corresponding to the prescribed data f 3 , g 3 only is denoted by u $ 3 part . The other data of the problem come from Eq. ( 33) and h 1 $ o 3 e $ 1 in Eq. ( 34). In order to give the form of the solution with respect to these data, it must be noticed here that the ®rst matrix on the right side of Eq. ( 33) is identical to the data matrix (21) of the preceding P 0 cell problem, except that the superscripts have increased by one. As a consequence, the set of these six data, namely o 3 û2 i z 3 , o 33 û1 a z 3 , o 3 u 2 z 3 , leads to the same displacement solutions as those obtained by solving the P 0 cell problem. Thus, besides u $ 3 part , the only new unknowns of the current problem are the solutions corresponding to the derivatives of the ®rst order macroscopic strains de®ned in Eq. ( 24). Consequently, the displacement ®eld solution of the P 1 cell problem can be formally written as follows:

u $ 3 u $ 3 z 3 ; y $ v 1 y $ Á e $ 2 z 3 v 2 y $ Á o 3 e $ 1 z 3 u $ 3 part z 3 ; y $ 35 where u $ 3 z 3 ; y $ û3 i z 3 Á e i $ u 3 z 3 y 1 Á e 2 $ À y 2 Á e 1 $ À y a Á o 3 û2 a z 3 Á e 3 $ ; 36 e $ 2 z 3 t fo 3 û2 3 z 3 ; o 33 û1 1 z 3 ; o 33 û1 2 z 3 ; o 3 u 2 z 3 g; 37 v 2 y $ v $ 2E y $ ; v $ 2C 1 y $ ; v $ 2C 2 y $ ; v $ 2T y $ ; 38
o 3 e $ 1 z 3 t fo 33 û1 3 z 3 ; o 333 û0 1 z 3 ; o 333 û0 2 z 3 ; o 33 u 1 z 3 g: 39
In expression (35), the ®elds grouped in v 2 y $ are the solutions of the problem (31) with the data contained in o 3 e $ 1 z 3 as the only nonzero data: i.e. the body forces h 1 $ o 3 e $ 1 , no traction on oY b , and an initial strain state which is restricted to the last matrix of Eq. ( 33). The vector e $ 2 z 3 stands for the second order macroscopic strains. The ®elds v 1 y $ have already been de®ned in Section 4.3.

Remark 4.3. The ®eld v 2 has been introduced in [START_REF] Trabucho | Mathematical modelling of rods[END_REF], [START_REF] Duva | The usefulness of elementary theory for the linear vibrations of layered, orthotropic elastic beams and corrections due to two-dimensional end eects[END_REF] and [START_REF] Fan | Re®ned engineering beam theory based on the asymptotic expansion approach[END_REF] for beams with constant cross-section. In the homogeneous and isotropic case, analytical solution is available for v $ 2E , and for v $ 2C a for some cross-sections. For heterogeneous and periodic beams, v $ 2E and v $ 2T appear in [START_REF] Kolpakov | The asymptotic theory of thermoelastic beams[END_REF]. See also some related work in the case of periodic plates in Lewi nski (1991a).

The stress ®eld r 2 solution of the P 1 cell problem can also be formally expressed as follows:

r 2 s 1E y $ Á o 3 û2 3 z 3 s 1Ca y $ Á o 33 û1 a z 3 s 1T y $ Á o 3 u 2 z 3 s 2E y $ Á o 33 û1 3 z 3 s 2Ca y $ Á o 333 û0 a z 3 s 2T y $ Á o 33 u 1 z 3 s 2 part z 3 ; y $ s 1 y $ Á e $ 2 z 3 s 2 y $ Á o 3 e $ 1 z 3 s 2 part z 3 ; y $ 40 with s 2 ij a ijk3 v 1 k a ijkh o y h v 2 k 41
The stress ®elds contained in the ®rst brackets have been determined by solving the P 0 cell problem, while those in the second brackets are four new elementary solutions of the P 1 cell problem, when the four data of o 3 e $ 1 z 3 are prescribed. s 2 part is given by s 2 part ay $ :grad s y u $ 3 part .

Generalization: formal expression of the outer displacement ®eld

By now, we have gone far enough to see how to proceed the formal construction of the displacement ®eld u $ e . Inserting Eq. ( 35) and Eq. ( 40) in the equations of the P 2 cell problem, it is not dicult to see that the macroscopic data of this cellular problem will involve the third order macroscopic strains e $ 3 z 3 , the ®rst gradient of the second order macroscopic strains (i.e. o 3 e $ 2 z 3 ), plus the second gradient of the ®rst order macroscopic strains (i.e. o 33 e $ 1 z 3 ). Furthermore, the loadings f a e a $ and g a e a $ have to be added to these data, according to assumptions (6).

In a recursive manner, the number of data involved in a cellular problem P k cell will increase, starting from the k 1th order macroscopic strains e $ k1 z 3 until the kth gradient of the ®rst order macroscopic strains,

o k 3 e $ 1 z 3 .
Thus, assuming that the data of each cellular problem verify the compatibility condition ( 14), the asymptotic expansion of the displacement ®eld u $ e takes the form:

u $ e û0 a z 3 e a $ e 1 u $ 1 z 3 ; y $ e 2 u $ 2 z 3 ; y $ v 1 y $ Á e $ 1 z 3 e 3 u $ 3 z 3 ; y $ v 1 y $ Á e $ 2 z 3 v 2 y $ Á o 3 e $ 1 z 3 u $ 3 part e 4 u $ 4 z 3 ; y $ v 1 y $ Á e $ 3 z 3 v 2 y $ Á o 3 e $ 2 z 3 v 3 y $ Á o 33 e $ 1 z 3 u $ 4 part e 5 Á Á Á 42
The expression ( 42) is similar to the asymptotic expansion obtained in the case of 3D periodic media (see relation ( 21) in [START_REF] Gambin | Higher-order terms in the homogenized stress strain relation of periodic elastic media[END_REF]).

One can also express the asymptotic expansion of the stress ®eld r e under a similar recursive form. The generalization of expression (40) leads also to the following expansion:

r e e 1 s 1 y $ Á e $ 1 z 3 e 2 s 1 y $ Á e $ 2 z 3 s 2 y $ Á o 3 e $ 1 z 3 s 2 part e 3 s 1 y $ Á e $ 3 z 3 s 2 y $ Á o 3 e $ 2 z 3 s 3 y $ Á o 33 e $ 1 z 3 s 3 part e 4 Á Á Á 43
Relation (42) (and consequently Eq. ( 43)) constitutes a formal expression of the solution ®eld in the sense that, by now, only the microscopic parts v i y $ have been determined by solving in series the cellular problems. The macroscopic part of Eq. ( 42), characterized by the ®elds u $ i as well as their successive gradients, has now to be found. The way of obtaining it will be explained in the Section 5.

Remark 4.4. As in the treatment of the P 1 cell problem, it is necessary to take into account the compatibility conditions of the P k cell problem. Therefore, one has to solve P k cell in a similar way as made in the case of P 1 cell (see Sections 4.4 and 5.1.5). In that way, the elementary problems corresponding to each data of P k cell are well posed.

The set of macroscopic homogenized problems P k hom

As already mentioned in Section 4.1, the equilibrium equations corresponding to the unknown displacement ®elds u $ i are obtained from the compatibility condition (14). Expressing this condition for the cellular problems P k cell and P k1 cell leads indeed to the formulation of the homogenized 1D problems P k hom . This process will be applied in the next subsections: the way of deriving the equations of the ®rst homogenized problem, denoted by P 1 hom , will be developed in detail in Section 5.1. A generalization will then be outlined in Section 5.2 in order to give the form of the general homogenized problem P k hom , with k corresponding to an arbitrary power of e.

Formulation of the ®rst homogenized problem P 1 hom

Equilibrium equations

Firstly, it must be noted that the compatibility condition ( 14) is satis®ed identically for the ®rst two cellular problems ( 16) and (20). As a consequence, the ®rst homogenized problem occurs at order k 1.

Let us ®rst focus our attention on the derivation of the macroscopic equilibrium equations of the ®rst homogenized problem P 1 hom . The P 1 cell cellular problem (31) admits a solution provided that the data div $ z 3 r 1 , 0; 0; f 3 and 0; 0; g 3 satisfy the relation ( 14). In particular, if we choose as test functions v $ the four 'elementary' functions of R: v3 z 3 e 3 $ , va z 3 e a $ and y 1 e 2 $ À y 2 e 1 $ , condition (14) leads to the four following equations:

o oz 3 Z Y Ã r 1 33 dY Z Y Ã f 3 dY Z oY b g 3 dC 0 44 o oz 3 Z Y Ã r 1 a3 dY 0 45 o oz 3 Z Y Ã y 1 r 1 23 À y 2 r 1 13 dY 0 46
Furthermore, putting w $ y a e 3 $ in the variational formulation of the P 1 cell problem, given by relation ( 13) with k 1, we have:

Z Y Ã r 2 a3 dY Z Y Ã y a or 1 33 oz 3 y a f 3 dY Z oY b y a g 3 dC; a 1; 2 47
In the same manner, if we express the condition ( 14) for the P 2 cell problem, choosing now the two test functions va z 3 e a $ with a 1 or 2, we obtain:

o oz 3 Z Y Ã r 2 a3 dY Z Y Ã f a dY Z oY b g a dC 0; a 1; 2 48
Let us introduce the following notations:

N 1 z 3 r 1 33 ; T 2 a z 3 r 2 a3 M 1 a z 3 y a r 1 33 ; M 1 3 z 3 y 2 r 1 13 y 1 r 1 23 49
with Á h i 1=l 3 R Y Ã dY and where l 3 stands for the scaled length of period Y (see Fig.

( 2)). The beam stresses N 1 z 3 , T 2 a z 3 , M 1 a z 3 and M 1 3 z 3 respectively correspond to the ®rst order macroscopic axial force, the second order transverse shearing forces, the ®rst order bending moments and the ®rst order twisting moment. They are simply the average of their local corresponding quantity over the period length.

Remark 5.1. The de®nition of the bending moments according to Eq. ( 49) do not obey the classical conventions used in strength of material. Following Eq. ( 49), M 1 1 z 3 and ÀM 1 2 z 3 are about the e 2

$

-and e 1 $ -axis respectively, (see Fig. 3).

With notations (49), it becomes obvious that relations (44), and (46) (48) can be written, respectively, as:

3 N 1 hf 3 i hg 3 i oY b 0 o 3 T 2 a hf a i hg a i oY b 0; a 1; 2 ÀT 2 a À o 3 M 1 a hy a f 3 i hy a g 3 i oY b 0; a 1; 2 o 3 M 1 3 0 50 with h:i oY b 1=l 3 R oY b dY .
Note that the set of the Eqs. (50) corresponds to the classical equilibrium relations of a beam theory problem and hence constitutes the local equations of the P 1 hom problem. The Eqs. (50) (®rst, second and third equations) and beam stresses representation are illustrated Fig. 3, considering the equilibrium of a beam element of length dz 3 .

Remark 5.2. It appears from Eq. ( 50) (third equation) that the ®rst order bending moments M 1 a are not related to the transverse shearing forces of the same order but to the second order ones T 2 a . As a matter of fact, the ®rst order shearing resultants T 1 a are equal to zero, and thus Eq. ( 45) is identically satis®ed. This remarkable result can be easily established as follows: we ®rst notice that T 1 a can be de®ned as:

l 3 Á T 1 a Z Y Ã r 1 a3 dY Z Y Ã r 1 : grad s y w $ dY with w $ y a e 3 $
Green's formula can then be applied, so that:

l 3 Á T 1 a Z oY Ã r 1 3j n j y a dS À Z Y Ã or 1 3j oy j y a dY 51
where oY Ã is constituted of the lateral boundaries oY b and oY c and of the left and right sides of the period. The ®rst integral in Eq. ( 51) vanishes by virtue of the y 3 -periodicity of r 1 and y a , of the absence of prescribed surface force at this order on the lateral outer boundary oY b , and of the stress-free condition on the holes boundary oY c . In the same way, the second integral vanishes too, according to the equilibrium equation of the P 0 cell problem.

Constitutive relations

After obtaining the equilibrium equations of the P 1 hom problem, we focus now our attention on the constitutive stress strain relations of P 1 hom . Grouping the `eective' (i.e. nonequal to zero) ®rst order macroscopic beam stresses in a vector, the constitutive relations of the P 1 hom problem can be de®ned as:

N 1 z 3 M 1 1 z 3 M 1 2 z 3 M 1 3 z 3 8 > > > < > > > : 9 > > > = > > > ; A hom1 Á o 3 û1 3 z 3 o 33 û0 1 z 3 o 33 û0 2 z 3 o 3 u 1 z 3 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; 52
and for later consistency of notations, relation (52) will be written in the form:

r $ 1 A hom1 Á e $ 1 53
The components of the 4 Â 4 matrix A hom1 are de®ned as follows:

A hom1 1m s 1 33m ; A hom1 2m y 1 s 1 33m A hom1 3m y 2 s 1 33m ; A hom1 4m y 2 s 1 13m y 1 s 1 23m 54
with 1 6 m 6 4. The quantities s 1 ijm , obtained after solution of the P 0 cell cellular problem, have been de®ned in Eq. ( 30).

Consequently, A hom1 11 is the stretching stiness, A hom1 22 and A hom1 33 the two bending stinesses, A hom1 44 the twisting stiness and the extra-diagonal quantities are the dierent coupling terms. Note that the eective stiness matrix A hom1 is determined from the solution of the ®rst order cellular problem.

Following [START_REF] Sanchez-Hubert | Introduction aux M ethodes Asymptotiques et a l'Homog en eisation Masson[END_REF] for transversely nonhomogeneous rods or [START_REF] Caillerie | Thin elastic and periodic plates[END_REF] for periodic plates, it can be proved that A hom1 ful®lls the symmetry conditions A hom1 mn A hom1 nm and is positive de®nite.

Boundary conditions

To complete the formulation of the P 1 hom problem, it still remains to give the boundary conditions corresponding to the both ends z 3 0, L. More precisely, one has to derive from the exact boundary conditions expressed on the 2D end sections, S e 0 and S e L , the prescribed data for the ®rst order macroscopic functions for z 3 0 and z 3 L.

As a ®rst step, let us deal with the clamped condition on S e 0 . Writing the boundary conditions u e i 0 at each power of e leads to:

u $ m 0; y $ 0 $ ; m P 0 55
which yields to the following conditions on the ®rst two terms of the asymptotic expansion: û0 a 0 0 and

û1 1 0 À y 2 u 1 0 0 û1 2 0 y 1 u 1 0 0 Vy a : û1 3 0 À y a o 3 û0 a 0 0 8 > < > : 56
Thus, relations (56) can be identically satis®ed provided that:

û0 a 0 o 3 û0 a 0 û1 3 0 u 1 0 0 57
Relation (57) hence corresponds to the displacement boundary conditions of the ®rst order homogenized problem P 1 hom . Let us deal now with the other end section S e L . Firstly, recalling that the initial 3D conditions are r e i3 r e i3 x 1 ; x 2 , and taking into account the order of magnitude of the prescribed stress data (6) yields:

r 1 a3 0 r 2 a3 r a3 y 1 ; y 2 r 1 33 r 33 y 1 ; y 2 r 2 33 0 ( r k i3 0 k P 2 58
However, the stresses r 1 and r 2 obtained from the cellular problems depend on the microscopic variables y a , and are not able to satisfy arbitrary prescribed edge data r i3 y 1 ; y 2 . Therefore, a speci®c study is necessary in order to derive the appropriate boundary conditions on this end section. This will be treated in Part II of this paper, in which a rigorous justi®cation of Saint Venant's principle is provided. The initial 3D boundary conditions are thus written as: R For the P 1 hom problem, the boundary conditions have to be expressed as a function of the macroscopic stresses fN 1 ; T 2 a ; M $ 1 g, and Eq. ( 59) leads to:

N 1 L R S L r 33 dS L ; T 2 a L R S L r a3 dS L M 1 a L R S L Ày a r 33 dS L ; M 1 3 L 0 ( 60
The proof of Eq. ( 60) is based on the property that the macroscopic stresses r 1 and T 2 a , which are obtained following an average process over the period Y, are also equal to the resultant beam forces on the right side of the period. Hence the boundary conditions (60), assuming that the structure is constituted of a whole number of periods. The relations (60) have been proposed in Cimeti ere et al. (1988), and [START_REF] Trabucho | Mathematical modelling of rods[END_REF].

Remark 5.3. It can be seen in Eq. ( 60) that there is no torque applied to the end-section for the P 1 hom problem. This result follows directly from the assumption r e a3 x 1 ; x 2 e 2 Á r a3 y 1 ; y 2 , which produces a torque at the second order. Indeed, this last assumption has been made in order to lead to a zero ®rst order shearing force at the beam end, which is compatible with the result T 1 a 0. However, this assumption might be relaxed and one might consider a distribution of r e a3 , such that the resultant shearing force remains zero but now with a nonzero resultant torque, so that M 1 3 T 0.

Summary

To summarize, the ®rst order homogenized problem, P 1 hom , consists in ®nding the macroscopic stresses fN 1 ; T 2 a ; M $ 1 g and the four macroscopic displacements fû 0 a ; û1 3 ; u 1 } such that:

o 3 N 1 hf 3 i hg 3 i oY b 0 o 3 T 2 a hf a i hg a i oY b 0 ÀT 2 a o 3 M 1 a hy a f 3 i hy a g 3 i oY b 0 o 3 M 1 3 0 N 1 z 3 M 1 1 z 3 M 1 2 z 3 M 1 3 z 3 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; A hom1 Á o 3 û1 3 z 3 o 33 û0 1 z 3 o 33 û0 2 z 3 o 3 u 1 z 3 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; û0 a 0 o 3 û0 a 0 û1 3 0 u 1 0 0 N 1 L R S L r 33 dS L ; T 2 a L R S L r a3 dS L M 1 a L R S L Ày a r 33 dS L ; M 1 3 L 0 8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : 61
Due to the positive-de®niteness of A hom1 , it can be proved that the problem ( 61) is well posed. The P 1 hom problem (61) generalizes and justi®es the Euler Bernoulli Navier's beam model, initially proposed for homogeneous isotropic rods. In the case of periodic heterogeneity, a coupled stretching bending torsion model is generally obtained. Its mathematical justi®cation, using convergence results, can be found in [START_REF] Kolpakov | Calculation of the characteristics of thin elastic rods with a periodic structure[END_REF].

It must be noticed that the equilibrium equation for the torque with the boundary condition at the end z 3 L leads to M 1 3 z 3 0. Thus, if the torsion is not coupled with stretching or bending, u 1 z 3 0 due to the boundary condition at z 3 0.

Note also that the P 1 hom problem is a 1D beam problem that can easily be solved analytically. Only the construction of the constitutive matrix A hom1 requires generally a numerical solution of the cellular problem P 0 cell posed on the period Y.

5.1.5. Treatment of the equilibrium equation of the P 1 cell problem Since the relations between the macroscopic quantities r 1 and e $ 1 are established, let us now come back to the P 1 cell cellular problem. As already explained in Section 4.4, one has to take into account the compatibility conditions when solving this problem, and the way of proceeding will be presented here.

Expressing the ®ctive volume force involved in P 1 cell in terms of the quantities o 3 e 1 p ; p P 1; 4, the equilibrium equation (31) (®rst equation) yields:

div $ y r 2 s 1 i3p y $ o 3 e 1 p z 3 e i $ f 3 e 3 $ 0 $ 62
Furthermore, recalling that A hom1 is positive de®nite, the stress strain relation ( 53) can be written as:

o 3 e $ 1 z 3 S hom1 Á o 3 r $ 1 z 3 63
where S hom1 denotes the inverse matrix of A hom1 . As proved in Section 5.1.1, the compatibility conditions for the P 1 cell problem reduce to Eq. ( 50) (®rst and fourth equation). Inserting them into Eq. ( 63), we get:

o 3 e 1 p z 3 ÀS hom1 p1 hf 3 i hg 3 i oY b S hom1 p2 o 3 M 1 1 z 3 S hom1 p3 o 3 1 2 z 3 64
As a consequence, the equilibrium equation ( 62) has to be written as:

div $ y r 2 f 3 e 3 $ À s 1 i3p y $ S hom1 p1 hf 3 i hg 3 i oY b e i $ s 1 i3p y $ fS hom1 p2 A hom1 2m S hom1 p3 A hom1 3m g o 3 e 1 m e i $ 0 $ 65
with summation on the repeated indices, i P 1; 3 and p; m P 1; 4 2 .

Remark 5.4. The relation ( 65) gives the exact de®nition of the functions h 1

$ , h 2 $ , h 3 $
, introduced in Eq. ( 34) in Section 4.4. Thus, it is obvious that h

2 $ f 3 Às 1 i3p y $ S hom1 p1 hf 3 ie i $ , h 3 $ g 3 Às 1 i3p y $ S hom1 p1 hg 3 i oY b e i $
and that h 1 $ is given by the last terms of Eq. ( 65).

In the case of a constitutive law A hom1 without any coupling, the relation (65) can be simpli®ed in the following manner:

div $ y r 2 f 3 e 3 $ À s 1E i3 y $ A hom1 11 À1 hf 3 i hg 3 i oY b e i $ s 1Ca i3 y $ o 333 û0 a z 3 e i $ 0 $ 66
5.2. Formulation of the kth homogenized problem P k hom

Equilibrium equations

In the preceding section, it has been shown how to derive the ®rst order homogenized problem P 1 hom . By applying exactly the same method for each order k > 1, one obtains the formulation of the higher-order homogenized problems, P k hom . Hence the equilibrium equations of the P k hom problems:

o 3 N k hf k 3 i hg k1 3 i oY b 0 67 o 3 T k1 a hf k1 a i hg k2 a i oY b 0 68 ÀT k1 a À o 3 M k a hy a f k 3 i hy a g k1 3 i oY b 0 69 o 3 M k 3 hy 1 f k 2 À y 2 f k 1 i hy 1 g k1 2 À y 2 g k1 1 i oY b 0 70 with N k z 3 hr k 33 i; T k1 a z 3 hr k1 a3 i M k
a z 3 hÀy a r k 33 i; M k 3 z 3 hÀy 2 r k 13 y 1 r k 23 i

71

N k , M k b and M k 3 correspond respectively to the macroscopic axial force, bending moments and twisting moment of order k, while T k1 a represent the macroscopic shearing forces of order k 1. We recall that the prescribed volume force f $ k and surface force g $ k1 satisfy assumptions (6).

Constitutive relations

Let us now study the macroscopic stress strain relation of order k. We have seen in Eq. ( 43) that the stress ®eld r k , solution of the P kÀ1 cell problem, is a linear function of the kth order macroscopic strain e $ k , the ®rst-gradient of the k À 1th order macroscopic strain (i.e. o 3 e $ kÀ1 ), and so on until the k À 1th-gradient of the ®rst order macroscopic strain (i.e. o kÀ1 3 e $ 1 ). Therefore, the macroscopic stress strain relation at any order k with k P 1 can be written as:

r $ k A hom1 Á e $ k A hom2 Á o 3 e $ kÀ1 A hom3 Á o 33 e $ kÀ2 Á Á Á A hom k Á o kÀ1 3 e $ 1 r $ k part with r $ k z 3 t fN k ; M k 1 ; M k 2 ; M k 3 g 72
where strain vectors e $ p vanish when p 6 0, and where o kÀ1 cell problem. This solution is obtained considering as data the volume and surface forces involved in the current problem, if any, as well as the derivatives of the particular solutions obtained at the preceding orders, grad s 54). In a similar way, the components of the 4 Â 4 matrix A hom k are deduced from the four elementary stress solutions of the k À 1th order cellular problem, namely s kE , s kCa and s kT , grouped in s k . We recall that these stress tensors correspond to the solution of P kÀ1 cell when the components of the k À 1th gradient of e $ 1 are respectively considered as data, i.e. o k 3 û1

3 z 3 , o k1 3 û0 a z 3 , and o k 3 u 1 z 3 . Thus, A hom k is de®ned as:

A hom k 1m hs k 33m i; A hom k 2m hÀy 1 s k 33m i A hom k 3m hÀy 2 s k 33m i; A hom k 4m hÀy 2 s k 13m y 1 s k 23m i 73
Contrary to the ®rst order eective stiness matrix A hom1 , the higher-order stress strain matrices A hom k , k P 2, are not necessarily symmetric or positive de®nite tensors. Especially, the second order one, A hom2 , appears to be antisymmetric in 3D periodic media and even equal to zero following certain symmetry properties of the period Y [START_REF] Boutin | Microstructural eects in elastic composites[END_REF][START_REF] Triantafyllidis | The in¯uence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models[END_REF].

Remark 5.5. Solving the kth order homogenized problem P k hom implies that the lower-order macroscopic problems have already been solved. Therefore, when considering the P k hom problem, the macroscopic strains e $ 1 ; . . . ; e $ kÀ1 are known and so are their successive gradients. Therefore, the only unknown strain ®eld in the right-hand side of relation ( 72) is the kth order strain vector e $ k . All the other terms constitute data for the P k hom problem and can be considered as ®ctive initial stress states for the current macroscopic problem. As a matter of fact, following the method presented in [START_REF] Boutin | Microstructural eects in elastic composites[END_REF], the ®rst equilibrium equation (67) of the P k hom problem may be written as:

o 3 A hom1 1m Á e k m Àhf k 3 i À hg k1 3 i oY b À o 3 A hom2 1m Á o 3 e kÀ1 m A hom3 1m
Á o 33 e kÀ2 m Á Á Á with 1 6 m 6 4: 74

Writing all the equilibrium equations of P k hom in a similar way shows that this problem may be regarded as the coupled stretching bending torsion model of Section 5.1.4, the higher-order eects arising under the form of ®ctive volume loadings. It becomes also clear that the displacement unknowns of the P k hom problem are the four macroscopic quantities fû kÀ1 a ; ûk 3 ; u k g.

Boundary conditions

To complete the formulation of the P k hom problem, one must add to equilibrium equations (67) (70) and constitutive relations (72) the boundary conditions which have to be expressed on the displacements fû kÀ1 a ; ûk 3 ; u k g for z 3 0 on one hand, and on the kth order macroscopic stresses fN k ; T k1 a ; M $ k g for z 3 L on the other hand.

These conditions, which are obtained from the initial 3D conditions on the two end sections S e 0 , S e L , are given in Eqs. ( 55) and ( 58). Since we are interested in the P k hom problem with k P 2, it can be seen from Eqs. ( 42) and ( 43) that it is impossible to ful®ll these conditions exactly, so that boundary layers arise at the two ends of the beam. This is a classical problem in asymptotic analysis of slender structures. In Part II of this paper, a method is proposed to derive the macroscopic boundary conditions at each order, so that wellposed P k hom problems are obtained.

Summary

Let us summarize here the results provided by the formal asymptotic method. The solution in series of the ®rst k cellular problems, P À1 cell to P kÀ1 cell , leads to the determination of the y 3 -periodic displacement ®elds fv 1 y $ ; . . . ; v k y $ g, as well as the associated periodic stress ®elds fs kE ; s kCa ; s kT g. Then, following the average process given in Eq. ( 73), the ®rst k eective matrices A hom k can be calculated.

Treating in parallel the ®rst k macroscopic problems, P 1 hom to P k hom , leads to the macroscopic parts of the asymptotic expansions (42) and (43). Especially, the solution of the homogenized problems up to the kth order, P k hom , gives the macroscopic axial displacement ûp 3 z 3 and the macroscopic torsion rotation u p z 3 up to order k, as well as the macroscopic de¯ections ûp a z 3 up to order k À 1. Thus, after having solved in series the cellular and the homogenized problems, one obtains from Eqs. ( 42) and ( 43), both local and global information on the solutions u $ e , r $ e of the initial problem. Particularly, the macroscopic description of the displacements of the structure is given by:

ûa z 3 û0 a z 3 eû 1 a z 3 Á Á Á û3 z 3 û1 3 z 3 eû 2 3 z 3 Á Á Á uz 3 u 1 z 3 eu 2 z 3 Á Á Á 75

Concluding remarks

In this paper, it is shown that the asymptotic expansion method provides a rigorous and systematic way to derive the overall response of a periodic heterogeneous beam. Especially, the macroscopic description of the displacement ®eld is given by u $ e , de®ned as:

u $ e z 3 ; y $ u $ 0 z 3 ; y $ e u $ 1 z 3 ; y $ e 2 u $ 2 z 3 ; y $ Á Á Á 76
so that the components of u $ e are given by: u e 1 z 3 ; y

$ û1 z 3 À ey 2 uz 3 u e 2 z 3 ; y $ û2 z 3 ey 1 uz 3 u e 3 z 3 ; y $ eû 3 z 3 À ey a o 3 ûa z 3 8 > > > > < > > > > : 77
with ûi z 3 and uz 3 de®ned in Eq. ( 75).

The determination of the global ®eld u $ e may be achieved through a rational calculation of the successive terms of the interior expansions. Thus it is necessary to solve in series several 3D microscopic problems as well as 1D homogenized problems to ®nd u $ e up to a certain desired order. The cellular problems allow us to characterize the beam response at the period scale, under dierent macroscopic loadings corresponding to macroscopic strains and their derivatives. Thus the eective beam behavior is obtained.

Nevertheless, it should be more judicious to de®ne one homogenized problem which would enable us the derivation in a single step of u $ e up to the desired order. To this end, let us derive from the successive homogenized problems P k hom the 1D equations involving the unknowns ûi z 3 and uz 3 of u $ e . Introducing the ®eld u $ e into the expansion (42), we see that the displacement ®eld solution of the initial problem (3) can be written as:

u $ e z 3 ; y $ u $ e z 3 ; y $ ev 1 y $ Á e $ u $ e e 2 v 2 y $ Á o 3 e $ u $ e e 3 v 3 y $ Á o 33 e $ u $ e Á Á Á u $ part with e $ u $ e e $ 1 e e $ 2 Á Á Á t fo 3 û3 ; o 33 û1 ; o 33 û2 ; o 3 ug and u $ part e 3 u $ 3 part e 4 u $ 4 part Á Á Á 78
In the same way, Eq. ( 43) can be written as follows:

r e es 1 Á e $ u $ e e 2 s 2 Á o 3 e $ u $ e e 3 s 3 Á o 33 e $ u $ e Á Á Á r part 79
Moreover, the expansion of the macroscopic beam stresses r $ e is de®ned by: r $ e r $ 1 e r $ 2 Á Á Á 80

and one has for the transverse shearing forces:

T e a T 2 a eT 3 a Á Á Á 81
Thus, from Eq. ( 72) and from the addition of the equilibrium equations at each order, the macroscopic ®elds u $ e , r $ e and T e a are found to satisfy:

o 3 N e hf 3 i hg 3 i oY b 0 o 3 T e a hf a i hg a i oY b 0 ÀT e a À o 3 M e a hy a f 3 i hy a g 3 i oY b 0 o 3 M e 3 hy 1 f 2 À y 2 f 1 i hy 1 g 2 À y 2 g 1 i oY b 0 8 > > > < > > > : 82 r $ e t fN e z 3 ; M e 1 z 3 ; M e 2 z 3 ; M e 3 z 3 g A hom1 Á e $ u $ e eA hom2 Á o 3 e $ u $ e e 2 A hom3 Á o 33 e $ u $ e Á Á Á r $ part 83
It is interesting to note that the macroscopic stress strain relation (83) contains strain gradients up to in®nite order. As a consequence, the macroscopic description obtained when taking into account higher-order terms can be regarded as a higher-order gradient theory. This result was already pointed out in [START_REF] Gambin | Higher-order terms in the homogenized stress strain relation of periodic elastic media[END_REF] in the case of 3D elastic periodic media. Indeed, for such problems, the asymptotic expansion method including higher-order terms brings out the contribution of nonlocal terms under the form of the successive strain gradients, exactly as in Eq. ( 83). Furthermore, if we restrict the study to the ®rst three terms of the expansion of r $ e , then the stress strain relation (83) appears to generalize the well-known second gradient theory. Moreover if A hom2 0 (which is obtained when the period Y e presents certain elastic symmetries), we recover exactly the latter theory. This comparison between the higher order theory derived from the asymptotic expansion method and the second gradient theory is widely discussed in [START_REF] Boutin | Microstructural eects in elastic composites[END_REF] for 3D periodic media.

Another interesting point of view is to draw a parallel between the global model given by Eqs. ( 82) and ( 83) and re®ned beam theories, i.e. more sophisticated 1D beam theories than Euler Bernoulli's one. In that way, in the case of homogeneous isotropic rods, Timoshenko's model can be recovered and thus justi®ed via the asymptotic expansion method. This justi®cation is given in [START_REF] Fan | Re®ned engineering beam theory based on the asymptotic expansion approach[END_REF] or [START_REF] Trabucho | Mathematical modelling of rods[END_REF], where a generalization of Timoshenko's theory is also established for the isotropic nonhomogeneous case. Let us outline here the way of proceeding to recover Timoshenko's theory from the general asymptotic model Eqs. ( 82) and (83). To this end, we consider the case of bending of a homogeneous isotropic rod in one of its principle planes e 1 $ À e 3 $ , by the external forces f e 1 ; 0; 0 and g e 1 ; 0; 0 verifying (6), and with a clamping condition at the both ends S e 0 , S e L . Our aim is to derive the second order model associated to the approximation of the expansions u $ e , r $ e up to the second nonzero term. The ®rst two eective terms of the macroscopic beam forces are found to be r $ e r $ 1 e 2 r $ 3 , and from Eq. ( 82),

o 3 N 1 e 2 N 3 0 o 3 T 2 1 e 2 T 4 1 R S f 1 dS R oS g 1 dc 0 T 2 1 e 2 T 4 1 o 3 M 1 1 e 2 M 3 1 0 8 > < > : 84
since, for the beam under consideration here, the operators hÁi and hÁi oY b can be reduced to R S Á dS and R oS Á dc, where S and oS stand for the scaled beam cross-section and its lateral boundary respectively. Considering moreover the case where the bending does not give rise to either torsion or tension eects (symmetric bending without any coupling), and given that A hom2 is zero for a homogeneous beam, the stress strain relation (83) leads to: where k denotes the shear correction factor introduced by Timoshenko. Therefore, the comparison between Eqs. ( 87) and ( 88) enables us the de®nition of a shear coecient k from A hom1 22 and e 2 A hom3 22 (which corresponds to the descaled eective behavior). Nevertheless, it must be noticed that several terms in the asymptotic expansion (42) of the complete de¯ection u e a z 3 ; y a have to be neglected in order that Eq. ( 87) reduces to Timoshenko's theory. Especially, Poisson's eects as well as geometrical torsional eects are neglected. The second order model ( 82) and (83) thus incorporates 3D eects which are not taken into account in the classical Timoshenko theory.

Generally speaking, the asymptotic expansion method has the advantage of taking into account, in a consistent and systematic way, nonclassical eects such as cross-sectional warping, as well as transverse shear and normal stresses and strains. This is a major dierence from existing higher-order beam theories (see e.g. [START_REF] Kosmatka | Transverse vibrations of shear-deformable beams using a general higher order theory[END_REF], [START_REF] Reddy | Relationships between bending solutions of classical and shear deformation beam theories[END_REF], [START_REF] Soldatos | A general theory for the accurate stress analysis of homogeneous and laminated composite beams[END_REF] and references herein) which are based on a priori assumptions regarding stress and displacement variations. These theories are found to capture only a part of the correction due to higher-order eects, as it was proved previously for the second order Timoshenko theory.

Moreover, since approximate boundary conditions have to be considered, edge eects are an important source of errors in re®ned engineering theory [START_REF] Duva | The usefulness of elementary theory for the linear vibrations of layered, orthotropic elastic beams and corrections due to two-dimensional end eects[END_REF]. On the contrary, the asymptotic expansion method enables us to obtain an outer solution which is valid far from the edges (see Part II of this paper).

  Fig. 1. 3D slender periodic structure X e .

$

  linear functions. The latter expression is such that if in the problem (31) only the data involving f 3 and g 3 , i.e. the body forces h 2 $ f 3 h 3 $ g 3 f 3 e 3 $ and tractions g 3 e 3 $ on oY b , then a well-posed problem is found. In the same way, the problems involving the other data, i.e. the body forces h 1 also well posed. A more complete treatment of that question will be given later (in Section 5.1.5), once the compatibility relations of the problem (31) have been expressed. Let us study now the form of the solutions u $ 3 and r 2 of the problem (

Fig. 3 .

 3 Fig. 3. Element of the equivalent beam (all loads and stress resultants are shown in their positive directions).

  o

3

  denotes the partial derivative o kÀ1 =oz kÀ1 3 . The kth order stress vector r $ k part contains the beam forces deduced from the stress state s $ k part , i.e. the particular solution of the well-posed P kÀ1

  The 4 Â 4 matrix A hom1 has already been de®ned in Eq. (

  where E denotes the Young's modulus. Consequently, the equilibrium equations (84) (second and third equations) yield: relation (87) is found to be exactly of the form of the dierential equation for Timoshenko's beam theory (u T 1 stands for the beam de¯ection):

Acknowledgements

The authors gratefully acknowledge Professor J.G. Simmonds (University of Virginia) for fruitful discussions, useful advices, and encouragements during the preparation of this work. The authors also wish to express their appreciation to Dr. S. Bourgeois (LMA Marseille, France), for helpful ideas and comments.