Higher-order effective modelling of periodic heterogeneous beams - Part I : Asymptotic expansion method

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Higher-order effective modelling of periodic

heterogeneous beams - Part I : Asymptotic expansion

method

Natacha Buannic, Patrice Cartraud

To cite this version:

Natacha Buannic, Patrice Cartraud. Higher-order effective modelling of periodic heterogeneous beams

- Part I : Asymptotic expansion method. International Journal of Solids and Structures, Elsevier, 2001,

38 (40 41), pp.7139-7161. �10.1016/S0020-7683(00)00422-4�. �hal-01006929�

Higher-order e€ective modeling of periodic heterogeneous

beams. I. Asymptotic expansion method

Natacha Buannic, Patrice Cartraud

Laboratoire de Mecanique et Materiaux, Ecole Centrale de Nantes, 1 Rue de la Noe, BP 92101, 44321 Nantes Cedex 3, France eceived pril in revised form ovember

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 sys-tematic 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 e€ects is thus underlined. Ó lsevier cience td. ll rights reserved.

Keywords: Asymptotic analysis; Beams; Constitutive model; E€ective property; Homogenization methods; Timoshenko

1. Introduction

Heterogeneous structures consisting of elements arranged periodically are widely used in civil engi-neering 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 homo-geneous 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 be-havior. Such structures possess two small parameters: e, which measures the ratio of the width of the cross-section 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 Geymonat et al. (1987). The authors established that for a beam with a variable cross-section (transversal heterogeneity), di€erent one-dimensional (1D) homogenized models are obtained when letting e ! 0 then e ! 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 di€erent e€ective sti€-nesses. The method consisting in letting ®rst e tend to zero and then e amounts to study a Bernoulli's beam with rapidly varying properties (Cioranescu and Saint Jean Paulin, 1999; Miller, 1994). Following the other method, we take the limit with respect to e ®rst (which corresponds to averaging the e€ect 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 Caillerie (1984) for periodic plates, and extended to the case of periodic beams in Kolpakov (1991) and Kalamkarov and Kolpakov (1997). 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 ex-plained in Kolpakov (1991), 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 di€erent examples shows that its domain of validity can be enlarged (Buannic and Cartraud, 1999). 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 ! 0 then e ! 0) and method 2 (e ' e ! 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 (Bourgeois, 1997). This is in the same line as the conclusion drawn in Lewinski (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 pe-riodic 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 Cimetiere et al. (1988) in the nonlinear case, Rigolot (1976), Fan and Widera (1990) or Trabucho and Via~no (1996) in the linear case for an arbitrary cross-section, and Duva and Simmonds (1991) for a narrow rectangular cross-section treated in plane stress analysis. The case of transversely nonhomogeneous isotropic rods is also treated in Trabucho and Via~no (1996). 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 e€ects 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=oz3, o2=oz23

and o3_=oz3

3will be denoted o3, o33and o333.

2. The initial three-dimensional problem

The 3D slender structure Xe_{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 x1ˆ x2ˆ 0.

The periodicity cell Ye _{is de®ned by (see Fig. 2):}

Ye_{ˆ x}   ˆ …xi†=le1 …x2; x3† < x1< le1‡…x2; x3†; le2 …x1; x3† < x2< le2‡…x1; x3†; l e 3 2 < x3< le 3 2  …1† where the functions le

aare assumed to be periodic in x3with period le3. Let Yebe the solid part of the cell

with boundary oYe_{(see Fig. 2) such that oY}e_{ˆ oY}e

a [ oYbe[ oYcewith oYaethe plane surfaces perpendicular

to the e_3 direction, oYe

b the lateral outer boundary of the cell and oYceits inner boundary (cell holes).

The elastic moduli of the beam, ae

ijkl…x†, are periodic in x3 with period le3, and satisfy the following

classical relations: …i† ae

ijkl…x† ˆ aejikl…x† ˆ aeklij…x†; x2 Xe

…ii† 9 m > 0 such that 8s=sijˆ sji; msijsij6 aeijkl…x†sijskl

…iii† 9 M such that M ˆ sup ae

ijkl…x†; x2 Xe

…2† The boundary of the domain Xe _{is de®ned by oX}e_{ˆ S}e

0[ SLe[ Ceb[ Cec, with S0e and SLe the two end

sec-tions of the beam and Ce

b, Cec obtained from the periodic repetition of oYbe, oYce respectively (see Fig. 1).

Fig. 1. 3D slender periodic structure Xe_.

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 Ce

c is supposed to be free of traction. The left end Se0 is clamped and stress data

re

3i…x1; x2† are prescribed on the right end SLe.

The static problem Pe_{of linear elasticity consists in ®nding the ®elds r}e_{, e}e_{and u}

e, such that: div_{ x}re_{ˆ f} e re_{ˆ a}e_x † : ee…ue† ee_u e† ˆ gradsx…ue† re
_n ˆ g_e on Ceb re
_n ˆ 0 on Cec re
_e 3 ˆ re3i…x1; x2†
e_i on SLe u eˆ 0 on S0e 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : …3† where grads

xand div xcorrespond 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 Pe_{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 re_{, 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…x1; x2† are suciently smooth, and the boundary oXeregular.

3. 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 Caillerie (1984) and Kolpakov (1991), 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:

…z1; z2; z3† ˆ x_e1;x_e2; x3   ; …y1; y2; y3† ˆ z1; z2;z_e3   ˆ1_e…x1; x2; x3† since e ˆ e …4† Consequently, z3represents the slow or large scale or macroscopic variable of the problem and y_ˆ 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 fol-lowing manner:

grads

x: ˆ gradsz3: ‡ 1 egradsy:

div_{ x}: ˆ div_{ z}3: ‡1ediv y:

(

…5† where grads

z3and div z3 correspond to partial di€erentiations with respect to the only variable z3, while gradsy

and div_{ y} are the di€erential operators with regard to the three microscopic variables yi.

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:

fe 3…x1; x2; x3† ˆ e1
f3…z3; y1; y2† ge3…x1; x2; x3† ˆ e2
g3…z3; y1; y2† fe a…x1; x2; x3† ˆ e2
fa…z3; y1; y2† gea…x1; x2; x3† ˆ e3
ga…z3; y1; y2† re 33…x1; x2† ˆ e1
r33…y1; y2† rea3…x1; x2† ˆ e2
ra3…y1; y2† …6† Furthermore, the elasticity moduli ae

ijkl are assumed to be independent of e, so we have:

ae_x

† ˆ a…y_† …7†

Remark3.1. The homogenized limit 1D model depend on the orders of magnitude of the applied loadings with respect to e, for example Karwowski (1990) for the theory of asymptotic modeling of rods, or Caillerie (1980) and Millet (1997) for the case of plates. The assumptions (6) and (7) are usually made in order that the limit behavior (e ! 0) of the 3D slender structure is that of a beam (Trabucho and Via~no, 1996; Cimetiere et al., 1988).

Third, following a standard technique, the solution u_e_{of the …P}e_{† problem is sought in the form used in}

Kolpakov (1991): u

e…x† ˆ ^u0a…z3†e_a‡ eu_1…z3; y_† ‡ e2u_2…z3; y_† ‡


…8†

where every function u_k_{z3; yi† is periodic in the variable y3with period l3…l3ˆ l}e

3=e†, which will be denoted

y3-periodic in the following.

Consequently, using Eq. (5) (®rst equation) and the constitutive relations, the strains and stresses ex-pansions are given by:

ee_x

† ˆ e0…z3; y_† ‡ ee1…z3; y_† ‡ e2e2…z3; y_† ‡

re_x

† ˆ r0…z3; y_† ‡ er1…z3; y_† ‡ e2r2…z3; y_† ‡

…9† Remark3.2. The form of the ®rst term of the expansion (8), which is composed of only the de¯ections ^u0

a…z3†, is not an assumption, in the sense that applying the asymptotic method with the relations (6) leads to

that expression of u_e _{(Trabucho and Via~no, 1996). A similar result has been established in the case of}

periodic plates (Caillerie, 1984), where the ®rst term of the expansion is found to be reduced to the de-¯ection of the middle plane (i.e. u0

3…za†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 (Trabucho and Via~no, 1996; Cioranescu and Saint Jean Paulin, 1999). For example, in Trabucho and Via~no (1996), the authors associate with the displacement ®eld u_e_x

†

the scaled functions u_…z3; y_†…e† through the following scalings:

ue a…x† ˆ e 1ua…z3; y_†…e† ue 3…x† ˆ u3…z3; y_†…e† 8 < : …10†

and these scaled functions are then assumed to have the asymptotic expansion: ui…z3; y_†…e† ˆ u0i…z3; y_† ‡ eui1…z3; y_† ‡ e2u2i…z3; y_† ‡

The justi®cation of the scalings (10) is purely mathematical and is related to convergence results of the asymptotic method as e ! 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 Pe_{problem (3) and equating the terms of a same order}

with respect to e, we replace the problem Pe _{by a family of problems. The ®elds involved in the latter are}

functions of the two kinds of variables yiand z3, but no longer depend on the small parameter e. So, when

treating the z3- and yi-coordinates as independent, and considering the ®elds function of the only variable z3

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 Pk

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 Pk

cell problems, we obtain the formulation of homogenized 1D-macroscopic problems,

denoted by Pk

hom, the solution of which gives the macroscopic (i.e. nonperiodic) parts of the ®elds (8) and (9).

The formulation and solution of the Pk

hom problems are treated in Section 5.

4. The set of cellular problems Pk 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

cellproblems, with

k starting from 1.

4.1. General formulation of the kth cellular problem Pk cell

For an arbitrary power k of the small parameter e, the Pk

cellproblem is posed on the ®xed period Y de®ned

as:

Y ˆ y





ˆ …yi†=l1 …y2; y3† < y1< l1‡…y2; y3†; l2 …y1; y3† < y2< l2‡…y1; y3†; l₂3< y3<l₂3

 with la…yb; y3† ˆ lea…eyb; ey3†=e and l3ˆ le3=e;

and with boundary oY _{such that oY}_{ˆ oYa[ oYb[ oYc. oYa, oYb, oYc denote the scaled lateral boundary}

surface obtained from oYe

a, oYbe, oYcerespectively.

The Pk

cellproblem consists in ®nding the ®elds rk‡1, ek‡1 and uk‡2satisfying the following equations:

div_{ y}rk‡1_{ˆ f} k div z3rk rk‡1_{ˆ a…y} † : e k‡1 ek‡1_{ˆ grad}s

y…u_k‡2† ‡ gradsz3…uk‡1

rk‡1
_n ˆ g_k‡1 on oYb rk‡1
_n ˆ 0 on oYc rk‡1 i3 and uk‡2 y3-periodic 8 > > > > > > > > > > > > < > > > > > > > > > > > > : …12†

with k P 1 and where the negative powers of rk _{and e}k _{vanish. f}

k and g_k correspond respectively to the

body and surface densities of forces which occur at the order ek_{. We recall that these forces are assumed to}

obey the relations (6), so that fk

boundary conditions (12) (sixth equation) result from the structure periodicity, see Eq. (8), and due to the opposite values of rk‡1
_n

on opposite sides of oYa where nˆ e_3.

Remark4.1. When solving the cellular problem Pk

cellat order k, we consider that the preceding Pcellk 1problem

has already been solved and thus that the ®elds rk _{and u}

k‡1 have been determined. Consequently, the

parameters div_{ z}3rkand gradsz3…uk‡1† constitute macroscopic given ®elds for the current problem Pcellk : 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 † ˆ fw

 2 ‰H

1_{Y †Š}3_{; y}

3-periodicg. The Pcellk problem is equivalent to: ®nd the

dis-placement ®eld u_k‡2 _{belonging to W …Y † such that}

8w  2 W …Y †; Z Yr k‡1_{: grad}s y…w_†dY ˆ Z Y…div z3r k_{‡ f} k†
w_dY ‡ Z oYb g  k‡1
_w dC …13†

where the stress ®eld rk‡1_{is related to the displacement ®eld u}

k‡2following Eq. (12) (second equation) and

where dY ˆ dy1dy2dy3. According to the variational form Eq. (13), it is easy to show that the Pcellk problem

possesses a solution provided that the data div_{ z3}rk_{, f}

k, g_k‡1 verify the following relation:

8v_2 R; Z Y…div z3r k_{‡ f} k†
vdY ‡ Z oYb g  k‡1
_v dC ˆ 0 …14†

where R corresponds to the set of the y3-periodic rigid body motions for the period Y, and is given by:

R ˆ fv_…z3; y_†=v_ˆ ^vi…z3†
e_i‡ u…z3†‰y1
e_2 y2
e_1Šg …15†

Under the necessary condition (14), the solutions rk‡1_{, e}k‡1_{and u}

k‡2(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.}

4.2. Solution of the cellular problem P 1 cell

The ®rst cellular problem occurs for k ˆ 1. Since we have assumed that no force f_ 1_{or g} 

0_{is applied at}

this order, it can be written as follows: div_{ y}r0_{ˆ 0}  r0_{ˆ a…y} † : e 0 e0_{ˆ grad}s

y…u1† ‡ gradsz3…u0†

r0
_n ˆ 0 on oYb[ oYc r0 i3 and u1 y3-periodic 8 > > > > > > > > < > > > > > > > > : …16†

The only data of the problem are thus contained in the tensor grads

z3…u0† and, according to the form of the

®eld u_0_{, we have:} grads z3…u0† ˆ 0 0 1 2o3^u01…z3† 0 1 2o3^u02…z3† sym 0 2 6 6 4 3 7 7 5 …17†

where o3^u0a…z3† are the two macroscopic data of the Pcell1problem 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

1partˆ ya
o3^u0a…z3†
e_3 and r0ˆ e0ˆ 0 …18†

In that sense, the two data o3^u0a…z3† do not constitute e€ective data, since the associated solution

corre-sponds to a zero deformation state (Sanchez-Hubert and Sanchez-Palencia, 1992).

The displacement ®eld given in Eq. (18) is obtained up to an element of R, so the complete solution of the P 1

cellproblem has to be written:

u

1ˆ ^u1i…z3†
e_i‡ u1…z3†‰y1
e_2 y2
e_1Š ya
o3^u0a…z3†
e_3 u_1…z3; y_† …19†

4.3. Solution of the zeroth order cellular problem P0 cell

Since r0_{ˆ 0, the P}0

cell problem consists in ®nding the ®elds r1, e1and u2 which satisfy:

div_{ y}r1_{ˆ 0}  r1_{ˆ a…y} † : e 1 e1_{ˆ grad}s

y…u2† ‡ gradsz3…u1†

r1
_n ˆ 0 on oYb[ oYc r1 i3 and u2 y3-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: grads z3…u1†  0 0 1 2…o3^u11…z3† y2o3u1…z3†† 0 1 2…o3^u12…z3† ‡ y1o3u1…z3†† sym …o3^u13…z3† yao33^u0a…z3†† 2 6 6 6 4 3 7 7 7 5 …21†

The two data o3^u1a…z3† will provide a direct solution u_2part similar to expression (18). The four other data

which are contained in grads

z3…u1†, namely o3^u13…z3†, o33^u0a…z3†, o3u1…z3†, correspond respectively to a

mac-roscopic extension, two macmac-roscopic curvatures and a macmac-roscopic 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 e€ective data.}

Adding the direct solution u_2

part provided by the two other data o3^u1a…z3† as well as the rigid motion, the

complete displacement ®eld at the second order assumes the following form: u

2ˆ u2‡ v_1E…y_†
o3^u13…z3† ‡ v_1Ca…y_†
o33^u0a…z3† ‡ v_1T…y_†
o3u1…z3† …22†

where u_2_z

3; y_† ˆ ^u2i…z3†
e_i‡ 2…z3†‰y1
e_2 y2
e_1Š ya
o3^u1a…z3†
e_3. For later consistency of notations,

we introduce the four-components vector e_1_z

3† and the 3  4 matrix v1…y_† so that we have:

u

2ˆ u 2_z

3; y_† ‡ v1…y_†
e_1…z3† …23†

e 1…z3† ˆt fo3^u13…z3†; o33^u10…z3†; o33^u02…z3†; o3u1…z3†g …24† v1_y † ˆ ‰v 1E_y †; v 1C1_y †; v 1C2_y †; v 1T_y †Š …25†

In Eq. (24), the four e€ective data have been grouped in the vector e_1…z3†, with the result that the latter

represents the ®rst order macroscopic strain vector.

Remark4.2. The problem (20) does not have an analytical form solution for the unknown v1_y

† in general,

except in the case of homogeneous rods, see for example Trabucho and Via~no (1996) for an isotropic material.

In the same manner as the displacement ®eld, the stress ®eld r1_{solution of P}0

cell has a linear expression

with regard to the data: r1_{ˆ s}1E_y †
o3^u 1 3…z3† ‡ s1Ca…y_†
o33^u0a…z3† ‡ s1T…y_†
o3u1…z3† …26† with s1E ij ˆ aij33‡ aijkhoyhv1Ek s1Ca ij ˆ yaaij33‡ aijkhoyhv1Ck a s1T

ij ˆ y2aij13‡ y1aij23‡ aijkhoyhv1Tk

8 > > < > > : …27†

which will be formally denoted as: r1_{ˆ s}1_y

†
e 1_z

3† …28†

where s1_y

† corresponds to the regrouping of the four elementary stress tensors s

1E_{, s}1Ca, s1T so that:

r1

ijˆ s1ij1…y_†
o3^u13…z3† ‡ s1ij2…y_†
o33^u01…z3† ‡ sij31 …y_†
o33^u02…z3† ‡ s1ij4…y_†
o3u1…z3† ˆ s1ijme1m; m 2 ‰1; 4Š

…29† with

s1

ij1ˆ s1Eij; s1ij2 ˆ s1Cij1; s1ij3 ˆ s1Cij2; s1ij4 ˆ s1Tij …30†

4.4. Solution of the ®rst order cellular problem P1 cell

It follows from Eqs. (6), (22) and (26) that the P1

cellproblem comprises the following relations:

div_{ y}r2_{ˆ div}  z3r1 f3
e_3 r2_{ˆ a…y} † : e 2 e2_{ˆ grad}s

y…u3† ‡ gradsz3…u2†

r2
_n ˆ g3e3 on oYb r2
_n ˆ 0 on oYc r2 i3 and u3 y3-periodic 8 > > > > > > > > > > > > < > > > > > > > > > > > > : …31† with

div_{ z}3r1ˆ …s1Ei3…y_†
o33^u13…z3† ‡ s1Ci3a…y_†
o333^u0a…z3† ‡ s1Ti3…y_†
o33u1…z3††
e_i …32† and grads z3…u2† ˆ 0 0 1 2…o3^u21…z3† y2o3u2…z3†† 0 1 2…o3^u22…z3† ‡ y1o3u2…z3†† sym …o3^u23…z3† yao33^u1a…z3†† 2 6 6 6 4 3 7 7 7 5 ‡ 0 0 1

2…v1E1 …y_†o33^u13…z3† ‡ v1C1 a…y_†o333^u0a…z3† ‡ v1T1 …y_†o33u1…z3††

0 1

2…v1E2 …y_†o33^u13…z3† ‡ v1C2 a…y_†o333^u0a…z3† ‡ v1T2 …y_†o33u1…z3††

sym …v1E

3 …y_†o33^u13…z3† ‡ v1C3 a…y_†o333^u0a…z3† ‡ v1T3 …y_†o33u1…z3††

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}3r1, …0; 0; f3† and …0; 0; g3†

satisfy the relation (14). From Eq. (32), div_{ z}3r1 can be expressed as a function of o3e_

1_{, which is the ®rst}

gradient of the strains e_1…z3†. Thus, the compatibility conditions (14) lead to a relation between o3e_1and f3

and g3, which enables us to express the ®ctive volume force div_{ z}3r1 in the form:

div_{ z}3r1ˆ h_1…o3e_1† ‡ h_2…f3† ‡ h_3…g3† …34†

where h_1; h_2; h_3are linear functions. The latter expression is such that if in the problem (31) only the data

involving f3and g3, i.e. the body forces h_2…f3† ‡ h_3…g3† ‡ f3e_3 and tractions g3e_3on oYb, then a well-posed

problem is found. In the same way, the problems involving the other data, i.e. the body forces h_1…o3e_1† and

the initial strain state grads z3…u

2

†, are 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 (14), which can be linearly expressed with}

respect to the data, in the same manner as at the preceding orders.

Firstly, the solution of the well posed elementary problem corresponding to the prescribed data f3, g3

only is denoted by u_3

part. The other data of the problem come from Eq. (33) and h_1…o3e_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 P0

cellproblem, except that the

superscripts have increased by one. As a consequence, the set of these six data, namely o3^u2i…z3†, o33^u1a…z3†,

o3u2…z3†, leads to the same displacement solutions as those obtained by solving the Pcell0 problem. Thus,

besides u_3

part, the only new unknowns of the current problem are the solutions corresponding to the

de-rivatives of the ®rst order macroscopic strains de®ned in Eq. (24). Consequently, the displacement ®eld solution of the P1

cellproblem can be formally written as follows:

u 3ˆ u3…z3; y_† ‡ v1…y_†
e 2_z 3† ‡ v2…y_†
o3e_1…z3† ‡ u_3part…z3; y_ …35† where u 3…z3; y_† ˆ ^u3i…z3†
e_i‡ u3…z3†‰y1
e_2 y2
e_1Š ya
o3^u2a…z3†
e_3; …36† e 2…z3† ˆtfo3^u23…z3†; o33^u11…z3†; o33^u12…z3†; o3u2…z3†g; …37†

v2_y † ˆ ‰v 2E_y †; v 2C1_y †; v 2C2_y †; v 2T_y †Š; …38† o3e_1…z3† ˆtfo33^u13…z3†; o333^u01…z3†; o333^u02…z3†; o33u1…z3†g: …39†

In expression (35), the ®elds grouped in v2_y

† are the solutions of the problem (31) with the data contained

in o3e_1…z3† as the only nonzero data: i.e. the body forces h_1…o3e_1†, no traction on oYb, and an initial strain

state which is restricted to the last matrix of Eq. (33). The vector e_2…z3† stands for the second order

macroscopic strains. The ®elds v1_y

† have already been de®ned in Section 4.3.

Remark4.3. The ®eld v2_{has been introduced in Trabucho and Via~no (1996), Duva and Simmonds (1991)}

and Fan and Widera (1990) for beams with constant cross-section. In the homogeneous and isotropic case, analytical solution is available for v



2E_{, and for v} 

2Cafor some cross-sections. For heterogeneous and periodic

beams, v



2E_{and v} 

2T _{appear in Kolpakov (1995). See also some related work in the case of periodic plates in}

Lewinski (1991a).

The stress ®eld r2_{solution of the P}1

cellproblem can also be formally expressed as follows:

r2_{ˆ ‰s}1E_y †
o3^u

2

3…z3† ‡ s1Ca…y_†
o33^u1a…z3† ‡ s1T…y_†
o3u2…z3†Š ‡ ‰s2E…y_†
o33^u13…z3†

‡ s2Ca…y †
o333^u 0 a…z3† ‡ s2T…y †
o33u 1_z 3†Š ‡ s2part…z3; y †  s 1_y †
e 2_z 3† ‡ s2…y †
o3e 1_z 3† ‡ s2part…z3; y † …40† with s2 ij ˆ aijk3v1k ‡ aijkhoyhv2k …41†

The stress ®elds contained in the ®rst brackets have been determined by solving the P0

cell problem, while

those in the second brackets are four new elementary solutions of the P1

cellproblem, when the four data of

o3e_1…z3† are prescribed. s2part is given by s2partˆ a…y_†:gradsy…u3part†.

4.5. 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 P2

cell problem, it is not dicult 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. o3e_2…z3†), plus the second gradient of the ®rst order

macroscopic strains (i.e. o33e_1…z3†). Furthermore, the loadings fae_aand gae_ahave to be added to these data,

according to assumptions (6).

In a recursive manner, the number of data involved in a cellular problem Pk

cellwill increase, starting from

the …k ‡ 1†th order macroscopic strains e_k‡1…z3† until the kth gradient of the ®rst order macroscopic strains,

ok 3e

1_z 3†.

Thus, assuming that the data of each cellular problem verify the compatibility condition (14), the as-ymptotic expansion of the displacement ®eld u_e_{takes the form:}

u eˆ ^u0a…z3†e_a‡ e1 u_1…z3; y_†   ‡ e2 _u 2…z3; y_  ‡ v1_y †
e 1_z 3†  ‡ e3 _u 3…z3; y_†  ‡ v1_y †
e 2_z 3† ‡ v2_y †
o3e 1_z 3† ‡ u_3part  ‡ e4 _u 4…z3; y_  ‡ v1_y †
e 3_z 3† ‡ v2…y_†
o3e_2…z3† ‡ v3_y †
o33e 1_z 3† ‡ u_4part  ‡ e5<sub>‰


Š …42†</sub>

The expression (42) is similar to the asymptotic expansion obtained in the case of 3D periodic media (see relation (21) in Gambin and Kroner (1989)).

One can also express the asymptotic expansion of the stress ®eld re_{under a similar recursive form. The}

generalization of expression (40) leads also to the following expansion: re_{ˆ e}1_‰s1_y

†
e 1_z

3†Š ‡ e2‰s1…y_†
e_2…z3† ‡ s2…y_†
o3_e1…z3† ‡ s2partŠ ‡ e3‰s1…y_†
e3…z3†

‡ s2_y †
o3e

2_z

3† ‡ s3…y_†
o33e_1…z3† ‡ s3partŠ ‡ e4‰


Š …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 vi_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}

gra-dients, has now to be found. The way of obtaining it will be explained in the Section 5. Remark4.4. As in the treatment of the P1

cellproblem, it is necessary to take into account the compatibility

conditions of the Pk

cellproblem. Therefore, one has to solve Pcellk in a similar way as made in the case of Pcell1

(see Sections 4.4 and 5.1.5). In that way, the elementary problems corresponding to each data of Pk cell are

well posed.

5. The set of macroscopic homogenized problems Pk hom

As already mentioned in Section 4.1, the equilibrium equations corresponding to the unknown dis-placement ®elds u_i _{are obtained from the compatibility condition (14). Expressing this condition for the}

cellular problems Pk

celland Pcellk‡1leads indeed to the formulation of the homogenized 1D problems Phomk . This

process will be applied in the next subsections: the way of deriving the equations of the ®rst homogenized problem, denoted by P1

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 Pk

hom, with k corresponding to

an arbitrary power of e.

5.1. Formulation of the ®rst homogenized problem P1 hom

5.1.1. 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 ˆ 1.

Let us ®rst focus our attention on the derivation of the macroscopic equilibrium equations of the ®rst homogenized problem P1

hom.

The P1

cell cellular problem (31) admits a solution provided that the data div z3r1, …0; 0; f3† and …0; 0; g3†

satisfy the relation (14). In particular, if we choose as test functions v_the four 'elementary' functions of R: ^v3…z3†e_3, ^va…z3†e_a and …y1e_2 y2e_1†, condition (14) leads to the four following equations:

o oz3 Z Yr 1 33dY ‡ Z Yf3dY ‡ Z oYb g3dC ˆ 0 …44† o oz3 Z Yr 1 a3dY ˆ 0 …45† o oz3 Z Y…y1r 1 23 y2r113†dY ˆ 0 …46† Furthermore, putting w

 ˆ yae3 in the variational formulation of the P 1

cell problem, given by relation (13)

with k ˆ 1, we have: Z Yr 2 a3dY ˆ Z Y ya or1 33 oz3  ‡ yaf3  dY ‡ Z oYb yag3dC; a ˆ 1; 2 …47†

In the same manner, if we express the condition (14) for the P2

cell problem, choosing now the two test

functions ^va…z3†e_awith a ˆ 1 or 2, we obtain:

o oz3 Z Yr 2 a3dY ‡ Z YfadY ‡ Z oYb gadC ˆ 0; a ˆ 1; 2 …48†

Let us introduce the following notations: N1_z 3† ˆ r133  ; T2 a…z3† ˆ r2a3  M1 a…z3† ˆ yar133  ; M1 3…z3† ˆ … y2r113‡ y1r123†  …49† with
h i  …1=l3†R_Y dY and where l3 stands for the scaled length of period Y (see Fig. (2)).

The beam stresses N1_z

3†, Ta2…z3†, Ma1…z3† and M31…z3† respectively correspond to the ®rst order

macro-scopic 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.

Remark5.1. The de®nition of the bending moments according to Eq. (49) do not obey the classical con-ventions used in strength of material. Following Eq. (49), M1

1…z3† and M21…z3† 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:

o3N1‡ hf3i ‡ hg3ioYb ˆ 0 o3Ta2‡ hfai ‡ hgaioYb ˆ 0; a ˆ 1; 2 T2 a o3Ma1‡ hyaf3i ‡ hyag3ioYb ˆ 0; a ˆ 1; 2 o3M31ˆ 0 …50† with h:i_oYb …1=l3†R_oYbdY .

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 P1

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 dz3.

Remark5.2. It appears from Eq. (50) (third equation) that the ®rst order bending moments M1

a are not

related to the transverse shearing forces of the same order but to the second order ones T2

a. As a matter of

fact, the ®rst order shearing resultants T1

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 T1

a can be de®ned as:

l3
Ta1ˆ Z Yr 1 a3dY  Z Yr 1_{: grad}s y…w_†dY with w_ ˆ yae_3

Green's formula can then be applied, so that: l3
Ta1ˆ Z oYr 1 3jnjyadS Z Y or1 3j oyj yadY …51†

where oY_{is constituted of the lateral boundaries oY}

band oYc and of the left and right sides of the period.

The ®rst integral in Eq. (51) vanishes by virtue of the y3-periodicity of r1 and ya, of the absence of

prescribed surface force at this order on the lateral outer boundary oYb, and of the stress-free condition on

the holes boundary oYc. In the same way, the second integral vanishes too, according to the equilibrium

equation of the P0

cellproblem.

5.1.2. Constitutive relations

After obtaining the equilibrium equations of the P1

hom problem, we focus now our attention on the

constitutive stress strain relations of P1 hom.

Grouping the `e€ective' (i.e. nonequal to zero) ®rst order macroscopic beam stresses in a vector, the constitutive relations of the P1

hom problem can be de®ned as:

N1_z 3† M1 1…z3† M1 2…z3† M1 3…z3† 8 > > > < > > > : 9 > > > = > > > ; ˆ Ahom1
o3^u13…z3† o33^u01…z3† o33^u02…z3† o3u1…z3† 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; …52†

and for later consistency of notations, relation (52) will be written in the form: r

1ˆ Ahom1
e

1 _53†

Ahom1 1m ˆ s133m  ; Ahom1 2m ˆ y1s133m  Ahom1 3m ˆ y2s133m  ; Ahom1 4m ˆ y2s113m‡ y1s123m  …54†

with 1 6 m 6 4. The quantities s1

ijm, obtained after solution of the Pcell0 cellular problem, have been de®ned in

Eq. (30).

Consequently, Ahom1

11 is the stretching sti€ness, Ahom122 and Ahom133 the two bending sti€nesses, Ahom144 the

twisting sti€ness and the extra-diagonal quantities are the di€erent coupling terms. Note that the e€ective sti€ness matrix Ahom1 _{is determined from the solution of the ®rst order cellular problem.}

Following Sanchez-Hubert and Sanchez-Palencia (1992) for transversely nonhomogeneous rods or Caillerie (1984) for periodic plates, it can be proved that Ahom1_{ful®lls the symmetry conditions A}hom1

mn ˆ

Ahom1

nm and is positive de®nite.

5.1.3. Boundary conditions

To complete the formulation of the P1

hom problem, it still remains to give the boundary conditions

corresponding to the both ends z3ˆ 0, L. More precisely, one has to derive from the exact boundary

conditions expressed on the 2D end sections, Se

0and SeL, the prescribed data for the ®rst order macroscopic

functions for z3ˆ 0 and z3ˆ L.

As a ®rst step, let us deal with the clamped condition on Se

0. Writing the boundary conditions uei ˆ 0 at

each power of e leads to: u

m…0; y_† ˆ 0; m P 0 …55†

which yields to the following conditions on the ®rst two terms of the asymptotic expansion: ^u0 a…0† ˆ 0 and ^u1 1…0† y2u1…0† ˆ 0 ^u1 2…0† ‡ y1u1…0† ˆ 0 8ya: ^u1 3…0† yao3^u0a…0† ˆ 0 8 > < > : …56†

Thus, relations (56) can be identically satis®ed provided that: ^u0

a…0† ˆ o3^u0a…0† ˆ ^u13…0† ˆ u1…0† ˆ 0 …57†

Relation (57) hence corresponds to the displacement boundary conditions of the ®rst order homogenized problem P1

hom.

Let us deal now with the other end section Se L.

Firstly, recalling that the initial 3D conditions are re

i3ˆ rei3…x1; x2†, and taking into account the order of

magnitude of the prescribed stress data (6) yields: r1 a3ˆ 0 r2a3ˆ ra3…y1; y2† r1 33ˆ r33…y1; y2† r233ˆ 0 ( rk i3ˆ 0 k P 2 …58†

However, the stresses r1_{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 ri3…y1; y2†.

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 Se Lr e i3dSeLˆ R Se L r e i3dSeL R Se Lxaea^ r e i3e_idSLe ˆ R Se Lxaea^ r e i3e_idSLe 8 < : …59†

For the P1

homproblem, the boundary conditions have to be expressed as a function of the macroscopic

stresses fN1_{; T}2

a; M1g, and Eq. (59) leads to:

N1_{L† ˆ}R SLr33dSL; T 2 a…L† ˆ R SLra3dSL M1 a…L† ˆ R SL yar33dSL; M 1 3…L† ˆ 0 ( …60† The proof of Eq. (60) is based on the property that the macroscopic stresses r1_{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 Cimetiere et al. (1988), and Trabucho and Via~no (1996).

Remark5.3. It can be seen in Eq. (60) that there is no torque applied to the end-section for the P1 hom

problem. This result follows directly from the assumption re

a3…x1; x2† ˆ e2
ra3…y1; y2†, 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 T1

a ˆ 0. However, this assumption might

be relaxed and one might consider a distribution of re

a3, such that the resultant shearing force remains zero

but now with a nonzero resultant torque, so that M1 3 6ˆ 0.

5.1.4. Summary

To summarize, the ®rst order homogenized problem, P1

hom, consists in ®nding the macroscopic stresses

fN1_{; T}2

a; M1g and the four macroscopic displacements f^u0a; ^u13; u1} such that:

o3N1‡ hf3i ‡ hg3ioYb ˆ 0 o3Ta2‡ hfai ‡ hgaioYbˆ 0 T2 a ‡ o3Ma1‡ hyaf3i ‡ hyag3ioYbˆ 0 o3M31ˆ 0 N1_z 3† M1 1…z3† M1 2…z3† M1 3…z3† 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; ˆ Ahom1
o3^u13…z3† o33^u01…z3† o33^u02…z3† o3u1…z3† 8 > > > > > < > > > > > : 9 > > > > > = > > > > > ; ^u0 a…0† ˆ o3^u0a…0† ˆ ^u13…0† ˆ u1…0† ˆ 0 N1_{L† ˆ}R SL r33dSL; T 2 a…L† ˆ R SL ra3dSL M1 a…L† ˆ R SL yar33dSL; M 1 3…L† ˆ 0 8 > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > : …61†

Due to the positive-de®niteness of Ahom1_{, it can be proved that the problem (61) is well posed.}

The P1

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 Kolpakov (1991).

It must be noticed that the equilibrium equation for the torque with the boundary condition at the end z3ˆ L leads to M31…z3† ˆ 0. Thus, if the torsion is not coupled with stretching or bending, u1…z3† ˆ 0 due to

the boundary condition at z3ˆ 0.

Note also that the P1

hom problem is a 1D beam problem that can easily be solved analytically. Only the

construction of the constitutive matrix Ahom1 _{requires generally a numerical solution of the cellular}

problem P0

5.1.5. Treatment of the equilibrium equation of the P1

cell problem

Since the relations between the macroscopic quantities r1_{and e} 

1_{are established, let us now come back to}

the P1

cellcellular 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 P1

cell in terms of the quantities o3e1p; p 2 ‰1; 4Š, the

equi-librium equation (31) (®rst equation) yields: div_{ y}r2_{‡ s}1

i3p…y_†o3e1p…z3†e_i‡ f3e_3ˆ 0_ …62†

Furthermore, recalling that Ahom1 _{is positive de®nite, the stress strain relation (53) can be written as:}

o3e_1…z3† ˆ Shom1
o3r_1…z3† …63†

where Shom1_{denotes the inverse matrix of A}hom1_.

As proved in Section 5.1.1, the compatibility conditions for the P1

cellproblem reduce to Eq. (50) (®rst and

fourth equation). Inserting them into Eq. (63), we get: o3e1p…z3† ˆ Sp1hom1…hf3i ‡ hg3ioYb† ‡ S

hom1

p2 o3M11…z3† ‡ Shom1p3 o3 21…z3† …64†

As a consequence, the equilibrium equation (62) has to be written as: div_{ y}r2_{‡ f}

3e_3 s1i3p…y_†Sp1hom1…hf3i ‡ hg3ioYb†e_i‡ s 1

i3p…y_†fShom1p2 Ahom12m ‡ Shom1p3 Ahom13m go3e1me_iˆ 0_ …65†

with summation on the repeated indices, i 2 ‰1; 3Š and …p; m† 2 ‰1; 4Š2_.

Remark5.4. 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…f3† ˆ s1i3p…y_†Shom1p1 hf3ie_i, h_3…g3† ˆ s1i3p…y_†Sp1hom1hg3ioYbe_iand that

h_1 is given by the last terms of Eq. (65).

In the case of a constitutive law Ahom1_{without any coupling, the relation (65) can be simpli®ed in the}

following manner: div_{ y}r2_{‡ f}

3e_3 s1Ei3…y_†…Ahom111 † 1…hf3i ‡ hg3ioYb†e_i‡ s 1Ca

i3 …y_†o333^u0a…z3†e_iˆ 0_ …66†

5.2. Formulation of the kth homogenized problem Pk hom

5.2.1. Equilibrium equations

In the preceding section, it has been shown how to derive the ®rst order homogenized problem P1 hom. By

applying exactly the same method for each order k > 1, one obtains the formulation of the higher-order homogenized problems, Pk

hom.

Hence the equilibrium equations of the Pk

hom problems: o3Nk‡ hf3ki ‡ hgk‡13 ioYbˆ 0 …67† o3Tak‡1‡ hfak‡1i ‡ hgak‡2ioYbˆ 0 …68† Tk‡1 a o3Mak‡ hyaf3ki ‡ hyagk‡13 ioYb ˆ 0 …69† o3M3k‡ hy1f2k y2f1ki ‡ hy1g2k‡1 y2gk‡11 ioYbˆ 0 …70†

with Nk_z 3† ˆ hrk33i; Tak‡1…z3† ˆ hrk‡1a3 i Mk a…z3† ˆ h yark33i; M3k…z3† ˆ h… y2rk13‡ y1rk23†i …71† Nk_{, M}k

b and M3k correspond respectively to the macroscopic axial force, bending moments and twisting

moment of order k, while Tk‡1

a represent the macroscopic shearing forces of order …k ‡ 1†. We recall that the

prescribed volume force f_k _{and surface force g} 

k‡1 _{satisfy assumptions (6).}

5.2.2. Constitutive relations

Let us now study the macroscopic stress strain relation of order k. We have seen in Eq. (43) that the stress ®eld rk_{, solution of the P}k 1

cell problem, is a linear function of the kth order macroscopic strain e k_{, the}

®rst-gradient of the …k 1†th order macroscopic strain (i.e. o3e_k 1), and so on until the …k 1†th-gradient of

the ®rst order macroscopic strain (i.e. ok 1 3 e

1_{). Therefore, the macroscopic stress strain relation at any order}

k with k P 1 can be written as: r

kˆ Ahom1
ek‡ Ahom2
o3ek 1‡ Ahom3
o33ek 2‡


‡ Ahom k
ok 13 e1‡ rkpart

with r_k_z

3† ˆtfNk; M1k; M2k; M3kg

…72† where strain vectors e_p _{vanish when p 6 0, and where o}k 1

3 denotes the partial derivative …ok 1=ozk 13 †.

The kth order stress vector r_k

part contains the beam forces deduced from the stress state s_kpart, i.e. the

particular solution of the well-posed Pk 1

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 so-lutions obtained at the preceding orders, grads

 z3…u k

part† and div_{ z}3…s_k 1part†. The 4  4 matrix Ahom1has

al-ready been de®ned in Eq. (54). In a similar way, the components of the 4  4 matrix Ahom k _{are deduced}

from the four elementary stress solutions of the …k 1†th order cellular problem, namely skE_{, s}kCa and skT,

grouped in sk_{. We recall that these stress tensors correspond to the solution of P}k 1

cell when the components of

the …k 1†th gradient of e_1are respectively considered as data, i.e. ok

3^u13…z3†, ok‡13 ^ua0…z3†, and ok3u1…z3†. Thus,

Ahom k _{is de®ned as:}

Ahom k

1m ˆ hsk33mi; Ahom k2m ˆ h y1sk33mi

Ahom k

3m ˆ h y2sk33mi; Ahom k4m ˆ h y2sk13m‡ y1sk23mi

…73† Contrary to the ®rst order e€ective sti€ness matrix Ahom1_{, 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, Ahom2_,

appears to be antisymmetric in 3D periodic media and even equal to zero following certain symmetry properties of the period Y (Boutin, 1996; Triantafyllidis and Bardenhagen, 1996).

Remark5.5. Solving the kth order homogenized problem Pk

hom implies that the lower-order macroscopic

problems have already been solved. Therefore, when considering the Pk

homproblem, the macroscopic strains

e

 1_{; . . . ; e}



k 1_{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}

Pk

homproblem and can be considered as ®ctive initial stress states for the current macroscopic problem.

As a matter of fact, following the method presented in Boutin (1996), the ®rst equilibrium equation (67) of the Pk

homproblem may be written as:

o3…Ahom11m
ekm† ˆ hf3ki hgk‡13 ioYb o3…A hom2

1m
o3ek 1m ‡ Ahom31m
o33ek 2m ‡


† with 1 6 m 6 4:

Writing all the equilibrium equations of Pk

homin 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 e€ects arising under the form of ®ctive volume loadings. It becomes also clear that the displacement unknowns of the Pk

homproblem

are the four macroscopic quantities f^uk 1

a ; ^uk3; ukg.

5.2.3. Boundary conditions

To complete the formulation of the Pk

homproblem, one must add to equilibrium equations (67) (70) and

constitutive relations (72) the boundary conditions which have to be expressed on the displacements f^uk 1

a ; ^uk3; ukg for z3ˆ 0 on one hand, and on the kth order macroscopic stresses fNk; Tak‡1; Mkg for z3ˆ L

on the other hand.

These conditions, which are obtained from the initial 3D conditions on the two end sections Se

0, SeL, are

given in Eqs. (55) and (58). Since we are interested in the Pk

homproblem 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 well-posed Pk

homproblems are obtained.

6. Summary

Let us summarize here the results provided by the formal asymptotic method. The solution in series of the ®rst k cellular problems, P 1

cell to Pcellk 1, leads to the determination of the y3-periodic displacement ®elds

fv1_y †; . . . ; v

k_y

†g, as well as the associated periodic stress ®elds fs

kE_{; s}kCa; skTg.

Then, following the average process given in Eq. (73), the ®rst k e€ective matrices Ahom k _{can be}

cal-culated.

Treating in parallel the ®rst k macroscopic problems, P1

homto Phomk , leads to the macroscopic parts of the

asymptotic expansions (42) and (43). Especially, the solution of the homogenized problems up to the kth order, Pk

hom, gives the macroscopic axial displacement ^up3…z3† and the macroscopic torsion rotation up…z3† up

to order k, as well as the macroscopic de¯ections ^up

a…z3† up to order …k 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}

eof the initial problem. Particularly,

the macroscopic description of the displacements of the structure is given by: ^ua…z3† ˆ ^u0a…z3† ‡ e^u1a…z3† ‡

^u3…z3† ˆ ^u13…z3† ‡ e^u23…z3† ‡

u…z3† ˆ u1…z3† ‡ eu2…z3† ‡

…75†

7. 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…z3; y_† ˆ u0…z3; y_† ‡ eu1…z3; y_† ‡ e2u2…z3; y_† ‡


…76†

so that the components of u_eare given by: ue 1…z3; y_† ˆ ^u1…z3† ey2u…z3† ue 2…z3; y_† ˆ ^u2…z3† ‡ ey1u…z3 ue 3…z3; y_† ˆ e^u3…z3† eyao3^ua…z3† 8 > > > > < > > > > : …77†

with ^ui…z3† and u…z3† 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 di€erent macroscopic loadings corresponding to macroscopic strains and their derivatives. Thus the e€ective 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 Pk

homthe 1D equations involving the unknowns ^ui…z3† and u…z3† 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…z3; y_† ˆ ue…z3; y_† ‡ ev1…y_†
e…ue† ‡ e2v2…y_†
o3e…ue† ‡ e3v3…y_†
o33e…ue† ‡


‡ upart with

e …u e_{† ˆ e}  1_{‡ ee}  2<sub>‡


ˆ</sub>t_fo

3^u3; o33^u1; o33^u2; o3ug and u_partˆ e3u_3part‡ e4u_4part‡


…78†

In the same way, Eq. (43) can be written as follows: re_{ˆ es}1
_e

…ue† ‡ e2s2
o3e…ue† ‡ e3s3
o33e…ue† ‡


‡ rpart …79†

Moreover, the expansion of the macroscopic beam stresses r_e _{is de®ned by:}

r

eˆ r1‡ er2‡


…80†

and one has for the transverse shearing forces: Te

aˆ Ta2‡ eTa3‡


…81†

Thus, from Eq. (72) and from the addition of the equilibrium equations at each order, the macroscopic ®elds u_e_{, r}

e and Taeare found to satisfy:

o3Ne‡ hf3i ‡ hg3ioYbˆ 0 o3Tae‡ hfai ‡ hgaioYbˆ 0 Te a o3Mae‡ hyaf3i ‡ hyag3ioYbˆ 0 o3M3e‡ hy1f2 y2f1i ‡ hy1g2 y2g1ioYbˆ 0 8 > > > < > > > : …82† r eˆtfNe…z3†; M1e…z3†; M2e…z3†; M3e…z3†g ˆ Ahom1
_e …u e_{† ‡ eA}hom2
_o

3e_…u_e† ‡ e2Ahom3
o33e_…u_e† ‡


‡ 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-higher-order gradient theory. This result was already pointed out in Gambin and Kroner (1989) in the case of 3D elastic periodic media. Indeed, for such problems, the asym-ptotic 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 Ahom2_{ˆ 0 (which is}

obtained when the period Ye_{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 (Boutin, 1996) 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 Fan and Widera (1990) or Trabucho and Via~no (1996), where a generalization of Timoshenko's theory is also established for the isotropic nonho-mogeneous 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 homoge-neous isotropic rod in one of its principle planes …e_1 e_3†, by the external forces …f1e; 0; 0† and …ge1; 0; 0†

verifying (6), and with a clamping condition at the both ends Se

0, SLe. Our aim is to derive the second order

model associated to the approximation of the expansions u_e_{, r}

eup to the second nonzero term. The ®rst two

e€ective terms of the macroscopic beam forces are found to be r_e_{ˆ r}

1‡ e2r3, and from Eq. (82),

o3…N1‡ e2N3† ˆ 0 o3…T12‡ e2T14† ‡ R Sf1dS ‡ R oSg1dc ˆ 0 …T2 1 ‡ e2T14† ‡ o3…M11‡ e2M13† ˆ 0 8 > < > : …84†

since, for the beam under consideration here, the operators h
i and h
i_oYb can be reduced to R_S
dS and R

oS
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 e€ects (symmetric bending without any coupling), and given that Ahom2 _{is zero for a homogeneous beam, the}

stress strain relation (83) leads to: M1 1 ˆ Ahom122 o33^u01ˆ EI1o33^u01 M3 1 ˆ Ahom122 o33^u21‡ Ahom322 o43^u01 ( …85† with I1ˆRSy12dS and where E denotes the Young's modulus.

Consequently, the equilibrium equations (84) (second and third equations) yield: Ahom1

22 o43…^u01‡ e2^u21† ‡ e2Ahom322 o63…^u01†

Z

Sf1dS

Z

oSg1dc ˆ 0 …86†

Moreover, from the ®rst-order homogenized problem, the de¯ection ^u0

1is the solution of the di€erential

equation Ahom1

22 o43…^u01† ‡

R

Sf1dS ‡

R

oSg1dc ˆ 0, so that Eq. (86) can be written under the form:

Ahom1

22 o43^u2e1 ‡ e2Ahom322 …Ahom122 † 1o23

Z S f1dS  ‡ Z oSg1dc  Z Sf1dS Z oSg1dc ˆ 0 …87† with ^u2e 1 ˆ ^u01‡ e2^u21.

The relation (87) is found to be exactly of the form of the di€erential equation for Timoshenko's beam theory (uT

EI1o43uT1 ‡ …kGS† 1EI1o23 Z S f1dS  ‡ Z oSg1dc  Z Sf1dS Z oSg1dc ˆ 0 …88†

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 coecient k from Ahom1

22 and e2Ahom322 (which corresponds to the descaled e€ective behavior). Nevertheless, it must be

noticed that several terms in the asymptotic expansion (42) of the complete de¯ection ue

a…z3; ya† have to be

neglected in order that Eq. (87) reduces to Timoshenko's theory. Especially, Poisson's e€ects as well as geometrical torsional e€ects are neglected. The second order model (82) and (83) thus incorporates 3D e€ects 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 e€ects such as cross-sectional warping, as well as transverse shear and normal stresses and strains. This is a major di€erence from existing higher-order beam theories (see e.g. Kosmatka (1993), Reddy et al. (1997), Soldatos and Watson (1997) 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 e€ects, as it was proved previously for the second order Timoshenko theory.

Moreover, since approximate boundary conditions have to be considered, edge e€ects are an important source of errors in re®ned engineering theory (Duva and Simmonds, 1991). 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).

Acknowledgements

The authors gratefully acknowledge Professor J.G. Simmonds (University of Virginia) for fruitful dis-cussions, 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. References

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