1 Introduction

The aim of this work was to establish the modeling of a thin elastic beam made of elastic inclusions periodically distributed along its length.More precisely letδandεbe two small parameters respectively the thickness of the beam and the size of the inclusions,and both are supposed to vanish.Several works have shown how to derive rigorously the reduced onedimensional models of elastic beams,starting from the three-dimensional equilibrium equations and letting the sizeδof the cross-section tend to zero(just to mention some of the pioneer papers for plates and rods(see[2,13]));on the other hand the limit model of structures made of periodically heterogeneous elastic material has been established by the homogenization approach,when the number of inclusions tends to infinity,i.e.,ε tends to zero(just to mention some of the pioneer papers in elasticity(see[1,11])).In this paper we let both parameters tend to zero simultaneously,which gives rise to three different one-dimensional models of homogeneous beams depending upon the limit of the ratio

In the classical Euclidean space the Cartesian coordinate system attached to the beam is denoted Ox1x2x3 and we associate an orthonormal basis(e1,e2,e3)(this will be defined more precisely later).The three-dimensional body is thin in the direction e1,e2;letδbe a small parameter which takes into account the thinness of the beam.The scaled cross-section of the beam occupies the bounded domain(with Lipschitz boundary)ω⊂R2.Hence the straight beam occupies the cylinderof length L and section ωδ= δω.A generic point inΩδis denoted by x=(x1,x2,x3)with(x1,x2)∈ ωδ,x3∈ (0,L),and therefore

Let1 Greak indices or exponents,exceptεand δ,take their values in the set{1,2},Latin indices(except e and y)take their values in the set{1,2,3}.The Einstein summation convention of repeated indices is applied. u=(u i)be the three-dimensional displacement field attached to an elastic solid under the action of applied body forces;it can be decomposed as the sum of a displacement field(rigid in the cross-sections)and a warpingThe elementary displacement field U e is the sum of a Bernoulli-Navier displacement described by a vectorand a scalarΘ and a contributionto complete the centerline displacement U(see(2.11)):

The structure is clamped on a partΓ0,δ= ωδ ×{0}of the boundary ∂Ωδ.

依据修正净雨成果，采用推理公式方法重新进行设计洪峰流量计算，并与崇礼水文站流量推求成果进行比较，成果如表7。

where the limit strain tensors are given in(3.5),(5.5),(5.7)and(5.9).In Section 6 we study the convergence of the sequencein the three cases: θ=+∞,θis finite and θ=0.However,it could be of interest to compute a lower dimensional approximation of the“real displacement field”,i.e.,with a finite value of the small(but not “equal to zero”)parameters ε and δ;this is done in Section 7 where the first terms of an asymptotic expansion of uε,δ (see(7.1))are given.

where e(u)represents the linear strain tensor.

Under the action of applied volume force Fδ,the beam,made of an elastic material characterized by its elastic tensor aδ,undergoes a displacement field uδ solution to the variational problem(the regularities of aδand Fδare detailed later):

Findsuch that2 The “dot” notation is for vector product and the “colon” notation is for the tensor product:where A=(A ij k l)is a symmetric fourth-order tensor and E=(E ij),F=(F ij)are two second-order symmetric tensors.

We assume that the applied forces have a specific dependence(2.14)with respect toδ,so that the strain tensor is of order δ2(see(2.18)),thus we are in a position to infer that the sequence uδconverges in an appropriate space.

For all u∈H 1(Ωδ;R3)and δ≤ L,we have

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where the symmetric third-order tensors3 The subscript y stands for derivative with respect to(y1,y2,y3),the subscript y′is for derivative with respect to(y1,y2),and the subscript y3 is for derivative with respect to y3. E and E y3 are the strain tensors(3.4),(4.1)(the functional spaces are completely described in Section 3).

In Section 5 we show that the convergence of the second part of the strain tensordepends upon the ratioand we study the three possibilities taken byMore precisely we establish the existence of limit fields and in appropriate spaces such that the following weak convergences take place:

In Section 2 we begin to recall Korn’s type inequalities(2.13)for thin beams(see[7,10]):

Finally we mention the continuity of the functionin an appropriate functional space.

2 Displacement Field in a Thin Structure

Letδbe a small parameter which takes into account the thinness of a straight beam.The beam occupies the cylinderof length L and section ωδ= δω (where ωis a bounded domain of R2 with Lipschitz boundary).The Cartesian coordinate system attached toωhas the gravity center of the structure for origin and the direction of its main inertia axes as the orthonormal basis(e1,e2),i.e.,

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and I1,I2 are the two principal moments of inertia.

The beam is supposed to be fixed on its extremity Γδ,0= ωδ ×{0}.

The structure is made of a material characterized by its elasticity tensor aδ =(a ij kl,δ)with the classical properties of symmetry,boundedness and coercivity,i.e.,for all symmetric second order tensor e there exist two positive constants c,C such that

Under the action of applied volume forces Fδthe beam undergoes a displacement field uδ ∈ V(Ωδ)solution to the variational problem

posed in the functional space

In this space the classical Korn’s inequality reads:For all v ∈ V(Ωδ),there exists a positive constant C(Ωδ)which depends upon the domain Ωδsuch that

Hence,for Fδ ∈ [L 2(Ωδ)]3 and aδ ∈ [L∞(Ωδ)]3×3×3×3,we can apply Lax-Milgram theorem to obtain the existence and uniqueness of the solution uδ ∈ V(Ωδ).

2.1 Decomp osition of the displacement field in a cylinder

In order to study the behavior of the sequence{uδ}δ when δgoes to zero it is of interest to introduce the fixed bi-dimensional domainω⊂R2(the reference cross-section of the beam),henceωδ= δω,and the three-dimensional cylinderΩ =ω×(0,L).We also consider the decomposition 4 The case of a curved beam was consider in[7]. of any displacement field u ∈ [L 1(Ωδ)]3 as the sum of an elementary displacement field U e ∈ [L 1(Ωδ)]3 and a warping(see[5,12]):

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where satisfies

To sum up,we get

The last equality above means that the warping u does not capture the couple of torsion forces(see Section 2.4).The same approach was considered for plates(see[8]).The elementary displacement U e is given by the displacement of the middle line U∈[L 1(0,L)]3 and the small rotation along the vector R∈[L 1(0,L)]3:

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We recall here the Definitions 2–3 of[9–10].

where the two principal moments of inertia I1,I2 are given by(2.1).

Remark 2.1 From now on,we assume the displacements to be in V(Ωδ).Asa consequence,for every u ∈ V(Ωδ),the terms U and R of the decomposition belong to[H 1(0,L)]3,while the warping u belongsto V(Ωδ).The terms of the decomposition also satisfy the boundary clamping condition

We recall the bounds on the displacement fields given in[10,Theorem 3.1,p.206].

In Section 3 we introduce the second small parameter ε which is also supposed to tend to zero.We describe the thin beam as made of an heterogeneous material whose elasticity tensor aε,δ depends also upon ε;the heterogeneities are distributed along the e3 axis with periodicity ε.In order to study the displacement field,now denoted uε,δ,we introduce the unfolding operator Tε,δ;for all p ∈ [1,+∞]and ϕ ∈ L p(Ωδ)it associates a functionwhereis the dilated domain andis the unit cell occupied by the heterogeneities.In Section 4 we state the existence of a unique limit displacement field U e and a displacement corrector and we establish a weak convergence result(4.3),

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Corollary of[10,Theorem 3.1](Korn’s Inequality with Boundary Conditions)For all v ∈ V(Ωδ),we have

We note that in this expression of Korn’s inequality the bound depends explicitly on the thickness of the thin beam(compare to(2.4)).More precisely,one has

The constant C is independent of δ.

2.2 Introduction of a new decomposition

We introduce a new decomposition of the elementary displacement part U e,in order to simplify the expression of the strain tensor

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More precisely let(U,u)be the new functions defined by

Then we can eliminate the first two components of the rotation and get

and U3≡0.From now on,we denote Θ=R3 the third component of the rotation;therefore we have a new decomposition of the field U e:

or component-wise

This decomposition yields the new expression of the strain tensor as

where the positive constant C depends neither onδnor on the length L of the beam.

We note that the clamping condition u=0 on Γ0,δ implies boundary conditions on the decomposition:

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We also note that,since Rα∈H 1(0,L),one has Uα∈H 2(0,L).

2.3 First bounds

In the sequel C represents any positive constant which depends neither onδnor on the length L of the beam.

When u belongs to V(Ωδ)then,taking into account the boundary conditions(2.12),one obtains the bounds on the new fields

2.4 Assumption on the ap plied volume forces

Let v be a general elastic displacement field belonging to V(Ωδ)decomposed as follows:

Moreover,and

(1)Influence of a volume force

(2)Influence of a moment

The quantity represents the torsion forces acting on the beam.

(3)Influence ofwithgiven by the solution of the ODE,where Iαis given by(2.1):

Now let us combine all those elementary forces to define the global one

or component-wise

Note that,due to the definition(2.6)ofone hasWith the decomposition(2.11)of the displacement field and assumption(2.14)on the forces we get the expression of the potential energy

and the bound follows:

Remark 2.2 The specific choice(2.14)is made so that every component of the force contributes equally to the total elastic energy as we can see in(2.16)and,consequently,will appear through(6.2)in the expression of the limit models.

The variational problem:Find uδ ∈ V(Ωδ).

has a unique solution and the a priori bound on the strain tensor e(uδ)can be derived from(2.16):

which in turns yields

where the constant C>0 does not depend uponδ.

2.5 A priori estimates

Any displacement field uδsatisfying(2.18)can be decomposed as presented in the previous subsections.Therefore,the appropriate choice of the applied forces and moments(2.14)leads to the following bounds on the different fieldsassociated to(2.13):

(1)Bound on the total displacement uδ:

(2)Bound on the principal flexion Uδ:

(3)Bound on the stretching or complementary flexion

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(4)Bound on the angle of torsion of the sections(around the middle straight line)Θδ:

(5)Bound on the warping

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In the sequel,we consider periodically heterogeneous thin beams:More precisely,the material is described by an elasticity tensor(with the same kind of coercivity and symmetry properties asin(2.2))whose components are periodic along the e3 direction and depend,now,upon two small parameters δ and ε(the applied forces are independent ofε).The variational problem:Find uε,δ ∈ V(Ωδ)such that

has a unique solution uε,δ.This paper aims at studying the behaviour of the sequence of solutions{uε,δ}ε,δ when both parameters εand δgo to zero independently.

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3 Introduction of Another Small Parameter:The Sizeεof the Inclusions

3.1 The unfolding op erator Tε,definition and first prop erties

The beam is made of periodic cells distributed along the direction e3,in such a manner that each of these identical cells occupies a domain of thin section ωδand of small length ε.In order to simplify the presentation we assume that the macroscopic domain(0,L)is covered by an integer number of elementary cells5 The domainΩδdoes not depend uponε.:ε=L/N,N ∈N∗.

We define the unique decomposition of almost every real number z∈R as the sum of its integer part[z](also called the“slow”evolving part)and the remainder{z}(also called the“fast” evolving part)which belongs to the microscopic domain(0,1)):

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The unfolding operator Tεmaps L p(0,L)into L p((0,L)×(0,1))for all p∈[1,+∞]:

An immediate property of this linear operator is that for all functionsϕ,φ∈L 1(0,L)we have

Let us introduce the spaces

In the sequel we will make use of important results of convergence gathered in the following three theorems(see[3,p.1599,p.1603]),in which the variable x lies in(0,L)and y in(0,1).

Theorem 3.1 Let{vε}ε be a sequence in L 2(0,L)satisfyingThere exists a subsequence,still denoted{ε},and limits v∈ L 2(0,L)andsuch that

We rely on the bounds(2.13)and(2.10)and we consider,for simplicity,the scaled forces and moments.

Theorem 3.2 Let{vε}ε be a sequence in H 1(0,L)satisfyingThere exists a subsequence,still denoted{ε},v∈ H 1(0,L)and(0,L;such that

Theorem 3.3 Let{vε}ε be a sequence in H 2(0,L)satisfyingThere exists a subsequence,still denoted{ε},v∈ H 2(0,L)and(0,L;such that

Wearethen in a position to extend these results to the study of sequences{wε,δ}ε,δ according to the limit of the ratiowhen both the two small parameters converge to zero.

In the lemma below,we consider a sequence{(ε,δ)}converging to(0,0).

Lemma 3.1 Letbe a sequence converging weakly to w in L 2(0,L)and satisfying

If then there exists in L 2((0,L)×(0,1))satisfying such that the following convergences hold:

Ifthen there exists in L 2(0,L;(0,1))such that the following convergences hold:

Proof We begin with the obvious estimates

Step 1 We prove(i)and(ii).Convergence(i)is an immediate consequence of the first part of Theorem 3.1.Since the sequenceis bounded from above we get

From the above estimates and the fact that we deduce

And

gives convergence(ii).

Step 2 We prove(iii)and(iv).In this second case,sinceis bounded from above,from Theorem 3.1,up to a subsequence,there exists W∈L 2(0,L;(0,1))such that

Taking into account the fact thatwe getthen W is independent of y.And for a.e.x∈(0,L)gives W=0 and convergence(iii).

Applying Theorem 3.2 to the sequence{δwε,δ}ε,δ,which is uniformly bounded in H 1(0,L),and weakly convergent to 0 in H 1(0,L),we obtain a function in L 2(0,L;(0,1))such that(up to a subsequence)

whence convergence(iv).

3.2 The dilation op erator Πδ and the unfolding op erator Tε,δ

Associated to the scaled domainΩδwe introduce another unfolding operatorL 2(Ω)defined for allφ ∈ L 2(Ωδ)by

and we note that

Moreover,for every ϕ ∈ H 1(Ωδ)one has

Next,we combine the two previous scalings due to the thinness of the geometry and the periodicity of the elasticity tensor and we introduce the third unfolded operator

From now on,the reference microscopic domain is denoted Y=ω×(0,1)and we get the definition

and the main properties

Remark 3.1 Let us observe that when dealing with the operator Tε,only the variable x3 is unfolded(ω is just a set of parameters),when dealing with the operatorΠδ,only the variables x1,x2 are unfolded,while with the operator Tε,δ the three variables(x1,x2,x3)are unfolded.

3.3 The p eriodically heterogeneous beam

In order to take into account the periodicity of the elasticity tensor aε,δ we assumethat there exists a tensor A ∈ L∞(Y)3×3×3×3(with the classical properties of symmetry,boundedness and coercivity)such that

This yields

The problem we have to solve now is to find the limit of the sequence{uε,δ}ε,δ whose elements are solution to the variational problem:Find uε,δ ∈ V(Ωδ),

As in(2.5),(2.11)we have the decomposition

3.4 First convergences

Let us introduce the following vector spaces:

For every,we define the symmetric tensor E by

and for everywe define the symmetric tensor e y′by

where the subscript y′emphasizes the derivation with respect to variables(y1,y2)only.

Then we can state the first convergence result.

Lemma 3.2 There exists a subsequence of(ε,δ),still denoted(ε,δ),and limit displacement fields

such that

Moreover,one has

where the limit symmetric tensorsandare given by(3.4)and(3.5).

Proof The convergences(3.7)and the boundary conditions(3.6)are the immediate consequences of the estimates(2.19),(3.1)–(3.2)and the boundary conditions satisfied by the terms of the decomposition of uε,δ.Moreover,since the fieldverifies the equalities(2.6);dividing them byδ2,then transforming withΠδand passing to the limit show that the limit field u belongs toThen the convergences(3.8)1 and(3.8)2 are the consequence of the convergences in(3.7)and the decomposition of uε,δ.

A straightforward computation yields the symmetric tensor6 To save space,the star∗indicates a symmetric term.

Then(3.8)3 is the consequence of the convergences in(3.7)1,2,3.

A similar computation yields

Then(3.8)4 is the consequence of the convergences in(3.7)4,5.

In the next sections we study separately the two components of the complete unfolded strain tensor

4 Expression of the Unfolded Strain Tensorand Convergence Result

Let

and for every,we define the symmetric tensorby

In an obvious way,to the displacement fieldwea ssociate thene wunknownsand a straightforward computation yields the unfolded scaled symmetric tensor

and we get the following convergence result.

Lemma 4.1 There exists a subsequence of(ε,δ),still denoted(ε,δ),the limit of Bernoulli-Navier displacement fieldand the correctors

such that

where the symmetric strain tensorsare given by(3.4)and(4.1).

Proof Again,from the bounds(2.19),and now with the help of Theorem 3.2,we infer the existence of a subsequence of(ε,δ)and of limit corrector fields such that the following weak convergences hold:

and

Hence,the convergences in(4.3)are obtained.

Remark 4.1 The limit(U,Θ,u)is the same as the one obtained in Lemma 3.2.

5 Expression of the Unfolded Strain Tensorand Convergence Results

Thissection is devoted to the contribution of the warping part of the displacement field uε,δ,for which we use the bounds given in(2.19),

Let us recall the chain rule which gives the transformation of the gradient for any vector-field ϕ ∈ H 1(Ωδ),

Then,a straightforward computation yields

Because of the chain rule(5.2)and estimates(3.1)and(5.1),we deduce that there exists a constant C>0 such that we have the bounds

Therefore,the limit of the unfolded strain tensorand,consequently that of the unfolded tensorwill depend upon the relationship between εand δ.Let us denotethere exist three possible cases which are

•θ=+∞,

•θis finite,

•θ=0.

The remain of this section is devoted to the study of these three possibilities.

5.1 The case

In order to study the convergence of the unfolded strain tensorin the case θ=+∞,we introduce the vector space:

For any v ∈ [L 2(ω;H 1(0,1))]3,let be the strain tensor defined by

Lemma 5.1 There existssuch that(up to a subsequence)

where was introduced in Lemma 3.2,with e y′defined by(3.5)and where is given by(5.5).

Proof We still have the bounds as given in(2.19),

and the conditions given by(2.6).Therefore,on the one hand,recall(3.7)4−5,

On the other hand,the sequencesatisfies the assumption of Lemma 3.1(second part):

Hence,there existssuch that the weak convergences(5.6)hold.

5.2 The case

Denote

For any v be the strain tensor defined by

Lemma 5.2 There existssuch that(up to a subsequence)

where is introduced in Lemma 3.2 with e y′ defined by(3.5)and where is given by(5.7).

Proof Proceeding as in Lemma 5.1,we introduce the sequenceFrom(5.4),the sequence Tε(wε)ε isuniformly bounded in[L 2(0,L;H 1(Y))]3,satisfing the assumption of Theorem 3.1.Hence,there existssuch that the weak convergences(5.8)hold.

5.3 The case

Denote

For every we defineby

Lemma 5.3 There exists such that(up to a subsequence)

where is introduced in Lemma 3.2 with e y′defined by(3.5)and whereis given by(5.9).

Proof We proceed as in the two previous Lemmas 5.1–5.2,but here using part 1 of Lemma 3.1.

The following section is devoted to the study of the whole limit field u according to the different values ofθ.

6 The Limit Problems

Let us make more precise the framework in which the convergences and,consequently,the limit problems are obtained.

(1)First,based on the decomposition

we can represent this strain tensor in a more compact form(recall that Y=ω×(0,1))

with an obvious definition of the matrices A i and unknowns g i,i=1,···,6,formed with the unknowns

(2)Under the assumptions on the forces given in Subsection 2.4,and for all displacement field we have introduced the potential energy

In the sequel we will also make use of another set of unknowns,namely,for the displacement fieldwith we will associate the equivalent formulation of the potential energy

with

Even though the treatment of the three different values ofθis quite similar,we decided to present them in three self-contained sub-sections.

6.1 The case

To anywe associate the corrector field

belonging to

The spaces(endowed with their usual norms)and are isometric.To anywe associate the symmetric tensor(see(4.1)and(5.5))

By collecting the convergence results given by Lemmas 4.1 and 5.1 we can state the following result.

Corollary 6.1 There exists a subsequence of(ε,δ),still denoted(ε,δ),limit displacement fields and correctors and such that

where the strain tensors are given by(3.4)–(3.5)and(6.6).

We endow with the following norm:

Lemma 6.1 On the space,the norm is equivalent to the usual norm of this product spaces.

Proof Due to the fact that for a.e.(x3,y1,y2)∈Ω,we first have

The 3D Korn’s inequality,the periodicity of and its mean value property imply the existence of a constant C>0 such that

Then we easily prove(see[7])

The existence and uniqueness of the limit fields and correctorsand are given in the following theorem.

Theorem 6.1 The limit fields solve the coupled variational problems:

where the potential energy is given by(6.2).

Proof Based on the decomposition(2.11)we choosevanishing on Γ0,δ and we select the test-function

A straightforward calculation leads to the convergences

Now,dividing(2.17)by δ2 then transforming with Tε,δ (the left hand-side)leads to

and by passing to the limit,we get

Let us introduce the correctors and consider the second testfunctions as

The simple computation yields the strong convergence

which,sinceachieves establishing(6.8).

Set C∞ the 6×6 symmetric matrix defined by its elements

where the second order tensors are given in(6.13).

Theorem 6.2 Under the assumptions on the forces given in Section 2.4,[L 2(0,L)]9,the limit displacement field is the solution of the variational problem

where the symmetric matrix C∞is given in(6.9)and where

Proof For all and denote

We rewrite(6.8)with that as:

Find such that for all

We consider the decomposition of the strain tensor E as given by(6.1),and we introduce the 6 auxiliary fields,as the unique solution to the following six independent variational problems:

The solution of problem(6.11)is under the form

HenceLet us return to the limit homogeneous problem(6.8)with b v=0,b V=0:

where h is formed with the unknownsLetting

we get

6.2 The case

To anywe associate the corrector field

belonging to

The spacesand(endowed with their usual norms)are isometric.To anywe associate the symmetric tensor(see(4.1)and(5.7)),

By collecting the convergence results given by Lemmas 4.1 and 5.2 we can state the following result.

Corollary 6.2 There exists a subsequence of(ε,δ),still denoted(ε,δ),limit displacement fields and correctors such that

where the limit of the symmetric strain tensor is obtained using(3.4)–(3.5)and(6.15).

We endow with the following norm:

Lemma 6.2 On the spacethe norm is equivalent to the usual norm of this product spaces.

Proof Due to the fact that for a.e.we first have

Now,consider the change of variableω×(0,θ)which transforms the scaled beam of sectionω and length 1 to a beam of sectionω and length θ.We define the new function associated to given by(6.14),

One has

The 3D Korn’s inequality,the periodicity of and the mean value property of imply the existence of a constant C(which depends onθ)such that

Hence,there exists a constant C(which depends onθ)such that

We easily prove(see[7]),

The existence and uniqueness of the limit displacement fields and correctorsand are given in the following theorem whose proof is obtained as in the previous section.

Theorem 6.3 The limit fields solve the coupled variational problems: Z

where the potential energy is given in(6.2).

Then we are in the position to state the main result in the caseθ∈]0,∞[.

Theorem 6.4 Under the assumptions on the forces given in Section 2.4,[L 2(0,L)]9,the limit displacement field is the solution of the variational problem

and the 6×6 symmetric matrix Cθis given in(6.22)and where

Proof The proof follows the same schemeas before.For all,we setdenoteand introduce the 6 auxiliary fieldsas the unique solution to the following six independent variational problems:

The limit problem(6.20)is obtained by defining the 6×6 matrix Cθby its elements,as

with

6.3 The case

To any we associate the corrector field

belonging to

The spaces(endowed with their usual norms) and are isometric.To anywe associate the symmetric tensor(see(4.1)and(5.9))

By collecting the convergence results given by Lemmas 4.1 and 5.3 we can state the following result.

Corollary 6.3 There exists a subsequence of(ε,δ),still denoted(ε,δ),limit displacement fields and correctors and such that

where the strain tensors are given by(3.4)–(3.5)and(6.24).

We endow with the following norm:

Lemma 6.3 On the spacethe norm is equivalent to the usual norm of this product spaces.

Proof Due to the fact that for a.e.we first have

The 3D Korn’s inequality,the periodicity of and the mean value property of imply the existence of a constant C.Hence,there exists a constant C such that

We easily prove that(see[7])

The existence and uniqueness of the limit fields and correctorsand are given in the following lemma.

Theorem 6.5 The limit fields solve the coupled variational problems:

where the potential energy is given by(6.2).

Then we are in the position to state the main result in the caseθ=0.

Theorem 6.6 Under the assumptions on the forces given in Section 2.4,[L 2(0,L)]9,the limit displacement field is the solution of the variational problem

where the 6×6 symmetric matrix C0 is given in(6.30)and where

Proof The proof follows the same scheme as before.For all,setand denote

We rewrite(6.26)with as:Find such that for all

We consider the decomposition of the strain tensor E as given by(6.1)and introduce the 6 auxiliary fieldsastheuniquesolution to the following six independent variational problems:

The solution of problem(6.28)is under the form

HenceLet us return to the limit homogeneous problem(6.26)with

where h is formed with the unknownsLetting

we get

The limit problem(6.27)is obtained by defining the 6×6 matrix C0 by its elements

6.4 Strong formulation

Let us remark that the six order unknown obtained for solve respectively the limit problems(6.10),(6.20),(6.27)which can be written in the form

where

and where is given in(6.4),and the fourth order symmetric positive elasticity tensor takes respectively the value C∞,Cθ,C0 associated to the fields

Since,whatever the value ofwe have the boundary condition.One can also write these problems as

Remark 6.1 Even though the initial elasticity tensor aε,δ is diagonal and depending only upon the variable x3,generally,the homogenized six-order elasticity tensor C∞,Cθand C0,given respectively by(6.9),(6.22)and(6.30),will be coupled so that the limit fields will be the solution of coupled systems.

7 Asymptotic Expansion

We recall the decomposition(2.11),

With appropriate definition of the vectors and of the functional spaces(according to the value of the limitθ),we have established in(4.3),(6.7),(6.16)and(6.25)the existence of limits

such that

with

This suggest an asymptotic expansion up to the second order in ε,δ,

with

It is of interest to note that the corrector appears in the expansion at the same orderδ2 as

Remark 7.1 Following the lines of[4,Chapter 10,Section 9]we obtain these two results.

(1)The function from[0,+∞]into R6×6 is continuous and uniformly elliptic.And as an immediate consequence we can establish that the function from[0,+∞]into V M is continuous.

(2)We can also prove that the limit problem corresponding toθ=0 is the one obtained when firstεgoes to 0 and thenδwhile the limit problem forθ=+∞is the one obtained when firstδgoes to 0 and thenε(see also[6]).

Acknowledgements Professor Miara wishes to thank Professor Yamamoto very deeply for his kind hospitality at the University of Tokyo where part of this work was done.

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