Asymptotic Behavior of Stable Structures Made of Beams

In this paper, we study the asymptotic behavior of an ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document}-periodic 3D stable structure made of beams of circular cross-section of radius r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r$\end{document} when the periodicity parameter ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document} and the ratio r/ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${r/\varepsilon }$\end{document} simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

tends to 0, in the framework of the linear elasticity. By "thin", we mean that the radius r of the beams is much smaller than the periodicity parameter ε and that we deal with the case where ε and r/ε simultaneously tend to 0.
It is well known to engineers that for wire trusses, lattices made of very thin beams, bending dominates the stretching-compression. A contrario, if the same structures are made of thick beams the stretching-compression dominates. This is what several mathematical studies of recent decades have obtained for periodic structures made of beams. For such structures, from the mathematical point of view, this means that the processes of homogenization and dimension reduction do not commute (see the pioneer works [5,11,12] and also [1,6,8,24,25,27,28,31]). Our aim is to investigate between these extreme cases. More precisely, we consider the case for which the ratios diam( )/ε and ε/r are of the same order ( is the 3D domain covered by the beam structure). In Sects. 5 and following, we show that the ratio r/ε 2 and its limit κ ∈ [0, +∞] play an important role in the estimates and the asymptotic behaviors. It worth to notice that in our analysis, κ = 0 also corresponds to the case where first the dimension reduction is done and then the homogenization, while κ = +∞ is for the vice-versa case. In the convergences (7.12) of Theorem 2, we show that the rescaled global displacement depends on κ. If κ ∈ (0, +∞), its limit is a combination of a global displacement (a pure stretching-compression) and a local bending; if κ = +∞ it is just a global displacement and if κ = 0 it is a local bending.
Our analysis relies on a displacement decomposition for a single beam introduced in [13][14][15]. According to those studies, a beam displacement is the sum of an elementary displacement and a warping. The elementary displacement has two components. The first one is the displacement of the beam centerline while the second stands for the small rotation of the beam cross-sections (see [13,15]). This decomposition has been extended for structures made of a large number of beams in [14] (see [4] for the structures made of beams in the nonlinear elasticity framework). Here, similar displacement decompositions are obtained, these decompositions are used for stable beam structures (see Lemma 5) and then for periodic 3D stable structures made of beams. It is important to note that estimate (4.5) 1 is the key point of this paper. It characterizes the stable structures. In a forthcoming paper, we will investigate the unstable and auxetic 3D periodic structures made of beams and we will see that all the estimates of Lemma 5 will remain except (4.5) 1 . These decompositions allow to obtain Korn type inequalities as well as relevant estimates of the centerline displacements.
To study the asymptotic behavior of periodic stable structures and derive limit problem we use the periodic unfolding method introduced in [9] and then developed in [10]. This method has been applied to a large number of different types of problems. We mention only a few of them which deal with periodic structures in the framework of the linear elasticity (see [3,[16][17][18][19][20][21]26]). As general references on the theory of beams or structures made of beams, we refer to [2,7,22,23,29,30].
The paper is organized as follows. Section 2 introduces structures made of segments and remind properties of Sobolev spaces defined on these structures. Furthermore, in this section we give a simple definition of stable and unstable structures and present several examples. In Sect. 3 we remind known results concerning the decomposition of a beam displacement into an elementary displacement and a warping. This section also gives estimates with respect to the L 2 -norm of the strain tensor of the terms appearing in the decomposition. In Sect. 4 we extend the results of the previous section to structures made of beams. Complete estimates of our decomposition terms and Korn-type inequalities are obtained for stable structures.
In Sect. 5 we deal with an ε-periodic stable 3D structure made of r-thin beams, S ε,r . For this structure we introduce a linearized elasticity problem and specify the assumptions on the applied forces. Using results from the previous section we decompose every displacement of S ε,r as the sum of an elementary displacement and a warping and provide estimates of the terms of this decomposition. The scaling of the applied forces are given with respect to ε and r. That leads to an upper bound for the L 2 -norm of the strain tensor of the solution of the elasticity problem of order 1.
In Sect. 6 we introduce different types of unfolding operators, mainly one for the centerline beams and another for the cross-sections. This last operator concerns the dimension reduction. Several results on these operators are given in this section and Appendix C.
Sect. 7, deals with the asymptotic behavior of a sequence of displacements and their strain tensors. Then, in Sect. 8, in order to obtain the limit unfolded problem we split it into three problems: the first involving the limit warpings (these fields are concentrated in the cross-sections, this step corresponds mainly to the process of dimension reduction), the second involving the local extensional and inextensional limit displacements posed on the skeleton structure and the third involving the macroscopic limit displacement posed in the homogeneous domain .
In Sect. 9 we complete this analysis by giving the homogenized limit problem (Theorem 4). We obtain a linear elasticity problem with constant coefficients calculated using the correctors.
In Sect. 10 we apply the previously obtained results in the case where the periodic 3D beam structure is made of isotropic and homogeneous material. We present an approximation to the solution of the linearized elasticity problem which can be explicitly computed using the solution of the homogenized problem.
In the Appendix we give the most technical results.

Structures Made of Segments
In this paper we consider structures made up of a large number of segments.
, be a set of segments and K the set of the extremities of these segments.
S is a structure if • S is nonincluded in a plane, • S is connected, • a common point to two segments is a common extremity of these segments, • if an element of K belongs to only two segments then the directions of these segments are noncollinear, • for every segment γ we denote t 1 a unit vector in the direction of γ , ∈ {1, . . . , m}.
For every ∈ L 1 (S) define Observe that the right-hand side of the above equality does not depend on the choice of a unit vector in the directions of the segments. The space L 2 (S) is endowed with the norm , ∀ ψ ∈ H 1 (S).

Stable Structures
The space of all rigid displacements is denoted by R We define the space U S as follows: 3 | for every segment γ ⊂ S, U |γ is an affine function, ∈ {1, . . . , m} .
If the above condition is not satisfied, S is an unstable structure.
Remark 1 1. The structure made of the edges of a tetrahedron is stable (see Fig. 1.a). If we remove one edge then the structure becomes unstable (see Fig. 1.b). 2. The structure made of 12 edges and 6 diagonals of the faces of a cube is stable (see Fig. 1.c). If we remove one diagonal then the structure becomes unstable (see Fig. 1.d).

Fig. 1 Stable and unstable structures
We equip U S with the following bilinear form: and the associated semi-norm Lemma 1 Let S be a stable structure. There exists a constant C, which depends on S, such that for every U in U S there exists r ∈ R such that Proof Let R ⊥ be the orthonormal of R in U S for the scalar product If U belongs to R ⊥ and satisfies U S = 0 then, since S is a stable structure, U belongs to R. Therefore U is equal to 0. The semi-norm · S is a norm on the space R ⊥ . Since R ⊥ is a finite dimensional vector space, all the norms are equivalent. Thus (2.4) is proved.

Decomposition of Beam Displacements
In this section, we remind some results concerning the decomposition of a beam displacement. These results will be used later and can be found in [15]. For the sake of simplicity these results are formulated for the beam B l,r . = (0, l) × D r whose cross-sections are disc of radius r (r ≤ l). The beam is referred to the orthonormal frame (O; e 1 , e 2 , e 3 ) (e 1 is the direction of the centerline). In this frame the running point is denoted Any displacement u ∈ H 1 (B l,r ) 3 of the beam B l,r is uniquely decomposed as follows where U e is called elementary displacement and it stands for the displacement of the centerline of the beam and the small rotation of the cross-section at every point of the centerline (see Fig. 2): 3 is the warping (the deformation of the cross-sections), it satisfies (for more details see [15]) Taking into account the decomposition (3.1) and the representation for the elementary displacement given by (3.2) the strain tensor e(u) has the following form: (3.4) Below is a lemma proven in [13,15]. It gives estimates for the warping and the terms from U e in the above strain tensor (3.4).
Lemma 2 Let u be in H 1 (B l,r ) 3 decomposed as (3.1)-(3.2)-(3.3). The following estimates hold: The constants are independent of l and r ≤ l.
The function U , defined in (3.1), is decomposed into the sum of two functions U h and U , where U h coincides with U in the extremities of the centerline and is laffine between them (see Fig. 2), and U = U − U h is the residual part, i.e., In the same way the function R, defined in (3.1), is decomposed into the sum of two functions R h and R. It is obvious, but important to note that

Lemma 3
The following estimates hold: The constants do not depend on l and r. Proof Then, the Poincaré and the Poincaré-Wirtinger inequalities together with the above estimates yield from which we derive the other estimates in (3.6).

Decomposition of the Displacements of a Beam Structure
From now on, S is a stable structure. The beam structure S 1,r is defined as follows: For ∈ {1, . . . , m}, denote P ,r the straight beam with centerline γ = [A , B ] and reference cross-section the disk D r . = D(O, r) of radius r, 0 < r ≤ l (the disk D 1 for simplicity will be denoted D). The straight beam P ,r is referred to the orthonormal frame (A ; t 1 , t 2 , t 3 ) By definition, the whole structure S 1,r contains the straight beams P ,r , ∈ {1, . . . , m} and the balls of radius r centered in the points of K, more precisely one has The set of junction domains is denoted by J r . There exists c 0 which only depends on S such that The set J r is defined in such a way that S 1,r \ J r only consists of disjoint straight beams.
Definition 3 An elementary beam-structure displacement is a displacement U e belonging to H 1 (S 1,r ) 3 whose restriction to each beam is an elementary displacement and whose restriction to each junction is a rigid displacement In [14] it is shown that every displacement u ∈ H 1 (S 1,r ) 3 can be decomposed as where U e is an elementary beam-structure displacement and where u ∈ H 1 (S 1,r ) 3 is the warping. Here, the pair (U e , u) is not uniquely determined. Furthermore, the warping satisfies the conditions (3.3) "outside" the domain J r (see [14,15]), more precisely, one has The following lemma is proved in [14,Lemma 3.4]: There exists a decomposition of u, u = U e + u for which U e is an elementary beam-structure displacement. The terms of this decomposition satisfy The constants do not depend on r.
Here, again we split the field U into the sum of two fields U h and U , where U h coincides with U in the nodes of S and is affine between two contiguous nodes and U = U − U h is the residual part.
In the same way the fields R h and R are introduced. The field U h describes the displacement of the nodes, i.e. the global behavior of the structure, whereas U stands for the local displacement of the beams.
By construction the fields U h and R h belong to U S . Furthermore one has Lemma 5 For every u ∈ H 1 (S 1,r ) 3 the following estimates hold: The constants do not depend on r.
Proof Estimates (4.4) are the immediate consequences of the Lemmas 3 and 4. Since S is a stable structure, Lemma 1 and again (4.4) yield a rigid displacement r ∈ R (r(x) = a+b∧x) such that (4.5) 1 holds. Besides, from the Poincaré-Wirtinger inequality and (4.4) 4 , there exists b ∈ R 3 such that The constant does not depend on r. Then, (4.5) 1 and the above estimate give Since the structure has more than two segments with non-collinear directions, this yields Hence, (4.5) 2 is proved.
Let S be a stable structure such that S ∪ (S + e 1 ) is a stable structure. For every displacement u ∈ H 1 (S 1,r ∪ (S 1,r + e 1 )) 3 , Lemma 5 gives two rigid displacements r 0 , r 1 such where G is the center of mass of S.

Lemma 6
Let S be a stable structure such that S ∪ (S + e 1 ) is also a stable structure. The following estimate holds: The constant does not depend on r.
Proof From Lemma 5, there exists a rigid displacement r such that The constant does not depend on r. Hence The above estimates yield (4.7) since in R the norms · H 1 (S) , · H 1 (S+e 1 ) and · H 1 (S∪(S+e 1 )) are equivalent.

A Periodic Beam Structure as 3D-Like Domain
From now on, in all the estimates, we denote by C a strictly positive constant which does not depend on ε and r.

Notations and Statement of the Problem
Below we consider periodic structures S included in a closed parallelotope.

Definition 4
A structure S is a 3D-periodic structure if for every i ∈ {1, 2, 3} the set S ∪ S + e i is a structure in the sense of Definition 1.
Definition 5 A 3D-periodic structure S is a 3D-periodic stable structure (briefly 3-PSS) if S and S ∪ S + e i , i ∈ {1, 2, 3}, are stable structures in the sense of Definition 2.
Remark 2 1. The structure made of 12 edges and 6 diagonals of the faces of a cube is a 3D-periodic stable structure ( Fig. 3.a). Let be a bounded domain in R 3 with a Lipschitz boundary and be a subset of ∂ with nonnull measure. We assume that there exists an open set with a Lipschitz boundary such that ⊂ and ∩ ∂ = . Denote The open sets ε , ε , ε , int ε and int ε are connected. Moreover, the following inclusions hold The running point of S ε is denoted s. Let S ε,r be a beam structure consisting of balls of radius r centered on the points of K ε and beams, whose cross-sections are discs of radius r and their centerlines are the segments The parametrization of the beam P ξ ε ,r ( ∈ {1, . . . , m}) is given by (see (4.1)) The junction domains (the common parts of the beams) is denoted J ε,r . One has The structure S ε,r is included in ε . The space of all admissible displacements is denoted V ε,r V ε,r = u ∈ H 1 (S ε,r ) 3 | ∃u ∈ H 1 (S ε,r ) 3 such that u |Sε,r = u and u = 0 in S ε,r \ S ε,r .
It means that the displacements belonging to V ε,r "vanish" on a part ε,r included in ∂S ε,r ∩ ∂ . We assume that S ε,r is made of isotropic and homogeneous material. For a displacement u ∈ V ε,r , we denote by e the strain tensor (or symmetric gradient) e(u) .
We have two coordinate systems. The first one is the global Cartesian system (x 1 , x 2 , x 3 ) and is related to the frame (O; e 1 , e 2 , e 3 ). The second one is the local coordinate system (s 1 , s 2 , s 3 ) defined for every beam and related to the frame (εξ + εA 2 ; t 1 , t 2 , t 3 ), ∈ {1, . . . , m}. The orthonormal transformation matrix from the basis (t 1 , t 2 , t 3 ) to the basis (e 1 , e 2 , e 3 ) is T = t 1 | t 2 | t 3 , this matrix belongs to SO (3).
The coefficients a ε ij kl are given via the functions a ij kl ∈ L ∞ (S × D) The constitutive law for the material occupying the domain S ε,r is given by the relation between the linearized strain tensor and the stress tensor = a ε,r ij kl e s,kl (u), ∀ u ∈ V ε,r .
The unknown displacement u ε 1 : S ε,r → R 3 is the solution to the linearized elasticity system: on ∂S ε,r \ ε,r , where ν ε is the outward normal vector to ∂S ε,r \ , f ε is the density of volume forces. The variational formulation of problem (5.7) is

Final Decomposition of the Displacements of a Periodic Beam Stable Structure as a 3D-Like Domain
Let u be a displacement belonging to V ε,r . As proved in [14], we can decompose u as the sum of an elementary displacement and a warping.
The decompositions introduced in Sect. 4, the estimates of Lemma 5 lead to the following estimates: Lemma 7 For every u ∈ V ε,r the following estimates hold: Moreover, one has Proof We apply Lemma 5 to the structure ε(ξ + S 1,r ). Replacing r by r ε and then summing over all ξ ∈ ε give the estimates (5.9) and (5.10).

Lemma 9
Let be a function defined on ε and extended using the classical Q 1 interpolation procedure in a function denoted and belonging to W 1,∞ ( ε ) then we have Moreover, there exists a rigid displacement r such that The estimates (5.13) 1,2 and Lemma 9 yield And (5.16) 1,2 are proved. From which we get which also read (5.16) 3 . Lemma 8 allows to apply the 3D-Korn inequality in the domain int ε using estimate (5.16) 3 . That gives (5.17).

Proposition 2 Let S be a 3-PSS.
For every u in V ε,r , the following estimates of the elementary displacement holds:

Moreover, one has the Korn type inequalities
Proof This proposition is a consequence of Proposition 1 and two lemmas postponed in Appendix A.

Assumptions on the Applied Forces
We distinguish two types of applied forces. The first ones are applied in the beams (between the junctions) and the second ones are applied in the junctions.
The applied forces f ε in the set of beams ξ ∈ ε m =1 P ξ ε ,r .
For simplicity, we choose these applied forces constant in the cross-sections and equal to The applied forces F r,Kε in the junctions.
These forces are defined in the balls centered in the nodes with radius r

Lemma 10 Taking the applied forces as
Proof The proof is postponed in Appendix B.
As a consequence of the above lemma one obtains Proof In order to obtain apriori estimate of u ε , we test (5.8) with v = u ε . From (5.21), we obtain which leads to (5.22).

The Unfolding Operators
The classical unfolding operator T ε is developed in [9,10]. Here, we will use similar opera- in the context of the domains ε , S ε and S ε,r .
Definition 6 (Classical unfolding-operator) For a measurable function φ on , the unfolding operator T ε is defined as follows: Definition 7 (Unfolding-operator) For a measurable function φ on ε , the unfolding operator T ext ε is defined as follows: Proof Inequality (6.1) is an immediate consequence of the definitions of these operators.
As a consequence of the above lemma, the properties of the operator T ext ε are similar to those of the classical unfolding operator T ε . For the main properties of the unfolding operator T ε , we refer the reader to [10, Chap. 1].
Below, we introduce two new unfolding operators. The first one is used for the centerlines of beams and the second one is used for the small beams (it concerns the reduction of dimension).
In the definitions below, ε x ε represents a macroscopic coordinate (the same coordinate for all the points in the cell ε x ε + εY ) while S is the coordinate of a point belonging to S. Hence, ε x ε + εS represents the coordinate of a point belonging to S ε . In order to get a map (x, S) −→ ε x ε + εS almost one to one, we have to restrict the set S. This is why from now on, to introduce the unfolding operator, in lieu of S we consider the set For simplicity we still refer to it as S. The set of new nodes is always denoted K and the number of beams of S is still denoted m. Definition 8 (Centerlines unfolding) For a measurable function φ on S ε , the unfolding operator T S ε is defined as follows: Definition 9 (Beams unfolding) For a measurable function u on S ε,r , the unfolding operator A is an extremity of the segment γ ⊂ S and D = D 1 is the disc of radius 1.

Lemma 12 (Properties of the operators T
The constant only depends on S.
From now on, every function belonging to L p ( ) (p ∈ [1, +∞]) will be extended by 0 in containing the functions which are the Q 1 interpolations of their values at the vertices of the parallelotope Y .

Lemma 13 For every in
Then |Sε belongs to W 1,∞ (S ε ) and it satisfies Let { ε } ε be a sequence of functions belonging to W 1,∞ ( ε ) satisfying (6.9) and Moreover, if one also has Proof The proof is given in Appendix C.
First convergence results for sequences in H 1 (S ε ).
Then, up to a subsequence, there exists φ ∈ L 2 ( ; H 1 per (S)) such that If we only have then, up to a subsequence, there exists φ ∈ L 2 ( ; H 1 per (S)) such that Proof The proof is postponed in Appendix C.

Definition 10 The local average operator
A second lemma for sequences in H 1 (S ε ).
Proof The proof is postponed in Appendix C. Denote Corollary 2 Let {φ ε } ε be a sequence of functions belonging to H 1 (S ε ) 3 ∩ V ε,r and satisfying the following Then, up to a subsequence, there exists Proof Since {φ ε } ε belongs to V ε,r , these functions equal to 0 in S ε \ S ε . Applying Lemma 15 with S ε instead S ε and with instead give the result.

Asymptotic Behavior of a Sequence of Displacements
From now on, we assume that r is a function of ε satisfying the following conditions: In addition, every field appearing in the decomposition introduced in the previous sections will be denoted with only the index ε.
In this section we consider a sequence {u ε } ε of displacements belonging to V ε,r and satisfying e(u ε ) L 2 (Sε,r ) ≤ C.
function on every segment of S and the following convergences hold: and Proof Below, every convergence is up to a subsequence of {ε} still denoted {ε}.
In order to show convergence (7.7) 2 , note that from (5.9) 2 , (6.7) 1 and (6.5) it follows Therefore, convergence (7.7) 2 is proved, since Remark 3 Due to (4.2), the warping u satisfies a.e. in × γ , ∀ ∈ {1, . . . , m}. 3 |A ∧ t 1 is an affine function on every segment γ , ∈ {1, . . . , m} , The field U is in L 2 ( ; D Ex ) while the pair U, R belongs to L 2 ( ; D I n ). It worth to notice that a field A belonging to H 1 per,0 (S) 3 3 which is the first component of an element belonging to D I n . We endow D Ex (resp. D I n ) with the semi-norm

is a local extensional displacement if and only if
).

Lemma 16
On D Ex the semi-norm · S is a norm equivalent to the norm of H 1 (S) 3 . On D I n the semi-norm (·, ·) D In is a norm equivalent to the norm of H 1 (S) 3 × H 1 (S) 3 .
Proof The proof is given in Appendix D.

The Limit Unfolded Problem
To obtain the limit unfolded problem, we will choose test displacements v in V ε,r which vanish in the junction domain J ε,r or which are equal to rigid displacements in J ε,r . In doing so, we will have The step-by-step construction of the unfolded limit problem (8.12) is considered in Lemmas 17, 18, 19.
and r/ε goes to 0, the support of the above test-displacement is only included in the beams whose centerline is εξ + εγ . Moreover, this displacement vanishes in the neighborhood of the extremities of this beam, it means that this displacement vanishes in the junction domain J ε,r .
One has We apply the unfolding operator T b, ε and pass to the limit, this gives Using (5.20) and then unfolding and passing to the limit yield The above convergences lead to ×γ ×D 3 we obtain (8.1). Step 1. The test displacement. Set
The convergences (8.6) yield The presence of v ε,r in the test displacement is just to eliminate ε 3 Then, again using the convergences (8.6), we obtain Hence, Unfolding the left-hand side of (5.8) and passing to the limit give Step 3. Limit of the RHS. Now, we consider the right-hand side of (5.8) Let's take the first term in the right-hand side of (8.8). Taking into account the symmetries of the ball B(εξ + εA , r) and the fact that B(O,r) |x| 2 dx = 4πr 5 5 . After a straightforward calculation, one obtains Since |Y | = 1, one has Hence, Now, we take the second term in the right-hand side of (8.8). Due to (6.6), we only need to consider r 2 Assumptions (7.1) and convergence (8.6) 1 lead to Hence,

Lemma 24 and the density of
Besides, since v belongs to L 2 ( × S; H 1 (D)) 3 equality (8.1) together with the one above yield (8.5).

10)
where 2 |K| is the number of points of K and S the measure of S.

Proof
Step 1. Limit of the LHS of (5.8).
Let V be in D(R 3 ) 3 such that V = 0 in \ . We define V ε,r using F. This function is extended as in Step 1 of the proof of Lemma 18. Set v ε,r = ε r V ε,r ∈ V ε,r .
We have r ε a.e. in × S × D.

Convergence (8.11) leads to
Step 2. Limit of the RHS. Now we consider the right-hand side of (5.8). By (5.20), firstly we have and secondly, due to (6.6), we pass to the limit in Since the set of functions belonging to D(R 3 ) 3

and vanishing in
Taking into account that v belongs to L 2 ( × S; H 1 (D)) 3 and using (8.1), equality (8.10) is proved. Theorem 3 (The unfolded limit problem) Let u ε be the solution to (5.8). There exist U ∈ H 1 ( ) 3 , (U, U , R) ∈ L 2 ( ; D Ex × D I n ) and u ∈ L 2 ( × S; D w ) such that U , U, U , R, u is the solution to the following unfolded problem: (8.12) Moreover, the following convergences hold ( ∈ {1, . . . , m}): Proof From Lemmas 17, 18, 19 we obtain that (U, U , U, R, u) satisfies (8.12) for every test The coercivity of this problem is given by Lemma 26. Since the problem (8.12) admits a unique solution, the whole sequences in Theorems 1, 2 and (8.13) converge to their limits. Now, we prove the strong convergence (8.13). First, observe that due to the inclusion of J ε,r in A∈Kε B(A, c 0 r) given by (5.1), the portions of beams which correspond to S 1 ∈ (2c 0 r, l − 2c 0 r) are all disjoint. Furthermore, since σ (u ε ) : e(u ε ) is non-negative, one has From (7.13) and the fact that r goes to 0, one obtains ( ∈ {1, . . . , m}) Hence, choosing u ε as a test function in (5.8) and using a weak lower semi-continuity of convex functionals, one has Thus, all inequalities above are equalities and which in turn leads to the strong convergence (8.13).

Expression of the Warping u
In this subsection we give the expression of the warping u in terms of the macroscopic displacement U and the microscopic fields U , U , R.
To this end, we use the variational formulation (8.1). For every ∈ {1, . . . , m} one has This shows that u can be expressed in terms of the elements of the tensors E and E S .
We write a.e. in × S × D. (9.1) Now, we introduce 4 correctors which are the solutions to the following cell problems: Since a ij kl 's belong to L ∞ (S × D), then χ q ∈ L ∞ (S; D w ), q ∈ {1, . . . , 4}. Hence, we have

Expression of the Microscopic Fields U , U , R
In this subsection we give the expression of the microscopic fields U , U , R in terms of the macroscopic displacement U . To this end, as before, we use the variational formulation (8.12). Thus, taking V = 0, v = 0 in (8.12), then replacing u by its expression, using the following equality: and the variational problem (9.3) has the following form: a.e. in S since χ q 's verify (9.2).

At this step, the unfolded problem becomes
.
Hence, one has In problem (9.5), we replace (U, U , R) by (9.7) and we choose (V, V, B) = (0, 0, 0). That gives (9.8) Now, taking into account the definition of the corrector χ ij . = χ ij , χ ij , χ ij ), the left-hand side becomes where B hom is a symmetric bilinear form associated to the definite positive quadratic form Now, we simplify the right-hand side of (9.8). Set Thus, the limit field U ∈ H 1 ( ) 3 is the solution to the homogenized problem b hom ij kl e ij (U) e kl (V) dx

Lemma 20
The components of the homogenized elasticity tensor b ij kl ∈ R satisfy the usual symmetry and positivity conditions Proof By definition of the b hom ij kl 's, the symmetry of matrices M ij = M ji and correctors χ ij = χ ji we obtain the symmetries of the b hom ij kl 's. From equality (9.9), Lemma 27 and estimate (G.4) we have Theorem 4 (The homogenized limit problem) The limit field U ∈ H 1 ( ) 3 is the unique solution to the homogenized problem b hom ij kl e ij (U) e kl (V) dx where the b hom ij kl are given by (9.10) and the c hom ij q by (9.11).

The Case of an Isotropic and Homogeneous Material
We consider an isotropic and homogeneous material for which the relation between the linearized strain tensor and the stress tensor is given as follows σ (u) = λ Tr(e(u)) I 3 + 2μ e(u), (10.1) where I 3 is the unit 3 × 3 matrix and λ, μ are the material Lamé constants. The correctors χ q ∈ L ∞ (S; D w ), q ∈ {1, 2, 3, 4}, have the following form (see [13]) where ν = λ 2(μ+λ) is the Poisson coefficient. Due to the symmetries of the elasticity coefficients and cross-sections, we have immediately Hence, we obtain (10. 2) The matrix A becomes where E = μ(3λ+2μ) λ+μ is the Young's modulus.

That gives
and then It means that χ ij · t 1 is affine on every segment of S. The function χ ij belongs to U S . Set For every (i, j ) ∈ {1, 2, 3} 2 one has Denote M ij the restriction to S of the linear field x ∈ R 3 −→ M ij x ∈ R 3 . It belongs to U S .

Problem (10.4) becomes
The corrector χ ij is the projection on U S,per,0 of the field M ij ∈ U S for the scalar product < ·, · > 1 (see (2.2) and Lemma 1).

Conclusion
We conclude, that for our ε-periodic r-thin structure, the solution to the linearized elasticity problem (5.7) (in the strong), or (5.8) (in the weak/variational form) can be reconstructed in the following form: for a.e. x ∈ S ε,r . (11.1) It is illustrated on Fig. 4. The strain tensor in the global coordinates can be obtained using (5.3). Then, we can reconstruct the local stress field for P ξ ε ,r beam as follows a.e. in × S × D. (11.2) Funding Note Open Access funding enabled and organized by Projekt DEAL.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Open Access
Proof Since u belongs to V ε,r , by definition, it is equal to 0 in S ε,r \ S ε,r . Then, there exists a rigid displacement r (x) = a + b ∧ x, (a , b ) ∈ R 3 × R 3 such that (using (5.17) with instead of ) As a consequence U = R = 0 a.e. in O. Hence, The constants do not depend on ε and r. Therefore, where the constant C 0 only depends on the volume and diameter of . Finally,

Appendix B: The Applied Forces
First, note that the number of elements in K ε , which is denoted by |K ε | is less than where |K| is the number of elements in K.
Proof of Lemma 10 Let u be in V ε,r . By the estimates of Proposition 2, we have Now, taking into account that for every node A ∈ K ε the following decomposition holds: Thus, the second and third terms in the right-hand side of (B.3) vanish. Then, using the Cauchy-Schwarz inequality, (5.9) 1 and (B.1), the last two integrals in (B.3) are estimated as follows and , then using the fact that U h , R h are affine functions between two contiguous nodes Then, the remaining two integrals in the right-hand side of (B. and The above estimates, those of Lemma 22 and the fact that r ≤ ε end the proof of Lemma 10.

Proof of Lemma 14
Using the properties of the unfolding operator T S ε (6.3)-(6.4) and the estimates for φ ε , we obtain Thus, Hence, up to a subsequence ε, there exists φ ∈ L 2 ( ; H 1 (S)) such that (6.14) holds.
In both cases, the periodicity of φ is obtained proceeding in the same way as to prove [10,Theorem 4.28].
Observe that ε also belongs to W 1,∞ ( int ε ). Proceeding as in [10,Chap. 4] we obtain the following estimates: [16] gives ∈ H 1 ( ) such that (up to a subsequence) Besides, by definition of ε , ε| ε ∩Sε belongs to W 1,∞ ( ε ∩ S ε ) and Lemma 13 gives By (6.16), (C.1)-(C.2), we obtain Therefore, Lemma 14 gives a function φ ∈ L 2 ( ; H 1 per (S)) such that (up to a subsequence) Due to estimate (C.4), there exist a subsequence of {ε} and F ∈ L 2 (S) such that Let O be an open set strictly included in . If ε is small enough, one has Applying Lemma 13 in the context of the open set O leads to (up to a subsequence) Hence, As a consequence F = ∇ · t 1 a.e. in × S and (6.17) are proved. Now, from (C.3) and (C.5) we obtain Hence,

Appendix D: Proof of Lemma 16
Step 1. We show that the semi-norm · S is a norm in D Ex . Indeed, if dA dS · t 1 L 2 (S) = 0 then A is a rigid displacement (remind that S is a stable structure). The periodicity of A implies that A is a constant field. Since the mean value of A is equal to zero then A = 0. Hence, the semi-norm · S is a norm in D Ex .
Step 2. We show that the norm · S is equivalent to the norm · H 1 (S) .
First, we have The map where A Aff is defined by Lemma 1 claims that there exists a rigid displacement r such that .
Since S is a 3-periodic structure and A Aff is a periodic function, we can choose r constant. Hence, . (D.1) The function A−A Aff vanishes on all the nodes. Therefore by the definitions of the functions A and A Aff we obtain A − A Aff ∧ t 1 is an affine function on all the segments γ , ∈ {1, . . . , m}. Hence, and, therefore, As a consequence of (D.1)-(D.3), one obtains .
Remind that since A belongs to H 1 per,0 (S) 3 , the Poincaré-Wirtinger inequality gives Thus, Both norms are equivalent.
Step 3. We show that the semi-norm (·, ·) D In is a norm in D I n .
Indeed, if d B dS L 2 (S) = 0, then B is a constant field. Remind that A vanishes on all the nodes, therefore one has B ∧ t 1 = 0 in S. Since every node is a common extremity of at least two segments with non-collinear direction, then B vanishes on every node and thus B = 0 in S. Hence, we have A = 0 on S and the semi-norm (·, ·) D In is a norm in D I n .
First, we have ∀( A, B) ∈ D I n , We prove by contradiction that there exists a constant C strictly positive such that and then A n −→ 0 strongly in H 1 per (S) 3 .
Finally ( A n , B n ) H 1 (S)×H 1 (S) → 0 which gives us a contradiction.

Appendix E: A Density Result for H 1 per (S) and D I n
Let r and a be two constants such that 0 < 4r < a. H 1 (0, a), we define F r,a (φ) ∈ H 1 (0, a) by

(E.2)
The constant does not depend on a and r.
Proof From (E.1) 1 we have By definition of F r,a and again using the Cauchy-Schwartz inequality we have In the same way we obtain and (E.2) holds.
Moreover, there exists a strictly positive constant C such that Observe that Hence, we obtain (G.1). Then (G.2) follows from the definition of D w , the Poincaré-Wirtinger and the Korn inequality.

Lemma 26
There exists a strictly positive constant C such that

As in
Step 2 of Lemma 16 and since the structure S is stable, we obtain a rigid displacement r such that Remind that S is also a 3-periodic structure. Therefore, comparing the values of V ζ + W − r on the opposite faces of Y ∩ S gives where b = ∇r is a 3 × 3 antisymmetric matrix. Hence, .
Since W belongs to H 1 per,0 (S) 3 , we obtain .