Junctions between two plates and a family of beams

The aim of this paper was to study the junction between a periodic family of beams and two thin plates. This structure depends on 3 small parameters. We use the decompositions of the displacement fields in every beam and plate to obtain a priori estimates. Then in the case for which the displacements of both plates match, we derive the asymptotic behavior of this structure.


INTRODUCTION
This paper concerns the asymptotic behavior of an elastic multistructure composed by a family of elastic beams having the same length of order  and as cross-section a disk of radius r.The beams are -periodically distributed between two plates of different thicknesses of order .We assume r < ∕2 and r ≤  so as to deal with a family of distinct beams.The lateral boundary of the lower plate is clamped, and the other parts of the boundary are free of forces.The mechanical model is the linear isotropic elasticity.In this paper, the main novelty is to propose a way to obtain sharp estimates.
The aim of this paper was to introduce a simplified model of the skin.So the top layer stands for the epidermis, while the lower one is the hypoderm.The beams periodically distributed between these two layers stand for the collagen fibers in the dermis (for more details, see Blasselle and Griso 1 ).
When we want to study an elastic multistructure the first difficulty is as follows: how can we obtain sharp estimates of the displacements?The Korn inequalities for a plate, a beam, or a bounded 3D regular domain are unfortunately not sufficient.To overcome this difficulty, here we use the decomposition of plates and beams displacements introduced by Griso 2,3 for straight and curved rods, shells, or plates.These decompositions have been extended to the structures made of beams or plates by Griso. 4,5A beam displacement is written as the sum of an elementary displacement and a warping.The elementary displacement is affine in the cross-sections; it depends on two vector fields define on the centerline of the beam (see (3.2).The warping stands for the deformation of the cross-sections.Similarly, a plate displacement is also written as the sum of an elementary displacement and a warping.Here, the elementary displacement is affine in the fibers (see (3.13).In Sections 3.1 and 3.2, for both decompositions, we recall the full estimates of the warpings and the estimates of the strain tensors of the elementary displacements with respect to the strain tensors of the displacements.We use these decompositions for the set of beams and for the two plates.Then it remains to obtain the full estimates of the elementary displacements.To do that we compare the terms of the elementary displacements in the small portions of beams included in the plates.In particular, we prove that the estimates of the displacements in both plates differ by the factor 1 ) . Hence, if we want to deal with displacements of the same order in both plates, we assume that the above factor is uniformly bounded.We link the small parameters , , and r:  =  0   , r =  1   ,  > 0,  > 0.
Under the previous assumptions on these small parameters, we show that the couple (, ) must belong to a convex polygon.We introduce two unfolding operators Π  and Π r ; they make possible both reductions of dimension  → 0 and r → 0.
The paper is organized as follows: In Section 2 we describe the structure, and we introduce some notations; we also present the elasticity problem.Section 3 concerns the estimates of the admissible displacements of the structure.In Section 4 we link the small parameters , , and r.We opt to devote the next sections to the general case that corresponds to the interior of a polygon.Section 5 is dedicated to the applied forces.For the sake of simplicity we do not choose surface forces.In Section 6 we introduce the unfolding operators Π  and Π r , and we give their first properties.In Section 7, Theorem 1 gives the weak limits of the different terms involved in the decompositions.We show that the limit displacements are of Kichhoff-Love type in both plates and also in the set of beams.In Sections 8.1 to 8.3 we obtain the limits of the strain and stress tensors.Section 8.4 is concerned by the limit problem, which links the bending and the membrane displacements of the plates.The convergence of the total elastic energy is given in Section 9. Finally, Section 10 is dedicated to the proof of a Poincaré-Wirtinger type inequality.

Throughout this paper
• the Greek indexes  and  belong to {1, 2}, while the Latin indexes i, j, k, l belong to {1, 2, 3}, • the constants, which are denoted by C, do not depend on , , and r, and • we use the Einstein convention of summation over repeated indices.

THE GEOMETRY OF THE STRUCTURE AND THE ELASTICITY PROBLEM
The structure is composed of two plates Ω a  and Ω b  whose thicknesses are 2 a , 2 b ; they are connected by a family of beams (see Figure 1), regularly spaced, whose thicknesses are of order r and lengths 2 − ( a +  b ), which also represents the distance between the two plates, where  a +  b < 1.
Set 2 the reference midsurface of the plates, ) , or equivalently by its components The plates and the beams are made of homogeneous and isotropic elastic materials; for the sake of simplicity, one chooses the same Lamé constants for the plates.Set where  pl ,  pl ,  be , and  be are the Lamé's constants of the materials.They are strictly positive constants.Let {u  }  be a sequence of displacements belonging to H 1 (Ω ,,r ; R 3 ).* The Cauchy stress tensor   in Ω ,,r is linked to the symmetric gradient (∇u  ) S through the standard Hooke's law: where In the domain Ω ,,r , consider the standard problem of elasticity, and the equations of equilibrium in Ω ,,r are where f  ∈ L 2 (Ω ,,r ; R 3 ) denotes the applied forces.
*For the sake of clarity, in this section, we decide to omit the dependence of the fields with respect to the parameters  and r.In Section 7, we will link the parameters , , and r, and then we will only use the parameter  for any fields.
To specify the boundary conditions on Ω ,,r , one assumes that the 3d plate Ω b  is clamped on its lateral boundary and that the boundary Ω ,,r ⧵ Γ b  is free of forces: where  denotes the exterior unit normal vector to Ω ,,r .
Remark 1.The boundary condition (2.5) means that the applied surface forces on the boundary Ω ,,r ⧵ Γ b  are null.This assumption is not necessary to carry on the analysis, but it is natural as far as the family of beams is concerned.
The variational formulation of (2.3) to (2.5) is standard.If V ,,r denotes the space of admissible displacements We equip H 1 (Ω ,,r ; R 3 ) with the seminorm: and throughout the paper and for every v ∈ V ,,r we denote by the total elastic energy of the displacement v. Indeed, choosing v = u  in (2.6) leads to the usual energy relation: For a.e.z ∈ R 2 , we denote [z] ∈ Z 2 the integer part of z and {z} ∈ Y its fractional part, hence z = [z] + {z}.

ESTIMATES FOR THE STRUCTURE DISPLACEMENTS
To obtain a priori estimates on the displacements of the whole structure, one needs a Korn's inequality for this kind of domain.Here, we are concerned with a multistructure, and it is not convenient to estimate the constant in Korn's type inequality with respect to , , and r.To overcome this difficulty, we use decompositions of the beams displacements and of the plates displacements.

Estimates for the set of beams
In this paper, one considers the following assumptions: With the first assumption, one claims that the structure is made of distinct beams, with the second one, one only wants to deal with a set of beams between the two plates.† The space H 1 (P  ; R 3 ) is equipped with the seminorm: Let u be an element belonging to H 1 (Ω ,,r ; R 3 ), theorem 3.1 in Griso 3 gives a decomposition of the restrictions u  of u to the beam  + P  ,  ∈ Ξ  .
For a.e.x = x 1 e 1 + x 2 e 2 + x 3 e 3 ∈  + P  , we write (x ′ = x 1 e 1 + x 2 e 2 ): where U  ∈ H 1 (I  ; R 3 ), R  ∈ H 1 (I  ; R 3 ) and ü  ∈ H 1 (P  ; R 3 ).The residual displacement ü  (named the warping) satisfies for a. e. x 3 ∈ I  , The following estimates of the terms of the decomposition (3.2) are proven in theorem 3.1 in Griso 3 : The strain tensor field of u  is Set From the expression (3.2) of u  and after a straightforward calculation one derives ) .
The estimates (3.4) and the above one lead to Recall the following consequences of the Poincaré-Wirtinger inequality (d ∈ {a, b}): and (3.9) The constants do not depend on , , and r.
Let  be an open subset of R l , l ∈ N * .For every measurable function  on Ξ  × , denote φ the piecewise constant function defined on  ×  by φ(z 1 , z 2 , x) =   (x) for all (z 1 , z 2 ) ∈  + Y ,  ∈ Ξ  and for a.e.x ∈ . (3.11) The fields associated to the decomposition (3.2) of u  are denoted: As a consequence of (3.4), one has

Decomposition of the plate displacements
Let u be in H 1 (Ω ,,r ; R 3 ).In the plates Ω a  and Ω b  , the displacement u is decomposed as (see the decomposition of the plate displacements introduced in Griso 3,5 ) where Moreover, the following estimates hold, d ∈ {a, b}, (see thoerem 4.1 of Griso 3 ): where The strain tensor field of the displacement u is given by (d ∈ {a, b}): where From now on, one assumes that the displacements belong to V ,,r .

𝛿
Observe that if u ∈ V ,,r , then all the terms of the decomposition of u vanish on Γ b  .In particular, one has Hence, because of the 2D-Korn inequality and (3.15), one obtains (3.17) Again, (3.15) together with (3.17) 1 leads to Apply the Poincaré inequality that gives Hence, from (3.15), (3.17), (3.18), and (3.19) one derives the classical estimates for a plate clamped on its lateral boundary:

Estimates for the plate
We know that there exists a rigid displacement r a such that There also exists a second rigid displacement R a such that This equality and (3.23) lead to |c| ≤ C  3∕2 ||u|| V , which, in turn, with (3.22) give (3.24)

Comparison of the terms of the plate decompositions
Set The constants do not depend on , , and r.
Proof.From the expression (3.13) of u in the plate Ω a  , one expresses ∇u − R a pl .Then the estimates (3.15), (3.21), and In the same way one shows (3.25) 2 .
For  ∈ L 1 (), define the piecewise constant function  r () belonging to L ∞ () by Recall that for every (3.26) (3.27) The constants do not depend on , , and r. ‡ In Appendix we give a short proof of these classical results.
Lemma 4.There exists a constant C (independent of , , and r), such that and (3.34) Proof.
As a consequence of the above two lemmas, one obtains the estimates of the restriction of u to the plate Ω a  : (3.41)

MAIN CASES
In view of (3.33), (3.34), and in order that both midsurface displacements  a and  b match, one assumes that  2 r 2 is uniformly bounded from above and Now, the 3 small parameters , r, and  are linked.Set Conditions (4.1) and assumptions (3.1) lead to The couple (, ) must belong to the convex polygon (without the edge  = 2) whose vertexes are Thus, there are 6 cases to analyze.They correspond to 2 vertexes, 3 edges, and the interior of the polygon.The interior of this convex polygon corresponds to the most general situation.We will analyze this case in the next sections.
From now on, one assumes (3.1) and (4.1).Now, we rewrite the estimates (3.40) and (3.41) obtained in the previous section:

ASSUMPTIONS ON THE APPLIED FORCES
In view of the energy relation (2.7) and the estimates (3.20) and ( 4.3), one can scale the applied forces: • in the plate Ω b  and Ω a  , the applied forces are given by (d ∈ {a, b}): where • in the set of beams B ,,r , the applied forces are given by for a.e.x in where f be ∈ (; L 2 (B be ; R 3 )).
As a consequence of (3.20) to (4.3), one obtains the following bound of the total elastic energy: where C is a constant independent of , , and r.Taking into account to (2.1) and (2.2), there exists a constant c > 0 independent of , , and r such that (5.4)

First rescaling operator
Let  be a measurable function on Ω d  , d ∈ {a, b}.Define the measurable function Π  () by The linear operator Π  also satisfies for every For every  ∈ L 2 (; ) .
As a consequence of the above equality one gets . (6.2)

Second rescaling operator
For  measurable on  × P  , define the measurable function Π r () by For every Φ ∈ L 2 ( × P  ), one has Hence, From these equalities one derives (6.4) From now on, the couple(, )belongs to the interior of the polygon.In this case observe that

THE LIMIT FIELDS
Let {u  }  be a sequence of displacements, u  ∈ V , satisfying Using (6.3) and (6.4), now the estimates (3.12) become As a consequence of the above estimates one obtains the following theorem.
Theorem 1.Let {u  }  be a sequence of displacements belonging to V ,,r and satisfying (7.1).There exist a subsequence of {}, still denoted {} and ü ∈ L 2 ( × I; weakly in L 2 (; H 1 (I)), weakly in L 2 ( × I). (7.5) 1 The limit fields , Ũ and   , are linked by the following junction conditions: a.e. in  × I. (7.8) Proof.The relations (6.5) are extensively used in the proof of this theorem even if this fact is not always specified.
Passing to the limit gives  d  =  I d ( Ũ ) and then with (7.11) Then one deduces the expression of Ũ (•, X 3 ) in  × I and  d  in  (see (7.8).
As a consequence of the above theorem and the decompositions (3.2) to (3.13), one has The limit displacement is of Kirchhoff-Love type.

Weak convergences of the strain tensor fields
As immediate consequence of the convergences in Theorem 1 and the expressions (3.5) to (3.16) of the symmetric gradient in the beams and in the plates, we obtain the following proposition.
Proposition 1.Under the hypotheses of Theorem 1, in the set of beams one has the weak convergence r  Π r (( ∇u  ) where the components of the symmetric matrix Γ are given by In the plates one obtains the following weak convergences (d ∈ {a, b}): where the components of the symmetric matrix Γ d are given by ( m =  1 e 1 +  2 e 2 ):

Determination of the strain tensor in the set of beams
To determine the Z 's and the warping ü one proceeds as in section 6.1 of Blanchard et al 10 and section 8.1 of Blanchard et al, 11 one first derives ü and Z .That gives (a.e. in  × B be ), where  be =  be 2( be + be ) is the Poisson coefficient of the material of the beams.From the expression (2.2) of the stress tensor field, the convergence (8.3) and the expressions (8.6) of ü and Z one obtains where a.e. in  × B be .(8.8) E be =  be (3 be +2 be ) be + be is the Young modulus of the elastic material of the beams.In Section 9, we will prove that Σ = 0 in  × B be .

Determination of the
Recall that from (3.14), one has ∫ I d u d 3 (x ′ , X 3 )dX 3 = 0, a.e. in .This equality allows to derive the function u d 3 in terms of the fields  m and  3 .But these expressions are useless; to give the limit of the stress tensors, we only need the knowledge of the partial derivative of u d 3 with respect to X 3 .Again, from the expression (2.2) of the stress tensor field and now using the convergence (8.4) one gets Inserting the above expression of u d X 3 and taking into account that Z d  = 0 lead to where is the Young modulus of the elastic material of the plates.

The limit problem
Set One has f i ∈ L 2 ().
Proof.Let  1 ,  2 , and Φ be in ().The test displacement v  is defined by where  ,,r , Φ,r are defined in Lemma 5.Because  = 1 in D 1 , in every beam the displacement v  coincides with a rigid displacement.Hence (∇v  ) S = 0 a.e. in B ,,r .
In Ω d  one has Applying the operator Π  , then using Lemma 5, and passing to the limit give

CONVERGENCE OF THE TOTAL ELASTIC ENERGY
In this section one proves that the total elastic energy Convergences (7.5), (7.6), (8.7) to (8.9), equalities (7.8), and the fact that the convex functional v  → (v) is lower-semicontinuous on V ,,r allow to pass to the limit in the above equalities.One obtains ∑ d∈{a,b} and also ∫ ×B be [  be ( Γ kk ) As immediate consequence of the above equality, one has  = 0in × B be .Moreover, the weak convergences (8.3) and (8.4) are strong convergences.