1 Introduction
Abstract

We consider a thin heterogeneous layer consisted of the thin beams (of radius ) and we study the limit behavior of this problem as the periodicity , the thickness and the radius of the beams tend to zero. The decomposition of the displacement field in the beams developed in [1] is used, which allows to obtain a priori estimates. Two types of the unfolding operators are introduced to deal with the different parts of the decomposition. In conclusion we obtain the limit problem together with the transmission conditions across the interface.

Homogenization via unfolding in domains separated by the thin layer of the thin beams

Georges Griso, Anastasia Migunova, Julia Orlik

1 Introduction

In this paper a system of elasticity equations in the domains separated by a thin heterogeneous layer is considered. The layer is composed of periodically distributed vertical thin, compared to their length, beams, whose diameter and height tend to zero together with the period of the structure. The structure is clamped on the bottom. We consider the case of an isotropic linearized elasticity system.

The elasticity problems involving thin layers of periodic fiber–networks appear in many technical applications, where special constraints on stiffness of technical textiles or composites are required, depending on a type of the application. For example, drainage and protective wear, working for outer–plane compression, should provide certain stiffness against external mechanical loading.

Thin layers were considered in number of papers (see e.g. [9, 10, 11, 12, 13, 14]). In particular, [9] deals with a layer composed of the holes scaled with additional small parameter; [10, 11] consider a case of a soft layer, whose stiffness is scaled by the thickness of the layer. Thin beams and their junction with 3D structures were studied in [1, 2, 3, 4]: [1] deals with the homogenization of a single thin body; in [2] a structure made of these bodies is considered. [3], [4] study the limit behavior of structures composed of rods in junction with a plate.

Our problem contains 3 small parameters: the thickness of the layer (and the height of the beams at the same time), the radius of the rods and the period of the layer . Obviously, the choice of an appropriate scaling of the problem defines what limit will be reached. For example, in [12, 13, 14] 3D periodic fiber–networks were considered and it is investigated, that if is of the order the bending moments in beams enter the homogenized macroscopic equation as micro–polar rotational degrees of freedom.

Considering the structure made of thin beams the first difficulty arises when we obtain estimates on the displacements. To overcome this problem, we use the decomposition of thin beam’s displacement into a displacement of a mean line and a rotation of its cross–section, introduced in [1]. After deriving the estimates on the decomposed components, we get bounds for the minimizing sequence which depend on . 3 critical cases were obtained with different ratios between small parameters. Two of them are considered in the present paper and lead to the same kind of the limit problem. The third one corresponds no longer to the thin beams but to the small inclusions and therefore is not studied in the present paper.

In order to obtain the limit problem, periodic unfolding method,[6], is applied to the components of the decomposition. Two additional types of unfolding operators are introduced to deal with the mean displacements and rotations, depending only on component , and the warping depending on all . In the limit, a 3D elasticity problem for two domains is obtained, where the domains are separated by the interface with an inhomogeneous Robin–type condition. The coefficients in the Robin condition are obtained from an auxiliary 1D bending problem for a beam. An important result is that the displacements are continuous in a direction normal to the interface and have a jump only in a tangential direction.

The paper is organized as follows. In Section 2, geometry and weak and strong formulations of the problem are introduced. Section 3 presents decomposition of a single beam and the preliminary estimates. Section 4 is devoted to derivation of a priori estimate in all subdomains of . In Section 5, the periodic unfolding operators are introduced and their properties are defined. Also the limit fields for the beams based on the estimates from Section 4 are defined. Section 6 deals with passing to the limit and obtaining the variational formulation for the limit problem. In Section 7, the results are summarized: the strong formulation for the limit problem is given and the final result on the convergences of the solutions is introduced. Section 8 contains additional information. Section 9 provides an auxiliary lemma, used in the proofs.

2 The statement of the problem

2.1 Geometry

In the Euclidean space let be a connected domain with Lipschitz boundary and let be a fixed real number. Define the reference domains:

 Ω− = ω×(−L,0), Ω+ = ω×(0,L), Σ = ω×{0}.

Moreover, (see Figure 1b) is defined by

 Ω=Ω+∪Ω−∪Σ=ω×(−L,L). (2.1)

For the domains corresponding to the structure with the layer of thickness introduce the following notations:

 Ω+δ = ω×(δ,L), Σ+δ = ω×{δ}.

In order to describe the configuration of the layer, for any we define the rod by

 Br,d=Dr×(0,d)

where is the disc of center and radius .

The set of rods is

 Ωir,ε,δ=⋃i∈ˆΞε×{0}{x∈R3|x∈iε+Br,δ}, (2.2)

where

 ˆΞε={ξ∈Z2|ε(ξ+Y)⊂ω},Y=(−12;12)2. (2.3)

Moreover, we set:

 ˆωε=interior⋃i∈ˆΞεε(i+¯¯¯¯Y). (2.4)

The physical reference configuration (see Figure 1a) is defined by :

 Ωr,ε,δ=interior(¯¯¯¯¯¯¯Ω−∪¯¯¯¯¯¯¯¯¯¯¯¯Ωir,ε,δ∪¯¯¯¯¯¯¯Ω+δ). (2.5)

The structure is fixed on a part with non null measure of the boundary .

We make the following assumptions:

 r<ε2,rδ≤C. (2.6)

Here, the first assumption (2.6) is a non penetration condition for the beams while with the second one, we want to eliminate the case which needs the use of tools for plates (see [1]).

2.2 Strong formulation

Choose an isotropic material with Lamé constants for the beams and another isotropic material with Lamé constants for and . Then we have the following values for the Poisson’s coefficient of the material and Young’s modulus:

 νm=λm2(λm+μm),νb=λb2(λb+μb), Em=μm(3λm+2μm)λm+μm,Eb=μb(3λb+2μb)λb+μb.

The symmetric deformation field is defined by

 (∇u)S=∇u+∇Tu2.

The Cauchy stress tensor in is linked to through the standard Hooke’s law:

 σr,ε,δ={λb(Tr(∇ur,ε,δ)S)I+2μb(∇ur,ε,δ)S in Ω−∪Ω+δ,λm(Tr(∇ur,ε,δ)S)I+2μm(∇ur,ε,δ)S in Ωir,ε,δ.

We consider the standard linear equations of elasticity in . The unknown displacement satisfies the following problem:

 ⎧⎪ ⎪⎨⎪ ⎪⎩∇⋅σr,ε,δ=−fr,ε,δin Ωr,ε,δ,ur,ε,δ=0on Γ,σr,ε,δ⋅ν=0on ∂Ωr,ε,δ∖Γ. (2.7)

2.3 Weak formulation

If denotes the space

 Vr,ε,δ={v∈H1(Ωr,ε,δ,R3)|v=0 on Γ},

the variational formulation of (2.7) is

 ⎧⎪⎨⎪⎩Find ur,ε,δ∈Vr,ε,δ,∫Ωr,ε,δσr,ε,δ:(∇φ)Sdx=∫Ωr,ε,δfr,ε,δ⋅φdx,∀φ∈Vr,ε,δ. (2.8)

Throughout the paper and for any we denote by

 σ(v)=λ(Tr(∇v)S)I+2μ(∇v)S={λb(Tr(∇v)S)I+2μb(∇v)S in Ω−∪Ω+δ,λm(Tr(∇v)S)I+2μm(∇v)S % in Ωir,ε,δ.

and

 E(v)=∫Ωr,ε,δσ(v):(∇v)Sdx

the total elastic energy of the displacement . Indeed choosing in (2.8) leads to the usual energy relation

 E(ur,ε,δ)=∫Ωr,ε,δfr,ε,δ⋅ur,ε,δdx. (2.9)

We equip the space with the following norm:

 ∥u∥V=∥(∇u)S∥L2(Ωr,ε,δ).

It follows from the 3D–Korn inequality for domain :

 ∥u∥H1(Ω−)≤C∥(∇u)S∥L2(Ω−). (2.10)

3 Decomposition of the displacements in Ωir,ε,δ

3.1 Displacement of a single beam. Preliminary estimates

To obtain a priori estimates on and we will need Korn’s inequalities for this type of domain. However, for a multi-structure like this, it is not convenient to estimate the constant in a Korn’s type inequality, because the order of each component of the displacement field may be very different. To overcome this difficulty, we will use a decomposition for the displacements of beams. A displacement of the beam is decomposed as the sum of three fields, the first one stands for the displacement of the center line, the second stands for the rotations of the cross sections and the last one is the warping, it takes into account the deformations of the cross sections.

We recall the definition of the elementary displacement from [1].

Definition 3.1.

The elementary displacement , associated to , is given by

 Ue(x1,x2,x3)=U(x3)+R(x3)∧(x1e1+x2e2),for a.e. x=(x1,x2,x3)∈Br,d, (3.1)

where

 (3.2)

We write

 ¯u=u−Ue. (3.3)

The displacement is the warping. Note that

 ∫Dr¯u(x1,x2,⋅)dx1dx2=0, (3.4) ∫Dr(x1¯u2(x1,x2,⋅)−x2¯u1(x1,x2,⋅))dx1dx2=0, ∫Drx1¯u3(x1,x2,⋅)dx1dx2=∫Drx2¯u3(x1,x2,⋅)dx1dx2=0.

The following theorem is proved in [1].

Theorem 3.1.

Let be in and the decomposition of given by (3.1)–(3.3). There exists a constant independent of and such that the following estimates hold:

 (3.5)

We set

 Yε=εY,Vε=Yε×(−ε,0),B′r,ε=Dr×(−ε,0),V′r,ε,δ=Vε∪Dr×(−ε,δ).
Lemma 3.1.

Let be in and the decomposition of the restriction of to the rod given by (3.1)–(3.3). There exists a constant independent of , and such that the following estimates hold:

 |R(0)|2≤Cr3∥∇u∥2L2(Vε),∥R∥2L2(0,δ)≤Cδr3∥∇u∥2L2(Vε)+Cδ2r4∥(∇u)S∥2L2(Br,δ),∥∥∥dUαdx3∥∥∥2L2(0,δ)≤Cδr3∥∇u∥2L2(Vε)+Cδ2r4∥(∇u)S∥2L2(Br,δ),∥U3−U3(0)∥2L2(0,δ)≤Cδ2r2∥(∇u)S∥2L2(Br,δ),∥Uα−Uα(0)∥2L2(0,δ)≤Cδ3r3∥∇u∥2L2(Vε)+Cδ4r4∥(∇u)S∥2L2(Br,δ),∥uα(⋅,⋅,0)−Uα(0)∥2L2(Yε)≤Cε∥∇u∥2L2(Vε)+Cε2r∥(∇u)S∥2L2(Vε),∥u3(⋅,⋅,0)−U3(0)∥2L2(Yε)≤Cε∥∇u3∥2L2(Vε)+Cε2r∥(∇u)S∥2L2(Vε). (3.6)
Proof.

Applying the 2D-Poincaré-Wirtinger’s inequality we obtain the following estimate:

 ∥u−U∥L2(B′r,ε)≤Cr∥∇u∥L2(B′r,ε) (3.7)

The constant does not depend on and .

Step 1. Estimate of .

Recalling the definition of from (3.2) and since , we can write

 ∀x3∈[−ε,0],R1(x3)=4πr4∫Drx2(u3(x)−U3(x3))dx1dx2.

By Cauchy’s inequality

 ∀x3∈[−ε,0],|R1(x3)|2 ≤16π2r8∫Drx2dx1dx2×∫Dr(u3(x)−U3(x3))2dx1dx2 ≤Cr4∫Dr(u3(x)−U3(x3))2dx1dx2.

Integrating with respect to gives

 ∫0−ε|R1(x3)|2dx3≤Cr4∫Br,ε(u(x)−U(x3))2dx.

Using (3.7) we can write

 ∥R1∥L2(−ε,0)≤Cr∥∇u∥L2(B′r,ε). (3.8)

The derivative of is equal to for a.e. . Then proceeding as above we obtain for a.e.

 ∣∣∣dR1dx3(x3)∣∣∣2≤Cr4∫Dr∣∣∣∂u3(x)∂x3∣∣∣2dx1dx2.

Hence

 ∥∥∥dR1dx3∥∥∥L2(−ε,0)≤Cr2∥∥∥∂u3∂x3∥∥∥L2(B′r,ε)≤Cr2∥∇u∥L2(B′r,ε). (3.9)

We recall the following classical estimates for ()

 |ϕ(0)|2≤2a∥ϕ∥2L2(−a,0)+a2∥ϕ′∥2L2(−a,0), (3.10) ∥ϕ∥2L2(−a,0)≤2a|ϕ(0)|2+a2∥ϕ′∥2L2(−a,0).

Due to (3.8)-(3.9), (3.10) with and since that gives for

 |R1(0)|2≤Cr3∥∇u∥2L2(B′r,ε).

The estimates for , are obtained in the same way. Hence we get (3.6).

Step 2. Estimate of .

 ∥R−R(0)∥L2(0,δ)≤δ∥∥∥dRdx3∥∥∥L2(0,δ).

From (3.5), (3.10) and (3.6) we get

 ∥R∥2L2(0,δ)≤2δ|R(0)|2+δ2∥∥∥dRdx3∥∥∥2L2(0,δ)≤Cδr3∥∇u∥2L2(B′r,ε)+Cδ2r4∥(∇u)S∥2L2(Br,δ). (3.11)

Hence (3.6) is proved.

Step 3. Estimate of .

Applying inequality (3.5) from Theorem 3.1 the following estimates on hold:

 ∥∥∥dU3dx3∥∥∥L2(0,δ)≤Cr∥(∇u)S∥L2(Br,δ),∥∥∥dUαdx3∥∥∥L2(0,δ)≤∥R∥L2(0,δ)+Cr∥(∇u)S∥L2(Br,δ). (3.12)

Combining (3.12) with (3.11) gives

 ∥∥∥dUαdx3∥∥∥2L2(0,δ)≤Cδr3∥∇u∥2L2(B′r,ε)+Cδ2r4∥(∇u)S∥2L2(Br,δ)+Cr2∥(∇u)S∥2L2(Br,δ).

Taking into account the assumption (2.6), we obtain (3.6). Then by (3.6), (3.12) and the Poincaré’s inequality (3.6), (3.6) follow.

Step 4. We prove the estimates (3.6)-(3.6).

By Korn inequality there exists rigid displacement

 r(x)=a+b∧(x+ε2e3),a=1ε3∫Vεu(x)dx,b=6ε5∫Vε(x+ε2e3)∧u(x)dx.

such that

 ∥u−r∥L2(Vε)≤Cε∥(∇u)S∥L2(Vε),∥∇(u−r)∥L2(Vε)≤C∥(∇u)S∥L2(Vε). (3.13)

Besides by Poincaré-Wirtinger inequality we have

 ∥ui−ai∥L2(Vε)≤Cε∥∇ui∥L2(Vε),i=1,2,3. (3.14)

The Sobolev embedding theorems give ()

 ∥φ∥L4(Y)≤C∥φ∥H1/2(Y)≤C(∥φ∥L2(V)+∥∇φ∥L2(V)),∀φ∈H1(V).

By a change of variables we obtain

 ∥φ∥L4(Yε)≤C(1ε∥φ∥L2(Vε)+ε∥∇φ∥L2(Vε)),∀φ∈H1(Vε).

Therefore, (3.13) and the above inequality lead to

 ∥u−r∥L4(Yε)≤C∥(∇u)S∥L2(Vε). (3.15)

From the identity

 1πr2∫Dr(u(x′,0)−r(x′,0))dx′=U(0)−a−b∧ε2e3,

estimate (3.15) and the Hölder inequality we get

 ∣∣∣U(0)−a−b∧ε2e3∣∣∣≤1πr2(∫Dr14/3dx′)3/4(∫Dr|u(x′,0)−r(x′,0)|4dx′)1/4≤Cr1/2∥(∇u)S∥L2(Vε). (3.16)

As a first consequence, we obtain

 |U3(0)−a3|≤Cr1/2∥(∇u)S∥L2(Vε). (3.17)

From the Cauchy-Schwarz inequality and taking into account (3.14), we derive

 |b|≤Cε5(∫Vε∣∣∣x+ε2e3∣∣∣2dx)1/2(∫Vε|u(x)−a|2dx)1/2≤Cε5⋅ε⋅ε3/2∥u−a∥L2(Vε)≤Cε5/2ε∥∇u∥L2(Vε)≤Cε3/2∥∇u∥L2(Vε). (3.18)

Using (3.16) and (3.18) we have

 |U(0)−a|≤∣∣∣U(0)−a−b∧ε2e3∣∣∣+∣∣∣b∧ε2e3∣∣∣≤Cr1/2∥(∇u)S∥L2(Vε)+Cε1/2∥∇u∥L2(Vε). (3.19)

Estimates (3.10) and (3.14) yield

 ∥ui(⋅,⋅,0)−ai∥2L2(Yε)≤Cε∥∇ui∥2L2(Vε),i=1,2,3. (3.20)

Combining (3.19), (3.20) gives

 ∥uα(⋅,⋅,0)−Uα(0)∥2L2(Yε) ≤C(∥uα(⋅,⋅,0)−aα∥2L2(Yε)+∥Uα(0)−aα∥2L2(Yε)) ≤Cε∥∇u∥2L2(Vε)+Cε2r∥(∇u)S∥2L2(Vε)+Cε∥∇uα∥2L2(Vε) ≤Cε∥∇u∥2L2(Vε)+Cε2r∥(∇u)S∥2L2(Vε),

and from (3.17) and again (3.20) we obtain

 ∥u3(⋅,⋅,0)−U3(0)∥2L2(Yε)≤Cε∥∇u3∥2L2(Vε)+Cε2r∥(∇u)S∥2L2(Vε),

Hence we get (3.6)-(3.6). ∎

4 A priori estimates

In this section all the constants do not depend on and . We denote the running point of .

4.1 Decomposition of the displacements in Ωir,ε,δ

We decompose the displacement in each beam , as in the Definition 3.1. The components of the elementary displacement are denoted , , where .

Now we define the fields , and for a.e. by

 ˜U(s1,s2,x3)=⎧⎨⎩Uξ(x3),ifξ=[sε]∈ˆΞε0,ifξ∉ˆΞε,˜R(s1,s2,x3)=⎧⎨⎩Rξ(x3),ifξ=[sε]∈ˆΞε0,ifξ∉ˆΞε,
 ˜¯u(s1,s2,x)=⎧⎨⎩¯uξ(x),ifξ=[sε]∈ˆΞε0,ifξ∉ˆΞε.

We have

 ˜U,˜R∈L2(ω,H1((0,δ),R3)),˜¯u∈L2(ω,H1(Br,δ,R3)).

Moreover,

 ∥˜U∥2L2(ω×(0,δ))=ε2∑ξ∈ˆΞε∥Uξ∥2L2(0,δ),∥˜R∥2L2(ω×(0,δ))=ε2∑ξ∈ˆΞε∥Rξ∥2L2(0,δ), ∥˜¯u∥2L2(ω×Br,δ)=ε2∑ξ∈ˆΞε∥¯uξ∥2L2(Br,δ).

As a consequence of the Theorem 3.1 and Lemma 3.1 we get

Lemma 4.1.

Let be in . The following estimates hold:

 ∥∥∂˜R∂x3∥∥L2(ω×(0,δ))≤Cεr2∥u∥V, (4.1) ∥∥∂˜U∂x3−˜R∧e3∥∥L2(ω×(0,δ))≤Cεr∥u∥V, ∥∇x˜¯u∥L2(ω×Br,δ)≤Cε∥u∥V, ∥˜¯u∥L2(ω×Br,δ)≤Cεr∥u∥V, ∥˜R∥L2(ω×(0,δ))≤Cεδr2∥u∥V, ∥∥∂˜Uα∂x3∥∥L2(ω×(0,δ))≤Cεδr2∥u∥V.

Moreover,

 ∥˜R(⋅,⋅,0)∥2L2(ˆωε)≤Cε2r3∥u∥2V, (4.2) ∥˜R(⋅,⋅,δ)∥2L2(ˆωε)≤Cε2r3∥∇u∥2L2(Ω+δ), ∥˜U3−˜U3(⋅,⋅,0)∥L2(ω×(0,δ))≤Cδεr∥u∥V, ∥˜Uα−˜Uα(⋅,⋅,0)∥L2(ω×(0,δ))≤Cδ2εr2∥u∥V,where α=1,2.
Proof.

Estimates (4.1) – (4.1) follow directly from (2.10), (3.5), (3.5) and (3.6)–(3.6) and estimates (4.2) – (4.2) are the consequences of the estimates in Lemma 3.1 and (2.10). ∎

4.2 Estimates of the interface traces

Lemma 4.2.

There exists a constant independent of