# A Study of the Anisotropic Static Elasticity System in Thin Domain

Authors: Yassine Letoufa; Salah Mahmoud Boulaaras; Hamid Benseridi; Mourad Dilmi; Asma Alharbi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We study the asymptotic behavior of solutions of the anisotropic heterogeneous linearized elasticity system in thin domain ofℝ3 which has a fixed cross-section in the ℝ2 plane with Tresca friction condition. The novelty here is that stress tensor has given by the most general form of Hooke’s law for anisotropic materials. We prove the convergence theorems for the transition 3D-2D when one dimension of the domain tends to zero. The necessary mathematical framework and (2D) equation model with a specific weak form of the Reynolds equation are determined. Finally, the properties of solution of the limit problem are given, in which it is confirmed that the limit problem is well defined.


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## Body

## 1. Introduction

In this paper, we are interested of the asymptotic behavior of the linear elasticity system in a domain ofℝ3 with a Tresca friction condition where the boundary of this domain has a fixed cross-section in dimension 2 and a small thickness. One of the objectives of this study is to obtain two-dimensional equation that allows a reasonable description of the phenomenon occurring in the three-dimensional domain by passing the limit to 0 on the small thickness of the domain (3D). Let us mention for example [1–8] in which the authors worked on the asymptotic behavior for the linearized elasticity system with different boundary conditions. Some problems of Newtonian or non-Newtonian fluids are considered in [9–11] where the authors proved a limit problem that gives a distribution of velocity and pressure through the weak form of the Reynolds equation. In [6, 7], the authors demonstrate the transition 3D-1D in anisotropic heterogeneous linearized elasticity; so, we mention here that this phenomenon has been studied only about strong solutions, without friction law. Benseridi in [2] investigated the asymptotic analysis of a dynamical problem of linear elasticity with Tresca’s friction. The static case with a nonlinear term for linear elastic materials has been considered in [3]. See another situation in [4] where the paper concerns asymptotic derivation of frictionless contact models for elastic rods on a foundation with normal compliance. Recently, the authors in [5, 12] have proved the asymptotic behavior of a frictionless contact problem between two elastic bodies, when the vertical dimension of the two domain reaches zero. However, all these papers have been only restricted in a homogeneous and isotropic case of elastic materials.The present work is a follow-up of [2, 3, 5] to study the heterogeneous and anisotropic situation with Tresca’s friction. Here, the stress tensor with its components is given by the generalized Hooke’s law (see [13]): σε=Aεeuε, where uε denotes the displacement vector, euε is the linearized strain tensor, and Aε is the fourth order tensor which describes the elastic properties of the material. Many materials that follow the linear elastic model, although they are well made, are not subject to the assumptions of isotropy, for example, wood, reinforced concrete, composite materials, and many biological materials, where the mechanical properties of these materials differ according to the directions of space; in that case, the elasticity operator depends on the location of the point (see [14, 15]). Necas in [7] and Sofonea in [16] established the existence of a weak solution for the static frictional contact problem involving linearly elastic and viscoelastic materials, by using a results of convex optimization [17], and numerical approximation of this problem was studied in [18]. For the variational analysis of various contact problems, we mention excellent references in [14, 15]. Mathematically, the asymptotic analysis is more difficult since in general, the limit problem involves an equation that takes into account the anisotropy of the medium, and it is thus important to identify the elastic components of Aε that appear in the (2D) equation model.The paper is organized as follows; in section2, the strong and weak formulation of the problem is given in terms of uε and also the related existence and uniqueness of the weak solution. In section 3, we introduce a scaling, and we find some estimates on the displacement which are independent of the parameter ε. In section 4, we state the main results concerning the existence of a weak limit u∗ of uε, the (2D) equation model with a specific weak form of the Reynolds equation is proved, the limit form of the Tresca boundary conditions is formulated, and finally, the uniqueness of u∗ is given.


## 2. Mathematical Formulation

Letω be an open set in ℝ with Lipschitz boundary, and we consider a smooth function h:ω⟶ℝ be a class C1 such that 0<hmin≤hx≤hmax, for all x∈ω, where hmin and hmax are constants. We define the smooth bounded domain Ω whose boundary has a flat part ω,
(1)Ω=x,z∈ℝ3,x∈ω,0<z<hx.We denote byΓ1 is the upper boundary of the equation z=hx, and ΓL is the lateral boundary.Letε>0 be a small parameter, and we define Ωε be the change of scale z=x3/ε and the points of Ω,
(2)Ωε=x,x3∈ℝ3,x∈ω,0<x3<εhx.We haveΓε=ω¯∪Γ¯1ε∪Γ¯Lε which its boundary of Ωε and where Γ1ε is the upper surface defined by x3=εhx, and ΓLε is the lateral boundary. The unit outward normal to Γε is denoted by ν. It follows that there is correspondence between the functions ϕ:Ωε⟶ℝn and ϕ^:Ω⟶ℝnn=1,2,3 given by ϕ^x,z=ϕx,x3.LetH1/2Γ3 be the space of traces of functions on Γ of functions from H1Ω3, and we use the vector function g∈H1/2Γ3 such that
(3)∫Γg.νds=0.We denote bySn the space of symmetric tensors on ℝn and . the Euclidean norm on ℝn and Sn. Here and below, the indices i,j,k,l run between 1 and 3, and the summation convention overrepeated indices is adopted.The basic equations of frictionless contact problem for the anisotropic heterogeneous elastic body occupy the domainΩε as follows:The equations of equilibrium are as follows:(4)∂σijε∂xj+fiε=0inΩε,(5)σijεuε=AijklεekluεinΩε,where the vector fε=f1ε,f2ε,f3ε represents the forces of density, uε=u1ε,u2ε,u3ε is the displacement field, the elements Aijklε denote the components of elasticity tensor Aε, and eijuε is the rate of deformation operator,
(6)eijuε=euεij=12∂uiε∂xj+∂ujε∂xi.OnΓLε, the displacement is known:
(7)uε=gεonΓLε.OnΓ1ε, we assume that the elastic body is held fixed:
(8)uε=gε=0onΓ1ε.On the surfaceω, we assume that the contact is bilateral:
(9)uε.ν=gε.ν=0and satisfies the Tresca boundary condition [7] with friction function kε;
(10)στε<kεuτε=s,στε=kε∃λ≥0suchthatuτε=s−λστε,where s=gε on ω. uτε, στε, and σνε are the tangential displacement, the tangential, and the normal stress tensor, respectively, with
(11)uτiε=uiε−ujενjνi,στiε=σijε.νj−σνε.νi,σνε=σε.ν.ν.Consider now the following closed convex subset ofH1Ωε3 given by
(12)Kε=ϕ∈H1Ωε3:ϕ=gεonΓ1ε∪ΓLε,ϕ.ν=0onω.Let us introduce the forma:Kε×Kε⟶ℝ and the functional Jε:Kε⟶ℝ+ defined by
(13)auε,ϕ=∫ΩεAijklεekluεeijϕdxdx3,Jεϕ=∫ωkεϕ−sdx.In the study of the mechanical problem (3)–(10), we assume that all components Aijklε belong to L∞Ωε and satisfy the usual properties of symmetry and ellipticity [19], i.e.,
(14)Aijklε=Ajiklε=Aklijε∈L∞Ωε,and there exists a constant μ>0 such that
(15)Aijklεyξklξij≥μξ2∀ξ∈S3,a.e.y∈Ωε.Remark 1.
It follows from previous properties and by Korn’ s inequality (see [16], pp. 79), that the bilinear form a is coercive and continuous, i.e.,
(16)aϕ,ϕ≥μCK∇ϕL2Ωε2∀ϕ∈Kε,(17)aφ,ψ≤M∇ϕL2Ωε∇ψL2Ωε∀ϕ,ψ∈Kε,where M=max1≤i,j,k,l≤3AijklεL∞Ωε and CK denoting a positive constant depends on Ωε,Γ1ε, and ΓLε.Lemma 2.
Assuming thatfε∈L2Ωε3 and kε∈L∞ω, the variational formulation of problem (3)–(10) is equivalent to
Finduε∈Kε satisfying
(18)auε,ϕ−uε+Jεϕ−Jεuε≥∫Ωεfε.ϕ−uεdxdx3∀ϕ∈Kε,for every ε small fixed.Moreover, if the assumptions of (14) and (15) hold, then the variational inequality (18) has a unique solution uε∈Kε.Remark 3.
A problem of the form (18) is called an elliptic variational inequality of the second kind ([17]). The following theorem (see [19], Theorem 6) allows us to replace the variational inequality (18) by a minimization problem. Thus, we will not repeat the proof, but our goal is to study the asymptotic behavior.


## 3. Some Estimates in Fixed Domain

To be able to study the asymptotic behavior of the solutions of (18), we use the change of variable z=x3/ε, to return to the fixed domain Ω, and then we define the following functions in Ω:
(19)u^iεx,z=uiεx,x3fori=1,2,3.For the dataA^ijkl, f^i, and k^, we have the following relations:
(20)A^ijklx,z=Aijklεx,x3,f^ix,z=ε2fiεx,x3andk^=εkε,(for1≤i,j,k,l≤3).Let(21)K=v∈H1Ω3:v=gonΓL∪Γ1,v.ν=0onω,Vz=v=v1,v2,v3∈L2Ω3:∂vi∂z∈L2Ω,i=1,2,3;v=0onΓ1.Vz is a Banach space for the following norm:
(22)vVz=∑i=13viL2Ω2+∂vi∂zL2Ω21/2.Everywhere in the sequel, the indexesα,β,γ and δ run from 1 to 2, and summation over repeated indices is implied. Follow the same steps as in 6,12, passing to the fixed domain Ω, and using the symmetry of σijε and Aijklε, after multiplication by ε, we have (18) that is equivalent toFindu^ε∈K, such that
(23)a^u^ε,ϕ^−u^ε+∫ωk^ϕ^−sdx−∫ωk^u^ε−sdx≥∑i=13∫Ωf^iϕ^i−u^iεdxdz,∀ϕ^∈K,where
(24)a^u^ε,ϕ^=ε2∫ΩA^αβγθe^γθu^ε∂ϕ^α∂xβdxdz+2ε∫ΩA^α3γθe^γθu^ε∂ϕ^α∂zdxdz+2ε2∫ΩA^αβγ3e^γ3u^ε∂ϕ^α∂xβdxdz+4ε∫ΩA^α3γ3e^γ3u^ε∂ϕ^α∂zdxdz+ε2∫ΩA^αβ33e^33u^ε∂ϕ^α∂xβdxdz+ε∫ΩA^33αβe^αβu^ε∂ϕ^3∂zdxdz+2ε∫ΩA^α333e^33u^ε∂ϕ^α∂zdxdz+2ε∫ΩA^33α3e^α3u^ε∂ϕ^3∂zdxdz+ε∫ΩA^3333e^33u^ε∂ϕ^3∂zdxdz,and e^u^ε=e^iju^εij is given by the relations
(25)e^iju^ε=12∂u^iε∂xj+∂u^jε∂xi,i,j=1,2,e^i3u^ε=e^3iu^ε=121ε∂u^iε∂z+∂u^3ε∂xi,i=1,2.e^33u^ε=1ε∂u^3ε∂z,Lemma 4.
Under the assumptions of Lemma2, there exists a constant C>0 independent of ε, such that
(26)∑i=13∂u^iε∂zL2Ω2+ε2∑i,j=12∂u^iε∂xjL2Ω2+ε2∑j=12∂u^3ε∂xjL2Ω2≤C.Proof.
Assume thatuε is a solution of 2.12. As Jεuε≥0, then
(27)auε,uε≤auε,ϕ+Jεϕ+∫Ωεfε.uεdxdx3−∫Ωεfε.ϕdxdx3,∀ϕ∈Kε.
Using the Young’s inequality(28)ab≤η2a22+η−2b22in (17) for η=μCK/2, we find
(29)auε,ϕ≤μCK4∇uεL2Ωε2+MμCK∇ϕL2Ωε2.
Also, by the Cauchy-Schwarz and Poincaré’s inequalities, we get(30)∫Ωεfε.ϕdxdx3≤εhmaxfεL2Ωε∇ϕL2Ωε,then using Young’s inequality for η=1/μCK to obtain
(31)∫Ωεfε.uεdxdx3≤εhmax22μCKfεL2Ωε2+μCK2∇ϕL2Ωε2.
Using (16), (29), and (31) in (27), we get
(32)μCK4∇uεL2Ωε2≤MμCK∇ϕL2Ωε2+εhmax2μCKfεL2Ωε2+μCK2∇ϕL2Ωε2+∫ωkεϕ−sdx.
Taking into account theg function introduced in (5) and using [20] (lemma 2 pp.24), there exists a function g~∈H1Ω3 such that
(33)g~=gonΓLandg~.ν=0onω∪Γ1.Thus, choosing ϕ^=g~ in 3.6, then multiplying product inequality by ε, and the fact that g=s on ω, we obtain
(34)μCK4ε∇uεL2Ωε2≤hmax2μCKf^L2Ω2+MμCK+μCK2∇g~L2Ω2.
From6, we can see the constant Korn CK contained in Remark 1 does not depend on ε and ϕ, for ε∈0,1; moreover, by changing the data of Aε, remark that μ and M are independent of ε. Therefore, passing to the fixed domain Ω, we get
(35)ε∇uεL2Ωε2=∑i=13∂u^iε∂zL2Ω2+ε2∑i,j=12∂u^iε∂xjL2Ω2+∑j=12∂u^3ε∂xjL2Ω2≤C,with
(36)C=4μCKhmax2μCKf^L2Ω2+MμCK+μCK2∇g~L2Ω2.☐Lemma 5.
Under the assumptions of Lemma4, there exists u∗=u1∗,u2∗,u3∗∈Vz such that
(37)u^ε⇀u∗weaklyinVz,(38)ε∂u^iε∂xα⇀0weaklyinL2Ωi=1,2,3andα=1,2,(39)εe^αβu^ε⇀0,weaklyinL2Ωα,β=1,2,(40)εe^γ3u^ε⇀12∂uγ∗∂zweaklyinL2Ωγ=1,2,(41)εe^33u^ε⇀∂u3∗∂zweaklyinL2Ω.Proof.
From (26), there exists a fixed constant C>0 such that
(42)∂u^iε∂zL2Ω2≤C,fori=1,2,3.
Using Poincaré’s inequality in the domainΩ(43)u^iεL2Ω≤hmax∂u^iε∂zL2Ω,fori=1,2,3,we deduce that u^ε is bounded in Vz. From the last two estimates, there exists u∗=u1∗,u2∗,u3∗∈Vz and satisfies (37). From (26), we can extract a subsequence such that ε∂u^iε/∂xα⇀η in L2Ω; on the other hand, from (37), we deduce (38). Also, (39)–(41) follow from (37) and (38).☐


## 4. Limit Problem and Main Result

At the limitε=0, we give the satisfactory equations of u∗ and the properties of solution of the limit problem for the system (3)-(10).Theorem 6.
With the same assumptions as Lemma5, u∗ satisfies
(44)A∗∂u∗∂z,∂∂zϕ^−u∗L2Ω+∫ωk^ϕ^−s−u∗−sdx≥∫Ωf^.ϕ^−u∗dxdz,∀ϕ^∈K,where the symmetric matrix A∗ is given by
(45)A∗=4A^13134A^13232A^13334A^23134A^23232A^23332A^33132A^3323A^3333.
Moreover, we have(46)−∂∂z4A^α3γ3∂uγ∗∂z+2A^α333∂u3∗∂z=f^α,α=1,2−∂∂z2A^33α3∂uα∗∂z+A^3333∂u3∗∂z=f^3inL2Ω.Proof.
As (23) can be written,
(47)a^u^ε,ϕ^+∫ωk^ϕ^−sdx−∑i=13∫Ωf^iϕ^i−u^iεdxdz≥a^u^ε,u^ε+∫ωk^u^ε−sdx.
Since the forma^.,. is a symmetry and K-elliptic, and the fact that ϕ^⟶∫ωk^ϕ^−sdx is convex and lower semicontinuous, we deduce
(48)liminfε⟶0a^u^ε,u^ε+∫ωk^u^ε−sdx≥a^u∗,u∗+∫ωk^u∗−sdx.
Using Lemma5, we let ε tend to 0 in (47), to obtain
(49)4∫ΩA^α3γ3∂uγ∗∂z∂∂zϕ^α−uα∗dxdz+2∫ΩA^α333∂u3∗∂z∂∂zϕ^α−uα∗dxdz+2∫ΩA^33α3∂uα∗∂z∂∂zϕ^3−u3∗dxdz+∫ΩA^3333∂u3∗∂z∂∂zϕ^3−u3∗dxdz+∫ωk^ϕ^−s−u∗−sdx≥∑i=13∫Ωf^iϕ^i−ui∗dxdz.
This completes the proof of (44) if we cross (49) in the matrix form A∗. We choose in the variational inequation (49) ϕ^i=ui∗±ψi, where ψi∈H01Ω(for i=1,2,3), and using Green’s formula, we find
(50)−∫Ω∂∂z4A^α3γ3∂uγ∗∂z+2A^α333∂u3∗∂zψαdxdz−∫Ω∂∂z2A^33α3∂uα∗∂z+A^3333∂u3∗∂zψ3dxdz=∑i=13∫Ωf^iψidxdz,choosing ψ3=0 and ψα∈H01Ω; then, ψα=0 and ψ3∈H01Ω, we get (59).☐Theorem 7.
Under the assumptions of Theorem6 then, the solution of the limit problem (44)–(46) is unique in Vz.Proof.
Suppose that there exists two solutionsu∗ and v∗ of the variational inequality (44), and taking ϕ^=v∗ in (44), then ϕ^=u∗ in the inequality relating to v∗. By subtracting the two obtained inequalities, we have
(51)A∗.∂∂zv∗−u∗,∂∂zv∗−u∗L2Ω≤0.
We must now check thatA∗ is ellipticity. So, we return to the properties of A^ mentioned in (14) and (15); in particular, we choose symmetric tensors ζ that are given by ξαβ=0 (for α,β=1,2); otherwise, the rest of the components ξi3 let it be whatever. Putting ηi=ξi3, for i=1,2,3, we will get
(52)A^ijklξklξij=4A^α3β3ηβηα+2A^α333η3ηα+2A^33α3ηαη3+A^3333η3η3=Aij∗ηjηi.
Consequently, and asξ2≥η2, there exists a positive constant μ, and for all vectors η in ℝ3, we have
(53)Aij∗ηjηi≥μη2.
So,A∗ is ellipticity. Thus, the relation (44) implies that
(54)μ∂∂zv∗−u∗L2Ω2≤0.
Using Poincaré’s inequality, we obtain(55)v∗−u∗L2Ω2≤hmax2∂∂zv∗−u∗L2Ω2=0,
and the proof of uniqueness ofu∗ is complete.☐Theorem 8.
Under the assumptions of Theorem7, the traces s∗,π∗ with s∗=si∗1≤i≤3 and π∗=πi∗1≤i≤3 defined by
(56)si∗x=ui∗x,0,πα∗x=4A^α3γ3∂uγ∗∂z+2A^α333∂u3∗∂zx,0;π3∗x=2A^33γ3∂uγ∗∂z+A^3333∂u3∗∂zx,0,satisfy the following limit form of the Tresca boundary conditions:
(57)∫ωk^ψ+s∗−s−s∗−sdx−∫ωπ∗.ψdx≥0∀ψ∈L2ω3,(58)π∗<k^⇒s∗=sπ∗=k^⇒∃λ>0suchthats∗=s+λπ∗a.e.inω.
Moreover, if the coefficientsA^i3j3for1≤i,j≤3, depending only on the variable x, we have the following weak form of the Reynolds equation:
(59)∫ωF~−h2s∗+∫0hu∗x,zdz.∇ψxdx=0,∀ψ∈H1ω,
whereInvA∗x denotes the inverse of A∗x and
(60)F~x=∫0hF~x,ρdρ−h2F~x,h,(61)F~x,ρ=InvA∗x.∫0ρ∫0θf^x,ydydθ.Proof.
We now choose in the variational inequality (49) ϕ^i=ui∗+ψi, where ψi∈HΓ1∪ΓL1Ω for i=1,2,3, and then using Green’s formula, we obtain
(62)−∫Ω∂∂z4A^α3γ3∂uγ∗∂z+2A^α333∂u3∗∂zψαdxdz−∫ω4A^α3γ3∂uγ∗∂z+2A^α333∂u3∗∂zψαx,0dx−∫Ω∂∂z2A^33γ3∂uγ∗∂z+A^3333∂u3∗∂zψ3dxdz−∫ω2A^33γ3∂uγ∗∂z+A^3333∂u3∗∂zψ3x,0dx+∫ωk^ψ+s∗−s−u∗−sdx≥∑i=13∫Ωf^iψidxdz.
On the other hand, from (46), we have
(63)∫ωk^ψ+s∗−s−s∗−sdx−∫ω∑απα∗ψα+π3∗ψ3dx≥0.By density theorems, we find (57). For (58), we use the analogue of [10].
To prove (59), we use those similar steps as in [2, 5, 9–11], by integrating (46) from 0 to z, and taking into account A^i3j3 depending only on x, we obtain
(64)−4A^α3γ3x∂uγ∗∂zx,z+2A^α333x∂u3∗∂zx,z+πα∗x=∫0zf^αx,ydy,−2A^33γ3x∂uγ∗∂zx,z+A^3333x∂u3∗∂zx,z+π3∗x=∫0zf^3x,ydy,(65)−A∗x.∂u∗∂z+π∗x=∫0zf^x,ydy.
It follows from (51) that it is a invertible matrix A∗x, for almost every x∈ω. Therefore,
(66)−∂u∗∂z+InvA∗x.π∗x=InvA∗x.∫0zf^x,ydy.
By integrating between0 and z, we obtain
(67)−u∗x,z+s∗x+zInvA∗x.π∗x=F~x,z.
Asui∗x,hx=0, we have
(68)s∗x+hxInvA∗x.π∗x=∫0hxF~x,ydy.
We integrate (67) from 0 to hx, and we obtain
(69)−∫0hxu∗x,zdz+hs∗x+hx22InvA∗x.π∗x=∫0hxF~x,ydy,and by (68), we deduce that
(70)−∫0hxu∗x,zdz+hx2s∗x−F~x=0,such that F~ is already defined in (61), and let us finally get the weak form (59) after multiplying (70) by ∇ψx and integrate it in ω.☐


## 5. Conclusions

We were able to find a framework to conclude that solving our original problem leads to solving a well-defined problem as in (44),(46) and (57)–(59) for the “small” parameter ε.The key of the problem lies in the relation between the matricesAε and A∗. Note that they have the same properties despite the difference in dimensions, therefore it played a key role in the transition from uε to u∗.Indeed, the special case(71)Aijklε=μεδikδjl+δilδjk+λεδijδklwhere λε, με>0 are the Lamé coefficients (see [13] pp. 102-103) corresponds to the homogeneous and isotropic case of elastic materials, and has been studied in [2, 3, 5]. Thus also, the Stokes flow in [11] can be recovered when λε tends to 0.



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