Mathematical Justification of an Obstacle Problem in the Case of a Plate?

Yan GUAN

1 Introduction

In this paper,we consider the so-called Signorini problem,also called unilateral contact problem,of an elastic body in contact with a rigid support.One of the major interests of this modeling is to keep the full elastic tensor,namely,there is no assumption on the elastic isotropy(see[1–2]).

Bilateral models for plates and shells were studied by formal asymptotic methods or by variational analysis(see[3–4]and the references therein).The contact problem can be stated as the minimization of some energy functional under an inequality constraint.The modelling of unilateral contact problems of elastic bodies was established by Signorini in 1933.The first mathematical properties of the solution to such a problem can be found in[5–6].Later Paumier gave,by an asymptotic approach,the model of an elastic Kirchho ff-Love plate in unilateral contact(see[7]).L′eger and Miara generalized Paumier’s work to elastic shallow shell.They obtained the limit model written in terms of a variational inequality in the framework of Cartesian(see[1])and curvilinear coordinates(see[2]),respectively.

In this paper we properly justify the modeling of a thin plate in unilateral contact with a rigid plane.By an asymptotic approach,we study the convergence of the displacement field as the thickness of the plate goes to zero.We establish that the transverse mechanical displacement field decouples from the in-plane components and solves an obstacle problem.In Section 2,we study a Signorini problem arising in the case of three-dimensional plate.In Section 3,by using appropriate scalings,we give the new scaled variational inequality problem.In Section 4,we prove the convergence of the solution when the thickness of the plate tends to zero and establish the limit problem of a elastic plate in unilateral contact.

2 Thin Plate

In this paper,Latin indices take their values in the set{1,2,3},Greek indices take their values in the set{1,2};and the Einstein summation convention is used.Bold letters are used for vectors or vector spaces.We denote by a·b the vector product between two vectors a and b.stand for the classical norms in L2(?),H1(?),respectively,for both scalar-valued and vector-valued functions.Moreover,for simplicity let c denote different positive constants.

In order to establish properly the bi-dimensional model of a thin plate in contact with a rigid plane,we take the reference configuration to be a cylinder with middle surface ω and thickness 2ε.More precisely,let ε>0 be a small parameter and ω be an open bounded and connected subset of R2with Lipschitz-continuous boundary γ.Then the reference configuration of the plate under consideration is denoted by,where.We define a new partition of the boundarywith the upper and lower facesand the lateral boundary

2.1 Three dimensional problem

We consider a family of plates with reference configuration,made of elastic material with elastic characteristic tensors C=(Cijkl).There exists a positive number c such that,for every second order 3×3 symmetric tensor M=(Mij)we have

We consider the situation that the body is clamped on the whole lateral surface,and is subjected to applied surface forces with density gεon the upper surface and is in mechanical contact with the lower face

2.2 The boundary conditions for a body in contact with a plane

We focus now on the unilateral contact with a horizontal plane set at level?ε.Letbe a point on the lower face.The unilateral contact conditions first mean that the displacement onmust satisfy a nonpenetrability conditionother words,

where e3=(0,0,1).

The so-called Signorini conditions which give the full description of the unilaterality are classically obtained by adding the following constraints to the nonpenetrability condition:

(1)No tensile forces but only compressive forces are exerted on the boundary by the obstacle;

(2)all points in contact are onso that conditions(2.4)is an equality.

These constraints read

2.3 The variational inequality in ?ε

The natural functional framework for(2.9)is the vector space,where is a convex set.Hence the weak solution uεto(2.9)is given by the following variational inequality

where daεis the area element of the boundary ??ε.

Based on classical arguments,(2.9)and(2.11)are equivalent.Moreover,(2.11)has a unique solution for any fi xed ε>0,and the weak solution associated to(2.9)is given by this unique solution(for proof,see[8]).

Let us now introduce the scaling procedure in order to establish the convergence theorem as ε→ 0.

3 Scaling and Equilibrium Equation on the Fixed Domain ?

3.1 Scalings of the unknowns and test functions

We now change the domain ?εhaving the middle surface ω and the thickness 2ε into a fixed domain ? with the same middle surface and the thickness 2 independent of ε by means of the simple geometrical transformation defined as follows:Letbe a generic point on.The corresponding pointThis inducesBy analogy,the boundary of the domain ? is divided into three parts:

We give the scaled displacement u(ε)and the scaled test functions v defined onas

where

3.2 Assumptions on the data

In order to obtain a nontrivial limit problem by asymptotic analysis,it is essential to scale the data in accordance with the scalings of the unknowns.More precisely,we assume that there exist functionsindependent of ε,such that

3.3 Contact condition on the fixed domain

After the scaling process,the non-penetrability condition holds now on Γ?and reads

3.4 Equilibrium problem on the fixed domain ? = ω × (?1,1)

Replacing uεand vεby their scaled values u(ε)and v given by(3.1)in the problem(2.11),respectively,we get the following problem posed over the fixed domain ?

where dais the area element of the boundary??.

Classical arguments(see[8])can be applied to prove the existence and uniqueness of the weak solution to the variational inequality problem(3.5).We have the following theorem.

Theorem 3.1 For any fixed ε>0,the problem(3.5)has a unique weak solution.

4 Convergence

The aim of this section is to show that when ε tends to zero,the sequence{u(ε)}converges to a limit u which solves a two-dimensional obstacle problem.An important preliminary point here is the following lemma,which is a version of Korn’s inequality.

Lemma 4.1 For all v∈H1(?),the mappingis a norm over the set K(?),which is equivalent to the norm induced by k ·k1,?

Proof The proof follows from the fact that the set K(?)is a closed subset of the vector space.

Theorem 4.1 Assume thatThen

(i)As ε tends to 0,the family{u(ε)}converges strongly in the set K(?)to a limit u.

(ii)The limit u is a Kirchhoff-Love displacement field,namely,there existssuch that

Proof The proof is divided into four steps.In the first step,we introduce a new scaled strain tensor R(ε).By means of some boundness results we get that the sequence{u(ε)}converges weakly to a limit u which is a Kirchhoff-Love field.The second step deals with certain technical results about the components of this strain tensor.In the third step we show that the convergence of the family{u(ε)}towards the Kirchhoff-Love field u is actually strong.The fourth step completes the proof by deducing the variational problem.

Step I.Let us introduce the following symmetric tensor

Using the coerciveness properties of tensors C,it follows from this inequality that

Moreover,(4.9)now reads

and

To show that this system has a unique solution, first by the symmetry and positivity of tensor C in(2.1),for every second order 3×3 symmetric tensor A=(Aij),we have

Next we note that the system(4.10)can be written asand the determinant of this linear system is

With Akα=0 we get,therefore the system(4.10)has a unique solution

whereare given by(4.4).

Step III.The whole familyconverges strongly.

Introduce the notation RR for all second order symmetric tensor A.We have

so we haveconverges strongly toBy the definition of R(ε),we have

Remark 4.1 It is interesting that(4.1)consists of an equality in a vector space for the ζαcomponents,namely,for the membrane part of the solution,and an inequality in a cone for the ζ3component,namely,for the bending part of the solution.Thus,the obstacle condition deals only with the bending part.

AcknowledgementThe author is greatly indebted to Professor Li Tatsien and Professor Zhou Yi for their instructive questions,corrections,encouragement,and help.

[1]L′eger,A.and Miara,B.,Mathematical justification of the obstacle problem in the case of a shallow shell,J.Elasticity,90,2008,241–257.

[2]L′eger,A.and Miara,B.,The obstacle problem for shallow shells:A curvilinear approach,Intl.J.of Numerical Analysis and Modeling,Ser.B,2,2011,1–26.

[3]Ciarlet,P.G.and Destuynder,P.,A justification of the two dimensional plate model,J.M′ecanique,18,1979,315–344.

[4]Ciarlet,P.G.,Mathematical Elasticity,Vol II,Theory of Plates,North-Holland,Amsterdam 1997.

[5]Fichera,G.,Problemi elastostatici con vincoli unilaterali:il problema di Signorini con ambigue condizioni al contorno,Mem.Accad.Naz.Lincei Ser.,VIII 7,1964,91–140.

[6]Duvaut,G.and Lions J.-L.,Les In′equations en M′ecanique et en Physique,Dunod 1972.

[7]Paumier,J.C.,Mod′elisation asymptotique d’un probl`eme de plaque mince en contact unilat′eral avec frotte-ment contre un obstacle rigide,Pr′epublication L.M.C.,http://www-lmc.imag.fr/paumier/signoplaque.ps,2002.

[8]Lions,J.-L.,Quelques M′ethodes de R′esolution des Probl?emes aux Limites Non Lin′eaires,Dunod-Gauthier-Villars,Paris,1969.