Variational analysis of a Signorini problem with friction S. Drabla Institut de Mathematiques,´ Universite´ de Setif´ Avenue Scipion, 19000 Setif,´ Algeria Keywords: Elastic material, Signorini’s conditions, The paper is structured as follows. In Section 2 we intro- Coulomb’s law of friction, variational inequality, fixed point, duce the notation and some preliminary material. In Section weak solution. 3 we describe the model for the process (problem P ) and we set it in two variational formulations, P1 and P2. Problem P1 is obtained from P using a Green type formula and the Abstract constitutive law. Since in this way the stress field is elimi- nated, the unknown of this problem is the displacement field This paper deals with the study of a nonlinear problem of u. Problem P2 is obtained from P using a similar method frictional contact between an elastic body and a rigid foun- and it involves as unique unknown the stress field σ. In Sec- dation. The elastic constitutive law is assumed to be non- tion 4 we present our main existence and uniqueness results linear and the contact is modeled with Signorini’s conditions (Theorems 4.1 and 4.2). They state that for sufficiently small and a version of Coulomb’s law of dry friction. We present coefficient of friction there exists a unique solution u to the two weak formulations of the problem and we establish ex- problem P1 and there exists a unique solution σ to the prob- istence and uniqueness results, using arguments of elliptic lem P2. The proofs are based on arguments from elliptic variational inequalities and a fixed point property. Moreover, variational inequalities and fixed point properties of certain we prove some equivalence results. maps. Then, we study the link between the solutions u and σ obtained in Theorems 4.1 and 4.2 and we state that u and σ 1 Introduction are related by the elastic constitutive law (Theorem 4.3). In this work we consider the process of frictional contact be- 2 Notation and preliminaries tween an elastic body, which is acted upon by volume forces and surface tractions and it is in contact with a rigid founda- In this short section we present the notation we shall use and tion. Situations which involve such type of problems abound some preliminary material. For further details we refer the in industry, especially in engines, motors and transmissions. reader to Duvaut & Lions (1972), Hlava´cekˇ & Necasˇ (1981), We assume that the forces and the tractions change slowly Panagiotopoulos (1985), Ionescu & Sofonea (1993). in time so that the accelerations in the system are negligible. We denote by SN the space of second order symmetric Neglecting the inertial terms in the equations of motion leads tensors on IRN (N = 2; 3), while “ ” and will represent · j · j N to a static approximation for the process. The material’s con- the inner product and the Euclidean norm on IR and SN . stitutive law is assumed to be nonlinear. The contact is fric- Let Ω IRN be a bounded domain with a Lipschitz bound- tional and we assume that the loss of contact may occur. It is ary Γ and⊂ let ν denote the unit outer normal on Γ. We shall modeled with Signorini’s conditions and a nonlocal version use the notation of Coulomb’s law of dry friction. The nonlocal character H = u = (u ) u L2(Ω) ; arise from the regularization of the normal contact stress. f i j i 2 g Existence and uniqueness results for static contact prob- = σ = (σ ) σ = σ L2(Ω) ; lems using a nonlocal friction law were obtained in Duvaut H f ij j ij ji 2 g (1982), Cocu (1984), Kikuchi & Oden (1988). In these pa- H = u = (u ) u H1(Ω) ; 1 f i j i 2 g pers linear elastic materials and variational formulations in- = σ σ H : volving the displacement field were considered. H1 f 2 H j ij;j 2 g In the present paper we consider the case of a nonlinear Here and below i; j = 1; 2; :::; N and summation over re- peated indices is implied. H, , H and are real Hilbert elastic constitutive law and we replace the friction law stud- H 1 H1 ied in the previous quotated papers by a more general one. spaces endowed with the inner products given by Moreover, we obtain a new variational formulation of the problem involving the stress field. We prove new existence u; v = u v dx; h iH i i and uniqueness results and we obtain an equivalence result. ZΩ We also study the behaviour of the solution with respect to σ; τ = σij τij dx; the coefficient of friction. h iH ZΩ u; v H1 = u; v H + "(u);"(v) ; in which F is a given nonlinear function. Here and below, in h i h i h iH order to simplify the notation, we do not indicate explicitly σ; τ = σ; τ + Div σ; Div τ H ; h iH1 h iH h i the dependence of various functions with respect the spatial respectively, where " : H1 and Div : 1 H are variable x. the deformation and the divergence!H operators,H respectively,! Next, we describe the contact condition on the surface Γ3. defined by We assume that the normal displacement uν and the normal 1 stress σν satisfy the Signorini’s contact conditions "(v) = (" (v));" (v) = (v + v ); ij ij 2 i;j j;i u 0; σ 0; σ u = 0: (3:2) ν ≤ ν ≤ ν ν Div σ = (σij;j ): The associated friction law is The associated norms on the spaces H; ;H and are H 1 H1 denoted by H ; ; H1 and 1 , respectively. στ µp( Rσν ); j · j 1 j · jNH j · j j · jH j j ≤ j j Let HΓ = H 2 (Γ) and let γ : H1 HΓ be the trace στ < µp( Rσν ) = uτ = 0 9 map. For every element v H we use,! when no confusion j j j j ) (3:3) 2 1 > is likely, the notation v for the trace γv of v on Γ and we de- στ = µp( Rσν ) = λ 0 > j j j j ) 9 ≥ = note by vν and vτ the normal and the tangential components such that στ = λuτ : − > of v on Γ given by > Here p is a nonnegative function, the so-called;> friction vν = v ν; vτ = v vν ν: (2:1) bound, uτ denotes the tangential displacement, στ represents · − the tangential force on the contact boundary and µ 0 is the Let H0 be the dual of H and let ; denote the duality ≥ Γ Γ h· ·i coefficient of friction. This is a static version of Coulomb’s pairing between H0 and H . For every σ let σν be the Γ Γ 2 H1 law of dry friction and should be seen either as a mechani- element of HΓ0 given by cal model suitable for the proportional loadings case or as a first approximation of a more realistic model, based on a fric- σν; γv = σ; "(v) + Div σ; v H v H1: (2:2) h i h iH h i 8 2 tion law involving the time derivative of u (see for instance We also denote by σν and στ the normal and tangential Shillor & Sofonea (1997), Rochdi et al: (1998)). traces of σ (see for instance Panagiotopoulos (1985)). We The friction law (3.3) states that the tangential shear can- 1 recall that if σ is a regular function (say C ), then not exceed the maximum frictional resistance µp( Rσν ). Then, if the inequality holds, the surfaces adheresj to thej σν; γv = σν v da (2:3) foundation and is in the so-called stick state, and when the h i · ZΓ equality holds there is relative sliding, the so-called slip state. for all v H1, where da is the surface measure element, and Therefore, the contact surface Γ3 is divided into three zones 2 : the stick zone, the slip zone and the zone of separation in σν = (σν) ν ; στ = σν σν ν: (2:4) · − which uν < 0 and there is no contact. The boundaries of In the sequel R will represent a normal regularization op- these zones are unknown a priory, are part of the problem, 1 and form free boundaries. We also remark that the introduc- erator i.e. a linear and continuous operator R : H− 2 (Γ) L2(Γ) (see e.g. Duvaut (1982), Cocu (1984)). We shall need! tion of the nonlocal smoothing operator R in (3.3) is done for it to regularize the normal trace of the stress tensor on Γ. technical reasons, since the trace of the stress tensor on the boundary is too rough. In Duvaut (1982), Cocu (1984) and Cocu (1992), the fric- 3 Problem statement and variational formula- tion law (3.3) was used with tions p(r) = r: (3:4) In this section we describe a model for the process, present its variational formulations and list the assumptions on the Recently, a new version for Coulomb’s law of friction was problem data. derived in Stromberg¨ et al: (1996), from thermodynamic The setting is as follows. An elastic body occupies the do- consideration. It consists of using in (3.3) the friction bound main Ω with surface Γ that is partitioned into three disjoint function p(r) = r(1 αr) (3:5) measurable parts Γ1; Γ2 and Γ3, such that meas Γ1 > 0. − + The body is clamped on Γ1 and thus the displacement field where α is a small positive coefficient related to the wear and vanishes there.