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 · | · | 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. { i | i ∈ } 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 { ij | ij ji ∈ } (1982), Cocu (1984), Kikuchi & Oden (1988). In these pa- H = u = (u ) u H1(Ω) , 1 { i | i ∈ } pers linear elastic materials and variational formulations in- = σ σ H . volving the displacement field were considered. H1 { ∈ H | ij,j ∈ } 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 + ε(u), ε(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 ε : 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 ε(v) = (ε (v)), ε (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 ), | · | 1 | · |NH | · | | · |H | | ≤ | | Let HΓ = H 2 (Γ) and let γ : H1 HΓ be the trace στ < µp( Rσν ) = uτ = 0  map. For every element v H we use,→ when no confusion | | | | ⇒ (3.3) ∈ 1  is likely, the notation v for the trace γv of v on Γ and we de- στ = µp( Rσν ) = λ 0  | | | | ⇒ ∃ ≥  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 ∈ 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, ε(v) + Div σ, v H v H1. (2.2) h i h iH h i ∀ ∈ 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 adheres| to the| 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 ∈ : 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. We also assume that a volume force of den- hardness of the surface and r = max 0, r . This friction + { } sity ϕ1 acts in Ω and that surface tractions ϕ2 act on Γ2. The law means that when the normal stress is too large i.e. it solid is in unilateral contact with a rigid foundation on Γ3. 1 exceeds α , the surface disintegrates and offers no resistance We denote by u the displacement vector, σ represents the to the motion. stress field and ε = ε(u) is the small strain tensor. The elastic constitutive law of the material is assumed to be Using (3.1)-(3.3), the mechanical problem of frictional contact of the elastic body may be formulated classically as σ = F (ε(u)) (3.1) follows : Problem P : Find a displacement field u :Ω IRN and Let us remark that condition (3.14) is fulfilled in the case a stress field σ :Ω S such that → of the function p given by (3.4) or (3.5). → N σ = F (ε(u)) in Ω (3.6) Remark 3.1. Using (3.13) we obtain that for all τ the function x F (x, τ(x)) belongs to and hence∈ we H may Div σ + ϕ1 = 0 in Ω (3.7) consider F 7→as an operator defined onH with the range on H u = 0 on Γ1 (3.8) . Moreover, F : is a strongly monotone Lipschitz H H → H σν = ϕ on Γ (3.9) continuous operator and therefore F is invertible and his in- 2 2 1 verse F − : is also a strongly monotone Lipschitz u 0, σ 0, σ u = 0 H → H ν ≤ ν ≤ ν ν continuous operator. σ µp( Rσ ),  | τ | ≤ | ν | We also suppose that the forces and the tractions have the  on Γ3. (3.10) στ < µp( Rσν ) = uτ = 0  regularity | | | | ⇒  2 N σ = µp( Rσ ) = λ 0  ϕ1 H, ϕ2 L (Γ2) (3.15) | τ | | ν | ⇒ ∃ ≥ ∈ ∈ such that στ = λuτ  −  while the coefficient of friction µ is such that To obtain variational formulations for problem (3.6)-  (3.10) we need additional notation. Let V denote the closed µ L∞(Γ ), µ(x) 0 a.e. on Γ . (3.16) ∈ 3 ≥ 3 subspace of H1 defined by Next, we denote by f the element of V given by V = v H v = 0 on Γ . { ∈ 1 | 1 } f, v V = ϕ1, v H + ϕ2, γv L2(Γ )N v V Since meas Γ1 > 0, Korn’s inequality holds : h i h i h i 2 ∀ ∈

ε(v) C v H v V, (3.11) and let j : 1 V IR be the functional | |H ≥ | | 1 ∀ ∈ H × → see, e.g., Hlava´cekˇ & Necasˇ (1981) p. 79. Here and below C denotes a strictly positive generic constant which may de- j(σ, v) = µp( Rσν ) vτ da. (3.17) Γ3 | | | | pend on Ω, Γ, F and p but does not depend on the input data Z 2 ϕ1, ϕ2 or µ and whose value may vary from place to place. Since Rσν lies in L (Γ), using (3.14) and (3.16) it follows On V we consider the inner product given by that the integral in (3.17) is well defined. Finally, we denote in the sequel by U the set of geometri- u, v V = ε(u), ε(v) (3.12) h i h iH cally admissible displacement fields defined by and let be the associated norm. It follows from (3.11) | · |V that and are equivalent norms on V . Therefore U = v V vν 0 on Γ3 | · |H1 | · |V { ∈ | ≤ } (V, V ) is a real Hilbert space. | · | and, for all g , let Σ(g) denote the set of statically In the study of the mechanical problem (3.6)-(3.10) we as- ∈ H1 sume that the elasticity operator F :Ω S S satisfies admissible stress fields given by × N → N (a) there exists M > 0 such that Σ(g) = σ σ, ε(v) + j(g, v) { ∈ H | h iH ≥ F (x, ε ) F (x, ε ) M ε ε f, v V v U . 1 2 1 2  h i ∀ ∈ } | ε , ε −S , a.e. in|Ω; ≤ | − | ∀ 1 2 ∈ N   Using (2.1)–(2.4) we have the following result. (b) there exists m > 0 such that   (F (x, ε )) F (x, ε )) (ε ε )  Lemma 3.2. If u, σ are sufficiently regular functions 1 − 2 · 1 − 2  { } m ε ε 2 ε , ε S ,  (3.13) satisfying (3.6)–(3.10), then: ≥ | 1 − 2| ∀ 1 2 ∈ N  a.e. in Ω;  u U, σ Σ(σ), (c) x F (x, ε) is Lebesgue measurable  ∈ ∈ 7→  on Ω, for all ε SN ;  ∈  σ, ε(v) ε(u) + j(σ, v) j(σ, u)  h − iH − ≥ (d) x F (x, 0) .  f, v u V v U, 7→ ∈ H  h − i ∀ ∈  The friction bound function p :Γ IR IRsatisfies τ σ, ε(u) 0 τ Σ(σ). (3.24) 3 × → + h − iH ≥ ∀ ∈ (a) there exists L > 0 such that Lemma 3.2, (3.6) and Remark 3.1 lead us to consider the p(x, r ) p(x, r ) L r r following two variational problems. | 1 − 2 | ≤ | 1 − 2|  r1, r2 IR, a.e. on Γ3; N ∀ ∈  Problem P1 : Find a displacement field u :Ω IR  → (b) x p(x, r) is Lebesgue  (3.14) such that 7→  measurable on Γ ,  3  u U, F (ε(u)), ε(v) ε(u) for all r IR; ∈ h − iH ∈  +j(F (ε(u)), v) j(F (ε(u)), u)  − ≥  (c) p(x, 0) = 0 a.e. on Γ3.  f, v u V v U.   h − i ∀ ∈    Problem P2 : Find a stress field σ :Ω SN such that Remark 4.4. The mechanical interpretation of the results → in Theorem 4.3 is the following. Let the coefficient of friction 1 σ Σ(σ), F − (σ), τ σ 0 τ Σ(σ). be small enough. Then : ∈ h − iH ≥ ∀ ∈ 1) if the displacement field u is the solution of the varia- Remark 3.3. Problems P1 and P2 are formally equivalent tional problem P1 then the stress field σ associated to u by to the mechanical problem P . Indeed, if u represents a suf- the elastic constitutive law σ = F (ε(u)) is the solution of ficiently regular solution of the variational problem P1 and the variational problem P2; σ is defined by σ = F (ε(u)), then, using the arguments of 2) if the stress field σ is the solution of the variational prob- Duvaut & Lions (1972), it follows that u, σ is a solution of lem P then there exists a displacement field u V associ- { } 2 problem P . Similarly, if σ represents a regular solution of the ated to σ by the elastic constitutive law σ = F∈(ε(u)) and variational problem P2 and u V is such that σ = F (ε(u)) moreover u is the solution of the variational problem P1. then, using the same arguments,∈ it follows that u, σ is a so- lution of the mechanical problem P . For this reason{ } we may Remark 4.5. Under the assumptions of Theorem 4.3 it also results that if the displacement field u is the solution consider problems P1 and P2 as variational formulations of the mechanical problem P . of problem P1 and the stress field σ is the solution of the problem P2, then u and σ are connected by the elastic con- stitutive law σ = F (ε(u)). For this reason we shall consider 4 Statement of the Main Results in the sequel the couple u, σ given by Theorems 4.1 and 4.2 as a weak solution for{ the} problem (3.6)–(3.10) and we

In the study of problems P1 and P2 we obtained the follow- conclude that this mechanical problem has a unique weak so- ing existence and uniqueness results. lution, provided µ is sufficiently small. Theorem 4.1. Let conditions (3.13)–(3.16) hold. Then there exists µ0 > 0 which depends only on Ω, Γ, F and p References such that if µ L∞(Γ3) < µ0 then there exists a unique solu- | | [1] COCU, M., 1984. Existence of solutions of Signorini tion to problem P1. problems with friction. Int. J. Engng. Sci., 22, 567–581. Theorem 4.2. Let conditions (3.13)–(3.16) hold and let [2] COCU, M., 1992. Unilateral contact problems with fric- µ0 > 0 be defined as in Theorem 4.1. Then, if µ L (Γ ) < | | ∞ 3 tion for an elastoviscoplastic material with internal state µ0, there exists a unique solution to problem P2. The proof of Theorems 4.1 and 4.2 are carried out in sev- variable. Proc. Contact Mechanics Int. Symp. Edt. A. eral steps, based on fixed point arguments similar to those Curnier, PPUR, 207–216. used in Shillor & Sofonea (1997), Sofonea (1997). Details [3] DUVAUT, G., 1982. Loi de frottement non locale. J. can be found in Drabla & Sofonea (1999). Mec.´ The.´ Appl. Special issue, 73–78. Keeping in mind Remark 3.3, by Theorem 4.1 we obtain [4] DUVAUT, G. & LIONS, J. L., 1972. Les inequations´ en the existence and the uniqueness of a displacement field u in mecanique´ et en physique. Dunod, Paris. the study of the mechanical problem (3.6)–(3.10), provided the coefficient of friction is small enough. Using again Re- [5] HLAVA´ CEKˇ , I. & NECASˇ , J., 1981. Mathematical the- mark 3.3, by Theorem 4.2 we obtain, under the same assump- ory of elastic and elastoplastic bodies : an introduction. tions, the existence and the uniqueness of the stress field σ in Elswier, Amsterdam. the study of the same problem (3.6)–(3.10). For this reason we investigate in the sequel the link between the solutions u [6] IONESCU, I. R. & SOFONEA, M., 1993. Functional and and σ obtained in Theorems 4.1 and 4.2. numerical methods in viscoplasticity. Oxford Univer- The following result was proved in Drabla & Sofonea sity Press, Oxford. (1999). [7] KIKUCHI, N. & ODEN, T.J., 1988. Contact problems in Theorem 4.3. Let conditions (3.13)–(3.16) hold and let elasticity : a study of variational inequalities and finite us suppose that the inequality µ < µ is satisfied, element methods SIAM, Philadelphia. | |L∞(Γ3) 0 where µ0 is defined as in Theorem 4.1 or 4.2. [8] PANAGIOTOPOULOS, P.D., 1985. Inequality problems 1) Let u denote the solution of problem P obtained in 1 in mechanics and applications. Birkhauser,¨ Basel. Theorem 4.1 and let σ be the function given by [9] ROCHDI, M., SHILLOR, M. & SOFONEA, M., 1998. σ = F (ε(u)). (4.1) A Quasistatic viscoelastic contact problem with normal compliance and friction. Journal of Elasticity, 51 105– Then σ is a solution of problem P2. 126. 2) Conversely, let σ be the solution of problem P2 obtained in Theorem 4.2. Then there exists an unique element u V [10] SHILLOR, M. & SOFONEA, M., 1997. A Quasistatic ∈ such that (4.1) hold and u is a solution of problem P1. viscoelastic contact problem with friction, preprint. [11] STROMBERG¨ , N., JOHANSSON, L. & KLARBRING, A., 1996. Derivation and analysis of a generalized standard model for contact friction and wear. Int. J. Solids Struc- tures, 33 (13), 1817–1836.