A variational-hemivariational inequality in contact problem for locking materials and nonmonotone slip dependent friction

Acta Mathematica Scientia 2017,37B(6):1639–1652 http://actams.wipm.ac.cn

A VARIATIONAL-HEMIVARIATIONAL INEQUALITY IN CONTACT PROBLEM FOR LOCKING MATERIALS AND NONMONOTONE SLIP DEPENDENT FRICTION∗ ´ Stanislaw MIGORSKI Chair of Optimization and Control, Jagiellonian University in Krak´ ow, ul. Lojasiewicza 6, 30348 Krak´ ow, Poland E-mail : [email protected]

Justyna OGORZALY Faculty of Mathematics and Computer Science, Jagiellonian University in Krak´ ow, ul. Lojasiewicza 6, 30348 Krak´ ow, Poland E-mail : [email protected] Abstract We study a new class of elliptic variational-hemivariational inequalities arising in the modelling of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modelled by the nonmonotone multivalued subdifferential condition which depends on the slip. The problem is governed by a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set which describes the locking constraints and a nonconvex locally Lipschitz friction potential. The result on existence and uniqueness of solution to the inequality is shown. The proof is based on a surjectivity result for maximal monotone and pseudomonotone operators combined with the application of the Banach contraction principle. Key words

variational-hemivariational inequality; Clarke subdifferential; locking material; unilateral constraint; nonmonotone friction

2010 MR Subject Classification

1

47J20; 47J22; 49J53; 74M10; 74M15

Introduction

In this paper we study a class of contact problems for elastic ideally locking materials. The contact is assumed to be static and it is described by the Signorini unilaterial contact condition with a nonmonotone friction condition between a locking body and a rigid foundation. The constitutive law is modelled by the (convex) subdifferential of the indicator function of a convex set which characterizes the locking constraints. On the other hand, the friction is described by ∗ Received

September 14, 2016; revised June 13, 2017. Research supported by the National Science Center of Poland under the Maestro 3 Project No. DEC-2012/06/A/ST1/00262, and the project Polonium “Mathematical and Numerical Analysis for Contact Problems with Friction” 2014/15 between the Jagiellonian University and Universit´ e de Perpignan Via Domitia.

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the (Clarke) subdifferential boundary condition involving a locally Lipschitz function which, in general, is nonsmooth, nonconvex and nondifferentiable. Moreover, the multivalued frictional contact depends on the slip which is important in many applications. In consequence, the convex analysis approach to such problem is not adequate and the weak formulation has a form of variational-hemivariational inequality of elliptic type. Variational-hemivariational inequalities represent a special class of inequalities, in which both convex and nonconvex functions are involved. Very recently, they were studied in several papers (cf. [7, 11, 12]) in the context of modelling of various problems of Contact Mechanics. In [7] Han et al. studied existence and uniqueness of a solution for static variationalhemivariational inequalities and they showed continuous dependence of the solution on the data. Using the numerical approximation they proved a convergence result and derived error estimates. Finally, they applied abstract results in the analysis of static contact problem with friction for elastic materials. In [11], the authors dealt with a viscoelastic problem on infinite time interval in which the contact is frictionless and is modeled with a boundary condition which describes both the instantaneous and the memory effects of the foundation. They proved that this problem leads to a variational-hemivariational inequality with the history-dependent operators. In turn, in [12] Mig´ orski et al. considered a class of elliptic variational-hemivariational inequalities in reflexive Banach spaces. They stated and proved an existence and uniqueness result and two convergence results, one of them concerns the continuous dependence of the solution with respect to the data, while the second one is obtained by means of penalization method. Using abstract results, they studied an elastic contact problem with unilateral constraint. In the present paper, we use variational-hemivariational inequalities to investigate a new static contact problem for elastic ideally locking materials. To the best of our knowledge, this is the first work on variational-hemivariational inequalities for locking materials. The theory of locking materials was initiated by Prager [16–18]. The variational problems encountered in the theory of locking materials were studied in [2], where the equivalence between the statical and the kinematical methods for such materials were shown. Locking materials belong to a class of hyperelastic bodies for which the strain tensor is constrained to stay in a convex set. Let B be a closed, convex subset of Sd with 0Sd ∈ B, where Sd stands for the space of second order symmetric tensors on Rd . The elastic ideally locking materials are characterized by the following law   σ = σ e + σ l , σ e = a ε (u), ij ijkl kl ij ij ij (1.1)  ε(u) ∈ B, σl · (ε∗ − εij (u)) ≤ 0 for all ε∗ = (εij ) ∈ B. ij

ij

e l Here σij and σij are elastic and locking components of the stress tensor σij and ε(u) = (εij (u)) is the infinitesimal strain tensor defined by εij (u) = 12 (ui,j + uj,i ). The indices i, j, k, l run between 1 and d, and the summation convention over repeated indices is used. In the one dimensional case, the typical strain-stress law of the type (1.1) is of the form   0, if ε < 0,      aε, if 0 ≤ ε < ε0 , σ=    [aε0 , +∞), if ε = ε0 ,    ∅, if ε > ε0

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and it describes the behaviour of the body which for ε < ε0 is linear (Hooke’s law) and for ε = ε0 presents an infinite jump called a locking effect (cf. [16]). Here ε0 and a are positive constants. In this paper we consider the elastic materials for which the constitutive law has the form σ(u) ∈ A(ε(u)) + ∂IB (ε(u)) in Ω,

(1.2) d

where A : Ω × Sd → Sd is a nonlinear elasticity operator and ∂IB : Sd → 2S denotes the subdifferential of the indicator function of the set B, i.e., IB : Sd → {0, +∞},  0, if ε ∈ B, IB (ε) = for ε ∈ Sd , +∞, if ε ∈ /B

and Ω is a connected bounded open set in Rd with a Lipschitz boundary ∂Ω. If the operator ε 7→ A(x, ε) is linear with the elasticity tensor aijkl , then (1.2) reduces to (1.1). Alternatively, in this case, the law (1.2) can be written in the subdifferential form σ(u) ∈ ∂w(ε(u)) in Ω with the potential w : Sd → R given by w(ε) =

1 aijkl εij εkl + IB (ε) for ε ∈ Sd . 2

The set B ⊂ Sd describes the locking constraints and it determines the properties of the material. Several forms of B are met in the literature for perfectly locking material under consideration. For instance, let B = {ε ∈ Sd | Q(ε) ≤ 0}, where the locking function Q : Sd → R is a convex, continuous function such that Q(0) ≤ 0. In this way the strains are supposed to be restricted by an inequality Q(ε) ≤ 0. The choice Q(ε) =

1 D D ε ε − κ2 , 2 ij ij

1 D where εD = (εD ij ) denotes the strain deviator, εij = εij − d tr(ε) I, tr(ε) = εii denotes the trace of the tensor ε(u), and I is the identity matrix was considered by Prager in [17] to model an ideal-locking effect (cf. [9, 16]). The choice

Q(ε) = tr(ε) − κ2 leads to the class of materials called materials with limited compressibility (cf. Prager [17]). In both cases, κ > 0 is a material constant. This choice of Q can be used to describe fairly the behaviour of rubber and some of other types of plastic materials (cf. [6]). We remark that in this case, the set of admissible displacements (cf. (3.18) below) is as follows K = {v ∈ V | ε(v(x)) ∈ B a.e. x ∈ Ω} = {v ∈ V | div v(x) ∈ [−κ2 , κ2 ] a.e. x ∈ Ω}. Note that in the limiting case κ = 0, the set K corresponds to incompressible elastic materials (cf. [2]). Finally, in the modelling of torsion of a cylindrical bar made of a locking material, the set B is chosen to be the ball of center at zero and a radius r > 0 (cf. [2]). In the onedimensional models with idealized strain-stress law for rubber, we can choose Q(ε) = ε − ε0 (cf. [14, 15]), which shows that the stress can take at the locking criterion any arbitrary large value without any change of the strain state.

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The paper is divided into four sections. In Section 2 we recall the notation and some preliminary materials from nonlinear analysis. Section 3 provides the classical and variational formulations of the contact problem under investigation. The main result on existence and uniqueness of weak solution to the problem is established in Section 4. The paper is concluded with examples of nonmonotone friction conditions.

2

Preliminaries

In this section we recall basic materials on subdifferentials, multivalued operators, and single-valued operators which we will use in this paper. The material presented in this section can be found, for instance, in [1, 3, 4, 10, 13]. Given a Banach space X with norm k · kX , the dual space to X is denoted by X ∗ and ∗ h·, ·i stands for the duality pairing of X ∗ and X. The symbol 2X denotes the collection of all subsets of X ∗ . First, we recall the definitions of the convex and the Clarke subdifferentials. Definition 2.1 Let ϕ : X → R ∪ {+∞} be a proper, convex and lower semicontinuous ∗ function. The mapping ∂ϕ : X → 2X defined by ∂ϕ(x) = {x∗ ∈ X ∗ | hx∗ , v − xi ≤ ϕ(v) − ϕ(x) for all v ∈ X} is called the subdifferential of ϕ. Any element x∗ ∈ ∂ϕ(x) is called a subgradient of ϕ in x.

Definition 2.2 Let h : X → R be a locally Lipschitz function. The (Clarke) generalized directional derivative of h at the point x ∈ X in the direction v ∈ X is defined by h(y + λv) − h(y) . λ λ↓0

h0 (x; v) = lim sup y→x,

The generalized subdifferential (gradient) of h at x is a subset of the dual space X ∗ given by ∂h(x) = { ζ ∈ X ∗ | h0 (x; v) ≥ hζ, vi for all v ∈ X }. A locally Lipschitz function h is said to be regular (in the sense of Clarke) at the point x ∈ X if for all v ∈ X the one-sided directional derivative h′ (x; v) exists and h0 (x; v) = h′ (x; v). ∗

Next, we present definitions and theorems for multivalued operators. Let T : X → 2X be an S operator. We recall that D(T ) = {x ∈ X | T x 6= ∅} is the domain of T , R(T ) = {T x | x ∈ X} is the range of T and Gr(T ) = {(x, x∗ ) ∈ X × X ∗ | x∗ ∈ T x} is the graph of T . ∗

Definition 2.3 An operator T : X → 2X is called (a) monotone, if hu∗ − v ∗ , u − vi ≥ 0 for all (u, u∗ ), (v, v ∗ ) ∈ Gr(T ); (b) maximal monotone, if it is monotone and maximal in the sense of inclusion of graphs ∗ in the family of all monotone operators from X to 2X ; (c) u0 -coercive, if for given u0 ∈ X, there exists a function α : R+ → R with α(r) → +∞, as r → +∞ such that for all (u, u∗ ) ∈ Gr(T ), we have hu∗ , u − u0 i ≥ α(kukX ) kukX ; (d) coercive, if it is 0X -coercive, where 0X denotes the zero element in X. ∗

Definition 2.4 Let X be a reflexive Banach space. A multivalued operator T : X → 2X is pseudomonotone, if (a) for every u ∈ X, the set T u ⊂ X ∗ is nonempty, closed and convex; (b) T is upper semicontinuous from each finite dimensional subspace of X into X ∗ endowed with its weak topology;

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(c) for any sequences {un } ⊂ X and {u∗n } ⊂ X ∗ such that un → u weakly in X, u∗n ∈ T un for all n ≥ 1 and lim suphu∗n , un − ui ≤ 0, we have that for every v ∈ X, there exists u∗ (v) ∈ T u such that hu∗ (v), u − vi ≤ lim inf hu∗n , un − vi. ∗

Definition 2.5 Let X be a reflexive Banach space. A multivalued operator T : X → 2X is generalized pseudomonotone, if for any sequences {un } ⊂ X and {u∗n } ⊂ X ∗ such that un → u weakly in X, u∗n ∈ T un for n ≥ 1, u∗n → u∗ weakly in X ∗ and lim suphu∗n , un − ui ≤ 0, we have u∗ ∈ T u and lim hu∗n , un i = hu∗ , ui. ∗

Proposition 2.6 Let X be a reflexive Banach space and T : X → 2X . (a) If T is pseudomonotone, then it is generalized pseudomonotone. (b) If T is a bounded generalized pseudomonotone operator such that for all u ∈ X, T u is a nonempty, closed and convex subset of X ∗ , then T is pseudomonotone.

Next, we recall the surjectivity result (cf. Theorem 2.12 in [13]), which plays an important ∗ role in the proof of Theorem 3.4 in Section 4. Let u0 ∈ X be given, T : X → 2X and Tu0 (v) = T (v + u0 ) for all v ∈ X. ∗

Theorem 2.7 Let X be a reflexive Banach space, T1 : X → 2X be a pseudomonotone ∗ operator, T2 : X → 2X be a maximal monotone operator, and u0 ∈ D(T2 ). Assume that T1 is u0 -coercive in the sense of Definition 2.3(c), and either T1u0 or T2u0 is bounded. Then T1 + T2 is surjective, i.e., R(T1 + T2 ) = X ∗ . Finally, we turn to the case of single-valued operators. Definition 2.8 A single-valued operator A : X → X ∗ is called (a) monotone, if for all u, v ∈ X, we have hAu − Av, u − vi ≥ 0; (b) maximal monotone, if it is monotone, and the conditions (v, w) ∈ X × X ∗ and hAu − w, u − vi ≥ 0 for any u ∈ X entail w = Av; (c) bounded, if A maps bounded sets of X into bounded sets of X ∗ ; (d) coercive, if hAu, ui ≥ α(kukX ) kukX for all u ∈ X, where α : R+ → R is a function such that with lim α(r) = +∞; r→+∞

(e) pseudomonotone, if it is bounded and for any sequence {un } → u weakly in X with lim suphAun , un − ui ≤ 0, we have hAu, u − vi ≤ lim infhAun , un − vi for all v ∈ X.

3

Classical and Variational Formulations

In this section we study a contact problem for elastic idealy locking body with unilateral constraints and nonmonotone friction condition with the slip depenedent friction coefficient. The classical model under consideration is as follows. Problem 3.1 Find a displacement field u : Ω → Rd and a stress field σ : Ω → Sd such that σ(u) ∈ A(ε(u)) + ∂IB (ε(u)) Div σ(u) + f 0 = 0 u=0 σ(u)ν = f N

in Ω,

(3.1)

in Ω,

(3.2)

on ΓD ,

(3.3)

on ΓN ,

(3.4)

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uν ≤ 0, σν (u)uν = 0

on ΓC ,

(3.5)

−στ (u) ∈ µ(kuτ k) ∂j(uτ )

on ΓC .

(3.6)

We provide below the description of Problem 3.1. The set Ω is supposed to be an open, bounded, connected in Rd , d = 2, 3 with a Lipschitz boundary ∂Ω = Γ, and it represents the elastic body in its reference configuration. The boundary Γ is partitioned into three disjoint measurable parts ΓD , ΓN and ΓC such that meas(ΓC ) > 0. The notation ν = (νi ) is used for the outward unit normal at the boundary. The normal and tangential components of a vector field v on the boundary are given by vν = v · ν and v τ = v − vν ν. Moreover, σν and σ τ represent the normal and tangential components of the stress field σ on the boundary, that is, σν = (σν) · ν and σ τ = σν − σν ν. The notation Sd stands for the space of second order symmetric tensors on Rd . On Rd and Sd we use the inner products and the Euclidean norms defined by u · v = ui vi , kvk = (v · v)1/2 for all u = (ui ), v = (vi ) ∈ Rd , σ · τ = σij τij , kτ k = (τ · τ )1/2 for all σ = (σij ), τ = (τij ) ∈ Sd , respectively. The indices i, j, k, l run between 1 and d and the summation convention over repeated indices is applied. Equation (3.1) represents the constitutive law for elastic materials with locking constraints in which A is the elasticity operator, ε(u) denotes the linearized strain tensor, and ∂IB stands for the convex subdifferential of the indicator function of a set B. The possible forms of the set B are presented in Section 1. Equation (3.2) is the equilibrium equation in which f 0 represents the density of the body forces. It is assumed that the process is static and, therefore, the inertial term in the equation of motion is neglected. Conditions (3.3) and (3.4) represent the classical displacement and traction boundary conditions. They mean that the body is fixed on ΓD and surface tractions of density f N act on ΓN . Relation (3.5) represents the Signorini contact condition without gap which is stated in its classical complementarity form. This unilateral contact condition assumes that the foundation is perfectly rigid and it holds on a part ΓC of the boundary. Condition (3.6), also formulated on the surface ΓC , represents the friction law, in which ∂j denotes the Clarke subdifferential of the given function j, and µ denotes a positive function, the coefficient of friction. The function µ is supposed to depend on the slip, i.e., the tangential displacement. More details on mechanical interpretation on static contact models with elastic materials could be found in the books [8, 10, 19]. In the study of Problem 3.1 we assume that the elasticity operator A and the locking constraint set B satisfy the following hypotheses.    A : Ω × Sd → Sd is such that      (a) A(·, ε) is measurable on Ω for all ε ∈ Sd ;       (b) A(x, ·) is continuous on Sd for a.e. x ∈ Ω;      2    (c) there exists a0 ∈ L (Ω), a0 ≥ 0 and a1 > 0 such that kA(x, ε)k ≤ a0 (x) + a1 kεk for all ε ∈ Sd , a.e. x ∈ Ω;     (d) there exists mA > 0 such that       (A(x, ε1 ) − A(x, ε2 )) · (ε1 − ε2 ) ≥ mA kε1 − ε2 k2       for all ε1 , ε2 ∈ Sd , a.e. x ∈ Ω;      (e) A(x, 0 d ) = 0 d for a.e. x ∈ Ω. S S

(3.7)

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B is a closed, convex subset of Sd with 0Sd ∈ B.

(3.8)

In hypotheses (3.7)(e) and (3.8), the symbol 0Sd denotes the zero element in Sd . The potential function j and the coefficient of friction µ satisfy the following conditions.   j : ΓC × Rd → R is such that       (a) j(·, ξ) is measurable on ΓC for all ξ ∈ Rd and there       exists e ∈ L2 (ΓC ; Rd ) such that j(·, e(·)) ∈ L1 (ΓC );   (3.9) (b) j(x, ·) is locally Lipschitz on Rd for a.e. x ∈ ΓC ;      (c) k∂j(x, ξ)k ≤ cj for all ξ ∈ Rd , a.e. x ∈ ΓC with cj > 0;     (d) j 0 (x, ξ ; ξ − ξ ) + j 0 (x, ξ ; ξ − ξ ) ≤ α kξ − ξ k2   j 1 2 1 2 1 2 1 2     d for all ξ1 , ξ 2 ∈ R , a.e. x ∈ ΓC with αj ≥ 0.   µ : ΓC × R+ → R+ is such that       (a) there exists Lµ > 0 such that       |µ(x, r1 ) − µ(x, r2 )| ≤ Lµ |r1 − r2 |   for all r1 , r2 ∈ R+ , a.e. x ∈ ΓC ;     (b) µ(·, r) is measurable on ΓC for all r ∈ R;       (c) there exists µ0 > 0 such that µ(x, r) ≤ µ0      for all r ∈ R+ , a.e. x ∈ ΓC .

(3.10)

Finally, we assume that the densities of body forces and surface tractions have the regularity f 0 ∈ L2 (Ω; Rd ),

f N ∈ L2 (ΓN ; Rd ).

(3.11)

Remark 3.2 The condition (d) in hypothesis (3.9) is equivalent to the following one which is called relaxed monotonicity condition h∂j(x, ξ 1 ) − ∂j(x, ξ 2 ), ξ 1 − ξ 2 i ≥ −αj kξ1 − ξ 2 k2

(3.12)

for all ξ 1 , ξ 2 ∈ R , a.e. x ∈ Ω. In the case when j(x, ·) is a convex function, then condition (3.12) reduces to the monotonicity of the (convex) subdifferential, i.e., (3.12) holds with αj = 0. Examples of functions which satisfy the relaxed monotonicity condition are presented in [10]. d

Next, we introduce the following spaces V = { v ∈ H 1 (Ω; Rd ) | v = 0 on ΓD },

H = L2 (Ω; Rd ),

H = L2 (Ω; Sd ).

The space H is a Hilbert space with the inner product defined by Z hσ, τ iH = σij (x) τij (x) dx, Ω

and the associated norm denoted by k · kH . On the space V we consider the inner product hu, viV = hε(u), ε(v)iH

for u, v ∈ V

and the associated norm k · kV . Recall that, since meas(ΓD ) > 0, it follows that V is a real Hilbert space. We always denote by v the trace of an element v ∈ V . By the continuity of the trace operator, it follows that kvkL2 (∂Ω;Rd ) ≤ kγk kvkV for all v ∈ V,

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where kγk denotes the norm of the trace operator γ : V → L2 (∂Ω; Rd ). We derive now the variational formulation of Problem 3.1. To this end, let (u, σ) be a pair of smooth functions which satisfies (3.1)–(3.6). Let v ∈ V . Multiplying the equilibrium equation (3.2) by v − u and using the Green formula, we have Z hσ(u), ε(v) − ε(u)iH = hf 0 , v − uiH + σ(u)ν · (v − u) dΓ. ∂Ω

Using the boundary conditions (3.3) and (3.4), we have Z hσ(u), ε(v) − ε(u)iH = hf , v − ui + (σν (u)(vν − uν ) + σ τ (u) · (v τ − uτ )) dΓ,

(3.13)

ΓC

where the element f ∈ V ∗ is defined by

hf , vi = hf 0 , viH + hf N , viL2 (ΓN ;Rd )

for all v ∈ V.

(3.14)

Next, we introduce the set of admissible displacement fields K1 defined by K1 = { v ∈ V | vν ≤ 0 on ΓC }.

(3.15)

Then, for v ∈ K1 , we have σν (u)(vν − uν ) = σν (u)vν − σν (u)uν ≥ 0 on ΓC . On the other hand, from (3.6), by the definition of the Clarke subdifferential, we obtain Z Z − σ τ (u) · (v τ − uτ ) dΓ ≤ µ(kuτ k)j 0 (uτ ; v τ − uτ ) dΓ for all v ∈ V. (3.16) ΓC

ΓC

Therefore, it follows that Z Z (σν (u)(vν − uν ) + σ τ (u) · (v τ − uτ )) dΓ ≥ − ΓC

ΓC

µ(kuτ k)j 0 (uτ ; v τ − uτ ) dΓ

(3.17)

for all v ∈ K1 . Next, we introduce

K2 = { v ∈ V | ε(v(x)) ∈ B a.e. x ∈ Ω }.

(3.18)

From the constitutive law (3.1), we have σ(u) = A(ε(u)) + ζ(u) and ζ(u) ∈ ∂IB (ε(u)) in Ω. The latter, for v, u ∈ K2 , implies ζ(u) : (ε(v) − ε(u)) ≤ IB (ε(v)) − IB (ε(u)) ≤ 0 in Ω. Hence, we obtain hσ(u), ε(v) − ε(u)iH ≤ hA(ε(u)), ε(v) − ε(u)iH .

(3.19)

Inserting (3.17) and (3.19) into (3.13), we obtain the following variational formulation for Problem 3.1. Problem 3.3 Find a displacement field u ∈ K1 ∩ K2 such that Z hA(ε(u)), ε(v) − ε(u)iH + µ(kuτ k)j 0 (uτ ; v τ − uτ ) dΓ ΓC

≥ hf , v − uiV ∗ ×V

for all v ∈ K1 ∩ K2 .

We have the following existence and uniqueness result whose proof will be provided in the next section. Theorem 3.4 Assume hypotheses (3.7)–(3.11) and the following smallness condition (µ0 αj + cj Lµ )kγk2 < mA are valid. Then Problem 3.3 has a unique solution u ∈ K1 ∩ K2 .

(3.20)

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Proof of Theorem 3.4

The proof consists of three main steps and it is based on the surjectivity result of Theorem 2.7 and the Banach contraction principle. Let K = K1 ∩ K2 . It is clear that K is a closed and convex set with 0V ∈ K. We define the operator A : V → V ∗ and functional J : V × V → R by hAu, vi = hA(ε(u)), ε(v)iH for u, v ∈ V, Z J(z, v) = µ(kz τ k)j(v τ ) dΓ for z, v ∈ V.

(4.1) (4.2)

ΓC

We show that under hypotheses (3.7), (3.9) and (3.10) properties (4.3) and (4.4) below hold.   (a) A is pseudomonotone.      (b) For all v ∈ V, we have hAv, vi ≥ m kvk2 . A V (4.3)   (c) A is strongly monotone, i.e., for all v , v ∈ V, we have 1 2     hAv 1 − Av 2 , v 1 − v 2 i ≥ mA kv 1 − v 2 k2V .    (a) J(z, ·) is locally Lipschitz (in fact, Lipschitz on bounded      subsets of V ) for all z ∈ V.       (b) k∂J(z, v)kV ∗ ≤ µ0 cj meas(ΓC ) for all z, v ∈ V.       (c) For all z, v, w ∈ V, we have   Z  0 J (z, v; w) ≤ µ(kz τ k)j 0 (v τ ; wτ ) dΓ.  ΓC     (d) For all z 1 , z 2 , v 1 , v 2 ∈ V, we have       J 0 (z 1 , v 1 ; v 2 − v 1 ) + J 0 (z 2 , v 2 ; v 1 − v 2 )   
   ≤ αJ kv 1 − v 2 k2V + kz 1 − z 2 kV kv1 − v 2 kV      with αJ = max{µ0 αj , Lµ cj }kγk2.

First, we observe that from (3.7)(c) and the H¨older inequality, we have √
kAvkV ∗ ≤ 2 ka0 kL2 (Ω) + a1 kvkV for all v ∈ V,

(4.4)

which yields the boundedness of the operator A. The strong monotonicity condition (4.3)(c) is an easy consequence of (3.7)(d). Then, using the hypothesis (3.7)(b), the H¨older inequality and the Lebesgue dominated convergence theorem, similarly as in Theorem 7.3 of [10], we deduce that the operator A is continuous. Since A is bounded, monotone and hemicontinuous, by Theorem 3.69(i) in [10], we infer that A is pseudomonotone. The coercivity property (4.3)(b) follows from (3.7)(d) and the fact that A0 = 0. This completes the proof of condition (4.3). Next, we establish condition (4.4). Based on condition (3.9)(b) and Theorem 3.47(iv) in [10], we deduce that (4.4)(a) and (4.4)(c) hold. By Proposition 3.37 in [10], conditions (3.9)(c) and (3.10)(c), we obtain Z k∂J(z, v)kV ∗ ≤ µ(kz τ k)k∂j(v)k dΓ 6 µ0 cj meas(ΓC ) ΓC

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for all z, v ∈ V which implies (4.4)(b). We show property (4.4)(d). In order to obtain this condition, we need the following inequality which is a consequence of (3.9)(c) and Proposition 3.23(iii) in [10] j 0 (v 2 ; v 1 − v 2 ) = max{hξ, v 1 − v 2 i | ξ ∈ ∂j(v 2 )} ≤ k∂j(v2 )kkv 1 − v 2 k ≤ cj kv 1 − v 2 k. Hence, from the conditions (3.9)(d), (3.10)(a), (c) and the H¨older inequality, we have J 0 (z 1 , v 1 ; v 2 − v 1 ) + J 0 (z 2 , v 2 ; v 1 − v 2 ) Z Z 0 = µ(kz 1τ k)j (v 1 ; v 2 − v 1 ) dΓ + µ(kz 2τ k)j 0 (v 2 ; v 1 − v 2 ) dΓ ΓC ΓC Z = µ(kz 1τ k)(j 0 (v 1 ; v 2 − v 1 ) + j 0 (v 2 ; v 1 − v 2 )) dΓ ΓC Z
+ µ(kz 2τ k) − µ(kz 1τ k) j 0 (v 2 ; v 1 − v 2 ) dΓ Z ΓC Z ≤ µ0 αj kv1 − v 2 k2 dΓ + Lµ cj kz 1τ k − kz 2τ k kv 1 − v 2 k dΓ ΓC

ΓC

≤ µ0 αj kγk2 kv1 − v 2 k2V + Lµ cj kγk2 kz 2 − z 1 kV kv 1 − v 2 kV , which implies that condition (4.4)(d) holds. This proves condition (4.4). Step 1 Let η ∈ K. Consider the auxiliary problem.

Problem 4.1 Find uη ∈ K such that Z hAuη − f , v − uη i + µ(kη τ k)j 0 (uητ ; v τ − uτ ) dΓ ≥ 0 for all v ∈ K. ΓC

Lemma 4.2 Problem 4.1 has a unique solution uη ∈ K. First, we observe that Problem 4.1 is equivalent to the following problem. Problem 4.3 Find uη ∈ V such that hAuη − f , v − uη i + ϕ(v) − ϕ(uη ) +

Z

ΓC

µ(kη τ k)j 0 (uητ ; v τ − uτ ) dΓ ≥ 0

for all v ∈ V . Here ϕ : V → {0, +∞} is the indicator function of the set K, that is,  0, if v ∈ K, ϕ(v) = IK (v) = for v ∈ V. +∞, if v ∈ /K We introduce the following operator inclusion.

Problem 4.4 Find uη ∈ V such that Auη + ∂ϕ(uη ) + ∂J(η, uη ) ∋ f . In what follows, we state and prove the following claims. Claim 1 Problem 4.4 has a solution. Claim 2 Every solution to Problem 4.4 is a solution to Problem 4.3. Claim 3 The solution to Problem 4.1 is unique. ∗ Proof of Claim 1 We introduce two multivalued operators T1 , T2 : V → 2V which appear in Problem 4.4 and are defined by T1 v = Av + ∂J(η, v), T2 v = ∂ϕ(v) for v ∈ V, respectively.

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Now, we prove that the operator T1 is bounded, coercive in the sense of Definition 2.3(d) and pseudomonotone. The boundedness of T1 follows from the boundedness of the operators A and ∂J(η, ·) (cf. (4.4)(b)). We observe that condition (4.4)(d) is equivalent to the following one h∂J(z 1 , v 1 ) − ∂J(z 2 , v 2 ), v 1 − v 2 i ≥ −αJ (kv 1 − v 2 k2V + kz 1 − z 2 kV kv 1 − v 2 kV )

(4.5)

for all z 1 , z 2 , v 1 , v 2 ∈ V . Exploiting condition (4.4)(b), the Cauchy inequality and condition (4.5), we have hT1 v, vi = hAv, vi + h∂J(η, v) − ∂J(η, 0), vi + h∂J(η, 0), vi ≥ (mA − αJ ) kvk2V − µ0 cj meas(ΓC )kvkV

for all v ∈ V . Taking hypothesis (3.20) into account, the coercivity of T1 follows. Next, we prove that the operator T1 is pseudomonotone. It is clear that the set Av+∂J(η, v) is nonempty, closed and convex in V ∗ for all v ∈ V , so, based on Proposition 2.6(b), we need to show that the operator T1 is generalized pseudomonotone. From hypotheses (3.20), (4.3)(c) and (4.5), it is clear that the operator T1 is strongly monotone, that is, hT1 v 1 − T1 v 2 , v 1 − v 2 i ≥ (mA − αJ ) kv1 − v 2 k2V for all v 1 , v 2 ∈ V.

(4.6)

To prove the generalized pseudomonotonicity of T1 , we suppose that un ∈ V , un → u weakly in V , u∗n ∈ T1 un , u∗n → u∗ weakly in V ∗ and lim suphu∗n , un − ui ≤ 0. We prove that u∗ ∈ T1 u and hu∗n , un i → hu∗ , ui. Exploiting (4.6), from the relation (mA − αJ )kun − uk2V ≤ hu∗n , un − ui − hT1 u, un − ui, we deduce that un → u in V . Since u∗n ∈ T1 un , we get u∗n = w n + z n , where w n = Aun and z n ∈ ∂J(η, un ). Using the boundedness of operators A and ∂J(η, ·), by passing to a subsequence, if necessary, we may assume that wn → w and z n → z both weakly in V ∗ with some w, z ∈ V ∗ . Therefore, from u∗n = wn + z n , we have u∗ = w + z. Now, we recall (cf. e.g. Proposition 3.66 in [10]) the following equivalent condition for the pseudomonotonicity. The operator A : V → V ∗ is pseudomonotone, if and only if it is bounded and ξ n → ξ weakly in V together with lim sup hAξ n , ξ n − ξi ≤ 0 yielding lim hAξ n , ξ n − ξi = 0 and Aξ n → Aξ weakly in V ∗ . Using this charaterization of pseudomonotonicity, we deduce Aun → Au weakly in V ∗ , ∗ and hence that w = Au. On the other hand, from the fact that V ∋ v 7→ ∂J(η, v) ∈ 2V has a closed graph with respect to the strong topology in V and weak topology in V ∗ (cf. e.g. Proposition 3.23 in [10]), we conclude that z ∈ ∂J(η, u). These observations lead to the following conclusion u∗ = w + z ∈ Au + ∂J(η, u) = T1 u. Since u∗n → u∗ weakly in V ∗ and un → u in V , it is easily seen that hu∗n , un i → hu∗ , ui. This implies that T1 is generalized pseudomonotone and also it is pseudomonotone. The function ϕ is proper, convex and lower semicontinuous with dom ϕ = K which is a consequence of hypothesis (3.8) and the definition of ϕ. It is well known (cf. e.g. [4, Theorem ∗ 1.3.19]) that the operator T2 = ∂ϕ : V → 2V is maximal monotone with D(∂ϕ) = K. We are now in a position to apply a surjectivity result of Theorem 2.7 to obtain a solution uη ∈ V of Problem 4.4. This completes the proof of Claim 1.  Proof of Claim 2

Let uη ∈ V be a solution to Problem 4.4. We have Auη + ξ η + θη = f

(4.7)

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with ξη , θη ∈ V ∗ , ξ η ∈ ∂ϕ(uη ) and θ η ∈ ∂J(η, uη ). Exploiting the definitions of convexity and the Clarke subdifferential, and (4.4)(c), we have hξ η , v − uη i ≤ ϕ(v) − ϕ(uη ), 0

hθ η , v − uη i ≤ J (η, uη ; v − uη ) ≤

Z

ΓC

µ(kη τ k)j 0 (uητ ; v τ − uητ ) dΓ

for all v ∈ V . Combining (4.7) with the last two inequalities, we obtain Z hAuη − f , v − uη iV ∗ ×V + ϕ(v) − ϕ(uη ) + µ(kη τ k)j 0 (uητ ; v τ − uτ ) dΓ ≥ 0 ΓC

for all v ∈ V . This implies that uη ∈ V solves Problem 4.3. We conclude that Problem 4.1 has at least one solution uη ∈ K which completes the proof of Claim 2.  Proof of Claim 3 For the uniqueness part, let u1 , u2 ∈ K be solutions to Problem 4.1 corresponding to the same η ∈ K, i.e., Z hAu1 − f , v − u1 i + µ(kη τ k)j 0 (u1τ ; v τ − u1τ ) dΓ ≥ 0, ΓC

Z

hAu2 − f , v − u2 i +

ΓC

µ(kη τ k)j 0 (u2τ ; v τ − u2τ ) dΓ ≥ 0

for all v ∈ K. We put v = u2 in the first inequality and v = u1 in the second one. Adding obtained inequalities, we get hAu1 − f , u1 − u2 i − hAu2 − f , u1 − u2 i Z
≤ µ(kη τ k) j 0 (u1τ ; u2τ − u1τ ) + j 0 (u2τ ; u1τ − u2τ ) dΓ. ΓC

From property (3.9)(d) and the strong monotonicity of A, we have Z Z mA ku1 − u2 k2V ≤ µ0 αj ku1τ − u2τ k2 dΓ ≤ µ0 αj ΓC

ΓC

ku1 − u2 k2 dΓ

= µ0 αj ku1 − u2 k2L2 (ΓC ;Rd ) ≤ µ0 αj kγk2 ku1 − u2 k2V .

The latter, due to the smallness condition (3.20), implies u1 = u2 . This shows that the solution to Problem 4.1 is unique and completes the proof of Claim 3. Step 2 Define the operator Λ : K → K by Λη = uη

for η ∈ K,

where uη ∈ K is the unique solution to Problem 4.1. We show that Λ has a unique fixed point. Let η 1 , η 2 ∈ K and ui = uηi ∈ K be the unique solutions to Problem 4.1 for i = 1, 2. Thus Z hAu1 − f , v − u1 i + µ(kη1τ k)j 0 (u1τ ; v τ − u1τ ) dΓ ≥ 0, ΓC

hAu2 − f , v − u2 i + for all v ∈ K. Hence hAu1 − Au2 , u1 − u2 i

Z

ΓC

µ(kη2τ k)j 0 (u2τ ; v τ − u2τ ) dΓ ≥ 0

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≤

Z

ΓC



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 µ(kη1τ k)j 0 (u1τ ; u2τ − u1τ ) + µ(kη2τ k)j 0 (u2τ ; u1τ − u2τ ) dΓ.

Consequently, using (3.9)(c), (3.10)(a),(c) and (4.3)(c), it follows that

mA ku1 − u2 k2V ≤ cj Lµ kγk2 kη1 − η 2 kV ku1 − u2 kV + µ0 αj kγk2 ku1 − u2 k2V and ku1 − u2 kV ≤

cj Lµ kγk2 kη − η 2 kV . mA − µ0 αj kγk2 1

From the smallness condition (3.20), we obtain that Λ is a contraction. Hence, there exists a unique η ∗ ∈ K such that η∗ = Λη ∗ = uη∗ ∈ K. Step 3 We denote by u the solution of Problem 4.4 for η = η ∗ , i.e., u = uη∗ . Since ∗ η = Λη∗ , we have η ∗ = uη∗ . Hence, we deduce that u is a solution of Problem 3.3. The uniqueness of a solution of Problem 3.3 is a consequence of Claim 3 and the uniqueness of the fixed point of Λ. This concludes the proof of the theorem.  Finally, we elaborate on the friction condition (3.6) by providing the following two examples. Example 4.5 Let j : ΓC × Rd → R be defined as follows  a(1 − a)   if kξk ≥ a, akξk + 2 j(x, ξ) =    a − 1 kξk2 + kξk if kξk ≤ a 2a

(4.8)

for all ξ ∈ Rd , a.e. x ∈ ΓC , where 0 ≤ a < 1. For simplicity, we skip the dependence on the x variable. Then  ξ  a if kξk ≥ a,    kξk    if ξ = 0, ∂j(x, ξ) = B(0, 1)        a − 1 kξk + ξ if 0 < kξk ≤ a a kξk

for all ξ ∈ Rd , where B(0, 1) denotes the closed unit ball in Rd . It is easy to see that j(x, ·) is nonconvex and it satisfies hypotheses (3.9) with cj = 1 and αj = 1 (cf. [10]). In this case, the friction law (3.6) takes the following form    if uτ = 0, kστ k ≤ µ(0)       a−1 uτ −στ = µ(kuτ k) kuτ k + if 0 < kuτ k ≤ a, (4.9) a kuτ k     uτ   if kuτ k ≥ a −στ = µ(kuτ k)a kuτ k

on ΓC . We also remark that for the particular case a = 1 in (4.8), the function j reduces to the convex potential j(x, ξ) = kξk for all ξ ∈ Rd , a.e. x ∈ ΓC and the associated friction law (4.9) becomes the classical version of Coulomb’s law of dry friction. The latter was studied in several monographs (see e.g. [5, 8, 10, 19, 20]). p Example 4.6 Consider the function j : Rd → R defined by j(ξ) = kξk2 + ρ2 − ρ for ξ ∈ Rd , where ρ > 0 is a parameter. The function j is convex, continuously differentiable with

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its subdifferential given by ξ ∂j(ξ) = j ′ (ξ) = p for ξ ∈ Rd . kξk2 + ρ2

(4.10)

It is clear that j satisfies (3.9)(b), (3.9)(c) with cj = 1 and ∂j is monotone, i.e., (3.9)(d) holds with αj = 0. We remark that with this choice of j, condition (3.6) reduces to the so-called static version of the regularized Coulomb friction law with slip of the following form uτ −στ (u) = µ(kuτ k) p on ΓC . kuτ k2 + ρ2 Other examples of the friction condition (3.6) can be found in Chapter 7 in [10]. References [1] Clarke F H. Optimization and Nonsmooth Analysis. New York: Wiley, Interscience, 1983 [2] Demengel F, Suquet P. On locking materials. Acta Appl Math, 1986, 6: 185–211 [3] Denkowski Z, Mig´ orski S, Papageorgiou N S. An Introduction to Nonlinear Analysis: Theory. Boston, Dordrecht, London, New York: Kluwer Academic/Plenum Publishers, 2003 [4] Denkowski Z, Mig´ orski S, Papageorgiou N S. An Introduction to Nonlinear Analysis: Applications. Boston, Dordrecht, London, New York: Kluwer Academic/Plenum Publishers, 2003 [5] Eck C, Jaruˇsek J, Krbeˇ c M. Unilateral Contact Problems: Variational Methods and Existence Theorems. Pure and Applied Mathematics 270. New York: Chapman/CRC Press, 2005 [6] Haines D W, Wilson W D. Strain-energy density function for rubber-like materials. J Mech Phys Solids, 1979, 27: 345–360 [7] Han W, Mig´ orski S, Sofonea M. A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J Math Anal, 2014, 46: 3891–3912 [8] Han W, Sofonea M. Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Studies in Advanced Mathematics 30. Providence, RI: Amer Math Soc, Somerville, MA: International Press, 2002 [9] Maier G. A quadratic programming approach for certain classes of nonlinear structural problems. Meccanica, 1968, 3: 121–130 [10] Mig´ orski S, Ochal A, Sofonea M. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics 26. New York: Springer, 2013 [11] Mig´ orski S, Ochal A, Sofonea M. History-dependent variational-hemivariational inequalities in contact mechanics. Nonlinear Anal Real World Appl, 2015, 22: 604–618 [12] Mig´ orski S, Ochal A, Sofonea M. A class of variational-hemivariational inequalities in reflexive Banach spaces. J Elasticity, 2017, 127: 151–178 [13] Naniewicz Z, Panagiotopoulos P D. Mathematical Theory of Hemivariational Inequalities and Applications. New York, Basel, Hong Kong: Marcel Dekker, Inc, 1995 [14] Panagiotopoulos P D. Inequality Problems in Mechanics and Applications. Boston: Birkh¨ auser, 1985 [15] Panagiotopoulos P D. Hemivariational Inequalities, Applications in Mechanics and Engineering. Berlin: Springer-Verlag, 1993 [16] Prager W. On ideal-locking materials. Trans Soc Rheol, 1957, 1: 169–175 [17] Prager W. Elastic solids of limited compressibility//Proceedings of the 9th International Congress of Applied Mechanics, vol 5. Brussels, 1958: 205–211 [18] Prager W. On elastic, perfectly locking materials//G¨ ortler H, ed. Applied Mechanics, Proceedings of the 11th International Congress of Applied Mechanics, Munich, 1964. Berlin, Heidelberg: Springer-Verlag, 1966: 538–544 [19] Shillor M, Sofonea M, Telega J J. Models and Analysis of Quasistatic Contact. Lect Notes Phys 655. Berlin, Heidelberg: Springer, 2004 [20] Sofonea M, Matei A. Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series 398. Cambridge University Press, 2012