Variational Analysis of a Frictional Contact Problem with Wear and Damage

. We study a quasistatic problem describing the contact with friction and wear between a piezoelectric body and a moving foundation. The material is modeled by an electro-viscoelastic constitutive law with long memory and damage. The wear of the contact surface due to friction is taken into account and is described by the diﬀerential Archard condition. The contact is modeled with the normal compliance condition and the associated law of dry friction. We present a variational formulation of the problem and establish, under a smallness assumption on the data, the existence and uniqueness of the weak solution. The proof is based on arguments of parabolic evolutionary inequations, elliptic variational inequalities and Banach ﬁxed point.


Introduction
The piezoelectric effect is characterized by the coupling between the mechanical and electrical behavior of the materials. It consists on the apparition of electric charges on the surfaces of some crystals after their deformation. Conversely, it was proved experimentally that the action of electric field on the crystals may generate strain and stress. A deformable material which presents such a behavior is called a piezoelectric material. Piezoelectric materials are used extensively as switches and actuators in many engineering systems, in radioelectronics, electroacoustics and measuring equipments. However, there are very few mathematical results concerning contact problems involving piezoelectric materials and therefore there is a need to extend the results on models for contact with deformable bodies which include coupling between mechanical and electrical properties. General models for elastic materials with piezoelectric effects can be found in [2,7]. A static frictional contact problem for electric-elastic materials was considered in [3,10,12]. A slip-dependent frictional contact problem for electro-elastic materials was studied in [21]. Contact problems with friction or adhesion for electro-viscoelastic materials were studied in [4,9,15,20,22,27] and recently in [8] for the case of an electrically conductive foundation. For works concerned with the frictional contact problems for electro-viscoelastic materials with long memory, we refer to [15,16] and the references therein. In the present paper we consider a mathematical model for the process of a frictional contact problem with normal compliance and wear for an electro-viscoelastic material with long memory, damage and a moving conductive foundation. The foundation is assumed to move steadily and only sliding contact takes places. A mathematical models which describe the equilibrium of an elastic or a viscoelastic body in frictional contact with a moving foundation were considered in [24,25,26]. In all these papers, the damage function β is restricted to have values between zero and one. When β = 1 there is no damage in the material, when β = 0 the material is completely damaged, when 0 < β < 1 there is partial damage and the system has a reduced load carrying capacity. Contact problems with damage have been investigated in [14,17,18,23].
The rest of the paper is structured as follows. In Section 2, we present the notation and some preliminaries. In Section 3, we present the mechanical problem, we list the assumptions on the data and give the variational formulation of the problem. In Section 4, we state our main existence and uniqueness result. It is based on arguments of classical results for elliptic variational inequalities, on parabolic inequalities and fixed point arguments.

Notation and preliminaries
In this section we present the notation we shall use and some preliminary material. For further details, we refer the reader to [5]. We denote by S d the space of second order symmetric tensors on R d (d = 2, 3), while "·" and | . | will represent the inner product and the Euclidean norm on S d and R d . Let Ω ⊂ R d be a bounded domain with a Lipschitz boundary Γ and let ν denote the unit outer normal on Γ . Everywhere in the sequel the index i and j run from 1 to d, summation over repeated indices is implied and the index that follows a comma represents the partial derivative with respect to the corresponding component of the independent spatial variable. We use the standard notation for Lebesgue and Sobolev spaces associated to Ω and Γ and introduce the spaces Here ε and Diυ are the deformation and divergence operators, respectively, defined by The spaces H, H, H 1 and H 1 are real Hilbert spaces endowed with the canonical inner products given by The associated norms on the spaces H, H, H 1 and H 1 are denoted by | . | H , | . | H , | . | H1 and | . | H1 , respectively. For every element υ ∈ H 1 we also use the notation υ for the trace of υ on Γ and we denote by υ ν and υ τ the normal and the tangential components of υ on Γ given by υ ν = υ ·ν, υ τ = υ −υ ν ν. We also denote by σ ν and σ τ the normal and the tangential traces of a function σ ∈ H 1 , we recall that when σ is a regular function then σ ν = (σν) · ν, σ τ = σν − σ ν ν, and the following Green's formula holds Let T > 0. For every real Banach space X we use the classical notation for the spaces C(0, T ; X), C 1 (0, T ; X), and we use the standard notation for the Lebesgue spaces and for the Sobolev spaces L p (0, T ; X) and W k,p (0, T ; X), 1 ≤ p ≤ ∞, 1 ≤ k. Moreover, if X 1 and X 2 are real Hilbert spaces then X 1 × X 2 denotes the product. Hilbert space endowed with the canonical inner product (., .) X1×X2 .

Mechanical and variational formulations
The physical setting is the following. An electro-viscoelastic body with long memory and damage occupies a bounded domain Ω ⊂ R d (d = 2, 3) with outer Lipschitz surface Γ. The body is submitted to the action of body forces of density f 0 and volume electric charges of density q 0 . It is also constrained mechanically and electrically on the boundary. We consider a partition of Γ into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 , on one hand, and a partition of Γ 1 ∪ Γ 2 into two disjoint measurable parts Γ a and Γ b , on the other hand, such that meas(Γ 1 ) > 0, meas(Γ a ) > 0. Let T > 0 and let [0, T ] be the time interval of interest. The body is clamped on Γ 1 , so the displacement field vanishes there. Surface tractions of density f 2 act on Γ 2 . We also assume that the electrical potential vanishes on Γ a and a surface free electrical charge of density q 2 is prescribed on Γ b . In the reference configuration the body may come in contact over Γ 3 with a conductive moving obstacle, which is also called the foundation. The contact is modeled with the normal compliance condition and a general version of Coulomb's law of dry friction. Also, there may be electrical charges on the part of the body which is in contact with the foundation and which vanish when contact is lost. We are interested in the evolution of the deformation of the body and of the electric potential on the time interval [0, T ]. The process is assumed to be isothermal, electrically static, i.e., all radiation effects are neglected, and mechanically quasistatic, i.e., the inertial terms in the momentum balance equations are neglected. To simplify the notation, we do not indicate explicitly the dependence of various functions on the variables x ∈ Ω ∪ Γ and t ∈ [0, T ] . Then, the classical formulation of the mechanical problem of sliding frictional contact problem with normal compliance and wear may be stated as follows.
Problem P. Find a displacement field u :

3)
Div in Ω. Here, µ = µ(ζ, |u τ − υ * |) and p ν = p ν (u ν − h − ζ). Equations (3.1) and (3.2) represent the constitutive law for a piezoelectric material with long memory and damage, where A and F are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively, M is a relaxation fourth order tensor, E(ϕ) = −∇ϕ is the electric field, E = (e ijk ) represents the third order piezoelectric tensor, E * is its transposed and B denotes the electric permittivity tensor. We use dots for derivatives with respect to the time variable t. Inclusion (3.3) describes the evolution of the damage field, governed by the source damage function S, where k e is a positive coefficient and ∂Ψ K is the subdifferential of indicator function of the set of admissible damage functions given by K = {ξ ∈ H 1 (Ω) / 0 ≤ ξ ≤ 1 a.e. in Ω}. Equations (3.4) and (3.5) represent the equilibrium equations for the stress and electric displacement fields. Equations (3.6) are the displacement and traction boundary conditions, respectively. Equation (3.8) describes a homogeneous Neumann boundary condition, where ∂β ∂ν is the normal derivative of β. Equations (3.9) represent the electric boundary conditions. In (3.11) u 0 is the given initial displacement, β 0 is the initial material damage and ζ(0) = 0 means that at the initial moment the body is not subject to any prior wear. The relations (3.7) represent the condition with normal compliance, friction and wear. The wear function ζ which measures the wear accumulated of the surface. The evolution of the wear of the contacting surface is governed by the differential form of Archard's law ( see, e.g., [1,19,28,29]), where k 1 > 0 is a wear coefficient, p ν is a prescribed function of the normal compliance, µ ≥ 0 is the coefficient of friction, h represents the gap in direction of ν, υ * is the tangential velocity of the foundation, |u τ − υ * | represents the slip rate between the contact surface and the foundation, and R * : R + → R + is the truncation operator R * (r) = r if r ≤ R and R * (r) = R if r > R, R being a fixed positive constant. We recall that in the case without wear, a general version of normal compliance is given by The difference u ν − h, when positive, represents the penetration of the surface asperities into those of the foundation. This condition was first introduced in [11] and used in a large number of papers, see for instance [6,13] and the references therein. Next, (3.10) is the electrical contact condition on Γ 3 , which is the main novelty of this work. This condition has been used in [9] for electroviscoelastic contact problem with short memory. It represents a regularized condition which may be obtained as follows. First, we assume that the foundation is electrically conductive and its potential is maintained at ϕ 0 . When there is no contact at a point on the surface (i.e., u ν < h), there are no free electrical charges on the surface and the normal component of the electric displacement field vanishes (i.e., D ·ν = 0). During the process of contact (i.e., when u ν ≥ h) the normal component of the electric displacement field or the free charge is assumed to be proportional to the difference between the potential of the foundation and the body's surface potential with k as the proportionality factor.
is the characteristic function of the interval [0, ∞). Since our process involves the wear of the contacting surfaces we need to take into account the change in the geometry by replacing the initial gap function h with h + ζ during the process, , this condition describes perfect electrical contact and is somewhat similar to the well-known Signorini contact condition. Both conditions may be over-idealizations in many applications. To make it more realistic, we regularize previous relation and write it as (3.10) in which kχ [0,∞) (u ν − h − ζ) is replaced with ψ which is a regular function which will be described below, and φ L is the truncation function where L is a large positive constant. We note that this truncation does not pose any practical limitations on the applicability of the model, since L may be arbitrarily large, higher than any possible peak voltage in the system, and therefore in To obtain a variational formulation of the problem (3.1)-(3.11) we introduce the closed subspace of H 1 defined byV = {υ ∈ H 1 /υ = 0 on Γ 1 }. Since meas(Γ 1 ) > 0, Korn's inequality holds and there exists a constant c K > 0 which depends only on Ω and Γ 1 such that On the space V we consider the inner product and the associated norm given by It follows from Korn's inequality that | . | H1 and | . | V are equivalent norms on V. Therefore (V, | . | V ) is a real Hilbert space. Moreover, by the Sobolev's trace theorem and Korn's inequality, there exists a constant c 0 > 0, depending only on Ω, Γ 1 and Γ 3 such that We also introduce the spaces where diυD = (D i,i ). The spaces W and W are real Hilbert spaces with the inner products given by The associated norms will be denoted by | . | W and | . | W , respectively. Moreover, when D ∈ W is a regular function, the following Green's type formula holds Notice also that, since meas(Γ a ) > 0, the following Friedrichs-Poincaré inequality holds where c F > 0 is a constant which depends only on Ω and Γ a . It follows from Friedrichs-Poincaré inequality that | . | H 1 (Ω) and | . | W are equivalent norms on W and therefore (W, | . | W ) is a real Hilbert space. Moreover, by the Sobolev's trace theorem, there exists a constant a 0 > 0, depending only on Ω, Γ a and Γ 3 such that In the study of the mechanical problem (3.1)-(3.11), we now list assumptions on the data. Assume that the operators A, F, E, B and the functions S, p ν , µ, ψ satisfy the following conditions with L A , m A , L F , L S , L ν , p * ν , L µ , µ * , m B , L ψ and N ψ being positive constants: The mapping x → A(x, 0) belongs to H.
on Ω for any ε ∈ S d and r ∈ R. (d) The mapping x → F(x, 0, 0) belongs to H.
x → S(x, ε, r) is Lebesgue measurable on Ω. (d) The mapping x → S(x, 0, 0) belongs to L 2 (Ω). (3.14) (c) e ijk = e ikj ∈ L ∞ (Ω). (3.17) The relaxation tensor M satisfiesM ∈ C(0, T ; H ∞ ), where H ∞ is the space of fourth order tensor field given by which is a real Banach space with the norm The density of volume forces, traction, volume electric charges and surface electric charges have the regularity Finally, we assume that the gap function h, the given potential of the foundation ϕ 0 , the initial displacement field u 0 and the initial damage field β 0 satisfy We define the bilinear form a : H 1 (Ω) × H 1 (Ω) → R by a(ς, ϑ) = k e Ω ∇ς · ∇ϑdx.
Using standard arguments we obtain the variational formulation of the mechanical problem ( The existence of the unique solution to Problem PV is stated and proved in the next section.

Existence and uniqueness result
The main result in this section is the following existence and uniqueness result. The functions u, σ, ϕ, D, β and ζ which satisfy (3.26)-(3.28) are called weak solution to contact problem P. We conclude that, under the assumptions (3.13)-(3.21) and if c 2 0 p * ν L µ < m A and N ψ < m B a 2 0 , the mechanical problem (3.1)-(3.11) has a unique weak solution satisfying (4.1)-(4.4). The proof of Theorem 1 is carried out in several steps. Everywhere in this section we suppose that assumptions of Theorem 1 hold. Below, c denotes a generic positive constant which may depend on Ω, Γ 1 , Γ 2 , Γ 3 , A, E, F, M, B, ψ, p ν , µ and T but does not depend on t nor of the rest of input data, and whose value may change from place to place.
Problem PV ζηgw . Find a velocity field v ζηgw : [0, T ] → V and a stress field σ ζηgw : [0, T ] → H such that for all t ∈ [0, T ], In the study of Problem PV ζηgw we have the following result.
Proof. We define the operator A : It follows from (4.7), (3.13)(b) and (3.13)(c) that A : V → V is Lipschitz continuous and a strongly monotone operator on V . Moreover using Riesz Representation Theorem we may define an element F ∈ C(0, T ; V ) by ( Since A is a strongly monotone and Lipschitz continuous operator on V and from (3.25), it follows from classical result on elliptic inequalities (see for example [5]) that there exists a unique function v ζηgw ∈ V which satisfies (Av We use the relation (4.5), the assumption (3.13) and the properties of the deformation tensor to obtain that σ ζηgw (t) ∈ H. Since v = v ζηgw (t) ± Φ satisfies (4.6), where Φ ∈ D(Ω) d is arbitrary, using the definition (3.22) we find With the regularity assumption (3.19) on f 0 we see that Divσ ζηgw (t) ∈ H. Therefore σ ζηgw (t) ∈ H 1 . Next we show that v ζηgw ∈ C(0, T ; V ). Let Using the relation (4.8) we find Then by using (3.23), (3.12), the conditions (3.13), (3.14) and (3.15) we obtain (4.9) This inequality and the regularity of the functions f , g, w, ζ and η show that v ζηgw ∈ C(0, T ; V ). From assumption (3.13), the relation (4.5) we have We have The regularity of the functions η, v, f 0 and the relations (4.10), (4.11) show that σ ζηgw ∈ C(0, T ; H 1 ).
Proof. Let w 1 , w 2 ∈ C(0, T ; V ). We use the notation v i = v ζηgwi for i = 1, 2. From the definition (4.12) we have Using similar arguments as those used in the proof of (4.9) we find Keeping in mind that c 2 0 p * ν L µ < m A , the two inequalities shows that the operator Λ ζηg is a contraction in the Banach space C(0, T ; V ), which concludes the proof.
In what follows we denote by w ζηg the fixed point given in Lemma 2 and let v ζηg ∈ C(0, T ; V ) be the function defined by v ζηg = v ζηgw ζηg . We have Λ ζηg w ζηg = w ζηg and Λ ζηg w ζηg = v ζηgw ζηg it follows that w ζηg = v ζηg . Therefore, choosing w = w ζηg in (4.8) and for all v ∈ V, t ∈ [0, T ], we see that v ζηg satisfies (4.13) We denote by u ζηg ∈ C 1 (0, T ; V ) the function and define the operator Λ ζη : C(0, T ; V ) → C(0, T ; V ) by Lemma 3. The operator Λ ζη has a unique fixed point g ζη ∈ C(0, T ; V ).
Proof. Let g 1 , g 2 ∈ C(0, T ; V ). We use the notation v i = v ζηgi and u i = u ζηgi for i = 1, 2. Using (4.14) and the estimates in the proof of Lemma 1 yield for all Using now (4.14)-(4.15) we obtain for all t ∈ [0, T ] , By reiterating this inequality m times, we obtain that a power of Λ ζη is a contraction mapping on C(0, T ; V ), which concludes the proof.
Problem PV ζη . Find a displacement field u ζη : In the study of the problem PV ζη we have the following result. Proof. For each ζ ∈ C(0, T ; L 2 (Γ 3 )) and η ∈ C(0, T ; H), we denote by g ζη ∈ C(0, T ; V ) be the fixed point guaranted by Lemma 3 and let u ζη be the function defined by (4.14), for g = g ζη . We have Λ ζη g ζη = g ζη . From (4.14) and (4.15) it follows that u ζη = g ζη . Therefore, taking g = g ζη in (4.13) and using (4.7) and (4.14) we see that u ζη is the unique solution to problem PV ζη satisfying the regularity expressed in (4.1).
In the third step, we let θ ∈ C(0, T ; L 2 (Ω)) be given and consider the following variational problem for the damage field.
Proof. To solve P V θ , we use the standard result for parabolic variational inequalities ( see for example the reference [23, p.47]).
The uniqueness part of Lemma 8 is a consequence of the uniqueness of the fixed point of the operator Λ ζ defined by (4.21)-(4.23) and the unique solvability of problems P V ζη ζ ,QV ζη ζ and P V θ ζ .
Let us now consider the operator T : (4.28) The last step in the proof of Theorem 1 is the next result.
Using the inequality 2ab ≤ a 2 + b 2 and integrating this inequality with respect to time, we obtain Since u 1 (0) = u 2 (0) = u 0 and using the previous inequality we obtain  It follows from (4.29) and the previous inequality that Therefore, for m large enough, T m is a contractive operator on the Banach space C(0, T ; L 2 (Γ 3 )). The operator T has a unique fixed point ζ * ∈ C(0, T ; L 2 (Γ 3 )).
Now we have all ingredients to prove Theorem 1.