A frictional contact problem with wear involving elastic-viscoplastic materials with damage and thermal effects

We consider a mathematical problem for quasistatic contact between a thermo-elastic-viscoplastic body with damage and an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. We employ the thermo-elastic-viscoplastic with damage constitutive law for the material. The damage of the material caused by elastic deformations. The evolution of the damage is described by an inclusion of parabolic type. The problem is formulated as a coupled system of an elliptic variational inequality for the displacement, a parabolic variational inequality for the damage and the heat equation for the temperature. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed point arguments.


Introduction
Scientific research and recent papers in mechanics are articulated around two main components, one devoted to the laws of behavior and other devoted to boundary conditions imposed on the body. The boundary conditions reflect the binding of the body with the outside world.
Recent researches use coupled laws of behavior between mechanical and electric effects or between mechanical and thermal effects. For the case of coupled laws of behavior between mechanical and electric effects, general models for electro-elastic materials can be found in G σ(s) − A(ε(u(s))), ε(u(s)), θ(s) ds, (1.1) 3 and S is a given constitutive function which describes the sources of the damage in the system. When β = 1, the material is undamaged, when β = 0, the material is completely damaged, and for 0 < β < 1 there is partial damage. General models of mechanical damage, which were derived from thermodynamical considerations and the principle of virtual work, can be found in [22] and references therein. The models describe the evolution of the material damage which results from the excess tension or compression in the body as a result of applied forces and tractions. Mathematical analysis of one-dimensional damage models can be found in [15]. Damage may be initiated and evolves in both the elastic and plastic deformation processes. Particularly, damage in the elastic deformation state is termed elastic damage which is the case for most brittle materials while damage in the plastic deformation state is termed plastic damage which is mainly for ductile materials. In this paper we use the damage caused by elastic deformations for mechanical and mathematical reasons. Mechanically we use elastic damage because the brittle materials are more susceptible to the damage and wear, mathematically it is easier to treat the case of an internal variable outside the integral compared to the case when the internal variable inside the integral term. The differential inclusion used for the evolution of the temperature field iṡ θ − k 0 ∆θ = ψ(σ, ε(u), θ) + q.
Dynamic and quasistatic contact problems are the topic of numerous papers, e.g. [1,2,13]. A model of damage coupled to wear was studied in [10]. However, the mathematical problem modelled the quasi-static evolution of damage in thermo-viscoplastic materials has been studied in [18], the dynamic evolution of damage in elastic-thermo-viscoplastic materials was studied in [15].
Most papers related with wear process use laws of behavior of mechanical kind or mechanical nature with electric effects. In this paper we deal the case of laws of behavior coupled between mechanical and thermal effects. In practice the thermal effect facilitates wear which makes this paper closer to the reality.
The paper is organized as follows. In Section 2 we introduce the notations and give some preliminaries. In Section 3 we present the mechanical problem, list the assumptions on the data, give the variational formulation of the problem. In Section 4 we state our main existence and uniqueness result Theorem 4.1.

Notations and preliminaries
In this short section, we present the notation we shall use and some preliminary material. For more details, we refer the reader to [8]. We denote by S d the space of second order symmetric tensors on R d (d = 2, 3), while "· " and · denotes the absolute value if it is applied to a scalar or the Euclidean norm if it applied to a vector on S d and R d , respectively.
Let Ω ⊂ R d be a bounded domain with outer Lipschitz boundary Γ and let ν denote the unit outer normal on ∂Ω = Γ. We shall use the notation Here and below, the indices i and j run from 1 to d, the summation convention over repeated indices is used and the index that follows a comma indicates a partial derivative with respect to the corresponding component of the independent variable. The spaces H, H, H 1 (Ω) d and H 1 are real Hilbert spaces endowed with the canonical inner products given by: The associated norms are denoted by · H , · H , · H 1 and · H 1 , respectively. Let H Γ = (H 1/2 (Γ)) d and γ : H 1 (Γ)) d → H Γ be the trace map. For every element v ∈ H 1 (Ω) d , we also use the notation v to denote the trace map γv of v on Γ, and we denote by v ν and v τ the normal and tangential components of v on Γ given by Similarly, for a regular (say C 1 ) tensor field σ : Ω → S d we define its normal and tangential components by (2.2) and for all σ ∈ H 1 the following Green's formula holds We recall the following standard result for parabolic variational inequalities used in Section 4 (see [4, p. 124]).
Let V and H be real Hilbert spaces such that V is dense in H and the injection map is continuous. The space H is identified with its own dual and with a subspace of the dual V of V. We write and we say that the inclusions above define a Gelfand triple. We denote by · V , · H and · V , the norms on the spaces V, H and V respectively, and we use (·, ·) V ×V for the duality pairing between V and V. Note that if f ∈ H then Theorem 2.1. Let V ⊂ H ⊂ V be a Gelfand triple. Let K be a nonempty, closed, and convex set of V. Assume that a(·, ·) : V × V → R is a continuous and symmetric bilinear form such that for some constants ζ > 0 and c 0 , Then, for every u 0 ∈ K and f ∈ L 2 (0, T; H), there exists a unique function u ∈ H 1 (0, T; H) ∩ L 2 (0, T; V) such that u (0) = u 0 , u (t) ∈ K for all t ∈ [0, T], and for almost all t ∈ (0, T), Finally, for any real Hilbert space X, we use the classical notation for the spaces L p (0, T; X) and W k,p (0, T; X), where 1 p ∞ and k > 1. For T > 0 we denote by C(0, T; X) and C 1 (0, T; X) the space of continuous and continuously differentiable functions from [0, T] to X, respectively, with the norms respectively. Moreover, we use the dot above to indicate the derivative with respect to the time variable and 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 (·, ·) X 1 ×X 2 .
The mechanical problem may be formulated as follows.

Mechanical and variational formulations
The physical setting is the following. A body occupies the domain Ω ⊂ R d (d = 2, 3) with outer Lipschitz surface Γ. The body undergoes the action of body forces of density f 0 and external heat source q. It also undergoes the mechanical and thermal constraint on the boundary. We consider a partition of Γ into three disjoint parts Γ 1 , Γ 2 and Γ 3 . We assume that meas(Γ 1 ) > 0. Let T > 0 and let [0, T] be the time interval of interest. We admit a possible external heat source applied in Ω × (0, T), given by the function q. The body is clamped on Γ 1 × (0, T) and so the displacement field vanishes there. Surface tractions of density f 2 act on Γ 2 × (0, T) and a volume force of density f 0 is applied in Ω × (0, T). Finally, on the part Γ 3 the body may come into frictional and bilateral contact with a moving rigid foundation which results in the wear of the contacting surface. We suppose that the body forces and tractions vary slowly in time, and therefore, the accelerations in the system may be neglected. Neglecting the inertial terms in the equation of motion leads to a quasistatic approach to the process. For the body we use a thermo-elastic-viscoplastic constitutive law with damage given by (1.1) to model the material's behavior.
We now briefly describe the boundary conditions on the contact surface Γ 3 , based on the model derived in [22]. We introduce the wear function w : Γ 3 × [0, T] → R + which measures the wear of the surface. The wear is identified as the normal depth of the material that is lost. Since the body is in bilateral contact with the foundation, it follows that Thus the location of the contact evolves with the wear. We point out that the effect of the wear is the recession on Γ 3 and therefore, it is natural to expect that u ν 0 on Γ 3 , which implies w 0 on Γ 3 .
The evolution of the wear of the contacting surface is governed by a simplified version of Archard's law (see [22]) which we now describe. The rate form of Archard's law iṡ where k > 0 is a wear coefficient, v * is the tangential velocity of the foundation and u τ − v * represents the slip speed between the contact surface and the foundation.
We see that the rate of wear is assumed to be proportional to the contact stress and the slip speed. For the sake of simplicity we assume in the rest of the section that the motion of the foundation is uniform, i.e., v * does not vary in time.
We assume that v * is large so that we can neglect in the sequelu τ compared with v * to obtain the following version of Archard's laẇ The use of the simplified law (3.2) for the evolution of the wear avoids some mathematical difficulties in the study of the quasistatic thermo-elastic-viscoplastic contact problem.
We can now eliminate the unknown function w from the problem. In this manner, the problem decouples, and once the solution of the frictional contact problem has been obtained, the wear of the surface can be obtained by integration of (3.2). Let ζ = kv * and α = 1 ζ . Using (3.1) and (3.2) we have We model the frictional contact between the thermo-elastic-viscoplastic body and the foundation with Coulomb's law of dry friction. Since there is only sliding contact, it where µ > 0 is the coefficient of friction. These relations set constraints on the evolution of the tangential stress; in particular, the tangential stress is in the direction opposite to the relative sliding velocity u τ − v * . Naturally, the wear increases in time, i.e.ẇ 0. Hence, it follows from (3.1) and (3.2) thaṫ u ν 0 and σ ν 0 on Γ 3 . Thus, the conditions (3.3) and (3.4) imply 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 a frictional bilateral contact with wear may be stated as follows. Div ) Here, equation (3.6) is the thermo-elastic-viscoplastic constitutive law with damage introduced in the first section. Equation (3.7) represents is the steady equation for the stress field. Equation (3.8) represents the energy conservation where ψ is a nonlinear constitutive function which represents the heat generated by the work of internal forces and q is a given volume heat source. Inclusion (3.9) describes the evolution of damage field, governed by the source damage function φ, where ∂ K ϕ(ς) is the subdifferential of indicator function of the set K of admissible damage functions.
Equalities (3.10) and (3.11) are the displacement-traction boundary conditions, respectively. (3.12) describes the frictional bilateral contact with wear described above on the potential contact surface Γ 3 . (3.13), (3.14) represent, respectively on Γ, a Fourier boundary condition for the temperature and a homogeneous Neumann boundary condition for the damage field on Γ. The functions u 0 , θ 0 and β 0 in (3.15) are the initial data.
In the study of the mechanical problem P, we consider the following assumptions.
(c) The mapping x → A(x, ε) is Lebesgue measurable on Ω, for any ε ∈ S d .
(c) The mapping x → S(x, 0, 0) belongs to L 2 (Ω). (3.20) In order to write a variational formulation of mechanical problem, we introduce the closed subspace of H 1 (Ω) d defined by Since meas(Γ 1 ) > 0, Korn's inequality holds and there exists a constant C k > 0, depending only on Ω and Γ 1 , such that A proof of Korn's inequality may be found in [20, p. 79]. On the space V we consider the inner product and the associated norm given by It follows that the norms · H 1 (Ω) d and · V are equivalent on V and, therefore, the space (V, (·, ·) V ) is a real Hilbert space. Moreover, by the Sobolev trace theorem and (3.22), there exists a constant C 0 > 0, depending only on Ω, Γ 1 and Γ 3 , such that

23)
A frictional contact problem with wear 9 The body forces, surface tractions, volume heat source and the functions α and µ, satisfy We denote by f(t) ∈ V the following element The use of (3.24) permits to verify that f ∈ C(0, T; V).

(3.30)
We introduce the following bilinear forms a i : Finally, we consider the wear functional j : (3.33) Using the above notation and Green's formula, we derive the following variational formulation of mechanical problem P.

Main results
The main results are stated by the following theorems. Theorem 4.1. Assume that (3.16)-(3.28) hold and, in addition, the smallness assumption , such that m A is defined in (3.16) and C 0 defined by (3.23). Then there exists a unique solution {u, σ, θ, β} to problem P V. Moreover, the solution has the regularity

Remark 4.2.
We remark that if v * is large enough then α = 1/(kv * ) is sufficiently small and therefore, the condition (4.1) for the unique solvability of Problem P V is satisfied. We conclude that the mechanical problem (3.6)-(3.15) has a unique weak solution if the tangential velocity of the foundation is large enough. Moreover, having solved the problem (3.6)-(3.15), we find that there exists a unique solution w ∈ C 1 (0, T; L 2 (Γ 3 )), for the auxiliary problem (3.2) and using the initial condition w(0) = 0 which means that at the initial moment the body is not subject to any prior wear. Moreover, by using the Cauchy-Lipschitz theorem, we find that there exists a unique solution w ∈ C 1 (0, T; L 2 (Γ 3 )), for an auxiliary problem satisfying (3.2) and w(0) = 0. The proof of Theorem 4.1 is carried out in several steps that we prove in what follows. Everywhere in this section we suppose that assumptions of Theorem 4.1 hold, and we consider that C is a generic positive constant which depends on Ω, Γ 1 and Γ 3 and may change from place to place. The proof is based on arguments of elliptic variational inequalities, classical and uniqueness results on parabolic inequalities and fixed point arguments.

First step
Let η ∈ C(0, T; H) and g ∈C(0, T; V) we consider the following variational problem.
Proof. Let g 1 , g 2 ∈ C(0, T; V) and let η ∈ C(0, T; H). We use the notation v η,g i = v i and σ η,g i = σ i for i = 1, 2. Using similar arguments as those in (4.12), we find From (4.14) and (4.15) we find that where α 0 is a positive constant which depends on Ω, Γ 1 , Γ 3 , and on the operator A. If (4.1) is satisfied we deduce from (4.16) that the operator Λ η is a contraction. From Banach's fixed point theorem we conclude that the operator Λ η has a unique fixed point g * η ∈ C(0, T; V).

Second step
For χ ∈ C(0, T; V ), we consider the following variational problem.  Proof. By an application of the Friedrichs-Poincaré inequality, we can find a constant B > 0 such that Thus, we obtain where c 1 = k 0 min(1, B )/2, which implies that a 0 is V-elliptic. Consequently, based on classical arguments of functional analysis concerning parabolic equations, the variational equation (4.18) has a unique solution θ χ satisfying (4.4).
We apply Theorem 2.1 to problem P V φ . Proof. The inclusion mapping of (H 1 (Ω) , . H 1 (Ω) ) into (L 2 (Ω) , . L 2 (Ω) ) is continuous and its range is dense. We denote by H 1 (Ω) the dual space of H 1 (Ω) and, identifying the dual of L 2 (Ω) with itself, we can write the Gelfand triple We use the notation (·, ·) (H 1 (Ω)) ×H 1 (Ω) to represent the duality pairing between H 1 (Ω) and H 1 (Ω). We have and we note that K is a closed convex set in H 1 (Ω). Then, using the definition (3.32) of the bilinear form a 1 , and the fact that β φ ∈ K in (3.25), it is easy to see that Lemma 4.6 is a consequence of Theorem 2.1.
By taking into account the above results and the properties of the operators B and G and of the functions ψ and S, we may consider the operator Λ : C(0, T; H × V × L 2 (Ω)) → C(0, T; H × V × L 2 (Ω)), defined by We have the following result.
Thus, for m sufficiently large, Λ m is a contraction on C(0, T; H × V × L 2 (Ω)), and so Λ has a unique fixed point in this Banach space. Now, we have all the ingredients to prove Theorem 4.1.

Uniqueness
The uniqueness of the solution is a consequence of the uniqueness of the fixed point of the operator Λ defined by (4.24)-(4.27) and the unique solvability of the Problem P V η,g , P V χ and P V φ which completes the proof.