ANALYSIS OF A DYNAMIC THERMO-ELASTIC-VISCOPLASTIC CONTACT PROBLEM

We consider a dynamic frictionless contact problem for thermo-elastic-viscoplastic materials with damage and adhesion. The contact is modeled with normal compliance condition. We derive a weak formulation of the system, then we prove existence and uniqueness of the solution. The proof is based on arguments of monotonicity and fixed point.


Introduction
Situations of contact between deformable bodies are very common in the industry and everyday life.Contact of braking pads with wheels, tires with roads, pistons with skirts or the complex metal forming processes are just a few examples.The constitutive laws with internal variables has been used in various publications in order to model the effect of internal variables in the behavior of real bodies like metal and rocks polymers.Some of the internal state variables considered by many authors are the spatial display of dislocation, the work-hardening of materials, the absolute temperature and the damage field.See for examples [6,26,27,28,29,35,36] for the case of hardening, temperature and other internal state variables and the references [18,20,27] for the case of damage field and the adhesion field which is denoted in this paper by β.It describes the pointwise fractional density of active bonds on the contact surface, and sometimes referred to as the intensity of adhesion.Following [15,16], the bonding field satisfies the restrictions 0 ≤ β ≤ 1.When β = 1 at a point of the contact surface, the adhesion is complete and all the bonds are active.When β = 0 all the bonds are inactive, severed, and there is no adhesion.When 0 < β < 1 the adhesion is partial and only a fraction β of the bonds is active.We refer the reader to the extensive bibliography on the subject in [31,33,34].
In this paper we deal with the study of a dynamic problem of frictionless adhesive contact for general thermo-elastic-viscoplastic materials.For this, we consider a rate-type constitutive equation with two internal variables of the form in which u, σ represent, respectively, the displacement field and the stress field where the dot above denotes the derivative with respect to the time variable, θ represents the absolute temperature, ς is the damage field, A and E are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively, and G is a nonlinear constitutive function which describes the visco-plastic behavior of the material.It follows from (1.1) that at each time moment, the stress tensor σ(t) is split into two parts: σ(t) = σ V (t) + σ R (t), where σ V (t) = A(ε( u(t))) represents the purely viscous part of the stress, whereas σ R (t) satisfies a rate-type elastic-viscoplastic relation with absolute temperature and damage When G = 0 in (1.1) reduces to the Kelvin-Voigt viscoelastic constitutive law given by The damage is an extremely important topic in engineering, since it affects directly the useful life of the designed structure or component.There exists a very large engineering literature on it.Models taking into account the influence of the internal damage of the material on the contact process have been investigated mathematically.General models for damage were derived in [17,18] from the virtual power principle.Mathematical analysis of one-dimensional problems can be found in [19].In all these papers the damage of the material is described with a damage function ς, restricted to have values between zero and one.When ς = 1 there is no damage in the material, when ς = 0 the material is completely damaged, when 0 < ς < 1 there is partial damage and the system has a reduced load carrying capacity.In this paper the inclusion used for the evolution of the damage field is where K denotes the set of admissible damage functions defined by represents the subdifferential of the indicator function of the set K and φ is a given constitutive function which describes the sources of the damage in the system.Examples and mechanical interpretation of elastic-viscoplastic can be found in [12,21].Dynamic and quasistatic contact problems are the topic of numerous papers, e.g.[1,2,4,11,14,32].More recently in [5], we study an electro-elastic-visco-plastic frictionless contact problem with damage and adhesion.The mathematical problem modelled the quasi-static evolution of damage in thermo-viscoplastic materials has been studied in [27].
We model the material's behavior with an elastic-viscoplastic constitutive law with damage.We derive a variational formulation of the problem and prove the existence of a unique weak solution.The paper is organized as follows.In Section 2 we present the mechanical problem of the dynamic evolution of damage and adhesion in thermo-elastic-viscoplastic materials.We introduce some notations and preliminaries and we derive the variational formulation of the problem.We prove in Section 3 the existence and uniqueness of the solution.

Statement of the Problem
Let Ω ⊂ R n (n = 2, 3) be a bounded domain with a Lipschitz boundary Γ, partitioned into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 such that meas(Γ 1 ) > 0. We denote by S n the space of symmetric tensors on R n .We define the inner product and the Euclidean norm on R n and S n , respectively, by Here and below, the indices i and j run from 1 to n and the summation convention over repeated indices is used.We shall use the notation Here ε : H 1 → H and Div : H 1 → H are the deformation and divergence operators, respectively, defined by The sets H, H, H 1 , H 1 and V are real Hilbert spaces endowed with the canonical inner products: The associated norms are denoted by • H , • H , • H1 , • H1 and • V .Since the boundary Γ is Lipschitz continuous, the unit outward normal vector field ν on the boundary is defined a.e.For every vector field v ∈ H 1 we denote by v ν and v τ the normal and tangential components of v on the boundary given by Let H Γ = (H 1/2 (Γ)) n and γ : H 1 → H Γ be the trace map.We denote by V the closed subspace of H 1 defined by We also denote by H Γ the dual of H Γ .Moreover, since meas(Γ 1 ) > 0, Korn's inequality holds and thus, there exists a positive constant C 0 depending only on Ω, Γ 1 such that On the space V we consider the inner product given by and let • V be the associated norm, defined by It follows from Korn's inequality that • H1 and is a real Hilbert space.Moreover, by the Sobolev trace theorem there exists a positive constant C 0 which depends only on Ω, Γ 1 and Γ 3 such that Furthermore, if σ ∈ H 1 there exists an element σν ∈ H Γ such that the following Green formula holds EJQTDE, 2013 No. 71, p. 3 In addition, if σ is sufficiently regular (say C 1 ), then where dΓ denotes the surface element.Similarly, for a regular tensor field σ : Ω → S n we define its normal and tangential components on the boundary by Moreover, we denote by V and V the dual of the spaces V and V , respectively.Identifying H, respectively L 2 (Ω), with its own dual, we have the inclusions We use the notation •, • V ×V , •, • V ×V to represent the duality pairing between V , V and V , V , respectively.Let T > 0. For every real space X, we use the notation C(0, T ; X), and C 1 (0, T ; X) for the space of continuous an continuously differentiable functions from [0, T ] to X respectively, C(0, T ; X) is a real Banach space with the norm While C 1 (0, T ; X) is a real Banach space with the norm Finally, for k ∈ N and p ∈ [1, ∞], we use the standard notation for the Lebesgue space L p (0, T ; X) and for the Sobolev spaces W k,p (0, T ; X).Moreover, for a real number r, we use r + to represent its positive part that is r + = max(0, r), and if X 1 and X 2 are real Hilbert spaces, than X 1 × X 2 denotes the product Hilbert space endowed with the canonical inner product (•, •) X1×X2 .
The physical setting is the following.A body occupies the domain Ω, and is clamped on Γ 1 and so the displacement field vanishes there.Surface tractions of density f 0 act on Γ 2 × (0, T ) and a volume force of density f is applied in Ω × (0, T ).We assume that the body is in adhesive frictionless contact with an obstacle, the so-called foundation, over the potential contact surface Γ 3 .We admit a possible external heat source applied in Ω × (0, T ), given by the function q.Moreover, the process is dynamic, and thus the inertial terms are included in the equation of motion.We use an elastic-viscoplastic constitutive law with damage to model the material's behaviour and an ordinary differential equation to describe the evolution of the adhesion field.The mechanical formulation of the frictionless problem with normal compliance is as follow. ) ) ) EJQTDE, 2013 No. 71, p. 4 (2.10) ) ) This problem represents the dynamic evolution of damage and adhesion in thermo-elastic-viscoplastic materials.Equation (2.4) is the thermo-elastic-viscoplastic constitutive law where A and E are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively, and G is a nonlinear constitutive function which describes the viscoplastic behavior of the material.(2.5) represents the equation of motion in which the dot above denotes the derivative with respect to the time variable and ρ is the density of mass.Equation (2.6) 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 (2.7) describes the evolution of damage field.Equalities (2.8) and (2.9) are the displacement-traction boundary conditions, respectively.Condition (2.10) represents the normal compliance condition with adhesion where γ ν is a given adhesion coefficient and p ν is a given positive function which will be described below.In this condition the interpenetrability between the body and the foundation is allowed, that is u ν can be positive on Γ 3 .The contribution of the adhesive to the normal traction is represented by the term γ ν β 2 R ν (u ν ) the adhesive traction is tensile and is proportional, with proportionality coefficient γ ν , to the square of the intensity of adhesion and to the normal displacement, but only as long as it does not exceed the bond length L. The maximal tensile traction is γ ν L. R ν is the truncation operator defined by Here L > 0 is the characteristic length of the bond, beyond which it does not offer any additional traction.The contact condition (2.10) was used in various papers, see e.g.[9,10,34,37].Condition (2.11) represents the adhesive contact condition on the tangential plane, in which p τ is a given function and R τ is the truncation operator given by This condition shows that the shear on the contact surface depends on the adhesion field and on the tangential displacement, but only as long as it does not exceed the adhesion length L. The frictional tangential traction is assumed to be much smaller than the adhesive one, and therefore omitted.
The introduction of the operator R ν , together with the operator R τ defined above, is motivated by mathematical arguments but it is not restrictive for physical point of view, since no restriction on the size of the parameter L is made in what follows.
Next, equation (2.12) represents the ordinary differential equation which describes the evolution of the adhesion field and it was already used in [9,34], see also [33] for more details.Here, besides γ ν , two a.e. on Γ 3 }.

Main Results
The existence of the unique solution to Problem PV is proved in the next section.To this end, we consider the following remark which is used in different places of the paper.Remark 3.1.We note that, in Problem P and in Problem PV, we do not need to impose explicitly the restriction 0 ≤ β ≤ 1.Indeed, (2.39) guarantees that β(x, t) ≤ β 0 (x) and, therefore, assumption (2.28) shows that β(x, t) ≤ 1 for t ≥ 0, a.e.x ∈ Γ 3 .On the other hand, if β(x, t 0 ) = 0 at time t 0 , then it follows from (2.39) that β(x, t) = β 0 (x) for all t ≥ t 0 , and therefore β(x, t) = 0 for all t ≥ t 0 , x ∈ Γ 3 .We conclude that 0 ≤ β(x, t) ≤ 1 for all t ≥ t 0 , x ∈ Γ 3 .Theorem 3.2 (Existence and uniqueness).Under assumptions (2.17)-(2.29),there exists a unique solution {u, σ, θ, ς, β} to problem PV.Moreover, the solution has the regularity ü ∈ L 2 (0, T ; V ), EJQTDE, 2013 No. 71, p. 8 σ ∈ L 2 (0, T ; H), ς ∈ L 2 (0, T ; V ), (3.8) A quintuple (u, σ, θ, ς, β) which satisfies (2.35)-(2.40) is called a weak solution to the compliance contact Problem P. We conclude that under the stated assumptions, problem (2.4)-(2.16)has a unique weak solution satisfying (3.1)-(3.9).We turn now to the proof of Theorem 3.2, which will be carried out in several steps and is based on arguments of nonlinear equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities and fixed-point arguments.To this end, we assume in the following that (2.17)-(2.29)hold.Below, C denotes a generic positive constant which may depend on Ω, Γ 1 , Γ 2 , Γ 3 , A, E, G, ψ, φ, p ν , p τ , γ ν , γ τ , L and T but does not depend on t nor on the rest of input data, and whose value may change from place to place.Moreover, for the sake of simplicity we suppress in what follows the explicit dependence of various functions on x ∈ Ω ∪ Γ.Let η ∈ L 2 (0, T ; V ) be given.In the first step we consider the following variational problem.

Proof. Let us introduce the operator
Therefore, (3.10) can be rewritten as follows where It follows from (2.1), (3.12) and hypothesis (2.17) that A is bounded, semi-continuous and coercive on V. We recall that by (2.31) we have F η ∈ L 2 (0, T ; V ).Then by using classical arguments of functional analysis concerning parabolic equations [8,24] we can easily prove the existence and uniqueness of w η satisfying In the second step we use the displacement field u η obtained in Lemma 3.3 and we consider the following initial value problem.
Lemma 3.4.There exists a unique solution Proof.We use a version of the classical Cauchy-Lipschitz theorem given in [38, p. 60].Proof.By an application of the Poincaré-Friedrichs inequality, we can find a constant α > 0 such that Thus, we obtain where C 1 = k 0 min(1, α )/2, which implies that a 0 is V -elliptic.Consequently, based on classical arguments of functional analysis concerning parabolic equations, the variational equation (3.21) has a unique solution θ λ satisfies (3.5) and (3.6).
Problem PV µ .Find the damage field ς Proof.We know that the form a 1 is not V -elliptic.To solve this problem we introduce the functions ςµ (t) = e −k1t ς µ (t), ξ(t) = e −k1t ξ(t).