A dynamic contact problem between elasto-viscoplastic piezoelectric bodies

We consider a dynamic contact problem with adhesion between two elasticviscoplastic piezoelectric bodies. The contact is frictionless and is described with the normal compliance condition. We derive variational formulation for the model which is in the form of a system involving the displacement field, the electric potential field and the adhesion field. We prove the existence of a unique weak solution to the problem. The proof is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed point arguments.


Introduction
The adhesive contact between deformable bodies, when a glue is added to prevent relative motion of the surfaces, has received recently increased attention in the mathematical literature.Analysis of models for adhesive contact can be found in [7,15,17] and recently in the monographs [18,19].The novelty in all these papers is the introduction of a surface internal variable, the bonding field, denoted in this paper by β, which describes the pointwise fractional density of adhesion of active bonds on the contact surface, and some times referred to as the intensity of adhesion.Following [10], the bonding field satisfies the restriction 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.In this paper we deal with the study of a dynamic frictionless contact problem with adhesion between two where D represents the electric displacement field, u the displacement field, σ and ε(u ) represent the stress and the linearized strain tensor, respectively.Here A is a given nonlinear function, F is the relaxation tensor, and G represents the elasticity operator.E(ϕ ) = −∇ϕ is the electric field, E = (e ijk ) represents the third order piezoelectric tensor, (E ) * is its transposition.In (1.1) and everywhere in this paper the dot above a variable represents derivative with respect to the time variable t.It follows from (1.1) that at each time moment, the stress tensor σ (t) is split into three parts: σ (t) = σ V (t) + σ E (t) + σ R (t), where σ V (t) = A ε( u (t)) represents the purely viscous part of the stress, σ E (t) = (E ) * ∇ϕ (t) represents the electric part of the stress and σ R (t) satisfies a rate-type elastic-viscoplastic relation Various results, examples and mechanical interpretations in the study of elastic-viscoplastic materials of the form (1.3) can be found in [8,11] and references therein.Note also that when F = 0 the constitutive law (1.1)becomes the Kelvin-Voigt electro-viscoelastic constitutive relation, σ (t) = A ε( u (t)) + G ε(u (t)) + (E ) * ∇ϕ (t). (1.4) Dynamic contact problems with Kelvin-Voigt materials of the form (1.4) can be found in [3,26].The normal compliance contact condition was first considered in [14] in the study of dynamic problems with linearly elastic and viscoelastic materials and then it was used in various references, see e.g.[13,20].This condition allows the interpenetration of the body's surface into the obstacle and it was justified by considering the interpenetration and deformation of surface asperities.
In this paper we consider a mathematical frictionless contact problem between two electroelastic-viscoplastics bodies for rate-type materials of the form (1.1).The contact is modelled with normal compliance and adhesion.The paper is organized as follows.In Section 2 we describe the mathematical models for the frictionless contact problem between two electroelastic-viscoplastics bodies.The contact is modelled with normal compliance and adhesion.In Section 3 we list the assumption on the data and derive the variational formulation of the problem.In Section 4 we state our main existence and uniqueness result, Theorem 4.1.The proof of the theorem is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities and fixed-point arguments.
disjoint measurable parts Γ 1 , Γ 2 and Γ 3 , on one hand, and on two 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 Ω body is submitted to f 0 forces and volume electric charges of density q 0 .The bodies are assumed to be clamped on Γ 1 × (0, T).The surface tractions f 2 act on Γ 2 × (0, T).We also assume that the electrical potential vanishes on Γ a × (0, T) and a surface electric charge of density q 2 is prescribed on Γ b × (0, T).The two bodies can enter in contact along the common part Γ 1 3 = Γ 2 3 = Γ 3 .The bodies are in adhesive contact with an obstacle, over the contact surface Γ 3 .With these assumptions, the classical formulation of the mechanical frictionless contact problem with adhesion between two electro-elastic-viscoplastic bodies is the following. ) ) ) ) ) ) First, equations (2.1) and (2.2) represent the electro-elastic-viscoplastic constitutive law of the material in which ε(u ) denotes the linearized strain tensor, E(ϕ ) = −∇ϕ is the electric field, where ϕ is the electric potential, A and G are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively.F is a nonlinear constitutive function describing the viscoplastic behaviour of the material.E represents the piezoelectric operator, (E ) * is its transpose, B denotes the electric permittivity operator, and D = (D 1 , . . ., D d ) is the electric displacement vector.Equations (2.3) and (2.4) are the equilibrium equations for the stress and electric-displacement fields, respectively, in which "Div"and "div"denote the T. Hadj ammar, B. Benyattou and S. Drabla divergence operator for tensor and vector valued functions, respectively.Next, the equations (2.5) and (2.6) represent the displacement and traction boundary condition, respectively.Condition (2.7) represents the normal compliance conditions with adhesion where γ ν is a given adhesion coefficient and [u ν ] = u 1 ν + u 2 ν stands for the displacements in normal direction.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 as long as it does not exceed the bond length L. The maximal tensile traction is γ ν β 2 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 introduction of the operator R ν , together with the operator R τ defined below, is motivated by mathematical arguments but it is not restrictive from the physical point of view, since no restriction on the size of the parameter L is made in what follows.Condition (2.8) represents the adhesive contact condition on the tangential plane, where stands for the jump of the displacements in tangential direction.R τ is the truncation operator given by This condition shows that the shear on the contact surface depends on the bonding field and on the tangential displacement, but as long as it does not exceed the bond length L. The frictional tangential traction is assumed to be much smaller than the adhesive one and, therefore, omitted.
Next, the equation (2.9) represents the ordinary differential equation which describes the evolution of the bonding field and it was already used in [6], see also [22,23] for more details.Here, besides γ ν , two new adhesion coefficients are involved, γ τ and ε a .Notice that in this model once debonding occurs bonding cannot be reestablished since, as it follows from (2.9), β ≤ 0. Equation (2.12) represents the initial displacement field and the initial velocity.Finally, (2.13) represents the initial condition in which β 0 is the given initial bonding field, (2.10) and (2.11) represent the electric boundary conditions.

Variational formulation and preliminaries
In this section, we list the assumptions on the data and derive a variational formulation for the contact problem.To this end, we need to introduce some notation and preliminary material.Here and below, S d represent the space of second-order symmetric tensors on R d .We recall that the inner products and the corresponding norms on S d and R d are given by Here and below, the indices i and j run between 1 and d and the summation convention over repeated indices is adopted.Now, to proceed with the variational formulation, we need the following function spaces: The spaces H , H 1 , H and H 1 are real Hilbert spaces endowed with the canonical inner products given by and the associated norms • H , and • H 1 respectively.Here and below we use the notation For every element v ∈ H 1 , we also use the notation v for the trace of v on Γ and we denote by v ν and v τ the normal and the tangential components of v on the boundary Γ given by 2 ,Γ denote the duality pairing between H Γ and H Γ .For every element σ ∈ H 1 let σ ν be the element of H Γ given by Denote by σ ν and σ τ the normal and the tangential traces of σ ∈ H 1 , respectively.If σ is continuously differentiable on Ω ∪ Γ , then where da is the surface measure element.
To obtain the variational formulation of the problem (2.1)-(2.13),we introduce for the bonding field the set and for the displacement field we need the closed subspace of H 1 defined by Since meas Γ 1 > 0, the following Korn's inequality holds: where the constant c K denotes a positive constant which may depend only on Ω , Γ 1 (see [18]).
Over the space V we consider the inner product given by and let • V be the associated norm.It follows from Korn's inequality (3.1) that the norms • H 1 and • V are equivalent on V .Then (V , • V ) is a real Hilbert space.Moreover, by the Sobolev trace theorem and (3.2), there exists a constant c 0 > 0, depending only on Ω , Γ 1 and We also introduce the spaces Since meas Γ a > 0, the following Friedrichs-Poincaré inequality holds: where c F > 0 is a constant which depends only on Ω , Γ a .Over the space W , we consider the inner product given by (ϕ , ψ ) W = Ω ∇ϕ .∇ψdx and let • W be the associated norm.It follows from (3.4) that • H 1 (Ω ) and • W are equivalent norms on W and therefore (W , • W ) is a real Hilbert space.Moreover, by the Sobolev trace theorem, there exists a constant c 0 , depending only on Ω , Γ a and Γ 3 , such that The space W is real Hilbert space with the inner product where div D = (D i,i ), and the associated norm • W .
In order to simplify the notations, we define the product spaces (3.6) The spaces V , W and W are real Hilbert spaces endowed with the canonical inner products denoted by (•, •) V , (•, •) W , and (•, •) W .The associate norms will be denoted by • V , • W , and • W , respectively.Finally, for any real Hilbert space X, we use the classical notation for the spaces L p (0, T; X), W k,p (0, T; X), where 1 ≤ p ≤ ∞, k ≥ 1.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, for a real number r, we use r + to represent its positive part, that is r + = max{0, r}.For the convenience of the reader, we recall the following version of the classical theorem of Cauchy-Lipschitz (see, [23, p. 48]).Theorem 3.1.Assume that (X, • X ) is a real Banach space and T > 0. Let F(t, •) : X → X be an operator defined a.e. on (0, T) satisfying the following conditions: 1.There exists a constant L F > 0 such that 2. There exists p ≥ 1 such that t → F(t, x) ∈ L p (0, T; X) ∀x ∈ X.
Then for any x 0 ∈ X, there exists a unique function x ∈ W 1,p (0, T; X) such that Theorem 3.1 will be used in Section 4 to prove the unique solvability of the intermediate problem involving the bonding field.
In the study of the Problem P, we consider the following assumptions: we assume that the viscosity operator A : Ω × S d → S d satisfies: The elasticity operator G : Ω × S d → S d satisfies:  The piezoelectric tensor (3.10) Recall also that the transposed operator (E ) * is given by (E ) * = (e , * ijk ) where e , * ijk = e kij and the following equality holds The normal compliance functions p ν : for all r ≤ 0, a.e.x ∈ Γ 3 . (3.12) The tangential compliance functions p τ : We suppose that the mass density satisfies ρ ∈ L ∞ (Ω ) and ∃ρ 0 > 0 such that ρ (x) ≥ ρ 0 a.e.x ∈ Ω , = 1, 2. (3.14) The following regularity is assumed on the density of volume forces, traction, volume electric charges and surface electric charges: The adhesion coefficients γ ν , γ τ and ε a satisfy the conditions and, finally, the initial data satisfy We will use a modified inner product on H, given by Using the notation (•, •) V ×V to represent the duality pairing between V and V we have Finally, we denote by • V the norm on V .Using the Riesz representation theorem, we define the linear mappings f : [0, T] → V and q : [0, T] → W as follows: Next, we denote by j ad : L ∞ (Γ 3 ) × V × V → R the adhesion functional defined by In addition to the functional (3.21), we need the normal compliance functional Keeping in mind (3.12)-(3.13),we observe that the integrals (3.21) and (3.22) are well defined and we note that conditions (3.15) imply f ∈ L 2 (0, T; V ), q ∈ C(0, T; W).
By a standard procedure based on Green's formula, we derive the following variational formulation of the mechanical (2.1)-(2.13).
Problem PV.Find a displacement field u : [0, T] → V , a stress field σ : [0, T] → H, an electric potential field ϕ : [0, T] → W, a bonding field β : [0, T] → L ∞ (Γ 3 ) and a electric displacement field D : [0, T] → W such that We notice that the variational Problem PV is formulated in terms of a displacement field, a stress field, an electrical potential field, a bonding field and a electric displacement field.The existence of the unique solution to Problem PV is stated and proved in the next section.
Below in this section β, β 1 , β 2 denote elements of L 2 (Γ 3 ) such that 0 ≤ β, β 1 , β 2 ≤ 1 a.e.x ∈ Γ 3 , u 1 , u 2 and v represent elements of V and C > 0 represents generic constants which may depend on Ω , Γ 3 , p ν , p τ , γ ν , γ τ and L. First, we note that the functional j ad and j νc are linear with respect to the last argument and, therefore, (3.30) Next, using (3.21), the properties of the truncation operators R ν and R τ as well as assumption (3.13) on the function p τ , after some calculus we find and, by (3.20), we obtain Similar computations, based on the Lipschitz continuity of R ν , R τ and p τ show that the following inequality also holds: We take now Also, we take u 1 = v and u 2 = 0 in (3.32) then we use the equalities R ν (0) = 0, R τ (0) = 0 and (3.30) to obtain Now, we use (3.22) to see that and therefore (3.12.b) and (3.3) imply We use again (3.22) to see that and therefore (3.12.c) implies We take u 1 = v and u 2 = 0 in the previous in equality and use (3.22) and (3.36) to obtain

Existence and uniqueness result
Now, we propose our existence and uniqueness result.
ϕ ∈ C(0, T; W), (4.2) 3) The functions u, ϕ, σ, D and β which satisfy (3.24)-(3.29)are called a weak solution to the contact Problem P. We conclude that, under the assumptions (3.7)- (3.18), the mechanical problem (2.1)-(2.13)has a unique weak solution satisfying (4.1)-(4.3).The regularity of the weak solution is given by (4.1)-( 4.3) and, in term of stresses, Indeed, it follows from (3.26) and (3.27) that ρ ü = Div σ (t) + f 0 (t), div D (t) −q 0 (t) = 0 for all t ∈ [0, T] and therefore the regularity 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 , Γ 3 , p ν , p τ , γ ν , γ τ and L and may change from place to place.Let η ∈ L 2 (0, T; V ) be given.In the first step we consider the following variational problem.
To solve Problem PV u η , we apply an abstract existence and uniqueness result which we recall now, for the convenience of the reader.Let V and H denote real Hilbert spaces such that V is dense in H and the inclusion map is continuous, H is identified with its dual and with a subspace of the dual V of V , i.e.V ⊂ H ⊂ V , and we say that the inclusions above define a Gelfand triple.The notations • V , • V and (•, •) V ×V represent the norms on V and on V and the duality pairing between V them, respectively.The following abstract result may be found in [23, p. 48].Theorem 4.2.Let V , H be as above, and let A : V → V be a hemicontinuous and monotone operator which satisfies ) for some constants w > 0, C > 0 and λ ∈ R.Then, given u 0 ∈ H and f ∈ L 2 (0, T; V ), there exists a unique function u which satisfies We have the following result for the problem.

Lemma 4.3.
There exists a unique solution to Problem PV u η and it has its regularity expressed in (4.1).

Proof. We define the operator
Using (4.10), (3.2) and (3.7) it follows that and keeping in mind the Krasnoselski theorem (see [12, p. 60]), we deduce that A : V → V is a continuous operator.Now, by (4.10), (3.2) and (3.7), we find where the positive constant which implies that A satisfies condition (4.8) with ω = m 2 and λ = − 1 2m Ao 2 V .Moreover, by (4.10) and (3.7) we find where . This inequality and (3.2) imply that A satisfies condition (4.9).Finally, we recall that by (3.15) Let u η : [0, T] → V be the function defined by It follows from (4.10) and (4.12)-(4.15) that u η is a unique solution to the variational problem PV u η and it satisfies the regularity expressed in (4.1).
In the second step we use the displacement field u η obtained in Lemma 4.3 to construct the following Cauchy problem for the stress field.

Problem PV σ
η .Find a stress field ) In the study of Problem PV σ η we have the following result.
Consider now η 1 , η 2 ∈ L 2 (0, T; V ) and, for i = 1, 2, denote u η i = u i , σ η i = σ i .We have and, using the properties (3.8) and (3.9) of F , and G we find Using now a Gronwall argument in the previous inequality we deduce (4.17), which concludes the proof.
In the third step, let η ∈ L 2 (0, T; V ), we use the displacement field u η obtained in Lemma 4.3 and we consider the following variational problem.
We have the following result.Proof.We define a bilinear form: We use (4.20), (3.4) and (3.11) to show that the bilinear form b(•, •) is continuous, symmetric and coercive on W.Moreover, using the Riesz representation theorem we may define an element q η : [0, T] → W such that We apply the Lax-Milgram theorem to deduce that there exists a unique element and the previous inequality, the regularity of u η and q imply that ϕ η ∈ C(0, T; W). )

.23)
We have the following result.
Lemma 4.6.There exists a unique solution Proof.For the simplicity we suppress the dependence of various functions on Γ 3 , and note that the equalities and inequalities below are valid a.e. on Γ 3 .Consider the mapping for all t ∈ [0, T] and β ∈ L 2 (Γ 3 ).It follows from the properties of the truncation operator R ν and R τ that F η is Lipschitz continuous with respect to the second variable, uniformly in time.

T. Hadj ammar, B. Benyattou and S. Drabla
Reiterating this inequality n times we obtain   Uniqueness.The uniqueness of the solution is a consequence of the uniqueness of the fixed point of the operator Λ defined by (4.25).