A dynamic contact problem between elasto-viscoplastic piezoelectric bodies

(1)

A dynamic contact problem between elasto-viscoplastic piezoelectric bodies

Tedjani Hadj ammar

^B¹

, Benabderrahmane Benyattou

²

and Salah Drabla

³

1Department of Mathematics, University of El-Oued, El-Oued 39000, Algeria

2Laboratory of Mathematics and Computer Sciences, University of Laghouat, Laghouat 03000, Algeria

3Department of Mathematics, University of Setif 1, Setif 19000, Algeria

Received 13 May 2014, appeared 13 October 2014 Communicated by Michal Feˇckan

Abstract. We consider a dynamic contact problem with adhesion between two elastic- viscoplastic 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.

Keywords: elastic-viscoplastic piezoelectric materials, normal compliance, adhesion, evolution equations, fixed point.

2010 Mathematics Subject Classification: 74M15, 74H20, 74H25.

1 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 litera- ture. 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 frac- tional 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 βof the bonds is active. In this paper we deal with the study of a dynamic frictionless contact problem with adhesion between two

BCorresponding author. Email: hadjammart@gmail.com

(2)

elastic-viscoplastic piezoelectric materials of the form σ^` =A^`ε(u˙^`) +G^`ε(u^`)+(E^`)^∗∇ϕ^`

+

Z _t

0

F^`σ^`(s)− A^`ε(u˙^`(s))−(E^`)^∗∇ϕ^`, ε(u^`(s))ds, (1.1) D^` =E^`ε(u^`)− B^`∇ϕ^`, (1.2) where D^` represents the electric displacement field, u^` the displacement field, σ^` andε(u^`) represent the stress and the linearized strain tensor, respectively. HereA^`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 trans- position. 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 tensorσ^`(t)is split into three parts: σ^`(t) =σ^`_V(t) +σ^`_E(t) +σ^`_R(t), whereσ_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 andσ^`_R(t)satisfies a rate-type elastic-viscoplastic relation

σ^`_R(t) =G^`ε(u^`(t)) +

Z _t

0

F^` σ^`_R(s),ε(u^`(s))ds. (1.3) 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 deforma- tion of surface asperities.

In this paper we consider a mathematical frictionless contact problem between two electro- elastic-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 electro- elastic-viscoplastics bodies. The contact is modelled with normal compliance and adhesion.

In Section3we list the assumption on the data and derive the variational formulation of the problem. In Section4 we state our main existence and uniqueness result, Theorem4.1. 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.

2 Problem statement

We consider the following physical setting. Let us consider two electro-elastic-viscoplastic bodies, occupying two bounded domainsΩ¹,Ω² of the spaceR^d (d= 2, 3). For each domain Ω^`, the boundary Γ^` is assumed to be Lipschitz continuous, and is partitioned into three

(3)

disjoint measurable parts Γ₁^`, Γ^`₂ and Γ^`₃, on one hand, and on two measurable parts Γ^`_a and Γ^`_b, on the other hand, such that measΓ^`₁ > 0, measΓ^`_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 Γ₁^`×(0,T). The surface tractions f^`₂ act on Γ^`₂×(0,T). We also assume that the electrical potential vanishes onΓ^`_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 Γ¹₃ = _Γ²₃ = _Γ₃. The bodies are in adhesive contact with an obstacle, over the contact surface Γ3. With these assumptions, the classical formulation of the mechanical frictionless contact problem with adhesion between two electro-elastic-viscoplastic bodies is the following.

Problem P. For ` = 1, 2, find a displacement field u^`: Ω^`×(0,T) −→ _R^d, a stress field σ^`: Ω^`×(_0,T) −→ _S^d, an electric potential field ϕ^`: Ω^`×(_0,T) −→ R, a bonding field β: Γ3×(0,T)−→_Rand a electric displacement fieldD^`: Ω^`×(0,T)−→_R^d such that

σ^` =A^`ε(u˙^`) +G^`ε(u^`)+(E^`)^∗∇ϕ^` +

Z _t

0

F^`σ^`(s)− A^`ε(u˙^`(s))−(E^`)^∗∇ϕ^`, ε(u^`(s))ds in Ω^`×(0,T), (2.1) D^` =E^`ε(u^`)− B^`∇ϕ^` inΩ^`×(0,T), (2.2) ρ^`u¨^`=Divσ^`+ f^`₀ in Ω^`×(0,T), (2.3) divD^`−q₀^` =₀ _in _Ω^`×(_0,T)_, _(2.4)

u^` =0 on Γ^`₁×(0,T), (2.5)

σ^`ν^` = f^`₂ on Γ^`₂×(0,T), (2.6) (

σ_ν¹ =σ_ν² ≡σν,

σν =−pν([uν]) +γνβ²Rν([uν]) ^on^Γ³×(0,T), (2.7) (

σ¹_τ = −σ²_τ ≡στ,

στ = p_τ(β)R_τ([u_τ]) ^on ^Γ³×(0,T), (2.8) β˙ =−β γν(Rν([uν]))²+γτ|Rτ([uτ])|²−εa

+ onΓ3×(0,T), (2.9)

ϕ^`=0 onΓ^`_a×(0,T), (2.10)

D^`.ν^`= q₂^` on Γ^`_b×(0,T), (2.11) u^`(₀) =u^`₀, u˙^`(₀) =v^`₀ inΩ^`, (2.12)

β(0) =β₀ onΓ3. (2.13)

First, equations (2.1) and (2.2) represent the electro-elastic-viscoplastic constitutive law of the material in whichε(u^`)denotes the linearized strain tensor,E(ϕ^`) =−∇ϕ^ìs the electric field, where ϕ^ìs the electric potential,A^àndG^` are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively. F^ìs 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, andD^` = (D₁^`, . . . ,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

(4)

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. Con- dition (2.7) represents the normal compliance conditions with adhesion where γν is a given adhesion coefficient and[u_ν] =u¹_ν+u²_νstands for the displacements in normal direction. The contribution of the adhesive to the normal traction is represented by the term γ_νβ²Rν([uν]), the adhesive traction is tensile and is proportional, with proportionality coefficientγν, to the square of the intensity of adhesion and to the normal displacement, but as long as it does not exceed the bond lengthL. The maximal tensile traction isγ_νβ²L. Rνis the truncation operator defined by

R_ν(s) =











L ifs <−L,

−s if −L≤s≤_0, 0 ifs >0.

Here L > 0 is the characteristic length of the bond, beyond which it does not offer any additional traction. The introduction of the operatorRν, together with the operatorRτ 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 parameterLis made in what follows. Condition (2.8) represents the adhesive contact condition on the tangential plane, where[uτ] = u¹_τ−u²_τ stands for the jump of the displacements in tangential direction. R_τ is the truncation operator given by

R_τ(v) =

(v if|v| ≤L, L_|^v_v_| if|v|>L.

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, besidesγ_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 β₀ is the given initial bonding field, (2.10) and (2.11) represent the electric boundary conditions.

3 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 onR^d. We recall that the inner products and the corresponding norms onS^d andR^dare given by

u^`.v^`= u^`_i.v^`_i, v^`

= (v^`.v^`)¹², ∀u^`,v^` ∈_R^d, σ^`.τ^` =σ_ij^`.τ_ij^`,

τ^`

= (τ^`.τ^`)¹², ∀σ^`,τ^` ∈_S^d.

Here and below, the indicesiandjrun between 1 anddand the summation convention over repeated indices is adopted. Now, to proceed with the variational formulation, we need the

(5)

following function spaces:

H^` ={v^` = (v^`_i); v^`_i ∈ L²(_Ω^`)}, H₁^` ={v^`= (v^`_i); v^`_i ∈ H¹(_Ω^`)}, H^`= {τ^` = (τ_ij^`); τ_ij^` =τ_ji^` ∈ L²(_Ω^`)},H^`₁={τ^` = (τ_ij^`)∈ H^`; divτ^` ∈ H^`}.

The spaces H^`, H₁^`, H^` and H^`₁ are real Hilbert spaces endowed with the canonical inner products given by

(u^`,v^`)_H` =

Z

Ω^`u^`.v^`dx, (u^`,v^`)_H`

1 =

Z

Ω^`u^`.v^`dx+

Z

Ω^`∇u^`.∇v^`dx, (σ^`,τ^`)_H` =

Z

Ω^`σ^`.τ^`dx, (σ^`,τ^`)_H`

1 =

Z

Ω^`σ^`.τ^`dx+

Z

Ω^`divσ^`. Divτ^`dx and the associated normsk · k_H`,k · k_H`

1,k · k_H`, andk · k_H`

1 respectively. Here and below we use the notation

∇u^` = (u^`_i,j), ε(u^`) = (ε_ij(u^`)), ε_ij(u^`) = ¹

2(u^`_i,j+u^`_j,i), ∀u^` ∈ H₁^`, Divσ^`= (σ_ij,j^` ), ∀σ^` ∈ H^`₁.

For every element v^` ∈ H₁^`, we also use the notation v^` for the trace of v^` on Γ^` and we denote byv^`_ν andv^`_τ thenormaland thetangentialcomponents ofv^` on the boundaryΓ^` given by

v^`_ν=v^`.ν^`, v^`_τ =v^`−v^`_νν^`. LetH_Γ⁰`be the dual ofH_Γ` = H¹²(_Γ^`)^d_{and let}(·_,·)₋1

2,¹₂,Γ^` denote the duality pairing between H_Γ⁰` andH_Γ^`. For every element σ^` ∈ H^`₁letσ^`ν^` be the element of H_Γ⁰` given by

(σ^`ν^`,v^`)₋1

2,¹₂,Γ^` = (σ^`,ε(v^`))_H`+ (Divσ^`,v^`)_H` ∀v^` ∈ H₁^`.

Denote byσ_ν^` andσ^`_τ thenormaland thetangentialtraces ofσ^` ∈ H₁^`, respectively. Ifσ^` is continuously differentiable onΩ^`∪_Γ^`, then

σ_ν^` = (σ^`ν^`).ν^`, σ^`_τ = σ^`ν^`−σ_ν^`ν^`, (σ^`ν^`,v^`)₋1

2,¹₂,Γ^` =

Z

Γ^`σ^`ν^`.v^`da fore allv^` ∈ H₁^`, wheredais the surface measure element.

To obtain the variational formulation of the problem (2.1)–(2.13), we introduce for the bonding field the set

Z =θ∈ L^∞ 0,T;L²(_Γ₃); 0≤θ(t)≤1 ∀t∈[0,T], a.e. onΓ3 , and for the displacement field we need the closed subspace of H₁^` defined by

V^` =ⁿv^` ∈ H₁^`; v^` =0 on Γ^`₁o . Since measΓ^`₁>0, the following Korn’s inequality holds:

kε(v^`)k_H` ≥ cKkv^`k_H`

1 ∀v^` ∈V^`, (3.1)

(6)

where the constantc_Kdenotes a positive constant which may depend only onΩ^`,Γ^`₁(see [18]).

Over the spaceV^` we consider the inner product given by

(u^`,v^`)_V` = (ε(u^`),ε(v^`))_H`, ∀u^`,v^` ∈V^`, (3.2) and letk · k_V` be the associated norm. It follows from Korn’s inequality (3.1) that the norms k · k_H`

1 andk · k_V` are equivalent onV^`. Then(V^`,k · k_V`)is a real Hilbert space. Moreover, by the Sobolev trace theorem and (3.2), there exists a constantc₀ > 0, depending only onΩ^`,Γ^`₁ andΓ3such that

kv^`k_L2(_Γ₃)^d ≤ c₀kv^`k_V` ∀v^`∈ V^`. (3.3) We also introduce the spaces

W^` =ⁿψ^`∈ H¹(_Ω^`); ψ^`=0 on Γ^`_ao ,

W^` =ⁿD^` = (D_i^`)_; D^`_i ∈ L²(_Ω^`)_{, div}D^` ∈ L²(_Ω^`)^o_. Since measΓ^`_a >0, the following Friedrichs–Poincaré inequality holds:

k∇ψ^`k_L2(_Ω^`)^d ≥c_Fkψ^`k_H1(_Ω^`) ∀ψ^` ∈W^`, (3.4) wherec_F >0 is a constant which depends only onΩ^`,Γ^`_a.

Over the spaceW^`, we consider the inner product given by (ϕ^`,ψ^`)_W` =

Z

Ω^`∇ϕ^`.∇ψ^`dx

and let k · k_W` be the associated norm. It follows from (3.4) that k · k_H1(_Ω^`)andk · k_W` are equivalent norms onW^` and therefore(W^`,k · k_W`)is a real Hilbert space. Moreover, by the Sobolev trace theorem, there exists a constantc0, depending only onΩ^`,Γ^`_aandΓ3, such that

kζ^`k_L2(_Ω^`) ≤c₀kζ^`k_W` ∀ζ^`∈W^`. (3.5) The spaceW^` is real Hilbert space with the inner product

(D^`,E^`)_W` =

Z

Ω^`D^`.E^`dx+

Z

Ω^`divD^`. divE^`dx, where divD^` = (D^`_i,i), and the associated normk · k_W`.

In order to simplify the notations, we define the product spaces

V =V¹×V², H= H¹×H², H₁= H¹₁×H²₁, H =H¹× H²,

H₁= H¹₁× H²₁, W =W¹×W², W =W¹× W². (3.6) The spacesV,W andW are real Hilbert spaces endowed with the canonical inner products denoted by (·,·)_V, (·,·)_W, and (·,·)_W. The associate norms will be denoted by k · k_V, k · k_W, andk · k_W, respectively.

Finally, for any real Hilbert spaceX, we use the classical notation for the spacesL^p(0,T;X), W^k,p(_0,T;X)_{, where 1} ≤ p ≤ _∞,k ≥ 1. We denote byC(_0,T;X) _andC¹(_0,T;X)the space of

(7)

continuous and continuously differentiable functions from [0,T] to X, respectively, with the norms

kfk_C₍_0,T;X₎= max

t∈[0,T]kf(t)k_X, kfk_C1(0,T;X)= max

t∈[0,T]kf(t)k_X+ max

t∈[0,T]kf^˙(t)k_X,

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,k · k_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 LF>0such that

kF(t,x)−F(t,y)k_X ≤ L_Fkx−yk_X ∀x,y∈ X, a.e. t ∈(0,T). 2. There exists p≥1such that t7→ F(t,x)∈L^p(0,T;X) ∀x∈ X.

Then for any x₀ ∈X,there exists a unique function x∈W^1,p(0,T;X)such that

˙

x(t) =F(t,x(t))_{, a.e.}t ∈(_0,T)_, x(0) =x₀.

Theorem3.1 will be used in Section4 to prove the unique solvability of the intermediate problem involving the bonding field.

In the study of the ProblemP, we consider the following assumptions:

we assume that theviscosity operatorA^`:Ω^`×_S^d→_S^d satisfies:











(a)There exists L_A^` >0 such that

|A^`(x,ξ₁)− A^`(x,ξ₂)| ≤ L_A^`|ξ₁−ξ₂|

∀_ξ₁,ξ₂∈_S^d, a.e.x∈ _Ω^`. (b)There existsm_A^` >0 such that

(A^`(x,ξ₁)− A^`(x,ξ₂))·(ξ₁−ξ₂)≥m_A^`|ξ₁−ξ₂|²

∀_ξ₁_,_ξ₂∈_S^d_{, a.e.}x∈ _Ω^`_.

(c) The mapping x7→ A^`(x,ξ)is Lebesgue measurable on Ω^`, for any ξ ∈_S^d.

(_d)The mapping x7→ A^`(x,0)is continuous on S^d, a.e.x∈_Ω^`_.

(3.7)

Theelasticity operator G^`: Ω^`×_S^d→_S^d satisfies:











(a)There existsL_G^` >0 such that

|G^`(x,ξ₁)− G^`(x,ξ₂)| ≤ L_G^`|ξ₁−ξ₂|

∀ξ₁,ξ₂∈_S^d, a.e.x ∈_Ω^`.

(b)The mappingx7→ G^`(x,ξ)is Lebesgue measurable onΩ^`, for anyξ ∈_S^d.

(c) The mappingx7→ G^`(x,0)belongs toH^`.

(3.8)

(8)

Theviscoplasticity operatorF^`: Ω^`×_S^d×_S^d→_S^d satisfies:











(a)There existsL_F^` >0 such that

|F^`(x,η₁,ξ₁)− F^`(x,η₂,ξ₂)| ≤L_F` |_η₁−_η₂|+|_ξ₁−_ξ₂|

∀η₁,η₂,ξ₁,ξ₂ ∈_S^d, a.e.x∈_Ω^`.

(b)The mappingx7→ F^`(x,η,ξ)is Lebesgue measurable on Ω^`, for anyη,ξ ∈_S^d_.

(c) The mappingx7→ F^`(x,0,0) belongs toH^`.

(3.9)

Thepiezoelectric tensorE^`: Ω^`×_S^d →_R^dsatisfies:

((a)E^`(x,τ) = (e^`_ijk(x)τ_jk), ∀τ= (τ_ij)∈_S^d a.e. x∈_Ω^`.

(b)e^`_ijk= e^`_ikj ∈ L^∞(_Ω^`), 1≤i,j,k ≤d. (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

E^`σ.v=σ.(E^`)^∗v ∀σ∈_S^d, ∀v∈_R^d. Theelectric permittivity operatorB^` = (b^`_ij): Ω^`×_R^d→_R^dverifies:











(a)B^`(x,E) = (b_ij^`(x)E_j) ∀E= (E_i)∈_R^d, a.e.x∈_Ω^`. (b)b_ij^` = b^`_ji, b^`_ij ∈ L^∞(_Ω^`), 1≤i,j≤d.

(c)There existsm_B^` >0 such that B^`E.E≥m_B^`|E|²

∀E= (E_i) ∈_R^d, a.e.x∈_Ω^`.

(3.11)

Thenormal compliance functions pν: Γ3×_R→_R+satisfies:











(a)∃Lν>0 such that|pν(x,r₁)−pν(x,r2)| ≤ Lν|r₁−r2|

∀r₁,r₂ ∈_{R, a.e.} x∈_Γ₃.

(b)The mappingx7→ p_ν(x,r)is measurable onΓ3, ∀r∈_R. (c) pν(x,r) =0, for all r ≤0, a.e.x ∈_Γ₃.

(3.12)

Thetangential compliance functions p_τ: Γ3×_R→_R₊_satisfies:











(a) ∃Lτ >0 such that|pτ(x,d₁)−pτ(x,d₂)| ≤ Lτ|d₁−d₂|

∀d₁,d₂ ∈_R, a.e.x∈ _Γ₃.

(b)∃Mτ >0 such that|pτ(x,d)| ≤ Mτ ∀d∈_{R, a.e.}x∈ _Γ₃. (c) The mappingx7→ p_τ(x,d)is measurable onΓ3, ∀d∈_R.

(d)The mappingx7→ p_τ(x, 0)∈ L²(_Γ₃).

(3.13)

We suppose that the mass density satisfies

ρ^`∈ L^∞(_Ω^`)and ∃ρ₀ >0 such that ρ^`(x)≥ρ₀ 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:

f^`₀ ∈ L²(0,T;L²(_Ω^`)^d), f^`₂∈ L²(0,T;L²(_Γ^`₂)^d),

q^`₀∈C(0,T;L²(_Ω^`)), q^`₂∈ C(0,T;L²(_Γ^`_b)), (3.15)

(9)

q^`₂(t) =0 on Γ3 ∀t∈ [0,T]. (3.16) The adhesion coefficients γν,γτ andε_a satisfy the conditions

γν,γτ ∈ L^∞(_Γ₃), εa ∈L²(_Γ₃), γν,γτ,εa ≥0, a.e. onΓ3, (3.17) and, finally, the initial data satisfy

u₀∈V, v₀∈ H, β₀∈ L²(_Γ₃), 0≤ β₀ ≤1, a.e. onΓ3. (3.18) We will use a modified inner product on H, given by

((u,v))_H =

∑

2

`=1

(ρ^`u^`,v^`)_H`, ∀u,v ∈H,

that is, it is weighted with ρ^`, and we let||| · |||_H be the associated norm, i.e.,

|||v|||_H = ((v,v))_H¹² , ∀v∈ H.

It follows from assumption (3.14) that ||| · |||_H andk · k_H are equivalent norms on H, and the inclusion mapping of (V,k · k_V) into(H,||| · |||_H)is continuous and dense. We denote by V⁰ the dual of V. IdentifyingH with its own dual, we can write the Gelfand triple

V ⊂ H⊂V⁰.

Using the notation(·,·)_V⁰_×_V to represent the duality pairing betweenV⁰ andV we have (u,v)_V⁰_×_V = ((u,v))_H, ∀u ∈H,∀v∈V.

Finally, we denote by k · k_V⁰ the norm on V⁰. Using the Riesz representation theorem, we define the linear mappingsf: [0,T]→V⁰ andq: [0,T]→W as follows:

(f(t),v)_V⁰_×_V =

∑

2

`=1

Z

Ω^`f^`₀(t)·v^`dx+

∑

2

`=1

Z

Γ2

f^`₂(t)·v^`da ∀v∈V, (3.19) (q(t),ζ)_W =

∑

2

`=1

Z

Ω^`q^`₀(t)ζ^`dx−

∑

2

`=1

Z

Γ^`_bq^`₂(t)ζ^`da ∀ζ ∈W. (3.20) Next, we denote by j_ad: L^∞(_Γ₃)×V×V →_Rthe adhesion functional defined by

j_ad(β,u,v) =

Z

Γ3

−γνβ²Rν([uν])[vν] +pτ(β)Rτ([uτ])[vτ]da. (3.21) In addition to the functional (3.21), we need the normal compliance functional

jνc(u,v) =

Z

Γ3

pν([_u_ν])[_v_ν]_da. _(3.22) 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∈ L²(0,T;V⁰), q∈C(0,T;W). (3.23) By a standard procedure based on Green’s formula, we derive the following variational formulation of the mechanical (2.1)–(2.13).

(10)

Problem PV. Find a displacement fieldu: [0,T]→V, a stress fieldσ: [0,T]→ H, an electric potential field ϕ: [0,T] →W, a bonding field β: [0,T] → L^∞(_Γ₃)and a electric displacement fieldD:[0,T]→ W such that

σ^` =A^`ε(u˙^`) +G^`ε(u^`)+(E^`)^∗∇ϕ^` +

Z _t

0

F^`σ^`(s)− A^`ε(u˙^`(s))−(E^`)^∗∇ϕ^`,ε(u^`(s))ds inΩ^`×(0,T), (3.24)

D^`= E^`ε(u^`)− B^`∇ϕ^` in Ω^`×(_0,T)_, _(3.25) (u,¨ v)_V⁰_×_V +

∑

2

`=1

(σ^`, ε(v^`))_H`+j_ad(β(t),u(t),v) +j_νc(u(t),v)

= (f(t),v)_V0×V ∀v∈ V,t ∈(0,T),

(3.26)

∑

2

`=1

(B^`∇ϕ^`(t),∇φ^`)_H`−

∑

2

`=1

(E^`ε(u^`(t)),∇φ^`)_H` = (q(t),φ)_W ∀φ∈W, t∈(0,T), (3.27) β˙(t) =−β(t) γ_ν(R_ν([u_ν(t)]))²+γ_τ|R_τ([u_τ(t)])|²−ε_a

+in a.e.(0,T), (3.28) u(0) =u₀, u˙(0) =v₀, β(0) =β₀. (3.29) We notice that the variational ProblemPV 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 ProblemPVis stated and proved in the next section.

Remark 3.2. We note that, in Problem P and in Problem PV, we do not need to impose explicitly the restriction 0 ≤ β ≤ 1. Indeed, equation (3.28) guarantees that β(x,t) ≤ β₀(x) and, therefore, assumption (3.18) shows that β(x,t) ≤ 1 for t ≥ 0, a.e. x ∈ _Γ₃. On the other hand, if β(x,t₀) = 0 at timet₀, then it follows from (3.28) that ˙β(x,t) = 0 for all t ≥ t₀ and therefore, β(x,t) = 0 for all t ≥ t₀, a.e. x ∈ _Γ₃. We conclude that 0 ≤ β(x,t) ≤ 1 for all t∈ [0,T], a.e. x∈ _Γ₃.

Below in this section β,β₁,β₂ denote elements of L²(_Γ₃) such that 0 ≤ β,β₁,β₂ ≤ 1 a.e.

x ∈ _Γ₃, u₁, u2 andv represent elements of V andC > 0 represents generic constants which may depend onΩ^`, Γ3, 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,

j_ad(β,u,−v) =−j_ad(β,u,v),

j_νc(u,−v) =−j_νc(u,v). (3.30) Next, using (3.21), the properties of the truncation operators RνandR_τ as well as assumption (3.13) on the functionpτ, after some calculus we find

j_ad(β₁,u₁,u₂−u₁) +j_ad(β₂,u₂,u₁−u₂)≤ C Z

Γ3

|β₁−β₂||u₁−u₂|da,

and, by (3.20), we obtain

j_ad(β₁,u₁,u₂−u₁) +j_ad(β₂,u₂,u₁−u₂)≤C|β₁−β₂|_L2(_Γ₃)|u₁−u₂|_V_. _(3.31)

(11)

Similar computations, based on the Lipschitz continuity of R_ν, R_τ and p_τ show that the following inequality also holds:

|j_ad(β,u₁,v)−j_ad(β,u₂,v)| ≤Cku₁−u₂k_Vkvk_V. (3.32) We take now β₁ = β₂= βin (3.31) to deduce

j_ad(β₁,u₁,u₂−u₁) +j_ad(β₂,u₂,u₁−u₂)≤_0. _(3.33) Also, we takeu₁ =vandu₂ =0 in (3.32) then we use the equalitiesRν(0) =0, Rτ(0) =0 and (3.30) to obtain

j_ad(β,v,v)≥0. (3.34)

Now, we use (3.22) to see that

jνc(u₁,v) +jνc(u2,v)≤

Z

Γ3

|pν([u_1ν])−pν([u2ν])||[vν]|da, and therefore (3.12.b) and (3.3) imply

|j_νc(u₁,v) +j_νc(u₂,v)| ≤Cku₁−u₂k_Vkvk_V. (3.35) We use again (3.22) to see that

j_νc(u₁,u₂−u₁)+j_νc(u₂,u₁−u₂) =−

Z

Γ3

(p_ν([u_1ν])−p_ν([u_2ν]))([u_1ν−u_2ν])da and therefore (3.12.c) implies

jνc(u₁,u₂−u₁) +jνc(u₂,u₁−u₂)≤0. (3.36) We takeu₁ =vandu₂ =0 in the previous in equality and use (3.22) and (3.36) to obtain

j_νc(v,v)≥0. (3.37)

Inequalities (3.31)–(3.37) and equality (3.30) will be used in various places in the rest of the paper.

4 Existence and uniqueness result

Now, we propose our existence and uniqueness result.

Theorem 4.1. Assume that(3.8)–(3.18) hold. Then there exists a unique solution{u,σ,ϕ,β,D}to ProblemPV. Moreover, the solution satisfies

u∈ H¹(0,T;V)∩C¹(0,T;H), u¨ ∈ L²(0,T;V⁰), (4.1)

ϕ∈C(0,T;W), (4.2)

β∈W^1,∞(0,T;L²(_Γ₃))∩ Z. (4.3)

(12)

The functions u,ϕ,σ,D and β which satisfy (3.24)–(3.29) are called a weak solution to the contact ProblemP. 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,

σ ∈L²(0,T;H), (4.4)

D∈C(0,T;W). (4.5)

Indeed, it follows from (3.26) and (3.27) thatρ^`u¨^` =Divσ^`(t) +f^`₀(t), divD^`(t)−q₀^`(t) =0 for allt ∈ [0,T]and therefore the regularity (4.1) and (4.2) ofuand ϕ, combined with (3.14)–

(3.16) implies (4.4)–(4.5).

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 Theorem4.1hold, and we consider that C is a generic positive constant which depends on Ω^`, Γ^`₁, Γ3, pν,pτ, γν, γτ and L and may change from place to place. Let η ∈ L²(0,T;V⁰)be given. In the first step we consider the following variational problem.

Problem PV^u_η. Find a displacement fieldu_η: [0,T]→V such that (u¨η(t),v)_V⁰_×_V +

∑

2

`=1

(A^`ε(u˙^`(t)), ε(v^`))_H`+ (η(t),v)_V⁰_×_V

= (f(t),v)_V0×V ∀v∈V, a.e.t ∈(0,T),

(4.6)

u^`(₀) =u^`₀, u˙^`(₀) =v^`₀ in Ω^`, (4.7) To solve ProblemPV^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 thatV is dense in H and the inclusion map is continuous, H is identified with its dual and with a subspace of the dualV⁰ of V, i.e. V ⊂ H ⊂ V⁰, and we say that the inclusions above define a Gelfand triple. The notationsk · k_V,k · k_V⁰ and(·,·)_V⁰_×_V represent the norms onV and onV⁰ and the duality pairing betweenV⁰ them, respectively. The following abstract result may be found in [23, p. 48].

Theorem 4.2. LetV,H be as above, and let A: V →V⁰ be a hemicontinuous and monotone operator which satisfies

(Av,v)_V0×V ≥ wkvk²_V+λ ∀v ∈V, (4.8) kAvk_V⁰ ≤ C(kvk_V +1) ∀v∈V, (4.9) for some constants w>0,C>0andλ∈_R.Then, givenu₀∈ H and f ∈ L²(0,T;V⁰),there exists a unique functionuwhich satisfies

u ∈L²(0,T;V)∩C¹(0,T;H), u˙ ∈ L²(0,T;V⁰),

˙

u(t) +Au(t) =f(t)a.e.t∈ (0,T), u(0) =u₀

We have the following result for the problem.

Lemma 4.3. There exists a unique solution to ProblemPV^u_η and it has its regularity expressed in(4.1).