Asymptotic analysis of quasistatic electro-viscoelastic problem with Tresca's friction Law

1 INTRODUCTION AND STATEMENT OF THE PROBLEM

Scientific research in mechanics are articulated around two main components: one devoted to the laws of behavior and the other on boundary conditions imposed on the body. For the constitutive law, we consider an quasistatic electro-viscoelastic body with Tresca free boundary friction conditions in the dynamic regime occupying a bounded homogeneous domain of 3D. The piezoelectric is characterized by the coupling between the mechanical and the electrical properties of the material: it was observed that the appearance of electric charges on some crystals was due to the action of body forces and surface tractions and, conversely, the action of the electric field generated strain or stress in the body (see the work of Batra and Yang1 and the references therein). This coupling leads to the appearance of electric potential when mechanical stress is present and, conversely, mechanical stress is generated when electric potential is applied. Piezoelectric materials for which the mechanical properties are elastic are called electro-elastic materials and those for which the mechanical properties are viscoelastic are called electro-viscoelastic materials. Antiplane shear deformations are one of the simplest examples of deformations that solids can undergo. In the work of Sofonea and Dalah,2 the authors studied antiplane frictional contact problem of electro-viscoelastic cylinders. Chougui and Drabla3 studied a frictional contact problem with adhesion between an elastic piezoelectric body and a deformable obstacle. Contact problems for electro-viscoelastic materials were considered in the works of Sofonea et al.4, 5 In most cases, the authors are interested in studying the weak formulation of the solution, then prove the existence and uniqueness of the solution.

Here, we continue this line of research and study the asymptotic analysis of the electro-viscoelastic quasistatic problem with friction modeled by the law of Tresca, in a thin domain , where 0 < ε < 1 is a small parameter that will tend to zero. The boundary of Ω^ε will be noted , such that is the upper boundary of equation x₃ = εh(x₁,x₂), is the lateral boundary and ω is a fixed bounded domain of of equation x₃ = 0, which is bottom of the domain Ω^ε.

In mathematical literature, there are many research papers regarding the asymptotic analysis of the problems that arise in a thin domain of which we mention; Benseridi and Dilmi6 studied the asymptotic analysis of a dynamical problem of isothermal elasticity with nonlinear friction of Tresca type. The asymptotic behavior of a dynamical problem of nonisothermal elasticity with nonlinear friction of Tresca type was studied in the work of Saadallah.7 Bayada and Lhalouani8 investigated the asymptotic and numerical analysis for a unilateral contact problem with Coulomb's friction between an elastic body and a thin elastic soft layer. Benseghir et al9 studied the theoretical analysis of a frictionless contact between two general elastic bodies in a stationary regime in a three-dimensional thin domain with Tresca friction law. The asymptotic analysis of some problem in mechanics of the fluids in a thin domain in the stationary case is found in the works of Bayada and Boukrouche10 and Dilmi et al.11

Our work is structured as follows. In Section 1, we present the model of the electro-viscoelastic, and we give some notation on the data of the problem. In Section 2, we derive the variational formulation for the problem and state our result of existence and uniqueness of the solution. In Section 3, we employ the change of variable z = x₃/ε to transform the initial problem posed in the domain Ω^ε at a new problem posed on a fixed domain Ω independent of parameter ε. We establish some estimates independent of parameter ε, for the displacement and the electric potential . Finally, we give the convergence results, the limit problem, and its uniqueness in Section 4.

Let h(.) be a function to class C¹ defined on ω such that

we note , . The domain Ω^ε is given by

with as its boundary.

Let u^ε(x,t) be the displacement field and φ^ε(x,t) be the electric potential in Ω^ε.

The electro-viscoelastic constitutive law is given by

where d(.) the strain tensor, which is given by

is the piezoelectric tensor, where is the piezoelectric coefficient, (E^ε)^T is the transposed of the tensor E^ε, μ^ε, λ^ε are the coefficients of Lamé, I₃ is the identity, θ^ε is the viscosity term, and β^ε is the electric coefficient of permitting.

Let n = (n₁,n₂,n₃) be the unit outward normal to Γ^ε, and

are, respectively, the normal and the tangential of the displacement u^ε, and the normal and tangential components of stress tensor field σ^ε.

The classical model for quasistatic electro-viscoelastic problem is as follows.

Problem ( P^E.V). Find the displacement field , and the electric potential such that

()

()

()

()

()

()

()

()

()

where 1 and 2 represent the equilibrium equations, 3-5 represent the boundary condition for the displacement field and normal velocity, 6 represent the boundary condition for the electric potential, and 7 represent the boundary condition for the electric charges, 8 represent the law of friction of Tresca, where |.| indicate the Euclidean norm, and k^ε is a given function,  f ^ε represent the densities of the forces, q^ε represent the densities of the electric charges, and 9 represent the initial condition of displacement.

We suppose that the coefficients of Lamé μ^ε, λ^ε and the viscosity term θ^ε satisfies

we also suppose that the electric coefficient of permitting and the piezoelectric parameter satisfies

2 WEAK FORMULATION

To get the weak formulation, we recall some standard spaces: L²(Ω^ε) is the space of Lebesgue for the norm , H¹(Ω^ε)³ is the space of Sobolev, which is given by

is the closing of D(Ω^ε)³ in H¹(Ω^ε)³, and H⁻¹(Ω^ε)³ is the dual space of .

We suppose that the problem of electro-viscoelastic (P^E.V) admits a solution (u^ε,φ^ε) regular, and let (v,ψ)∈(H¹(Ω^ε)³,H¹(Ω^ε)). We multiply Equation 1 by the element , in the same way for Equation 2 by the element ψ ∈ H¹(Ω^ε), then we integrate on Ω^ε and, using the Green's formula, we obtain the following variational problem.

Problem  . Find the displacement field , and the electric potential such that

()

()

()

where

()

and

()

Theorem 1.Under the assumptions

()

()

()

()

The problem 10-12 admits a unique solution (u^ε,φ^ε) ∈ K^ε × Q^ε, such that

()

Proof.The proof is based on the regularization method, which is based on an approximation of nondifferentaible term j^ε(.) by a family of differentaible onse , where

and we build a problem approximate

Using Galerkin's method, we show that there exists a unique solution of this last approximate problem (see the works of Duvaut and Lions12, 13). Then, the limit of when ζ tends to zero is a solution of 10-12.

3 CHANGE OF THE DOMAIN AND A PRIORI ESTIMATES

To study the asymptotic analysis of the problem, we will use the technique of change of the variable z = x₃/ε, thus, ∀(x ′,x₃) ∈ Ω^ε, we find (x ′,z) in Ω such that

with , its boundary. We define now on Ω the following functions:

For the data of problem 1-8, it is supposed that they depend of ε in the following manner:

with , , , , , , , and do not depend on ε.

Now, we define the function spaces on Ω.

Let

and

V_z is the Banach space with the norm

Then, the problem 10-11 is equivalent to the following problem.

Problem  . Find the displacement field , and the electric potential such that

()

()

where

and

Now, we establish the a priori estimates for the displacement and the electric potential .

Theorem 2.Under the same assumptions as in Theorem 1, there exists a constant c independent of ε such that

()

()

Proof.Let (u^ε,φ^ε) the solution of the problem 10-12, then we have

()

()

We derive 25 for t, then choose ψ = φ^ε and we use Korn's inequality , we find

this implies that

()

By integration of 26, we have

using Poincaré's inequality

we obtain

()

By applying the Hölder and Young inequalities, we find

()

()

and

()

By Poincaré's inequality

and Young's inequality, we obtain

()

From 27-31, we find

()

()

It is necessary to estimate ∇φ^ε(0). From 11, we deduce that

thus, ∀ψ ∈ Q^ε,

we are multiplying this inequality by , we obtain

where

On the other hand, the sum of 32 with 33 is given as

()

As

we multiply the inequality 34 by ε, we get

()

where

is a constant independent of ε.

Using Gronwall's lemma in inequality 35, we get

where c is a positive constant independent of ε. Consequently, we find

4 CONVERGENCE RESULTS AND LIMIT PROBLEM

Theorem 3.Under the assumptions of Theorem (2), there exists , i = 1,2 such that

()

()

()

Proof.According to Theorem 2, there is constant c independent of ε such that

Using these estimates with the Poincaré's inequality in the domain Ω, we obtain 36, for 37-38, according to 22-23, and 36.

Theorem 4.With the same assumptions of Theorem 3, (u^∗,φ^∗) satisfies the following relations:

()

()

and the limit problem

()

Proof.As J(.) is convex and lower semicontinuous

thus, by using the convergence results of the Theorem 3, we get

()

()

We choose now in the variational equation 42

we find

By using Green's formula and while choosing w₂ = 0 and , then w₁ = 0, and , we obtain

On ω, we have , thus we get

()

As , , then 44 is valid in L²(Ω).

Theorem 5.Let

Under the same assumptions of Theorem 4, the traces of the displacements τ^∗(x ′,t), s^∗(x ′,t) and electric potential l^∗(x ′,t), satisfy the following inequality:

()

and the following limit form of the Tresca boundary conditions:

()

Moreover, (u^∗,φ^∗) satisfies the following weak form of the Reynolds equation:

()

where

Proof.For 45 and 46, it is enough to follow the same techniques in the work of Bayada and Boukrouche.10 To prove 47, we integrate 41 between 0 and z, it is seen that

Integrating for the second time between 0 and z, we obtain

()

if z = h(x ′) in 48, we find

()

By integrating 48 from 0 to h(x ′), we get

()

According to 49 and 50, we deduce 47.

Proof.Let us suppose that there are two solution (u^∗1,φ^∗1) and (u^∗2,φ^∗2) of variational equations 39 and 40, we have

()

()

()

()

Take in 51 and in 52 (respectively), and in 53 and in 54 (respectively). By summing the two inequalities and two equations, we obtain

()

()

We put and , according to 55 and 56, we find

()

()

By the integration of 57 and 58, we get

()

()

the two inequalities 59 and 60, given

Now, by Gronwall's lemma, we find

using Poincaré's inequality, we deduce