On the asymptotic study of transmission problem in a thin domain

Abstract

In this paper, we study the theoretical analysis of a frictionless contact between two general elastic bodies in a stationary regime in a three-dimensional thin domain {\Omega^{\varepsilon}} with Tresca friction law. Firstly, the problem statement and variational formulation are presented. We then obtain the estimates on displacement independently of the parameter ε. Finally, we obtain the main results concerning the limit of a weak problem and its uniqueness.

On the asymptotic behavior of solutions of anisotropic viscoelastic body

“…Moreover, this law and its traces meet the basic equation of motion in (4.2)-(4.3) and the boundary conditions of the Tresca friction in (4.10)-(4.11). The phenomenon is described mathematically by the weak formula (4.12), known as the Reynolds equation, and has been proved in many papers for the particular cases, see [1,2,4,7,12]. Specifically, if A ε = 0, the model matches the case of the anisotropic linearized elasticity system and has been studied in [13,14].…”

“…Simulation and asymptotic behavior have been performed for many physical milieu that occupy thin layers of R 3 in a lot of mechanical and mathematical papers in [1,8,9,12], both for Newtonian and non-Newtonian fluids, including visco-plastic materials. Otherwise, the materials related to the linear theory of elasticity have been studied in research papers conducted by Benseridi et al in [4][5][6]16] and others in [2,14,16]. Here, the authors investigated several elasticity systems in thin layers in dimension 3 under various boundary conditions, through which they reached low-dimensional constitutive laws implicitly prevalent in many applications, particularly in lubrication problems, to describe the behavior of the phenomena that are already occurring in thin layers when the thickness is as small as 50 nm (e.g.…”

Study of a Transmission Problem with Friction Law and Increasing Continuous Terms in a Thin Layer

“…We would like readers to note that, in this study, the authors are interested in a very particular body that follows Hooke's law (an isotropic case of elastic materials). The asymptotic study of a transmission problem governed by an elastic body in a stationary regime with Tresca has been studied in [17]. Another work analogous to this present study, but relating only to the study of the existence and uniqueness of the weak solution of a frictionless contact problem between an elastic body and a rigid foundation, is given by [18].…”

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

“…Bayada and Lhalouani 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 al 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 Boukrouche and Dilmi et al…”