Estimates for the asymptotic convergence of a non-isothermal linear elasticity with friction

2019

Comp and Math Methods

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This paper is devoted to the study of the asymptotic analysis of a mathematical model involving a frictionless contact between an quasistatic electro‐viscoelastic and a conductive foundation in a three‐dimensional thin domain Ωε. The contact is modeled with Tresca friction law. We derive a variational formulation of the problem and prove its unique weak solution. Then, we study the asymptotic analysis when the small parameter of the domain tends to zero. Finally, the specific Reynolds limit equation and the limit of Tresca free boundary conditions are obtained.

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

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

2019

Comp and Math Methods

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“…In the context of the asymptotic analysis of problems posed in a thin domain, there are many papers that are interested in these problems but using the classical derivative, among them, the authors in the paper 13 studied the asymptotic analysis of a dynamical problem of linear elasticity with Tresca free boundary friction conditions. In Saadallah et al, 14 the authors studied the same problem but in the nonisothermal case. The authors in Benseghir et al 15 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.…”

Section: Introductionmentioning

confidence: 99%

Variational formulation and asymptotic analysis of viscoelastic problem with Riemann‐Liouville fractional derivatives

Math Methods in App Sciences

2019

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In this paper, we study the variational formulation and asymptotic analysis of a viscoelastic problem with fractional derivative in the Riemann‐Liouville sense and nonlinear friction of Tresca type in a three‐dimensional thin domain Ωε. We prove the existence and uniqueness of the weak solution; then, we study the asymptotic analysis when one dimension of the domain tends to zero. The limit problem, the limit of Tresca free boundary conditions, and the specific Reynolds equation are obtained.

“…For example, Benseridi et al 16,17 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 non-isothermal elasticity with nonlinear friction of Tresca type was studied in Saadallah et al 18 The authors in Benterki et al 19 established the asymptotic analysis of a nonlinear problem governed by the isothermal Bingham fluid in a 3-dimensional thin domain with Fourier and Tresca boundary conditions. In this present paper, we analyses the asymptotic behavior of the generalized Stokes system as (2) with ∈ 1, 2 , in a thin domain Ω ⊂ R 3 where a part of its boundary is subject to conditions of friction and another part is subjected to Dirichlet conditions, where 0 < < 1 is a positive real intended to tend towards zero.…”

Section: Introductionmentioning

confidence: 99%

Study of generalized Stokes operator in a thin domain with friction law (casep< 2)

Math Methods in App Sciences

2018

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MSC Classification: 35R35; 76F10; 78M35 In this work, we consider a mathematical model of an incompressible fluid governed by the generalized Stokes system in a stationary regime in a 3-dimensional thin domain Ω with Tresca friction law. The problem statement and variational formulation of the problem are formulated. Then the estimates on velocity and pressure are proved in a fixed domain. Finally, our main goal concerns the specific Reynolds equation and the uniqueness of the solution (û ,̂) are obtained. KEYWORDS a priori estimates, Reynolds equation, Stokes systems, Tresca law, variational formulationHowever, in this case, the Stokes principle is not satisfied.These types of models were studied intensively in the 1980s and 1990s by Necas et al, 8 in which the authors studied the particular types of fluids, and describe the shear thickening (p > 2) and shear thinning (p < 2) phenomena. The author in other works 9,10 establish regularity results up to the boundary for solutions to generalized Stokes and Navier-Stokes systems of equations in the stationary and evolutive cases. The asymptotic analysis of the type (1), in a thin domain with Tresca and Coulomb friction has been studied in previous works. 11,12 In Boukrouche and EL Mir, 13 the authors proved the asymptotic analysis of a non-Newtonian and incompressible fluid with stress tensor (u , ) = |D(u )| −2 D(u ) − I, in a thin domain. In Bayada et al, 14 the authors consider the Navier-Stokes equations in a thin moving boundary domain.