scholarly journals A Study of the Anisotropic Static Elasticity System in Thin Domain

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yassine Letoufa ◽  
Salah Mahmoud Boulaaras ◽  
Hamid Benseridi ◽  
Mourad Dilmi ◽  
Asma Alharbi
Keyword(s):  
Weak Form ◽  
Equation Model ◽  
Limit Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Mathematical Framework ◽  
Thin Domain ◽  
Elasticity System ◽  
Static Elasticity ◽  
Linearized Elasticity ◽  
Linearized Elasticity System

We study the asymptotic behavior of solutions of the anisotropic heterogeneous linearized elasticity system in thin domain of ℝ 3 which has a fixed cross-section in the ℝ 2 plane with Tresca friction condition. The novelty here is that stress tensor has given by the most general form of Hooke’s law for anisotropic materials. We prove the convergence theorems for the transition 3D-2D when one dimension of the domain tends to zero. The necessary mathematical framework and (2D) equation model with a specific weak form of the Reynolds equation are determined. Finally, the properties of solution of the limit problem are given, in which it is confirmed that the limit problem is well defined.

Homogenization of the anisotropic heterogeneous linearized elasticity system in thin reticulated structures

2004 ◽  
Vol 134 (6) ◽  
pp. 1041-1083 ◽  
Author(s):  
J. Casado-Díaz ◽  
M. Luna-Laynez
Keyword(s):  
Asymptotic Behaviour ◽  
Strain Tensor ◽  
Relative Size ◽  
Limit System ◽  
Elasticity System ◽  
Linearized Strain ◽  
Linearized Elasticity ◽  
Small Parameters ◽  
Linearized Elasticity System ◽  
Suitable Approximation

The aim of this paper is to study the asymptotic behaviour of the solutions of the linearized elasticity system, posed on thin reticulated structures involving several small parameters. We show that this behaviour depends on the relative size of the parameters. In each case, we obtain a limit system where the microstructure and macrostructure appear simultaneously. From it, we get a suitable approximation in L2 of the displacements and the linearized strain tensor.

scholarly journals ASYMPTOTIC BEHAVIOR OF STRUCTURES MADE OF PLATES

2005 ◽  
Vol 03 (04) ◽  
pp. 325-356 ◽  
Author(s):  
GEORGES GRISO
Keyword(s):  
Asymptotic Behavior ◽  
Variational Problem ◽  
A Priori Estimates ◽  
A Priori ◽  
Middle Surface ◽  
Rigid Displacement ◽  
Elasticity System ◽  
Convergence Results ◽  
Linearized Elasticity ◽  
Linearized Elasticity System

The aim of this paper is to study the asymptotic behavior of a structure made of plates of thickness 2δ when δ → 0. This study is carried out within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of displacements of the structure and on the passing to the limit in fixed domains.We begin by studying the displacements of a plate. We show that any displacement is the sum of an elementary displacement concerning the normal lines on the middle surface of the plate and a warping. An elementary displacement is linear with respect to the variable x3. It is written [Formula: see text] where [Formula: see text] is a displacement of the mid-surface of the plate. We show a priori estimates and convergence results when δ → 0. We characterize the limits of the unfolded displacements of a plate as well as the limits of the unfolded strained tensor.Then, we extend these results to structures made of plates. We show that any displacement of a structure is the sum of an elementary displacement of each plate and of a residual displacement. The elementary displacements of the structure (e.p.s.d.) coincide with elementary rod displacements in the junctions. Any e.p.s.d. is given by two functions belonging to H1( S ; ℝ3) where S is the skeleton of the structure (the set formed by the mid-surfaces of the plates constituting the surface). One of these functions, [Formula: see text], is the skeleton displacement. We show that [Formula: see text] is the sum of an extensional displacement and of an inextensional one. The first one characterizes the membrane displacements and the second one is a rigid displacement in the direction of the plates and it characterizes the flexion of the plates.Eventually, we pass to the limit as δ → 0 in the linearized elasticity system. On the one hand, we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem satisfied by the limit of inextensional displacements.

scholarly journals On singularities of solution of the elasticity system in a bounded domain with angular corner points

2021 ◽  
Author(s):  
Yasir Nadeem ◽  
Akhtar Ali
Keyword(s):  
Boundary Conditions ◽  
Weak Solutions ◽  
Singular Vector ◽  
Corner Singularities ◽  
Qualitative Properties ◽  
Elasticity System ◽  
Linearized Elasticity ◽  
Rigorous Description ◽  
Corner Points ◽  
Linearized Elasticity System

This paper aims to give a mathematically rigorous description of the corner singularities of the weak solutions for the plane linearized elasticity system in a bounded planar domain with angular corner points on the boundary. The qualitative properties of the solution including its regularity depend crucially on these corner points or such types of boundary conditions. In particular, the resulting expansion of the solutions of the underlying problem involves singular vector functions, inlines, depending on a certain parameter ξ

scholarly journals On the asymptotic behavior of solutions of anisotropic viscoelastic body

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yassine Letoufa ◽  
Hamid Benseridi ◽  
Salah Boulaaras ◽  
Mourad Dilmi
Keyword(s):  
Asymptotic Behavior ◽  
Three Dimensional ◽  
Viscoelastic Body ◽  
Weak Form ◽  
Uniqueness Result ◽  
Limit Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Quasistatic Problem ◽  
Rigid Plane ◽  
Tresca’S Friction

AbstractThe quasistatic problem of a viscoelastic body in a three-dimensional thin domain with Tresca’s friction law is considered. The viscoelasticity coefficients and data for this system are assumed to vary with respect to the thickness ε. The asymptotic behavior of weak solution, when ε tends to zero, is proved, and the limit solution is identified in a new data system. We show that when the thin layer disappears, its traces form a new contact law between the rigid plane and the viscoelastic body. In which case, a generalized weak form equation is formulated, the uniqueness result for the limit problem is also proved.

scholarly journals Asymptotic Behavior of Solutions to Free Boundary Problem with Tresca Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Abdelkader Saadallah ◽  
Nadhir Chougui ◽  
Fares Yazid ◽  
Mohamed Abdalla ◽  
Bahri Belkacem Cherif ◽  
...  
Keyword(s):  
Asymptotic Behavior ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Reynolds Equation ◽  
Limit Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Thin Domain

In this paper, we study the asymptotic behavior of an incompressible Herschel-Bulkley fluid in a thin domain with Tresca boundary conditions. We study the limit when the ε tends to zero, we prove the convergence of the unknowns which are the velocity and the pressure of the fluid, and we obtain the limit problem and the specific Reynolds equation.

scholarly journals Analysis for Flow of an Incompressible Brinkman-Type Fluid in Thin Medium with Friction

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Soumia Manaa ◽  
Salah Boulaaras ◽  
Hamid Benseridi ◽  
Mourad Dilmi ◽  
Sultan Alodhaibi
Keyword(s):  
Fluid Flow ◽  
Variational Formulation ◽  
Reynolds Equation ◽  
Three Dimensional ◽  
Stationary Regime ◽  
Limit Problem ◽  
Brinkman Equation ◽  
Asymptotic Convergence ◽  
Thin Domain ◽  
Uniqueness Of The Solution

In this paper, we consider the Brinkman equation in the three-dimensional thin domain ℚ ε ⊂ ℝ 3 . The purpose of this paper is to evaluate the asymptotic convergence of a fluid flow in a stationary regime. Firstly, we expose the variational formulation of the posed problem. Then, we presented the problem in transpose form and prove different inequalities for the solution u ε , p ε independently of the parameter ε . Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.

scholarly journals ASYMPTOTIC BEHAVIOR OF STRUCTURES MADE OF CURVED RODS

2008 ◽  
Vol 06 (01) ◽  
pp. 11-22 ◽  
Author(s):  
GEORGES GRISO
Keyword(s):  
Asymptotic Behavior ◽  
Variational Problem ◽  
Cross Section ◽  
A Priori Estimates ◽  
A Priori ◽  
The Other ◽  
Linearized Elasticity ◽  
Curved Rods ◽  
The Cross ◽  
Linearized Elasticity System

In this paper, we study the asymptotic behavior of a structure made of curved rods of thickness 2δ when δ tends to 0. This study is carried on within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of the structure displacements and on the passing to the limit in fixed domains. We show that any displacement of a structure is the sum of an elementary rods-structure displacement (e.r.s.d.) concerning the rods' cross sections and a residual one related to the deformation of the cross section. The e.r.s.d. coincides with rigid body displacements in the junctions. Any e.r.s.d. is given by two functions belonging to [Formula: see text] where [Formula: see text] is the skeleton structure (i.e. the set of rods with middle lines). One of this function [Formula: see text] is the skeleton displacement, the other [Formula: see text] gives the cross section rotation. We show that [Formula: see text] is the sum of an extensional displacement and an inextensional one. We establish a priori estimates and then, we characterize the unfolded limits of the rods-structure displacements. Eventually, we pass to the limit in the linearized elasticity system and using all results in [6], where on one hand, we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem coupling the limit of inextensional displacement and the limit of the rod torsion angles.

scholarly journals Dimension reduction and homogenization of random degenerate operators. Part I

2012 ◽  
Vol 15 ◽  
pp. 1-22
Author(s):  
Abdelaziz Aït Moussa ◽  
Loubna Zlaïji
Keyword(s):  
Boundary Conditions ◽  
Dimension Reduction ◽  
Random Function ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Limit Problem ◽  
Thin Domain ◽  
Small Thickness ◽  
Degenerate Equations ◽  
Degenerate Operators

AbstractOur aim in this paper is to identify the limit behavior of the solutions of random degenerate equations of the form −div Aε(x′,∇Uε)+ρεω(x′)Uε=F with mixed boundary conditions on Ωε whenever ε→0, where Ωε is an N-dimensional thin domain with a small thickness h(ε), ρεω(x′)=ρω(x′/ε), where ρω is the realization of a random function ρ(ω) , and Aε(x′,ξ)=a(Tx′ /εω,ξ) , the map a(ω,ξ) being measurable in ω and satisfying degenerated structure conditions with weight ρ in ξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the condition h(ε)/ε→0 , the sequence of solutions of them converges to a limit u0, where u0 is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.

Sign in / Sign up

Export Citation Format

Share Document