# Asymptotic study of a frictionless contact problem between two elastic bodies

Volume 16, Issue 3, pp 336-350
Publication Date: September 15, 2016 Submission Date: February 03, 2016
• 1559 Views ### Authors

Y. Letoufa - Applied Mathematics Laboratory, El Oued University, 39000, Algeria. H. Benseridi - Applied Mathematics Laboratory, Setif 1 University, 19000, Algeria. M. Dilmi - Applied Mathematics Laboratory, Setif 1 University, 19000, Algeria.

### Abstract

We consider a mathematical model which describes the bilateral, frictionless contact between two elastic bodies. We will establish a variational formulation for the problem and prove the existence and uniqueness of the weak solution. We then study the asymptotic behavior when one dimension of the domain tends to zero. In which case, the uniqueness result of the solution for the limit problem are also proved.

### Keywords

• A priori inequalities
• free boundary problems
• nonlinear equation
• transmission conditions
• Tresca law
• variational problem.

•  35R35
•  76F10
•  78M35

### References

•  A. Atangana, A novel model for the Lassa Hemorrhagic Fever: Deathly Disease for Pregnant Women , Neural Comput. Appl., 26 (2015), 1895-1903.

•  A. Atangana, E. F. Doungmo Goufo, A model of the groundwater flowing within a leaky aquifer using the concept of local variable order derivative , J. Nonlinear Sci. Appl., 8 (2015), 763-775.

•  H. M. Baskonus, H. Bulut , New hyperbolic function solutions for some nonlinear partial differential equation arising in mathematical physics, Entropy, 17 (2015), 4255-4270.

•  G. Bayada, M. Boukrouche , On a free boundary problem for the Reynolds equation derived from the Stokes systems with Tresca boundary conditions, J. Math. Anal. Appl., 282 (2003), 212-231.

•  G. Bayada, K. Lhalouani, Asymptotic and numerical analysis for unilateral contact problem with Coulomb's friction between an elastic body and a thin elastic soft layer , Asymptot. Anal., 25 (2001), 329-362.

•  H. Benseridi, M. Dilmi, Some inequalities and asymptotic behavior of a dynamic problem of linear elasticity , Georgian Math. J., 20 (2013), 25-41.

•  M. Boukrouche, R. El mir, On a non-isothermal, non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions, Nonlinear Anal., 9 (2008), 674-692.

•  M. Boukrouche, G. Lukaszewicz, On a lubrication problem with Fourier and Tresca boundary conditions , Math. Models Methods Appl. Sci., 14 (2004), 913-941.

•  M. Boukrouche, F. Saidi, Non-isothermal lubrication problem with Tresca fluid-solid interface law. , Part I, Nonlinear Anal., 7 (2006), 1145-1166.

•  M. Dilmi, H. Benseridi, A. Saadallah , Asymptotic analysis of a Bingham fluid in a thin domain with Fourier and Tresca boundary conditions, Adv. Appl. Math. Mech., 6 (2014), 797-810.

•  G. Duvaut, J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris (1972)

•  N. Hemici, A. Matei , A frictionless contact problem with adhesion between two elastic bodies, An. Univ. Craiova Ser. Mat. Inform., 30 (2003), 90-99.

•  J. Koko, Uzawa block relaxation domain decomposition method for the two-body contact problem with Tresca friction, Comput. Methods Appl. Mech. Engrg., 198 (2008), 420-431.

•  X. Li, Symmetric Coupling of the Meshless Galerkin boundary node and finite element methods for Elasticity , CMES Comput. Model. Eng. Sci., 97 (2014), 483-507.

•  X. Li, H. Chen, Y. Wang, Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method, Appl. Math. Comput., 262 (2015), 56-78.

•  A. Saadallah, H. Benseridi, M. Dilmi, S. Drabla, Estimates for the asymptotic convergence of a non- isothermal linear elasticity with friction , Georgian Math. J., 23 (2016), 435-446.