Asymptotic analysis of a signorini problem with Coulomb friction for shallow shells

Abstract

The objective of this thesis is to study the asymptotic modeling of threedimensional problems of nonlinearly elastic shallow shells, in dynamical case, with and without unilateral contact. Also, to study the numerical approximation of the generalized Marguerre-von Kármán equations. In the first Part, we consider a three-dimensional dynamical models for a nonlinearly elastic shallow shells with a specific class of boundary conditions of generalized Marguerrevon Kármán type, without unilateral contact. Using technics from asymptotic analysis, we justify two two-dimensional models. The first model in homogeneous and isotropic material case, called dynamical equations of generalized Marguerre-von Kármán shallow shells. The second one in nonhomogeneous and anisotropic material case, called dynamical equations of generalized nonhomogeneous anisotropic Marguerre-von Kármán shallow shells. In addition, we establish the existence of solution to the first model. In the second Part, we extend the two models in first part, to a Signorini contact with Coulomb friction case. To this end, we justify the dynamical contact equations of generalized Marguerre-von Kármán shallow shells. Also, we establish the existence of solution to these equations. Next, we justify the dynamical contact equations of generalized nonhomogeneous anisotropic Marguerre-von Kármán shallow shells. In addition, we justify the contact equations of generalized Marguerre-von Kármán shallow shells, in static case. In the third Part, we establish the convergence of a conforming finite element approximations to the generalized Marguerre-von Kármán equations.