Abstract : This PhD thesis treats the mathematical modelization of thin layers and the domain decomposition methods in contact mechanics.\\
The first part is dedicated to a quasistatic contact problem with non local Coulomb friction law between an elastic body and a thin layer. After establishing an existence theorem, we define a critical ratio between the geometrical and the elastic parameters. For this ratio, we establish rigorously a limit contact law by making the layer thickness tends to zero.\\
The second part is devoted to some ``natural'' domain decomposition methods in contact problems. This method consists of retaining the natural interface between two bodies as a numerical interface for the domain decomposition. Firstly, we study a contact problem without friction between two elastic bodies (Signorini problem) for which we propose and prove the convergence of a Neumann-Dirichlet algorithm. This result is then generalized to a contact problem with Coulomb friction. At last, we propose and prove the convergence of a ``Neumann-Neumann'' decomposition algorithm for a Signorini problem.
Some numerical results give confidence to the validity of the theoretical results.