We present in this work a unified framework for elliptic variational inequalities that gathers several problems in contact mechanics like the unilateral contact of one or two membranes or the Signorini problem. We study a family of Galerkin numerical schemes that discretize this framework. We prove the well-posedness of the discrete problem and we show that it is equivalent to a saddle-point mixed formulation containing complementarity constraints. To solve the nonlinear problem, we employ a semismooth Newton method and prove local convergence properties. The abstract framework is then applied to the discretization of the unileteral contact between two membranes. We propose to discretize this problem with a finite element (FEM), a discontinuous Galerkin (dG), and a hybrid high-order (HHO) methods. We also adapt the semismooth Newton algorithm, including a static condensation procedure for the HHO method. Finally, we run numerical experiments for the FEM and HHO discretizations and compare their behavior.