Use of stabilization methods is becoming an increasingly well-accepted technique due to their success in dealing with numerous numerical pathologies that arise in a variety of applications in computational mechanics. In this paper, a multi-scale finite element method technique to deal with pressure stabilization of incompressibility and nearly incompressibility problems in nonlinear solid mechanics, using low order finite elements, is presented. An Orthogonal Sub-Grid Scales (OSGS) method for both incompressible elasticity and J2-plasticity at finite deformations is proposed. Standard mixed finite element formulations, particularly those using low order interpolations, perform poorly or totally fail to perform for nearly incompressibility or incompressibility problems, producing results throughly polluted by spurious oscillations of the pressure. The goal here is to avoid this undesirable effect while retaining the use of low order elements. To achieve this goal we consistently derive, within the framework of the OSGS method, a modified variational mixed formulation of the original problem with enhanced stability properties. An assessment of the behavior of the formulation is presented. Results are compared with standard Galerkin and Q1P0 mixed large strain formulations.