This paper exploits the concept of orthogonal sub-grid scales to stabilize the behaviour of mixed linear/linear simplicial elements (triangles and tetrahedra) in incompressible or nearly incompressible situations. Both incompressible elastic and J2-plastic constitutive behaviours have been considered. The different assumptions and approximations used to derive the method are exposed. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the elements derived are free of volumetric locking and spurious oscillations of the pressure, and that the results obtained compare favourably with those obtained with the Q1P0 quadrilateral. © 2003 Elsevier B.V. All rights reserved.