In this work we consider the problem of numerical locking in composite materials featuring quasi-incompressible and compressible sections. More specifically, we start by extending a classical regularity estimate for the $H^1$-norm of the divergence of the displacement field to the heterogeneous case. The proof is based on a reformulation of the elasticity problem as a Stokes system with nonzero divergence constraint. This result is then used to design a locking-free discontinuous Galerkin method. The key point is to make sure that the multiplicative constant in the estimate of the convergence rate uniquely depend on this bounded quantity. Thanks to a fine tuning of the penalty term, the lower bound for the penalty parameter appearing in the method is simply expressed in terms of the space dimension. To conclude, numerical validation of the theoretical results is provided.
