We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Muller and the second author by dropping their assumption of periodicity. As a first application, we extend their Gamma-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Gamma-closure is local at identity for this class of energy densities. (C) 2011 Elsevier Masson SAS. All rights reserved.