This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components) or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation were further reviewed, and the approach was exemplified in the case of a material that was transversely isotropic almost surely. In a second stage, we made use of such derivations to propose a generalized model for random elasticity matrices that took into account, almost separately, constraints on both the level of stochastic anisotropy and the level of statistical fluctuations. An example was finally provided and showed the efficiency of the approach. Copyright © 2011 John Wiley & Sons, Ltd.