Spinodal decomposition is a process where an initial homogeneous but unstable mixture spontaneously separates into two or more stable phases with a characteristic arrangement termed spinodal structure. This process can be modeled with the well-known Cahn-Hilliard equation [1] which dates back to 1958. In the Cahn-Hilliard theory, spinodal decomposition can be obtained in a variationally consistent way by defining a chemical energy density composed by a non-convex bulk term and an interfacial gradient term, both depending on an order variable, i.e. the phase field, representing the smooth transition between the unstable initial mixture and the stable phases. Although the key role of elasticity in controlling the evolution of the microstructure was already outlined in this pioneering study, to date only little attention has been paid to coupling elasticity and spinodal decomposition in solid-solid or solid-liquid mixtures, e.g. metal alloys or liquids absorbed in a polymer matrix. In this work we couple the Cahn-Hilliard bulk energy density with the elastic strain energy density by introducing an additional coupling term in the free-energy functional, allowing to derive the governing equations while preserving the variational consistency of the original formulation. The model is first studied in 1D and the results show how the mechanical deformation controls both the composition of the stable phases and the spinodal structure, including its initial characteristic length and its coarsening. The formulation is then extended to the multi-dimensional setting and validated using the experimental results in [2], where the spinodal decomposition of a mixture composed of fluorinated oil in a Polydimethylsiloxane (PDMS) matrix is shown to be governed by the stiffness of the latter. The numerical results show an excellent agreement with the experimental evidence, especially in terms of initial characteristic length and pattern of the spinodal structure. [1] J. Cahn, and J. Hilliard, 1958, Free Energy of a Nonuniform System. I. Interfacial Free Energy. J. Chem. Phys. Vol 28, No 2. [2] C. Fern´andez-Rico, T. Sai, A. Sicher, Robert W. Style, and Eric R. Dufresne, 2022, Putting the Squeeze on Phase Separation. JACS Au. Vol 2, No 1.