We consider a variational model for composition of finitely many strongly elliptic homogenous elastic materials in linear elasticity. We give conditions for universal coercivity for the variational integrals which are independent of particular compositions of materials involved. We show that in the two-dimensional case all elliptic materials satisfy universal coercivity and hence can “in principle” be homogenized. In the case of two homogeneous materials in three-dimensional elasticity we give a complete answer to our universal coercivity problem. We also give sufficient conditions for coercivity in the three-dimensional case and give an example to show that there is a noncoercive variational integral for which there are infinitely many minimizers, which are nowhere $C^1$ on their supports.