Abtract: We discuss a differential inclusion arising in the context of bounding effective conductiv- ities of polycrystalline composites. The datum is a set of three positive numbers identified with a positive definite diagonal matrix S. The aim is to find suitable solutions to the inclusion DU ∈ K := {λR^t S R : λ ∈ R, R ∈ SO(3)}. We will show how to construct a class of so-called approximate solutions via infinite-rank laminations. The resulting average fields provide an inner bound for the quasi-convex hull of K, which is an improvement of the bounds that were previously established by Avellaneda et al. [1] and Milton & Nesi [2]. We will also discuss some open problems related to the lack of outer bounds and the existence of exact solutions to the differential inclusion. This is joint work with N. Albin and V. Nesi. References [1] M. Avellaneda, A. V. Cherkaev, K. A. Lurie, G. Milton. On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality. J. Appl. Phys. 63, 4989–5003, 1988. [2] V. Nesi, G. Milton. Polycrystalline configurations that maximize electrical resistivity. J. Mech. Phys. Solids no. 4, 525–542, 1991.