The relaxation of a double-well energy

This paper studies coherent, energy-minimizing mixtures of two linearly elastic phases with identical elastic moduli. We derive a formula for the relaxed or macroscopic energy of the system, by identifying microstructures that minimize the total energy when the volume fractions and the average strain are fixed. If the stress-free strains of the two phases are incompatible then the relaxed energy is nonconvex, with double-well structure. An optimal microstructure always exists within the class of layered mixtures. The optimal microstructure is generally not unique, however; we show how to construct a large family of optimal, sequentially laminated microstructures in many circumstances. Our analysis provides a link between the work of Khachaturyan and Roitburd in the metallurgical literature and that of Ball, James, Pipkin, Lurie, and Cherkaev in the recent mathematical literature. We close by explaining why the corresponding problem for three or more phases is fundamentally more difficult.Supported in part by ARO contract DAAL03-89-K-0039, DARPA contract F49620-87-C-0065, ONR grant N00014-88-K-0279, NSF grant DMS-8701895 and AFOSR grant 90-0090