Stress invariance and exact relations in the mechanics of composite materials: Extensions of the CLM result – A review

We focus on the remarkable result in mechanics of composite materials which is due to Cherkaev, Lurie, and Milton [Cherkaev, A., Lurie, K., Milton, G.W., 1992. Invariant properties in the stress in plane elasticity and equivalence classes in composites. Proc. R. Soc. Lond. A 438, 519–529]. It pointed out the invariance in the stress field in planar linear elastic materials, subjected to tractions, under a shift in planar compliances and showed that the effective elastic compliances of such materials undergo the same shift. These findings give rise to the reduced parameter dependence and exact relations for this class of materials. We summarize this result in a unified way and review its extensions to other classes of materials which include multi-phase materials with perfectly bonded and slipping interfaces in the contexts of planar linear elasticity, and to the planar elasticity with body forces and eigenstrains, planar micropolar elasticity, planar piezoelectricity, and three-dimensional linear elasticity.