Recently, Cherkaev, Lurie, and Milton (1992) established an invariance of stress field in planar linear anisotropic elasticity under a specific shift in bulk and shear moduli; this is now known as the CLM theorem. Motivated by the importance of micropolar models in mechanics of media with micropolar structures, Ostaja-Starzewski and Jasiuk (1995) generalized the CLM theorem to planar micropolar elastic materials and considered inhomogeneous simply-connected materials. The present study addresses inhomogeneous, multiply-connected materials (with holes), which require global compatibility conditions involving Cesa`ro integrals, as well as multi-phase simply-connected materials, where the interface conditions need to be considered. Just as in the previous paper, both of these cases display a reduction in the parameter space.

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