Stress invariance in planar Cosserat elasticity

Abstract

We extend the CLM theorem (Cherkaev, Lurie & Milton, Proc. R. Soc. Lond. A (1992) 438, 519-529) to planar linear elastic materials of Cosserat (micropolar) type, that is those having microrotation as an additional degree of freedom besides the two in-plane displacements. More specifically, it is shown that in the first planar problem of such media with smoothly varying material properties, a shift in three, out of four, compliances is possible without changing the stress field; the fourth coefficient represents a connection between the couple stress tensor and the torsion tensor, while the other three represent compliances relating traction-stress vector with the strains. Same shift holds for a locally anisotropic material, whereby the shift tensor is seen to be a multiple of a rotation by a right angle; a null-Lagrangian formulation is set up on this basis. These results are obtained for materials with smooth properties, with natural implications being drawn for their macroscopically effective moduli. Also, it is shown that no shift is possible in case of a pseudo-continuum – a material admitting couple stresses but restricted to have the same connection between rotation and displacement gradient as in classical elasticity. Finally, we establish that there is no shift in the second planar problem which represents a micropolar generalization of the classical out-of-plane elasticity.