The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

Abstract

In any geometrically nonlinear quadratic Cosserat‐micropolar extended continuum model formulated in the deformation gradient field and the microrotation field , the shear–stretch energy is necessarily of the form urn:x-wiley:00442267:media:zamm201500194:zamm201500194-math-0003where is the Lamé shear modulus and is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations as a function of and weights and . For , we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: and . In contrast to Grioli's theorem, we derive non‐classical minimizers for the parameter range in dimension . Currently, optimality results for are out of reach for us, but we contribute explicit representations for which we name and which arise for by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non‐classical optimal Cosserat rotations for simple planar shear.