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

In any geometrically nonlinear quadratic Cosserat‐micropolar extended continuum model formulated in the deformation gradient field F : = ∇ φ : Ω →GL + ( n )and the microrotation field R : Ω → SO ( n ) , the shear–stretch energy is necessarily of the formW μ , μ c( R ; F ): = μsym ( R T F − 1 ) 2 + μ cskew ( R T F − 1 ) 2,where μ > 0 is the Lamé shear modulus andμ c ≥ 0 is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotationsargmin R ∈SO ( n )W μ , μ c( R ; F )as a function of F ∈GL + ( n )and weights μ > 0 andμ c ≥ 0 . For n ≥ 2 , we prove a parameter reduction lemma which reduces the optimality problem to two limit cases:( μ , μ c ) = ( 1 , 1 )and( μ , μ c ) = ( 1 , 0 ) . In contrast to Grioli's theorem, we derive non‐classical minimizers for the parameter range μ > μ c ≥ 0 in dimension n = 2 . Currently, optimality results for n ≥ 3 are out of reach for us, but we contribute explicit representations for n = 2 which we namerpolarμ , μ c ± ( F ) ∈ SO ( 2 )and which arise for n = 3 by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non‐classical optimal Cosserat rotationsrpolarμ , μ c ± ( F γ )for simple planar shear.