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 $F := \nabla\varphi: \Omega \to \mathrm{GL}^+(n)$ and the microrotation field $R: \Omega \to \mathrm{SO}(n)$, the shear-stretch energy is necessarily of the form \begin{equation*} W_{\mu,\mu_c}(R\,;F) := \mu\,\left\lVert{\mathrm{sym}(R^T F - \boldsymbol{1})}\right\rVert^2 + \mu_c\,\left\lVert{\mathrm{skew}(R^T F - \boldsymbol{1})}\right\rVert^2\;, \end{equation*} where $\mu > 0$ is the Lam\'e shear modulus and $\mu_c \geq 0$ is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations $\mathrm{argmin}_{R\,\in\,\mathrm{SO}(n)}{W_{\mu,\mu_c}(R\,;F)}$ as a function of $F \in \mathrm{GL}^+(n)$ and weights $\mu > 0$ and $\mu_c \geq 0$. For $n \geq 2$, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: $(\mu, \mu_c) = (1,1)$ and $(\mu,\mu_c) = (1,0)$. In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range $\mu > \mu_c \geq 0$ in dimension $n\!=\!2$. Currently, optimality results for $n \geq 3$ are out of reach for us, but we contribute explicit representations for $n\!=\!2$ which we name $\mathrm{rpolar}^{\pm}_{\mu,\mu_c}(F) \in \mathrm{SO}(2)$ 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 rotations $\mathrm{rpolar}^\pm_{\mu,\mu_c}(F_\gamma)$ for simple planar shear.