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 $\mathrm{G}\mathrm{L}$^+(n)$andthemicrorotationfield$R: \Omega \to $\mathrm{S}\mathrm{O}$(n)$, the shear-stretch energy is necessarily of the form \begin{equation*} W_{\mu,\mu_c}(R,;F) := \mu,\left\lVert{$\mathrm{s}\mathrm{y}\mathrm{m}$(R^T F - \boldsymbol{1})}\right\rVert^2 + \mu_c,\left\lVert{$\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}$(R^T F - \boldsymbol{1})}\right\rVert^2;, \end{equation*} where $\mu >0$ is the Lam'e shear modulus and ${\mu }_c\ge 0$ is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations ${\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{R\in \mathrm{S}\mathrm{O}(n)}{W}_{\mu ,{\mu }_c}(R;F)$ as a function of $F\in {\mathrm{G}\mathrm{L}}^{+}(n)$ and weights $\mu >0$ and ${\mu }_c\ge 0$. 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 $\mu >{\mu }_c\ge 0$ in dimension $n\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}2$. Currently, optimality results for $n\ge 3$ are out of reach for us, but we contribute explicit representations for $n\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}2$ which we name ${\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}_{\mu ,{\mu }_c}^{\pm }(F)\in \mathrm{S}\mathrm{O}(2)$ and which arise for $n\text{\hspace{0.17em}⁣}=\text{\hspace{0.17em}⁣}3$ by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations ${\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}_{\mu ,{\mu }_c}^{\pm }(F_{\gamma })$ for simple planar shear.