Grioli's Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations

Let $F\in {\mathrm{G}\mathrm{L}}^{+}(3)$ and consider the right polar decomposition $F = R_p(F)\cdot U$intoanorthogonalfactor$R_p(F) \in {\rm SO}(3)$ and a symmetric, positive definite factor $U(F)=\sqrt{F^{T}F}\in \mathrm{P}\mathrm{s}\mathrm{y}\mathrm{m}(3)$. In 1940 Giuseppe Grioli proved that $${\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{R\in \mathrm{S}\mathrm{O}(3)}\mathrm{\mid }\mathrm{\mid }{R}^{T}F-1{\mathrm{\mid }\mathrm{\mid }}^2=\{R_p(F)\}={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{R\in \mathrm{S}\mathrm{O}(3)}\mathrm{\mid }\mathrm{\mid }F-R{\mathrm{\mid }\mathrm{\mid }}^2\mathrm{.}$$ This variational characterization of the orthogonal factor $R_p(F)\in \mathrm{S}\mathrm{O}(n)$ holds in any dimension $n\ge 2$ (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations $${\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{r}}_{\mu ,{\mu }_c}(F):={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{R\in \mathrm{S}\mathrm{O}(n)}\{\mu \mathrm{\mid }\mathrm{\mid }\mathrm{s}\mathrm{y}\mathrm{m}(R^{T}F-1){\mathrm{\mid }\mathrm{\mid }}^2+{\mu }_c\mathrm{\mid }\mathrm{\mid }\mathrm{s}\mathrm{k}\mathrm{e}\mathrm{w}(R^{T}F-1){\mathrm{\mid }\mathrm{\mid }}^2\}$$ for given weights $\mu >0$ and $\mu_c \geq 0$\mathrm{.}Weidentifyaclassicalparameterrange$\mu_c \geq \mu > 0$ for which Grioli’s Theorem is recovered and a non-classical parameter range $\mu > \mu_c \geq 0$ giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from $R_p(F)$. In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.