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Let $F \in {\rm GL}^+(3)$ and consider the right polar decomposition $F =R_p(F)\cdot U$ into an orthogonal factor $R_p(F) \in {\rm SO}(3)$ and asymmetric, positive definite factor $U(F) = \sqrt{F^TF} \in {\rm Psym}(3)$. In1940 Giuseppe Grioli proved that $$ {\rm argmin}_{R \in {\rm SO}(3)} ||R^TF -1{||}^2 \quad=\quad \{\,R_p(F)\,\} \quad=\quad {\rm argmin}_{R \in {\rm SO}(3)}||F - R{||}^2\;. $$ This variational characterization of the orthogonal factor$R_p(F) \in {\rm SO}(n)$ holds in any dimension $n \geq 2$ (a result due toMartins and Podio-Guidugli). In a similar spirit, we characterize the optimalrotations $$ {\rm rpolar}_{\mu,\mu_c}(F) \;:=\, {\rm argmin}_{R \in {\rmSO}(n)} \left\lbrace \mu\, ||{\rm sym}(R^TF - 1){||}^2 \;+\, \mu_c\, ||{\rmskew}(R^TF - 1){||}^2 \right\rbrace $$ for given weights $\mu > 0$ and $\mu_c\geq 0$. We identify a classical parameter range $\mu_c \geq \mu > 0$ for whichGrioli'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 whichcan substantially deviate from $R_p(F)$. In mechanics, the weighted energysubject to minimization appears as the shear-stretch contribution in anygeometrically nonlinear, quadratic, and isotropic Cosserat theory.

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