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

Abstract

Let $F\in {\mathrm{GL}}^{+}(3)$ and consider the right polar decomposition $F={\mathrm{R}}_{\mathrm{p}}(F)\cdot U$ into an orthogonal factor ${\mathrm{R}}_{\mathrm{p}}(F)\in \mathrm{SO}(3)$ and a symmetric, positive definite factor $U=\sqrt{F^{T}F}\in \mathrm{Psym}(3)$. In 1940 Giuseppe Grioli proved that

This variational characterization of the orthogonal factor ${\mathrm{R}}_{\mathrm{p}}(F)\in \mathrm{SO}(n)$ holds in any dimension $n\ge 2$ (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations

for given weights $\mu >0$ and ${\mu }_c\ge 0$. We identify a classical parameter range ${\mu }_c\ge \mu >0$ for which Grioli’s Theorem is recovered and a non-classical parameter range $\mu >{\mu }_c\ge 0$ giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from ${\mathrm{R}}_{\mathrm{p}}(F)$. In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.