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

## TL;DR

This paper generalizes Grioli's Theorem to weighted cases, characterizing optimal Cosserat rotations with weights, revealing new globally minimizing rotations that can differ significantly from classical polar factors.

## Contribution

It introduces a weighted variational characterization of optimal rotations, extending Grioli's Theorem and identifying conditions for new types of energy-minimizing rotations.

## Key findings

- Classical Grioli's Theorem recovered for specific weights
- New globally minimizing rotations identified for different weight ranges
- Application to Cosserat nonlinear elasticity models

## Abstract

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 a symmetric, positive definite factor $U(F) = \sqrt{F^TF} \in {\rm Psym}(3)$. In 1940 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 to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations $$ {\rm rpolar}_{\mu,\mu_c}(F) \;:=\, {\rm argmin}_{R \in {\rm SO}(n)} \left\lbrace \mu\, ||{\rm sym}(R^TF - 1){||}^2 \;+\, \mu_c\, ||{\rm skew}(R^TF - 1){||}^2 \right\rbrace $$ for given weights $\mu > 0$ and $\mu_c \geq 0$. We identify a classical parameter range $\mu_c \geq \mu > 0$ for 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.