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 $F=R_p(F)\cdot U$ into an orthogonal factor $R_p(F)\in \mathrm{SO}(3)$ and a symmetric, positive definite factor $U(F)=\sqrt{F^{T}F}\in \mathrm{Psym}(3)$. In 1940 Giuseppe Grioli proved that $${\mathrm{argmin}}_{R\in \mathrm{SO}(3)}||R^{T}F-1{||}^2=\{R_p(F)\}={\mathrm{argmin}}_{R\in \mathrm{SO}(3)}||F-R{||}^2.$$ This variational characterization of the orthogonal factor $R_p(F)\in \mathrm{SO}(n)$ holds in any dimension $n\ge 2$ (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations $${\mathrm{rpolar}}_{\mu ,{\mu }_c}(F):={\mathrm{argmin}}_{R\in \mathrm{SO}(n)}\left\{\mu ||\mathrm{sym}(R^{T}F-1){||}^2+{\mu }_c||\mathrm{skew}(R^{T}F-1){||}^2\right\}$$ 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 $R_p(F)$. In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.

This is an overview paper collecting results distributed over three preceding papers (without proofs)

References in corpus (1)

• Integrability conditions between the first and second Cosserat deformation tensor in geometrically nonlinear micropolar models and existence of minimizers