JournalsrlmVol. 28, No. 3pp. 573–600

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

  • Andreas Fischle

    Technische Universität Dresden, Germany
  • Patrizio Neff

    Universität Duisburg-Essen, Essen, Germany
Grioli’s Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations cover
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Abstract

Let FGL+(3)F \in \mathrm {GL}^+(3) and consider the right polar decomposition F=Rp(F)UF = \mathrm {R_p}(F) \cdot U into an orthogonal factor Rp(F)SO(3)\mathrm {R_p}(F) \in \mathrm {SO}(3) and a symmetric, positive definite factor U=FTFPsym(3)U = \sqrt{F^TF} \in \mathrm {Psym}(3). In 1940 Giuseppe Grioli proved that \begin{align*} \argmin{R\,\in\,\SO(3)}{\hsnorm{R^TF - \id}^2} \quad=\quad \{\,\polar(F)\,\} \quad=\quad \argmin{R\,\in\,\SO(3)}{\hsnorm{F - R}^2}\;. \end{align*} This variational characterization of the orthogonal factor Rp(F)SO(n)\mathrm {R_p}(F) \in \mathrm {SO}(n) holds in any dimension n2n \geq 2 (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations

rpolarμ,μc(F)  \eqdef  argminRSO(n){μ\hsnorm\sym(RTF\id)2  +  μc\hsnorm\skew(RTF\id)2}\mathrm {rpolar}_{\mu,\mu_c}(F) \; \eqdef \; \mathrm {arg min}{R\,\in\,\mathrm {SO}(n)} {\,\left\lbrace \mu \hsnorm{\sym(R^TF - \id)}^2 \;+\; \mu_c\hsnorm{\skew(R^TF - \id)}^2 \right\rbrace\,}

for given weights μ>0\mu > 0 and μc0\mu_c \geq 0. We identify a classical parameter range μcμ>0\mu_c \geq \mu > 0 for which Grioli’s Theorem is recovered and a non-classical parameter range μ>μc0\mu > \mu_c \geq 0 giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from Rp(F)\mathrm {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.

Cite this article

Andreas Fischle, Patrizio Neff, Grioli’s Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 3, pp. 573–600

DOI 10.4171/RLM/777