On Grioli’s minimum property and its relation to Cauchy’s polar decomposition

Abstract In this paper we rediscover Grioli’s important work on the optimality of the orthogonal factor in the polar decomposition in an euclidean distance framework. We also draw attention to recently obtained generalizations of this optimality property in a geodesic distance framework.

[1]  Victor A. Eremeyev,et al.  On natural strain measures of the non-linear micropolar continuum , 2009 .

[2]  P. Seppecher,et al.  Edge Contact Forces and Quasi-Balanced Power , 1997, 1007.1450.

[3]  Yuji Nakatsukasa,et al.  The minimization of matrix logarithms - on a fundamental property of the unitary polar factor , 2013, 1308.1122.

[4]  L. Autonne.  Sur les groupes linéaires, réels et orthogonaux , 1902 .

[5]  F. Cosserat,et al.  Sur la théorie de l'élasticité. Premier mémoire , 1896 .

[6]  Nicholas J. Higham,et al.  Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD , 2013, SIAM J. Sci. Comput..

[7]  Patrizio Neff,et al.  Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm , 2013 .

[8]  C. Truesdell,et al.  The Classical Field Theories , 1960 .

[9]  Patrizio Neff,et al.  Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom. , 2008 .

[10]  P. Neff,et al.  Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity , 2009 .

[11]  Patrizio Neff,et al.  Subgrid interaction and micro-randomness – Novel invariance requirements in infinitesimal gradient elasticity , 2009 .

[12]  P. Neff,et al.  A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via Γ-convergence , 2007 .

[13]  Zhaojun Bai,et al.  Optimizing Halley's Iteration for Computing the Matrix Polar Decomposition , 2010, SIAM J. Matrix Anal. Appl..

[14]  Nicholas J. Higham,et al.  Backward Stability of Iterations for Computing the Polar Decomposition , 2012, SIAM J. Matrix Anal. Appl..

[15]  PATRIZIO NEFF,et al.  A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms , 2014, SIAM J. Matrix Anal. Appl..

[16]  Francesco dell’Isola,et al.  The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power , 1995 .

[17]  L. Russo.  The forgotten revolution , 2004 .

[18]  Patrizio Neff,et al.  A Riemannian approach to strain measures in nonlinear elasticity , 2014 .

[19]  W. Pietraszkiewicz,et al.  Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition , 2008 .

[20]  S. Vidoli,et al.  Generalized Hooke's law for isotropic second gradient materials , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[21]  A. Hoffman,et al.  Some metric inequalities in the space of matrices , 1955 .

[22]  Gerrit Bakker.  Sur la théorie de l'élasticité , 1901 .

[23]  Ralph Byers,et al.  A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability , 2008, SIAM J. Matrix Anal. Appl..

[24]  Y. Nakatsukasa,et al.  A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm , 2013, 1302.3235.

[25]  Patrizio Neff,et al.  Existence of minimizers for a finite-strain micromorphic elastic solid , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[26]  Silvio Levy,et al.  The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Reborn , 2004 .

[27]  Patrizio Neff,et al.  A Geometrically Exact Micromorphic Model for Elastic Metallic Foams Accounting for Affine Microstructure. Modelling, Existence of Minimizers, Identification of Moduli and Computational Results , 2007 .

[28]  L. C. Martins,et al.  An Elementary Proof of the Polar Decomposition Theorem , 1980 .

[29]  P. Neff,et al.  Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations , 2012, 1210.1251.

[30]  P. Neff.  Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation , 2005 .

[31]  C. Bouby,et al.  Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient , 2005 .

[32]  The Hencky strain energy ‖log U‖2 measures the geodesic distance of the deformation gradient to SO(n) in the canonical left-invariant Riemannian metric on GL(n) , 2013 .

[33]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[34]  Gene H. Golub,et al.  Matrix computations , 1983 .

[35]  Victor A. Eremeyev,et al.  On vectorially parameterized natural strain measures of the non-linear Cosserat continuum , 2009 .