Journal of Elasticity 27: 267-279, 1992. 267 © 1992 Kluwer Academic Publishers. Printed in the Netherlands. On the local measures of mean rotation in continuum mechanics LUIZ C. MARTINS ~ and P. PODIO-GUIDUGLI z,* 1Department of Mechanical Engineering - COPPE, Federal University of Rio de Janeiro - C.P. 68503, 21945 Rio de Janeiro, Brazil; 2Dipartimento di Ingegneria Civile, Universitd~ di Roma 2 - via O. Raimondo, 00173 Roma, Italy (*author for correspondence) Received 17 August 1989 Abstract. We implement Cauchy's concept of a rotation-angle function on an oriented plane, and characterize situations when a rotation-angle function exists, and hence when measuring mean rotations in the manner of Cauchy or Novozhilov makes sense. We also discuss in passing the role of the skew part of the deformation gradient in measuring the mean deformation. AMS (MOS) Subject Classification: 73B15. 1. Introduction The theory of finite strain and rotation is essentially due to Cauchy [1], [2], [3]. A masterly modern exposition of the theory, accompanied by many illuminating historical notes, is found in Sections 25-39 of [4]. We read in the beginning of Section 36 that "Cauchy took as a measure of local rotation the mean values of the angles through which all elements in each of three perpendicular planes are turned". For {i, j, k} three orthogonal unit vectors along a given cartesian reference, let n~ = cos 9i + sin ~j be a unit vector perpendicular to the z-axis, and let ms = Fn~ be the image of nz in a deformation of gradient F. Cauchy considered the angle q~(n~) between nz and the projection Pkmz, with Pk-'=I-k®k, of m~ on the x-y plane; he then proposed to determine tp(nz) from an expression for tan tp(n~) that he was able to derive in terms of Pk FPk only, and to measure the mean rotation about the z-axis as <q,z>,=1 fo 2" 2~z ~O(Oz(0)). (1.1) Cauchy's measure of mean rotation was never used; as elegant as it is, it has various drawbacks, the main one being perhaps that the mean rotations about three perpendicular axes do not suffice to determine the rotation of a typical line element.