Some remarks on metric and deformation

Abstract

This work is aimed at emphasising the relationship between metric and deformation, under the light of a novel formalism for the Polar Decomposition Theorem. All results are first presented in the classical formalism of Cauchy’s celebrated theorem, and then in the proposed alternative formalism. Although the latter requires a little more work to be established, it allows for directly defining all strain tensors as “covariant”, i.e. with both feet being covectors. Emphasis is also placed on how, in the absence of the metric structure, the available mathematical tools are restricted to the deformation gradient alone. Along with these main results, and in the didactical intention that permeates this work, several hints are given, which could be useful in teaching Continuum Mechanics, e.g. the rigorous definition of the determinant of the deformation gradient in Riemannian manifolds, and a caveat on the definition of the spatial Hencky logarithmic strain. The setting is that of modern Continuum Mechanics, based on the description given by Differential Geometry in terms of differentiable manifolds. However, passing to the simpler case of affine spaces takes almost no effort, paying attention to keeping the distinction between vectors and covectors, and therefore allowing the matrices representing the metric tensors to differ from the unit matrix.