Objectivity in continuum mechanics, an introduction. Motions, Eulerian and Lagrangian variables and functions, deformation gradient, Lie derivatives, velocity-addition formula, Coriolis

Abstract

In classical mechanics, there are two objectivities: The ``isometric objectivity'' which concerns the constitutive laws of materials once expressed in a reference frame, and the ``covariant objectivity'' which concerns the universal laws of physics required to be observer independent. Covariant objectivity is the main topic in this manuscript. We therefore first introduce motions and Eulerian variables and functions, then we deduce Lagrangian variables and functions, then define the function ``deformation gradient'', from which we deduce simple definitions of the Lie derivatives. Then we detail the construction of the velocity addition formula and recall the proof of the objective covariance of the Lie derivatives. And constituvive laws built from Lie derivatives are introduced.