The Truesdell rate in Continuum Mechanics

What today we call Truesdell rate of the Cauchy stress was originally introduced by Truesdell, as a frame-indifferent measure of stress rate for his theory of hypoelasticity. This work aims at showing that the Truesdell rate is in fact a more general concept, which arises from the application of Reynolds’ Transport Theorem. In the customary three-dimensional setting, the Transport Theorem can be formulated in terms of differential forms, i.e., three-forms for volume integrals and two-forms for surface integrals, in which case the only differential operator involved is the Lie derivative with respect to the velocity field of the continuum body. Alternatively, the Transport Theorem can be formulated in terms of the pseudo-scalar density associated with a three-form and the pseudo-vector flux density associated with a two-form. In this alternative approach, the differential operator involved is precisely the Truesdell rate. As an example, we shall show how the “convected derivative” of the theory of Electromagnetism is in fact a Truesdell rate.