Objective rates as covariant derivatives on the manifold of Riemannian metrics

The re-foundation and geometrization of Continuum Mechanics was initiated by C. Truesdell and W. Noll at the beginning of the second half of the 20th century. The starting point of this formulation is an abstract manifold of dimension 3 (with boundary), the body, and equipped with a volume form (the mass measure). In line with Eringen, Green and Zerna, Benzecri, Noll, Epstein and Segev, Rougée has furthermore rightly understood the fundamental role played in Continuum Mechanics by the manifold of Riemannian metrics on the body. On the other hand, the objectivity principle, or material frame indifference, is nowadays a cornerstone for the formulation of rate-form constitutive equations for solids and fluids. So called objective time-derivatives (or objective rates) have been proposed in the literature in order to introduce some kind of elasticity for viscous fluids or to derive computationally efficient formulations of finite strain elasto-(visco-)plasticity. Since a lot of objective derivatives are available in the literature, the natural question arose of how to unify them all, as well as of better understanding their intrinsic nature. In other words, to clarify the mathematical concept underlined. Marsden and Hughes did claim that all objective derivatives of the stress tensor were in fact Lie derivatives, or more precisely, linear combinations of Lie derivatives of the (contravariant) stress tensor and of its different covariant and mixed forms (obtained by lowering subscripts thanks to the Euclidean metric). However, the discovery, by Fiala in 2004, of a new objective rate which cannot be written this way, reopened the subject of their correct mathematical interpretation. In this talk, I will show, finally, that all objective derivatives correspond in fact to covariant derivatives on the infinite dimensional manifold of Riemannian metrics. Moreover, a natural Leibniz rule, which allows to define an objective derivative on contravariant tensor fields given one on covariant tensor fields and vice versa makes the distinction between those of them who are of ``Lie type'' or of ``co-rotational type'' useless.