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Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2024

Objective rates as covariant derivatives on the manifold of Riemannian metrics

Boris Kolev

Résumé

The subject of so-called objective derivatives in Continuum Mechanics has a long history and has generated varying views concerning their true mathematical interpretation. Several attempts have been made to provide a mathematical definition that would at least partially unify the existing notions. In this paper, we demonstrate that, under natural assumptions, all objective derivatives correspond to covariant derivatives on the infinite-dimensional manifold Met(B) of Riemannian metrics on the body. Furthermore, a natural Leibniz rule enables canonical extensions from covariant to contravariant tensor fields and vice versa. This makes the sometimes-used distinction between objective derivatives of “Lie type” and “corotational type” unnecessary. For an exhaustive list of objective derivatives found in the literature, we exhibit the corresponding covariant derivative on Met(B).
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Dates et versions

hal-03246478 , version 1 (02-06-2021)
hal-03246478 , version 2 (17-07-2024)

Identifiants

Citer

Boris Kolev, Rodrigue Desmorat. Objective rates as covariant derivatives on the manifold of Riemannian metrics. Archive for Rational Mechanics and Analysis, 2024, 248 (4), pp.66. ⟨10.1007/s00205-024-02010-x⟩. ⟨hal-03246478v2⟩
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