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Objective rates as covariant derivatives on the manifold of Riemannian metrics

Boris Kolev, Rodrigue Desmorat

Abstract

The subject of so-called objective derivatives in Continuum Mechanics has along 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).

Objective rates as covariant derivatives on the manifold of Riemannian metrics

Abstract

The subject of so-called objective derivatives in Continuum Mechanics has along 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).

Paper Structure

This paper contains 26 sections, 18 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. The configuration space in Continuum Mechanics
  3. Strains and stresses
  4. The manifold of Riemannian metrics
  5. Weak Riemannian structure on $\mathrm{Met}(\mathcal{B})$
  6. Riemannian product structure on $\mathrm{Met}(\mathcal{B})$
  7. Covariant derivatives on $T\mathrm{Met}(\mathcal{B})$
  8. Geodesics and the Riemannian exponential mapping
  9. Covariant derivatives on $T^{\star}\mathrm{Met}(\mathcal{B})$
  10. General covariance -- Material frame indifference -- Objectivity
  11. Material and objective derivatives
  12. A converse theorem for local objective derivatives
  13. Objective rates in the literature
  14. Oldroyd objective rate
  15. Truesdell objective rate
  16. ...and 11 more sections

Key Result

Theorem 3.2

Along a deformation path $p(t)$ of embeddings in $\mathrm{Emb}^{\infty}(\mathcal{B},\mathcal{E})$, the Riemannian metric over $\mathcal{B}$, ${\bm{\gamma}}(t) = p(t)^{*}\mathbf{q}$, satisfies the evolution equation

Theorems & Definitions (78)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Theorem 3.2: Rougée, 1980, 1991
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 68 more