Vol. 3, No. 4, 2015

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ISSN: 2325-3444 (e-only)
ISSN: 2326-7186 (print)
Spatial and material stress tensors in continuum mechanics of growing solid bodies

Jean-François Ganghoffer

Vol. 3 (2015), No. 4, 341–363
Abstract

We presently derive generalized expressions of the stress tensor for continuum bodies with varying mass, considering both the Lagrangian and Eulerian viewpoints in continuum mechanics. We base our analysis and derivation of the expressions of both Cauchy and Eshelby stress tensors on an extension of the virial theorem for both discrete and continuous systems of material points with variable mass. The proposed framework is applicable to describe physical systems at very different scales, from the evolution of a population of biological cells accounting for growth to mass ejection phenomena occurring within a collection of gravitating objects at the very large astrophysical scales. As a starting basis, the field equations in continuum mechanics are written to account for a mass source and a mass flux, leading to a formulation of the virial theorem accounting for a varying mass within the considered system. The scalar and tensorial forms of the virial theorem are written successively in both Lagrangian and Eulerian formats, incorporating the mass flux. This delivers generalized formal expressions of Cauchy and Eshelby stress tensors versus the average tensor spatial and material virials respectively, incorporating the mass flux contribution.

Keywords
continuum bodies with changing mass, generalized virial theorem, discrete mechanics, mass flux, Eshelby stress
Mathematical Subject Classification 2010
Primary: 74B20
Secondary: 70FXX
Milestones
Received: 2 May 2015
Revised: 26 August 2015
Accepted: 19 October 2015
Published: 12 February 2016

Communicated by Francesco dell'Isola
Authors
Jean-François Ganghoffer
LEMTA
Université de Lorraine
2, Avenue de la Forêt de Haye, TSA 60604
54504 Vandoeuvre
France