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.

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Mathematical Subject Classification 2010

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