Abstract
The application to continuum mechanics of the general methods of the classical theory of fields is advocated and illustrated by the example of the static elastic field. The non-linear theory of elasticity is set up in the most convenient form (lagrangian coordinates and stress tensor). The appropriate energy-momentum tensor is derived, and it is shown that the integral of its normal component over a closed surface gives the force (as the term is used in the theory of solids) on defects and inhomogeneities within the surface. Other topics discussed are Günther's and related integrals, symmetrization of the energy-momentum tensor, and the Eulerian formulation. Some further extensions, existing and potential, are indicated.
Résumé
On signale les avantages de l'application à la méchanique des milieux continus des méthodes de la théorie classique des champs. A titre d'example la théorie de l'élasticité est construite sous la forme la plus convenable a cette fin (coordonées de Lagrange, composantes de tension de Boussinesq). On déduit ensuite le tenseur d'énergie-impulsion dont l'intégrale de la composante normale étendue sur une surface fermée donne la force (comme on l'entend en théorie des solides) agissant sur les défaults dans son intérieur. On discute aussi quelques intégrales semblables, entr'elles celles de Günther, la symétrisation du tenseur d'énergie-impulsion et la formulation en coordonées d'Euler. On récapitule par conclure certains autres résultats déjà connus, tout en indiquant quelques extensions possibles.
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Eshelby, J.D. The elastic energy-momentum tensor. J Elasticity 5, 321–335 (1975). https://doi.org/10.1007/BF00126994
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DOI: https://doi.org/10.1007/BF00126994