Dual integrals in small strain elasticity with body forces

Vlado A. Lubarda; Xanthippi Markenscoff

doi:10.1016/j.crme.2007.11.009

Dual integrals in small strain elasticity with body forces
[Intégrales duales en élasticité infinitésimale avec forces de masse]

Vlado A. Lubarda ¹ ; Xanthippi Markenscoff ¹

¹ Department of Mechanical and Aerospace Engineering, University of California, San Diego; La Jolla, CA 92093-0411, USA

Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 190-202.

Résumé
Abstract

Des intégrales duales en élasticité infinitésimale sont obtenues et reliées aux taux de restitution d'énergie associés au mouvement d'un défaut en présence de forces de masse. On définit un tenseur d'énergie–impulsion qui inclut un terme de travail des forces de masse, et qui fournit des expressions simples des forces configurationnelles en fonction des intégrales $J_{k}$ , $L_{k}$ et M. Du fait que l'énergie potentielle complémentaire n'inclut pas explicitement les forces de masse, le tenseur d'énergie-impulsion complémentaire a la même structure qu'en élasticité sans forces de masse. Les expressions des intégrales non-conservées $J_{k}$ , $L_{k}$ et M et de leurs duales sont reliées à des intégrales de volume dont les intégrandes dépendent des forces de masse et de leurs gradients. Si les forces de masse sont spatialement uniformes, les lois de conservation $J_{k} = {\hat{J}}_{k} = 0$ s'appliquent aux problèmes tant 2D que 3D, de même que la loi $L_{3} = {\hat{L}}_{3} = 0$ aux problèmes antiplans. La loi de conservation $M = \hat{M} = 0$ s'applique en l'absence de forces de masse ou si ce sont des fonctions homogènes de degré particulier des coordonnées.

Dual integrals of small strain elasticity are derived and related to the energy release rates associated with a defect motion in the presence of body forces. A modified energy momentum tensor is used, which includes a work term due to body forces, and which yields simple expressions for the configurational forces in terms of the $J_{k}$ , $L_{k}$ , and M integrals. Since the complementary potential energy does not include body forces explicitly, the complementary energy momentum tensor has the same structure as in the elasticity without body forces. The expressions for the nonconserved $J_{k}$ , $L_{k}$ , and M integrals, and their duals, are related to the volume integrals whose integrands depend on body forces and their gradients. If body forces are spatially uniform, the conservation laws $J_{k} = {\hat{J}}_{k} = 0$ hold for both 2D and 3D problems, and $L_{3} = {\hat{L}}_{3} = 0$ for the antiplane strain problems. The conservation law $M = \hat{M} = 0$ holds if body forces are absent, or if they are homogeneous functions of particular degree in spatial coordinates.

Publié le : 2008-01-01

DOI : 10.1016/j.crme.2007.11.009

Keywords: Computational solid mechanics, Complementary energy, Body force, Configurational force, Dual integrals, Energy momentum tensor, Potential energy
Mot clés : Mécanique des solides numérique, Énergie complémentaire, Forces de masse, Forces configurationnelles, Intégrales duales, Tenseur d'énergie–impulsion, Énergie potentielle

Affiliations des auteurs :

Vlado A. Lubarda ¹ ; Xanthippi Markenscoff ¹

¹ Department of Mechanical and Aerospace Engineering, University of California, San Diego; La Jolla, CA 92093-0411, USA

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     title = {Dual integrals in small strain elasticity with body forces},
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Vlado A. Lubarda; Xanthippi Markenscoff. Dual integrals in small strain elasticity with body forces. Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 190-202. doi : 10.1016/j.crme.2007.11.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.11.009/

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