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On spatial and material covariant balance laws in elasticity

Yavari, Arash and Marsden, Jerrold E. and Ortiz, Michael (2006) On spatial and material covariant balance laws in elasticity. Journal of Mathematical Physics, 47 (4). Art. No. 042903. ISSN 0022-2488. http://resolver.caltech.edu/CaltechAUTHORS:YAVjmp06

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Abstract

This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal (referential) variations are equivalent to those resulting from vertical (spatial) variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of [openface R]3. Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call configurational inviscidity. In the last part of the paper, the connection between Noether's theorem and covariance is investigated. It is shown that the Doyle-Ericksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density.


Item Type:Article
Additional Information:©2006 American Institute of Physics (Received 6 January 2006; accepted 23 February 2006; published online 28 April 2006) The authors have benefited from discussions with M. Arroyo, K. Bhattacharya, M. Desburn, E. Kanso, J. K. Knowles, J. Lu, P. M. Mariano, P. Papadopoulos, Y. Tong, and M. Zielonka. The research of one of the authors (J.E.M.) was partially supported by the California Institute of Technology and NSF-ITR Grant No. ACI-0204932.
Subject Keywords:elasticity; continuum mechanics; Lagrangian field theory; covariance analysis; geometry
Record Number:CaltechAUTHORS:YAVjmp06
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:YAVjmp06
Alternative URL:http://dx.doi.org/10.1063/1.2190827
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:4639
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:30 Aug 2006
Last Modified:26 Dec 2012 09:00

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