Abstract
In this paper we make a connection between covariant elasticity based on covariance of energy balance and Lagrangian field theory of elasticity with two background metrics. We use Kuchař’s idea of reparametrization of field theories and make elasticity generally covariant by introducing a “covariance field”, which is a time-independent spatial diffeomorphism. We define a modified action for parameterized elasticity and show that the Doyle-Ericksen formula and spatial homogeneity of the Lagrangian density are among its Euler–Lagrange equations.
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References
Anderson J.L.: Principles of Relativity Physics. Academic Press, New York (1967)
Isham C.J., Kuchař K.V.: Representations of spacetime diffeomorphisms 1. Canonical parametrized field-theories. Ann. Phys. 164, 288–315 (1985)
Kuchař K.: Canonical quantization of gravity. In: Israel W. (ed.) Relativity, Astrophysics and Cosmology, pp. 237–288 (1973)
Lopez M.C., Gotay M.J., Marsden J.E.: Parametrization and stress-energy-momentum tensors in metric field theories. J. Phys. A 41, 344002: 1–10 (2008)
Lopez, M.C., Gotay, M.J.: Covariantizing classical field theories. arXiv:1008.3170v1 (2010)
Marsden J.E., Hughes T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (1983)
Ozakin A., Yavari A.: A geometric theory of thermal stresses. J. Math. Phys. 51, 032902 (2010)
Post E.J.: Formal Structure of Electromagnetics General Covariance and Electromagnetics. Dover, New York (1997)
Simo J.C., Marsden J.E.: On the rotated stress tensor and the material version of the Doyle-Ericksen formula. Arch. Ration. Mech. Anal. 86, 213–231 (1984)
Yavari, A., Marsden, J.E., Ortiz, M.: On the spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903: 85–112 (2006)
Yavari A.: A geometric theory of growth mechanics. J. Nonlinear Sci. 20(6), 781–830 (2010)
Yavari, A., Goriely, A.: Riemann-Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. (to appear)
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Yavari, A., Marsden, J.E. Covariantization of nonlinear elasticity. Z. Angew. Math. Phys. 63, 921–927 (2012). https://doi.org/10.1007/s00033-011-0191-7
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DOI: https://doi.org/10.1007/s00033-011-0191-7