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Covariantization of nonlinear elasticity

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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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Authors and Affiliations

  1. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Arash Yavari

  2. Control and Dynamical Systems, California Institute of Technology, Pasadena, CA, 91125, USA

    Jerrold E. Marsden

Authors
  1. Arash Yavari
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  2. Jerrold E. Marsden
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Correspondence to Arash Yavari.

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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

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