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Cauchy Relations in Linear Elasticity: Algebraic and Physics Aspects

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Abstract

The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure’s deviation from the basic physical assumptions of Cauchy’s model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke’s law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters.

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Notes

  1. We particularly appreciate the anonymous reviewer who provided us with such significant information and accurate references.

  2. However, in general, a decomposition of a tensor space under \(GL(3,\mathbb{R})\) need not be unique. Some explicit examples are given in [21].

  3. Equations (11), (12), and (13) first appeared in Backus [2], whose starting point seems to be the harmonic decomposition of (totally) symmetric tensors.

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Acknowledgements

F.-W. Hehl (Cologne), R.L. Fosdick (Minnesota), V. I. Alshits (Moscow), M. Vianello (Milan), A.H. Norris (Rutgers), M.B. Rubin (Haifa), and R. Segev (Beer Sheva) deserve my heartfelt thanks. I would want to thank all of the elasticity and acoustics experts with whom I have had extremely useful discussions (both online and in person) about various aspects of the Cauchy relations over the years. I appreciate the reviewers’ high level of expertise and their extremely helpful recommendations.

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  1. Jerusalem College of Technology, Jerusalem, Israel

    Yakov Itin

  2. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Yakov Itin

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Itin, Y. Cauchy Relations in Linear Elasticity: Algebraic and Physics Aspects. J Elast (2023). https://doi.org/10.1007/s10659-023-10035-8

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