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Journal of Elasticity

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14-06-2022

A New Proof That the Number of Linear Elastic Symmetries in Two Dimensions Is Four

Authors: Jeremy Trageser, Pablo Seleson

Published in: Journal of Elasticity | Issue 2/2022

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Abstract

We present an elementary and self-contained proof that there are exactly four symmetry classes of the elasticity tensor in two dimensions: oblique, rectangular, square, and isotropic. In two dimensions, orthogonal transformations are either reflections or rotations. The proof is based on identification of constraints imposed by reflections and rotations on the elasticity tensor, and it simply employs elementary tools from trigonometry, making the proof accessible to a broad audience. For completeness, we identify the sets of transformations (rotations and reflections) for each symmetry class and report the corresponding equations of motions in classical linear elasticity.

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There does not appear to be a consensus in the literature for the naming of the symmetry classes in two dimensions. For instance, in [1] the symmetry classes are labeled monoclinic, orthotropic, tetragonal, and isotropic.

For (25a) and (25c), we employ the trigonometric identity \(\sin ^{4}(x) + \cos ^{4}(x) = 1 - \frac{1}{2}\sin ^{2}(2x)\) in combination with \(C_{1111} = C_{2222}\) to obtain (30).

Auffray, N., Kolev, B., Olive, M.: Handbook of bi-dimensional tensors: Part I: Harmonic decomposition and symmetry classes. Math. Mech. Solids 22(9), 1847–1865 (2017) MathSciNetCrossRef

Blinowski, A., Ostrowska-Maciejewska, J., Rychlewski, J.: Two-dimensional Hooke’s tensors - isotropic decomposition, effective symmetry criteria. Arch. Mech. 48(2), 325–345 (1996) MathSciNetMATH

Bóna, A., Bucataru, I., Slawinski, M.A.: Material symmetries of elasticity tensors. Q. J. Mech. Appl. Math. 57(4), 583–598 (2004) MathSciNetCrossRef

Chadwick, P., Vianello, M., Cowin, S.C.: A new proof that the number of linear elastic symmetries is eight. J. Mech. Phys. Solids 49(11), 2471–2492 (2001) MathSciNetCrossRef

Dummit, D.S., Foote, R.M.: Abstract Algebra, 3rd edn. Wiley, New York (2004) MATH

Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elast. 43(2), 81–108 (1996) MathSciNetCrossRef

Forte, S., Vianello, M.: A unified approach to invariants of plane elasticity tensors. Meccanica 49(9), 2001–2012 (2014) MathSciNetCrossRef

He, Q.-C., Zheng, Q.-S.: On the symmetries of 2D elastic and hyperelastic tensors. J. Elast. 43(3), 203–225 (1996) CrossRef

Timoshenko, S.: Theory of Elasticity. Engineering Societies Monographs. McGraw-Hill, New York (1934) MATH

10.

Ting, T.C.T.: Anisotropic Elasticity: Theory and Applications. Oxford University Press, London (1996) CrossRef

11.

Ting, T.C.T.: Generalized Cowin–Mehrabadi theorems and a direct proof that the number of linear elastic symmetries is eight. Int. J. Solids Struct. 40(25), 7129–7142 (2003) MathSciNetCrossRef

12.

Tsai, S.W., Pagano, N.J.: Invariant properties of composite materials. Tech. Rep. AFML-TR-67-349, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio (1968)

13.

Vannucci, P.: Plane anisotropy by the polar method. Meccanica 40(4), 437–454 (2005) MathSciNetCrossRef

14.

Vianello, M.: An integrity basis for plane elasticity tensors. Arch. Mech. 49(1), 197–208 (1997) MathSciNetMATH

Title: A New Proof That the Number of Linear Elastic Symmetries in Two Dimensions Is Four
Authors: Jeremy Trageser
Pablo Seleson
Publication date: 14-06-2022
Publisher: Springer Netherlands
Published in: Journal of Elasticity / Issue 2/2022
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-022-09902-7

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