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The Quarterly Journal of Mechanics and Applied Mathematics 1987 40(3):431-448; doi:10.1093/qjmam/40.3.431
© 1987 by Oxford University Press
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INVARIANTS OF ANISOTROPIC ELASTIC CONSTANTS

T. C. T. TING

( Department of Civil Engineering, Mechanics and Metallurgy, University of Illinois at Chicago PO Box 4348, Chicago, Illinois 60680, USA )

There are certain combinations of the elastic constants of anisotropic materials which remain invariant under an orthogonal transformation. When the transformation is completely arbitrary, we obtain eleven invariants, two of which are linear invariants known in the literature. The other nine are new and are nonlinear combinations of elastic constants up to sixth order. Each invariant breaks up into more than one invariant when the transformation is limited to a rotation about the x3-axis. In this case, explicit expressions are presented for five linear invariants and nine second-order invariants. Most of the nine second-order invariants are new. The five linear invariants which can be found in the literature were in many cases obtained by assuming that the material is orthotropic. Since no material symmetry is assumed in this paper, all invariants presented here are valid for any anisotropic material. For the rotation about the x3-axis the transformation can be shown to consist of six uncoupled transformations. The invariants for some of these transformations are shown to be complete. By considering the invariants due to coupling of these transformations, we obtain six more second-order invariants resulting in a total of 15 independent second-order invariants. The number of invariants increases dramatically for the higher-order invariants, most of which are from coupling of the transformations.


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T. Ting and Q-C He
Decomposition of elasticity tensors and tensors that are structurally invariant in three dimensions
Q J Mechanics Appl Math, August 1, 2006; 59(3): 323 - 341.
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