Abstract
We consider the old problem of finding a basis of polynomial invariants of the fourth rank tensorC of elastic moduli of an anisotropic material. DecomposingC into its irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), wherea andb are traceless symmetric second rank tensors, andD is completely symmetric and traceless fourth rank tensor (D ∈ T 4 ss).We obtain by reinterpreting the results of classical invariant theory a polynomial basis of invariants forD which consists of 9 invariants of degrees 2 to 10 in components ofD. Finally we use this result together with a well-known descriptin of joint invariants of a number of second-rank symmetric tensors to obtain joint invariants of the triplet (a, b, D) for ageneric D.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
H. Weyl,The Classical Groups, their Invariants and Representations. Princeton Univ. Press (1946).
E. T.Onat, Effective properties of elastic materials that contain penny shaped voids,Int. J. Eng. Sci. 22 (1984) 1013–1021.
J. H. Grace and A. Young,The Algebra of Invariants, Cambridge Univ. Press (1903) (reprinted by Chelsea Publ. Co. in 1965).
J. Sylvester,Tables of the generating functions and groundforms for the binary quantics of the first ten orders, Collected mathematical papers of J. J. Sylvester, vol. III, Cambridge Univ. Press (1909) pp. 283–311. J. Sylvester,Sur les covariants irréductibles du quantic binaire de huitième order, ibid., 481–488.
vonGall,Das vollständige Formensystem einer binären Form achter Ordnung.Matematische Annalen 17 (1880) 31–51, 139–152, 456.
ShiodaTetsuji,On the graded ring of the invariants of binary octavic.Am. J. Math 89 (1967) 1022–1046.
J. P.Boehler,On irreducible representations for isotropic scalar functions.Zeitschrift für Angewandte Mathematik und Mechanik 57 (1977) 323–327.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boehler, J.P., Kirillov, A.A. & Onat, E.T. On the polynomial invariants of the elasticity tensor. J Elasticity 34, 97–110 (1994). https://doi.org/10.1007/BF00041187
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00041187