A linear function defined on the space of elasticity tensors is a restricted invariant under a group of rotations G if it has an invariant restriction to a proper subspace which is larger than the set left fixed by the action of G itself. A necessary and sufficient condition for a function to be a restricted invariant is given using concepts related with isotypic decomposition, Haar integration and G -dependence. The result is applied to characterize isotropic and transversely isotropic restricted invariants.
Ting, T.Anisotropic elastic constants that are structurally invariants . Quarterly Journal of Mechanics and Applied Mathematics, 53, 511-523 (2000).
2.
Ting, T.Can a linear anisotropic elastic material have a uniform contraction under a uniform pressure ? Mathematics and Mechanics of Solids, 6, 235-243 (2001).
3.
Ahmad, F.Invariants and structural invariants of the anisotropic elasticity tensor . Quarterly Journal of Mechanics and Applied Mathematics, 55, 597-606 (2002).
4.
Golubitsky, M. , Stewart, I. and Schaeffer, D.Singularities and Groups in Bifurcation Theory Vol. 2, Springer, New York , 1985.
5.
Sattinger, D. and Weaver, O.Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics, Springer, Berlin , 1986.
6.
Gurtin, M.The linear theory of elasticity, in Mechanics of Solids II, ed. C. Truesdell , Vol. VIa/2 of Handbuch der Physik, pp. 1-295. Springer, Berlin , 1972.
7.
Forte, S. and Vianello, M.Symmetry classes for elasticity tensors . Journal of Elasticity, 43, 81-108 (1996).
8.
Gaffke, N. , Heiligers, B. and Offinger, R.Isotropic discrete orientation distributions on the 3D special orthogonal group . Linear Algebra and its Applications, 354, 119-139 (2002).
9.
Hochschild, G.The Structure of Lie Groups, Holden-Day, San Francisco, CA , 1965.
10.
Simon, B.Representations of Finite and Compact Groups, Graduate Studies in Mathematics Series. American Mathematical Society, Providence, RI , 1996.
11.
Backus, G.A geometrical picture of anisotropic elastic tensors . Reviews of Geophysics and Space Physics, 8, 633-671 (1970).
12.
Baerheim, R.Harmonic decomposition of the anisotropic elasticity tensor . Quarterly Journal of Mechanics and Applied Mathematics, 46, 391-419 (1993).
13.
Mochizuki, E.Spherical harmonic decomposition of an elastic tensor . Geophysical Journal, 93, 521-526 (1988).
14.
Onat, E.Effective properties of elastic materials that contain penny shaped voids . International Journal of Engineering Science, 22, 1013-1021 (1984).
15.
Spencer, A.A note on the decomposition of tensors into traceless symmetric tensors . International Journal of Engineering Science, 8, 475-481 (1970).
16.
Forte, S. and Vianello, M.Symmetry classes and harmonic decomposition for photoelasticity tensors . International Journal of Engineering Science, 35, 1317-1326 (1997).
17.
Forte, S. and Vianello, M.Functional bases for transversely isotropic and transversely hemitropic invariants of elasticity tensors . Quarterly Journal of Mechanics and Applied Mathematics, 51, 543-552 (1998).
