Summary
A newly developed approach, based on the spectral decomposition principle, which is especially useful in crystallography, is applied in this paper. The compliance fourth-rank tensor of crystalline media belonging to the monoclinic system is spectrally decomposed, its eigenvalues are evaluated, together which its elementary idempotent tensors, which expand uniquely the fourth-rank tensor space into orthogonal subspaces. Next, the compliance tensor is spectrally analysed for anisotropic media of the orthorhombic, tetragonal, hexagonal and cubic crystal systems, by regarding these decompositions as particular cases of the spectral decomposition for monolinic media. Consequently, the characteristic values and the idempotent fourth-rank tensors are derived, as well as the stress and strain second-rank eigentensors for all the above mentioned symmetries.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rychlewski, J.: On Hooke's law. Prikl. Matem. Mekhan.48, 303–314 (1984).
Thomson, W. K.: Elements of a mathematical theory of elasticity. Phil. Trans. R. Soc.166, 481–498 (1856).
Thomson, W. K.: Encyclopaedia Britannica. Edinburgh: Adam and Charles Black 1878.
Rychlewski, J.: Unconventional approach to linear elasticity. Arch. Mech. Stosow.47, 149–171 (1995).
Blinowski, A., Ostrowska-Maciejewska, J.: On the elastic orthotropy. Arch. Mech. Stosow.48, 129–141 (1996).
Theocaris, P. S., Philippidis, T. P.: Variational bounds on the eigenangle ω of transversely isotropic materials. Acta Mech.85, 13–26 (1990).
Theocaris, P. S., Philippidis, T. P.: Spectral decomposition of compliance and stiffness fourth-rank tensors suitable for orthotropic materials. ZAMM71, 161–171 (1991).
Mehrabadi, M. M., Cowin, S. C.: Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math.43, 15–41 (1990).
Cowin, S. C., Mehrabadi, M. M.: On the structure of the linear anisotropic elastic symmetries. J. Mech. Phys. Solids40, 1459–1472 (1992).
Sutcliffe, S.: Spectral decomposition of the elasticity tensor. J. Appl. Mech.59, 762–773 (1992).
Theocaris, P. S., Sokolis, D. P.: Spectral decomposition of the compliance tensor for anisotropic plates. J. Elasticity51, 89–103 (1998).
Theocaris, P. S., Sokolis, D. P.: Spectral decomposition of the linear elastic tensor for monoclinic symmetry. Acta Cryst.A55, 635–647 (1999).
Hooke, R.: Lecture de potentia restitutiva, or of spring, explaining the power of springing bodies to which are added some collections. London: Martin 1678.
Love, A. E. H.: A treatise on the mathematical theory of elasticity, 4th ed. Cambridge: University Press 1927.
Sokolnikoff, I. S.: Mathematical theory of elasticity. New York: McGraw-Hill 1946.
Timoshenko, S. P., Goodier, J.: Theory of elasticity, 2nd ed. New York: McGraw-Hill 1951.
Kittel, C.: Introduction to solid state physics, 2nd ed. New York: Wiley; London: Chapman and Hall 1956.
Voigt, W.: Lehrbuch der Kristallphysik. Leipzig: Teubner (1910, reprinted 1928).
Nye, J. F.: Physical properties of crystals: their representation by tensors and matrices. Oxford: Clarendon Press 1957.
Hearmon, R. F. S.: An introduction to applied anisotropic elasticity. Oxford: Clarendon Press 1961.
Lekhnitskii, S.: Theory of elasticity of an anisotropic elastic body. San Francisco: Holden-Day 1963.
Neumann, F.: Vorlesungen über die Theorie der Elastizität. Leipzig 1885.
Mason, W. P.: Crystal physics of interaction processes. New York London: Academic Press 1966.
Gurtin, M. E.: The linear theory of elasticity. Encylopaedia of Physics, vol. vla/2. Berlin: Springer 1972.
Thurston, R. N.: Waves in solids. Encyclopaedia of Physics, vol. vla/4. Berlin: Springer 1974.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Theocaris, P.S., Sokolis, D.P. Spectral decomposition of anisotropic elasticity. Acta Mechanica 150, 237–261 (2001). https://doi.org/10.1007/BF01181814
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01181814