Variational Methods for Eigenvalue Problems in Composites

TLDR: In this paper, a survey of various methods for effective estimation of the eigenvalues of a single discontinuous coefficient in a composite material mechanics problem is provided, where the main idea is to transform the one-dimensional Sturm-Liouville problems of concern to Liouville normal form.

Abstract: Eigenvalue problems with discontinuous coefficients occur naturally in many areas of composite material mechanics In previous work, based on mixed variational schemes, an approximation technique of Rayleigh-Ritz type applied to a modified “new quotient” has been developed by Nemat-Nasser and coworkers and applied in estimating eigenvalues and eigenfunctions for such problems in a wide variety of contexts Alternative approaches, resulting from modification of classical Sturm-Liouville theory, have been established recently by the present authors The central idea is to transform the one-dimensional Sturm-Liouville problems of concern to Liouville normal form This leads to a problem with a single discontinuous coefficient which moreover occurs in an undifferentiated term Eigenvalue estimates based on the transformed problem are established This paper provides a survey of these various methods for effective estimation of the eigenvalues of such problems Related issues arising in the area of eigenvalue optimization are briefly discussed