Upper and Lower Bounds on the N th Eigenvalue of Vibrating Anisotropic Solids

Abstract

Within the context of the linear theory of elasticity, it is shown that, for any arbitrary anisotropic material having up to 21 independent elastic constants, one can compute with relative ease so-called lower-bound and upper-bound isotropic moduli that consistently yield lower and higher eigenvalues, respectively, than the anisotropic material. Computations are carried out for a variety of common cubic, hexagonal, tetragonal, trigonal, and orthorhombic crystalline materials. It is found that the free-vibration frequencies of these materials can usually be predicted by the bounding isotropic moduli with an accuracy of 9.2% or better. The application of such bounds is particularly promising for the numerous boundary-value problems in wave propagation in which exact solutions currently exist only for the isotropic case.