Free vibrations of a polar elastic body

Abstract

This paper concerns certain features of the equations governing the time-harmonic free vibrations of a polar elastic body. The governing equations of micropolar elasticity are expressed in differential form; then, the uniqueness in their solutions is investigated. The conditions sufficient for the uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained for the free vibrations. The principle leads to all the governing equations of the free vibrations, as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh’s quotient is given, and the high-frequency vibrations of a micropolar plate [cf., Altay and Dkmeci, Int. J. Solids Struct. (2006)] are likewise treated. [Work supported by TUBA.]