Free vibrations of a polar body at elastic range

Abstract

<p>The purpose of this paper is to study certain features of the equations governing the time-harmonic free vibrations of a polar body at elastic range. The governing equations of micropolar elasticity are expressed in differential form, and then, the uniqueness of their solutions is investigated. The conditions sufficient for 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 that 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. As an application, the high-frequency vibrations of an elastic plate are treated.</p>