Dynamics of convecting elastic solids

The dynamics of a class of convecting elastic media is considered. The general field equation governing small oscillations is derived variationally. The mathematical structure of the field equation thus derived is examined. The origins of the `gyroscopic' and the `centrifugal' effects are traced. These can be associated with various terms in the expression for the Lagrangian density. In particular, terms in the kinetic energy density that are independent the velocity field, those that are linear in the velocity field and those that are quadratic in the velocity field are associated with the centrifugal, gyroscopic, and inertia terms in the equation of motion respectively. A close mathematical analogy between the dynamics of this class of continua and the dynamics of discrete gyroscopic-centrifugal systems having fixed material particles is noted. The free vibration problem is posed in its generality. An appropriate Rayleigh quotient is defined. The stationarity associated with the quotient can potentially be used for computational work. Illustrative examples and applications are discussed.