Dynamics of convecting elastic solids

Bhaskar, A (2010) Dynamics of convecting elastic solids. Days on Diffraction 2010 - International Conference, Saint Petersburg, Russian Federation. 07 - 10 Jun 2010.

Abstract

The dynamics of a class of convecting elastic media is considered. On the basis of an appropriate variational principle, the general field equation governing small oscillations is derived. The variational formulation demands (i) conservations of mass, (ii) conservation of energy, and (iii) conservation of the identity of particles. Of these, conservation of mass needs to be satisfied explicitly as a constraint. This is achieved by constraining the classical mechanical Lagrangian using a Lagrange multiplier with the continuity equation. Hamilton's principle modified for a control volume in this way then leads to the equation of motion for small oscillations of convecting gyroelastic solids. 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.

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