A Model for Polar Shells with Thickness Extension for Aeroacoustic Applications

The paper presents a novel exact formulation for the equilibrium equations of a shell-like body, whose material fibers, or directors, are free to distend and rotate. In this formulation the use of material coordinates, proper internal constraints added ad hoc, and the virtual-work approach, allows one to reformulate the Reissner-Mindlin model as a novel one that consists of the Kirchhoff-Love model plus an additional vector equation describing the difference between the two models. The ultimate aim of this approach is directed towards the actuation ofthe thickness distension feature of a piezoelectric shell for the control of the sound radiated from the shell surface itself. In the numerical application, in order to simplify the mathematical treatment of the problem, the case of a shell infinitely long in one direction experiencing only small deformations is considered.The numerical results and the equilibrium equations are limited to statics; furthermore, the shell is assumed to be loaded by means of an external electric field imposed across the boundaries, with the electric field constant across the thickness (ie a linear electric potential). As a consequence of this hypothesis, the first Maxwell's equation that governs the electrical unknowns of the problem, is considered to be independent of (but influencing) the displacement field. This is true as well for the constitutive equations that relate the electrical displacement field with the deformations and the electric potential. Consequently, one obtains an equivalent beam model capable of describing shear effects and thickness change. Finally, it should be noted that the results areintended to demonstrate the potentiality of the model foraero-acoustic control and so a special fiber distribution is assumed throughout the main surface.