Dynamics of Microstructured Shells with Thickness Extension

The analysis of the natural frequencies of a plate model is carried out in this paper starting from a model of shells which incorporates the effect of thickness extension and that is derived from the Virtual Work Theorem using material coordinates in the deformed configuration. Moreover, the shell is regarded as a micro-structured body whose fibers are free to rotate and distend. Finally, introducing proper internal constraints, and suitable stress resultant definitions, the equilibrium equations are reduced, in the framework of properly modified Reissner-Mindlin kinematical assumptions, to ones formally equivalent to that that can be obtained in the framework of properly modified Kirchhoff-Love hypotheses, but with additional equations describing the equilibrium in the fiber direction. Using a numerical approach based on a finite difference scheme, it is shown how the naturalfrequencies of the Reissner-Mindlin model reduce when the Kirchhoff-Love constraints are retained. In particular, results indicate that, in the limit in which the Kirchhoff-Love hypotheses tend to become valid, the numerical frequencies of the pure Reissner-Mindlin model are affected by some round-off error, whereas in the case corresponding to the formulation adopted, where the solution is sought in terms of transversal displacementand dierence between the RM and the KL rotation of the fiber, the numerical solution reproduces exactly the analytical solution, and, notably, this behavior is emphasized at the highest frequencies. Therefore, in the case in which the transverse shear is treated independently one obtains good results, whereas, in the case in which the transverse shear has to be obtained as the difference of the derivative of the transverse displacement and of the fiber rotation, the numerical solution introduces numerical errors due to the closenessof the present model to the KL kinematical hypotheses.