Frequency correspondence between membranes and functionally graded spherical shallow shells of polygonal planform

The present work presents further development of the linking relationships between vibration frequencies predicted by different theories, and they are extended from a flat plate to a spherical shallow shell. In analogy with the membrane vibration problem, exact correspondences are found for vibration frequencies of a functionally graded spherical shallow shell using the classical theory and the first-order and third-order shear deformation theories. Only the predominantly stretching and thickness-shear vibration of dilatational type and predominantly flexural vibration are considered in this work. They are decoupled from the predominantly stretching and thickness-shear vibration of rotational type. These results apply to a simply supported functionally graded spherical shallow shell of polygonal planform with arbitrarily varying material properties in the thickness direction. A Winkler–Pasternak elastic foundation and rotary inertias are incorporated. It is proved that the mathematical analogy warrants positive free vibration frequencies for the shallow shell. Mori–Tanaka's scheme is used to estimate the material properties in the numerical results.