Two-dimensional equations for thickness-stretch vibrations of plates

Two-dimensional equations for coupled extensional and thickness-stretch vibrations of elastic plates are simplified by eliminating the extensional displacements in a systematic manner; the result is a single equation governing thickness-stretch motions. A similar reduction is also performed for coupled extensional, thickness-stretch, and symmetric thickness-shear vibrations. The procedure is similar to that used in the thickness-shear approximation, wherein the flexural displacement is eliminated from the equations for coupled flexural and thickness-shear vibrations. The resulting equations are used to discuss the energy-trapped vibration of plates in thickness-stretch modes.