Nonlinear dynamics of shallow spherical caps subjected to peripheral loading

Abstract

The behavior of spherical caps under radial dynamic edge loading in the form of step loading of infinite duration in the time domain has been investigated. The aim here is to present a mathematical model of shallow shells of revolution which may undergo snap-through buckling as a result of the radial displacement of the circular boundary. The snap-through buckling under such a loading condition is contrary to intuition and it seems not to have been previously observed. In this search it is observed that snap-through buckling is also possible under peripheral dynamic loading conditions. The amount of snapping is remarkable when the cap has an opening around the apex. A technological application of the peripheral type of loading is seen in metal-ceramic composite transducers achieved by installing a piezoelectric ceramic disk between two metal end caps. The radial motion of the ceramic is converted into a flex-tensional motion in the spherical caps. As a result, a large displacement is obtained in the perpendicular direction, which may result in snap-through buckling. For the numerical solution of the problem a computer program, using a linearized finite element incremental-iterative approach based on updated Lagrangian formulation is developed, and the whole process is accomplished using the Newmark method as the time integration scheme.