Chaotic Behavior Resulting in Transient and Steady State Instabilities of Pressure-Loaded Shallow Spherical Shells
In this paper, the axisymmetric dynamic behavior and snap-through buckling of thin elastic shallow spherical shells under harmonic excitation is investigated. Based on Marguerre kinematical assumptions, the governing partial differential equations of motion for a pre-loaded cap are presented in the form of a compatibility equation and a transverse motion equation. The continuous model is reduced to a finite degree of freedom system using the Galerkin method and a Fourier-Bessel approach. Results show that pre-loaded shells may exhibit co-existing stable equilibrium states and that with the application of sufficiently large dynamic loads the structure escapes from the well corresponding to pre-buckling configurations to another. This escape load may be much lower than the corresponding quasi-static buckling load. Indeed, complex resonances can occur until the system snaps-through, often signalling the loss of stability. As parameters are slowly varied, steady state instabilities may occur; these can include jumps to resonance, subharmonic period-doubling bifurcations, cascades to chaos, etc. Moreover a sudden pulse of excitation may lead to a transient failure of the system. In this paper, we examine how spherical caps under harmonic loading may be assessed in an engineering context, with a view to design against steady state instabilities as well as the various modes of transient failure. Steady state and transient stability boundaries are presented in which special attention is devoted to the determination of the critical load conditions. From this theoretical analysis, dynamic buckling criteria can be properly established which may constitute a consistent and rational basis for design of these shell structures under harmonic loading.