On the dynamic snap-through of a non-linear elastic system

Abstract

The dynamic instability “in the large” of a non-linear elastic continuous conservative system with Rayleigh friction subjected to a load applied instantaneously at the same t = 0 and keeping a constant value for all t⩾0 /1–6/ is investigated. By using a potential theory analysis, the concepts of a well and equilibrium stability factor introduced by Myshkis /7, 8/, definitions are given of the dynamic stability of the system, the critical load of its dyanmic snap-through, and the astatic critical load. The latter yields a lower limit of those values of the load for which dynamic snap-through occurs. For the class of systems with a potential energy of the form of the square of the norm plus a weakly continuous functional /9/, among which are shallow elastic shells, for instance, /10–12/, it is proved that the existence of saddle points with negative index follows from the non-uniqueness of the stable equilibrium. Here at least one saddle point is found on the boundary of the well of each stable equilibrium. Therefore, the stability factor acquires a graphic meaning as the least crossing among the energetic peaks leading from the well of a given equilibrium to the wells of the other equilibria (or to infinity, which is impossible, it is true, for functionals increasing at infinity). The application of this property for systems with potential energy depending on the load parameter p is the fundametal effective calculation of the stability factor and the astatic critical load p a . A basis is presented for the applicability of the energetic approach for the non-linear vibrations equations of elastic shallow shells. In particular, the classical problem is examined of the dynamic snap-through of a shallow elastic spherical shell subjected to an instantaneously applied hydrostatic load, and an example is presented of the determination of the astatic ritical load p a in the case of ambiguity of the families of unstable equilibria. We note that the load p a for shallow spherical shells with different geometric parameters and boundary conditions has been found earlier in /13–17/. The good agreement between the values of p a and the critical load of dynamic snap-through, obtained by a direct numerical integration of the non-stationary problem /2–6, 13–18/ indicates the efficiency of using the energetic approach developed here in the theory of shells. The reasoning associated with estimating the height of the energetic barrier was used earlier in finite-dimensional models of the Galerkin method for the equations of the vibration of an arch /1, 19–21/. The considerations presented later understandably also include the case of systems with a finite number of degrees of freedom.