Dynamic snap-through of an elastic shell subjected to a pulsed load

Abstract

Dynamic instability “in the large” is investigated for an elastic shallow shell subjected to a pulsed load (PL) and a more general non-linear elastic continuous conservative system with potential energy of the “square of the norm plus a weakly continuous functional” kind with Rayleigh friction and a given initial velocity. An energy approach /1, 2/ developed in /3–6/ is used to analyse the dynamic snap-through (DS) of the non-linear elastic system subjected to a stationary step load. The problem is considered in an exact infinite-dimensional formulation. By using the concept of an equilibrium stability trough and reserve /7/, definitions are given of the dynamic stability of a system, the critical PL of its DS and the astatic critical PL. The latter is determined from the stationary problem and yields the lower bound for those PL values for which DS occurs. It is established that a necessary condition for the DS of a system subjected to a PL is the existence of a saddle point of the potential energy for the same load-free system at the boundary of a stable zero equilibrium trough. It is proved that this necessary condition is satisfied for sufficiently thin strictly convex shells of revolution with movable and fixed hinge-support as well as for a certain class of arbitrary strictly convex shells with a movable hinge support. Therefore, the stability reserve has a graphic mechanical meaning as an exact upper bound of the kinetic energy which can be added to the system at rest so that it does not reach the least saddle points among the energy heights leading from a zero equilibrium trough to troughs of other equilibria. Results of computer calculations of the critical PL of dynamic snap-through and astatic snap-through for spherical and conical shells are presented. Depending on the boundary conditions, the lower limits are determined for the ratio between the shell rise and its thickness, for which DS is possible under the action of a PL. Note that the reasoning associated with estimating the kinetic energy needed to overcome the energy barrier in the problem of DS under the action of a PL on a system with two degrees of freedom, obtained by the Bubnov-Galerkin method from the vibrations equations for an elastic arch, were first applied in /1/. The extensions to finite-dimensional models with a large number of degrees of freedom made by different authors are reflected in /2, 8, 9/.