Dynamic Buckling of an Elastic Ring in a Soap Film

Abstract

Dynamic buckling may occur when a load is rapidly applied to, or removed from, an elastic object at rest. In contrast to its static counterpart, dynamic buckling offers a wide range of accessible patterns depending on the parameters of the system and the dynamics of the load. To study these effects, we consider experimentally the dynamics of an elastic ring in a soap film when part of the film is suddenly removed. The resulting change in tension applied to the ring creates a range of interesting patterns that cannot be easily accessed in static experiments. Depending on the aspect ratio of the ring's cross section, high-mode buckling patterns are found in the plane of the remaining soap film or out of the plane. Paradoxically, while inertia is required to observe these nontrivial modes, the selected pattern does not depend on inertia itself. The evolution of this pattern beyond the initial instability is studied experimentally and explained through theoretical arguments linking dynamics to pattern selection and mode growth. We also explore the influence of dynamic loading and show numerically that, by imposing a rate of loading that competes with the growth rate of instability, the observed pattern can be selected and controlled.

Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses

“…(4) is modified by the ring's curvature κ = 1/a, with a being the ring radius. In a related study [22], quantitative agreement with experiment was found by replacing the force F in Eq. (4) with the compressive force induced by the pressure difference p combined with the ring curvature 1/a (Laplace's law), which gives F = pa.…”

“…The experiments described in Ref. [22] demonstrate that a circular elastic ring buckling under surface tension in this way can exhibit a range of modes, with higher modes selected dynamically as the importance of surface tension FIG. 1.…”

“…In particular, we shall show that the presence of inertia implies that higher modes are selected, and we identify the mode number as a function of the driving pressure. Experimentally, these modes can be obtained by following the dynamics of an elastic ring embedded in a soap film, as presented in a companion paper [22]. Initially, the elastic ring is maintained in equilibrium by the balance of surface tension between an internal and external soap film.…”

“…Schematic of an elastic ring subject to an external pressure p (indicated by the arrows). The experiments described in a companion paper [22] achieve this via an unbalanced soap film filling the shaded area within the elastic ring (see Fig. 2).…”

“…2. Time series from the companion paper [22] showing experimental images of the dynamic buckling of an elastic ring with square cross section. This dynamic buckling is achieved by placing an elastic ring (Young's modulus E = 42 kPa, thickness h = 1 mm, and radius a = 24 mm) in a soap film and bursting the outer soap film at time t = 0.…”