Asymptotic phenomena in pressurized thin films

Abstract

An asymptotic study of the wrinkling of a pressurized circular thin film is performed. The corresponding boundary-value problem is described by two nondimensional parameters; a background tension μ and the applied loading P. Previous numerical studies of the same configuration have shown that P tends to be large, and this fact is exploited here in the derivation of asymptotic descriptions of the elastic bifurcation phenomena. Two limiting cases are considered; in the first, the background tension is modest, while the second deals with the situation when it is large. In both instances, it is shown how the wrinkling is confined to a relatively narrow zone near the rim of the thin film, but the mechanisms driving the bifurcation are different. In the first scenario, the wrinkles are confined to a region which, though close to the rim, is asymptotically separate from it. By contrast, when μ is larger, the wrinkling is within a zone that is attached to the rim. Predictions are made for the value of the applied loading P necessary to generate wrinkling, as well as details of the corresponding wrinkling pattern, and these asymptotic results are compared to some direct numerical simulations of the original boundary-value problem.

On the asymptotic reduction of a bifurcation equation for edge-buckling instabilities

“…In keeping with the related work reported in [8] we have assumed 0 < ε 1, so that λ = O(1) and m = O(ε −3/2 ). As explained in that reference, for a low or moderate degree of initial stretching it follows that ε is no longer very small and, in fact, the limits ε 1 or ε = O(1) might become relevant (note that in the notation of [12] this corresponds to 0 < µ 1 and µ = O(1), respectively). In that case the roots of the characteristic equation (3.3) are rather different from what we found in §3, and it can be shown that the asymptotic reduction of §4 leads to contributions from both equations in (2.14), which can then be combined into a parabolic-cylinder type second-order differential equation.…”

“…However, we also draw attention to a number of "clues" that could have anticipated the main findings reported in this work. For instance, the more orthodox boundary-layer approach pursued in [9] and [12], which dealt with similar situations involving a vertical point-load and a uniform transverse pressure, respectively, revealed that the FvK bifurcation system asymptotically decoupled for µ 1, and its behaviour was driven by a hierarchy of second-order ordinary differential equation originating in our operator L 1 -see (2.15a). Furthermore, we note that the wrinkling scenario investigated by Coman & Haughton in [15] corresponds to letting L 2 ≡ 0 and setting the multiplicative factor λ to unity in (2.14); the effect of these changes leaves the two bifurcation equations trivially decoupled.…”

“…On the other hand, numerical simulations of the bifurcation problem (2.14)-(2.16) suggest that for µ 1 the eigenmodes are strongly localised near the rim of the plate and experience exponential decay within an extended region around ρ = 0 (cf. [8,9,12]). The upshot of this observation is that we can still use the expressions for W 1,2 found above as long as, instead of enforcing the restraints at ρ = 0, we apply them at ρ = η, for some arbitrary 0 < η 1.…”

“…In the present investigation we are motivated to explore the applicability of the aforementioned asymptotic reduction strategy to a rather different edge-wrinkling situation, recently studied by Coman et al [9,12] within the context of the Föppl-von Kármán (FvK) nonlinear plate theory. In this new case, a weakly clamped uniformly stretched circular elastic plate is subjected to a transverse pressure; due to the gradual development of compressive stresses near the edge of the plate, for a critical value of the loading the plate experiences a regular wrinkling pattern confined to that region.…”

“…The description of this bifurcation requires including finite rotations in the pre-buckling range, and hence leads to nonlinear equations for the basic state (assumed to possess radial symmetry). Furthermore, the out-of-plane character of the pre-buckling state introduces additional complications; for instance, the displacement buckling equations adopted in [9,12], and originally derived in [4], consisted of a coupled system involving two second-order and one fourth-order differential equations. In a more traditional form, these equations can also be cast as a system of two coupled fourth-order differential equations (e.g., [10]), and it is this latter formulation that is preferred in what follows.…”