Wrinkling Behavior of Highly Stretched Rectangular Elastic Films via Parametric Global Bifurcation

We consider the wrinkling of highly stretched, thin rectangular sheets – a problem that has attracted the attention of several investigators in recent years, nearly all of which employ the classical Föppl-von Kármán (F-K) theory of plates. (See Figure 1 below for a schematic diagram.) We first propose a rational model that correctly accounts for large mid-plane strain. We then carefully perform a numerical bifurcation/continuation analysis, identifying stable solutions (local energy minimizers). Our results in comparison to those from the F-K theory (also obtained herewith) show: (i) For a given fine thickness, only a certain range of aspect ratios admit stable wrinkling; for a fixed length (in the highly stretched direction), wrinkling does not occur if the width is too large or too small. In contrast, the F-K model erroneously predicts wrinkling in those very same regimes for sufficiently large applied macroscopic strain. (ii) When stable wrinkling emerges as the applied macroscopic strain is steadily increased, the amplitude first increases, reaches a maximum, decreases, and then returns to zero again. In contrast, the F-K model predicts an ever-increasing wrinkling amplitude as the macroscopic strain is increased. We identify (i), (ii) as global isola-center bifurcations – in terms of both the macroscopic-strain parameter and an aspect-ratio parameter. (iii) When stable wrinkling occurs, for fixed parameters, the transverse pattern admits an entire orbit of neutrally stable (equally likely) possibilities: These include reflection symmetric solutions about the mid-plane, anti-symmetric solutions about the mid-line (a rotation byπ radians about the mid-line leaves the wrinkled shape unchanged) and a continuously evolving family of shapes “in-between”, say, parametrized by an arbitrary phase angle, each profile of which is neither reflection symmetric nor antisymmetric.