Prototypical model for tensional wrinkling in thin sheets

Abstract

The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians, and engineers. This activity has been triggered by the growing interest in developing technologies at ever-decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. Although the most basic buckling instability of uniaxially compressed plates was understood by Euler more than two centuries ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length--a sheet under axisymmetric tensile loads. The first study of this geometry, which is attributed to Lamé, allows us to construct a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that the thinner the sheet is, the smaller is the compressive load above which the far-from-threshold regime emerges. This observation emphasizes the relevance of our analysis for nanomechanics applications.

Fig. 1.

The Lamé configuration: A mismatch between the inner and outer stresses yields a compression, which is relieved by wrinkling, in the region R_in < r < L.

Fig. 2.

A schematic phase diagram of wrinkling patterns in the Lamé geometry. The dimensionless parameters ϵ^-1, τ represent, respectively, bendability and confinement (see Eq. 8). Radial wrinkles emerge for ϵ, τ > τ_c(ϵ), where the threshold curve τ_c(ϵ), computed similarly to ref. , is marked with a black solid line. The NT analysis is valid below the blue dashed line (see text). After a cross-over region (purple), the sheet is under FFT conditions (red). The evolution of the hoop stress as τ increases is shown in the inset for ϵ^-1 = 10⁵ using three curves corresponding to points in the (ϵ,τ) phase space indicated in the main figure. Curves a and b show the stress profile as predicted by Eq. 6. However, curve c, which is well within the FFT region, shows that the hoop stress has collapsed in a manner compatible with Eq. 17. To emphasize the collapse of compressive stress, the red dashed line in the inset shows the hoop stress given by Eq. 6 for the same value of τ.

Fig. 3.

The mechanical energy U in the FFT limit (normalized by ) as a function of the wrinkle extent L (normalized by R_in) for τ = 5, R_out = 10R_in, and ν = 1/3. An inflection point exists at L_FFT = τ/2 = 2.5. ΔU is the energy gain for ϵ → 0 of the FFT energy (inflection point) with respect to the planar, Lamé solution (upper dashed line).