Wrinkling of Pressurized Elastic Shells

We study the formation of localized structures formed by the point loading of an internally pressurized elastic shell. While unpressurized shells (such as a ping-pong ball) buckle into polygonal structures, we show that pressurized shells are subject to a wrinkling instability. We study wrinkling in depth, presenting scaling laws for the critical indentation at which wrinkling occurs and the number of wrinkles formed in terms of the internal pressurization and material properties of the shell. These results are validated by numerical simulations. We show that the evolution of the wrinkle length with increasing indentation can be understood for highly pressurized shells from membrane theory. These results suggest that the position and number of wrinkles may be used in combination to give simple methods for the estimation of the mechanical properties of highly pressurized shells.

Figure 1

The wrinkling of a beach ball under indentation.

Figure 2

The transition from polygonal localizations to wrinkles with increasing internal pressure. Shading shows the mean curvature obtained in numerical simulations for a shell of radius $R=1\mathrm{m}$, thickness $h=2\mathrm{mm}$, Young’s modulus $E=70\mathrm{GPa}$, and internal pressures $p=0$, 2, 7, and 15 MPa. The corresponding values of the indentation are $w_0=0.1$, 0.1, 0.22, and 0.37 m, respectively.

Figure 3

The onset of instability. Results from numerical simulations (points) are shown for shells with radius $R=1\mathrm{m}$ and thicknesses $h=0.5$ (○), 1 (▵), 2 (×), and 5 mm (□) with a range of internal pressures. (a) The critical displacement at which asymmetric deformation patterns are first observed. The dotted line shows the result (5), valid for $\tau \ll 1$. The dashed line shows the prediction of membrane theory that $w_0^c\approx 2.52{\ell }_p^2/R$. (b) The wave number of the asymmetric deformation pattern $n$. The dashed line shows the result (6) valid for $\tau \gg 1$, with a prefactor 1.33. Experimental measurements of the critical wave number for a Pezzi ball ($R=18.5\mathrm{cm}$, $h=1\mathrm{mm}$, and $E=2.3\mathrm{MPa}$) inflated to different pressures are also shown (◆).

Figure 4

Evolution of wrinkle length with continuing indentation. The results from numerical simulations (points) are shown for shells with radius $R=1\mathrm{m}$ and thicknesses $h=1\mathrm{mm}$ (filled symbols) and $h=2\mathrm{mm}$ (open symbols). The internal pressure $p=8$ (●), 9 (□), and 15 MPa (△). The prediction from the numerical solution of the membrane-shell model ($\tau =\infty$) is shown (solid curve) along with the asymptotic result (7) (dashed line), which is obtained as the $w_{0}R/{\ell }_p^2\gg 1$ limit of the membrane-shell theory. The onset of the asymmetric instability is illustrated by the dotted vertical and horizontal lines.