Wrinkling reveals a new isometry of pressurized elastic shells

We consider the point indentation of a pressurized, spherical elastic shell. Previously it was shown that such shells wrinkle once the indentation reaches a threshold value. Here, we study the behaviour of this system beyond the onset of instability. We show that rather than simply approaching the classical “mirror-buckled” shape, the wrinkled shell approaches a new, universal shape that reflects a nontrivial type of isometry. For a given indentation depth, this “asymptotic isometry”, which is only made possible by wrinkling, is reached in the doubly asymptotic limit of weak pressure and vanishing shell thickness. Copyright c © EPLA, 2015 Introduction. – Just as one pokes an object to see how stiff it is, it is common to measure the mechanical properties of small-scale objects by indentation, often using an Atomic Force Microscope (AFM) [1]. In biology and medicine many objects of interest are relatively thin capsules swollen by osmotic pressure, e.g. viruses [2], bacteria [3], yeast cells [4] and the polymerosomes used in drug delivery [5]. Recent studies have focussed on modelling the indentation of pressurized elastic capsules [2,3,5,6] using numerous approaches including finite-element methods and analytical techniques. The problem of the indentation of a shell with no internal pressure (pressureless) has been extensively studied [7–10]. In this case, the classical picture is that a spherical shell of thickness t and radius R will accommodate indentation by “mirror-buckling”: it is as if a portion of the shell were sliced off, inverted and then reattached to the remainder of the shell (fig. 1(a)). In reality, the apparent discontinuity in slope is resolved by a boundary layer of width (tR) [9]. Since mirror-buckling represents a simple isometry of the shell (i.e. there is no strain except at isolated points or along curves), the energy of deformation is localized within this boundary layer and vanishes in the limit t → 0. This simple geometrical picture has recently been shown to hold even in the presence of an internal pressure, p, albeit under the assumption that the shell remains very close to an axisymmetric shape [6]. (a)E-mail: [email protected] (a)