The indentation of pressurized elastic shells: from polymeric capsules to yeast cells

Abstract

Pressurized elastic capsules arise at scales ranging from the 10 m diameter pressure vessels used to store propane at oil refineries to the microscopic polymeric capsules that may be used in drug delivery. Nature also makes extensive use of pressurized elastic capsules: plant cells, bacteria and fungi have stiff walls, which are subject to an internal turgor pressure. Here, we present theoretical, numerical and experimental investigations of the indentation of a linearly elastic shell subject to a constant internal pressure. We show that, unlike unpressurized shells, the relationship between force and displacement demonstrates two linear regimes. We determine analytical expressions for the effective stiffness in each of these regimes in terms of the material properties of the shell and the pressure difference. As a consequence, a single indentation experiment over a range of displacements may be used as a simple assay to determine both the internal pressure and elastic properties of capsules. Our results are relevant for determining the internal pressure in bacterial, fungal or plant cells. As an illustration of this, we apply our results to recent measurements of the stiffness of baker's yeast and infer from these experiments that the internal osmotic pressure of yeast cells may be regulated in response to changes in the osmotic pressure of the external medium.

Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

“…A similar type of geometric stiffness, which is determined by the underlying curvature rather than by the exerted loads, is known to govern the (unwrinkled) response of intrinsically curved elastic shells to loads (19). To demonstrate the geometric link between shells and sheets, consider the uniaxial compression of a cylindrical shell: An ordered pattern of diamond-like blocks is observed (31), whose characteristic size is proportional to the geometric mean of the radius (R) and thickness (t) of the shell, λ ∝ ffiffiffiffi ffi tR p .…”

“…Instead, it reflects the sheet's elastic response to the curved geometry alone. To our knowledge, the geometric stiffness of sheets, which resembles a shell's resistance (19), has not been noted before. We will show that it can have a dramatic effect on the wrinkle's wavelength.…”

Wrinkling reveals a new isometry of pressurized elastic shells

“…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].…”

“…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.…”

“…Deformation of the shell is resisted by a normal force that is proportional to the vertical displacement -effectively a Winkler foundation of stiffness κ w [6]. This foundation stiffness depends on the stretching stiffness of the shell, Et, and so on dimensional grounds we might expect that κ w ∼ Et/R 2 ; an analysis of the governing equations confirms this expectation [6,8]. By analogy with a liquid's meniscus, the combination of an intrinsic tension, σ ∞ ∼ pR, and a linear foundation stiffness, κ w ∼ Et/R 2 , gives rise to a characteristic horizontal length over which vertical deflections decay…”