Abstract
Experimentally measuring the elastic properties of thin biological surfaces is non-trivial, particularly when they are curved. One technique that may be used is the indentation of a thin sheet of material by a rigid indenter, while measuring the applied force and displacement. This gives immediate information on the fracture strength of the material (from the force required to puncture), but it is also theoretically possible to determine the elastic properties by comparing the resulting force–displacement curves with a mathematical model. Existing mathematical studies generally assume that the elastic surface is initially flat, which is often not the case for biological membranes. We previously outlined a theory for the indentation of curved isotropic, incompressible, hyperelastic membranes (with no bending stiffness) which breaks down for highly curved surfaces, as the entire membrane becomes wrinkled. Here, we introduce the effect of bending stiffness, ensuring that energy is required to change the shell shape without stretching, and find that commonly neglected terms in the shell equilibrium equation must be included. The theory presented here allows for the estimation of shape- and size-independent elastic properties of highly curved surfaces via indentation experiments, and is particularly relevant for biological surfaces.
Highlights
The experimental characterization of the elastic properties of a curved flexible shell is of interest within both biological and engineering contexts
Throughout biology, surfaces often grow with a three-dimensional structure, leading to complex curved shapes [1]. Such structures are not amenable to the majority of engineering techniques to determine elastic properties, such as vibration or tensile tests, because test-piece shapes must be controlled. Indentation tests are another classical technique, in which a rigid indenter is pushed into the specimen to generate a force–displacement curve
Little attention has been paid to curved surfaces at large indentation, with the context often that of atomic force microscopy (AFM) or nano-indentation, where the indentation depth and needle size are much smaller than the surface itself [11,12]; Deris & Nadler [13], who consider the indentation of a fluid-filled spherical membrane being a recent exception
Summary
The experimental characterization of the elastic properties of a curved flexible shell is of interest within both biological and engineering contexts. Little attention has been paid to curved surfaces at large indentation, with the context often that of atomic force microscopy (AFM) or nano-indentation, where the indentation depth and needle size are much smaller than the surface itself [11,12]; Deris & Nadler [13], who consider the indentation of a fluid-filled spherical membrane being a recent exception Without such a theoretical basis, shape-independent elastic properties cannot be extracted from the experimentally measured force–displacement curves, and the sole readout is the force required before puncture, which gives information on the strength of the material but not the elasticity
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