Previous studies demonstrate that, for nanostructures under transverse bending, the effective Young modulus is appreciably greater (in magnitude) than that of the same elements under axial loads. Therefore, in addition to the conventional residual surface tension and membrane stiffness, the curvature dependence of surface energy is desired for inhomogeneously strained nanostructures. In this paper, we aim to reevaluate the size-dependent nanoindentation hardness of an elastic half-space subjected to nanosized frictionless traction, through the use of both the curvature-independent Gurtin–Murdoch and the curvature-dependent Steigmann–Ogden models of surface elasticity. The nanoindentation problem is solved by the integration of Boussinesq’s method of displacement potentials and Hankel integral transforms. As examples, the effects of residual surface tension, membrane stiffness, and bending rigidity of the half-space boundary are parametrically analyzed in detail for a uniform circular pressure and a concentrated normal force. The observations in semianalytical calculations suggest a significant difference in the nanoindentation hardnesses predicted from the two popular models of surface mechanics. In most cases, the inclusion of bending rigidity results in smaller displacements and stresses, and therefore higher indentation hardness. Based on physically interpretable numerical values of surface material properties, we show that a curvature-dependent model of surface elasticity is required in order to characterize the size-dependent feature of nanoindentation problems correctly.
The Effect of Proteins and Lipids on Membrane Stiffness
