Analysis of two-dimensional contact problems considering surface effect

Two-dimensional contact problems including a Boussinesq model, a semi-infinite substrate punched by a rigid flat-ended indenter or a cylindrical one, are systematically investigated with a recently developed continuum theory, in which surface effect on mechanical properties of materials is considered based on the concept of surface energy density. The contact stress and displacement fields are analyzed. It is found that the surface energy density of the indented bulk substrate, as only one additional parameter, serves as an important factor to influence the contact properties in contrast to the classical contact models. All the results show that the semi-infinite substrate becomes hardened when the surface effect is considered. Scaling analysis further demonstrates that differences between the theoretical predictions with surface effect and the classical contact solutions without surface effect become significant only if the contact width is comparable with the ratio of the bulk surface energy density to the bulk shear modulus. Specially, in the two-dimensional cylindrical punch problem, the smaller the punch size or the external compressive load, the more serious the deviation of the nano-indentation hardness predicted by the theoretical model with surface effect and the classical contact one. The results should be helpful not only for precise measurement of nano-indentation hardness but also for accurate evaluation of service performance of nanomaterials and nano-devices.