Boussinesq problem with the surface effect based on surface energy density

Surface effect plays an important role in nanosized materials. In this paper, two-dimensional Boussinesq problem with the surface effect is systematically investigated based on Chen-Yao's surface elastic theory, in which surface effect on mechanical properties of materials is considered based on the concept of surface energy density. The Fourier integral transform method is adopted to derive the contact stress and displacement fields of the Boussinesq problem. As two examples, the deformations induced, respectively, by a uniform distributed pressure and a concentrated force are analyzed in detail. The theoretical result in this paper show 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. The numerical results show that the semi-infinite substrate becomes hardened when the surface effect is considered. Scaling analysis further indicates that only when the ratio of the contact width to the volume surface energy density to the shear modulus is equal, the difference between the theoretical prediction of surface effect and the classical contact solution without surface effect will be significant. The results provide a further understanding of the surface effect of nanomaterials, which should be helpful for the design and accurate evaluation of service performance of nanoscale devices or nanomaterials.