Boussinesq problem of a finite elastic layer with the surface effect

Both the thickness effect and surface effect should be important in nano-indentation behavior of coatings due to the finite thickness and small indentation size. As a basic solution, the two-dimensional Boussinesq problem of a finite elastic layer bonded to a rigid substrate is studied in this paper, employing the surface-energy-density-based elastic theory. The Airy stress function and Fourier integral transform methods are adopted to solve the problem. A nalytical solutions of both the stress and displacement fields are well achieved for a finite elastic layer under a concentrated force and a uniform pressure. Unlike the classical solutions, it is discovered that both the thickness effect and surface effect will show significant influences on the Boussinesq elastic behaviors. The surface effect would harden the finite elastic layer and induce a more uniformly distributing displacements and stresses. Only when the thickness is sufficiently large, the Boussinesq solution of an elastic half space may represent that of a finite elastic layer case. A generalized hardness is further defined to include the coupling effects of thickness and surface for the Boussinesq problem of a finite elastic layer. Such a study would assist in the design and property evaluation of coatings and micro-devices with layer-substrate structures.