On spherical nanoinhomogeneity embedded in a half-space analyzed with Steigmann–Ogden surface and interface models

Abstract

This paper determines the elastic fields in a positive half-space embedded with a spherical inhomogeneity under the context of Steigmann–Ogden surface/interface mechanical model. The half-space is loaded by an equal-biaxial far-field tension applied at infinity. While bulk domains are treated as linearly isotropic elastic solids, both the half-space plane boundary and the matrix/inhomogeneity interface are modeled by the Steigmann–Ogden theory. The well-developed method of Boussinesq displacement potentials is used to solve the elastostatic Navier’s equations of equilibrium. In view of the geometry and loading configurations, four sets of cylindrical and spherical harmonic potentials are carefully proposed to solve the problem. Implementation of the nonclassical Steigmann–Ogden boundary conditions at both the half-space free surface and the spherical interface results in a semianalytical solution in the form of infinite Legendre series. Extensive parametric studies are performed with respect to the surface and interface Steigmann–Ogden material parameters, shear moduli ratio between the spherical inhomogeneity and its surrounding matrix, and the inhomogeneity radius-to-depth ratio. Comparison and contrast with the well known Gurtin–Murdoch model reveals the significance of surface flexural rigidities at both boundaries of the considered mechanical model. When compared with the effects of matrix/inhomogeneity interface, the surface tension, Lamé constants and flexural rigidities at the half-space plane boundary are of secondary importance. This work is able to shed some lights on the modeling of self-organized adatoms and islands that is essential in the semiconductor industry.