Modified formulation and solution for an inclusion with Steigmann-Ogden model in plane deformation

The Steigmann-Ogden (S-O) model has been widely applied in the literature to study the surface/interface bending effects arising from surface/interface energy on the elastic behavior of nanostructures and nanocomposites. In the existing analytic solutions for an elastic inclusion-matrix system with linearized versions of the S-O model, however, the interface bending moment was consistently defined based on an incomplete curvature formula which neglects the contribution of interface stretch to the change in the curvature of the interface during deformation. In this paper, we are concerned with the plane deformation of an elastic interface-bulk system incorporating a modified linearized version of the S-O model in which the interface bending moment is defined by an accurate first-order formula describing the change in the curvature of the interface under arbitrary small deformation. We formulate the corresponding boundary condition for an arbitrary curved interface in terms of the complex potential functions of the surrounding bulk. We then apply this complex-variable formulation of the boundary condition to the plane deformation of a circular inclusion within an elastic infinite matrix undergoing a uniform far-field loading and derive a closed-form solution for the stress field around the inclusion and explicit expressions for the effective properties of the composite structure homogenized by the dilute and Mori-Tanaka methods. We identify a revised result that the stress distribution inside the circular inclusion keeps uniform for all types of uniform remote loading when the normalized interface stretching rigidity is six times the normalized interface bending rigidity. Numerical examples are given to compare the current results of the stress field and effective properties with those corresponding to the use of the incomplete curvature formula.

Automated summary

This paper investigates the plane deformation of an elastic interface-bulk system using a modified linearized version of the Steigmann-Ogden model, where the interface bending moment is accurately defined by a first-order formula. The corresponding boundary condition for an arbitrary curved interface is formulated in terms of complex potential functions, and applied to the plane deformation of a circular inclusion subjected to uniform far-field loading. Closed-form solutions for the stress field and effective properties of the composite structure are derived using the dilute and Mori-Tanaka methods. It is found that the stress distribution inside the circular inclusion remains uniform for all types of uniform remote loading when the normalized interface stretching rigidity is six times the normalized interface bending rigidity.