Circular inclusion with a refined linearized version of Steigmann–Ogden model

Abstract

The plane deformation of an infinite elastic matrix enclosing a single circular inclusion incorporating stretching and bending resistance for the inclusion–matrix interface is revisited using a refined linearized version of the Steigmann–Ogden model. This refined version of the Steigmann–Ogden model differs from other linearized counterparts in the literature mainly in that the tangential force of the interface defined in this version depends not only on the stretch of the interface but also on the bending moment and initial curvature of the interface (the corresponding bending moment relies on the change in the real curvature of the interface during deformation). Closed-form results are derived for the full elastic field in inclusion–matrix structure induced by an arbitrary uniform in-plane far-field loading. It is identified that with this refined version of the Steigmann–Ogden model a uniform stress distribution could be achieved inside the inclusion for any non-hydrostatic far-field loading when $R=\sqrt{3{\chi }_{\mathrm{int}}/{\lambda }_{\mathrm{int}}}$ (where R is the radius of the inclusion, while ${\lambda }_{\mathrm{int}}$ and ${\chi }_{\mathrm{int}}$ are the stretching and bending stiffness of the interface). Explicit expressions are also obtained for the effective transverse properties of composite materials containing a large number of unidirectional circular cylindrical inclusions using, respectively, the dilute and Mori–Tanaka homogenization methods. Numerical examples are presented to illustrate the differences between the refined version and two typical counterparts of the Steigmann–Ogden model in evaluating the stress field around a circular nanosized inclusion and the effective properties of the corresponding homogenized composites.