Analytical study of a circular inhomogeneity with homogeneously imperfect interface in plane micropolar elasticity

Abstract

In this paper we derive the full analytical solution for the problem of a circular micropolar inhomogeneity in an infinite micropolar plate subjected to a remote uni-axial tension. The interface between the inhomogeneity and the surrounding matrix is considered to be homogeneously imperfect. This model has been well known in classical elasticity and was validated experimentally [16] and verified analytically [25]. Mathematically it is expressed in the assumption that the stresses are continuous across the interface and proportional to the jumps in the corresponding displacements. This idea was extended to micropolar elasticity [20], where the additional assumption of continuous couple-tractions proportional to the jumps in the corresponding microrotations was introduced. In the present work we show the asymptotic derivation of the linear interface model in micropolar elasticity (plane-strain), based on the expansion of all fields in a thin "interphase" layer between the inhomogeneity and the matrix, [25], and link the interface parameters to the properties of the interphase layer. The problem is subsequently solved with the use of Eringen's stress functions, which allow to express all stresses/couple stresses and displacements/microrotation as a linear combination of the solutions of two governing equations and reduce the boundary conditions on the interface to a system of algebraic equations for the unknown coefficients. A parametric study is conducted to show that the stress concentration factors are significantly dependent on the micropolar material constants as well as the parameters characterizing the imperfect bonding between the inhomogeneity and the matrix. The solution is given in a ready-to-use form, freely downloadable, and can be further used, for example, for the analysis of interface failures or as a reference solution in numerical methods.