Study On The Eshelby Problem Of A Non-elliptcal Inclusion In Half-plane

The Eshelby problem is of great importance in various engineering and physical fields. The solution obtained by Eshelby for an ellipsoidal inclusion in a infinte domain has been used as a footstone of micromechanics. In practical applications, the shapes of most inclutions are non-ellipsoidal and the bodies under consideration are finite. In this paper, we study the problem of a non-elliptcal inclusion in half-plane. Based on the solutions of non-elliptcal inclusions in an infinite domain, the major conclusions includes:(ⅰ). A general approach proposed by Zou et al.(2012) is used to study the Eshelby problem of an arbitrarily inclusion in the half-plane. The analytical solution of Eshelby problem for a non-elliptical inclusion in an isotropic half-plane material is obtained. The explicit expressions of stress and strain fields for circular and square inclusions are given and illustrated.(ⅱ). For the planar elastic problem of piezoelectric material, the Lekhnitskii formalism is used to obtain the explicit expressions of eigenvectors of displacement and stress function. The resolution is easy to extend to the problem of multi-fields coupling of anisotropic material. As an example, the material parameters of GaAs are used to verify the coincidence between our results and the implicit ones from the Stroh formalism.(ⅲ). Further upon the results of eigenvalues and eigenvectors, we derive the solution of eigenfunctions for an arbitrary non-elliptical inclusion in an anisotropic half-plane and a numerical example is given.