Uniform strain fields inside multiple inclusions in an elastic infinite plane under anti-plane shear

This paper constructs multiple elastic inclusions with prescribed uniform internal strain fields embedded in an infinite matrix under given uniform remote anti-plane shear. The method used is based on the sufficient and necessary conditions imposed on the boundary values of a holomorphic function, which guarantee the existence of the holomorphic function in a multiply connected region. The unknown shape of each of the multiple inclusions is characterized by a polynomial conformal mapping with a finite number of unknown coefficients. With the aid of Cauchy's integral formula and Faber series, these unknown coefficients are determined by a system of nonlinear equations. Detailed numerical examples are shown for multiple inclusions with various prescribed uniform internal strain fields, for symmetrical inclusions and for inclusions whose shapes are independent of the remote loading, respectively. It is found that the admissible range of uniform internal strain fields for multiple inclusions is moderately larger than the admissible range of the uniform internal strain field for a single elliptical inclusion under the same remote loading. In particular, specific conditions on the prescribed uniform internal strain fields and elastic constants of the multiple inclusions are derived for the existence of symmetric inclusions and rotationally symmetrical inclusions. Moreover, for any two inclusions among multiple inclusions of shapes independent of the remote loading, it is shown that the ratio between the uniform internal strain fields inside the two inclusions equals a specific ratio determined by the shear moduli of the two inclusions and the matrix.