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

Abstract

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.

Periodic inclusions with uniform internal hydrostatic stress in an infinite elastic plane

“…The interest in such structures is due primarily to the fact that such inclusions eliminate the possibility of internal stress peaks thereby significantly reducing the probability of failure. Recently, Liu [6], Kang et al [8], Wang [9], and Dai et al [10,11] have identified shapes (and hence verified the existence) of multiple interacting inclusions with uniform internal stress fields inside an elastic plane subjected to either anti-plane shear or plane deformations. More recently, it has been shown that single or multiple non-elliptical inclusions can be designed to achieve uniform internal stress fields in an elastic half-plane or even in an elastic quarter-plane under anti-plane shear deformation [12].…”

“…Specifically, we can truncate the mapping into a polynomial involving finite coefficients a j ( j = 1… N ) which can be determined from the corresponding truncated Eq. for values of i = 1… N using the Newton‐Raphson method (see for a detailed implementation of Newton‐Raphson iteration). In particular, it is noted in the mapping that the final shape and orientation of the inclusion inside the strip in the z ‐plane are determined by only the shape (not the orientation) of the curve L ' in the z '‐plane, while it follows from Eqs.…”

“…. N using the Newton-Raphson method (see [10][11][12] for a detailed implementation of Newton-Raphson iteration). In particular, it is noted in the mapping (11) that the final shape and orientation of the inclusion inside the strip in the z-plane are determined by only the shape (not the orientation) of the curve L in the z -plane, while it follows from Eqs.…”

Non-circular nano-inclusions with interface effects that achieve uniform internal strain fields in an elastic plane under anti-plane shear

“…4, 5, 8) and non-circular hard inclusions with negative interface parameters γ (see Figs. 6,7,9), our extensive numerical results showed that it is much more difficult to construct convergent shapes of non-circular soft inclusions with negative interface parameters γ and non-circular hard inclusions with positive interface parameters γ which achieve uniform internal strain fields. In addition, for given interface shear modulus G s and residual interface tension τ , the extensive numerical examples (see also Figs.…”

“…Therefore, the present work aims to examine the existence of non-circular nano-inclusions with interface effects which achieve uniform internal strain fields in an elastic infinite plane subjected to uniform remote anti-plane shear loadings. It should be mentioned that the special shapes of inclusions with uniform internal fields constructed in [4][5][6][7][8][9] are restricted to the ideal case when the inclusion-matrix interfaces are perfectly bonded, whereas our present work focuses on the construction of nano-inclusions with uniform internal fields in the presence of imperfect interfaces incorporating interface tension and energy.…”