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### Strain Gradient Solution for the Eshelby-Type Problem of an Anti-Plane Strain Cylindrical Inclusion in a Finite Elastic Matrix

#### Abstract

Eshelby’s equivalent eigenstrain method and fourth-order strain transformation tensor [1] are essential for homogenization schemes including the Mori-Tanaka and self-consistent methods. However, Eshelby’s tensor originally provided in [1] is based on classical elasticity and is for an ellipsoidal inclusion embedded in an infinite elastic matrix. As a result, homogenization methods based on this classical Eshelby tensor cannot capture particle (inclusion) size effects or account for boundary effects. Hence, there has been a need to obtain Eshelby tensors for an inclusion in a finite matrix using higher-order (non-classical) elasticity theories. In this study, such an Eshelby tensor is provided for the finite-domain anti-plane strain inclusion problem of a finite elastic matrix containing a cylindrical inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient using a simplified strain gradient elasticity theory (SSGET) [2]. This SSGET involves only one material length scale parameter and has been applied to analytically solve several Eshelby-type inclusion problems [3–7]. In the current formulation, the SSGET-based Green’s function for an infinite anti-plane strain elastic body is first derived using the Fourier transform method. The extended Betti’s reciprocal theorem and Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems are then used to determine the displacement field in the finite matrix in terms of this Green’s function. The displacement solution reduces to that of the infinite-domain anti-plane inclusion problem when the boundary effect is suppressed.