STRAIN GRADIENT SOLUTION FOR THE ESHELBY-TYPE PROBLEM OF AN ANTI-PLANE STRAIN CYLINDRICAL INCLUSION IN A FINITE ELASTIC MATRIX

Eshelby's equivalent eigenstrain method and fourth-order strain transformation tensor [1] are essential for homogenization schemes including the Mod-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. The problem of a cylindrical inclusion embedded concentrically in a finite anti-plane strain cylindrical elastic matrix is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the anti-plane strain cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results (see Figure I) quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.