Solutions of Eshelby-Type Inclusion Problems and a Related Homogenization Method Based on a Simplified Strain Gradient Elasticity Theory

Abstract

Eshelby-type inclusion problems of an infinite or a finite homogeneous isotropic elastic body containing an arbitrary-shape inclusion prescribed with an eigenstrain and an eigenstrain gradient are analytically solved. The solutions are based on a simplified strain gradient elasticity theory (SSGET) that includes one material length scale parameter in addition to two classical elastic constants. For the infinite-domain inclusion problem, the Eshelby tensor is derived in a general form by using the Green’s function in the SSGET. This Eshelby tensor captures the inclusion size effect and recovers the classical Eshelby tensor when the strain gradient effect is ignored. By applying the general form, the explicit expressions of the Eshelby tensor for the special cases of a spherical inclusion, a cylindrical inclusion of infinite length and an ellipsoidal inclusion are obtained. Also, the volume average of the new Eshelby tensor over the inclusion in each case is analytically derived. It is quantitatively shown that the new Eshelby tensor and its average can explain the inclusion size effect, unlike its counterpart based on classical elasticity. To solve the finite-domain inclusion problem, an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET are proposed and utilized. The solution for the disturbed displacement field incorporates the boundary effect and recovers that for the infinite-domain inclusion problem. The problem of a spherical inclusion embedded concentrically in a finite spherical body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. It is demonstrated through numerical results that the newly obtained Eshelby tensor can capture the inclusion size and boundary effects, unlike existing ones. Finally, a homogenization method is developed to predict the effective elastic properties of a heterogeneous material using the SSGET. An effective elastic stiffness tensor is analytically derived for the heterogeneous material by applying the Mori-Tanaka and Eshelby’s equivalent inclusion methods. This tensor depends on the inhomogeneity size, unlike what is predicted by existing homogenization methods based on classical elasticity. Numerical results for a two-phase composite reveal that the composite becomes stiffer when the inhomogeneities get smaller