Two-dimensional nonlocal Eshelby’s inclusion theory: eigenstress-driven formulation and applications

Abstract

The classical Eshelby’s theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby’s inclusion theory to nonlocal elasticity. Starting from Eringen’s integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby’s inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media.