Explicit expression of Eshelby tensor for arbitrary weakly non-circular inclusion in two-dimensional elasticity

Any constant eigenstrain of an ellipsoidal inclusion in an infinite elastic medium results in uniform strain and stress fields in the inclusion, known as the Eshelby uniformity. The Eshelby tensor which characterizes the Eshelby uniformity plays a crucial role in micromechanics of matrix–particle composites. Since the Eshelby uniformity is not valid for any non-elliposidal inclusion and a general solution for the non-elliposidal inclusion is not available, herein we use the Fourier’s series to characterize the shape of the weakly non-circular inclusion in two-dimensional isotropic elasticity and obtain the explicit expression of its Eshelby tensor. We further give the expression of the average Eshelby tensor on the inclusion. The average Eshelby tensor depends upon only the second- and the fourth-shape coefficients of the Fourier’s series. Finally, we verify these expressions by comparing their computational results with the exact numerical results for various weakly non-circular inclusions.