The Eshelby inclusion problem in ageing linear viscoelasticity

The present paper focuses on the Eshelby inclusion problem which is revisited in the framework of ageing linear viscoelasticity. All known results established in linear viscoelasticity thanks to the correspondence principle are recovered as particular cases of a general solution extended to ageing. A closed form solution is presented for the Hill and Eshelby tensors related to an ellipsoidal inclusion embedded in an infinite anisotropic medium. The case of the isotropic medium is investigated in detail and related solutions are presented for spherical and ellipsoidal inclusions. A numerical procedure which operates in time domain and based on the trapezoidal rule to determine Volterra integral operators allows to efficiently calculate Hill and Eshelby tensors for a wide range of behaviours including in particular time-dependent Poisson ratios. Validation and verification of the new developed solution are presented for the ageing spherical inclusion on the basis of recent published results and it is completed by comparisons with finite element simulations for the general and novel case of an oblate spheroid embedded in an ageing linear viscoelastic matrix.