Validation of the tangent formulation for the solution of the non-linear Eshelby inclusion problem

An extension of the Eshelby problem for non-linear viscous materials is considered. An ellipsoidal heterogeneity is embedded in an infinite matrix. The material properties are assumed to be uniform within the ellipsoid and in the matrix. The problem of determining the average strain rate in the ellipsoid in terms of the overall applied strain rate is solved in an approximate way. The method is based on the non-incremental tangent formulation of the non-linear matrix behavior [Acta Metall. 35 (1987) 2983]. In the present work this approximate solution is verified with a good agreement by comparing to finite element calculations for various inclusion shapes and loading conditions.