Consideration of arbitrary inclusion shapes in the framework of isotropic continuum micromechanics: The replacement Eshelby tensor approach

The effective behavior of matrix-inclusion materials is governed by the properties and volume fractions of involved material phases as well as by the morphology of the inclusion domain. Eshelby's solutions on the ellipsoidal inclusion problem (Eshelby, 1957) paved the way to the development of a vast number of continuum micromechanics based homogenization schemes for evaluation of effective properties. This paper proposes an approach for utilizing these homogenization schemes also for non-ellipsoidal inclusions with particular focus on macroscopic isotropy. The key concept is to replace the originally-used Eshelby tensor by an analogous representation capturing the effect of non-spherical inclusions on strain fluctuation within the material domain. The so-called replacement Eshelby tensor is obtained by numerical evaluation of two adjustment factors being functions of the inclusion morphology and the stiffness contrast between the matrix and inclusion phase. These factors are evaluated and discussed for selected inclusion morphologies. Finally, the replacement Eshelby tensor approach is implemented in a homogenization procedure for effective elastic properties (cascade continuum micromechanics model (Timothy and Meschke, 2016)) as well as in a scheme for determining effective yield surfaces in case of plastic material behavior (Traxl and Lackner, 2015).