Self-consistent homogenization approach for polycrystals within second gradient elasticity

In this paper, we propose a generalized variant of Kröner’s self-consistent scheme for evaluation of the effective standard and gradient elastic moduli of polycrystalline materials within Mindlin-Toupin second-gradient elasticity theory. Assuming random orientation of crystallites (grains) we use an extended Eshelby’s equivalent inclusion method and mapping conditions between the prescribed linear distribution of macro-strain and corresponding micro-scale field variables averaged over the volume and all possible orientations of single grain. It is found that the developed self-consistent scheme predicts an absence of gradient effects at the macro-scale level for the model of ellipsoidal grains made of the first gradient (Cauchy-type) material. However, for the case of the second gradient crystallites, established approach allows to obtain a set of non-linear relations for determination of all effective standard and gradient elastic moduli of polycrystals. Example of calculations under simplified constitutive assumptions is presented for the model of anisotropic cubic Fe crystallites with spherical shape. It is shown that the presented approach predicts an increase of the effective length scale parameter of polycrystalline aggregates due to misorientation of grains axes and takes into account the size effects related to the mean diameter of grains.