A modulus gradient model for inhomogeneous materials with isotropic linear elastic constituents

The one-dimensional modulus gradient (E-grad) model proposed in Gulasik et al. (2018) is extended to more general three-dimensional inhomogeneous materials with isotropic linear elastic constituents. In addition to the constitutive equations and balance relations, length scale dependent differential relations for the material parameters of isotropic linear elasticity are provided. The finite element formulation for axisymmetric problems is derived and a model problem of a soft cylindrical rod with a stiff spherical inclusion is solved. It is seen that discontinuities and/or very sharp changes in the modulus, displacement, strain and stress fields that exist in local formulations are smoothed with the proposed gradient model. Furthermore, a graded interphase region around the stiff inclusion is obtained by the model as reported in the literature of polymer nanocomposites. The thickness of the graded interphase region is interpreted as a characteristic length scale of the polymer nanocomposite. It is also shown that increasing the internal length scale parameter increases the stiffness of the model. The proposed model is compared with a micromechanical model from literature and experiments conducted with polyimide/silica nanocomposites. The results obtained by the proposed approach capture the experimentally measured values of the nanocomposite modulus. Finally, the model is extended to obtain anisotropic macroscopic response by choosing different length scale parameters in different directions.