Centrosymmetric equilibrium of nested spherical inhomogeneities in first strain gradient elasticity

Abstract

The first strain gradient linear elasticity theory (FSGLET) is invoked to study an inhomogeneity problem. In the setting of this centrosymmetric problem, a set of concentric hollow spheres made of different materials is considered to serve as a model for a multi-coated nanometer-sized particle which in turn can be introduced as a representative volume element (RVE) of a nanocomposite. Perfect bonding is assumed between each layer of the RVE, and this system is subjected to uniform internal and external pressures. In the context of the FSGLET, the boundary value problem (BVP) for this model is formulated rigorously in spherical coordinates and its solution is obtained by means of analytical and numerical techniques. Employing the solutions of this BVP, we examine the important cases of a two-phase RVE consisting only of filler and matrix phases and a three-phase RVE where a homogeneous or a functionally graded interphase is also taken into account. We determine the associated elastic fields inside the RVE for each of these cases to study the impact of size effects, presence or absence of an interphase and its material properties distribution on the elastic fields variations. Furthermore, based on a simple and natural definition for the effective bulk modulus of the RVE, this effective modulus is calculated in each of the aforementioned cases at different filler volume fractions and characteristic lengths, and relevant comparisons are made.