The Design of Nano-Inhomogeneities with Uniform Internal Strain in Anti-Plane Shear Deformations of Composite Solids

Abstract

In the micromechanical analysis of composite materials, the objective is to predict the overall behavior of the composite from known properties of its individual constituents. When applied to fibrous composites, stress distributions in a composite material subjected to applied stresses can be modelled using inhomogeneity-matrix systems in which the fibers are represented by inhomogeneities embedded in a foreign matrix material. One of the most important challenges associated with inhomogeneity-matrix systems is concerned with the design of inhomogeneities in which the interior strain distribution remains uniform. The primary motivation for the interest in this class of problems lies in the optimal nature of an interior uniform strain field in that such a field does not give rise to stress peaks within the inhomogeneity and also effectively reduces the stress concentration in the surrounding matrix (it is well known that stress peaks are usually responsible for the mechanical failure of the inhomogeneity-matrix system). The main focus in addressing this challenge has been on designing the shape of the inhomogeneity and the properties of the material interface between the inhomogeneity and its surrounding matrix to achieve the desired uniform strain distribution inside the inhomogeneity. In the emerging area of nanocomposites, however, the presence of appreciable interface energy (known also as the “interface effect”) presents formidable challenges in the design of corresponding nano-inhomogeneities with uniform internal strain distributions. In this chapter, we present some new results in this area for anti-plane shear deformations of composite solids. In particular, we demonstrate the existence of a single nano-inhomogeneity with uniform internal strain distribution induced by a screw dislocation as well as that of periodic nano-inhomogeneities with uniform internal strain distributions when the composite is subjected to uniform remote (anti-plane shear) loading. Our method involves the identification of the corresponding unknown shape of the desired inhomogeneity via a conformal mapping whose unknown coefficients are determined from a system of nonlinear equations. Extensive numerical examples are given to verify the correctness of our method and to illustrate the size dependence of the shapes of the inhomogeneities. It is anticipated that these results will find extensive application in the optimal design of fibrous nanocomposites.