A new micromechanics approach to the application of Eshelby's equivalent inclusion method in three phase composites with shape memory polymer matrix

In the present work Eshelby's equivalent inclusion method is extended to compute the effective properties of two inclusions in a heterogeneous matrix. Generally Eshelby's problem involves a single inclusion under dilute distribution in two phase composites. Moreover, unlike the dual inclusion theory, which is based on the concept of one inclusion embedded in to the second inclusion, alternate problem is now addressed. In this paper Eshelby's single equivalent inclusion concept is extended to the case of composite consisting of two different inclusions embedded separately in a heterogeneous matrix. In the composite, first inclusion is fibre and second inclusion is carbon nanotube embedded in a shape memory polymer heterogeneous matrix. The solution is obtained through individual eigenstrains in both the inclusions. The general closed form relations for the effective moduli and inelastic strain tensors are derived under infinitesimal strains of constituent materials following a two-step homogenization procedure. Analytical solutions of Eshelby's problem with two cylindrical inclusions explicit of each other, having different properties and placed in the same inelastic matrix domain are obtained. The relations are then applied to characterise the shape memory polymer composite. The consistency of the novel relations are established by comparing the results with other micromechanics methods.