Eshelby’s inclusion problem in the gradient theory of elasticity: Applications to composite materials

We extend Eshelby’s integral representations for elastic inclusion problems to the case of gradient theory of elasticity. Gradient elastic effects associated with the existence of an interphase layer, within a simple and robust gradient model whose properties are described by the harmonic equation, are discussed. The decomposition of the corresponding solution into “classical” and “gradient” components is established. It is shown that the aforementioned Eshelby-type integral formulas for gradient elasticity can be expressed in the same form as in the standard theory of elasticity, but only for the “classical” part of the solution. The implementation of Eshelby’s approach in determining the effective properties of composites by the three-phase method requires the derivation of a complete solution for the gradient model. An example of application of the so-obtained generalized gradient method for determining the effective properties of composites with size effects due to cohesion and surface forces is given.