Uniform strain fields inside periodic inclusions incorporating interface effects in anti-plane shear

We incorporate the mechanics of the interface to construct optimal shapes of periodic inclusions which achieve uniform internal strain fields in an elastic plane subjected to uniform remote anti-plane shear loading. These shapes are determined by solving a problem of the existence of a holomorphic function which is defined outside the unit circle in an infinite imaginary plane with specific boundary value on the unit circle. We illustrate such shapes using several examples. We show that the incorporation of interface mechanics has a significant effect on the design of such shapes and hence on the existence of these inclusions at the nanoscale. In addition, we show that if the period of the inclusion–matrix system exceeds roughly seven times the inclusion size, such shapes can be treated essentially as being equivalent to those of a single inclusion enclosing the same uniform internal strain in the presence of identical bulk and interface material constants, inclusion size and remote loading.