Non-elliptical inclusions that achieve uniform internal strain fields in an elastic half-plane

Abstract

In existing literature, it remains an unexplored question whether any inclusion shape can achieve a uniform internal strain field in an elastic half-plane under either given uniform remote loadings or given uniform eigenstrains imposed on the inclusion. This paper examines the existence and construction of such single or multiple non-elliptical inclusions that achieve prescribed uniform internal strain fields in an elastic half-plane under given uniform anti-plane shear eigenstrains imposed on the inclusions. Such non-elliptical inclusion shapes in a half-plane can be determined by solving the original problem of an unknown holomorphic function in a multiply connected half-plane, which is transferred to an equivalent problem of an unknown holomorphic function in a multiply connected whole plane based on analytic continuation techniques. Extensive numerical examples are shown for single inclusion, multiple inclusions and two geometrically symmetrical inclusions, respectively. It is found that the inclusion shapes which achieve uniform internal strain fields depend on the given uniform eigenstrains, and the inclusion shapes that achieve uniform internal strain fields for arbitrarily given uniform eigenstrains do not exist. Moreover, specific conditions are derived on the given uniform eigenstrains and prescribed uniform internal strain fields for the existence of two geometrically symmetrical inclusions that achieve uniform internal strain fields.