On the Internal Field and Configuration of Harmonic Elastic Inclusions in Plane Deformation

Harmonic inclusions are defined as those that do not disturb the mean stress component of an initial stress field existing in a homogeneous elastic matrix when they are introduced into the matrix. The design of harmonic inclusions in the literature mainly focuses on the common cases in which the initial stress field has a constant mean stress component (while the corresponding deviatoric stress component may be either constant or non-constant). To identify the configuration of harmonic elastic inclusions in the common cases, researchers consistently assumed that the internal stresses inside the inclusions are hydrostatic and uniform although no rigorous justification was given. In this paper, we present a rigorous proof for the necessity of this assumption in the design of harmonic elastic inclusions in plane deformations. Specifically, we show that the internal stresses inside any elastic inclusion meeting the harmonicity condition must be uniform and (in-plane) hydrostatic (except for trivial cases in which the inclusion and matrix have the same shear modulus). We develop also a general analytic procedure to determine the desired shape for an isolated harmonic elastic inclusion for an arbitrary deviatoric component of the initial stress field, which is illustrated via a few numerical examples.