  In 1957 Eshelby showed that a homogeneous isotropic ellipsoidal inhomogeneity
embedded in a homogeneous isotropic host would feel uniform strains and
stresses when uniform strains or stresses are applied in the far-field. Of
specific importance is the uniformity of Eshelby's tensor S. Following this
paper a vast literature has been generated using and developing Eshelby's
result and ideas, leading to some beautiful mathematics and extremely useful
results in a wide range of application areas. In 1961 Eshelby conjectured that
for anisotropic materials only ellipsoidal inhomogeneities would lead to such
uniform interior fields. Although much progress has been made since then, the
quest to prove this conjecture is still not complete; numerous important
problems remain open. Following a different approach to that considered by
Eshelby, a closely related tensor P=S D^0 arises, where D^0 is the host medium
compliance tensor. The tensor P is associated with Hill and is of course also
uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host
phase. Two of the most fundamental and useful areas of applications of these
tensors are in Newtonian potential problems such as heat conduction,
electrostatics, etc. and in the vector problems of elastostatics.
Micromechanical methods established mainly over the last half-century have
enabled bounds on and predictions of the effective properties of composite
media. In many cases such predictions can be explicitly written down in terms
of the Hill, or equivalently the Eshelby tensor and can be shown to provide
excellent predictions in many cases. Here this classical problem is revisited
and a large number of results for problems that are felt to be of great utility
in a wide range of disciplines are derived or recalled.
