  New connections between static elastic cloaking, low frequency elastic wave
scattering and neutral inclusions are established in the context of two
dimensional elasticity. A cylindrical core surrounded by a cylindrical shell is
embedded in a uniform elastic matrix. Given the core and matrix properties, we
answer the questions of how to select the shell material such that (i) it acts
as a static elastic cloak, and (ii) it eliminates low frequency scattering of
incident elastic waves. It is shown that static cloaking (i) requires an
anisotropic shell, whereas scattering reduction (ii) can be satisfied more
simply with isotropic materials. Implicit solutions for the shell material are
obtained by considering the core-shell composite cylinder as a neutral elastic
inclusion. Two types of neutral inclusion are distinguished, \textit{weak} and
\textit{strong} with the former equivalent to low frequency transparency {and
the classical Christensen and Lo generalised self-consistent result for
in-plane shear from 1979. Our introduction of the \textit{strong neutral
inclusion} is an important extension of this result in that we show that
standard anisotropic shells can act as perfect static cloaks, contrasting
previous work that has employed "unphysical" materials.} The relationships
between low frequency transparency, static cloaking and neutral inclusions
provide the material designer with options for achieving elastic cloaking in
the quasi-static limit.
