  We consider plasmon resonances and cloaking for the elastostatic system in
$\mathbb{R}^3$ via the spectral theory of Neumann-Poincar\'e operator. We first
derive the full spectral properties of the Neumann-Poincar\'e operator for the
3D elastostatic system in the spherical geometry. The spectral result is of
significant interest for its own sake, and serves as a highly nontrivial
extension of the corresponding 2D study in [8]. The derivation of the spectral
result in 3D involves much more complicated and subtle calculations and
arguments than that for the 2D case. Then we consider a 3D plasmonic structure
in elastostatics which takes a general core-shell-matrix form with the
metamaterial located in the shell. Using the obtained spectral result, we
provide an accurate characterisation of the anomalous localised resonance and
cloaking associated to such a plasmonic structure.
