  (Abridged) It is well known that the tidal deformability of a compact star
carries important information about the interior equation-of-state (EOS) of the
star. The first gravitational-wave event GW170817 from a binary compact star
merger observed by the LIGO/VIRGO detectors have already put limits on the
tidal deformability and provided constraints on the ultra-high nuclear density
EOS. In view of this ground breaking discovery, we revisit and extend our
previous work [Phys. Rev. D 95, 101302(R) (2017)] which found that taking the
effect of elasticity into account in the calculation of the tidal deformability
of compact star models composed of crystalline color-superconducting (CCS)
quark matter can break the universal I-Love relation discovered for fluid
compact stars. In this paper, we present our formulation in detail and provide
more analysis to complement our previous findings. We focus and extend the
study of the screening effect on the tidal deformability, which we found
previously for hybrid star models, to various theoretical two-layer compact
star models. We show that the screening effect of these two-layer models in
general depends on the thickness of the envelope and the ratio between the
density gap and the core density at the core-envelope interface. However, for
models with a fluid envelope and a vanishing small density gap, the screening
effect remains strong even as the thickness of the envelops tends to zero if
the quark matter core has a fairly uniform density. The relevance of our study
to GW170817 is also discussed. Our study advocates that the tidal deformability
not only provides us information on the EOS, but may also give insights into
the multi-layer structure and elastic properties of compact star models
composed of CCS quark matter.
