LAUSR

(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.