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 has 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. Besides solid quark stars and hybrid stars, we also consider 1) solid quark stars dressed in a thin nuclear-matter crust and 2) quark stars with a fluid quark-matter core in the color-flavor-locked phase surrounded by a solid CCS quark-matter envelope. 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 vanishingly 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. We find that quark star models which are ruled out by the observation limits on the tidal deformability can be revived if the entire quark star is in a CCS phase instead of a fluid phase, thus complicating putting constraints on the quark star EOSs. In contrast, the screening effect causes the tidal deformability of a hybrid star with a CCS quark-matter core to agree with that of a corresponding stellar model with a fluid core to within less than 1% if the core size is less than about 70% of the stellar radius. The implication is that if a hybrid star EOS model is ruled out by the observation limits on the tidal deformability, the conclusion will hold no matter whether the quark matter is in a fluid or solid state, assuming that a large solid core comparable to the stellar radius is not favored in nature. Our study advocates that the tidal deformability not only provides us with information on the EOS, but may also give insights into the multilayer structure and elastic properties of compact star models composed of CCS quark matter.
