  Strange stars are one of the possible compact stellar objects that can be
formed after a supernova collapse. We consider a model of strange star having
an inner core in the color-flavor locked phase surmounted by a crystalline
color superconducting layer. These two phases constitute the {\it quarksphere},
which we assume to be the largest and heaviest part of the strange star. The
next layer consists of standard nuclear matter forming a ionic crust, hovering
on the top of the quarksphere and prevented from falling by a strong dipolar
electric field. The dipolar electric field arises because quark matter is
confined in the quarksphere by the strong interaction, but electrons can leak
outside forming a few hundreds Fermi thick electron layer separating the ionic
crust from the underlying quark matter. The ionic matter and the crystalline
color superconducting matter constitute two electromagnetically coupled crust
layers. We study the torsional oscillations of these two layers. Remarkably, we
find that if a fraction larger than $10^{-4}$ of the energy of a Vela-like
glitch is conveyed to a torsional oscillation, the ionic crust will likely
break. The reason is that the very rigid and heavy crystalline color
superconducting crust layer will absorb only a small fraction of the glitch
energy, leading to a large amplitude torsional oscillation of the ionic crust.
