Van
der Waals (vdW) screening or Faraday-cage-like screening of
vdW interaction by monolayer crystals has recently been observed in
experiments and understood from first-principles theories. Here, we
investigate the vdW screening by a bulky dielectric layer using the
Lifshitz theory. The ratio of vdW screening is found to depend on
not only the interobject distance but also the thicknesses of the
separated layers. Surprisingly, the screening ratio exhibits a nonmonotonous
distance dependence, first increasing, but beyond a critical distance
reducing, toward zero. The short-range trend coincides with that predicted
for graphene-like trilayers by the random phase approximation, while
the long-range trend poses a contrast to the increasing screening
with distance by graphene predicted by the many-body dispersion approach.
The positive correlation between the screening ratio and the dielectric
constant revealed for atomistic layers is reproduced for the bulky
dielectric layers.