  Theories which modify general relativity to explain the accelerated expansion
of the Universe often use screening mechanisms to satisfy constraints on Solar
System scales. We investigate the effects of the cosmic web and the local
environmental density of dark matter halos on the screening properties of the
Vainshtein and chameleon screening mechanisms. We compare the cosmic web
morphology of dark matter particles, mass functions of dark matter halos, mass
and radial dependence of screening, velocity dispersions and peculiar
velocities, and environmental dependence of screening mechanisms in $f(R)$ and
nDGP models. Using the ORIGAMI cosmic web identification routine we find that
the Vainshtein mechanism depends on the cosmic web morphology of dark matter
particles, since these are defined according to the dimensionality of their
collapse, while the chameleon mechanism shows no morphology dependence. The
chameleon screening of halos and their velocity dispersions depend on halo
mass, and small halos and subhalos can be environmentally screened in the
chameleon mechanism. On the other hand, the screening of halos in the
Vainshtein mechanism does not depend on mass nor environment, and their
velocity dispersions are suppressed. The peculiar velocities of halos in the
Vainshtein mechanism are enhanced because screened objects can still feel the
fifth force generated by external fields, while peculiar velocities of
chameleon halos are suppressed when the halo centers are screened.
