  Dust grains may be accelerated to relativistic speeds by radiation pressure,
diffusive shocks, and other acceleration mechanisms. Such relativistic grains
have been suggested as primary particles of ultrahigh energy cosmic rays
(UHECRs). In this paper, we first revisit the problem of acceleration by
radiation pressure and calculate maximum grain velocities achieved. We find
that grains can be accelerated to relativistic speeds with Lorentz factor
$\gamma < 2$ by powerful radiation sources, which is lower than earlier
estimates showing that $\gamma$ could reach $\sim 10$. We then investigate
different destruction mechanisms for relativistic grains traversing a variety
of environments. In solar radiation, we find that the destruction by thermal
sublimation and Coulomb explosions is important. We also quantify grain
destruction due to electronic sputtering by ions and grain-grain collisions.
Electronic sputtering is found to be rather inefficient, whereas the
evaporation following grain-grain collisions is shown to be an important
mechanism for which the $a \sim 0.01- 1\mu$m grains would be destroyed after
sweeping a gas column $N_{Coul}\sim 5\times 10^{19}-5\times 10^{20}cm^{-2}$.
Relativistic dust in the interstellar medium and intergalactic medium (IGM)
would be disrupted by Coulomb explosions due to collisional charging after
traversing a gas column $N_{Coul} \sim 10^{17}cm^{-2}$ unless grain material is
very strong. We show that photoelectric emission by optical and ultraviolet
background radiation is also significant for the destruction of relativistic
dust in the IGM. The obtained results indicate that relativistic dust from
galaxies would be destroyed before reaching the Earth's atmosphere and unlikely
to account for UHECRs.
