  In this paper, the acceleration of particles in astrophysical sources by the
Fermi mechanism is revisited under the assumption of Lorentz invariance
violation (LIV). We calculate the energy spectrum and the acceleration time of
particles leaving the source as a function of the energy beyond which the
Lorentz invariance violation becomes relevant. Lorentz invariance violation
causes significant changes in the acceleration of particles by the first and
second-order Fermi mechanisms. The energy spectrum of particles accelerated by
first-order Fermi mechanism under LIV assumption shows a strong suppression for
energies above the break. The calculations presented here complete the scenario
for LIV searches with astroparticles by showing, for the first time, how the
benchmark acceleration mechanisms (Fermi) are modified under LIV assumption.
