  Galaxy clusters grow by gas accretion, mostly from mergers of substructures,
which release powerful shock waves into cosmic plasmas and convert a fraction
of kinetic energy into thermal energy, amplification of magnetic fields and
into the acceleration of energetic particles. The modeling of the radio
signature of cosmic shocks, combined with the lack of detected gamma-rays from
cosmic ray (CR) protons, poses challenges to our understanding of how cosmic
rays get accelerated and stored in the intracluster medium. Here we review the
injection of CRs by cosmic shocks of different strengths, combining the
detailed "microscopic" view of collisionless processes governing the creation
of non-thermal distributions of electrons and protons in cluster shocks (based
on analytic theory and particle-in-cell simulations), with the "macroscopic"
view of the large-scale distribution of cosmic rays, suggested by modern
cosmological simulations. We discuss time dependent non-linear kinetic models
of particle acceleration by multiple internal shocks with large scale
compressible motions of plasma. The models produce soft CR spectra containing a
noticeable energy density in the super-thermal protons of energies below a few
GeV which are difficult to constrain by Fermi observations. We consider the
effect of plasma composition on CR injection and super-thermal particle
population in the hot intracluster matter which can be constrained by fine high
resolution X-ray spectroscopy of Fe ions.
