  Galaxy clusters are known to be reservoirs of Cosmic Rays (CRs), as inferred
from theoretical calculations or detection of CR-derived observables. CR
acceleration in clusters is mostly attributed to the dynamical activity that
produces shocks. Shocks in clusters emerge out of merger or accretion, but
which one is more effective in producing CRs? at which dynamical phase? and
why? To this aim, we study the production or injection of CRs through shocks
and its evolution in the galaxy clusters using cosmological simulations with
the {\sc enzo} code. Particle acceleration model considered here is primarily
the Diffusive Shock Acceleration (DSA) of thermal particles, but we also report
a tentative study with pre-existing CRs. Defining appropriate dynamical states
using the concept of virialization, we studied a sample of merging and
non-merging clusters. We report that the merger shocks (with Mach number
$\mathcal{M}\sim2-5$) are the most effective CR producers, while high-Mach
peripheral shocks (i.e. $\mathcal{M}>5$) are mainly responsible for the
brightest phase of CR injection in clusters. Clusters once merged, permanently
deviate from CR and X-ray mass scaling of non-merging systems, enabling us to
use it as a tool to determine the state of merger. Through a temporal and
spatial evolution study, we found a strong correlation between cluster merger
dynamics and CR injection. We observed that the brightest phase of X-ray and CR
injection from clusters occur respectively at about 1.0 and 1.5 Gyr after every
mergers, and CR injection peaks near to the cluster virial radius (i.e
$r_{200}$). Delayed CR injection peaks found in this study deserve further
investigation for possible impact on the evolution of CR-derived observables
from galaxy clusters.
