The origin of ultra-high-energy cosmic rays is still unknown. Despite strong evidence suggesting an extragalactic origin of the sources that accelerate the most energetic cosmic rays, they have not yet been identified. Galaxy cluster accretion shocks have been considered as a possible acceleration site for cosmic ray particles. External accretion shocks in galaxy clusters arise from the inflow of material in the cosmic web into their gravitational potential well. The size of these shocks can reach values on
the order of megaparsecs, placing them among the largest shocks found in nature. In this work, we investigate the possibility that the ultra-high-energy cosmic rays are accelerated in galaxy cluster accretion shocks. For that purpose, we develop a model considering a set of discrete sources, corresponding to nearby massive clusters (including Virgo), superimposed to a continuous distribution of sources which considers both low-mass and non-local massive clusters. We fit the cosmic ray energy spectrum and the composition profile measured by the Pierre Auger Observatory in order to obtain a set of parameters corresponding to the injection spectrum assumed in the model. The possibility of ultra-high-energy cosmic rays being accelerated in these astronomical objects is examined. The impact of the high-energy hadronic interaction model used to analyse the composition data is also presented.