Sulfate attack is an important cause of concrete durability deterioration. A multi-field peridynamic (PD) model is developed to investigate the expansion and cracking behaviour of concrete under sulfate attack. The deformation at the material point is decomposed into expansion and distortion, and an expression for expansion is derived in the PD framework. A novel pairwise force function is constructed to capture the expansion effect. Additionally, a distortion-based damage criterion is established to determine bond failure in expansion-dominated deformation. The mutually reinforcing effects between crack propagation and sulfate diffusion are explored. Several numerical examples are performed to verify the validity and accuracy of the PD model. The results of the coupled PD model exhibit in good agreement with those obtained from the analytical solutions, finite difference method, and experimental findings. The developed model further elucidates the profound interplay between concrete expansion cracking, chemical reaction and expansion process during sulfate attack.