The effects of the applied field ( B_a ) and a dimensionless parameter ( p ) on flux-induced stress distribution in a long cylindrical superconductor are studied based on the exponential model for zero-field cooling and field cooling magnetisation processes. The expressions of radial and hoop stresses for the descending field are deduced by the plane strain approach. The results indicate that the effects of Ba and p on stress distribution depend on the activation processes. In comparison, the results between the exponential model and the Kim model are generally similar, but there are some distinctions between peak value positions and plateau regions. This also means that the exponential model may be more helpful and applicable than the Kim model under the activation processes in the actual situation.