This paper presents a numerical study of creep crack growth (CCG) in a fracture mechanics specimen. The material properties used are representative of a carbon–manganese steel at 360 °C and the constitutive behaviour of the steel is described by a power law creep model. A damage-based approach is used to predict the crack propagation rate in a compact tension specimen and the data are correlated against an independently determined C ∗ parameter. Elastic–creep and elastic–plastic–creep analyses are performed using two different crack growth criteria to predict crack extension under plane stress and plane strain conditions. The plane strain crack growth rate predicted from the numerical analysis is found to be less conservative than the plane strain upper bound of an existing ductility exhaustion model, for values of C ∗ within the limits of the present CCG testing standards. At low values of C ∗ the predicted plane stress and plane strain crack growth rates differ by a factor between 5 and 30 depending on the creep ductility of the material. However, at higher loads and C ∗ values, the plane strain crack growth rates, predicted using an elastic–plastic–creep material response, approach those for plane stress. These results are consistent with experimental data for the material and suggest that purely elastic–creep modelling is unrealistic for the carbon–manganese steel as plastic strains are significant at relevant loading levels.