This study is to evaluate the effect of the heterogeneity on the failure processes and strength characterization of brittle rock containing the single pre-existing crack (or flaw) under uniaxial compression loadings. The numerical simulation reproduces the evolution of the stress and strain fields in flaw propagation process, the mode of acoustic emission related to the heterogeneity of rock and the phenomenon related to discontinuous. It is shown that the lower the value of the homogeneous index, the more influence of local variation on the propagation process of the pre-existing flaw, and there occurs more randomly distributed microfractures throughout the specimen. Studying the details of macrofracture formation in relatively homogeneous specimens, it is interesting to find that there exists a 'constant jump' propagation pattern of the wing crack, which is responsible for the formation of the pre-existing flaw. The numerical results also demonstrate that the stress-strain relation and strength characterization depends strongly on the heterogeneity of the specimen. The heterogeneous rock has a gentler post-peak behavior and lower strength, while the more homogeneous specimen has a higher strength, accordingly, the curve becomes more linear and the strength loss is also rapidly.