Molecular dynamics simulation was used to relax the heteroepitaxial Au islands in monolayer on Cu(001) substrate in the initial stage of film growth. The evolution of the morphology and local pressure of the heteroepitaxial Au islands as well as the adhesive energy to the substrate have been analyzed. The simulation results show that a pseudo-morphologically strained Au island is formed when the heteroepitaxial Au island is not larger than 7×7. When the heteroepitaxial Au island is as large as 8×8, however, a misfit dislocation induced by the stress, which results from the mismatch in the interface, appears. The number of the misfit dislocations increases with increasing scale of the heteroepitaxial Au island, and moreover, crossing dislocations and vacancies can be observed in large heteroepitaxial Au islands. The analysis results of the local pressure map show that the difference of the neighborhood results in the difference of the stress of the atoms in the heteroepitaxial Au islands. Furthermore, the psudo-morphologically strained Au islands are the results of the different local pressure of Au atoms in the heteroepitaxial Au islands. On the other hand, we found that the misfit dislocations lead to the decrease of the adhesive energy of the mismatched atoms to the substrate, and the adhesion of other atoms increases relatively.