Abstract: Using Fourier-integral transform technique，the paper studies the solution problem of the stress field around an axial moving crack with constant velocity in a cylindrical shell and obtains a small parameter solution of the stress field around crack tip. The result shows that the trength of the stress field around crack tip varies with crack propagation velocity，while
angular distribution of the stress field does not. Moreover，when crack moving velocity ar-rives a a certain value，branchy phenomena well be found in the running cracks.

Key words: Dynamic fracture, Stress field, Cylindrical shells