In this paper, fatigue life circular cross-section elastic bar under pure fatigue axial loading is studied through principles of linear elastic fracture mechanics (LEFM) coupled with the three-dimensional finite element technique for determination of critical crack size and residual lifetime. Three different initial notch depths are discussed. The relations between aspect ratio (b/c) and relative crack depth (b/D) are obtained, and it is shown that there is great difference in the growth of cracks with different front shapes and initial notch depths.