Fig. 1 Microstructures of the Ti-6Al-4V alloy: a phase map; b IPF; c BC map with twins. The green lines in c represent annealing twins

Fig. 2 a Picture of Shimadzu USF-2000A machine; b picture of installation of specimen tested under mean stress

Fig. 3 Geometry and size of specimen and defect under ultrasonic frequency fatigue test (unit: mm): a smooth specimen for R = − 1; b smooth specimen for R = 0.1; c schematic of surface defect

Fig. 4 Geometry and size of specimen for conventional frequency fatigue test (unit: mm)

Fig. 5 Maximum stress σmax as a function of fatigue life for smooth specimens and specimens with different defect sizes for the Ti-6Al-4V alloy at R = −1 and R = 0.1. The hollow symbols indicate that the specimens do not fail at the tested loading cycles

Fig. 6 Typical fracture surface morphologies of failed smooth specimens: a and b σmax = 775 MPa, R = −1, Nf = 7.57 × 105 cyc; c and d σmax = 750 MPa, R = −1, Nf = 2.49 × 108 cyc; e and f σmax = 900 MPa, R = 0.1, Nf = 1.63 × 107 cyc; g and h σmax = 750 MPa, R = 0.1, Nf = 1.47 × 108 cyc. b, d, f and h magnified images of RA in a, c, e and g, respectively

Fig. 7 Typical fracture surface morphologies of failed specimens with surface defect: a and b specimens with defect-A, σmax = 450 MPa, R = −1, Nf = 1.44 × 105 cyc; c and d specimens with defect-B, σmax = 600 MPa, R = −1, Nf = 1.57 × 105 cyc; e and f specimens with defect-A, σmax = 475 MPa, R = 0.1, Nf = 8.99 × 106 cyc; g and h specimens with defect-A, σmax = 600 MPa, R = 0.1, Nf = 8.76 × 104 cyc; i and j specimens with defect-B, σmax = 550 MPa, R = 0.1, Nf = 1.33 × 108 cyc; k and l specimens with defect-B, σmax = 700 MPa, R = 0.1, Nf = 8.78 × 104 cyc. b, d, f, h, j and l magnified images around defects in a, c, e, g, i and k, respectively

Fig. 8 Comparison of fatigue life of smooth specimens under variable amplitude loadings and constant amplitude loadings at R = −1 and R = 0.1

Fig. 9 Typical fracture surface morphologies of failed specimens under variable amplitude loadings: a and b σmax: 375 MPa (5.0 × 108 cyc) → 775 MPa (3.04 × 107 cyc, failure), R = −1; c and d σmax: 650 MPa (1.0 × 107 cyc) → 750 MPa (7.03 × 107 cyc, failure), R = 0.1. b and d Magnified images of crack initiation region in a and c, respectively

Fig. 10 Maximum stress as a function of fatigue life of smooth specimens under continuous loadings and intermittent loadings at R = −1. The hollow symbols indicate that the specimens do not fail at the tested loading cycles

Fig. 11 Variation of cumulative maximum strain with loading cycles for dog-bone shaped specimens. The dashed line indicates that the specimen fails

Fig. 12 EBSD results of the microstructure and damage evolution for the dog-bone shaped specimens subjected to 3000 cyc at the maximum stress of 775 MPa. a1-a3 and b1-b3 BC maps with twin, IPF and KAM map of the specimen at longitudinal section and transverse section for R = 0.1, respectively. c1-c3 and d1-d3 BC map with twin, IPF and KAM map of the specimen at longitudinal section and transverse section for R = −1, respectively

Fig. 13 Residual stresses on the minimum cross-section of specimens before and after a sequence of lower stress amplitude fatigue test at R = −1: a circumferential direction; b axial direction

Fig. 14 Fatigue strength of specimens at different fatigue lifetimes: a R = −1; b R = 0.1

Fig. 15 Haigh diagrams with fatigue strengths at 107 cyc, 108 cyc and 109 cyc: a smooth specimen; b specimen with defect-A; c specimen with defect-B