not allowed on xy symmetry plane and only radial    displacements

were allowed on yz symmetry plane. Additionally, displacement along y direction was not allowed for the nodes along central line in order to achieve an isostatic condition. The finite element model for F4 geometry tested on the Vitry test rig is reported in Fig. 4b. The boundary conditions are the same as previously pointed out. For Minden type test, xy plane symmetry was imposed only. Exter- nal surface of the wheel has been fastened (Fig. 4c). Two different analyses with different applied nominal stresses levels, selected on

of which is selected at the contact edge with an X axis parallel to the longitudinal axis of the axle and a Y axis in the radial direction, stress path from the last cycle was extracted (Fig. 5b). Cyclic stress at each node has been expressed by the mean stress tensor and amplitude  stress  tensor  with  respect  to  the  defined  coordinate

system.

3.2. F1 axle

The stress profile at distances 0, 2 and 20 mm away from the contact edge is shown in Fig. 6. A steep stress gradient is observed near T transition. Together with the mesh diagram presented in Fig. 3, computation of a mesh size dependent stress singularity is obviously seen. In order to achieve proper results application of proposed ‘‘local stress based” high cycle fatigue criteria to the sin- gularity region was avoided. For the assessment of the imposed limitation surface examination of the tested axles was made. MP inspection showed the distribution of the fretting fatigue initiation sites to range between 4 mm and 20 mm away from contact edge. Obtained results were also justified by the study presented in [7]. Accordingly, the proposed method was applied to the regions of reduced stress gradient starting from 2 mm away from the transi- tion edge (X P 2 mm). The presented approach could be extended

Table 3

Predicted allowable defect size (length) c. Subscripts: (DV) Dang Van Criterion, (LM) Liu–Mahadevan Criterion, ðexp; CPÞ experimentally observed defect size (length) with crack  propagation,  ðexp; NCPÞ  experimentally  observed  defect  size  (length)  with no

crack propagation, (–) no SEM observations.

the basis of ‘‘failure” and ‘‘run-out” stress levels observed in full-

scale tests, presented in Table 2, were conducted for each type of axle.

The stress data after two complete cycles of load application

was considered in order to obtain stress state after a stabilized con-

F4 run-out    120 540 510 – <120

dition have been attained (Fig. 5a). On a X-Y reference plane, origin

Fig. 9. Fractographic analysis for F1 axle tested at failure condition: (a) Axle’s sectioning, (b and c) cavitation and cracking on  tangential  direction  leading  to  crack propagation, (d) surface defects with non-propagating     cracks.

S. Foletti et al. / International Journal of Fatigue 86 (2016)  34–43 41

to the vicinity of contact edge by applying a criterion without local stress variables [10,30].

The estimations of non-propagating crack size and critical plane orientation are shown in Fig. 7 for the state of stress at the surface, (Y ¼ 0). In this region for both criteria the most sensitive region for crack propagation was predicted to be between 20 and 30 mm from transition edge, in full agreement with experimentally observed failure location, (Fig. 2a). At the failure stress level the critical defect size at the onset of crack propagation predicted by the criteria is in the order of 300–400 lm where the smaller value is the limit predicted by the Dang Van criterion. For the run-out axles the critical defect size increase to 390 lm and 520 lm by Dang Van criterion and Liu Mahadevan criterion,  respectively.