the present study. Using the proposed failure locus the fatigue limit condition for rh < 0 simply becomes:

eq ¼ 2 max ðsDV ðtÞÞ 6 rw ð10Þ

As already pointed out a prospective non-propagating crack, as well as a defect, is described both by its length and orientation. The previously described Dang Van model is able to predict the length but does not present any information about the orientation. For this reason, a critical plane based version of the Dang Van criterion

Fig. 4. Finite element model: (a) F1 axle tested on Vitry test rig, (b) F4 axle tested on Vitry test rig, (c) F4 axle tested on Minden test  rig.

(b)

Fig. 5.  (a) Load  step in finite  element  analysis, (b)  local reference  system.

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

was used in the study. When compared to the basic Dang Van rela- tion a negligible difference in terms of equivalent stress is expected.

Considering a material plane defined by its unit normal vector  n

the Dang Van shear stress can be defined as expressed in Eq.  (11):

sDV ð/; h; tÞ ¼ jjsð/; h; tÞ —  smð/; hÞjj ð11Þ

where /; h are the spherical angles used to express the unit normal vector  n in a Oxyz frame.  sð/; h; tÞ is the shear stress vector acting on the material plane under consideration,  sm ð/; hÞ is the mean shear stress vector and the bracket symbol (k k) represents the length

(measure) of the enclosed vector.

Computing the mean shear stress  sm ð/; hÞ on every plane pass- ing through a point of the body, the determination of the critical plane according to Dang Van criterion requires  the  solution  of the double maximization problem presented in Eq.   (12).

Stress [MPa] Stress [MPa]

Fig. 6.  Stress  gradient  along  the contact.

S. Foletti et al. / International Journal of Fatigue 86 (2016)  34–43 39

plane is defined as the plane which experiences the maximum nor- mal stress amplitude. The critical plane orientation may differ from

the fatigue fracture plane for different materials [15]. In the gen-

eral case the angle a, between the fracture and the critical plane,

: g ¼ 3 þ 1

ffi3ffi—1=s

p

can be obtained as a function of the ratio s ¼ sw=rw between the torsional and the uniaxial fully reversed fatigue   limit:

4 4 ffi3ffi—1

types of test rigs was carried out (Fig. 1b and c). The so called Min- den type test rig is a cantilever resonant machine where the axle is

where  ra;c ;  sa;c   and rH

are  the  normal  stress  amplitude, shear

constrained with a rigid wheel adaptor on one side and an  electric

stress amplitude and hydrostatic stress amplitude acting on the critical plane respectively. rm;c is  the  mean  normal  stress acting on the critical plane. b; g, and k are material parameters depending on the ratio s. For ductile  materials:

Allowable defect size [m] - Dang Van Criterion - F1 Axle

motor with rotating unbalanced masses fixed on the opposite side (Fig. 1b). The so called Vitry type test rig is a three point rotating bending device where the axle, rotated by a motor, is simply sup- ported by two journal axle bearing boxes and loaded in the middle by an actuator (Fig. 1c).