2.1.1. Modified Dang Van fatigue criterion

The basis of the Dang Van criterion is the application of the elas- tic shakedown principles at the mesoscopic scale (more details can

rw  ¼ rw0

pffiaffiffirffiffieffiffiaffiffiffi

þ pffiaffiffirffiffieffiffiaffiffiffi0ffiffi

ð2Þ

be found in Refs. [14,23]). The Dang Van criterion can be expressed by:

being rw0 the fatigue limit for smooth specimens, rw  the fatigue limit depending on defect size (expressed in terms of    Murakami’s

sDV

ðtÞþ aDV

rhðtÞ 6 sw

ð5Þ

pffiaffiffirffiffieffiffiaffiffiffi  parameter  [20])  and  the  fictitious  crack  length  parameter

where aDV

is  a material constant,  sw

is the fatigue limit in reversed

ð area0 Þ found by interpolating the fatigue limit experimentally obtained  for  different  defect sizes.

In the case of no crack formation, that is when the pffiaffiffirffiffieffiffiaffiffiffi term is

zero, equivalent stress level equals to the material fatigue limit for smooth specimen. Beyond this limit a fatigue crack would initiate and propagate spontaneously. For the cases of stress application lower than material fatigue limit a prospective defect size along the damaged area can be defined. That is, a reduced equivalent stress can be described by Eq. (3).

torsion,  rhðtÞ  is  the  instantaneous  hydrostatic  component  of  the

Finally, to obtain the crack size of a shallow 2D surface crack it is possible to use the relationship presented in Eq. (4)   [20]:

Fig. 3. Finite element mesh in the press fit  seat.

Fig. 2. Failure location: (a) F1 axle tested on Vitry test rig, (b) F4 axle tested on Vitry test rig, (c) F4 axle tested on Minden test   rig.

S. Foletti et al. / International Journal of Fatigue 86 (2016)  34–43 37

stress tensor and sDV ðtÞ is the instantaneous value of the Tresca shear  stress  expressed by  Eq. (6):

The  resulting failure locus  presented  by  a  line  on  sDV —rh   plane.  In

order to avoid non-conservative prediction in rolling contact fatigue

evaluated over a symmetrized stress deviator found at the meso- scopic scale, which is obtained by subtracting from the deviatoric stress sijðtÞ a constant tensor, sij;m :

problem, Desimone et al. [16], argued that the failure locus in the region  with  rh  < 0  should  be  modified  into   a   constant value sw ¼ 0:5rw  (proposed conservative locus):

^sijðtÞ¼ sijðtÞ— sij;m ð7Þ

where sij;m  is a residual stress deviator, fulfilling the condition of an

elastic shakedown state at the mesoscopic scale.

The constant aDV appearing in the expression of the Dang Van criterion is usually related with the tension–compression fatigue limit rw  and the pure torsion fatigue limit,    sw:

Further analyses and experimental fatigue tests on mild railway wheel steel subjected to out-of-phase multiaxial fatigue loading, simulating rolling contact fatigue conditions, have shown that the multiaxial fatigue limit does not depend on hydrostatic  stress  if the hydrostatic stress component is lower than zero [17]. Further- more, non-conservative results were achieved by the  application of the original locus concept. For this reason, the Dang–Van crite- rion will be applied in ‘‘modified” form in the present study. In fact, the stress analysis of the region under examination reveals the hydrostatic stress to remain negative during process of loading. The Dang Van criterion, in its modified forms, will be applied in