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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