Application of multiaxial fatigue parameters

Similarities between the features of cracking and observations of multiaxial fatigue and observations of the formation of cracks in fretting fatigue leads to application of multiaxial fatigue and damage concepts to the analysis of fretting problem [14,48].

Dang Van’s Criteria

Dang Van et al [49] has proposed a method for predicting fatigue life under high cycle fatigue conditions. This model was based on the observation that fatigue crack nucleation is a local process and begins within grains that have undergone plastic deformation.

The analysis is carried out by sub dividing a structure into sub divisions (similar to a mesh in finite element analysis). Analysis for fatigue failure can be carried out for any of the sub divisions to analyse the possibility of cracking at that point. First, the point concerned is confined to a macroscopic elementary volume and further that

FDP = r^S -(2.10)

FFDP = (a xxx ^ xy / l <?)_max

volume is sub divided into mesoscopic volumes. According to Dang Van et al [49], the stress tensor at this scale results from the macroscopic one and the local residual stresses.

Where, p h is the Hydrostatic stress, r(t) is Instantaneous mesoscopic shear stress and m, n are material constants evaluated experimentally.

The fatigue resistance is checked point by point analysing the quantity

fatigue cracking. Dang Van and Maitoumam [50] have applied this criterion to the fretting problem and have obtained satisfactory predictions.

Application of the SWT parameter

Smith, Watson and Topper [28] have proposed a model to describe the fatigue life in push pull tensile compression. It was expressed as

The Dang Van’s Criteria is given as an inequality related to the mesoscopic stresses at all instants (t) of a loading cycle as follows.

max {r(t) + aph{ (t) -b } < b -(2.12)

max {r(t) + aph{ (t) - n}

The positive value of the DV means occurrence of b

«,max -(2.13)

Where,

O f .Fatigue strength coefficient b - Fatigue strength exponent 6f - Fatigue ductility coefficient c- Fatigue ductility exponent E - Young’s modulus

Nf - Number of cycles to initiate a crack of a given length Ae - Strain range in the principal plane

On,Max - Maximum normal stress on the principal strain range plane

Szolwinski and Farris [14] applied a modified SWT parameter based on critical plane orientations successfully to predict the fretting crack initiation. They assumed that the crack nucleation occurs on a plane where the combination of strain amplitude and the maximum normal stress is maximum and not necessarily the maximum principal strain amplitude or the maximum normal stress at a given point.

The SWT parameter has been successfully applied to the fretting problem by several researchers assuming that the fretting damage process is a mode I cracking problem.

[4,14,58, 61, 64, 65, 66, 67, 68, 69,70].

Stress gradient effect considerations

Araujo and Nowell [61] applying the SWT parameter point out that the stress concentration is extremely localised and fretting cracks nucleate at points of high

localised stresses similar to a notch situation. But the multiaxial crack initiation parameters do not account for the stress gradients. In fretting situations a high stress field exists in the vicinity of the contact and rapidly decays when moving into the material. For more rapidly varying stress fields, the high localised (surface) maximum is not sufficient to fully describe the crack initiation and growth unless the parameter concerned is sustained over a considerable length or volume.

Therefore, they proposed an averaging methodology over a length or a volume.

The averaging methodology is basically based on the fact that the high stress must be sustained over a critical volume in order for a crack to rupture through the strongest micro-structural barrier.

Fouvry et al [71] have introduced a methodology of averaging parameters over a volume in order to take into account the high stress gradient exist in vicinity of the contact surface. Proudhon et al [67] have adopted this technique and analysed for 2024-T-351 alloy and found a good correlation for the average grain size of the material with the process volume. Further, Munoz et al [69] compared the process volume for 2024-T-351 and 7075-T651 alloys and found that the process volume cannot correlate with the grain size for 7075-T651 alloy. During the same period, Proudhon et al [68] proposed a variable process volume approach.

New advancements were made by the application of the SWT parameter for the fretting wear damage process by McColl [3 ], Ding et. al. [17 ], Ratsimba et al [72]

and Madge et al [73].

A modified version of the SWT parameter was proposed by Ding et al [74] in order to capture the type of damage in the mixed slip regime and the damage in the gross slip regime separately. A modified Ruiz parameter in conjunction with the SWT parameter was also proposed by Vidner and Leidich [75].

Fatemi and Socie (FS) parameter

Fatemi and Socie [16] proposed a model based on the shear strain amplitude in a critical plane, for mode II cracking. Their model was expressed as;

Where,

AyMax - Maximum shear strain range

ery - yield strength

crnMax " Maximum stress acting normal to the plane with maximum shear strain range

t\ - Shear fatigue strength coefficient

G - shear modulus

b7 - Shear fatigue strength exponent

c7- Shear fatigue ductility exponent

Nf - Number of cycles to initiate a crack of a given length a - is a constant

/t, max -(2.14)

2.3.1.3 Fracture mechanics approach

Fracture mechanics is an approach which can be used to estimate and predict fretting fatigue life [1,2,14, 23,47,51-60, 76-80].

Such methods are based on establishing the stress intensity factor AK for cracks initiated in the contact zone. Calculation of the number of loading cycles, from an initial crack to failure was integrated using a Paris type crack propagation law.

where,

af - is the crack size at failure

a - the initial crack size,_{i 7}

AK - is the range of the stress intensity factor (that depends on the crack size c) C, m - are material parameters.

In early work of Nix and Lindley [11, 60] a fracture mechanics based procedure was used to predict fretting fatigue damage. However, in their analyses they assumed that the cracks propagates normal to the contact surface although in practice the actual cracks are typically generated oblique to the surface an angle at about 15° - 45°.

Furthermore they only used mode I stress intensity factors in the analysis.

Faanes and Fernando [52] also have used LEFM approach to predict fretting crack growth. Crack propagation was evaluated by assuming a growth law of modified Paris type:

1 da -(2.15)

^ - = c[t±KeJf-hK lh(a)]' -(2.16) dN

Here, &Keff was calculated taking into the multiaxial stress state incorporating the bulk stress, normal and friction, and AKth {a) is the threshold stress intensity factor, modified to account for small cracks through the relationship proposed by El Haddad [93]. They have correlated the threshold stress intensity factor range for a short crack through an intrinsic crack length aQi as follows.

A K Ja ) = ^ M = L -(2.17)

^/1 + a0 / a

Where,

a - is the crack size

AKth. is the threshold stress intensity for a long crack.

The a0was evaluated from

a0 = —

n A*«(«>)

. Ao>i . - a , -(2.18)

Here,

ai is an “initial crack length” normally assumed to be the surface roughness

A<j fl is the uniaxial fatigue limit.

Mutoh and Xu [36] introduced a new concept based on the stress field near the contact. Based on experimental observations that fretting cracks initiate at very

shallow angles to the surface and follows the direction of the maximum shear stress, they assumed that the fretting crack initiation is in the direction of the maximum shear stress, and the crack changes direction to the direction of maximum normal stress, after propagating for a few micrometres.

In this method it is very important to ascertain the crack initiation location and the location is assumed to be the point where, the most severe stress concentrated point on the contact surface. The crack path is predicted in steps assuming that the crack will initiate in the maximum shear stress direction and propagate in the maximum normal stress direction.

2.4 Experimental Techniques in fretting fatigue.

Generally, experimental investigations are focused on unveiling the influence of certain parameters which influence fretting damage processes. One objective of fretting damage experiments is to monitor fretting crack initiation and propagation under different loading conditions.

Hills and Nowell [62, 63] categorise fretting experiments into three groups depending on the objective of the experiment

1. Simulation of real engineering problems - The objective of these tests are to reproduce as accurately as possible, either full size or possibly on a reduced scale, the contact problem. This type of experiment is of great advantage to designers, as the results obtained from tests, could be applied without modification to real engineering problems.

2. Material ranking tests - The objective of this type of tests is to assess a materials resistance to fretting damage.

3. Idealised fretting fatigue tests - These tests are carried out in certain conditions to understand the influence and contributions of individual parameters to fretting fatigue. The objective of these tests is to understand the mechanics and underlying processes of fretting fatigue phenomena.

Out of the above three categories, simulation of a real engineering problem is more difficult since in most of the cases new experimental rigs need to be designed. In some cases, it might be extremely difficult to simulate the real contact conditions and associated stress fields under laboratory conditions.