Fretting fatigue investigation on Al 7075-T651 alloy: experimental, analytical and numerical analysis

In this work, a fretting fatigue experimental campaign on Al 7075-T651 alloy specimens is simulated by using both analytical and numerical approaches. According to one of the classifications available in the literature, the theoretical approaches for fretting fatigue assessment may be divided into stress-based models or Stress-Intensity Factor (SIF) models. The analytical approach here applied falls in the group of the stress-based models, whereas two numerical methods fall in the group of the SIF-based models. The fatigue assessment is performed by evaluating both crack path and lifetime.

[18], such approaches may be classified into two main groups: (i) stress-based models, where a multiaxial fatigue criterion, generally proposed for plain fatigue, 5 is applied to a non-local representative stress tensor; (ii)

Fretting fatigue test set-up
The scheme of the fretting fatigue test device employed is shown in Figure 1(a). At the beginning of each test, a constant normal load N presses two cylindrical pads against a dog-bone specimen ( Figure   1(b)). Then, dog-bone specimen is subjected to a cyclic (harmonic) axial load P . Due to (1) the friction of the contact surfaces and (2) the compliance of the fretting device, an in-phase cyclic tangential load Q is produced. In such a case, both a constant normal stress distribution and a time-varying shear stress distribution are generated by N and Q at the contact surfaces. 7 In addition, the axial load P leads to a global stress  in the test specimen.
The device was constructed with two adjustable supports in order to modify the stiffness of the assembly, and thus producing different values for the tangential load Q by applying the same axial load P . With this configuration, the fretting fatigue tests have been carried out using multiple independent loading combinations ( N , Q , P ). In order to monitoring in real time the above fretting loads, the test device has been instrumented with a series of load cells, as is shown in Figure 1(a).

Figure 1.
The relevant geometry and sizes for the pads and test specimens are displayed in Figure 1 ) in the following. 9 In Figure 2( where the colour code represents the distance of the failure surface from a given reference plane parallel to the yz-plane, and the crack initiation point is marked with a white solid circle. The confocal microscope gives us the data (x,y,z cartesian coordinates) of the analysed crack surface. In the present work, these data are processed to create a (3D) mathematical surface represented by a 3D line approximation (spline) that resembles the analysed crack in the best way. In all cases, the experimental (2D) crack paths are obtained from these mathematical surfaces. In order to determine a certain crack path, its corresponding mathematical surface (a 3D spline) is intersected with several planes (see the sectioning planes in Figure 2b), thus producing a set of crack profiles. All these intersection planes (Figure 2(  A different trend is observed on the third crack path plotted in Figure 3(a): initially growing outside the contact zone, and then (after a few microns) turning towards the opposite direction to follow a crack path very similar to that described above, that is to say, turning towards the contact zone with an angle equal to about 22°. using a curved path which starts from the hot spot and is normal to an averaged maximum principal stress direction in each of its points, as is detailed in the following Sections.

Analytical stress field calculation
The experimental test set-up examined (represented by a cylindrical contact characterised by a constant normal load and in-phase tangential and bulk load), the relative sizes between tested specimen and contact pads, and the assumed linear elastic behaviour 12 of the material ensure that the stress fields in the vicinity of the contact zone can be accurately evaluated through a closed-form analytical solution, by exploiting both the solution by Hertz [40] and that by Mindlin [41], as is detailed in Refs [42,23].
A plane strain condition is assumed. For the fretting fatigue problem being examined, such a condition is verified, from a theoretical point of view, only in the xy-plane shown in Figure   1(b), but it can also be assumed for planes parallel to this, according to the strain behaviour observed for the tested specimens during the experimental campaign.

Brief description of the Carpinteri et al. criterion
Let us consider the critical point P , lying in the above xy-plane.
At a given instant, the principal stress directions 1, 2 and 3 (being where T is the period of the cyclic loading, and where k is the inverse slope of the S-N curve under fully reversed normal stress, k  is the inverse slope of the S-N curve under fully reversed shear stress, and The path proposed to compute the position of the critical point P is a curve lying in the xy-plane, starting from the above hot spot and normal to the 1 -direction in each of its points (Figure 4).
According to the Point Method by Taylor  is the threshold range of the stress intensity factor for long cracks, and af   is the fatigue limit range.
The point P coincides with the end of the above path, which is also assumed as the crack path.

Results in terms of crack path
The above approach is used to estimate the crack paths for the experimental Analogously, Figure 5(  Note that further investigation of the crack path beyond L /2 = 0.02455mm would not provide additional information useful to determine the critical point location.
As a matter of fact, a crack path with a length greater than L /2 would violet one of the assumptions of the proposed approach, that is, the PM assumption.

Results in terms of fatigue life
For  (Figure   1(b)).
In such a situation, a plane (bi-dimensional) FEM model (plane strain behaviour) is appropriate for the present analysis.  Finally, in order to simulate the crack growth process, the model is re-meshed after each new crack increment inc l by employing a user subroutine. In all FEM crack growth simulations, and between consecutive steps, the crack grows by a fixed length, inc l .
This crack length increment has the minimum value for which no significant difference has been found in the estimated crack path. 20

Description of the maximum ΔKI * approach
The first numerical approach herein used to estimate the crack path is based on the range of the Mode I SIF, I K   , for an infinitesimal kinked crack emanating from the actual crack (Figure 7(a)).
According to Ref. [ 7) and (8), the kinking angle that maximises I K   is found: this angle is named max  (Figure   7(b)).
In next step, the FEM model is re-meshed in order to take into account that the crack length is grown of the increment inc l in the direction dictated by

Description of the maximum SWT approach
The second numerical approach to predict the crack path is similar to that described in the previous Sub-section but, in the present procedure, the angle of the crack increment between consecutive steps is provided by that direction that, in front of the crack tip, maximises the SWT (Smith-Watson-Topper) parameter. Such a parameter is expressed as follows:   where  and r are the relative polar coordinates at crack tip.
Using the Hooke's law and Eqs 10(a)-(d), the asymptotic strain field can be obtained (not shown here for the sake of brevity).
It is straight forward to see that, for a given orientation (defined by a certain value of  ), the stress field defined by Eqs 10(a)-(d) only varies according to 1/ r and, thus, any value of r can be considered in order to search that material plane (defined by  ) which maximises the SWT parameter. However, to fulfil the asymptotic behaviour according to Eqs 10(a)-(d), the crack length l 23 must satisfy the condition r « l . The possibility to use any value of r eliminates the second drawback exposed above. Analogous to the previous numerical approach, the crack path can be obtained repeating the procedure again and again.

Results in terms of crack path
Now, by applying the above numerical approaches up to a crack depth equal to 155μm, the crack initiation paths for tests No.   The experimental fretting fatigue lives are listed in Table 3 for although it would be possible to introduce some geometric factor to 28 the 2D through-the-thickness crack SIF in order to take into account the actual semi-elliptical crack shape, it would not be clear how to apply the above geometric factor to a crack with a not-straight path.

CONCLUSIONS
In the present paper, three different approaches (one analytical        Poisson's ratio