TENSION FAILURE ASSESSMENT AT LUG HOLE EDGES

The lug is the critical component of lug joints, and one of its failure mode is the tension failure at hole edges. In the present paper, the fatigue crack growth simulation of a semi-elliptical crack located in the lug net cross-section, at one of the two hole edges, is carried out by employing a theoretical model based on the Paris law. The lug sizes, loading conditions and crack configurations are assumed in accordance with those related to some experimental tests available in the literature, performed on lugs of PolyMethylMethAcrylate under pulsating tension.

Semi-axes of the semi-elliptical crack  (a) tension failure at the hole edges; (b) bearing failure at the pin/lug interface; (c) shear failure at the lug edge, that is, the pin tries to push out through the edge of the lug (fracture along two planes); (d) tensile failure at the lug edge, that is, the pin tries to push out the edge of the lug (fracture on a single plane); (e) out-of-plane buckling failure of the lug.  including only the tensile part of the SIF range, whereas the mean stress effect has been accounted for by Noroozi et al. [21] analysing the elastic-plastic strain-stress history at the crack tip.

FCG in lug joint
In all the above empirical FCG equations, an accurate computation of have been recently applied to examine FCG in lug joints [31].
An accurate determination of the SIF values at the front of 3D surface cracks is the main task in order to predict fatigue life of 8 lug joints and, consequently, a simple and computationally efficient procedure avoiding complex meshes or remeshing techniques can be an interesting tool for practical applications.
In the present paper, the numerical procedure proposed in Refs   (Figure 3(a)).
For each specimen configuration, two values of the maximum applied load P (Figure 1) are analysed, as is listed in Table 1, being the loading ratio R equal to zero. The specimen acronyms used in Table 1 agree with those employed in Ref.
[7].  Table 1 The reference coordinate frame XYZ is assumed to have its origin O located at one of the two hole edges (Figure 3(a)). The X -axis is taken parallel to the loading direction coincident with the lug longitudinal one, the Y -axis is oriented along the lug width W and, hence, the Z -axis completes the right-hand frame (Figure 3

(b)).
A semi-elliptical crack is assumed to exist in correspondence to the lug net cross-section at one of the two hole edges (Figure 3(b)), and its initial semi-axes lengths (named  Table 2 for each sample. , and the Stress-Intensity Factor range in m MPa ). Table 3 3

. STEPS 1 AND 2: STRESS FIELD IN THE UNCRACKED LUG AND COEFFICIENT DETERMINATION
According to Step 1 of the procedure summarised in Figure 2, the stress field in the uncracked lug of the joint under a static tension loading P (Figure 1) is needed to be computed.
A two-dimensional FE analysis is initially performed, where the effect of the pin on the lug is simulated by using displacement radial constrains at the half hole, as is shown in Figure 4(a). The load P is applied directly to the lug through the uniform pressure   xr P W T   . Note that no friction is taken into account at the pinlug interface.
Such an assumption, that allows us to simplify the problem, is based on the numerical results provided by Naderi et al.

[31].
They analysed the friction between the lug and the pin modelling the lug loaded by either a uniform or a cosine pressure along the half hole, and also examined the whole joint without friction. Their SIF evaluations along the front of the crack in the lug net cross-section show that the frictionless model provides safety results and, for such a reason, the friction effect is neglected in the numerical model here employed. 11 The FE analysis is linear elastic, performed by using about 1650 eight-node plane strain quadrilateral elements (minimum element size equal to about 0.01 mm) with reduced integration. As an example, the numerical model discretisation related to the lug characterised by is shown in Figure 4(a).

Figure 4
To determine the Stress-Concentration Factor   Table 1.

Figure 5
Therefore, to evaluate the accuracy of the two-dimensional model previously employed to determine The stress distribution at the net cross-section along X -direction, xh  for 0 x  and 0 yt  (Figure 4(a)), is determined from the above 2D numerical model, and then normalised with respect to the remote stress, xr  (Figure 4(a)), by obtaining *  (Figure 3(b)). Note that the origin O of the waxis is located at ya  .
For each lug configuration examined, the expression of the 5th order polynomial fitting curve is reported in the Appendix, together with the corresponding Pearson correlation coefficient which measures the correlation between the FE stress values and the approximated ones.
As an example, the normalised stress distribution for the lug geometric configuration characterised by is plotted in Figure   5(b) (dot symbols) together with the corresponding 5th order polynomial fitting curve (continuous line).
In such a case, the Pearson coefficient is equal to about 1.0, proving the accuracy of the approximation proposed.

STEPS 3 AND 4: SIF COMPUTATION
According to the procedure summarised in Figure 2, a simplified SIF computation for the cracked lug is herein described (Steps 3 and 4).

STEP 5: FATIGUE ANALYSIS OF THE LUG
The two-parameter theoretical model proposed by Carpinteri For each initial crack configuration listed in Table 2, the crack aspect ratio  is plotted against the relative crack depth  ( Figure   16 7 shows an experimental FCG that is far from the numerical estimation, even if the proposed model is able to catch the experimental FCG of the PT11 specimen which presents geometry and loading condition very similar to those of PT10 specimen (see Table 1 and 2).
Moreover, for each initial crack configuration listed in Table 2, the relative crack depth  is plotted against the number of loading cycles N (Figure 8). Such results are compared with the experimental data in Ref. [7].

Figure 8
From Figure 8, we can observe a good agreement between the numerical results and the experimental data for all the geometric configurations except specimen PT10, as has been previously remarked.
Finally, the shape evolution of the crack front at some representative number of loading cycles is displayed in Figure 9, by comparing the numerical results with the experimental data presented in Ref. [7].

Figure 9
The quality of the numerical results can be evaluated through two error indexes computed at the examined number of loading cycles:

CONCLUSIONS
In the present paper, the fatigue behaviour of a lug joint made of PMMA, containing a semi-elliptical crack under pulsating cyclic tension, has been examined through a simplified procedure.
First of all, the uncracked lug has been modelled through a 2D FE mesh to determine the stress field in the lug net cross-section. Then such a stress field has been approximated through a power series expansion.
The                    NUM. EXP.