A 3D analysis of crack-front shape of asymmetric repaired aluminum panels with composite patches

A BSTRACT . Through this study, a numerical simulation based on 3D in order to investigate the effect of crack-front shape on the stress intensity factor and fatigue crack growth behavior of center cracked aluminum plate repaired asymmetrically with bonded composite patch. Consequently, skew degree it is a significant effect on stress intensity factor (SIF) distribution along the crack front in thick panels more than in thin panels. Moreover, fatigue life was calculated using different averaged stress intensity factor of patched panel determined from uniform crack front model and skew crack front model obtained from FEM and when comparing fatigue life values obtained from the finite element model with experimental values were shown a good agreement.

INTRODUCTION mong the methods of repairing damaged structures, bonding a composite patch is currently the most used. In particular, composite patch repair has shown its effectiveness in the field of aeronautics and maritime structures [1][2]. According to research, most studies address the problem of shape optimization, i.e the surface and the thickness of the patch [3][4][5][6][7][8][9][10]. Without omitting the effect of the parameters related to the adhesive such as the thickness and the type of the glue that have provoked several searches as for example in [11][12][13][14][15]. Allan Baker initially developed the technique of bonding a composite patch in a cracked structure in the late 1970s. This technique appeared to be an advantageous alternative to the more conventional techniques of repair, by riveting or welding metallic structures. Indeed, the bonding of a composite patch to the damaged structure significantly reduces the stress field on the damaged area. That leads to improve the structural strength and the extension of the lifetime of the structure under stress [16][17], without A introducing a new hole or changing the microstructure of the material. In fact, there are two patch repair techniques, namely, simple patch repair (asymmetric repair) and double side patch repair (symmetric repair) [16][17][18][19]. Generally, it is the double patch that is preferable, because of the deflection effects that the simple patch introduces into the charged structure [20]. Regardless of the type of repair (symmetrical or asymmetrical), parameters such as thickness, surface area, number of layers, fiber orientation and patch shape remain the most studied for their interdependent influence on repair efficiency. In this context, Ramji & all carried out a comparative study in [16]. These authors studied several forms of patch and concluded that an extended octagonal patch has better performance for stress intensity factor reduction. Recently, Bachir Bouiadjra & all concluded in [7] that a trapezoidal form is more powerful than a rectangular form. Furthermore, in [10], it was concluded that a butterfly shape with a suitable dimensions is more efficient than the rectangular shape. For the number of layer and the thickness of the patch, H. Hosseini and colleagues in [21] concluded that the performance of the patch is proportionately improved by increasing the number of layers and reducing the thickness of the plate. Furthermore, [22] established that increasing the life of the repaired plate is related to increasing the thickness of the patch. Also for the type of composite used in the repair, H. Hosseini and co-workers in [23] found that, for a 45° inclined crack in a thick aluminum panel repaired by an asymmetric glass / epoxy patch, the lifetime during the crack propagation is significantly improved. Among the positive effects on the repair, [24] studied the effect of overload during the fatigue test. As a result, it was found that overloading extends the service life of the repaired structure. Despite a large number of published researches on this topic, to our knowledge, few of them consider the value of the stress intensity factor to be used for estimating the lifetime of repaired structures. Indeed, D. C. Seo and J. J. Lee in [25] experimentally proved that the crack front is not always a straight line when the crack is propagated. These latter concluded that the stress intensity factor of thick specimen showed a large variation through thickness direction. Moreover, Woo-Yong Lee and Jung-Ju Lee in [26] stated that the stress intensity factor (SIF) calculated at the crack tip is much influenced by crack front shape. Consequently, they proposed to predict the actual crack front shape evolution and take it into account for the accurate analysis of fatigue behavior. This was confirmed later by H. Hosseini et al. in [27] and recently in [28], where it was noted that the crack growths non-uniformly from its initial location through the thickness of the repaired panel and the maximum crack length occur at un-patched surface of the panels. In this study, for the accurate investigation of fatigue crack growth behavior, a consideration is taken for the crack front inclined in its plan from the patched toward un-patched side by various angles. This is done to simulate the experimental fatigue crack-growth behavior in mode-I failure and examine the SIF's distribution along the crack front within centrally cracked aluminum panels. Moreover, trying to understand the physic of the crack growth difference between patched and un-patched side in two different panels thick and thin.

GEOMETRIES AND MECHANICAL PROPERTIES OF THE MODEL
ig. 1 shows the basic geometry of the cracked structure considered in this study. This model is adapted from that proposed in [21]. The aluminum alloy plate 2014-T3 has dimensions of 100 x 50 mm2 with two different thicknesses, namely 2.29 and 6.35 mm. the plates contain an initial crack length of 2a = 10 mm and perpendicular to the loading axis ( Fig. 1). Table 1   The objective of this work is to analyze the effect of the inclination of the crack front in its plane. For this purpose, the crack line is inclined from the patched side toward the un-patched side by different angles Ө = 0°, 15°, 30°, 45° and 60° respectively ( Fig. 2.a). Fig. 2.b shows the different inclinations of the crack front, where the mesh refinment indicates the front of the crack and the immediate adjacent region in the plane of the thickness. Then to compare the stress intensity distribution between two different thickness panels, i.e. thin (2.29mm) and thick (6.35mm). So, the plate is subjected to a maximum stress σ=118 MPa with a load ratio of R = 0.05. The analysis was carried out using Abaqus 6.13 software. A three-dimensional finite element method with automatic meshing was carried out on the structure using 8-nodes brick elements and refinement of the mesh was created in the vicinity of the crack tips as shown in figures 3.c. The composite patches ply orientation is parallel to the loading axis. So, as geometry and loading conditions are symmetrical, only one quarter of the specimens were simulated. Due to its simplicity, Paris law is usually used to predict the fatigue life, which can be expressed as [21]: T where C and m are properties constant for a material regrouped in tab.3, ΔK=K max -K min is the SIF range, N is number of cycles, a is crack length and da is change in crack length.   The adopted method was to evaluate SIFs with ABAQUS using J-integral. The energy approach states that fracture will occur when the energy release rate, reaches a critical value G c , and is consistent with the idea that failure occurs when the SIF in mode I, KI, reaches a critical value, called the fracture toughness (KIC). For linear elastic material, J (equivalent to G) is related to the SIF by [29]: where I, II and III denotes the modes of fracture, and E*=E / (1- 2 ) for plane strain case (3D configuration). The Jintegral in pure mode I loading condition, can then be correlated to KI using the relation: where E is Young's modulus and  is the Poisson's ratio. If an arbitrary contour is considered as illustrated in Fig.4, the J-integral is given by: where w is the strain energy density, T i are components of the traction vector, u i are the displacement vector components, and ds is a length increment along the contour Γ. The strain energy is defined as: where, σij and εij are the stress and strain tensors, respectively. The traction is a stress vector normal to the contour. To evaluate these integrals, ABAQUS defines the domain in terms of rings of elements surrounding the crack tip. The Von-Mises stress used in the analysis of stress distribution along the crack front can be written in terms of stress components as [30]: where, xx  , yy  and zz  are the normal stresses according to x, y and z direction respectively and xy  , yz  and zx  are shear stresses in the yz, xz and xy planes respectively (Fig. 5).

RESULTS AND DISCUSSION
igs 7.a and 7.b present the distribution of von-mises stresses in thin and thick repaired aluminum plates respectively. The crack length is taken to be 2a=10 mm. From fig 7.a, a non uniform distribution along the crack front can be observed. In fact, the stress transfer between the repaired plate and the patches mainly occurs in the patched side of the plate. The stress at the crack tip in the patched side of the plate is much smaller than that in the unpatched side of the plate, which accounts for SIF variation in thickness direction of the aluminum plate. Moreover, comparing the crack tip stresses distribution between figs7.a and 7.b, one can also observe that with the help of the patches, the stress at the crack tip in thin repaired plate is much smaller than that in the thicker one. This accounts for the significant SIF's difference between thin and thick plates.   fig) is with growth of the crack where this value is shifted from the free side toward the patched side and this applies to all panels, whether thin or thick. Accordingly, this finding will be highlighted, hereinafter by the stress intensity factor results in fig 9.a and 9.b.  Figures 9.a and 9.b shows the distribution of stress intensity factor along the crack front. It is noted through the curves trend that the maximum value of SIF is shifted from the free side of the plate towards the patched side. This suggests a cascade growth of the crack, obviously starting from the free surface going toward the patched surface. A differed of this behavior can be noted between the thin and the thick plates. Effectively, the maximum value in fig 8.a is resulted for the angle Ө=0° curve and the last point in crack front in the free side of the plate. This maximum value is in no case F exceeded for other configurations ( fig. 8.a). On the other side, fig. 8.b shows that the maximum value of the stress intensity factor increases with the growth of the angle and is located inside the front of the crack.     9 shows the distribution of the stress intensity along the crack front for the thick panels where it can be observed a shifted of maximum value from the free side of the plate towards the patched side. Accordingly, the maximum value is obtained for the angle Ө=60° and located at approximately 60% ( fig. 9.b) of the thickness from the patched side to the free side and this also makes it compatible with the shape of the crack in fig10.a. Fig. 10.a also shows the maximum length of crack whose position exactly corresponds to the position for the maximum value of stress intensity factor. Fig.10.b compares the normalized crack-front shapes between repaired plates with 2.29 mm and 6.35 mm thickness. As can be seen in the figure, the difference between the maximum length crack positions confirms the current results. Thus, it is useful to give a precise prediction of the crack behavior and to identify the adequate value of the stress intensity factor for an accurate estimate of the lifetime of the patch-repaired structure and this for the two thicknesses studied in this work. In this respect, Fig.11 shows a comparison between average of the stress intensity factors versus the crack front inclination for thin and thick panels. Incontestably, the obtained results indicate horizontal monotony of the average stress intensity factors for a thin panel. Accordingly, one can suggest that the ideal choice for the calculation of the crack propagation is the average of the SIF's values along the crack front. This will be confirmed, hereinafter through the analyzed fatigue life results. Conversely, the curve of the thick panel (t=6.35 mm) shows a change of the average stress intensity factor when the skew changed. Here, the averaged SIF icreases with the increase of the skew angle.  As expected above by analyzing the distribution of stress intensity factors, Fig. 12.a shows that the fatigue lifetime obtained from using of the stress intensity factor (avg-all) and to a lesser extent the results obtained from using the (avg-9.mid) , (K.mid) and (avg-K.rms) respectively, gives a precise results for the case of thin repaired plates .This is evident through a good agreement of FEM curve results obtained from (avg-all) with experimental curve. On another side, Fig.12.b shows that the values obtained from the root mean square stress intensity factor (avg-K.rms) and to a lesser extent obtained from the (K.mid), (avg-9 .mid) and (avg-all), respectively gives precise results for the case of the thick repaired plates. this is evidenced by the good agreement of the numerically obtained lifetimes with those obtained experimentally.

CONCLUSION
his study focused on the fatigue crack growth behavior of center cracked aluminum plate repaired asymmetrically with bonded composite patch .The analysis led to the following conclusions: -The stress at the crack tip in the patched side of the plate is much smaller than that in the un-patched side of the plate.
-The maximum length of crack whose position exactly corresponds to the position for the maximum value of stress intensity factor. -The crack front shape differs between the thin and thick plates.
-Reliance on the stress average all (avg-all) in the calculate the fatigue life of the thin plates repaired asymmetrically. Showed excellent accuracy compared with experimentally reverse thick plates which the stress root mean square (K.rms).