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Mixed-mode numerical and experimental fatigue crack growth analyses of thick aluminium panels repaired with composite patches
Composite Structures 91 (2009) 1–8
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Mixed-mode numerical and experimental fatigue crack growth analyses of thick aluminium panels repaired with composite patches H. Hosseini-Toudeshky a,*, B. Mohammadi b,1 a b
Aerospace Engineering Department and Center of Excellence in Computational Aerospace Engineering, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran Aerospace Engineering Department, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran
a r t i c l e
i n f o
Article history: Available online 18 April 2009 Keywords: Repair Composite Mixed-mode Fatigue crack growth Thick aluminium panel Finite element modelling
a b s t r a c t In this paper, fatigue crack growth of repaired thick aluminium panels containing a central inclined crack is studied. The cracked panels were repaired on one side with glass/epoxy composite patches. Effects of patch lay-up configuration on starting the crack growth after bonding and crack growth rate of the repaired panels were investigated. The finite element analyses were performed assuming uniform crack growth along the panel’s thickness to simplify the analyses. The predicted fatigue lives using both midplane and un-patched surface fracture parameters of the repaired panels are compared with the experimental results. It is shown that the predicted crack propagation lives using the un-patched surface fracture parameters are too conservative for large crack growth values. However, the obtained lives using the mid-plane results may lead to non-conservative life prediction. It is also shown that the predicted starting crack growth life and the crack growth rate at the early crack propagation stages using the un-patched surface fracture parameters are close to the experimental results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Bonded repair of aluminium structures using laminated composite patch was initiated by Baker [1] and followed by other researchers to evaluate the efficiency of that repair. They performed numerical and experimental investigations for single and double side repaired panels in mode-I condition and using boron/ epoxy and graphite epoxy patches [1–3]. In a single-side repaired panel, a non-uniform stress distribution occurs over the thickness of the cracked panel which is due to the existing out-of-plane bending. Many investigations have already been performed to avoid the three dimensional analyses of the repaired panels. Furthermore, in many simplified numerical analyses the stress and strain fields of the mid-plane have been used to calculate the fracture parameters [2–7]. However, the maximum stress and strain occur at the un-patched surface of the repaired panels [8,9]. In single-side repairs, the non-uniform stress distribution along the panel thickness leads to the non-uniform crack propagation along the panel’s thickness and forms a curved crack-front shape during the crack propagation [10,11]. Most of the previous investigations on fracture analysis of single-side repaired panels using composite patches were focused * Corresponding author. Tel.: +98 21 6454 3224; fax: +98 21 6640 4885. E-mail addresses: [email protected] (H. Hosseini-Toudeshky), Bijan_Moh@aut. ac.ir (B. Mohammadi). 1 Tel.: +98 21 6454 3224; fax: +98 21 6640 4885. 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.04.022
on the safety evaluation and life prediction of components under uniaxial loading in mode-I condition [2–11]. However, in many applications, components are subjected to various forms of loading conditions which the crack growth and failure of them occur due to the conjoiner of such loading. Research results were rarely published on the repaired panels in mixed-mode conditions [12,13]. These works investigate the fatigue crack growth of repaired panels in mixed-mode conditions using numerical and experimental methods. Chung and Yang [13] performed fatigue crack growth tests for repaired thick panels made of Al 6061-T6 containing the edge cracks. They only used one lay-up and patch thickness for panels with various inclined crack angles. They showed that, the crack grows non-uniformly through the panel’s thickness. They did not investigate the effect of patch layers orientations on the crack growth rate of the repaired panels. Wang and Rose also presented theoretical investigations on the fracture analyses of singleside repair in mode-I and in-plane mixed-mode condition [14,15]. They presented an analytical method for the combined tensile stretching and bending of one-sided repairs based on Reissner’s plate theory and classical plate theory within the frameworks of geometrically linear and nonlinear elasticity. They also presented fracture analysis of repaired cracks under in-plane mixed-mode loading [15], focusing on the special case of isotropic adherents having the same Poisson’s ratio. They performed a parametric study considering various factors that may influence the repair efficiency. They did not perform any fatigue crack propagation analysis in their works.
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Hosseini-Toudeshky et al. [16] performed uniform crack growth finite element analyses to study the effects of patch thickness and lay-up orientations on fatigue crack growth life and crack trajectory of the repaired thin panels with central inclined crack of 45°. They indicated that, for the repaired panel with the inclined crack of 45°, the patch fibres orientation of h = 90° (unidirectional in loading direction) is an optimum fibre angle leading to longer fatigue crack growth life extensions. However, the optimum patch layers angle is h = 105° for starting crack growth life extension. Mixed-mode fatigue crack growth of thin aluminium panels with single-side repair using experimental and numerical methods were also performed by the authors [17]. Hosseini-Toudeshky et al. performed experiments on fatigue crack-growth of repaired thick panels with a central inclined crack using glass/epoxy composite patch [18]. The numerical crack growth of repaired thick panels in mixed-mode condition has not yet been compared with the experimental results to the authors’ knowledge. In this paper, effects of various patch lay-ups on the life extension of repaired thick aluminium panels for very small (starting) and large crack growths are investigated. These cracked panels contain a typical central inclined crack of 45° and made of Al 2024T3 with single-side glass/epoxy composite patches. The predicted crack growth rates are compared with the experimental results. The accuracy and applicability of the obtained results from the simplified 3D finite element analysis (uniform crack growth modelling along the panel’s thickness) of the repaired panels can also be evaluated. Finite element crack propagation analysis of a single-side repaired panel in general mixed-mode conditions is really a cumbersome task. 2. Fatigue and fracture analysis Fig. 1 shows a typical geometry and loading of the single-side repaired panels containing a central inclined crack. The problem needs to be studied as a mixed-mode cracked problem in modes I and II conditions. Having the stress and strain fields around the crack-tip, fracture parameters for mixed-mode problems are calcu-
lated to predict the crack propagation path and fatigue life of the repaired panels. For this purpose, the fracture parameters such as KI, KII and J-integral can be used. J-integral definition considers a balance of mechanical energy for a translation in front of the crack along the local crack-tip xaxis, which is a path independent contour integral defined as [19]:
J¼
I
Wn1 rij nj
C
@ui ds @x
ð1Þ
where, W is strain energy density, rij are stress components, ui are the displacements corresponding to the local i-axis, s is the arc length of the contour, nj is the jth component of the unit vector outward normal to the contour C, which is any path of vanishing radius surrounding the crack-tip. Using the assumption of linear elastic fracture mechanics and the following equation, KI and KII parameters are decoupled from the calculated J values [20].
J ¼ K 2I =E0 þ K 2II =E0
ð2Þ
where, E0 is modulus of elasticity, E0 ¼ E for plane stress condition and E0 ¼ E=ð1 m2 Þ for plane strain condition problems. To solve the problem for KI and KII the ratio of KI/KII is also obtained from the ratio of the normal distance to the horizontal distance of two closest nodes to the crack-tip which they have been coincided before loading. Having the fracture parameters, a criterion is needed to predict the crack growth direction in a mixed-mode problem. Several criteria have already been proposed for this purpose. Previous researches [16] show that there are not significant differences between the predicted crack trajectories based on various crack propagation criteria. Using stress as a parameter, the maximum tangential stress (MTS) criterion was presented by Erdogan and Sih [21]. This criterion states that a crack propagates in a direction corresponding to the direction of maximum tangential stress along a constant radius around the crack-tip. Using the Westergaard stress field in the polar co-ordinates and applying the MTS-criterion, the following equation is obtained to predict the crack propagation direction in each incremental step [22]:
Fig. 1. Typical geometry and loading of single-side repaired panels.
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Fig. 2. Typical 3D finite element mesh of the repaired panels, (a) hole patched area, (b) initial deformed crack-tip area, (c) crack-tip area after a few steps growth.
tan2
a 2
l 2
tan
a 2
1 ¼0 2
ð3Þ
where, l, defined as the ratio of KI over KII (l = KIKII) and a is incremental crack propagation angle. To find the maximum root value of Eq. (3), the condition of @ 2 ra =@ a2 < 0 should be satisfied. Having the fracture parameters and crack propagation direction from the above formulation, fatigue life of each crack growth increment can be calculated. A number of crack growth laws have been developed relating the crack growth rate to the stress intensity factors. The well known Paris equation has been used in this study which is given as follows [23]:
da ¼ C DK m dN
ð4Þ
where DK ¼ K max K min is the stress intensity factor range in fatigue loading, N is the number of load cycles, da is the crack growth increment, and C and m are empirical material constants. In the mixed-mode problems the effective stress intensity factor range, DKeff, can be used instead of DK and it can be calculated as follows [24]:
DK eff ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK I þ jK III jÞ2 þ 2K 2II
ð5Þ
Table 1 Material properties of aluminium alloy 2024-T3 and adhesive [18]. Parameters
Aluminium 2024-T3
Glue FM 73
E
71.02 (GPa) 0.3
1.83 0.33
t
As far as the analyses are performed in the mixed fracture modes of I and II the KIII = 0.0 is substituted to this equation. Having the effective stress intensity factor and material data, Eq. (4) can be used to calculate the fatigue crack growth lives of the cracked panels in mixed-mode conditions. 3. Finite element analysis A finite element method (FEM) is widely employed for the solution of mixed-mode crack problems. Three dimensional elastic analyses were performed to obtain the stress and strain fields of the repaired panels using ANSYS finite element program. Typical geometry and loading of a repaired plate are shown in Fig. 1 and a typical finite element mesh is shown in Fig. 2. The panels are made of 2024-T3 aluminium alloy with the thickness of 6.35 mm containing an inclined sharp flaw of 2a = 10 mm length and 0.2 mm wide with the inclined angle of b = 45°. They are bonded with four layers glass/epoxy composites with various lay-up configurations of [90]4, [105]4, [45]4, [452/+452] and [902/02]. The lay-up angles are based on the defined h in Fig. 1. The patch layup configurations are selected based on the results of the previous finite element investigations presented by the authors [16]. These analyses showed that the patch lay-up of [105]4 is the best lay-up configuration among the considered lay-ups in terms of the obtained crack growth lives using the mid-plane finite element fracture parameters. Material properties of the aluminium cracked panel, adhesive, and composite patch are given in Table 1 and Table 2. Experimental fatigue crack growth of two un-repaired specimens with initial crack angle of b = 0° were used to obtain
Table 2 Material properties of glass/epoxy composite patch [7]. Elasticity modulus (GPa) E11 E22 E33
Shear modulus (GPa) 27.82 5.83 5.83
G12 G13 G23
Poisson’s ratio 2.56 2.56 2.24
m12 m13 m23
0.31 0.31 0.41
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Fig. 3. Typical de-bonding area around the crack for the repaired panel with [45]4 patch (a) at 19,000 cycles; X ctip ffi 7:5 mm (b) 25,000 cycles; X ctip ffi 11:5 mm.
20 18
Un-repaired Specimen
16
β =45
0
Crack Length (mm)
the material constants of Paris law using the ASTM E-647 method [18]. Therefore, the used material constants in Eq. (4) are: m = 3.66 and C = 4.57e14 in metric system. In these analyses, an isotropic 8-node-solid element was used to model the aluminium panel and adhesive layer. Furthermore, a layered 8-node-solid element was used to model the laminated composite patch. The used 8-node-solid elements are non-conforming elements which can be selected by the option of extra shape function in Ansys. The non-conforming elements have excellent performance in the bending situation when compared with the conforming elements. A typical mesh of the component is shown in Fig. 2. The in-plane elements aspect ratios near the crack tip are about 1.0. The obtained finite element results show that different J-integral contours and various refined meshes on the surfaces and through the thickness of the repaired panels (variation of 2–10 elements along the thickness) result in about 2% differences in the obtained fracture parameters [16]. Therefore, two elements in the thickness of each part, plate, adhesive and patch were used and a very fine mesh was generated for the region close to the crack-tip. Each model contained the total number of 12744 elements and the plane area of (1.0 mm 1.0 mm) close to the crack-tip contained about 900 elements as shown in Fig. 2. A Macro program was developed using ANSYS Parametric Design Language (APDL) to handle the crack growth modelling procedure as well as to find the crack trajectory analysis of each repaired panel. The main assumptions in the crack growth analysis are: (1) linear elastic fracture mechanics behaviour, (2) the crack-front line through the thickness, remains perpendicular to the plate surfaces during the crack propagation (uniform crack growth modelling) to simplify the modelling and analysis, and (3) no major de-bonding occurs between the panel surface and patch during the crack propagation. According to our experimental observations [18], for most of the cases which the surface preparation performed properly, debounding between the patch and aluminium panel occurred in a small area around the crack, only. It is worth to mention that, a small de-bounded area with the size of an element width has been considered in the FEM modelling. Typical de-bonding areas around the crack for the repaired panel with [45]4 patch are shown in Fig. 3. In this procedure a dynamic mesh generation was designed to generate automatic mesh of the repaired panels at each crack growth step. The J-integral value for the presented crack configuration was computed in each step, and then using Eq. (2) and the explained decoupling procedure in Section 2, the values of KI and KII were calculated. Using the obtained values of KI and KII and Eq. (3), the crack growth direction, a, is determined. The equations were numerically solved in each step and then the geometry is updated with the new crack length configuration. The above procedure was performed for the new configuration and repeated for several steps to find the crack trajectory in each repaired panel. Full details and
14 12 10 8 6 FEM
4
Exp-1 Exp-2
2 0 0
10000
20000
30000
40000
50000
N (Cycle) Fig. 4. Comparison between experimental and finite element crack growth rates (Xdirection) for un-repaired panels.
explanations of fatigue and fracture analysis, finite element modelling, and APDL macro programming are presented in Ref. [16]. 4. Results and discussions In this section, the obtained fatigue crack growth life of the repaired thick panels from the finite element analyses are compared with the experimental results performed by the authors in Ref. [18]. 4.1. Verification Fig. 4 compares the predicted finite element crack growth behaviour with the experimental results for the un-repaired specimen. The good agreement between the experimental and numerical results verifies the developed finite element crack growth procedure and the material constants used in the Paris law. The experimental and numerical crack propagation paths obtained at patched and un-patched surfaces of a typical repaired panel are also shown in Fig. 5. There is a good agreement between the predicted and experimental crack propagation paths at both patched and un-patched surfaces of the repaired panel. The experimental and numerical results also show that the major component of the crack growth is in the X-direction or approximately perpendicular to the loading direction. Therefore the X-component of the crack propagation path of the repaired panels is used in the crack growth results presentation and discussions in the following sections. For further information variation of KII/KI ratio versus crack tip X-coordinate for un-repaired and repaired panels with various
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H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 91 (2009) 1–8
(a)
(b)
Fig. 5. Typical experimental and numerical crack propagation path of repaired panel with the patch lay-up of [902/02], (a) un-patched surface, (b) patched surface.
(a)
1
patch lay-ups at both mid-plane and un-patched surface are depicted in Fig. 6. These variations show that the mixed-mode condition is mainly existing at the early stages of crack propagation. For crack growth length of higher than 1.0 mm (Xctip > 4.5 mm) these ratios vanished to almost zero value for mid-plane and lie between 0.05 and 0.1 at un-patched surface meaning that the mode II does not significantly affect the crack growth rate and path.
1
0.9 0.99
0.8 0.98
0.7
|
K II /K I|
at Mid-Plane 0.6
0.97
Un-repaired [90/90/90/90] [90/90/0/0] [105/105/105/105] [-45/-45/-45/-45] [-45/+45/-45/-45]
0.5 0.4 0.3
4.2. Crack growth behaviour
0.96
0.95
0.94 3
0.2
3.5
4
0.1 0 0
2
4
6
8
10
12
14
Crack tip X-coordinate (mm)
1
1
0.9
18 0.96
0.8
at Un-Patched surface
K II /K I|
0.7
|
20
0.98
Un-repaired [90/90/90/90] [90/90/0/0] [105/105/105/105] [-45/-45/-45/-45] [-45/+45/-45/-45]
0.6 0.5 0.4 0.3
16
0.94
Crack Length (mm)
(b)
The experimental and finite element crack growth rates of the repaired and un-repaired panels have been shown for [452/ +452], [45]4, [90]4, [105]4 and [902/02] patch lay-ups in Figs. 7– 11 respectively. These figures show that the obtained crack growth rates using the un-patched surface fracture parameters are comparable with the experimental results at the early stages of crack propagation. However, they behave considerably different for large crack growth values. It is worth to mention that the crack-front
0.92 0.9 0.88 0.86 0.84 0.82
14 12 10 8 6
0.8
0.2
3
3.5
4
0.1
Patch lay-up [-452/+452]
2
0 0
2
4
6 8 10 Crack tip X-coordinate (mm)
12
14
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
0 0
10000
20000
30000
40000
50000
N (Cycle) Fig. 6. Variation of KII/KI ratio versus crack tip X-coordinate for un-repaired and repaired panels with various patch lay-ups (a) at mid-plane (b) at un-patched surface.
Fig. 7. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [452/+452].
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 91 (2009) 1–8
20
20
18
18
16
16
14 12 10 8 6
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
Patch lay-up [-45]4
2 0 0
10000
20000
30000
40000
50000
N (Cycle)
20 18
Crack Length (mm)
16 14 12 10 8
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Exp-3 Un-repaired Exp.
4
Patch lay-up [90]4
2 0 0
10000
20000
30000
40000
50000
N (Cycle) Fig. 9. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [90]4.
20 18
Crack Length (mm)
16 14 12 10 8 6
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
Patch lay-up [105]4
2 0
0
10000
20000
14 12 10 8 6
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
Patch lay-up [902/02]
2 0 0
10000
20000
30000
40000
50000
N (Cycle)
Fig. 8. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [45]4.
6
Crack Length (mm)
Crack Length (mm)
6
30000
40000
50000
N (Cycle) Fig. 10. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [105]4.
shape is a major factor affecting the stress intensity factors and crack growth rate in the single-side repaired panels [8,9,11]. At the early stages of the experimental crack propagations where
Fig. 11. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [902/02].
the crack-front has not completely developed to a curved shape, the real crack-front is more compatible with the considered uniform crack-front modelling in the finite element analyses. Therefore, the obtained crack growth rates using the un-patched surface results are close to the experimental results for small crack growth values. These figures also show that, the predicted crack growth rate using the mid-plane fracture parameters are slower than those obtained from the experiments for all patch lay-up configurations except the patch lay-up of [90]4 in Fig. 9. This figure shows that the results of Exp-2 and Exp-3 are almost compatible but the exceptional crack growth rate in Exp-1 behaves differently and is close to the mid-plane finite element results. This test has been performed at the early stage of the experimental work and may prone to errors due to the test set-up difficulties such as initial blunted crack-tip, measuring of initial crack length and angle, decentred crack, and patch manufacturing and curing conditions. The scattering due to the initial blunted crack-tip affect the restarting crack growth life and in this case the initial crack growth life is significantly larger than the other two test results. The difference between the predicted crack growth rate using the mid-plane results and experiments for the repaired panel with the asymmetric patch lay-up of [902/02] in Fig. 11 are larger than the differences occur for the other patch lay-ups. It should be noted that the asymmetric lay-up condition in the [902/02] lay-up produces a secondary bending against the existing out-of-plane bending due to the geometry of the single-side repair of the structure. In this case, de-bonding of the patch was also observed at the edges of the patch after a few crack growth steps during the test. It could occur due to the above mentioned asymmetric patch lay-up bending and producing of pilling stress on the bonding area. If the bonding quality can be improved in this case, crack growth life extension of better than the other used patch lay-ups may be obtained for the repaired panels. 4.3. Small (restarting) crack growth life Table 3 shows the obtained restarting crack growth life after bonding of the panels from both finite element and experimental [18] methods. The crack restarting growth defined as the extension of the crack-tip from the initial value of 3.54 mm to the Xctip = 4.0 mm at un-patched surface of the panels. It is also worth to note that the crack at patched surface may start to growth with delay or with smaller growth rate due to the existing differences between the fracture parameters along the panel thickness. Table 3 shows considerable scattering in the obtained experimental
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 91 (2009) 1–8 Table 3 Comparison of FEM and experimental restarting crack growth life of repaired panels up to Xctip = 4.0 mm (cycles). Patch lay-up
Mid-plane (FEM)
Un-patched surface (FEM)
Exp. A [18]
Exp. B [18]
[90]4 [902/02] [105]4 [45]4 [452/+452] Un-repaired
11,480 15,982 12,133 10,683 10,751 3369
6658 7115 6839 7233 7176 3369
5700 5200 4900 5900 4600 5000
8100 4700 6300 6900 5500 7400
Table 4 Comparison of FEM and experimental crack growth life of repaired panels up to Xctip = 17 mm (cycles). Patch lay-up
Mid-plane (FEM)
Un-patched surface (FEM)
Exp. A [18]
Exp. B [18]
[90]4 [902/02] [105]4 [45]4 [452/+452] Un-repaired
37,100 42,692 37,167 31,462 33,200 19,000
20,790 22,205 20,732 21,879 22,300 19,000
28,740 30,600 27,300 28,564 29,020 20,770
39,040 28,370 32,700 29,224 27,060 20,100
restarting crack growth lives of the repaired panels with various lay-ups. It may be due to the difficulties in recognizing the initial 0.46 mm crack growth at un-patched surface of the panels during the tests and therefore the experimental data for crack restarting lives are prone to some errors. The finite element analyses results presented in this table show that, the predicted lives using the unpatched surface fracture parameters are closer to the experimental results than those obtained using mid-plane results. The differences between the experimental and un-patched surface finite element results may also be due to the assumption of uniform 0.46 mm initial crack propagation along the crack-front. The predicted lives using the mid-plane fracture parameters lead to very non-conservative results. Comparing of the predicted restarting crack growth lives using the un-patched surface fracture parameters with the experimental results are more realistic than those obtained using the mid-plane fracture parameters. The numerical lives also show that, using the un-patched surface results leads to the almost same restarting crack growth life prediction for the three patch lay-ups of [45]4, [452/452] and [902/02]. These layups are more effective than the other two considered lay-ups in term of the restarting crack growth life extension. 4.4. Crack growth life Table 4 shows the obtained crack propagation life of the repaired panels with various patch lay-ups from experimental and finite element methods. The crack growth lives were obtained for the crack length extension of 13.46 mm (between the initial crack-tip Xctip = 3.54 mm and Xctip = 17.0 mm). It is worth to mention that two experimental results were only available for the crack growth life up to the crack extension of Xctip = 17.0 mm. The experimental results in Table 4 shows that using the considered various patch lay-ups, crack propagation life of the repaired panels may increase by the order of about 30–85% with respect to the un-repaired panel life. The obtained experimental crack growth life extension for the repaired panels with [90]4 and [105]4 are larger than the other three patch lay-ups. Disregarding the crack growth life of the panel with the patch lay-up of [90]4 (Exp. B in Table 4), there are not considerable differences between the obtained lives for the repaired panels with various patch lay-ups. Finite element analyses results presented in Table 4 also show that, the obtained
7
crack propagation lives using the un-patched surface fracture parameters are too conservative and the lives obtained from the mid-plane results may lead to 50% non-conservative life predictions in some cases. The mid-plane finite element lives presented in this table shows that the patch lay-ups of [90]4, [902/02] and [105]4 are more effective than the other two considered lay-ups and the patch lay-up of [902/02] is the best lay-up configuration among the considered patch lay-ups in term of crack growth life extension. The differences between the experimental and finite element crack growth lives may be due to the assuming of uniform crack propagation along the panel thickness in the numerical analyses. It was already explained that in a single-side repaired panel, a non-uniform stress distribution occurs over the thickness of the cracked panel which is due to the existing out-of-plane bending. This non-uniformity of stress field leads to non-uniform crack propagation along the panel’s thickness and forms a curved crack-front shape during the crack propagation which affects the crack growth life significantly as shown by the authors for repaired panels in mode-I condition [9]. It is also worth to mention that the mixed fracture modes of I and II were considered in the analyses, however, the real condition is a general mixed fracture modes of I, II and III. A finite element crack propagation analysis in general mixed-mode conditions assuming a non-uniform 3D crack growth along the panel thickness may reduce the existing differences between the numerical and experimental results which is really a cumbersome task. 5. Conclusion In this paper, fatigue crack growth analysis of thick aluminium panels containing a central inclined crack with single-side glass/ epoxy composite patch was performed. Effects of patch lay-up configuration on the restarting crack growth life and crack growth rate of the repaired panels were studied. It was shown that the obtained restarting crack growth lives using the un-patched surface finite element results are closer to the experimental results, but, the calculated lives using the mid-plane fracture parameters leads to very non-conservative life prediction. It was also shown that the predicted crack growth life using the un-patched surface finite element results is too conservative and it may be 50% nonconservative when the mid-plane results are used. The experimental results showed that, the crack propagation life of the panels may increase by the order of 30–85% depending on the used patch lay-up. References [1] Baker A. Bonded composite repair of fatigue-cracked primary aircraft structure. Compos Struct 1999;67:421–43. [2] Sun CT, Klug J, Arendt C. Analysis of cracked aluminium plates repaired with bonded composite patches. AIAA J 1996;34(2):369–74. [3] Ratwani MM. Analysis of cracked adhesively bonded laminated structures. AIAA J 1979;17(4):988–94. [4] Okfar C, Singh N, Enemuoh UE, Rao SV. Design, analysis and performance of adhesively bonded composite patch repair of cracked aluminum aircraft panels. Compos Struct 2005;71(2):258–70. [5] Rose LRF. A cracked plate repaired by bonded reinforcements. Int J Fract 1982;18(2):135–44. [6] Lu J, Hu Y, Ju D. An assessing method of fatigue life on central-through cracked plate with adhesive bounded reinforcement, fatigue, fracture, and risk. ASME 1991;215:135–40. [7] Daghyani HR, Sayadi A, Hosseini-Toudeshky H. Fatigue crack propagation of aluminium panels repaired with adhesively bonded composite laminates. Proc Instn Mech Eng Part I: J Mater Des Appl 2003;217:291–3. [8] Lee WY, J Lee J. Successive 3D FE analysis technique for characterization of fatigue crack growth behavior in composite-repaired aluminum plate. Compos Struct 2004;66(1–4):513–20. [9] Hosseini-Toudeshky H, Mohammadi B. A simple method to calculate the crack growth life of adhesively repaired aluminum panels. Compos Struct 2007;79: 234–41.
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[10] Hosseini-Toudeshky H, Sadeghi G, Daghyani HR. Experimental fatigue crack growth and crack-front shape analysis of asymmetric repaired aluminium panels with glass/epoxy composite patches. Compos Struct 2005;71:401–6. [11] Hosseini-Toudeshky H, Mohammadi B, Sadeghi G, Daghyani HR. Numerical and experimental fatigue crack growth analysis in mode-I for repaired aluminium panels using composite material. Compos Part A 2007;38: 1141–8. [12] Bouiadjra B, Belhouari B, Serier B. Computation of the stress intensity factors for repaired cracks with bonded composite patch in mode I and mixed mode. Compos Struct 2002;56:401–6. [13] Chung KH, Yang WH. Mixed-Mode fatigue crack growth in aluminium plates with composite patches. Int J Fatigue 2003;25:325–33. [14] Wang CH, Rose LFR. Bonded repair of cracks under mixed mode loading. Int J Sol Struct 1998;35:2749–73. [15] Wang CH, Rose LRF. A crack bridging model for bonded plates subjected to tension and bending. Int J Sol Struct 1999;36:1985–2014. [16] Hosseini-Toudeshky H, Mohammadi B, Daghyani HR. Mixed-mode fracture analysis of repaired aluminium panels using composite patches. Compos Sci Technol 2006;66:188–98.
[17] Hosseini-Toudeshky H, Mohammadi B, Bakhshandeh S. Mixed-mode fatigue crack growth of thin aluminium panels with single-side repair using experimental and numerical methods. Fatigue Fract Eng M 2007;30:629–39. [18] Hosseini-Toudeshky H, Bakhshandeh S, Mohammadi B, Daghyani HR. Experimental investigations on fatigue crack growth of repaired thick aluminium panels in mixed-mode conditions. Compos Struct 2006;75:437–43. [19] Rice JA. A path independent integral and the approximate analysis of strain concentration by cracks and notches. J Appl Mech 1968;35:379–86. [20] Farahmand B, Bockrath G, Glassco J. Fatigue and fracture mechanics of high risk parts: application of LEFM and FMDM theory. International Thomson Publishing; 1997. [21] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Eng 1963;85:519–27. [22] Shafique Khan MA, Khraisheh MK. Analysis of mixed mode crack initiation angles under various loading conditions. Eng Fract Mech 2000;67:397–419. [23] Paris PC. The crack growth of fatigue cracks due to variation in load. Phd thesis, Lehigh University, Bethlehen, 1962. [24] Yaoming Mi. Three-dimensional analysis of crack growth. Southampton UK/ Boston, MA: Computational Mechanics Publications; 1996.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Mixed-mode numerical and experimental fatigue crack growth analyses of thick aluminium panels repaired with composite patches H. Hosseini-Toudeshky a,*, B. Mohammadi b,1 a b
Aerospace Engineering Department and Center of Excellence in Computational Aerospace Engineering, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran Aerospace Engineering Department, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran
a r t i c l e
i n f o
Article history: Available online 18 April 2009 Keywords: Repair Composite Mixed-mode Fatigue crack growth Thick aluminium panel Finite element modelling
a b s t r a c t In this paper, fatigue crack growth of repaired thick aluminium panels containing a central inclined crack is studied. The cracked panels were repaired on one side with glass/epoxy composite patches. Effects of patch lay-up configuration on starting the crack growth after bonding and crack growth rate of the repaired panels were investigated. The finite element analyses were performed assuming uniform crack growth along the panel’s thickness to simplify the analyses. The predicted fatigue lives using both midplane and un-patched surface fracture parameters of the repaired panels are compared with the experimental results. It is shown that the predicted crack propagation lives using the un-patched surface fracture parameters are too conservative for large crack growth values. However, the obtained lives using the mid-plane results may lead to non-conservative life prediction. It is also shown that the predicted starting crack growth life and the crack growth rate at the early crack propagation stages using the un-patched surface fracture parameters are close to the experimental results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Bonded repair of aluminium structures using laminated composite patch was initiated by Baker [1] and followed by other researchers to evaluate the efficiency of that repair. They performed numerical and experimental investigations for single and double side repaired panels in mode-I condition and using boron/ epoxy and graphite epoxy patches [1–3]. In a single-side repaired panel, a non-uniform stress distribution occurs over the thickness of the cracked panel which is due to the existing out-of-plane bending. Many investigations have already been performed to avoid the three dimensional analyses of the repaired panels. Furthermore, in many simplified numerical analyses the stress and strain fields of the mid-plane have been used to calculate the fracture parameters [2–7]. However, the maximum stress and strain occur at the un-patched surface of the repaired panels [8,9]. In single-side repairs, the non-uniform stress distribution along the panel thickness leads to the non-uniform crack propagation along the panel’s thickness and forms a curved crack-front shape during the crack propagation [10,11]. Most of the previous investigations on fracture analysis of single-side repaired panels using composite patches were focused * Corresponding author. Tel.: +98 21 6454 3224; fax: +98 21 6640 4885. E-mail addresses: [email protected] (H. Hosseini-Toudeshky), Bijan_Moh@aut. ac.ir (B. Mohammadi). 1 Tel.: +98 21 6454 3224; fax: +98 21 6640 4885. 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.04.022
on the safety evaluation and life prediction of components under uniaxial loading in mode-I condition [2–11]. However, in many applications, components are subjected to various forms of loading conditions which the crack growth and failure of them occur due to the conjoiner of such loading. Research results were rarely published on the repaired panels in mixed-mode conditions [12,13]. These works investigate the fatigue crack growth of repaired panels in mixed-mode conditions using numerical and experimental methods. Chung and Yang [13] performed fatigue crack growth tests for repaired thick panels made of Al 6061-T6 containing the edge cracks. They only used one lay-up and patch thickness for panels with various inclined crack angles. They showed that, the crack grows non-uniformly through the panel’s thickness. They did not investigate the effect of patch layers orientations on the crack growth rate of the repaired panels. Wang and Rose also presented theoretical investigations on the fracture analyses of singleside repair in mode-I and in-plane mixed-mode condition [14,15]. They presented an analytical method for the combined tensile stretching and bending of one-sided repairs based on Reissner’s plate theory and classical plate theory within the frameworks of geometrically linear and nonlinear elasticity. They also presented fracture analysis of repaired cracks under in-plane mixed-mode loading [15], focusing on the special case of isotropic adherents having the same Poisson’s ratio. They performed a parametric study considering various factors that may influence the repair efficiency. They did not perform any fatigue crack propagation analysis in their works.
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Hosseini-Toudeshky et al. [16] performed uniform crack growth finite element analyses to study the effects of patch thickness and lay-up orientations on fatigue crack growth life and crack trajectory of the repaired thin panels with central inclined crack of 45°. They indicated that, for the repaired panel with the inclined crack of 45°, the patch fibres orientation of h = 90° (unidirectional in loading direction) is an optimum fibre angle leading to longer fatigue crack growth life extensions. However, the optimum patch layers angle is h = 105° for starting crack growth life extension. Mixed-mode fatigue crack growth of thin aluminium panels with single-side repair using experimental and numerical methods were also performed by the authors [17]. Hosseini-Toudeshky et al. performed experiments on fatigue crack-growth of repaired thick panels with a central inclined crack using glass/epoxy composite patch [18]. The numerical crack growth of repaired thick panels in mixed-mode condition has not yet been compared with the experimental results to the authors’ knowledge. In this paper, effects of various patch lay-ups on the life extension of repaired thick aluminium panels for very small (starting) and large crack growths are investigated. These cracked panels contain a typical central inclined crack of 45° and made of Al 2024T3 with single-side glass/epoxy composite patches. The predicted crack growth rates are compared with the experimental results. The accuracy and applicability of the obtained results from the simplified 3D finite element analysis (uniform crack growth modelling along the panel’s thickness) of the repaired panels can also be evaluated. Finite element crack propagation analysis of a single-side repaired panel in general mixed-mode conditions is really a cumbersome task. 2. Fatigue and fracture analysis Fig. 1 shows a typical geometry and loading of the single-side repaired panels containing a central inclined crack. The problem needs to be studied as a mixed-mode cracked problem in modes I and II conditions. Having the stress and strain fields around the crack-tip, fracture parameters for mixed-mode problems are calcu-
lated to predict the crack propagation path and fatigue life of the repaired panels. For this purpose, the fracture parameters such as KI, KII and J-integral can be used. J-integral definition considers a balance of mechanical energy for a translation in front of the crack along the local crack-tip xaxis, which is a path independent contour integral defined as [19]:
J¼
I
Wn1 rij nj
C
@ui ds @x
ð1Þ
where, W is strain energy density, rij are stress components, ui are the displacements corresponding to the local i-axis, s is the arc length of the contour, nj is the jth component of the unit vector outward normal to the contour C, which is any path of vanishing radius surrounding the crack-tip. Using the assumption of linear elastic fracture mechanics and the following equation, KI and KII parameters are decoupled from the calculated J values [20].
J ¼ K 2I =E0 þ K 2II =E0
ð2Þ
where, E0 is modulus of elasticity, E0 ¼ E for plane stress condition and E0 ¼ E=ð1 m2 Þ for plane strain condition problems. To solve the problem for KI and KII the ratio of KI/KII is also obtained from the ratio of the normal distance to the horizontal distance of two closest nodes to the crack-tip which they have been coincided before loading. Having the fracture parameters, a criterion is needed to predict the crack growth direction in a mixed-mode problem. Several criteria have already been proposed for this purpose. Previous researches [16] show that there are not significant differences between the predicted crack trajectories based on various crack propagation criteria. Using stress as a parameter, the maximum tangential stress (MTS) criterion was presented by Erdogan and Sih [21]. This criterion states that a crack propagates in a direction corresponding to the direction of maximum tangential stress along a constant radius around the crack-tip. Using the Westergaard stress field in the polar co-ordinates and applying the MTS-criterion, the following equation is obtained to predict the crack propagation direction in each incremental step [22]:
Fig. 1. Typical geometry and loading of single-side repaired panels.
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Fig. 2. Typical 3D finite element mesh of the repaired panels, (a) hole patched area, (b) initial deformed crack-tip area, (c) crack-tip area after a few steps growth.
tan2
a 2
l 2
tan
a 2
1 ¼0 2
ð3Þ
where, l, defined as the ratio of KI over KII (l = KIKII) and a is incremental crack propagation angle. To find the maximum root value of Eq. (3), the condition of @ 2 ra =@ a2 < 0 should be satisfied. Having the fracture parameters and crack propagation direction from the above formulation, fatigue life of each crack growth increment can be calculated. A number of crack growth laws have been developed relating the crack growth rate to the stress intensity factors. The well known Paris equation has been used in this study which is given as follows [23]:
da ¼ C DK m dN
ð4Þ
where DK ¼ K max K min is the stress intensity factor range in fatigue loading, N is the number of load cycles, da is the crack growth increment, and C and m are empirical material constants. In the mixed-mode problems the effective stress intensity factor range, DKeff, can be used instead of DK and it can be calculated as follows [24]:
DK eff ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK I þ jK III jÞ2 þ 2K 2II
ð5Þ
Table 1 Material properties of aluminium alloy 2024-T3 and adhesive [18]. Parameters
Aluminium 2024-T3
Glue FM 73
E
71.02 (GPa) 0.3
1.83 0.33
t
As far as the analyses are performed in the mixed fracture modes of I and II the KIII = 0.0 is substituted to this equation. Having the effective stress intensity factor and material data, Eq. (4) can be used to calculate the fatigue crack growth lives of the cracked panels in mixed-mode conditions. 3. Finite element analysis A finite element method (FEM) is widely employed for the solution of mixed-mode crack problems. Three dimensional elastic analyses were performed to obtain the stress and strain fields of the repaired panels using ANSYS finite element program. Typical geometry and loading of a repaired plate are shown in Fig. 1 and a typical finite element mesh is shown in Fig. 2. The panels are made of 2024-T3 aluminium alloy with the thickness of 6.35 mm containing an inclined sharp flaw of 2a = 10 mm length and 0.2 mm wide with the inclined angle of b = 45°. They are bonded with four layers glass/epoxy composites with various lay-up configurations of [90]4, [105]4, [45]4, [452/+452] and [902/02]. The lay-up angles are based on the defined h in Fig. 1. The patch layup configurations are selected based on the results of the previous finite element investigations presented by the authors [16]. These analyses showed that the patch lay-up of [105]4 is the best lay-up configuration among the considered lay-ups in terms of the obtained crack growth lives using the mid-plane finite element fracture parameters. Material properties of the aluminium cracked panel, adhesive, and composite patch are given in Table 1 and Table 2. Experimental fatigue crack growth of two un-repaired specimens with initial crack angle of b = 0° were used to obtain
Table 2 Material properties of glass/epoxy composite patch [7]. Elasticity modulus (GPa) E11 E22 E33
Shear modulus (GPa) 27.82 5.83 5.83
G12 G13 G23
Poisson’s ratio 2.56 2.56 2.24
m12 m13 m23
0.31 0.31 0.41
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Fig. 3. Typical de-bonding area around the crack for the repaired panel with [45]4 patch (a) at 19,000 cycles; X ctip ffi 7:5 mm (b) 25,000 cycles; X ctip ffi 11:5 mm.
20 18
Un-repaired Specimen
16
β =45
0
Crack Length (mm)
the material constants of Paris law using the ASTM E-647 method [18]. Therefore, the used material constants in Eq. (4) are: m = 3.66 and C = 4.57e14 in metric system. In these analyses, an isotropic 8-node-solid element was used to model the aluminium panel and adhesive layer. Furthermore, a layered 8-node-solid element was used to model the laminated composite patch. The used 8-node-solid elements are non-conforming elements which can be selected by the option of extra shape function in Ansys. The non-conforming elements have excellent performance in the bending situation when compared with the conforming elements. A typical mesh of the component is shown in Fig. 2. The in-plane elements aspect ratios near the crack tip are about 1.0. The obtained finite element results show that different J-integral contours and various refined meshes on the surfaces and through the thickness of the repaired panels (variation of 2–10 elements along the thickness) result in about 2% differences in the obtained fracture parameters [16]. Therefore, two elements in the thickness of each part, plate, adhesive and patch were used and a very fine mesh was generated for the region close to the crack-tip. Each model contained the total number of 12744 elements and the plane area of (1.0 mm 1.0 mm) close to the crack-tip contained about 900 elements as shown in Fig. 2. A Macro program was developed using ANSYS Parametric Design Language (APDL) to handle the crack growth modelling procedure as well as to find the crack trajectory analysis of each repaired panel. The main assumptions in the crack growth analysis are: (1) linear elastic fracture mechanics behaviour, (2) the crack-front line through the thickness, remains perpendicular to the plate surfaces during the crack propagation (uniform crack growth modelling) to simplify the modelling and analysis, and (3) no major de-bonding occurs between the panel surface and patch during the crack propagation. According to our experimental observations [18], for most of the cases which the surface preparation performed properly, debounding between the patch and aluminium panel occurred in a small area around the crack, only. It is worth to mention that, a small de-bounded area with the size of an element width has been considered in the FEM modelling. Typical de-bonding areas around the crack for the repaired panel with [45]4 patch are shown in Fig. 3. In this procedure a dynamic mesh generation was designed to generate automatic mesh of the repaired panels at each crack growth step. The J-integral value for the presented crack configuration was computed in each step, and then using Eq. (2) and the explained decoupling procedure in Section 2, the values of KI and KII were calculated. Using the obtained values of KI and KII and Eq. (3), the crack growth direction, a, is determined. The equations were numerically solved in each step and then the geometry is updated with the new crack length configuration. The above procedure was performed for the new configuration and repeated for several steps to find the crack trajectory in each repaired panel. Full details and
14 12 10 8 6 FEM
4
Exp-1 Exp-2
2 0 0
10000
20000
30000
40000
50000
N (Cycle) Fig. 4. Comparison between experimental and finite element crack growth rates (Xdirection) for un-repaired panels.
explanations of fatigue and fracture analysis, finite element modelling, and APDL macro programming are presented in Ref. [16]. 4. Results and discussions In this section, the obtained fatigue crack growth life of the repaired thick panels from the finite element analyses are compared with the experimental results performed by the authors in Ref. [18]. 4.1. Verification Fig. 4 compares the predicted finite element crack growth behaviour with the experimental results for the un-repaired specimen. The good agreement between the experimental and numerical results verifies the developed finite element crack growth procedure and the material constants used in the Paris law. The experimental and numerical crack propagation paths obtained at patched and un-patched surfaces of a typical repaired panel are also shown in Fig. 5. There is a good agreement between the predicted and experimental crack propagation paths at both patched and un-patched surfaces of the repaired panel. The experimental and numerical results also show that the major component of the crack growth is in the X-direction or approximately perpendicular to the loading direction. Therefore the X-component of the crack propagation path of the repaired panels is used in the crack growth results presentation and discussions in the following sections. For further information variation of KII/KI ratio versus crack tip X-coordinate for un-repaired and repaired panels with various
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(a)
(b)
Fig. 5. Typical experimental and numerical crack propagation path of repaired panel with the patch lay-up of [902/02], (a) un-patched surface, (b) patched surface.
(a)
1
patch lay-ups at both mid-plane and un-patched surface are depicted in Fig. 6. These variations show that the mixed-mode condition is mainly existing at the early stages of crack propagation. For crack growth length of higher than 1.0 mm (Xctip > 4.5 mm) these ratios vanished to almost zero value for mid-plane and lie between 0.05 and 0.1 at un-patched surface meaning that the mode II does not significantly affect the crack growth rate and path.
1
0.9 0.99
0.8 0.98
0.7
|
K II /K I|
at Mid-Plane 0.6
0.97
Un-repaired [90/90/90/90] [90/90/0/0] [105/105/105/105] [-45/-45/-45/-45] [-45/+45/-45/-45]
0.5 0.4 0.3
4.2. Crack growth behaviour
0.96
0.95
0.94 3
0.2
3.5
4
0.1 0 0
2
4
6
8
10
12
14
Crack tip X-coordinate (mm)
1
1
0.9
18 0.96
0.8
at Un-Patched surface
K II /K I|
0.7
|
20
0.98
Un-repaired [90/90/90/90] [90/90/0/0] [105/105/105/105] [-45/-45/-45/-45] [-45/+45/-45/-45]
0.6 0.5 0.4 0.3
16
0.94
Crack Length (mm)
(b)
The experimental and finite element crack growth rates of the repaired and un-repaired panels have been shown for [452/ +452], [45]4, [90]4, [105]4 and [902/02] patch lay-ups in Figs. 7– 11 respectively. These figures show that the obtained crack growth rates using the un-patched surface fracture parameters are comparable with the experimental results at the early stages of crack propagation. However, they behave considerably different for large crack growth values. It is worth to mention that the crack-front
0.92 0.9 0.88 0.86 0.84 0.82
14 12 10 8 6
0.8
0.2
3
3.5
4
0.1
Patch lay-up [-452/+452]
2
0 0
2
4
6 8 10 Crack tip X-coordinate (mm)
12
14
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
0 0
10000
20000
30000
40000
50000
N (Cycle) Fig. 6. Variation of KII/KI ratio versus crack tip X-coordinate for un-repaired and repaired panels with various patch lay-ups (a) at mid-plane (b) at un-patched surface.
Fig. 7. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [452/+452].
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 91 (2009) 1–8
20
20
18
18
16
16
14 12 10 8 6
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
Patch lay-up [-45]4
2 0 0
10000
20000
30000
40000
50000
N (Cycle)
20 18
Crack Length (mm)
16 14 12 10 8
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Exp-3 Un-repaired Exp.
4
Patch lay-up [90]4
2 0 0
10000
20000
30000
40000
50000
N (Cycle) Fig. 9. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [90]4.
20 18
Crack Length (mm)
16 14 12 10 8 6
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
Patch lay-up [105]4
2 0
0
10000
20000
14 12 10 8 6
Un-patched surface (FEM) Mid-plane (FEM) Exp-1 Exp-2 Un-repaired Exp.
4
Patch lay-up [902/02]
2 0 0
10000
20000
30000
40000
50000
N (Cycle)
Fig. 8. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [45]4.
6
Crack Length (mm)
Crack Length (mm)
6
30000
40000
50000
N (Cycle) Fig. 10. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [105]4.
shape is a major factor affecting the stress intensity factors and crack growth rate in the single-side repaired panels [8,9,11]. At the early stages of the experimental crack propagations where
Fig. 11. Experimental and finite element crack growth rates in the X-direction for un-repaired and repaired panels with patch lay-up of [902/02].
the crack-front has not completely developed to a curved shape, the real crack-front is more compatible with the considered uniform crack-front modelling in the finite element analyses. Therefore, the obtained crack growth rates using the un-patched surface results are close to the experimental results for small crack growth values. These figures also show that, the predicted crack growth rate using the mid-plane fracture parameters are slower than those obtained from the experiments for all patch lay-up configurations except the patch lay-up of [90]4 in Fig. 9. This figure shows that the results of Exp-2 and Exp-3 are almost compatible but the exceptional crack growth rate in Exp-1 behaves differently and is close to the mid-plane finite element results. This test has been performed at the early stage of the experimental work and may prone to errors due to the test set-up difficulties such as initial blunted crack-tip, measuring of initial crack length and angle, decentred crack, and patch manufacturing and curing conditions. The scattering due to the initial blunted crack-tip affect the restarting crack growth life and in this case the initial crack growth life is significantly larger than the other two test results. The difference between the predicted crack growth rate using the mid-plane results and experiments for the repaired panel with the asymmetric patch lay-up of [902/02] in Fig. 11 are larger than the differences occur for the other patch lay-ups. It should be noted that the asymmetric lay-up condition in the [902/02] lay-up produces a secondary bending against the existing out-of-plane bending due to the geometry of the single-side repair of the structure. In this case, de-bonding of the patch was also observed at the edges of the patch after a few crack growth steps during the test. It could occur due to the above mentioned asymmetric patch lay-up bending and producing of pilling stress on the bonding area. If the bonding quality can be improved in this case, crack growth life extension of better than the other used patch lay-ups may be obtained for the repaired panels. 4.3. Small (restarting) crack growth life Table 3 shows the obtained restarting crack growth life after bonding of the panels from both finite element and experimental [18] methods. The crack restarting growth defined as the extension of the crack-tip from the initial value of 3.54 mm to the Xctip = 4.0 mm at un-patched surface of the panels. It is also worth to note that the crack at patched surface may start to growth with delay or with smaller growth rate due to the existing differences between the fracture parameters along the panel thickness. Table 3 shows considerable scattering in the obtained experimental
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 91 (2009) 1–8 Table 3 Comparison of FEM and experimental restarting crack growth life of repaired panels up to Xctip = 4.0 mm (cycles). Patch lay-up
Mid-plane (FEM)
Un-patched surface (FEM)
Exp. A [18]
Exp. B [18]
[90]4 [902/02] [105]4 [45]4 [452/+452] Un-repaired
11,480 15,982 12,133 10,683 10,751 3369
6658 7115 6839 7233 7176 3369
5700 5200 4900 5900 4600 5000
8100 4700 6300 6900 5500 7400
Table 4 Comparison of FEM and experimental crack growth life of repaired panels up to Xctip = 17 mm (cycles). Patch lay-up
Mid-plane (FEM)
Un-patched surface (FEM)
Exp. A [18]
Exp. B [18]
[90]4 [902/02] [105]4 [45]4 [452/+452] Un-repaired
37,100 42,692 37,167 31,462 33,200 19,000
20,790 22,205 20,732 21,879 22,300 19,000
28,740 30,600 27,300 28,564 29,020 20,770
39,040 28,370 32,700 29,224 27,060 20,100
restarting crack growth lives of the repaired panels with various lay-ups. It may be due to the difficulties in recognizing the initial 0.46 mm crack growth at un-patched surface of the panels during the tests and therefore the experimental data for crack restarting lives are prone to some errors. The finite element analyses results presented in this table show that, the predicted lives using the unpatched surface fracture parameters are closer to the experimental results than those obtained using mid-plane results. The differences between the experimental and un-patched surface finite element results may also be due to the assumption of uniform 0.46 mm initial crack propagation along the crack-front. The predicted lives using the mid-plane fracture parameters lead to very non-conservative results. Comparing of the predicted restarting crack growth lives using the un-patched surface fracture parameters with the experimental results are more realistic than those obtained using the mid-plane fracture parameters. The numerical lives also show that, using the un-patched surface results leads to the almost same restarting crack growth life prediction for the three patch lay-ups of [45]4, [452/452] and [902/02]. These layups are more effective than the other two considered lay-ups in term of the restarting crack growth life extension. 4.4. Crack growth life Table 4 shows the obtained crack propagation life of the repaired panels with various patch lay-ups from experimental and finite element methods. The crack growth lives were obtained for the crack length extension of 13.46 mm (between the initial crack-tip Xctip = 3.54 mm and Xctip = 17.0 mm). It is worth to mention that two experimental results were only available for the crack growth life up to the crack extension of Xctip = 17.0 mm. The experimental results in Table 4 shows that using the considered various patch lay-ups, crack propagation life of the repaired panels may increase by the order of about 30–85% with respect to the un-repaired panel life. The obtained experimental crack growth life extension for the repaired panels with [90]4 and [105]4 are larger than the other three patch lay-ups. Disregarding the crack growth life of the panel with the patch lay-up of [90]4 (Exp. B in Table 4), there are not considerable differences between the obtained lives for the repaired panels with various patch lay-ups. Finite element analyses results presented in Table 4 also show that, the obtained
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crack propagation lives using the un-patched surface fracture parameters are too conservative and the lives obtained from the mid-plane results may lead to 50% non-conservative life predictions in some cases. The mid-plane finite element lives presented in this table shows that the patch lay-ups of [90]4, [902/02] and [105]4 are more effective than the other two considered lay-ups and the patch lay-up of [902/02] is the best lay-up configuration among the considered patch lay-ups in term of crack growth life extension. The differences between the experimental and finite element crack growth lives may be due to the assuming of uniform crack propagation along the panel thickness in the numerical analyses. It was already explained that in a single-side repaired panel, a non-uniform stress distribution occurs over the thickness of the cracked panel which is due to the existing out-of-plane bending. This non-uniformity of stress field leads to non-uniform crack propagation along the panel’s thickness and forms a curved crack-front shape during the crack propagation which affects the crack growth life significantly as shown by the authors for repaired panels in mode-I condition [9]. It is also worth to mention that the mixed fracture modes of I and II were considered in the analyses, however, the real condition is a general mixed fracture modes of I, II and III. A finite element crack propagation analysis in general mixed-mode conditions assuming a non-uniform 3D crack growth along the panel thickness may reduce the existing differences between the numerical and experimental results which is really a cumbersome task. 5. Conclusion In this paper, fatigue crack growth analysis of thick aluminium panels containing a central inclined crack with single-side glass/ epoxy composite patch was performed. Effects of patch lay-up configuration on the restarting crack growth life and crack growth rate of the repaired panels were studied. It was shown that the obtained restarting crack growth lives using the un-patched surface finite element results are closer to the experimental results, but, the calculated lives using the mid-plane fracture parameters leads to very non-conservative life prediction. It was also shown that the predicted crack growth life using the un-patched surface finite element results is too conservative and it may be 50% nonconservative when the mid-plane results are used. The experimental results showed that, the crack propagation life of the panels may increase by the order of 30–85% depending on the used patch lay-up. References [1] Baker A. Bonded composite repair of fatigue-cracked primary aircraft structure. Compos Struct 1999;67:421–43. [2] Sun CT, Klug J, Arendt C. Analysis of cracked aluminium plates repaired with bonded composite patches. AIAA J 1996;34(2):369–74. [3] Ratwani MM. Analysis of cracked adhesively bonded laminated structures. AIAA J 1979;17(4):988–94. [4] Okfar C, Singh N, Enemuoh UE, Rao SV. Design, analysis and performance of adhesively bonded composite patch repair of cracked aluminum aircraft panels. Compos Struct 2005;71(2):258–70. [5] Rose LRF. A cracked plate repaired by bonded reinforcements. Int J Fract 1982;18(2):135–44. [6] Lu J, Hu Y, Ju D. An assessing method of fatigue life on central-through cracked plate with adhesive bounded reinforcement, fatigue, fracture, and risk. ASME 1991;215:135–40. [7] Daghyani HR, Sayadi A, Hosseini-Toudeshky H. Fatigue crack propagation of aluminium panels repaired with adhesively bonded composite laminates. Proc Instn Mech Eng Part I: J Mater Des Appl 2003;217:291–3. [8] Lee WY, J Lee J. Successive 3D FE analysis technique for characterization of fatigue crack growth behavior in composite-repaired aluminum plate. Compos Struct 2004;66(1–4):513–20. [9] Hosseini-Toudeshky H, Mohammadi B. A simple method to calculate the crack growth life of adhesively repaired aluminum panels. Compos Struct 2007;79: 234–41.
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