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A simple method to calculate the crack growth life of adhesively repaired aluminum panels
Composite Structures 79 (2007) 234–241 www.elsevier.com/locate/compstruct
A simple method to calculate the crack growth life of adhesively repaired aluminum panels H. Hosseini-Toudeshky *, B. Mohammadi Aerospace Engineering Department, Amirkabir University of Technology, Hafez Avenue, 424 Tehran, Iran Available online 21 February 2006
Abstract In this paper, a simple method is developed to predict the crack growth life of single-side repaired aluminum panels. The patches are considered to be made of glass/epoxy, boron/epoxy and graphite/epoxy composite materials. For this purpose, crack growth life of the repaired panels are calculated using two different approaches: (1) The crack growths along the panel thickness uniformly. In this simplified approach, crack-front line remains perpendicular to the plate surfaces during the crack propagation. (2) The crack growths along the panel thickness non-uniformly. In this approach, initial crack-front line changes to a curved shape according to the variation of stress intensity factor during the crack propagation (real crack-front shape). Then, a new simple method is developed based on a parametric study using both approaches. Using this simple method, it is not necessary to perform the complex crack growth analysis based on the real crack-front shape. Therefore, the results of 3D analysis using the first approach or 2D analysis similar to the three layers technique may be used to estimate the crack growth life of the repaired panels. The obtained results are verified with the available experimental results. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Repair; Aluminum panels; Composite; Simple method
1. Introduction Service life of cracked components may be extended using repair technologies. Adhesively bounded composite patches have been recognized as an efficient and cost-effective method to repair cracked components in advanced aerospace structures. A double-sided repair increases considerably the fatigue life of a cracked plate, although, owing to the manufacturing difficulties, a single-side repair is normally preferred. Fig. 1 shows a typical configuration of a single-side repaired panel. One of the most challenging aspects of the bounded composite repair technology has been the stress analysis of repaired panels and the calculations of subsequent fracture parameters. The difficulty arises from the fact that a thin metallic panel under the in-plane loading may fail *
Corresponding author. Tel.: +98 21 6640 5032; fax: +98 21 6640 4885. E-mail address: [email protected] (H. Hosseini-Toudeshky).
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.01.005
under highly complicated three-dimensional stresses, if composite patches are bounded to its surface asymmetrically (single-sided repair) [1–5]. The stress variations over the thickness of a cracked plate in asymmetric repair present a greater challenge in modeling due to the existence of out-of-plane bending. This causes stress variations over the thickness of the cracked panel and therefore causes nonuniform crack propagation along the crack-front. Fig. 2 shows a typical experimental crack-front shape in fatigue crack growth of the repaired panels [6]. Furthermore, in many studies the stresses and strains fields of the mid-plane have been used to calculate the fracture parameters, however, the maximum stresses and strains occur at un-patched surface of the cracked panels [7]. Recently, a few researchers published papers considering the non-uniform crack propagation along the crackfront of the repaired panels [8,9]. Seo and Lee [8] performed finite elements analysis of the asymmetric repaired panels assuming skew crack-front model. However, the
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235
Fig. 1. Typical geometry and loading of single-side repaired panels.
Fig. 2. Typical crack-front shape of a repaired panel in fatigue crack growth.
single-sided repair under cyclic loading produces a curvilinear crack-front shape in the real applications. HosseiniToudeshky et al. [9,10] predicted the actual crack-front shape using experiments and 3D finite elements fatigue crack growth analysis of the asymmetric repaired panels in mode-I condition. They predicted the actual crack-front shape and crack growth life of the repaired panels for a certain crack growth length and for various patch thicknesses. In the present paper, three-dimensional finite elements analyses are performed for the repaired aluminum panels, containing a central crack with single-side composite patches in mode-I condition. The patches are made of glass/epoxy, boron/epoxy and graphite/epoxy. Fatigue crack growth life of the repaired panels are calculated using two different approaches: (1) The crack growths along the panel thickness uniformly (UCG). In this simplified approach, the crack-front line remains perpendicular to the plate surfaces during the crack propagation. (2) The crack growths along the panel thickness non-uniformly (NUCG). In this approach, initial crack-front line changes to a curved shape according to the variation of stress intensity factor during the crack propagation (real crack-front shape analysis). Finite elements crack growth analyses based on the NUCG modeling is a complex and time consuming task. Then, a new simple method is developed
based on a parametric study using both approaches. In this method a position along the panel thickness is purposed, so that, if the stress intensity factors of that position during the crack propagation are used, almost the same fatigue crack growth life as the one calculated in the real crackfront shape analysis is obtained. Using this simple method, it is not necessary to perform the complex crack growth analysis based on the real crack-front shape. Therefore, the results of 3D analysis based on the first simplified approach or 2D analysis results similar to the three layers technique [11] can be used to estimate the crack growth life of the repaired panels. The obtained fatigue crack growth lives from the real crack-front shape analyses and the simple method are compared with those obtained from the experiments in some cases. 2. Fatigue and fracture analysis Crack closure technique may be used to calculate the fracture parameters and fatigue crack growth life of the repaired panels. The crack closure technique used here is based on Irwin’s notation of reversing crack growth to compute the energy release rate (G). Modified crack closure technique (MCCT) is also used to reduce the computational efforts. It is applicable for linear elastic problems only and orthogonal mesh is required to model the crackfront in the finite elements analysis. Equivalent nodal forces and their conjugate nodal displacements in elements around the crack-front computed by FEM are used in calculation of G. Nodal displacement are primary variables in the FEM, and equivalent nodal forces will be computed more accurate than the crack-front stresses [12]. For 3D problems the mode-I strain energy release rate are calculated using the following expression [12] GI ¼
1 Z Li ðwLi wLi Þ 2DA
ð1Þ
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in which DA is the area of element, ZLi and wLi are nodal forces and displacements, respectively. Strain energy release rate and stress intensity factor are related by [13] pffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ K I ¼ GI =E in which E* = E for plane stress and E* = E/(1 t2) for plane strain conditions. Having the stress intensity factor, the Paris equation is used to relate the crack growth rate to the stress intensity factor [13] da ¼ CDK n dN
ð3Þ
where C and n are empirical material constants, DK = Kmax Kmin is the stress intensity factor range in fatigue loading, N is number of cycles, and da is crack extension length. 3. Finite elements analysis Typical geometries of the aluminum panels, adhesive layer and composite patch are shown in Fig. 1. The panels are made of 2024-T3 aluminum alloy with various thicknesses containing an initial crack of 2a = 10 mm length. Specimens’ dimensions followed the ASTM E-647 standard. Fig. 3 shows a typical finite elements mesh of the repaired panels. A wide range of three-dimensional finite element crack growth analyses are performed for the repaired panels considering both the uniform (crack-front perpendicular to the panel surfaces) and non-uniform (real crack-front shape) crack growth assumptions. Fig. 4 shows the typical crack-front configuration in both approaches. The analyses are carried out for panels with three different patch materials, viz. boron/epoxy, graphite/epoxy and glass/epoxy with various thicknesses. The thickness of patch in each model is assumed to have the nominal stiffness ratio of less than 1.5 according to the plate thickness and does not exceed the plate thickness. The stiffness ratio, S, is defined as S¼
ðEr tr Þ ðEp tp Þ
ð4Þ
Fig. 4. Crack-front configuration of the repaired panels. (a) Schematic of the real fracture surface, (b) crack-front at uniform crack growth, (c) crack-front at non-uniform crack growth.
where E is Young’s modulus, t is thickness, and ‘r’ and ‘p’ are subscript designating the repair patch and plate, respectively. It will be shown that the fatigue crack growth lives of the repaired thick panels are not significantly increased for the patch thicknesses of more than eight layers later in this paper. Therefore the maximum patch layers of 16 layers are considered for them. The used adhesive material is FM77. The applied remote stress is 118 MPa in all repaired panels. Tables 1 and 2 show the dimensions and material properties of the aluminum cracked panels, adhesive layers, and composite patches. Unidirectional lay-up perpendicular to the crack length is used for the patches in all models. The stress and strain fields of the repaired panels are obtained using the ANSYS finite elements
Fig. 3. Typical finite elements mesh of the repaired panels.
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 79 (2007) 234–241 Table 1 Dimensions of the panels, adhesive layers and patches
L (mm) W (mm) t (mm)
Aluminum panel
Adhesive
Patch
100 50 1.5, 2.29, 3.5, 5, 6.35, 7.5
40 35 0.1
40 35 0.18 per layer
Table 2 Material properties
E11 (GPa) E22 (GPa) E33 (GPa) G12 (GPa) G13 (GPa) G23 (GPa) m12 m13 m23
Glass/epoxy [10]
Graphite/epoxy [8]
Boron/epoxy [1]
50 14.5 14.5 2.56 2.56 2.24 0.33 0.33 0.33
134 10.3 10.3 5.5 5.5 3.2 0.33 0.33 0.53
208 25.4 25.4 7.2 7.2 4.9 0.1677 0.1677 0.035
program. In the three-dimensional analysis, an isotropic 8-node-solid element is used to model the aluminum panel and adhesive layer. Furthermore, a layered 8-nodesolid element is employed to model the laminated composite patch. Two elements in the thickness of adhesive and patch and 10 elements in the thickness of plate are used and a fine mesh is generated for the regions close to the crack-front. The material constants in Paris equation were calculated based on the ASTM E-647 method and the required experimental data were obtained from the tests of un-repaired panels containing a central crack [6,10]. The obtained constants m and c were 3.2828 and 3.63 E-13, respectively for thin panels and 4.224 and 1.51 E-15 for thick panels. In the crack growth analyses based on both uniform crack growth rate and non-uniform crack growth (see Fig. 4), it is assumed that the crack growths only in the aluminum panel with no failure and debounding on the composite patch. This assumption is mainly based on our experimental evidence on fatigue crack growth of repaired panels containing a crack in mode-I condition. A Macro program is developed using ANSYS Parametric Design Language (APDL) to handle the crack growth modelling procedure for any cracked panel geometry and patch configuration. In this procedure, a dynamic mesh generation using automesh capability of the code is used to generate automatic mesh of the repaired panels for each crack growth step. The used procedures to find the crack growth rates in both uniform and non-uniform crack growth modeling are explained in the following sections. 3.1. Uniform crack growth modeling In these analyses it is assumed that the crack-front remains perpendicular to the panel’s surfaces during the
237
crack propagation. Performing linear elastic solution for the repaired panels with an existed pre-crack the stresses and displacements at the crack-front are obtained. Then, the GI value for the presented crack configuration is computed in each step using the modified crack closure technique presented by Eq. (1). Then using Eq. (2), the values of KI along the crack-front are calculated. Using the obtained value of KI at un-patched surface and Eq. (3), crack growth length for a certain number of load cycles is calculated or vice versa. The equations are solved in each step and then the geometry is updated for the repaired panel with the new crack length. The above procedure is performed for the new configuration and repeated for several steps to find the crack growth rate of the repaired panels. 3.2. Non-uniform crack growth modeling In these analyses the real crack-front shape is obtained using the variation of stress intensity factor along the crack-front at each crack propagation step. The major steps of the developed Macro program to handle the crack growth modeling procedure as well as to find the crack growth life of the cracked panel are as follows: (i) Geometry and finite element mesh generation of the problem. (ii) Defining the loading, constraints, and material properties. (iii) Performing the linear elastic solution. (iv) Calculation of GI and KI at the crack-front using the modified crack closure technique. (v) Considering 0.1 mm crack length increment at unpatched surface for each step and calculating the loads cycles (DN) for this increment using the Paris Law. (vi) Having DN and the values of KI at crack-front nodes the ratio of crack growth at each node with respect to the 0.1 mm growth at un-patched surface are calculated and the new crack-front shape is generated. (vii) Updating the finite element model based on the new crack length and crack-front configuration. (viii) If KI 6 KIC or crack length at un-patched surface is less than a defined value, return to step (iii). (ix) Process the results and stop the solution.
4. Results and discussion Table 3 compare the lives obtained from the experiments with those obtained from the finite elements analyses using mid-plane and un-patched surface results in uniform crack growth (UCG) and non-uniform crack growth (NUCG) modeling for two panels thicknesses of 2.29 mm and 6.35 mm with various 4, 8, and 16 layers glass/epoxy patches. This table shows that the lives obtained from
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Finite elements crack growth analyses using the NUCG modeling is a complex and quite time consuming task. Therefore, a simple method is developed to use the UCG modeling and predict the reasonable crack growth life of the single-side repaired panels. In this method a position along the panel thickness, Ze, is purposed, so that, if the stress intensity factors of that position in the UCG modeling are used, almost the same fatigue crack growth life as the one calculated in the NUCG modeling is obtained. Tables 4 and 5 show the lives obtained from the NUCG analyses and the non-dimensional coordinate of Ze/tp on the crack-front line in the UCG analyses of the repaired panels for thin and thick panels respectively. It is noted that, if the ratio of the thickness over the wide of the panel is less than 0.05 (tp/(2W) < (1/20)), they are categorized as thin panels. It is also mentioned that, Ze is measured from the un-patched surface of the panels as indicated in Figs. 3 and 4 and tp is the plate thickness. Ze/tp is the point that if the KI values of that point in UCG analyses are used, the same life as the NUCG analyses of the repaired panel is obtained. Tables 4 and 5 show that there are not large scattered between the values of Ze/tp obtained for panels with various number of patch layers for each patch composite material. Therefore the average values of Ze/tp for each patch composite material and for both thin and thick panels are calculated and depicted in Fig. 5 against the ratio of Young’s modulus of elasticity of aluminum panel over the patch material. These data presented two well linear curves for thin and thick repaired panels. The equations fitted on these points using least square method are as follows:
Table 3 Comparison between experimental and uniform and non-uniform crackfront finite elements modeling lives of the repaired panels with glass/epoxy patches No. of layer
t = 2.29 mm
t = 6.35 mm
4
8
16
Mid-plane results (UCG) Un-patched surface results (UCG) NUCG Experiment 1 Experiment 2
54,010 16,369
68,724 21,696
88,148 31,307
34,458 33,800 34,700
41,140 38,000 39,200
63,757 68,900 65,300
Mid-plane results Un-patched surface results NUCG Experiment 1 Experiment 2
25,583 9863 17,818 17,800 19,800
30,752 9396 18,799 17,500 19,200
35,569 8930 19,383 18,100 19,800
UCG modeling using un-patched surface results are too conservative and the lives calculated using the mid-plane results are non-conservative by the order of 35–90% comparing with the experimental results. The lives obtained from NUCG modeling are close to those obtained from the experiments with the differences of less than 10%. The results presented in this table show that the lives obtained from NUCG modeling are reliable with an acceptable difference from experimental results. But the lives obtained from UCG modeling using mid-plane stress intensity factors are too non-conservative; therefore, they cannot be used for design and life extension purpose of the repaired panels.
Table 4 NUCG lives and the values of Ze/tp for thin plates with various patch layers and composite material Patch layers
Patch
Glass/epoxy
tp (mm)
1.5
2.29
3.5
1.5
2.29
3.5
1.5
2.29
3.5
1
NUCG Ze/tp
33,210 0.359
– –
– –
37,850 0.357
– –
– –
42,100 0.349
– –
– –
2
NUCG Ze/tp
35,350 0.362
31,575 0.365
29,210 0.387
43,120 0.352
36,314 0.350
32,880 0.349
49,850 0.358
39,630 0.345
35,315 0.340
4
NUCG Ze/tp
41,507 0.370
34,458 0.366
30,979 0.368
56,550 0.365
41,110 0.354
35,127 0.348
71,239 0.365
46,976 0.351
38,187 0.342
6
NUCG Ze/tp
51,074 0.377
37,520 0.367
32,240 0.363
– –
47,763 0.358
37,049 0.347
– –
57,953 0.357
41,261 0.344
8
NUCG Ze/tp
63,270 0.385
41,140 0.370
33,361 0.362
–
56,822 0.364
39,323 0.348
– –
– –
45,201 0.346
10
NUCG Ze/tp
– –
45,461 0.373
34,544 0.362
–
–
42,236 0.351
– –
– –
– –
12
NUCG Ze/tp
–
50,511 0.377
35,889 0.363
–
–
45,892 0.354
–
–
–
14
NUCG Ze/tp
–
56,288 0.382
37,440 0.365
– –
– –
– –
– –
– –
– –
,
Graphite/epoxy
Boron/epoxy
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Table 5 NUCG lives and the values of Ze/tp for thick plates with various patch layers and composite material Patch layers
Patch
Glass/epoxy
Graphite/epoxy
Boron/epoxy
tp (mm)
5
6.35
7.5
5
6.35
7.5
5
6.35
7.5
2
NUCG Ze/tp
17,986 0.375
16,823 0.375
16,071 0.372
20,455 0.344
18,848 0.339
17,799 0.342
22,100 0.333
20,173 0.333
18,916 0.326
4
NUCG Ze/tp
19,164 0.362
17,818 0.361
16,937 0.358
21,948 0.338
20,153 0.336
18,967 0.333
23,785 0.331
21,597 0.337
20,186 0.324
6
NUCG Ze/tp
19,888 0.355
18,430 0.352
17,480 0.350
22,714 0.334
20,732 0.329
19,478 0.325
24,843 0.328
22,262 0.322
20,714 0.318
8
NUCG Ze/tp
20,360 0.350
18,799 0.347
17,807 0.344
23,292 0.332
21,009 0.326
19,669 0.321
25,892 0.328
22,703 0.320
20,959 0.315
10
NUCG Ze/tp
20,714 0.348
19,021 0.343
17,990 0.340
23,947 0.332
21,210 0.324
19,736 0.319
27,210 0.329
23,174 0.320
21,149 0.314
12
NUCG Ze/tp
21,037 0.347
19,161 0.341
18,081 0.337
24,797 0.333
21,452 0.324
19,783 0.318
28,915 0.331
23,786 0.320
21,391 0.313
14
NUCG Ze/tp
21,385 0.346
19,271 0.339
18,122 0.334
25,896 0.335
21,792 0.324
19,869 0.317
31,076 0.333
24,592 0.321
21,735 0.314
16
NUCG Ze/tp
21,789 0.347
19,383 0.338
18,143 0.333
27,273 0.336
22,260 0.325
20,024 0.317
33,745 0.335
25,622 0.323
22,209 0.314
0.375 0.370 0.365
(Ze /tp) = 0.0166(EAl /Epatch ) + 0.3453
(Ze/tp)average
0.360 0.355 0.350 0.345 0.340 0.335
Thin Plates Thick Plates
0.330
(Ze /tp) = 0.0235(EAl /Epatch ) + 0.3159
0.325 0.320 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
EAl /EPatch Fig. 5. Variation of the average values of Ze/tp against the ratio of Young’s modulus of elasticity of aluminum panel over the patch material.
Ze EAl ¼ 0:0166 þ 0:3453 tp EPatch Ze EAl ¼ 0:0235 þ 0:3159 tp EPatch
for thin panels
ð5Þ
for thick panels
ð6Þ
Having the Young’s modulus of elasticity of both panel and patch materials and the thickness of the panel and using the above equations, the required position at the crack-front, Ze, is obtained to use in the simple method. Tables 6 and 7 compare the predicted lives obtained from the UCG analyses and using the values of Ze calcu-
lated from the simple method (Eqs. (5) and (6)) with those obtained from the NUCG analyses for thin and thick panels respectively. Comparison of the predicted lives with those obtained from the NUCG analyses shows the maximum difference percentage of 4.5% in thin panels and 7.5% for thick panels. The maximum difference belongs to the panel with 5.0 mm thickness and 4 layers glass/epoxy patch. It is noted that using the midplane results of the UCG analysis of the same repaired panel leads to the non-conservative life prediction with 43.6% difference from the life obtained from NUCG analysis. To verify the predicted repaired panels’ lives obtained from the simple method, they are compared with those obtained from the NUCG analyses and experimental results performed in [10]. Table 8 shows that the maximum difference between the predicted lives using the simple method (UCG analyses) with those obtained from the NUCG analyses is about 6.2% which belongs to the panel thickness of 6.35 mm with 4 layers patch. The maximum difference between the predicted life using the simple method with those obtained from the experiments is about 11.1% which also belongs to the panel thickness of 6.35 mm with 4 layers patch. It is noted that the difference percentage between the life obtained from the NUCG analysis and experiment for the same case is about 5.3%. The presented results in Tables 6–8 shows that the purposed location of Ze in the simple method can be used to calculate the stress intensity factor from the UCG analysis and predict the fatigue crack growth life of the single-side repaired panels with an acceptable engineering accuracy in the obtained results.
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Table 6 Comparison between the predicted lives obtained from simple method and UCG analyses with those obtained from NUCG analyses for thin panels Patch layers
Patch
Glass/epoxy
Graphite/epoxy
Boron/epoxy
tp (mm)
1.5
2.29
3.5
1.5
2.29
3.5
1.5
2.29
3.5
1
NUCG Predicted Error (%)
33,210 34,135 2.8
– – –
– – –
37,850 38,750 2.4
– – –
– – –
42,100 42,750 1.5
– – –
– – –
2
NUCG Predicted Error (%)
35,350 35,950 1.7
31,575 31,950 1.2
29,210 28,300 3.1
43,120 42,930 0.4
36,314 35,950 1.0
32,880 31,450 4.3
49,850 49,032 1.6
39,630 41,215 4.0
35,315 36,350 2.9
4
NUCG Predicted Error (%)
41,507 41,381 0.3
34,458 34,752 0.9
30,979 31,069 0.3
56,550 54,546 3.5
41,110 41,173 0.2
35,127 35,841 2.0
71,239 68,027 4.5
46,976 47,050 0.2
38,187 39,453 3.3
6
NUCG Predicted Error (%)
51,074 49,915 2.3
37,520 37,776 0.7
32,240 32,756 1.6
– – –
47763 47,098 1.4
37,049 37,969 2.5
– – –
57,953 56,766 2.0
41,261 42,050 1.9
8
NUCG Predicted Error (%)
63,270 60,663 4.1
41,140 41,320 0.4
33,361 34,067 2.1
– – –
56,822 55,046 3.1
39,323 40,085 1.9
– – –
– – –
45,201 46,032 1.8
10
NUCG Predicted Error (%)
– –
45,461 44,900 1.2
34,544 35,273 2.1
– –
– –
42,236 42,661 1.0
– – –
– – –
– – –
12
NUCG Predicted Error (%)
– –
50,511 49,351 2.3
35,889 36,530 1.8
– –
– –
45,892 45,881 0.0
– –
– –
– –
14
NUCG Predicted Error (%)
– – –
56,288 54,449 3.3
37,440 37,921 1.3
– – –
– – –
– – –
– – –
– – –
– – –
Table 7 Comparison between the predicted lives obtained from simple method and UCG analyses with those obtained from NUCG analyses for thick panels Patch layers
Patch
Glass/epoxy
tp (mm)
5
6.35
7.5
Graphite/epoxy 5
6.35
7.5
Boron/epoxy 5
6.35
7.5
2
NUCG Predicted Error (%)
17,986 16,720 7.0
16,823 15,910 5.4
16,071 15,020 6.5
20,455 19,410 5.1
18,848 18,758 0.5
17,799 18,210 2.3
22,100 23,500 6.3
20,173 21,360 5.9
18,916 19,050 0.7
4
NUCG Predicted Error (%)
19,164 17,723 7.5
17,818 16,709 6.2
16,937 16,086 5.0
21,948 21,118 3.8
20,153 19,662 2.4
18,967 18,714 1.3
23,785 23,060 3.0
21,597 21,340 1.2
20186 20,223 0.2
6
NUCG Predicted Error (%)
19,888 18,651 6.2
18,430 17,532 4.9
17,480 16,834 3.7
22,714 22,244 2.1
20,732 20,727 0.0
19,478 19,752 1.4
24,843 24,380 1.9
22,262 22,491 1.0
20,714 21,336 3.0
8
NUCG Predicted Error (%)
20,360 19,283 5.3
18,799 18,099 3.7
17,807 17,365 2.5
23,292 22,957 1.4
21,009 21,312 1.4
19,669 20,331 3.4
25,892 25,405 1.9
22,703 23,156 2.0
20,959 21,931 4.6
10
NUCG Predicted Error (%)
20,714 19,742 4.7
19,021 18,488 2.8
17,990 17,736 1.4
23,947 23,592 1.5
21,210 21,677 2.2
19,736 20,655 4.7
27,210 26,532 2.5
23,174 23,678 2.2
21,149 22,298 5.4
12
NUCG Predicted Error (%)
21,037 20,115 4.4
19,161 18,760 2.1
18,081 17,991 0.5
24,797 24,322 1.9
21,452 21,977 2.4
19,783 20,855 5.4
28,915 27,942 3.4
23,786 24,234 1.9
21391 22,600 5.7
14
NUCG Predicted Error (%)
21,385 20,464 4.3
19,271 18,963 1.6
18,122 18,165 0.2
25,896 25,233 2.6
21,792 22,302 2.3
19,869 21,014 5.8
31,076 29,733 4.3
24,592 24,917 1.3
21,735 22,930 5.5
16
NUCG Predicted Error (%)
21,789 20,829 4.4
19,383 19,131 1.3
18,143 18,288 0.8
27,273 26,375 3.3
22,260 22,706 2.0
20,024 21,188 5.8
33,745 31,966 5.3
25,622 25,781 0.6
22,209 23,342 5.1
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 79 (2007) 234–241 Table 8 Comparison between the crack growth lives obtained from the simple method and UCG analysis with those obtained from NUCG analyses and experiments Patch layers
t = 2.29 mm
t = 6.35 mm
4
8
16
Predicted NUCG Experiment 1 Experiment 2
34,752 34,458 33,800 34,700
41,320 41,140 38,000 39,200
60,196 63,757 68,900 65,300
Predicted NUCG Experiment 1 Experiment 2
16,709 17,818 17,800 19,800
18,099 18,799 17,500 19,200
19,131 19,383 18,100 19,800
5. Conclusions In this paper, a simple method was developed to predict the crack growth life of single-side repaired aluminum panels. The patches were considered to be made of glass/epoxy, boron/epoxy and graphite/epoxy composite materials. Crack growth life of the repaired panels were calculated using two different approaches: (1) The crack growths along the panel thickness uniformly. (2) The crack growths along the panel thickness non-uniformly (real crack-front shape analyses). It was shown that, the lives obtained from non-uniform crack growth analyses (NUCG) are close to those obtained from the experiments with the differences of less than 10%. The lives calculated using the mid-plane results in uniform crack growth analyses (UCG) are nonconservative by the order of 35–90% comparing with the experimental results. Then, a new simple method was developed to find a position along the panel thickness, so that, if the stress intensity factors of that position during the crack propagation are used, almost the same fatigue crack growth life as the one calculated in the real crackfront shape analysis is obtained. Comparison of the predicted lives using the simple method with those obtained from the NUCG analyses showed the maximum difference percentage of 4.5% in thin and 7.5% for thick panels. The maximum difference between the predicted lives using the simple method with those obtained from the available experiments is 11.1%. Using this simple method, it is not
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necessary to perform the complex crack growth analysis based on the real crack-front shape. Therefore, the results of 3D analysis using the simplified approach (UCG) or 2D analysis results similar to the three layers technique can be used to estimate the crack growth life of the repaired panels. References [1] Baker A, Jones R. Bonded repair of aircraft structures. Dordrecht, Netherlands: Martinus Nijhoff; 1988. [2] Ratwani M. Analysis of cracked adhesively bonded laminated structures. AIAA J 1979;17(4):988–94. [3] Rose LRF. A cracked plate repaired by bonded reinforcements. Int J Fract 1982;18(2):135–44. [4] Lu J, Hu Y, Ju D. An assessing method of fatigue life on centralthrough cracked plate with adhesive bounded reinforcement. Fatigue, Fract Risk, ASME 1991;215:135–40. [5] Hosseini-Toudeshky H, Shahverdi H, Daghyani HR. Fatigue life assessment of repaired panels with adhesively bounded composite plates (second Asian Australian conference on composite materials, ACCM—2000, Kyonju, Korea, 18–20 August 2000). [6] Hosseini-Toudeshky H, Sadeghi Gh, 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(3–4):401–6. [7] Hosseini-Toudeshky H, Mohammadi B, Daghyani HR. Effects of patch lay-up configuration on fracture parameters and fatigue life of single-sided repaired panels in mixed-mode conditions. In 4th Iranian Aerospace Society conference, Tehran, 23–27 January 2003. [8] Seo DC, Lee JJ. Fatigue crack growth behavior of cracked aluminum plate repaired with composite patch. Compos Struct 2002;57(1–4): 323–30. [9] Hosseini-Toudeshky H, Sadeghi Gh, Daghyani HR. Three dimensional approach to fatigue crack propagation for aluminum panels repaired with single-sided composite laminates. Fourth Asian–Australasian conference on composite materials (ACCM-4), Sydney, Australia, July 6–9, 2004. [10] Hosseini-Toudeshky H, et al. Numerical and experimental fatigue crack growth analysis in Mode-I for repaired aluminum panels using composite material. DFC-8 & ETDCM-7 conference, 3–6 April 2005, Sheffield, UK. Composite A [special issue], accepted for publication. [11] Naboulsi S, Mall S. Modeling of a cracked metallic structure with bonded composite patch using three layer technique. Compos Struct 1996;35(3):295–308. [12] Krueger R. The virtual crack closure technique: history, approach and applications. NASA/CR-2002-211628. ICASE Report no. 200210, April 2002. [13] Farahmand B, Bockrath G, Glassco J. Fatigue and fracture mechanics of high risk parts: application of LEFM and FMDM theory. International Thomson Publishing; 1997.
A simple method to calculate the crack growth life of adhesively repaired aluminum panels H. Hosseini-Toudeshky *, B. Mohammadi Aerospace Engineering Department, Amirkabir University of Technology, Hafez Avenue, 424 Tehran, Iran Available online 21 February 2006
Abstract In this paper, a simple method is developed to predict the crack growth life of single-side repaired aluminum panels. The patches are considered to be made of glass/epoxy, boron/epoxy and graphite/epoxy composite materials. For this purpose, crack growth life of the repaired panels are calculated using two different approaches: (1) The crack growths along the panel thickness uniformly. In this simplified approach, crack-front line remains perpendicular to the plate surfaces during the crack propagation. (2) The crack growths along the panel thickness non-uniformly. In this approach, initial crack-front line changes to a curved shape according to the variation of stress intensity factor during the crack propagation (real crack-front shape). Then, a new simple method is developed based on a parametric study using both approaches. Using this simple method, it is not necessary to perform the complex crack growth analysis based on the real crack-front shape. Therefore, the results of 3D analysis using the first approach or 2D analysis similar to the three layers technique may be used to estimate the crack growth life of the repaired panels. The obtained results are verified with the available experimental results. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Repair; Aluminum panels; Composite; Simple method
1. Introduction Service life of cracked components may be extended using repair technologies. Adhesively bounded composite patches have been recognized as an efficient and cost-effective method to repair cracked components in advanced aerospace structures. A double-sided repair increases considerably the fatigue life of a cracked plate, although, owing to the manufacturing difficulties, a single-side repair is normally preferred. Fig. 1 shows a typical configuration of a single-side repaired panel. One of the most challenging aspects of the bounded composite repair technology has been the stress analysis of repaired panels and the calculations of subsequent fracture parameters. The difficulty arises from the fact that a thin metallic panel under the in-plane loading may fail *
Corresponding author. Tel.: +98 21 6640 5032; fax: +98 21 6640 4885. E-mail address: [email protected] (H. Hosseini-Toudeshky).
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.01.005
under highly complicated three-dimensional stresses, if composite patches are bounded to its surface asymmetrically (single-sided repair) [1–5]. The stress variations over the thickness of a cracked plate in asymmetric repair present a greater challenge in modeling due to the existence of out-of-plane bending. This causes stress variations over the thickness of the cracked panel and therefore causes nonuniform crack propagation along the crack-front. Fig. 2 shows a typical experimental crack-front shape in fatigue crack growth of the repaired panels [6]. Furthermore, in many studies the stresses and strains fields of the mid-plane have been used to calculate the fracture parameters, however, the maximum stresses and strains occur at un-patched surface of the cracked panels [7]. Recently, a few researchers published papers considering the non-uniform crack propagation along the crackfront of the repaired panels [8,9]. Seo and Lee [8] performed finite elements analysis of the asymmetric repaired panels assuming skew crack-front model. However, the
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235
Fig. 1. Typical geometry and loading of single-side repaired panels.
Fig. 2. Typical crack-front shape of a repaired panel in fatigue crack growth.
single-sided repair under cyclic loading produces a curvilinear crack-front shape in the real applications. HosseiniToudeshky et al. [9,10] predicted the actual crack-front shape using experiments and 3D finite elements fatigue crack growth analysis of the asymmetric repaired panels in mode-I condition. They predicted the actual crack-front shape and crack growth life of the repaired panels for a certain crack growth length and for various patch thicknesses. In the present paper, three-dimensional finite elements analyses are performed for the repaired aluminum panels, containing a central crack with single-side composite patches in mode-I condition. The patches are made of glass/epoxy, boron/epoxy and graphite/epoxy. Fatigue crack growth life of the repaired panels are calculated using two different approaches: (1) The crack growths along the panel thickness uniformly (UCG). In this simplified approach, the crack-front line remains perpendicular to the plate surfaces during the crack propagation. (2) The crack growths along the panel thickness non-uniformly (NUCG). In this approach, initial crack-front line changes to a curved shape according to the variation of stress intensity factor during the crack propagation (real crack-front shape analysis). Finite elements crack growth analyses based on the NUCG modeling is a complex and time consuming task. Then, a new simple method is developed
based on a parametric study using both approaches. In this method a position along the panel thickness is purposed, so that, if the stress intensity factors of that position during the crack propagation are used, almost the same fatigue crack growth life as the one calculated in the real crackfront shape analysis is obtained. Using this simple method, it is not necessary to perform the complex crack growth analysis based on the real crack-front shape. Therefore, the results of 3D analysis based on the first simplified approach or 2D analysis results similar to the three layers technique [11] can be used to estimate the crack growth life of the repaired panels. The obtained fatigue crack growth lives from the real crack-front shape analyses and the simple method are compared with those obtained from the experiments in some cases. 2. Fatigue and fracture analysis Crack closure technique may be used to calculate the fracture parameters and fatigue crack growth life of the repaired panels. The crack closure technique used here is based on Irwin’s notation of reversing crack growth to compute the energy release rate (G). Modified crack closure technique (MCCT) is also used to reduce the computational efforts. It is applicable for linear elastic problems only and orthogonal mesh is required to model the crackfront in the finite elements analysis. Equivalent nodal forces and their conjugate nodal displacements in elements around the crack-front computed by FEM are used in calculation of G. Nodal displacement are primary variables in the FEM, and equivalent nodal forces will be computed more accurate than the crack-front stresses [12]. For 3D problems the mode-I strain energy release rate are calculated using the following expression [12] GI ¼
1 Z Li ðwLi wLi Þ 2DA
ð1Þ
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in which DA is the area of element, ZLi and wLi are nodal forces and displacements, respectively. Strain energy release rate and stress intensity factor are related by [13] pffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ K I ¼ GI =E in which E* = E for plane stress and E* = E/(1 t2) for plane strain conditions. Having the stress intensity factor, the Paris equation is used to relate the crack growth rate to the stress intensity factor [13] da ¼ CDK n dN
ð3Þ
where C and n are empirical material constants, DK = Kmax Kmin is the stress intensity factor range in fatigue loading, N is number of cycles, and da is crack extension length. 3. Finite elements analysis Typical geometries of the aluminum panels, adhesive layer and composite patch are shown in Fig. 1. The panels are made of 2024-T3 aluminum alloy with various thicknesses containing an initial crack of 2a = 10 mm length. Specimens’ dimensions followed the ASTM E-647 standard. Fig. 3 shows a typical finite elements mesh of the repaired panels. A wide range of three-dimensional finite element crack growth analyses are performed for the repaired panels considering both the uniform (crack-front perpendicular to the panel surfaces) and non-uniform (real crack-front shape) crack growth assumptions. Fig. 4 shows the typical crack-front configuration in both approaches. The analyses are carried out for panels with three different patch materials, viz. boron/epoxy, graphite/epoxy and glass/epoxy with various thicknesses. The thickness of patch in each model is assumed to have the nominal stiffness ratio of less than 1.5 according to the plate thickness and does not exceed the plate thickness. The stiffness ratio, S, is defined as S¼
ðEr tr Þ ðEp tp Þ
ð4Þ
Fig. 4. Crack-front configuration of the repaired panels. (a) Schematic of the real fracture surface, (b) crack-front at uniform crack growth, (c) crack-front at non-uniform crack growth.
where E is Young’s modulus, t is thickness, and ‘r’ and ‘p’ are subscript designating the repair patch and plate, respectively. It will be shown that the fatigue crack growth lives of the repaired thick panels are not significantly increased for the patch thicknesses of more than eight layers later in this paper. Therefore the maximum patch layers of 16 layers are considered for them. The used adhesive material is FM77. The applied remote stress is 118 MPa in all repaired panels. Tables 1 and 2 show the dimensions and material properties of the aluminum cracked panels, adhesive layers, and composite patches. Unidirectional lay-up perpendicular to the crack length is used for the patches in all models. The stress and strain fields of the repaired panels are obtained using the ANSYS finite elements
Fig. 3. Typical finite elements mesh of the repaired panels.
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 79 (2007) 234–241 Table 1 Dimensions of the panels, adhesive layers and patches
L (mm) W (mm) t (mm)
Aluminum panel
Adhesive
Patch
100 50 1.5, 2.29, 3.5, 5, 6.35, 7.5
40 35 0.1
40 35 0.18 per layer
Table 2 Material properties
E11 (GPa) E22 (GPa) E33 (GPa) G12 (GPa) G13 (GPa) G23 (GPa) m12 m13 m23
Glass/epoxy [10]
Graphite/epoxy [8]
Boron/epoxy [1]
50 14.5 14.5 2.56 2.56 2.24 0.33 0.33 0.33
134 10.3 10.3 5.5 5.5 3.2 0.33 0.33 0.53
208 25.4 25.4 7.2 7.2 4.9 0.1677 0.1677 0.035
program. In the three-dimensional analysis, an isotropic 8-node-solid element is used to model the aluminum panel and adhesive layer. Furthermore, a layered 8-nodesolid element is employed to model the laminated composite patch. Two elements in the thickness of adhesive and patch and 10 elements in the thickness of plate are used and a fine mesh is generated for the regions close to the crack-front. The material constants in Paris equation were calculated based on the ASTM E-647 method and the required experimental data were obtained from the tests of un-repaired panels containing a central crack [6,10]. The obtained constants m and c were 3.2828 and 3.63 E-13, respectively for thin panels and 4.224 and 1.51 E-15 for thick panels. In the crack growth analyses based on both uniform crack growth rate and non-uniform crack growth (see Fig. 4), it is assumed that the crack growths only in the aluminum panel with no failure and debounding on the composite patch. This assumption is mainly based on our experimental evidence on fatigue crack growth of repaired panels containing a crack in mode-I condition. A Macro program is developed using ANSYS Parametric Design Language (APDL) to handle the crack growth modelling procedure for any cracked panel geometry and patch configuration. In this procedure, a dynamic mesh generation using automesh capability of the code is used to generate automatic mesh of the repaired panels for each crack growth step. The used procedures to find the crack growth rates in both uniform and non-uniform crack growth modeling are explained in the following sections. 3.1. Uniform crack growth modeling In these analyses it is assumed that the crack-front remains perpendicular to the panel’s surfaces during the
237
crack propagation. Performing linear elastic solution for the repaired panels with an existed pre-crack the stresses and displacements at the crack-front are obtained. Then, the GI value for the presented crack configuration is computed in each step using the modified crack closure technique presented by Eq. (1). Then using Eq. (2), the values of KI along the crack-front are calculated. Using the obtained value of KI at un-patched surface and Eq. (3), crack growth length for a certain number of load cycles is calculated or vice versa. The equations are solved in each step and then the geometry is updated for the repaired panel with the new crack length. The above procedure is performed for the new configuration and repeated for several steps to find the crack growth rate of the repaired panels. 3.2. Non-uniform crack growth modeling In these analyses the real crack-front shape is obtained using the variation of stress intensity factor along the crack-front at each crack propagation step. The major steps of the developed Macro program to handle the crack growth modeling procedure as well as to find the crack growth life of the cracked panel are as follows: (i) Geometry and finite element mesh generation of the problem. (ii) Defining the loading, constraints, and material properties. (iii) Performing the linear elastic solution. (iv) Calculation of GI and KI at the crack-front using the modified crack closure technique. (v) Considering 0.1 mm crack length increment at unpatched surface for each step and calculating the loads cycles (DN) for this increment using the Paris Law. (vi) Having DN and the values of KI at crack-front nodes the ratio of crack growth at each node with respect to the 0.1 mm growth at un-patched surface are calculated and the new crack-front shape is generated. (vii) Updating the finite element model based on the new crack length and crack-front configuration. (viii) If KI 6 KIC or crack length at un-patched surface is less than a defined value, return to step (iii). (ix) Process the results and stop the solution.
4. Results and discussion Table 3 compare the lives obtained from the experiments with those obtained from the finite elements analyses using mid-plane and un-patched surface results in uniform crack growth (UCG) and non-uniform crack growth (NUCG) modeling for two panels thicknesses of 2.29 mm and 6.35 mm with various 4, 8, and 16 layers glass/epoxy patches. This table shows that the lives obtained from
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Finite elements crack growth analyses using the NUCG modeling is a complex and quite time consuming task. Therefore, a simple method is developed to use the UCG modeling and predict the reasonable crack growth life of the single-side repaired panels. In this method a position along the panel thickness, Ze, is purposed, so that, if the stress intensity factors of that position in the UCG modeling are used, almost the same fatigue crack growth life as the one calculated in the NUCG modeling is obtained. Tables 4 and 5 show the lives obtained from the NUCG analyses and the non-dimensional coordinate of Ze/tp on the crack-front line in the UCG analyses of the repaired panels for thin and thick panels respectively. It is noted that, if the ratio of the thickness over the wide of the panel is less than 0.05 (tp/(2W) < (1/20)), they are categorized as thin panels. It is also mentioned that, Ze is measured from the un-patched surface of the panels as indicated in Figs. 3 and 4 and tp is the plate thickness. Ze/tp is the point that if the KI values of that point in UCG analyses are used, the same life as the NUCG analyses of the repaired panel is obtained. Tables 4 and 5 show that there are not large scattered between the values of Ze/tp obtained for panels with various number of patch layers for each patch composite material. Therefore the average values of Ze/tp for each patch composite material and for both thin and thick panels are calculated and depicted in Fig. 5 against the ratio of Young’s modulus of elasticity of aluminum panel over the patch material. These data presented two well linear curves for thin and thick repaired panels. The equations fitted on these points using least square method are as follows:
Table 3 Comparison between experimental and uniform and non-uniform crackfront finite elements modeling lives of the repaired panels with glass/epoxy patches No. of layer
t = 2.29 mm
t = 6.35 mm
4
8
16
Mid-plane results (UCG) Un-patched surface results (UCG) NUCG Experiment 1 Experiment 2
54,010 16,369
68,724 21,696
88,148 31,307
34,458 33,800 34,700
41,140 38,000 39,200
63,757 68,900 65,300
Mid-plane results Un-patched surface results NUCG Experiment 1 Experiment 2
25,583 9863 17,818 17,800 19,800
30,752 9396 18,799 17,500 19,200
35,569 8930 19,383 18,100 19,800
UCG modeling using un-patched surface results are too conservative and the lives calculated using the mid-plane results are non-conservative by the order of 35–90% comparing with the experimental results. The lives obtained from NUCG modeling are close to those obtained from the experiments with the differences of less than 10%. The results presented in this table show that the lives obtained from NUCG modeling are reliable with an acceptable difference from experimental results. But the lives obtained from UCG modeling using mid-plane stress intensity factors are too non-conservative; therefore, they cannot be used for design and life extension purpose of the repaired panels.
Table 4 NUCG lives and the values of Ze/tp for thin plates with various patch layers and composite material Patch layers
Patch
Glass/epoxy
tp (mm)
1.5
2.29
3.5
1.5
2.29
3.5
1.5
2.29
3.5
1
NUCG Ze/tp
33,210 0.359
– –
– –
37,850 0.357
– –
– –
42,100 0.349
– –
– –
2
NUCG Ze/tp
35,350 0.362
31,575 0.365
29,210 0.387
43,120 0.352
36,314 0.350
32,880 0.349
49,850 0.358
39,630 0.345
35,315 0.340
4
NUCG Ze/tp
41,507 0.370
34,458 0.366
30,979 0.368
56,550 0.365
41,110 0.354
35,127 0.348
71,239 0.365
46,976 0.351
38,187 0.342
6
NUCG Ze/tp
51,074 0.377
37,520 0.367
32,240 0.363
– –
47,763 0.358
37,049 0.347
– –
57,953 0.357
41,261 0.344
8
NUCG Ze/tp
63,270 0.385
41,140 0.370
33,361 0.362
–
56,822 0.364
39,323 0.348
– –
– –
45,201 0.346
10
NUCG Ze/tp
– –
45,461 0.373
34,544 0.362
–
–
42,236 0.351
– –
– –
– –
12
NUCG Ze/tp
–
50,511 0.377
35,889 0.363
–
–
45,892 0.354
–
–
–
14
NUCG Ze/tp
–
56,288 0.382
37,440 0.365
– –
– –
– –
– –
– –
– –
,
Graphite/epoxy
Boron/epoxy
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Table 5 NUCG lives and the values of Ze/tp for thick plates with various patch layers and composite material Patch layers
Patch
Glass/epoxy
Graphite/epoxy
Boron/epoxy
tp (mm)
5
6.35
7.5
5
6.35
7.5
5
6.35
7.5
2
NUCG Ze/tp
17,986 0.375
16,823 0.375
16,071 0.372
20,455 0.344
18,848 0.339
17,799 0.342
22,100 0.333
20,173 0.333
18,916 0.326
4
NUCG Ze/tp
19,164 0.362
17,818 0.361
16,937 0.358
21,948 0.338
20,153 0.336
18,967 0.333
23,785 0.331
21,597 0.337
20,186 0.324
6
NUCG Ze/tp
19,888 0.355
18,430 0.352
17,480 0.350
22,714 0.334
20,732 0.329
19,478 0.325
24,843 0.328
22,262 0.322
20,714 0.318
8
NUCG Ze/tp
20,360 0.350
18,799 0.347
17,807 0.344
23,292 0.332
21,009 0.326
19,669 0.321
25,892 0.328
22,703 0.320
20,959 0.315
10
NUCG Ze/tp
20,714 0.348
19,021 0.343
17,990 0.340
23,947 0.332
21,210 0.324
19,736 0.319
27,210 0.329
23,174 0.320
21,149 0.314
12
NUCG Ze/tp
21,037 0.347
19,161 0.341
18,081 0.337
24,797 0.333
21,452 0.324
19,783 0.318
28,915 0.331
23,786 0.320
21,391 0.313
14
NUCG Ze/tp
21,385 0.346
19,271 0.339
18,122 0.334
25,896 0.335
21,792 0.324
19,869 0.317
31,076 0.333
24,592 0.321
21,735 0.314
16
NUCG Ze/tp
21,789 0.347
19,383 0.338
18,143 0.333
27,273 0.336
22,260 0.325
20,024 0.317
33,745 0.335
25,622 0.323
22,209 0.314
0.375 0.370 0.365
(Ze /tp) = 0.0166(EAl /Epatch ) + 0.3453
(Ze/tp)average
0.360 0.355 0.350 0.345 0.340 0.335
Thin Plates Thick Plates
0.330
(Ze /tp) = 0.0235(EAl /Epatch ) + 0.3159
0.325 0.320 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
EAl /EPatch Fig. 5. Variation of the average values of Ze/tp against the ratio of Young’s modulus of elasticity of aluminum panel over the patch material.
Ze EAl ¼ 0:0166 þ 0:3453 tp EPatch Ze EAl ¼ 0:0235 þ 0:3159 tp EPatch
for thin panels
ð5Þ
for thick panels
ð6Þ
Having the Young’s modulus of elasticity of both panel and patch materials and the thickness of the panel and using the above equations, the required position at the crack-front, Ze, is obtained to use in the simple method. Tables 6 and 7 compare the predicted lives obtained from the UCG analyses and using the values of Ze calcu-
lated from the simple method (Eqs. (5) and (6)) with those obtained from the NUCG analyses for thin and thick panels respectively. Comparison of the predicted lives with those obtained from the NUCG analyses shows the maximum difference percentage of 4.5% in thin panels and 7.5% for thick panels. The maximum difference belongs to the panel with 5.0 mm thickness and 4 layers glass/epoxy patch. It is noted that using the midplane results of the UCG analysis of the same repaired panel leads to the non-conservative life prediction with 43.6% difference from the life obtained from NUCG analysis. To verify the predicted repaired panels’ lives obtained from the simple method, they are compared with those obtained from the NUCG analyses and experimental results performed in [10]. Table 8 shows that the maximum difference between the predicted lives using the simple method (UCG analyses) with those obtained from the NUCG analyses is about 6.2% which belongs to the panel thickness of 6.35 mm with 4 layers patch. The maximum difference between the predicted life using the simple method with those obtained from the experiments is about 11.1% which also belongs to the panel thickness of 6.35 mm with 4 layers patch. It is noted that the difference percentage between the life obtained from the NUCG analysis and experiment for the same case is about 5.3%. The presented results in Tables 6–8 shows that the purposed location of Ze in the simple method can be used to calculate the stress intensity factor from the UCG analysis and predict the fatigue crack growth life of the single-side repaired panels with an acceptable engineering accuracy in the obtained results.
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Table 6 Comparison between the predicted lives obtained from simple method and UCG analyses with those obtained from NUCG analyses for thin panels Patch layers
Patch
Glass/epoxy
Graphite/epoxy
Boron/epoxy
tp (mm)
1.5
2.29
3.5
1.5
2.29
3.5
1.5
2.29
3.5
1
NUCG Predicted Error (%)
33,210 34,135 2.8
– – –
– – –
37,850 38,750 2.4
– – –
– – –
42,100 42,750 1.5
– – –
– – –
2
NUCG Predicted Error (%)
35,350 35,950 1.7
31,575 31,950 1.2
29,210 28,300 3.1
43,120 42,930 0.4
36,314 35,950 1.0
32,880 31,450 4.3
49,850 49,032 1.6
39,630 41,215 4.0
35,315 36,350 2.9
4
NUCG Predicted Error (%)
41,507 41,381 0.3
34,458 34,752 0.9
30,979 31,069 0.3
56,550 54,546 3.5
41,110 41,173 0.2
35,127 35,841 2.0
71,239 68,027 4.5
46,976 47,050 0.2
38,187 39,453 3.3
6
NUCG Predicted Error (%)
51,074 49,915 2.3
37,520 37,776 0.7
32,240 32,756 1.6
– – –
47763 47,098 1.4
37,049 37,969 2.5
– – –
57,953 56,766 2.0
41,261 42,050 1.9
8
NUCG Predicted Error (%)
63,270 60,663 4.1
41,140 41,320 0.4
33,361 34,067 2.1
– – –
56,822 55,046 3.1
39,323 40,085 1.9
– – –
– – –
45,201 46,032 1.8
10
NUCG Predicted Error (%)
– –
45,461 44,900 1.2
34,544 35,273 2.1
– –
– –
42,236 42,661 1.0
– – –
– – –
– – –
12
NUCG Predicted Error (%)
– –
50,511 49,351 2.3
35,889 36,530 1.8
– –
– –
45,892 45,881 0.0
– –
– –
– –
14
NUCG Predicted Error (%)
– – –
56,288 54,449 3.3
37,440 37,921 1.3
– – –
– – –
– – –
– – –
– – –
– – –
Table 7 Comparison between the predicted lives obtained from simple method and UCG analyses with those obtained from NUCG analyses for thick panels Patch layers
Patch
Glass/epoxy
tp (mm)
5
6.35
7.5
Graphite/epoxy 5
6.35
7.5
Boron/epoxy 5
6.35
7.5
2
NUCG Predicted Error (%)
17,986 16,720 7.0
16,823 15,910 5.4
16,071 15,020 6.5
20,455 19,410 5.1
18,848 18,758 0.5
17,799 18,210 2.3
22,100 23,500 6.3
20,173 21,360 5.9
18,916 19,050 0.7
4
NUCG Predicted Error (%)
19,164 17,723 7.5
17,818 16,709 6.2
16,937 16,086 5.0
21,948 21,118 3.8
20,153 19,662 2.4
18,967 18,714 1.3
23,785 23,060 3.0
21,597 21,340 1.2
20186 20,223 0.2
6
NUCG Predicted Error (%)
19,888 18,651 6.2
18,430 17,532 4.9
17,480 16,834 3.7
22,714 22,244 2.1
20,732 20,727 0.0
19,478 19,752 1.4
24,843 24,380 1.9
22,262 22,491 1.0
20,714 21,336 3.0
8
NUCG Predicted Error (%)
20,360 19,283 5.3
18,799 18,099 3.7
17,807 17,365 2.5
23,292 22,957 1.4
21,009 21,312 1.4
19,669 20,331 3.4
25,892 25,405 1.9
22,703 23,156 2.0
20,959 21,931 4.6
10
NUCG Predicted Error (%)
20,714 19,742 4.7
19,021 18,488 2.8
17,990 17,736 1.4
23,947 23,592 1.5
21,210 21,677 2.2
19,736 20,655 4.7
27,210 26,532 2.5
23,174 23,678 2.2
21,149 22,298 5.4
12
NUCG Predicted Error (%)
21,037 20,115 4.4
19,161 18,760 2.1
18,081 17,991 0.5
24,797 24,322 1.9
21,452 21,977 2.4
19,783 20,855 5.4
28,915 27,942 3.4
23,786 24,234 1.9
21391 22,600 5.7
14
NUCG Predicted Error (%)
21,385 20,464 4.3
19,271 18,963 1.6
18,122 18,165 0.2
25,896 25,233 2.6
21,792 22,302 2.3
19,869 21,014 5.8
31,076 29,733 4.3
24,592 24,917 1.3
21,735 22,930 5.5
16
NUCG Predicted Error (%)
21,789 20,829 4.4
19,383 19,131 1.3
18,143 18,288 0.8
27,273 26,375 3.3
22,260 22,706 2.0
20,024 21,188 5.8
33,745 31,966 5.3
25,622 25,781 0.6
22,209 23,342 5.1
H. Hosseini-Toudeshky, B. Mohammadi / Composite Structures 79 (2007) 234–241 Table 8 Comparison between the crack growth lives obtained from the simple method and UCG analysis with those obtained from NUCG analyses and experiments Patch layers
t = 2.29 mm
t = 6.35 mm
4
8
16
Predicted NUCG Experiment 1 Experiment 2
34,752 34,458 33,800 34,700
41,320 41,140 38,000 39,200
60,196 63,757 68,900 65,300
Predicted NUCG Experiment 1 Experiment 2
16,709 17,818 17,800 19,800
18,099 18,799 17,500 19,200
19,131 19,383 18,100 19,800
5. Conclusions In this paper, a simple method was developed to predict the crack growth life of single-side repaired aluminum panels. The patches were considered to be made of glass/epoxy, boron/epoxy and graphite/epoxy composite materials. Crack growth life of the repaired panels were calculated using two different approaches: (1) The crack growths along the panel thickness uniformly. (2) The crack growths along the panel thickness non-uniformly (real crack-front shape analyses). It was shown that, the lives obtained from non-uniform crack growth analyses (NUCG) are close to those obtained from the experiments with the differences of less than 10%. The lives calculated using the mid-plane results in uniform crack growth analyses (UCG) are nonconservative by the order of 35–90% comparing with the experimental results. Then, a new simple method was developed to find a position along the panel thickness, so that, if the stress intensity factors of that position during the crack propagation are used, almost the same fatigue crack growth life as the one calculated in the real crackfront shape analysis is obtained. Comparison of the predicted lives using the simple method with those obtained from the NUCG analyses showed the maximum difference percentage of 4.5% in thin and 7.5% for thick panels. The maximum difference between the predicted lives using the simple method with those obtained from the available experiments is 11.1%. Using this simple method, it is not
241
necessary to perform the complex crack growth analysis based on the real crack-front shape. Therefore, the results of 3D analysis using the simplified approach (UCG) or 2D analysis results similar to the three layers technique can be used to estimate the crack growth life of the repaired panels. References [1] Baker A, Jones R. Bonded repair of aircraft structures. Dordrecht, Netherlands: Martinus Nijhoff; 1988. [2] Ratwani M. Analysis of cracked adhesively bonded laminated structures. AIAA J 1979;17(4):988–94. [3] Rose LRF. A cracked plate repaired by bonded reinforcements. Int J Fract 1982;18(2):135–44. [4] Lu J, Hu Y, Ju D. An assessing method of fatigue life on centralthrough cracked plate with adhesive bounded reinforcement. Fatigue, Fract Risk, ASME 1991;215:135–40. [5] Hosseini-Toudeshky H, Shahverdi H, Daghyani HR. Fatigue life assessment of repaired panels with adhesively bounded composite plates (second Asian Australian conference on composite materials, ACCM—2000, Kyonju, Korea, 18–20 August 2000). [6] Hosseini-Toudeshky H, Sadeghi Gh, 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(3–4):401–6. [7] Hosseini-Toudeshky H, Mohammadi B, Daghyani HR. Effects of patch lay-up configuration on fracture parameters and fatigue life of single-sided repaired panels in mixed-mode conditions. In 4th Iranian Aerospace Society conference, Tehran, 23–27 January 2003. [8] Seo DC, Lee JJ. Fatigue crack growth behavior of cracked aluminum plate repaired with composite patch. Compos Struct 2002;57(1–4): 323–30. [9] Hosseini-Toudeshky H, Sadeghi Gh, Daghyani HR. Three dimensional approach to fatigue crack propagation for aluminum panels repaired with single-sided composite laminates. Fourth Asian–Australasian conference on composite materials (ACCM-4), Sydney, Australia, July 6–9, 2004. [10] Hosseini-Toudeshky H, et al. Numerical and experimental fatigue crack growth analysis in Mode-I for repaired aluminum panels using composite material. DFC-8 & ETDCM-7 conference, 3–6 April 2005, Sheffield, UK. Composite A [special issue], accepted for publication. [11] Naboulsi S, Mall S. Modeling of a cracked metallic structure with bonded composite patch using three layer technique. Compos Struct 1996;35(3):295–308. [12] Krueger R. The virtual crack closure technique: history, approach and applications. NASA/CR-2002-211628. ICASE Report no. 200210, April 2002. [13] Farahmand B, Bockrath G, Glassco J. Fatigue and fracture mechanics of high risk parts: application of LEFM and FMDM theory. International Thomson Publishing; 1997.