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Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching
Composite Structures 78 (2007) 602–609 www.elsevier.com/locate/compstruct
Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching M.R. Ayatollahi *, R. Hashemi Fatigue and Fracture Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran Available online 28 December 2005
Abstract To increase the operational life of defected structures, a repairing method using composite patches has been used to reinforce cracked components. Due to various advantages of composite materials, this method has received much attention from researchers and engineers. Considerable investigations have been performed to highlight the effect of bonded composite patches on the fracture parameters such as stress intensity factors (SIF) and J-integral. However the effect of composite patches on the T-stress, the constant stress term acting parallel to the crack, has not been investigated in the past. In this paper, the finite element method is carried out to analyze the effect of bonded composite patches for repairing cracks in pure mode I and also mixed mode I/II conditions, by computing the stress intensity factors and the T-stress, as functions of the crack length, the crack inclination angle and the type of composite material. In pure mode I condition, the finite element analysis is carried out for three different specimens: centre crack, double edge crack and single edge crack specimens. For mixed mode I/II condition the analysis is conducted on an inclined central crack of various slant angles. For both pure mode I and mixed mode I/II, the numerical results show that composite patching has considerable effect on the T-stress. 2005 Elsevier Ltd. All rights reserved. Keywords: Single composite patch; Crack; Stress intensity factor; T-stress; Pure mode I; Mixed mode I/II
1. Introduction Service life enhancement of damaged structures is the main subject of many studies in recent decades, so that various investigations have been carried to increase the durability and damage tolerance of cracked metallic structures, efficiently and economically. A repair method using a composite patch to reinforce the cracked structure has been shown to be very promising owing to the light weight, high stiffness and strength of composites [1–3]. In early 1970s, Baker and Jones [4], as the pioneers of bonded patch repairing method, performed intensive research studies on this method. They elaborated many advantages of employing composite material patches for the bonded repair of cracked and damaged metallic structures. Numerous models have also been developed for analysis of repairs *
Corresponding author. Tel.: +98 21 73912922; fax: +98 21 7454050. E-mail address: [email protected] (M.R. Ayatollahi).
0263-8223/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.11.024
using various calculation techniques including the collocation method, boundary element method and finite element method. Due to its well established role in fracture mechanics, the stress intensity factor is an important measure for analyzing the performance of bonded patch in repairing cracks. The finite element method (FEM) can be used with a great accuracy to evaluate stress intensity factors and other fracture parameters at the crack tip. Finite element analysis of composite reinforcement by Mitchell et al. [5] appears to be the first thorough attempt towards analytical understanding of this class of problems. Later the finite element method was used by several other authors, among them Jones and Callinan [6], Ting et al. [7], Bachir et al. [8] and Turaga and Ripudaman [9]. In these studies, the effect of composite patch only on the values of stress intensity factors and occasionally on the J-integral was investigated. Meanwhile, the effect of composite patching on the T-stress has not been studied in the past. As described later, the T-stress is the second term in the Williams series
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solution for elastic stresses around the crack tip. It is independent of the distance from the crack tip. The conducted studies in fracture mechanics show that the T-stress has an important role in analyzing the strength and stability of brittle fracture in both linear elastic and elastic–plastic conditions. Hence more knowledge of the T-stress is important for investigating brittle fracture in cracked structures repaired by composite patches. The scope of this paper covers the investigation of composite reinforcement effects on both the stress intensity factors and the T-stress for several cracked specimens subjected to pure mode I and mixed mode I/II loading conditions. The specimens are reinforced by single sided composite patches. For this purpose the three-dimensional finite element model of each repaired sheet is provided in the code Abaqus V5.8 [10] and a unidirectional load is applied to the specimen. As recommended by Turaga and Ripudaman [9], for reducing the error in this type of problems, the geometry nonlinearity is considered in our analysis. Furthermore, two different types of materials (Gr/E and Br/E) are defined for the composite patch and its effects on the values of KI, KII and T-stress are investigated. 2. T-stress and its importance For linear elastic materials, the stress state near the crack tip can be determined from an asymptotic series solution. This method was developed originally by Williams [11] in the conventional crack tip coordinates r and h. He showed that the stress field in an isotropic elastic material containing a crack can be expressed as an pinfinite power ffiffi series, where the leading term exhibits a 1= r singularity, the second term pffiffi is independent of r, the third term is proportional to r, and so on. The classical theory of fracture mechanic normally neglects all except the singular term which results in a single-parameter description of the near-tip fields. Although the third and higher order terms in William’s solution, vanish very near the crack tip, the second term remains finite. For a crack in an isotropic elastic material subjected to plane strain mixed mode I/II loading, the first two terms of William’s solution are: 2 3 T 0 0 1 6 7 rij ¼ pffiffiffiffiffiffiffi K I fij ðhÞ þ K II gij ðhÞ þ 4 0 0 0 5 ð1Þ 2pr 0 0 mT where KI and KII are the mode I and II stress intensity factors respectively and T is a uniform stress in x direction (rzz is mT for plane strain). It can be shown that there are cases where the value of T is large relative to the singular term in Eq. (1). This occurs for example, for a centre crack, when the loading direction is parallel to the crack. In this case the singular term vanishes and it is only T, which describes the stress field in the body. To normalize the effect of T relative to the effective stress intensity factor, Leevers and Radon [12] proposed a dimensionless parameter called the biaxiality ratio B:
pffiffiffiffiffiffi T pa B ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2I þ K 2II
603
ð2Þ
where a is the crack length, for edge cracks and semi-crack length for internal cracks. The T-stress has an important role in strength and stability analysis of cracks in both linear elastic and elastic– plastic fracture mechanics. The importance of T-stress in brittle fracture for linear elastic materials and for mixed mode loading is emphasized by Smith et al. [13]. The Tstress is shown to have an influence on brittle fracture, so that the combination of singular term (characterized by KI or/and KII) and non-singular T-stress controls the brittle fracture processes in pure mode I or mixed mode I/II conditions. Indeed, the generalized maximum tensile stress (GMTS) criterion is formulated in terms of the mode I and II stress intensity factors and the T-stress [13]. In elastic–plastic problems it is now known that the T-stress can have a significant effect on the shape of plastic zone and the stresses inside the plastic zone. Betegon and Hancock [14] showed that T can be used as a second parameter in conjunction with J-integral to characterize fully the crack tip field in contained yielding problems. The T-stress is also a major factor for predicting the stability of fracture trajectory. For example it has been observed that when the Tstress is negative, the crack tends to grow along its initial plane and when the T-stress is positive, the crack deviates from original plane of growth [15,16]. Hence due to the important role of T-stress in crack behavior, it is very useful to determine T in defected structures that are reinforced by composite patching. There are various methods for calculating T from finite element analysis. Some of these methods were reviewed by Ayatollahi et al. [17] and Sherry et al. [18]. In this study, a stress-based method is used to calculate the T-stress for our three dimensional cracked specimens. The method is an extension to the technique used by Ayatollahi et al. [17] for two-dimensional crack problems. For using this method, the parallel-to-the-crack component of stress in Eq. (1) is rewritten to give the following equation for mixed mode loading: K I ðzÞ h h h rxx ðr; h; zÞ ¼ pffiffiffiffiffiffiffi cos 1 sin sin 3 2 2 2 2pr K II ðzÞ h h h þ pffiffiffiffiffiffiffi sin 2 cos cos 3 2 2 2 2pr þ T ðzÞ ð3Þ where r, h, x and y are coordinates in the conventional polar and Cartesian systems with origin at the crack tip and z is the third coordinate (parallel to the crack front) in both systems. According to the stress-method, for pure mode I, the T-stress can be determined along either of the crack faces (h = +p or p) where singular term of rxx vanishes, i.e.: T ¼ rxx
ð4Þ
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and for mixed mode I/II, T can be calculated from:
1 T ¼ ðrxx Þh¼p þ ðrxx Þh¼p 2
ð5Þ
provided the stress rxx is taken from nodal points close to the crack front where the contribution of higher order stress terms are negligible. The validity of the results obtained from this method has already been verified [17].
Table 1 Geometric dimensions of CC, DEC and SEC specimens (all dimensions are in mm) Crack type
Wp
Hp
tp
Wr
Hr
tr
ta
Centre Double-edge Single-edge
120 120 240
120 120 240
3 3 3
180 180 180
90 90 90
1 1 1
0.2 0.2 0.2
p: cracked plate; r: reinforcement; a: adhesive.
3. Crack specimens To investigate the effect of composite patching on the crack tip parameters, the values of KI, KII and T, were calculated via a set of 3D finite element analyses. For pure mode I, three different cracked specimens i.e. (1) the centre crack, (2) the single-edge crack and (3) the double-edge crack specimens made of an aluminum alloy were simulated (see Fig. 1a). The composite patches are connected to the cracked specimens through a film-adhesive. The dimensions of plates, patches and film-adhesives are given in Table 1. In order to determine the stress intensity factor and T-stress as a function of crack length for pure mode I, the analysis was performed for various crack lengths of 0.1 < a/Wp < 0.6 where Wp is the plate width. Also due to symmetry in geometry and loading conditions, only one quarter of the centre crack and double edge crack specimens and one half of the single edge crack specimen were simulated in our study. For mixed mode loading, the previ-
ous centre crack specimen of various inclination angle was used for simulation. The necessary data for geometry of plate and patch are given in Table 1. The fiber composite Graphite/Epoxy (Gr/E) and Boron/Epoxy (Br/E) have been frequently used in the past for patching or reinforcing the cracked plates. Therefore, both of these composite patches were considered for finite
Table 2 Material properties Material
E1
Aluminum Film adhesive Boron/epoxy Graphite/epoxy
72 0.97 208 172.4
E2, E3
m12, m13
m23
G12, G13
G23
25.4 10.34
0.33 0.32 0.17 0.3
0.04 0.18
7.24 4.82
4.94 3.1
E, G values are in GPa. Direction: 1-normal to crack (fiber), 2-along crack, 3-thickness.
Fig. 1a. Various configurations of cracked specimens under pure mode I.
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605
remote stress of r = 70 MPa taken arbitrarily as a reference value. 4. Finite element modeling
Fig. 1b. Configurations of centre cracked specimen under mixed mode I/ II.
element modeling and the effects of patch type on the values of crack parameters were investigated separately. The material properties for aluminum plate, Graphite/Epoxy, Boron/Epoxy and adhesive film are given in Table 2. The composite patch was modeled using eight layers of unidirectional ply that are oriented in the loading direction. As shown in Fig. 1, the cracked specimens are subjected to a uniaxial tensile load giving a uniform
Fig. 2, shows a typical three-dimensional finite element model used in our analysis. The 20-node iso-parametric brick elements were used in the commercial code Abaqus V.5.8 for simulation. The quarter point crack tip singular elements were considered for the crack tip region. The repairing patch, adhesive film and the cracked plate were modeled with one, one and four layers of elements respectively. It is well known that the single sided repair causes out-of-plane bending due to a shift in the neutral axis of the plate under mechanical loading. This out-of-plane bending could lead to large deflections of the repaired crack, therefore geometrically nonlinear analyses were performed to obtain more accurate results. The crack parameters KI, KII and T were determined from stresses around the crack tip. Since the maximum crack opening displacement occurs on the free side of single patched plates, the stress intensity factor on this side is maximum. Therefore the crack parameters have been calculated at the free side of single patched specimens. To validate the method used for calculating the crack tip parameters, first the finite element analysis was performed for un-patched centre cracked specimens of different crack length ratio a/Wp under pure mode I condition. Table 3 displays a comparison between the finite element results for KI and T-stress and theoretical results given by Sih [19] and Sherry et al. [18]. It is seen that the present results are in good agreement with previous theoretical results.
Fig. 2. Typical mesh pattern for a quarter of centre crack specimen: (a) 3-D finite element method model; (b) top view of elements around the crack tip.
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50 45 40
100
35 30 25 20 15
80 60 40
10
20
5 0 0.0
Without patch With patch (Boron/E) With patch (Graphite/E)
120
KI (MPa m1/2)
KI (MPa m1/2)
140
Without patch With patch (Boron/E) With patch (Graphite/E)
0 0.1
0.2
0.3
0.4
0.5
0.6
0
0.7
0.1
0.2
0.3
a/W p
0.4
0.5
0.6
0.7
a/WP
Fig. 3a. Variation of KI versus the crack length for patched and unpatched centre cracked specimen.
Fig. 3c. Variation of KI versus the crack length for patched and unpatched single-edge cracked specimen.
5.1. Pure mode I 40
Without patch With patch (Boron/E) With patch (Graphite/E)
35
K I (MPa m1/2)
30 25 20 15 10 5 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/W p Fig. 3b. Variation of KI versus the crack length for patched and unpatched double-edge cracked specimens.
5. Results and discussion In this section, the finite element results obtained for the crack tip parameters in the patched specimens for pure mode I and mixed mode I/II are presented and discussed for each specimen individually.
Fig. 3(a–c) shows the variations of mode I stress intensity factor (KI) as a function of crack length and patch material properties (Gr/E and Br/E), for centre crack (CC), double edge crack (DEC) and single edge crack (SEC) specimens, respectively. It can be seen from these figures that the patching reinforcement, considerably decreases the stress intensity factor (SIF). This effect is because the patch, in various reinforced configurations, carries the load in the crack region. As the crack length increases, asymptotic behavior is noted for all cracked configurations. This tendency is in agreement with the results reported in earlier investigations like [1–4]. The Br/E patch decreases KI more than the Gr/E patch. It means that stiffer patch materials are desirable for all configurations. It is also noted from Fig. 3(a–c) that the reduction in the stress intensity factor KI is more significant when the crack becomes longer. The variations of T-stress as a function of the crack length and the type of composite patch, are shown in Fig. 4(a–c), for three various cracked specimens. Here the T-stress has been normalized by the applied external stress(r). Fig. 4(a–c) shows that the composite patching has a considerable effect on the T-stress. In all reinforced cracked specimens, i.e. centre, double edge and single edge
Table 3 Comparison of SIF and T-stress for un-patched centre cracked specimen. Crack length (mm)
a/Wp
Sih [19] KI (MPa m1/2)
Present KI (MPa m1/2)
Deviation %D
Sherry [18] T (MPa)
Present T (MPa)
Deviation %D
12 24 36 48 60 72
0.1 0.2 0.3 0.4 0.5 0.6
13.67 19.70 24.93 30.21 36.13 43.40
13.22 19.13 24.33 29.45 35.36 42.83
3.3 2.9 2.4 2.5 2.1 1.3
69.68 71.94 75.32 79.54 85.36 94.52
66.22 68.43 72.15 76.39 82.67 91.3
5.0 4.9 4.2 4.0 3.2 3.4
M.R. Ayatollahi, R. Hashemi / Composite Structures 78 (2007) 602–609
0.0 0.0
0.1
0.2
a/WP 0.3 0.4
0.5
0.6
0.7
-0.2
T/σ
-0.4 -0.6 -0.8 -1.0 -1.2 -1.4
Without patch With patch (Boron/E) With patch (Graphite/E)
-1.6
Fig. 4a. Variation of T-stress versus the crack length for patched and unpatched centre cracked specimens.
0.0 0.1 0.0
0.1
0.2
a/WP 0.3 0.4
0.5
0.6
0.7
607
more stable for specimens having more negative T-stresses. Since composite patching in the specimen studied shifts the T-stress up toward positive values, it is expected that composite patching increases the deviation of fracture path from the initial crack line. Fig. 4 also shows that the effect of Br/E patch on the T-stress is more considerable than the Gr/E patch. The variation of KI through the thickness of the repaired centre cracked specimen is shown in Fig. 5a. The crack length for this certain configuration is a = 60 mm (a/ Wp = 0.5). Here, z is the distance from the patched side (through the thickness) and t is the specimen thickness. Previous studies [20,21] suggest that the variation of KI across the thickness of the sheet is almost linear. As can be seen in Fig. 5a, the variation of KI in our study is indeed nearly linear, but with a slight drop close to the free end. The value of KI in the reinforced side of the cracked plate is approximately zero, but in un-patched (free) side of plate, KI has its maximum value through the thickness. Fig. 5b, shows the variation of T-stress as a function of
Without patch With patch (Boron/E) With patch (Graphite/E)
25
T/σ
-0.1
20
-0.2 -0.3
15 KI
-0.4
10
-0.5 -0.6
5 With patch (Boron/E) With patch (Graphite/E)
Fig. 4b. Variation of T-stress versus the crack length for patched and unpatched double edge cracked specimens.
0
0
0.2
0.4
0.6
0.8
1
z/t
0.0 0.3 0.2 0.1
0.1
0.2
a/WP 0.3 0.4
0.5
0.6
0.7
Fig. 5a. Thickness-wise variation of KI along crack front for reinforced centre crack (mode I).
Without patch With patch (Boron/E) with patch (Graphite/E)
0
0.0
0.2
0.4
z/t
0.6
0.8
1
0
T/σ
-0.1 -0.2
-0.2 -0.3
-0.5
-0.4
T/σ
-0.4
-0.6
-0.6 -0.7 Fig. 4c. Variation of T-stress versus the crack length for patched and unpatched single edge cracked specimens.
-0.8 -1
With patch (Boron/E) With Patch (Graphite/E)
-1.2
cracked plates. The composite patching increases the Tstress. As mentioned earlier, the path of crack growth is
Fig. 5b. Thickness-wise variation of T-stress along crack front for reinforced centre crack (mode I).
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normalized distance from the patched side of the plate. Similar to KI, the variation of T-stress through the thickness is almost linear, such that its minimum magnitude is at the free side and its maximum value is at the patched end of the plate. Indeed near the patched side of crack, the T-stress has a very significant contribution in the stress field around the crack tip, because at this point, the absolute value of T-stress is maximum (jT/rj 0.8), but stress intensity is not considerable (KI 0). 5.2. Mixed mode I/II Fig. 6a and b shows the variations of mode I and mode II stress intensity factors as a function of the crack inclination angle and the patch material properties (Gr/E and Br/ E) for a centre crack under mixed mode I/II condition. The crack length ratio for the mixed mode I/II configuration is a/Wp = 0.4. It is clear from these figures that the composite patching substantially decreases KI and KII in all crack angles. However, a comparison between Fig. 6a and b shows that the mode II stress intensity factors is less influenced by composite patching than mode I stress intensity factor. It means that the crack opening is more affected
by composite reinforcement than the crack sliding. This point became more clear from Fig. 7a and b which shows the plots of SIF reduction factor with respect to a. The reduction factor is defined as: Rf K i ¼ 1
K ip K iu
i ¼ I; II
ð6Þ
where subscripts p and u stand for patched and un-patched cases. The mode I stress intensity factor of patched plates are reduced about 24% and 38% for Gr/E and Br/E composites respectively. However, these reductions for the mode II stress intensity factor are about 20% and 33%. In other words the effect of patching on mode I behavior of crack is more than the mode II behavior. The influence of composite patching on the T-stress is illustrated in Fig. 6c as a function of the crack slant angle and the type of composite material. It is seen that in general the patching has a considerable effect on the T-stress when the specimen is subjected to mixed mode loading. This effect is different for various inclination angles. For example, when the crack angle is less than 50, the effect of
1.5
35
Without Patch With Patch (Graphite/E) With patch (Boron/E)
1.0 0.5
25 T/σ
1/2
KI (MPa m )
30
20 15
0.0 -0.5
10
Without Patch With Patch (Graphite/E) With Patch (Boron/E)
-1.0
5 -1.5
0
0
0
10
20
30 40 50 60 70 Crack angle α (Degrees)
80
10
20
90
30 40 50 60 70 Crack angle α (Degrees)
80
90
Fig. 6c. Variation of T-stress with crack angle a. Fig. 6a. Variation of KI with crack angle a.
0.45 Reduction of SIF (1-KP/KU)
16 14 KII (MPa m1/2)
12 10 8 6 4
Without Patch With Patch (Graphite/E) With Patch (Boron/E)
2
0.40 0.35 0.30 0.25 0.20 0.15 0.10 With Patch (Boron/E) With Patch (Graphite/E)
0.05 0.00 0
0 0
10
20
30
40
50
60
70
Crack angle α (Degrees) Fig. 6b. Variation of KII with crack angle a.
80
90
10
20
30 40 50 60 70 Crack angle α (Degrees)
80
90
Fig. 7a. Reduction of mode I stress intensity factor with respect to crack angle.
M.R. Ayatollahi, R. Hashemi / Composite Structures 78 (2007) 602–609
• A stiffer patch material results in more reductions in SIFs and T. This implies that more considerable effects on the crack parameters take place when Br/E is used for composite patching instead of Gr/E. • A three-dimensional study of center crack plate under mode I loading showed that the variations of T-stress and KI through the thickness are almost linear. Close to the patched side of the plate, KI is a small number and the crack-tip stresses are dominated by the T-stress.
0.35 0.30 Reduction of SIF (1-Kp/KU)
609
0.25 0.20 0.15 0.10 With Patch (Boron/E) With Patch (Graphite/E)
0.05
References
0.00 0
10
20
30 40 50 60 70 Crack angle α (Degrees)
80
90
Fig. 7b. Reduction of mode II stress intensity factor with respect to crack angle.
composite patching on the T-stress becomes more significant. The maximum effect of patching on the T-stress occurs when the crack line is normal to the loading direction (a = 0). Fig. 6c also indicates that the composite reinforcement in the angled crack specimen generally reduces the absolute value of T-stress. When the crack line approaches the loading direction (i.e. when a is typically more than 75), the stress intensity factors attain small values and the T-stress describes predominantly the stresses around the crack tip. Previous studies [13] show that for such loading conditions, the Tstress has an important role in predicting the onset of brittle fracture. For a < 75, the composite patching affects significantly the T-stress. Therefore, ignoring the effect of patching on the T-stress can introduce considerable errors in strength analysis of reinforced cracks. 6. Conclusions In this study a three-dimensional finite element analysis was used to investigate the effect of asymmetric composite reinforcement on the crack tip parameters KI, KII and Tstress. It was shown that: • For three various mode I crack specimens studied here, the patching reinforcement greatly reduces KI. The decreased stress intensity factor exhibits an asymptotic behavior as the crack length increases. • In the angled crack specimen, the composite patching reduces both KI and KII but its effect on mode I stress intensity factor is more considerable. • For both pure mode I and mixed mode I/II, the composite reinforcement of cracks decreases the absolute values of T-stress. Fore mode I dominated conditions, the reinforcement shifts the T-stress from negative values toward zero or positive values.
[1] Baker AA, Callinan RJ, Davis MJ, Jones R, Williams JG. Repair of mirage III aircraft using BERP crack patching technology. Theor Appl Fract Mech 1984;2:1–16. [2] Baker AA. Repair of cracked or defective metallic components with advanced fiber composites an overview of Australian work. Compos Struct 1984;2:153–81. [3] Baker AA. Bonded composite repair for fatigue–cracked primary aircraft structure. Compos Struct 1999;74:431–43. [4] Baker AA, Jones R, editorsBonded repair of aircraft structures. Dordrecht: Martinus Nijoff Publishers; 1988. [5] Mitchel RA, Woolley RJ, Chwirut DJ. Analysis of composite reinforced cut-outs and cracks. AIAA J 1975;13:744–9. [6] Jones R, Callinan RJ. Finite element analysis of patched cracks. J Struct Mech 1979;7:107–30. [7] Ting T, Jones R, Chiu WK, Marshall IH, Greer JM. Composites repairs to rib stiffened panels. Compos Struct 1999;47:737–43. [8] Bachir BB, Belhouari M, Serier BC. Computation of the stress intensity factors for patched cracks with bonded composite repairs in mode I and mixed. Compos Struct 2002;56:401–6. [9] Turaga VRS, Ripudaman S. Modeling of patch repairs to a thin cracked sheet. Eng Fract Mech 1999;62:267–89. [10] ABAQUS Finite Element Program, ABAQUS/Standard 5.8. Hibbit, Karlsson and Sorensen, Inc. Pawtuket, USA, 2003. [11] Williams ML. On the stress distribution at the base of a stationary crack. J Appl Mech 1957;24:109–14. [12] Leevers PS, Radon JS. Inherent stress biaxiality in various fracture specimens. Int J Fract 1982;19:311–25. [13] Smith DJ, Ayatollahi MR, Pavier MJ. The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract Eng Mater Struct 2001;24:137–50. [14] Betegon C, Hancock JW. Two parameter characterization of elastic– plastic crack tip fields. J Appl Mech 1991;58:104–10. [15] Selvarathinam AS, Goree JG. T-stress based fracture model for cracks in isotropic materials. Eng Fract Mech 1998;60:543–61. [16] Richardson DE, Goree JG. Experimental verification of new two parameter fracture model. In Fracture Mechanics: 23th symposium, 1993. p. 738–50. [17] Ayatollahi MR, Pavier MJ, Smith DJ. Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading. Int J Fract 1998;91:283–98. [18] Sherry AH, France CC, Goldthorpe MR. Compendium of T-stress solutions for two and three dimensional cracked geometries. Fatigue Fract Eng Mater Struct 1995;18:141–55. [19] Sih GC. Handbook of stress intensity factors. Bethleham, bethleham PA: Institute of Fracture and Solid Mechanics, Lehigh University; 1973. [20] Callinan RJ, Rose LRF, Wang CH. Three dimensional stress analysis of crack patching. Int Conf Fract 1997;ICF-9:2151–8. [21] Kumar AM, Singh Ripudaman. 3D finite element modeling of a composite patch repair. Int Conf Fract 1997;ICF-9:2159–66.
Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching M.R. Ayatollahi *, R. Hashemi Fatigue and Fracture Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran Available online 28 December 2005
Abstract To increase the operational life of defected structures, a repairing method using composite patches has been used to reinforce cracked components. Due to various advantages of composite materials, this method has received much attention from researchers and engineers. Considerable investigations have been performed to highlight the effect of bonded composite patches on the fracture parameters such as stress intensity factors (SIF) and J-integral. However the effect of composite patches on the T-stress, the constant stress term acting parallel to the crack, has not been investigated in the past. In this paper, the finite element method is carried out to analyze the effect of bonded composite patches for repairing cracks in pure mode I and also mixed mode I/II conditions, by computing the stress intensity factors and the T-stress, as functions of the crack length, the crack inclination angle and the type of composite material. In pure mode I condition, the finite element analysis is carried out for three different specimens: centre crack, double edge crack and single edge crack specimens. For mixed mode I/II condition the analysis is conducted on an inclined central crack of various slant angles. For both pure mode I and mixed mode I/II, the numerical results show that composite patching has considerable effect on the T-stress. 2005 Elsevier Ltd. All rights reserved. Keywords: Single composite patch; Crack; Stress intensity factor; T-stress; Pure mode I; Mixed mode I/II
1. Introduction Service life enhancement of damaged structures is the main subject of many studies in recent decades, so that various investigations have been carried to increase the durability and damage tolerance of cracked metallic structures, efficiently and economically. A repair method using a composite patch to reinforce the cracked structure has been shown to be very promising owing to the light weight, high stiffness and strength of composites [1–3]. In early 1970s, Baker and Jones [4], as the pioneers of bonded patch repairing method, performed intensive research studies on this method. They elaborated many advantages of employing composite material patches for the bonded repair of cracked and damaged metallic structures. Numerous models have also been developed for analysis of repairs *
Corresponding author. Tel.: +98 21 73912922; fax: +98 21 7454050. E-mail address: [email protected] (M.R. Ayatollahi).
0263-8223/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.11.024
using various calculation techniques including the collocation method, boundary element method and finite element method. Due to its well established role in fracture mechanics, the stress intensity factor is an important measure for analyzing the performance of bonded patch in repairing cracks. The finite element method (FEM) can be used with a great accuracy to evaluate stress intensity factors and other fracture parameters at the crack tip. Finite element analysis of composite reinforcement by Mitchell et al. [5] appears to be the first thorough attempt towards analytical understanding of this class of problems. Later the finite element method was used by several other authors, among them Jones and Callinan [6], Ting et al. [7], Bachir et al. [8] and Turaga and Ripudaman [9]. In these studies, the effect of composite patch only on the values of stress intensity factors and occasionally on the J-integral was investigated. Meanwhile, the effect of composite patching on the T-stress has not been studied in the past. As described later, the T-stress is the second term in the Williams series
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solution for elastic stresses around the crack tip. It is independent of the distance from the crack tip. The conducted studies in fracture mechanics show that the T-stress has an important role in analyzing the strength and stability of brittle fracture in both linear elastic and elastic–plastic conditions. Hence more knowledge of the T-stress is important for investigating brittle fracture in cracked structures repaired by composite patches. The scope of this paper covers the investigation of composite reinforcement effects on both the stress intensity factors and the T-stress for several cracked specimens subjected to pure mode I and mixed mode I/II loading conditions. The specimens are reinforced by single sided composite patches. For this purpose the three-dimensional finite element model of each repaired sheet is provided in the code Abaqus V5.8 [10] and a unidirectional load is applied to the specimen. As recommended by Turaga and Ripudaman [9], for reducing the error in this type of problems, the geometry nonlinearity is considered in our analysis. Furthermore, two different types of materials (Gr/E and Br/E) are defined for the composite patch and its effects on the values of KI, KII and T-stress are investigated. 2. T-stress and its importance For linear elastic materials, the stress state near the crack tip can be determined from an asymptotic series solution. This method was developed originally by Williams [11] in the conventional crack tip coordinates r and h. He showed that the stress field in an isotropic elastic material containing a crack can be expressed as an pinfinite power ffiffi series, where the leading term exhibits a 1= r singularity, the second term pffiffi is independent of r, the third term is proportional to r, and so on. The classical theory of fracture mechanic normally neglects all except the singular term which results in a single-parameter description of the near-tip fields. Although the third and higher order terms in William’s solution, vanish very near the crack tip, the second term remains finite. For a crack in an isotropic elastic material subjected to plane strain mixed mode I/II loading, the first two terms of William’s solution are: 2 3 T 0 0 1 6 7 rij ¼ pffiffiffiffiffiffiffi K I fij ðhÞ þ K II gij ðhÞ þ 4 0 0 0 5 ð1Þ 2pr 0 0 mT where KI and KII are the mode I and II stress intensity factors respectively and T is a uniform stress in x direction (rzz is mT for plane strain). It can be shown that there are cases where the value of T is large relative to the singular term in Eq. (1). This occurs for example, for a centre crack, when the loading direction is parallel to the crack. In this case the singular term vanishes and it is only T, which describes the stress field in the body. To normalize the effect of T relative to the effective stress intensity factor, Leevers and Radon [12] proposed a dimensionless parameter called the biaxiality ratio B:
pffiffiffiffiffiffi T pa B ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2I þ K 2II
603
ð2Þ
where a is the crack length, for edge cracks and semi-crack length for internal cracks. The T-stress has an important role in strength and stability analysis of cracks in both linear elastic and elastic– plastic fracture mechanics. The importance of T-stress in brittle fracture for linear elastic materials and for mixed mode loading is emphasized by Smith et al. [13]. The Tstress is shown to have an influence on brittle fracture, so that the combination of singular term (characterized by KI or/and KII) and non-singular T-stress controls the brittle fracture processes in pure mode I or mixed mode I/II conditions. Indeed, the generalized maximum tensile stress (GMTS) criterion is formulated in terms of the mode I and II stress intensity factors and the T-stress [13]. In elastic–plastic problems it is now known that the T-stress can have a significant effect on the shape of plastic zone and the stresses inside the plastic zone. Betegon and Hancock [14] showed that T can be used as a second parameter in conjunction with J-integral to characterize fully the crack tip field in contained yielding problems. The T-stress is also a major factor for predicting the stability of fracture trajectory. For example it has been observed that when the Tstress is negative, the crack tends to grow along its initial plane and when the T-stress is positive, the crack deviates from original plane of growth [15,16]. Hence due to the important role of T-stress in crack behavior, it is very useful to determine T in defected structures that are reinforced by composite patching. There are various methods for calculating T from finite element analysis. Some of these methods were reviewed by Ayatollahi et al. [17] and Sherry et al. [18]. In this study, a stress-based method is used to calculate the T-stress for our three dimensional cracked specimens. The method is an extension to the technique used by Ayatollahi et al. [17] for two-dimensional crack problems. For using this method, the parallel-to-the-crack component of stress in Eq. (1) is rewritten to give the following equation for mixed mode loading: K I ðzÞ h h h rxx ðr; h; zÞ ¼ pffiffiffiffiffiffiffi cos 1 sin sin 3 2 2 2 2pr K II ðzÞ h h h þ pffiffiffiffiffiffiffi sin 2 cos cos 3 2 2 2 2pr þ T ðzÞ ð3Þ where r, h, x and y are coordinates in the conventional polar and Cartesian systems with origin at the crack tip and z is the third coordinate (parallel to the crack front) in both systems. According to the stress-method, for pure mode I, the T-stress can be determined along either of the crack faces (h = +p or p) where singular term of rxx vanishes, i.e.: T ¼ rxx
ð4Þ
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and for mixed mode I/II, T can be calculated from:
1 T ¼ ðrxx Þh¼p þ ðrxx Þh¼p 2
ð5Þ
provided the stress rxx is taken from nodal points close to the crack front where the contribution of higher order stress terms are negligible. The validity of the results obtained from this method has already been verified [17].
Table 1 Geometric dimensions of CC, DEC and SEC specimens (all dimensions are in mm) Crack type
Wp
Hp
tp
Wr
Hr
tr
ta
Centre Double-edge Single-edge
120 120 240
120 120 240
3 3 3
180 180 180
90 90 90
1 1 1
0.2 0.2 0.2
p: cracked plate; r: reinforcement; a: adhesive.
3. Crack specimens To investigate the effect of composite patching on the crack tip parameters, the values of KI, KII and T, were calculated via a set of 3D finite element analyses. For pure mode I, three different cracked specimens i.e. (1) the centre crack, (2) the single-edge crack and (3) the double-edge crack specimens made of an aluminum alloy were simulated (see Fig. 1a). The composite patches are connected to the cracked specimens through a film-adhesive. The dimensions of plates, patches and film-adhesives are given in Table 1. In order to determine the stress intensity factor and T-stress as a function of crack length for pure mode I, the analysis was performed for various crack lengths of 0.1 < a/Wp < 0.6 where Wp is the plate width. Also due to symmetry in geometry and loading conditions, only one quarter of the centre crack and double edge crack specimens and one half of the single edge crack specimen were simulated in our study. For mixed mode loading, the previ-
ous centre crack specimen of various inclination angle was used for simulation. The necessary data for geometry of plate and patch are given in Table 1. The fiber composite Graphite/Epoxy (Gr/E) and Boron/Epoxy (Br/E) have been frequently used in the past for patching or reinforcing the cracked plates. Therefore, both of these composite patches were considered for finite
Table 2 Material properties Material
E1
Aluminum Film adhesive Boron/epoxy Graphite/epoxy
72 0.97 208 172.4
E2, E3
m12, m13
m23
G12, G13
G23
25.4 10.34
0.33 0.32 0.17 0.3
0.04 0.18
7.24 4.82
4.94 3.1
E, G values are in GPa. Direction: 1-normal to crack (fiber), 2-along crack, 3-thickness.
Fig. 1a. Various configurations of cracked specimens under pure mode I.
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605
remote stress of r = 70 MPa taken arbitrarily as a reference value. 4. Finite element modeling
Fig. 1b. Configurations of centre cracked specimen under mixed mode I/ II.
element modeling and the effects of patch type on the values of crack parameters were investigated separately. The material properties for aluminum plate, Graphite/Epoxy, Boron/Epoxy and adhesive film are given in Table 2. The composite patch was modeled using eight layers of unidirectional ply that are oriented in the loading direction. As shown in Fig. 1, the cracked specimens are subjected to a uniaxial tensile load giving a uniform
Fig. 2, shows a typical three-dimensional finite element model used in our analysis. The 20-node iso-parametric brick elements were used in the commercial code Abaqus V.5.8 for simulation. The quarter point crack tip singular elements were considered for the crack tip region. The repairing patch, adhesive film and the cracked plate were modeled with one, one and four layers of elements respectively. It is well known that the single sided repair causes out-of-plane bending due to a shift in the neutral axis of the plate under mechanical loading. This out-of-plane bending could lead to large deflections of the repaired crack, therefore geometrically nonlinear analyses were performed to obtain more accurate results. The crack parameters KI, KII and T were determined from stresses around the crack tip. Since the maximum crack opening displacement occurs on the free side of single patched plates, the stress intensity factor on this side is maximum. Therefore the crack parameters have been calculated at the free side of single patched specimens. To validate the method used for calculating the crack tip parameters, first the finite element analysis was performed for un-patched centre cracked specimens of different crack length ratio a/Wp under pure mode I condition. Table 3 displays a comparison between the finite element results for KI and T-stress and theoretical results given by Sih [19] and Sherry et al. [18]. It is seen that the present results are in good agreement with previous theoretical results.
Fig. 2. Typical mesh pattern for a quarter of centre crack specimen: (a) 3-D finite element method model; (b) top view of elements around the crack tip.
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50 45 40
100
35 30 25 20 15
80 60 40
10
20
5 0 0.0
Without patch With patch (Boron/E) With patch (Graphite/E)
120
KI (MPa m1/2)
KI (MPa m1/2)
140
Without patch With patch (Boron/E) With patch (Graphite/E)
0 0.1
0.2
0.3
0.4
0.5
0.6
0
0.7
0.1
0.2
0.3
a/W p
0.4
0.5
0.6
0.7
a/WP
Fig. 3a. Variation of KI versus the crack length for patched and unpatched centre cracked specimen.
Fig. 3c. Variation of KI versus the crack length for patched and unpatched single-edge cracked specimen.
5.1. Pure mode I 40
Without patch With patch (Boron/E) With patch (Graphite/E)
35
K I (MPa m1/2)
30 25 20 15 10 5 0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
a/W p Fig. 3b. Variation of KI versus the crack length for patched and unpatched double-edge cracked specimens.
5. Results and discussion In this section, the finite element results obtained for the crack tip parameters in the patched specimens for pure mode I and mixed mode I/II are presented and discussed for each specimen individually.
Fig. 3(a–c) shows the variations of mode I stress intensity factor (KI) as a function of crack length and patch material properties (Gr/E and Br/E), for centre crack (CC), double edge crack (DEC) and single edge crack (SEC) specimens, respectively. It can be seen from these figures that the patching reinforcement, considerably decreases the stress intensity factor (SIF). This effect is because the patch, in various reinforced configurations, carries the load in the crack region. As the crack length increases, asymptotic behavior is noted for all cracked configurations. This tendency is in agreement with the results reported in earlier investigations like [1–4]. The Br/E patch decreases KI more than the Gr/E patch. It means that stiffer patch materials are desirable for all configurations. It is also noted from Fig. 3(a–c) that the reduction in the stress intensity factor KI is more significant when the crack becomes longer. The variations of T-stress as a function of the crack length and the type of composite patch, are shown in Fig. 4(a–c), for three various cracked specimens. Here the T-stress has been normalized by the applied external stress(r). Fig. 4(a–c) shows that the composite patching has a considerable effect on the T-stress. In all reinforced cracked specimens, i.e. centre, double edge and single edge
Table 3 Comparison of SIF and T-stress for un-patched centre cracked specimen. Crack length (mm)
a/Wp
Sih [19] KI (MPa m1/2)
Present KI (MPa m1/2)
Deviation %D
Sherry [18] T (MPa)
Present T (MPa)
Deviation %D
12 24 36 48 60 72
0.1 0.2 0.3 0.4 0.5 0.6
13.67 19.70 24.93 30.21 36.13 43.40
13.22 19.13 24.33 29.45 35.36 42.83
3.3 2.9 2.4 2.5 2.1 1.3
69.68 71.94 75.32 79.54 85.36 94.52
66.22 68.43 72.15 76.39 82.67 91.3
5.0 4.9 4.2 4.0 3.2 3.4
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0.0 0.0
0.1
0.2
a/WP 0.3 0.4
0.5
0.6
0.7
-0.2
T/σ
-0.4 -0.6 -0.8 -1.0 -1.2 -1.4
Without patch With patch (Boron/E) With patch (Graphite/E)
-1.6
Fig. 4a. Variation of T-stress versus the crack length for patched and unpatched centre cracked specimens.
0.0 0.1 0.0
0.1
0.2
a/WP 0.3 0.4
0.5
0.6
0.7
607
more stable for specimens having more negative T-stresses. Since composite patching in the specimen studied shifts the T-stress up toward positive values, it is expected that composite patching increases the deviation of fracture path from the initial crack line. Fig. 4 also shows that the effect of Br/E patch on the T-stress is more considerable than the Gr/E patch. The variation of KI through the thickness of the repaired centre cracked specimen is shown in Fig. 5a. The crack length for this certain configuration is a = 60 mm (a/ Wp = 0.5). Here, z is the distance from the patched side (through the thickness) and t is the specimen thickness. Previous studies [20,21] suggest that the variation of KI across the thickness of the sheet is almost linear. As can be seen in Fig. 5a, the variation of KI in our study is indeed nearly linear, but with a slight drop close to the free end. The value of KI in the reinforced side of the cracked plate is approximately zero, but in un-patched (free) side of plate, KI has its maximum value through the thickness. Fig. 5b, shows the variation of T-stress as a function of
Without patch With patch (Boron/E) With patch (Graphite/E)
25
T/σ
-0.1
20
-0.2 -0.3
15 KI
-0.4
10
-0.5 -0.6
5 With patch (Boron/E) With patch (Graphite/E)
Fig. 4b. Variation of T-stress versus the crack length for patched and unpatched double edge cracked specimens.
0
0
0.2
0.4
0.6
0.8
1
z/t
0.0 0.3 0.2 0.1
0.1
0.2
a/WP 0.3 0.4
0.5
0.6
0.7
Fig. 5a. Thickness-wise variation of KI along crack front for reinforced centre crack (mode I).
Without patch With patch (Boron/E) with patch (Graphite/E)
0
0.0
0.2
0.4
z/t
0.6
0.8
1
0
T/σ
-0.1 -0.2
-0.2 -0.3
-0.5
-0.4
T/σ
-0.4
-0.6
-0.6 -0.7 Fig. 4c. Variation of T-stress versus the crack length for patched and unpatched single edge cracked specimens.
-0.8 -1
With patch (Boron/E) With Patch (Graphite/E)
-1.2
cracked plates. The composite patching increases the Tstress. As mentioned earlier, the path of crack growth is
Fig. 5b. Thickness-wise variation of T-stress along crack front for reinforced centre crack (mode I).
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normalized distance from the patched side of the plate. Similar to KI, the variation of T-stress through the thickness is almost linear, such that its minimum magnitude is at the free side and its maximum value is at the patched end of the plate. Indeed near the patched side of crack, the T-stress has a very significant contribution in the stress field around the crack tip, because at this point, the absolute value of T-stress is maximum (jT/rj 0.8), but stress intensity is not considerable (KI 0). 5.2. Mixed mode I/II Fig. 6a and b shows the variations of mode I and mode II stress intensity factors as a function of the crack inclination angle and the patch material properties (Gr/E and Br/ E) for a centre crack under mixed mode I/II condition. The crack length ratio for the mixed mode I/II configuration is a/Wp = 0.4. It is clear from these figures that the composite patching substantially decreases KI and KII in all crack angles. However, a comparison between Fig. 6a and b shows that the mode II stress intensity factors is less influenced by composite patching than mode I stress intensity factor. It means that the crack opening is more affected
by composite reinforcement than the crack sliding. This point became more clear from Fig. 7a and b which shows the plots of SIF reduction factor with respect to a. The reduction factor is defined as: Rf K i ¼ 1
K ip K iu
i ¼ I; II
ð6Þ
where subscripts p and u stand for patched and un-patched cases. The mode I stress intensity factor of patched plates are reduced about 24% and 38% for Gr/E and Br/E composites respectively. However, these reductions for the mode II stress intensity factor are about 20% and 33%. In other words the effect of patching on mode I behavior of crack is more than the mode II behavior. The influence of composite patching on the T-stress is illustrated in Fig. 6c as a function of the crack slant angle and the type of composite material. It is seen that in general the patching has a considerable effect on the T-stress when the specimen is subjected to mixed mode loading. This effect is different for various inclination angles. For example, when the crack angle is less than 50, the effect of
1.5
35
Without Patch With Patch (Graphite/E) With patch (Boron/E)
1.0 0.5
25 T/σ
1/2
KI (MPa m )
30
20 15
0.0 -0.5
10
Without Patch With Patch (Graphite/E) With Patch (Boron/E)
-1.0
5 -1.5
0
0
0
10
20
30 40 50 60 70 Crack angle α (Degrees)
80
10
20
90
30 40 50 60 70 Crack angle α (Degrees)
80
90
Fig. 6c. Variation of T-stress with crack angle a. Fig. 6a. Variation of KI with crack angle a.
0.45 Reduction of SIF (1-KP/KU)
16 14 KII (MPa m1/2)
12 10 8 6 4
Without Patch With Patch (Graphite/E) With Patch (Boron/E)
2
0.40 0.35 0.30 0.25 0.20 0.15 0.10 With Patch (Boron/E) With Patch (Graphite/E)
0.05 0.00 0
0 0
10
20
30
40
50
60
70
Crack angle α (Degrees) Fig. 6b. Variation of KII with crack angle a.
80
90
10
20
30 40 50 60 70 Crack angle α (Degrees)
80
90
Fig. 7a. Reduction of mode I stress intensity factor with respect to crack angle.
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• A stiffer patch material results in more reductions in SIFs and T. This implies that more considerable effects on the crack parameters take place when Br/E is used for composite patching instead of Gr/E. • A three-dimensional study of center crack plate under mode I loading showed that the variations of T-stress and KI through the thickness are almost linear. Close to the patched side of the plate, KI is a small number and the crack-tip stresses are dominated by the T-stress.
0.35 0.30 Reduction of SIF (1-Kp/KU)
609
0.25 0.20 0.15 0.10 With Patch (Boron/E) With Patch (Graphite/E)
0.05
References
0.00 0
10
20
30 40 50 60 70 Crack angle α (Degrees)
80
90
Fig. 7b. Reduction of mode II stress intensity factor with respect to crack angle.
composite patching on the T-stress becomes more significant. The maximum effect of patching on the T-stress occurs when the crack line is normal to the loading direction (a = 0). Fig. 6c also indicates that the composite reinforcement in the angled crack specimen generally reduces the absolute value of T-stress. When the crack line approaches the loading direction (i.e. when a is typically more than 75), the stress intensity factors attain small values and the T-stress describes predominantly the stresses around the crack tip. Previous studies [13] show that for such loading conditions, the Tstress has an important role in predicting the onset of brittle fracture. For a < 75, the composite patching affects significantly the T-stress. Therefore, ignoring the effect of patching on the T-stress can introduce considerable errors in strength analysis of reinforced cracks. 6. Conclusions In this study a three-dimensional finite element analysis was used to investigate the effect of asymmetric composite reinforcement on the crack tip parameters KI, KII and Tstress. It was shown that: • For three various mode I crack specimens studied here, the patching reinforcement greatly reduces KI. The decreased stress intensity factor exhibits an asymptotic behavior as the crack length increases. • In the angled crack specimen, the composite patching reduces both KI and KII but its effect on mode I stress intensity factor is more considerable. • For both pure mode I and mixed mode I/II, the composite reinforcement of cracks decreases the absolute values of T-stress. Fore mode I dominated conditions, the reinforcement shifts the T-stress from negative values toward zero or positive values.
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