Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 4, pp. 937-946, Warsaw 2014 PREDICTION OF STATIC CRACK PROPAGATION IN ADHESIVE JOINTS Reza Hedayati, Meysam Jahanbakhshi Young Researchers and Elite Club, Najaf Abad Branch, Islamic Azad University, Najaf Abad, Isfahan, Iran e-mail: rezahedayati@gmail.com; jahanbakhshimeysam@gmail.com Saeid Ghorbani Khouzani Department of Mechanical Engineering, Khomeini Shahr Branch, Islamic Azad University, Isfahan, Iran e-mail: saeid.ghorbani@iaukhsh.ac.ir In this work, fracture mechanics methodology is used to predict crack propagation in the adhesive joining of aluminum and composite plates. Three types of loadings and two types of glass-epoxy composite sequences: [0/90]2s and [0/45/-45/90]s are considered for the com- posite plate. Therefore 2×3= 6 cases are considered and their results are compared. The debonding initiation load, complete debonding load, crack face profile and load-displacement diagram have been compared for the six cases. Keywords: fracture, adhesive joint, debonding, APDL, LEFM 1. Introduction With the increase in the number of bonded composite aircraft components and in the number of bonded repairsmade to crackedmetallic structures, knowledge of adhesive bonding is becoming crucial to aircraft design and life extension. Design and analysis of adhesively bonded joints has traditionally been performed using a variety of stress-based approaches (Tomblin et al., 1998). The use of fracturemechanics has become increasingly popular for the analysis of metallic com- ponents but has been oflimited use in bonded structure joints. Durability and damage tolerance guidelines, already in existence for metallic aircraft structures, need to be developed for bonded structures, and fracture mechanics provides one method for doing so(Tomblin et al., 1998). Previous research in the field of bonded joint analysis and design may be grouped into two major areas of emphasis (Tomblin et al., 1998). The first, a stress-based approach, was initiated byGoland andReissner (1944) andhas beenused extensively byHart-Smith (1983), Hart-Smith andThrall (1985), andothers.Thisapproachhas focusedondetermining thedistributionof shear and normal (or peel) stresses within the adhesive bond line under static loading conditions. In their seminal work, Goland and Reissner (1944) investigated single lap shear joints with thin (inflexible) and thick (flexible) adhesive layers. Their results indicated that both shear and normal stresses approach maxima at or near the free edge of the joint (Tomblin et al., 1998). Adams (1991) confirmed this observation and, using a finite element analysis, proposed that failure of the adhesive layer occurs in tension due to high peel stresses rather than in shear as suggested by the lap shear joint name. Because of the importance of the peel stresses, they have been incorporated into bonded joint design and current criteria call for their elimination or drastic reduction (Adams, 1991; Hart-Smith and Thrall, 1985). The presence of stress con- centrations at the edges of a joint combined with a lightly loaded though useful region of the adhesive at the center has led to techniques, such as increased overlaps and tapered adherents, which reduce the magnitude of the near-edge stresses. In addition, several stress-based failure criteria have been proposed. One of the most notable is Hart-Smith’s approach (Hart-Smith, 1973) which states that the bond strength is limited by the adhesive shear strain energy per 938 R. Hedayati et al. unit bonded area.To date, the stress-based approach to bonded joint design has functionedwell. It has been incorporated into computerized design programs used in the aerospace industry and has contributed to the success of theUSAF’sPABSTprogramandsubsequentadhesively bonded designs (Tomblin et al., 1998). Azari et al. (2011a,b) studied the effect of bondline thickness on the fatigue and fracture of aluminum adhesive joints bonded using a rubber-toughened epoxy adhesive using finite element analysis. The fracture data illustrates the relation between the adhesive thickness and the quasi- static crack initiation and steady-state critical strain energy release rates. Rabinovitch (2008) studied a linear elastic fracturemechanics (LEFM) approach and a cohesive interface (cohesive zone) modeling approach to the debonding analysis of concrete beams strengthened with exter- nally bonded fiber-reinforced-polymer (FRP) strips. The LEFMmodel combined stress analysis using a high order theory and fracture analysis using the concepts of the energy release rate and the J-integral. Bocciarelli et al. (2009) investigated the debonding strength of axially loaded double shear lap specimens between steel plates and a carbon fibre reinforced plastic. Failure of the steel-adhesive interface has been identified as the dominant failure mode and fracture mechanics, and a stress based approach has been presented in order to estimate the relevant failure load. Lenwari et al. (2012) addresses the debonding strength of adhesive-bonded double-strap steel joints. A fracture-based criterion has been formulated in terms of a stress singularity parameter. The test results showed that the interfacial failure near the steel/adhesive corner was the dominant failure mode. The failure was brittle and the debonding life was governed by the crack initiation stage. The finite element analysis was employed to calculate the stress intensity factors and investigate the effects of the adhesive layer thickness, lap length and joint stiffness ratio on the debonding strength. In this work, fracture mechanics is used to predictcrack propagation in the joint between aluminum and composite plates. The setup considered in this work is shown in Fig. 1. Three types of loadings: λ = 0, λ = 0.5 and λ = 1 (the parameter λ is defined as the fraction of the lateral loading from the total loading; i.e. λ = 0 means the loading is completely in plane and λ = 1 means the loading is completely lateral) are considered, while two types of glass-epoxy composite sequences: [0/90]2s and [0/45/-45/90]s are considered for the composite plate. Therefore, 2×3 = 6 cases are considered in this study, and their results are compared. Afterwards, the sequence [0/90]2s is called sequence 1, and [0/45/-45/90]s is called sequence 2. Half of a typical crack face shape is shown in Fig. 2 for the symmetrical problem considered in this work. Fig. 1. The composite/aluminum joint studied Prediction of static crack propagation in adhesive joints 939 Fig. 2. Shape of a typical crack face 2. The crack propagation criteria The main parameter in analyzing crack propagation is called the stress intensity factor which is respectively shown by KI, KII and KIII for opening, shearing, and out of plane shearing fracturemodes.The stress intensity factor is the representative of stress intensity arounda crack or crack tip or face. In first fracturemode (the openingmode), the stress around a crack tip can be calculated using the following equations σ11 = KI √ 2πr cos θ 2 ( 1− sin θ 2 sin 3θ 2 ) σ22 = KI √ 2πr cos θ 2 ( 1+sin θ 2 sin 3θ 2 ) σ12 = KI √ 2πr sin θ 2 cos θ 2 cos 3θ 2 (2.1) By considering θ=0 σ22 = KI √ 2πr →KI =σ22 √ 2πr (2.2) In the second fracture mode (the shearingmode), the relationship between the stresses and KII is σ11 = −KII √ 2πr sin θ 2 ( 2+cos θ 2 cos 3θ 2 ) σ22 = −KII √ 2πr sin θ 2 cos θ 2 cos 3θ 2 σ12 = −KII √ 2πr cos θ 2 ( 1− sin θ 2 sin 3θ 2 ) (2.3) By considering θ=0 σ12 = −KII √ 2πr →KII =−σ12 √ 2πr (2.4) In this paper, crack propagation in the adhesively joined composite and aluminum plates is studied. For investigating crack propagation in static loading, two criteria are needed: initiation criteria and propagation criteria. The crack propagates through the middle of the adhesive layer, relatively distant from either adhesive-adherent interface, leaving an adhesive layer on both adherents (Tomblin et al., 1998). In order to findthe shapeand size of the initial crack, a simplemethod is used.After applying the load, initial debonding is made at the locations in which the YZ shear stress is higher than 940 R. Hedayati et al. the adhesive shear yield stress. The crack initiation load is the load which (after creating the initial debonding) satisfies the propagation criteria on the nodes located on the crack front to a small extent. The crack propagation criteria used in this study is (Hosseini-Toudeshky et al., 2006) GI GIc + GII GIIc + GIII GIIIc =1 (2.5) Using G=K2/E and by considering only the first and the second mode of fracture (KI KIc )2 + (KII KIIc )2 =1 (2.6) In order to calculate KI and KII, the following substitutions are made to equations (2.2) and (2.4) σ22 =σzz σ12 =σyz (2.7) where the X, Y and Z directions are shown in Fig. 1. In this study, the adherent chosen for bonding the aluminum and composite plates is FM73. Thematerial properties of the FM73 adhesive are listed in Table 1 (Tomblin et al., 1998). Table 1.Material properties of the FM73 adhesive Property Amount Elasticity modulus 1.83GPa Yield stress 43MPa KIc 2263600Pa √ m KIIc 2530810Pa √ m 3. Finite element modeling In this project, a macro program is developed using ANSYS Parametric Design Language (APDL) to model the debonding growth. At each step, the debonding face propagation, which is non-uniform, is calculated. Then the elements are completely cleared and a newmodel which consists of the updated crack face is created and thenmeshed. Thismesh deletion and creation is done ateach propagation step in order to keep the accuracy of calculations well. The major steps of the developed macro programare as follows: (1) Define material properties of the model. (2) Define the crack initiation load. (3) Generate geometry andmesh of the composite and aluminum plates and the adhesive. (4) Define the loading and constraints. (5) Perform the linear elastic solution. (6) Predict the initial debonding area using the crack initiation criterion. (7) Move all the nodes located on the crack front by 0.2mm in the positive Y direction. (8) Perform the linear elastic solution. (9) Calculate the stress intensity factors (KI and KII) at each node located on the crack face. (10) Move back the nodes which have not satisfied the propagation criterion to their previous location (move them back by 0.2mm in the negative Y direction). Prediction of static crack propagation in adhesive joints 941 (11) If none of the nodes located on the crack front satisfy the propagation criterion, increase the applied load. (12) If the crack has reached its end (it has moved 300mm), stop the solution. (13) Return to step (8). The finite elementmodel of the problem is shown in Fig. 3, and is zoomed in at the adhesive interface in Fig. 4. For the composite plate 6000 8-noded SOLID46 elements, for the aluminum plate 22000 8-noded SOLID45 elements and for the adhesive 8000 SOLID45 elements have been used. For the composite plate, the aluminumplate and the adhesive, one, four and two elements through the thickness have been used. The elements at the two interfaces are glued. In other words, the composite and the adhesive share the same nodes at their interface. The same is true about the aluminum and the adhesive interface. This can be better seen in Fig. 4. Since the structure is symmetrical with respect to a plane perpendicular to the X direction, only half of themodel is created. The nodes located at the symmetry plane position are not allowed tomove in the X direction. For discretizing the entire model, mappedmeshing has been used. Fig. 3. Finite element model of the aluminum/composite joint Fig. 4. Finite element model of the aluminum/composite joint (zoomed in) Using a Core2Due 2.26 GHz CPU, each crack propagation step takes about two minutes. For each load step, about 400 propagation steps are needed. It must be noted that the crack initially moves quickly, but near the end of crack propagation at each load step, a long time is taken to move forward and backward most of the nodes on the crack face. Considering 4 to 6 load steps, solving each problem (in this study 6 cases are considered) takes about 40 hours. 942 R. Hedayati et al. 4. Results and discussion 4.1. Crack initiation load The load versus crack propagation for λ = 0 and for two composite sequences is shown in Fig. 5. As it can be seen, the crack initiation loads for the first and the second sequence are 232kN/m and 224kN/m, respectively. It can also be seen that the complete debonding loads for the first and the second sequence are 656kN/m and 496kN/m, respectively. Therefore, it can be concluded that the crack initiation load is close for the two sequences, but as the crack propagates, the load necessary for crack propagation is lower in sequence 2 than that in sequence 1. The load versus crack propagation for λ=0.5 and λ=1 and for two composite sequences are shown in Fig. 5b. The initiation and the complete debonding load for the two composite sequences and for three values of λ are listed in Tables 2 and 3. The following conclusions can bemade: • when λ=0.5 or 1, by increasing the load slightly over the crack initiation load, the crack propagates immediately by about 150mm(Fig. 5b). But on the other hand, it can be seen from Fig. 5a that when λ = 0, by increasing the load slightly over the crack initiation load, the crack propagates immediately only by about 50mm which is much lower than 150mm; • for all values of λ, the initiation and the complete debonding load is lower in the cases with composite sequence 2; • in both composite sequences, the initiation and the complete debonding load is higher for λ=0.5 than for λ=1. Fig. 5. Load versus crack propagation for: (a) two composite sequences (λ=0) and (b) four composite sequences (λ=0.5 and λ=1) Table 2.Crack initiation load for the six cases considered λ=1 λ=0.5 λ=0 Sequence 1 2.08kN/m 4.32kN/m 232kN/m Sequence 2 1.76kN/m 3.52kN/m 224kN/m Table 3.Complete debonding load for the six cases considered λ=1 λ=0.5 λ=0 Sequence 1 3.52kN/m 7.12kN/m 656kN/m Sequence 2 3.04kN/m 6kN/m 496kN/m Prediction of static crack propagation in adhesive joints 943 4.2. Crack face profile The crack face profile in different crack propagations can be seen in Fig. 6(a-d) for the two sequences and λ=0 and 1. The difference between the debonding propagations of the ends and the middle of the debonding front is listed for the six cases in Table 4. The oscillations visible in the profiles are because of two reasons: • Firstly, for better visibility of the crack face profiles for different crack propagations, the crack dimension in the Y direction (parallel to the direction in which the crack moves) is scaled by about 10 times in Fig. 6(a-d). Therefore, the real oscillations in the crack profile are exaggerated in plots. • Secondly, since each node is allowed to move forward and backward only by 0.2mm, the crack face profile cannot be completely smooth. By decreasing the value of themovement, a smoother crack face profile can be obtained. Fig. 6. Comparison of the crack face profile for different crack propagations: (a) sequence 1 and λ=0, (b) sequence 2 and λ=0, (c) sequence 2 and λ=0, (d) sequence 2 and λ=1 Table 4.Difference of crack propagation between the ends andmiddle parts of the crack face λ=0 λ=0.5 λ=1 Sequence 1 10mm 7.2mm 5.9mm Sequence 2 26mm 15mm 17mm It can be seen in Fig. 6(a-d) and Table 4 that: • For both sequence types and with λ = 0, the ends of the crack face propagates forward more than its middle part, while with λ=0.5 or λ=1, themiddle part of the crack face moves forwardmore than its ends. 944 R. Hedayati et al. • For all values of λ, the difference between the debonding propagations of the ends and the middle of the crack face for composite sequence of 2 is higher than that for sequence 1. This can bemore recognized when λ=0. • For both sequence types, the difference between the debonding propagations of the ends and themiddle of the crack face with λ=0 is higher than that in the corresponding case with λ=0.5 or λ=1. • Regardless of the sequence type, when λ=0 the debonding face profile can be divided in three regions: (a) at the beginning of debonding propagation, the difference between the debondingpropagations of the ends and themiddle of the crack face is small, (b)when the maximum propagation of the crack face is higher than 50mm, the difference between the debondingpropagationsof the endsand themiddleof the crack face gets larger and remains almost constant until near the end of propagation, and (c),when the crack face has reached near the end of the adhesive film, the difference between the debonding propagations of the ends and the middle of the crack face gets small again. • Regardless of the sequence type, when λ = 0.5 or λ = 1, the difference between the debondingpropagationsof theendsandthemiddleof the crack facegets larger consistently. In other words, the difference between the debonding propagations of the ends and the middle of the crack face is small at the beginning, then for a large range of the debonding propagation remains almost constant, and finally at the end of propagation gets large. • For any value of λ, the difference between the debonding propagations of the ends and the middle of the crack face is very close for λ=0.5 and λ=1. 4.3. Load-displacement diagrams For plotting the load-displacement diagram, the displacement at the end of the composite plate is measured. For λ = 0, the horizontal displacement and for λ = 0.5 and 1, the vertical displacement is measured. This is also true for λ=0.5, because the horizontal displacement of the composite end is negligible as compared to its vertical displacement. The load-displacement diagrams for the six cases are plotted in Fig. 7(a-f). The following conclusions can bemade: • The change in the load-displacement slope from the initial debonding until the complete debonding in the cases with λ=0.5 and 1 is more than that in the cases with λ=0. • Regardless of the composite sequence, if λ = 0, the complete debonding happens when the displacement at the end of the composite plate reaches 1cm. • Regardless of the composite sequence, if λ=0.5 and 1, the complete debonding happens when the displacement at the end of the composite plate reaches 16cm. The load-displacement diagrams at the complete debonding is compared for the two sequen- ces and λ=0.5 and 1 inFig. 8. It is interesting to see that the sequence type does not affect the load-displacement slope very much. On the other hand, the value of λ has a significant effect on the slope of the load-displacement diagram. 5. Conclusions In thiswork, fracturemechanicsmethodology isused topredict crackpropagation in theadhesive joining of aluminum and composite plates. Three types of loadings: λ=0, λ=0.5 and λ=1 are considered while two types of glass-epoxy composite sequences: [0/90]2s and [0/45/-45/90]s are considered for the composite plate. Therefore, 2×3= 6 cases are considered in this study, and their results are compared. It is observed that the crack initiation load is close for the two sequences, but as the crack propagates, the load necessary for crack propagation is lower in Prediction of static crack propagation in adhesive joints 945 Fig. 7. Load-displacement diagrams for: (a) sequence 1 and λ=0, (b) sequence 2 and λ=0, (c) se- quence 1 and λ=0.5, (d) sequence 2 and λ=0.5, (e) sequence 1 and λ=1, (f) sequence 2 and λ=1 Fig. 8. Comparison of load-displacement diagrams near the complete debonding for the two sequences and λ=0.5 and 1 946 R. Hedayati et al. sequence [0/45/-45/90]s than that in sequence [0/90]2s. As for the debonding front profile, it has been seen that for both sequence types, in the cases with λ=0, the side parts of the debonding front propagates more forward than its middle part, while in the cases with λ=0.5 or λ=1, themiddle part of the debonding frontmoves forwardmore than its side parts. It has been also seen that regardless of the λ value, the difference between the debonding propagations of the side and themiddle parts of the debonding front is very close for λ=0.5 and λ=1. In the load- displacement diagram, it is seen that the sequence type does not affect the load-displacement slope very much. On the other hand, the value of λ has a significant effect on the slope of the load-displacement diagram. References 1. Adams R.D., 1991, Testing of adhesives-useful or not? [In:] Adhesion 15: Proceedings of the 28th Annual Conference on Adhesion and Adhesives, K.W. Allen (Edit.), Elsevier Applied Science Pu- blishers, London, UK 2. 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Tomblin J., Seneviratne W., Escobar P., Yoon-Khian Y., 1998, Applications of fracture mechanics to the durability of bonded composite joints,DOT/FAA/AR-97/56 Manuscript received January 28, 2014; accepted for print April 29, 2014