- Email: [email protected]
Analysis of the plastic zone under mixed mode fracture in bonded composite repair of aircraft structures
Accepted Manuscript Analysis of the Plastic Zone under Mixed Mode Fracture in Bonded Composite Repair of Aircraft Structures
Wahid Oudad, Djamal Eddine Belhadri, Hamida Fekirini, Malika Khodja
PII: DOI: Reference:
S1270-9638(17)30191-8 http://dx.doi.org/10.1016/j.ast.2017.07.001 AESCTE 4095
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
29 January 2017 3 July 2017 4 July 2017
Please cite this article in press as: W. Oudad et al., Analysis of the Plastic Zone under Mixed Mode Fracture in Bonded Composite Repair of Aircraft Structures, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.07.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of the Plastic Zone under Mixed Mode Fracture in Bonded Composite Repair of Aircraft Structures Oudad Wahid1, Belhadri Djamal Eddine1, Fekirini Hamida2,*, Khodja Malika2, 3 1
Smart Structures Laboratory (SSL), University Centre of Ain Temouchent Po Box 284, 46000, Algeria.
2
Mechanical Physical Materials Laboratory (LMPM), Mechanical Engineering Department, University of Sidi Bel-Abbes 22000, Algeria. 3
CSIR Materials Science and Manufacturing, Meiring Naude Road Pretoria, 0184, South Africa *
Corresponding author: [email protected]
Abstract Material fracture by opening (mode I) is not the only failure criteria responsible for fracture propagation. Many industrial examples show the presence of mode II and mixed mode I + II. In the present work, numerical analyses of the three-dimensional and non-linear finite element method are used to estimate the performance of the bonded composite repair of metallic aircraft structures with a pre-existent damage by analyzing the plastic zone size ahead of repaired cracks under mixed mode loading, to assess the effect of the composite repair system on the plastic zone. The Von Mises stress is used to predict yielding of materials under this loading condition. The extension of the plastic zone, which takes place at the tip of a crack, strictly depends on many variables, such as the yield stress of the material, the loading conditions, the crack size and the thickness of the cracked component. The obtained results have demonstrated that the plastic zone ahead of the crack is significantly reduced by the presence of composite patch materials. Furthermore, parametric analysis has been carried out to evaluate the effect of lay-up and material system variation on the J integral. Keywords : J integral; plastic zone; crack; patch; Von Mises stress, crack inclination angle, peel stresses, composites bonded repair, Adhesive bonding, Boron/epoxy, Transversely graded material (TGM).
1. Introduction Bonded composite repairs of locally damaged metallic structures has gained considerable interest in aircraft structural maintenance and life extension solution in the last two decades [1, 2]. These repairs provide an efficient method for restoring the ultimate load capability of the structure [3]. The analysis of the effects of the geometrical properties of the composite on the repair performance has great interest in the literature. A good way to design a patch repair is to maximize the safety to cost ratio by finding the optimal composite patch shape [4, 5]. Riccio et al. [6] presented repair design tool aimed to help the designer by suggesting different repair typologies and proper repair size by means of optimization analyses that can provide the best repair solution with minimal adhesive shear stress and size of the repair patch. Their work has been tested against a literature case study on multistep composite-metal joints. They showed that minimization of the overlap length (which can be a fundamental requirement for repair design, especially for components with complex geometry) has been proven possible within an optimization process in order to provide practical repair configurations. The bonded repair reduces stresses in the cracked region and prevents the crack from opening and therefore from growing. The fiber composite patches have improved directional stiffness, high durability under cyclic loading, low density and excellent formability. Khodja et al. [7], have investigated, using finite element analysis, variation of the integral J depending on the crack size for different fiber orientations and different number of plies of repaired cracks in AA7075-T6 structures subjected to biaxial tensile stresses. Their study was carried out in order to estimate numerically the effect of biaxial tensile loading on the behavior of cracks in the presence of the bonded boron/epoxy repair in aircraft structures. The results show the beneficial effect of the patch composite in those cases and the relationship between the fiber orientations and it was noted that the best results are given by the orientation of 0 ° where the fibers are perpendicular to the crack direction. Baker [1] has initiated repair of aircraft aluminum structures using composite patch in the early 1970s mainly in order to enhance fatigue life of cracked components. From geometrical consideration, bonded repairs fall into two categories: double-sided (symmetric) and singlesided (asymmetric). In most of the practical cases, both sides of the cracked panels are not available to perform a symmetrical repair. Therefore, single-sided repair is often adopted such as in case of aircraft wings. This asymmetric repair causes a significant bending field which increases the stress intensity factor (SIF) at the crack tip beyond the value compared to unrepaired panel. This bending stress reduces the repair efficiency, hence static strength or fatigue life of the repaired model gets lower. Umamaheswar and Ripudiman Singh [8]
performed finite element modeling and analysis of single-sided composite patch repairs applied to thin aluminum sheets. They showed the SIF variation through the thickness of the panel assuming straight crack front. Chukwujekwu Okafor [9] developed a finite element model for analyzing the stress distribution of cracked plates repaired with a single-sided octagonal patch. They found that the zone of maximum stress shifted from the crack front (for unpatched specimen) to the edge of the patch (for the patched specimen) because of high peel stress development at overlay edge. Pastor et al. [10] have conducted experimental and numerical investigation of damaged and undamaged specimens patched with carbon epoxy composite material. They observed that the failure of the repaired model occurs when the maximum shear stress in the adhesive is close to its maximum shear strength. There are only a few investigations available on repairing panels in mixed-mode condition by the linear elastic and nonlinear fracture mechanics [11-17]. Hosseini-Toudeshky performed fatigue crack growth tests of single-sided repaired thick and thin panels containing center inclined cracks with various patch lay-ups configurations and various composite patch thicknesses [12, 13]. Ayatollahi and Hashemi [14, 15] used a finite element analysis to investigate the effect of composite patching on the SIF reduction for an inclined center crack panel under different mixed loading case. Bachir Bouiadjra et al. [16] have conducted FEA to estimate SIF in single and double-sided repairs in mode I and mixed mode edge-cracked panels. They have shown that the adhesive and composite patch properties have a significant and beneficial effect on the symmetrical patch. All these previous work do not thoroughly investigate the effect of bending in case of asymmetric repair, which in turn causes the peaking of stresses at the unpatched side generating higher SIF. Recently, Ramji et al. [17] have investigated that in the case of an asymmetrical patch, at the unpatched surface, the SIF value exceeds the value of SIF obtained in unrepaired panel. Therefore, the static strength of the repaired panel is reduced. The peel stresses in bonded joints normally peaks at the end of the overlap, which in turn can cause failure of the adhesive layer; thereby, reducing the performance of the repair. To avoid the severity of these peel stresses occurring at the overlapped ends, Duong [18] suggested usage of tapered patch. In the present work, the non-linear three-dimensional finite element method is used to compute the contour and the size of the plastic zone ahead of repaired cracks with bonded composite composite patch. The effects of the patch properties and the crack orientation on the plastic zone size are highlighted.
1.1. Stress field near the crack tip Irwin [19] has shown that the stress distribution near the crack tip can be described by the stress intensity factors (KI, KII, KIII) where each stress intensity factor is associated with a fracture mode. The relation that gives the stresses at the crack tip can be put in the form: ߪ ൌ
݂ ሺߠሻ ξଶగ
(1)
For the case of a mixed mode (I + II), the relation (1) can be written as ߪ ൌ
ξଶగ
ቀܭூ ݂ூ ሺߠሻ ܭூூ ݂ூூ ሺߠሻቁ
(2)
Where r and θ are the polar coordinates of the considered point, KI and KII are the stress intensity factors in mode I and II, and fij is a function dependent upon the polar angle. 1.2. Form and size of the plastic zone Due to the fact that the stress field at the crack tip reaches significant high values, it contributes to the creation of a plastic zone at the crack tip which depends more or less on the ductility of the material. The computation of the size depends on both the loads and the state of the stresses. Many authors have attempted to determine the form and to evaluate the size of the plastic zone at the crack tip on the basis of the classical criteria of elasticity or by calculation using the finite element method. In mode I, Irwin [20] proposed that the form of the plastic zone of dimension ry is circular and computed it as: ݎ௬ ൌ
ଵ
ଶ
(3)
ቀ ቁ
ఈగ ఙೊ
Where ߪ is the yield stress, α = 2 for plane stress and α= 6 for plane strain. ݎ௬ ൌ ܽ ܭூଶ ܾ ܭூூଶ ܿ ܭூ ܭூூ
(4)
Where ar, br and cr are the constants function of the Poisson coefficient ν and the yield stressߪ . ܽ Ͷሺͳ െ ʹ߭ሻଶ ሺͳ ܿߠݏሻ ͵ሺͳ െ ܿߠʹݏሻ ଵ ൈ ቐͶሺͳ െ ʹ߭ሻଶ ሺͳ െ ܿߠݏሻ ͻሺͳ ܿߠʹݏሻቑ ൝ܾ ൡ ൌ ଵగோమ ܿ െͺሺͳ െ ʹ߭ሻଶ ߠ݊݅ݏ ͳʹߠʹ݊݅ݏ
(5)
The J-integral value is evaluated using domain integral method [21] as shown in equation (6): ܬൌ ቂܹ݊ଵ െ ߪ ݊
డ௨ డ௫
ቃ ݀ݏ
(6)
2. Geometrical and materials properties The basic geometry of the cracked structure considered in this study is shown in Figure 1. Consider rectangular elastic-plastic aluminum 2024-T3 plate with dimensions of 39x160x3 mm3, with an inclined center crack ‘2a’ of length 10 mm. The crack is inclined at an angle of β = 45° with the horizontal as shown in Figure 1. The plate is subjected to a uniaxial load of 17.5 kN (σ =150MPa). The boron–epoxy patch of dimensions 25x25x1.5 mm3 is bonded asymmetrically using 0.1 mm thin film FM73 structural adhesive. The layer thickness of the laminate is taken as 0.375 mm. The composite had unidirectional lay-up where the fibers were oriented along the specimen length direction (parallel to the direction of load). The general material properties of aluminum panel, composite patch and adhesive are given in Table 1. The specimen dimensions follow the ASTM E-647 standard [12]. Standard tensile tests were carried out on Aluminum 2024-T3 and FM73 adhesive. The obtained stress–strain curves are presented in Figures 2 and 3 respectively. 3. Finite element model The analysis involved a three-dimensional finite element method by using a commercially available finite element package code ABAQUS [22]. The finite element model consists of three subsections to model the cracked plate, the adhesive, and the composite patch. The plate had four layers of elements in the thickness direction, the adhesive had only one layer of elements through thickness, and the composite patch had two layers of elements through thickness. The structure made up of the plate, the adhesive and the patch is subjected with uniform uniaxial tensile amplitude of 150MPa. The boundary conditions of fixing, subordinated to the conditions of tensile uniaxial loading of the geometry, were introduced into the initial phase. These conditions are represented in a 3D model as shown in Figure 1 and described as follows: - One edge of the plate has been clamped. - Uniform traction displacements are applied to the free edge of plate in Y direction. The grid used in the 3D model of finite elements is represented in Figure 4 and shows the overall mesh of the specimen and mesh refinement at the crack tip region. The mesh was generated using 3D hexagonal shape and was modeled with 20 node reduced integration quadratic brick elements (C3D20R). The mesh was refined near the crack tip area, a circular zone of radius 3 mm around the crack front was modeled with an element dimension of 0.05 mm using at least 15 such fine elements in the front and back of the crack tip. The
procedure used in the finite element analysis involved the following steps: (i) the tensile stress was applied to the gripped specimen; (ii) general static ‘STEP’-option was used for analysis with ABAQUS; (iii) Automatic increment of steps was used with maximum increments number of 100. Minimum increment size was 10-5 and maximum increment size was 1. Nevertheless, the ABAQUS solver code could override matrix solver choice according to the ‘STEP’- option. Abaqus does not recognize mechanical contact between part instances or regions of an assembly unless that contact is specified in the Interaction module. The mere physical proximity of two surfaces in an assembly is not enough to indicate any type of interaction between the surfaces. Therefore the ‘‘TIE’’-option was used for adhesive-plate and adhesivepatch interactions. A tie constraint allows fusing together two regions even though the meshes created on the surfaces of the regions may be dissimilar. This allows Abaqus to use the default meshing techniques and automatically generate tie constraints across the incompatible interfaces. Abaqus automatically chooses one side of the interface as the slave surface and the other as the master surface for the automatically generated tie constraint, but creates common (merged) nodes on the perimeter of the incompatible interface. Abaqus generally selects the surface with the finer mesh to be the slave surface. The computation for the depth of the slave node adjustment zone for the tie constraint is based on the bounding dimensions of the interfacing regions. The code default value was used for the position of tolerance. Rotational degrees of freedom (DOF) of nodes between the different sub-sections of the model were tied together. The Von Mises yield criterion is used to predict plastic deformation. Incremental plasticity theory is introduced to model the material nonlinearity. The Newton–Raphson iterative approach is used for resolving nonlinear finite element equations. [22] 4. Analysis and results 4.1. Reduction of J integral For a measure of the fracture mechanic safety and patching efficiency criterion at the repaired crack, the non-dimensionalised reduction of the J integral can be used: ܬ כܬൌ ͳ െ ൘ܬ ௨
(7)
Whereܬ ,ܬ௨ are ܬintegrals for the patched and unpatched crack plate respectively. The reduction of J integral is important to the design of the repaired cracked plate because this value
defines the patch efficiency. As כܬincreases, the crack propagation decreases. On the other hand, as כܬdecreases, the possibility of fracture increases. Figure 5 shows the variation of J integral according to the inclined crack angle for patched and unpatched crack plate. Two behaviors are observed: The first is where the maximum value of J integral is recorded for crack inclination of 0° corresponding to the pure mode I and the J integral value is null (zero) for an angle of 90° corresponding to the case of a crack parallel to the loading axis (unpatched crack plate). The second behavior is when the, J integral value is maximal for 45°. This last value corresponds to the high shear stresses (patched crack plate). The beneficial effect of the application of the repair technique can be seen. For the best estimate of the advantages of the presence of the composite patch, the variation of the J* as a function of the inclined crack angle is illustrated in Figure 6. The reduction ratio of the J integral for the repaired and unrepaired plate is between of 38 and 87%. This reduction does not have any sense when the crack inclination tends towards 90° because the J integral for repaired and unrepaired tend towards zero. This fact is due the closing of crack under this loading case. One can thus confirm that the use of a composite patch offers an enormous advantage for repaired inclined cracks and consequently increasing the service life. On the other hand, the decrease in repair efficiency for angle of 45 ° is essentially due to the dominance of mode II of rupture, which generates high shear stresses, as shown by Hosseini [13] for an elastic plate. To overcome this drawback, another type of composite patch must be used to improve this technique. 4.2. Effect of the crack orientation on the shape of plastic zone A plastic zone is characterized by a Von Mises stress value higher than 350 N/mm2, i.e. the yield stress value of the considered material. The shape and size of the plastic zone is shown in Figure 7 for unrepaired plate, for β = 0° (pure mode I), 45° and 60° respectively. It can be seen that the size of the plastic zone is significantly affected by the variation of the crack orientation. The size of the plastic zone decreases with the increment of the crack inclination angle β between 0° and 90°). The crack tip plastic zone in mode I present a “butterfly-like” shape. The plastic zone has a symmetrical distribution to the initial crack plane. The results are consistent with those shown by Lu et al. and Gao Xin et al. [23, 24].
To examine the efficiency of the patch, the shape and size of the plastic zone were plotted. It can be observed from Figure 8 that the plastic zone has a non-symmetrical distribution to this crack plane β = 45o and the size of the plastic zone for repaired plate is smaller by 40% than the size in the case of unrepaired plate at the crack tip. This behavior is due to the fact that the shear stress field at the crack tip under the patch becomes smaller and it contributes to minimize the creation of plastic deformation and a plastic zone around the crack tip. In correlation with Figure 5, it has been concluded that the energy approach of the nonlinear mechanics of the fracture based on the integral J affects the plastic zone size for the repaired and unrepaired structure. The reduction in the plastic zone is due to the beneficial effect of the patch on the absorption of the stresses at the crack tip vicinity and that improves the fatigue life of the structure. 4.3. Effect of the material properties of composite patch A three dimensional finite element (FE) analysis was performed to understand further the variation of the J integral through the thickness. In particular, the behavior of the J integral for unrepaired plate and repaired plate with various composite patch configurations was investigated. The details are presented in the following sections. Transversely graded material (TGM) as the patch material is proposed in this section. This material is new and still in the development stage. TGM is an orthotropic composite material whose Young's modulus varies linearly along the thickness [25]. The transversely graded material was used as a sequence of homogeneous layers for 1.5 mm thickness of composite patch for the bonded repair, each having a constant elastic modulus. The overall elastic modulus gradient was used by assigning different modulus values to different layers. Two transversely graded materials with different Young's modulus gradient variations were considered namely TGM1 and TGM2. Figure 9 shows the variation of Elastic modulus profile (E) of both transversely graded materials across the composite thickness where TGM1 varies linearly with the Young's modulus gradient from 200 GPa to 250 GPa and TGM2 varies linearly with the Young's modulus gradient from 300 GPa to 450 GPa. In this paragraph the effects of different parameters such as composite layup and composite material on the J integral is highlighted. Six configurations of patch are used: transversely graded material TGM1 & TGM2 and Boron/epoxy composite having different layup orientations such as [+45]2/[-45]2, [+90]2/ [-45]2, [+90]2/[0]2 and [0]4 (unidirectional). The
comparison of J integral throughout the plate thickness between repaired and unrepaired face is shown in Figure 10. The transversely graded patched model and the balanced patched layup [+90]2/[0]2 have an efficient effect to significantly reduce the value of J integral in both faces of the repaired plate compared to the unidirectional composite [0]4. In addition, in the repaired face a reduction of J integral has been observed in the TGM configuration compared to other configurations, because the transversely graded composite TGM absorbs more energy than the others. The fiber orientation based on the crack direction is important to consider to ensure good absorption of the stresses at the crack tip and the repairing material strength under mixed mode loading as shown by Khodja et al. [7]. Of further importance is material choice and the bonding durability to ensure better mechanical behavior of bonded composite repairs and proposed material models with the use of finite element assessment and the rich background in the literature to provide detailed and reliable information on the stress distribution, failure onset, and crackpath propagation as demonstrated by A. Riccio et al. [26]. To clearly illustrate the previous results in Figure 10, Figure 11 shows the data in the form of bar charts where one can see and compare easily the results of the various configurations of repair. Distinctly, the TGM1 and 2 materials composite (patch) reduce the J integral considerably compared to the other configurations of repairs. This is due to the variation across the thickness that is influenced by the strength of gradation where the gradation in TGM1 is lower than TGM2 due to differences in the Young modulus gradient. TGM2 gives better results, which influences the plastic zone shape and size. 4.4. Peel stress analysis The adhesive is the weak point of reinforcement by composite patch materials. It is a material having very low mechanical shear characteristics and it is known that de-bonding is triggered by the development of the peel stress concentrations (σz) at the overlap ends. The peel stresses distribution and variation in the repaired face and the overlap area along a virtual path line [AB] drawn to plot the results as shown in Figure 12, with different composite patches configurations. Three behaviors have been noted on the peel stresses along the area of interest, starting from the point A towards the point B. The first area covers the repaired face where the patch is bonded and covers the plate and low peel stresses are found in the second area in the middle, peak values of peels stresses are high as a side effect of the bond line between the patch
composite and the adherent. For this reason, tapered patch design is recommended to reduce the edge effect. Finally, the third area of the free edge has low values of peel stresses at the end of bond line and goes towards zero where the undamaged area of the plate is reached. The plots shows similar peel stresses behavior has been noticed with various composite patch configurations with different levels but it is clear that the use of a TGM material generates low peel stresses in the proximity of the patch free edge and consequently decreases the debonding probabilities. This is due to the influence of elastic gradient properties that vary along the thickness compared to the use of the boron /epoxy with different lay-up orientation. The use of TGM is recommended to extend the service life of cracked structural components. This result is also confirmed by Ramji and Srilakshmi [27]. 5. Conclusion The nature of the plastic zone that is formed ahead of a crack tip plays a very important role in the determination of the type of failure that occurs. In this study the shape and size of crack tip plastic zones have been estimated by three finite element analyses in an aluminum 2024-T3 plate repaired and unrepaired under mixed mode loading according to Von Mises yield criteria. The J integral was also calculated at the crack tip. It was found that: x
The reduction ratio of the J integral for the repaired and unrepaired plate is between 38 and 87% depending on the crack type. This reduction does not have any sense when the crack inclination tends towards 90° because the J integral for repaired and unrepaired tend towards zero. This fact is due the closing of crack under this loading case.
x
The plastic zone size and shape are influenced by the crack inclination which decreases with the increment of the crack inclination angle β (β = 0° corresponding to the pure mode I). The shear stress field at the crack tip under the patch composite decreases; it contributes to minimize the creation of plastic deformation and a plastic zone around the crack tip.
x
The transversely graded patched model and the balanced patched layup [+90]2/[0]2 have an efficient effect to significantly reduce the value of the J integral in both faces of repaired plate compared to the unidirectional composite (patch) repaired in these cases.
x
The use of a TGM generates low peel stress in the proximity of the patch and consequently decreases the debonding probability.
x
The energy approach based on the integral J of fracture of nonlinear mechanics affect the plastic zone size for repaired and unrepaired structure. The reduction in the plastic zone is due to the beneficial effect of the patch repair on the absorption of the stresses at the crack tip, which improves the fatigue life of the structure.
x
The variation across the thickness is influenced by the strength of gradation of transversely graded material properties. Optimal gradation with respect to applied load and crack direction must be investigated.
References
[1] Baker Alan A. A proposed approach for certification of bonded composite repair to flightcritical airframe structure, Applied Composite Materials. 18 (2011) 337–69. [2] Ahn JS, Basu PK and Woo KS. Analysis of cracked aluminum plates with one-sided patch repair using p-convergent layered model, Finite Elements in Analysis and Design. 46 (2010) 438–48. [3] Bakuckas John G, Westerman Bud. Fatigue and residual strength performance of bonded repair to metallic fuselage, Structural Integrity: Influence of Efficiency and Green Imperatives. 8 (2011) 735–52. [4] Breitzman TD, Iarve EV, Cook BM, Schoeppner GA, Lipton RP. Optimization of a composite scarf repair patch under tensile loading, Composites Part A: Applied Science and Manufacturing. 40 (2009) 1921–30. [5] Ouinas D, Hebbar A, Bachir Bouiadjra B, Belhouari M, Serier B. Numerical analysis of the stress intensity factors for repaired cracks from a notch with bonded composite semicircular patch, Composites Part B: Engineering (2009) 804–10. [6] A. Riccio, G. Di Felice, F. Scaramuzzino, A. Sellitto. A Practical Tool for the Preliminary Design of Bonded Composite Repairs. Applied Composite Materials. 2014; 21(3):495-509. [7] M. Khodja,H. Fekirini, G. Corderley, S. Govender, Repaired crack in AA7075-T6 structures subjected to biaxial tensile stresses, Insights and Innovations in Structural Engineering, Mechanics and Computation, Taylor & Francis Group, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 CRC Press 2016, 563-566.
[8] Umamaheswar Turaga VRS and Ripudaman Singh.Modeling of a patch repair to a thin cracked sheet, Engineering Fracture Mechanics. 62 (1999) 267–289. [9] Chukwujekwu Okafor A, Navdeep Singh, Enemouh UE, Design, analysis and performance of adhesively bonded composite patch repair of cracked aluminium aircraft Panels, Composite Structures. 71 (2005) 258–270. [10] Marie-Laetitia Pastor, Xavier Balandraud, Michel Grediac, On the fatigue response of aluminium specimens reinforced with carbon epoxy patches, Composite Structures. 83 (2008) 237–246. [11] A.O Randolph, M.F. Clifford, Journal of Reinforced Plastics and Composites. 4 (2002) 311–332. [12] Hosseini-Toudeshky H, Mohammadi B and Daghyani HR. Mixed mode fracture analysis of aluminium repaired panels using composite patches, Composites Science and Technology. 66 (2006) 188–198. [13] Hosseini-Toudeshky H, Mohammadi B and Bakhshandeh S. Mixed-mode fatigue crack growth of thin aluminium panels with single-side repair using experimental and numerical methods, Fatigue & Fracture of Engineering Materials & Structures. 30 (2007) 629–639. [14] Ayatollahi MR and Hashemi R. Mixed mode fracture in an inclined center crack repaired by composite patching, Composite Structures. 81 (2007) 264–273. [15] Ayotollahi MR and Hashemi R. Computation of stress intensity factors (KI, KII) and Tstress for cracks reinforced by composite patching, Composite Structures. 78 (2007) 602–609. [16] Bouiadjra Bachir B, Fekirini H, Belhouari M. SIF for double and single sided composite repair in mode I and mixed mode, Journal of Reinforced Plastics and Composites. 29 (2010) 1463–1477. [17] Ramji M and Srilakshmi R. Finite element modeling of composite patch repair. In: Proceedings of 5th international conference on theoretical, applied computational and experimental mechanics, IIT Kharagpur, India. December 27–29 (2010) 286–288. [18] Duong CH. A unified approach to geometrically nonlinear analysis of tapered bonded joints and doublers, International Journal of Solids and Structures. 43 (2005) 3498–3526.
[19] G.R. Irwin, Fracturing of Metals, ASM, Cleveland, Chio (1949), p 147. [20] H. Tada, P.C. Paris, G.R. Irwin, The stress analysis of cracks handbook, seconded., Del Research, Paris Production, Inc., St. Louis (1985). [21] Deryagin B.V. ET Krotova N.A. Doklady Akademii Nauk SSSR, vol. 61 (1948), pp 849. [22] ABAQUS/CAE Ver 6.9. User’s manual. Hibbitt, Karlsson & Sorensen, Inc (2007). [23] Lu, Y.B., Guan, B.T.,Lei,J.P. Influence of anisotropic material constants on plastic zone of crack tip and fracture characteristic. Wuhan JIAO TONG Sci. Tech. Univ.20 (1996), 273278. [24] Gao Xin, Wang Hangong, Kang Xingwu, Jiang Liangzhou, Analytic solutions to crack tip plastic zone under various loading conditions, European Journal of Mechanics A/Solids. 29 (2010), 738–45. [25] Erdogon F, Fracture mechanics of functionally graded material, J Comp Eng (1995); 5: 753–770. [26] A. Riccio, R. Ricchiuto, F. Di Caprio, A. Sellitto, A. Raimondo. Numerical investigation of constitutive material models on bonded joints in scarf repaired composite laminates. Engineering Fracture Mechanics. 2017; 173:91-106. [27] M Ramji, R Srilakshmi, Design of composite patch reinforcement applied to mixed-mode cracked panel using finite element analysis, Journal of Reinforced Plastics and Composites (2012) 31: 585-595. [28] Carl David Mortimer Liljedahl. Modelling the interfacial degradation in adhesively bonded joints. Submitted for a degree of Doctor of Philosophy in February 2006. University of Surrey Guildford Surrey United Kingdom.
Table 1. Elastic properties of the different materials Properties Longitudinal Young’s modulus E1 (GPa) Transversal Young’s modulus E2 (GPa) Transversal Young’s modulus E3 (GPa) Longitudinal Poisson’s ratio ߥ12 Transversal Poisson’s ratio ߥ13 Transversal Poisson’s ratio ߥ23 Longitudinal shear modulus G12 (GPa) Transversal shear modulus G13 (GPa) Transversal shear modulus G23 (GPa)
Materials Aluminum alloy T3 72 0.33 -
Boron/epoxy
Adhesive (FM-73)
200 25 25 0.21 0.21 0.21 7.2 5.5 5.5
2.55 0.32 -
Fig 1: Geometry of the repaired model (a) Front view (b) side view of asymmetrical patch (c) 3D Finite element model
500
Stress (MPa)
400
300
200
100
0
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
Strain (%)
Fig 2: Stress–strain curve of aluminum alloy 2024-T3.
50
stress (MPa)
40
30
20
10
0 0
1
2
3
4
5
strain (
Fig 3: Stress–strain curve of FM-73 adhesive [28]
(a)
(b)
Fig 4: Typical mesh model by finite elements of the plate, the patch and in the vicinity of the crack. (a) The global structure and (b) near the crack tip
5
4
without patch with patch
J (N/mm)
3
2
1
0 0
20
40
60
80
crack orientation Eq
Fig 5: J integral vs crack orientation β°
1,0 0,9 0,8
J*
0,7 0,6 0,5 0,4 0,3
0
20
40
60
80
crack orientation Eq
Fig 6: J* vs crack orientation for repaired face
0,6
PZS for crack angle 0° PZS for crack angle 45° PZS for crack angle 60°
0,4
Y(mm)
0,2 0,0 -0,4
-0,3
-0,2
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
-0,2 -0,4
X(mm)
-0,6 crack frond
Fig 7: Plastic zone shape (PZS) of unrepaired plate with different β
0,20 0,15
PZS for repaired plate PZS for unrepaired plate
crack frond
0,10 0,05
y (mm)
0,00 -0,2
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
-0,05 -0,10
x (mm)
-0,15 -0,20 -0,25 -0,30
Fig 8: Plastic zone shape (PZS) of repaired and unrepaired plate at β = 45 °
450
TGM2 TGM1
400
Young's modulus of the patch (GPa)
350 300 250 200 150
100 0,0
0,2
0,4
0,6 0,8 1,0 Thikness of the patch (mm)
1,2
1,4
1,6
Fig 9: Elastic modulus profile of the transversely graded material. (Variation of Young’s modulus (E) along the thickness of the patch)
3,0
2,5
1,5
unpatched face
patched face
J (N/mm)
2,0
[0]4 TGM1 TGM2 Composite [+45] /[-45]2
1,0
Composite [90]/[-45]2 Composite[ 90]/[0]2
0,5
Unreperaid plate
0
1
2
3
4
5
6
7
8
9
10
11
Normalised distance of thickness
Fig 10: Comparison of J integral variation through the thickness for the unrepaired and repaired plate having an inclined crack (2a = 10 mm and β = 45 °) with different configurations of patch
1,0
J integral (N/mm)
0,8
0,6
0,4
0,2
0,0
TGM1
TGM2
[90]2/[0]2
[90]2/[-45]2
[+45]2/[-45]2
[0]4
composite configuration
Fig 11: Comparison of the different configurations of repaired patch (a=5mm with β=45°).
14 plate
12 10
virtual line
Composite
A
8
Peel stress Vz (MPa)
free edge
B
6
[0]4
4
TGM2 TGM1 [90]2/[45]2
2
crack
0 -2 -4 -6 0,0
0,2
0,4
0,6
0,8
1,0
Path line [A-B] (mm)
Fig 12: Variation of peel stress along the path [A-B] of the repaired face
Wahid Oudad, Djamal Eddine Belhadri, Hamida Fekirini, Malika Khodja
PII: DOI: Reference:
S1270-9638(17)30191-8 http://dx.doi.org/10.1016/j.ast.2017.07.001 AESCTE 4095
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
29 January 2017 3 July 2017 4 July 2017
Please cite this article in press as: W. Oudad et al., Analysis of the Plastic Zone under Mixed Mode Fracture in Bonded Composite Repair of Aircraft Structures, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.07.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of the Plastic Zone under Mixed Mode Fracture in Bonded Composite Repair of Aircraft Structures Oudad Wahid1, Belhadri Djamal Eddine1, Fekirini Hamida2,*, Khodja Malika2, 3 1
Smart Structures Laboratory (SSL), University Centre of Ain Temouchent Po Box 284, 46000, Algeria.
2
Mechanical Physical Materials Laboratory (LMPM), Mechanical Engineering Department, University of Sidi Bel-Abbes 22000, Algeria. 3
CSIR Materials Science and Manufacturing, Meiring Naude Road Pretoria, 0184, South Africa *
Corresponding author: [email protected]
Abstract Material fracture by opening (mode I) is not the only failure criteria responsible for fracture propagation. Many industrial examples show the presence of mode II and mixed mode I + II. In the present work, numerical analyses of the three-dimensional and non-linear finite element method are used to estimate the performance of the bonded composite repair of metallic aircraft structures with a pre-existent damage by analyzing the plastic zone size ahead of repaired cracks under mixed mode loading, to assess the effect of the composite repair system on the plastic zone. The Von Mises stress is used to predict yielding of materials under this loading condition. The extension of the plastic zone, which takes place at the tip of a crack, strictly depends on many variables, such as the yield stress of the material, the loading conditions, the crack size and the thickness of the cracked component. The obtained results have demonstrated that the plastic zone ahead of the crack is significantly reduced by the presence of composite patch materials. Furthermore, parametric analysis has been carried out to evaluate the effect of lay-up and material system variation on the J integral. Keywords : J integral; plastic zone; crack; patch; Von Mises stress, crack inclination angle, peel stresses, composites bonded repair, Adhesive bonding, Boron/epoxy, Transversely graded material (TGM).
1. Introduction Bonded composite repairs of locally damaged metallic structures has gained considerable interest in aircraft structural maintenance and life extension solution in the last two decades [1, 2]. These repairs provide an efficient method for restoring the ultimate load capability of the structure [3]. The analysis of the effects of the geometrical properties of the composite on the repair performance has great interest in the literature. A good way to design a patch repair is to maximize the safety to cost ratio by finding the optimal composite patch shape [4, 5]. Riccio et al. [6] presented repair design tool aimed to help the designer by suggesting different repair typologies and proper repair size by means of optimization analyses that can provide the best repair solution with minimal adhesive shear stress and size of the repair patch. Their work has been tested against a literature case study on multistep composite-metal joints. They showed that minimization of the overlap length (which can be a fundamental requirement for repair design, especially for components with complex geometry) has been proven possible within an optimization process in order to provide practical repair configurations. The bonded repair reduces stresses in the cracked region and prevents the crack from opening and therefore from growing. The fiber composite patches have improved directional stiffness, high durability under cyclic loading, low density and excellent formability. Khodja et al. [7], have investigated, using finite element analysis, variation of the integral J depending on the crack size for different fiber orientations and different number of plies of repaired cracks in AA7075-T6 structures subjected to biaxial tensile stresses. Their study was carried out in order to estimate numerically the effect of biaxial tensile loading on the behavior of cracks in the presence of the bonded boron/epoxy repair in aircraft structures. The results show the beneficial effect of the patch composite in those cases and the relationship between the fiber orientations and it was noted that the best results are given by the orientation of 0 ° where the fibers are perpendicular to the crack direction. Baker [1] has initiated repair of aircraft aluminum structures using composite patch in the early 1970s mainly in order to enhance fatigue life of cracked components. From geometrical consideration, bonded repairs fall into two categories: double-sided (symmetric) and singlesided (asymmetric). In most of the practical cases, both sides of the cracked panels are not available to perform a symmetrical repair. Therefore, single-sided repair is often adopted such as in case of aircraft wings. This asymmetric repair causes a significant bending field which increases the stress intensity factor (SIF) at the crack tip beyond the value compared to unrepaired panel. This bending stress reduces the repair efficiency, hence static strength or fatigue life of the repaired model gets lower. Umamaheswar and Ripudiman Singh [8]
performed finite element modeling and analysis of single-sided composite patch repairs applied to thin aluminum sheets. They showed the SIF variation through the thickness of the panel assuming straight crack front. Chukwujekwu Okafor [9] developed a finite element model for analyzing the stress distribution of cracked plates repaired with a single-sided octagonal patch. They found that the zone of maximum stress shifted from the crack front (for unpatched specimen) to the edge of the patch (for the patched specimen) because of high peel stress development at overlay edge. Pastor et al. [10] have conducted experimental and numerical investigation of damaged and undamaged specimens patched with carbon epoxy composite material. They observed that the failure of the repaired model occurs when the maximum shear stress in the adhesive is close to its maximum shear strength. There are only a few investigations available on repairing panels in mixed-mode condition by the linear elastic and nonlinear fracture mechanics [11-17]. Hosseini-Toudeshky performed fatigue crack growth tests of single-sided repaired thick and thin panels containing center inclined cracks with various patch lay-ups configurations and various composite patch thicknesses [12, 13]. Ayatollahi and Hashemi [14, 15] used a finite element analysis to investigate the effect of composite patching on the SIF reduction for an inclined center crack panel under different mixed loading case. Bachir Bouiadjra et al. [16] have conducted FEA to estimate SIF in single and double-sided repairs in mode I and mixed mode edge-cracked panels. They have shown that the adhesive and composite patch properties have a significant and beneficial effect on the symmetrical patch. All these previous work do not thoroughly investigate the effect of bending in case of asymmetric repair, which in turn causes the peaking of stresses at the unpatched side generating higher SIF. Recently, Ramji et al. [17] have investigated that in the case of an asymmetrical patch, at the unpatched surface, the SIF value exceeds the value of SIF obtained in unrepaired panel. Therefore, the static strength of the repaired panel is reduced. The peel stresses in bonded joints normally peaks at the end of the overlap, which in turn can cause failure of the adhesive layer; thereby, reducing the performance of the repair. To avoid the severity of these peel stresses occurring at the overlapped ends, Duong [18] suggested usage of tapered patch. In the present work, the non-linear three-dimensional finite element method is used to compute the contour and the size of the plastic zone ahead of repaired cracks with bonded composite composite patch. The effects of the patch properties and the crack orientation on the plastic zone size are highlighted.
1.1. Stress field near the crack tip Irwin [19] has shown that the stress distribution near the crack tip can be described by the stress intensity factors (KI, KII, KIII) where each stress intensity factor is associated with a fracture mode. The relation that gives the stresses at the crack tip can be put in the form: ߪ ൌ
݂ ሺߠሻ ξଶగ
(1)
For the case of a mixed mode (I + II), the relation (1) can be written as ߪ ൌ
ξଶగ
ቀܭூ ݂ூ ሺߠሻ ܭூூ ݂ூூ ሺߠሻቁ
(2)
Where r and θ are the polar coordinates of the considered point, KI and KII are the stress intensity factors in mode I and II, and fij is a function dependent upon the polar angle. 1.2. Form and size of the plastic zone Due to the fact that the stress field at the crack tip reaches significant high values, it contributes to the creation of a plastic zone at the crack tip which depends more or less on the ductility of the material. The computation of the size depends on both the loads and the state of the stresses. Many authors have attempted to determine the form and to evaluate the size of the plastic zone at the crack tip on the basis of the classical criteria of elasticity or by calculation using the finite element method. In mode I, Irwin [20] proposed that the form of the plastic zone of dimension ry is circular and computed it as: ݎ௬ ൌ
ଵ
ଶ
(3)
ቀ ቁ
ఈగ ఙೊ
Where ߪ is the yield stress, α = 2 for plane stress and α= 6 for plane strain. ݎ௬ ൌ ܽ ܭூଶ ܾ ܭூூଶ ܿ ܭூ ܭூூ
(4)
Where ar, br and cr are the constants function of the Poisson coefficient ν and the yield stressߪ . ܽ Ͷሺͳ െ ʹ߭ሻଶ ሺͳ ܿߠݏሻ ͵ሺͳ െ ܿߠʹݏሻ ଵ ൈ ቐͶሺͳ െ ʹ߭ሻଶ ሺͳ െ ܿߠݏሻ ͻሺͳ ܿߠʹݏሻቑ ൝ܾ ൡ ൌ ଵగோమ ܿ െͺሺͳ െ ʹ߭ሻଶ ߠ݊݅ݏ ͳʹߠʹ݊݅ݏ
(5)
The J-integral value is evaluated using domain integral method [21] as shown in equation (6): ܬൌ ቂܹ݊ଵ െ ߪ ݊
డ௨ డ௫
ቃ ݀ݏ
(6)
2. Geometrical and materials properties The basic geometry of the cracked structure considered in this study is shown in Figure 1. Consider rectangular elastic-plastic aluminum 2024-T3 plate with dimensions of 39x160x3 mm3, with an inclined center crack ‘2a’ of length 10 mm. The crack is inclined at an angle of β = 45° with the horizontal as shown in Figure 1. The plate is subjected to a uniaxial load of 17.5 kN (σ =150MPa). The boron–epoxy patch of dimensions 25x25x1.5 mm3 is bonded asymmetrically using 0.1 mm thin film FM73 structural adhesive. The layer thickness of the laminate is taken as 0.375 mm. The composite had unidirectional lay-up where the fibers were oriented along the specimen length direction (parallel to the direction of load). The general material properties of aluminum panel, composite patch and adhesive are given in Table 1. The specimen dimensions follow the ASTM E-647 standard [12]. Standard tensile tests were carried out on Aluminum 2024-T3 and FM73 adhesive. The obtained stress–strain curves are presented in Figures 2 and 3 respectively. 3. Finite element model The analysis involved a three-dimensional finite element method by using a commercially available finite element package code ABAQUS [22]. The finite element model consists of three subsections to model the cracked plate, the adhesive, and the composite patch. The plate had four layers of elements in the thickness direction, the adhesive had only one layer of elements through thickness, and the composite patch had two layers of elements through thickness. The structure made up of the plate, the adhesive and the patch is subjected with uniform uniaxial tensile amplitude of 150MPa. The boundary conditions of fixing, subordinated to the conditions of tensile uniaxial loading of the geometry, were introduced into the initial phase. These conditions are represented in a 3D model as shown in Figure 1 and described as follows: - One edge of the plate has been clamped. - Uniform traction displacements are applied to the free edge of plate in Y direction. The grid used in the 3D model of finite elements is represented in Figure 4 and shows the overall mesh of the specimen and mesh refinement at the crack tip region. The mesh was generated using 3D hexagonal shape and was modeled with 20 node reduced integration quadratic brick elements (C3D20R). The mesh was refined near the crack tip area, a circular zone of radius 3 mm around the crack front was modeled with an element dimension of 0.05 mm using at least 15 such fine elements in the front and back of the crack tip. The
procedure used in the finite element analysis involved the following steps: (i) the tensile stress was applied to the gripped specimen; (ii) general static ‘STEP’-option was used for analysis with ABAQUS; (iii) Automatic increment of steps was used with maximum increments number of 100. Minimum increment size was 10-5 and maximum increment size was 1. Nevertheless, the ABAQUS solver code could override matrix solver choice according to the ‘STEP’- option. Abaqus does not recognize mechanical contact between part instances or regions of an assembly unless that contact is specified in the Interaction module. The mere physical proximity of two surfaces in an assembly is not enough to indicate any type of interaction between the surfaces. Therefore the ‘‘TIE’’-option was used for adhesive-plate and adhesivepatch interactions. A tie constraint allows fusing together two regions even though the meshes created on the surfaces of the regions may be dissimilar. This allows Abaqus to use the default meshing techniques and automatically generate tie constraints across the incompatible interfaces. Abaqus automatically chooses one side of the interface as the slave surface and the other as the master surface for the automatically generated tie constraint, but creates common (merged) nodes on the perimeter of the incompatible interface. Abaqus generally selects the surface with the finer mesh to be the slave surface. The computation for the depth of the slave node adjustment zone for the tie constraint is based on the bounding dimensions of the interfacing regions. The code default value was used for the position of tolerance. Rotational degrees of freedom (DOF) of nodes between the different sub-sections of the model were tied together. The Von Mises yield criterion is used to predict plastic deformation. Incremental plasticity theory is introduced to model the material nonlinearity. The Newton–Raphson iterative approach is used for resolving nonlinear finite element equations. [22] 4. Analysis and results 4.1. Reduction of J integral For a measure of the fracture mechanic safety and patching efficiency criterion at the repaired crack, the non-dimensionalised reduction of the J integral can be used: ܬ כܬൌ ͳ െ ൘ܬ ௨
(7)
Whereܬ ,ܬ௨ are ܬintegrals for the patched and unpatched crack plate respectively. The reduction of J integral is important to the design of the repaired cracked plate because this value
defines the patch efficiency. As כܬincreases, the crack propagation decreases. On the other hand, as כܬdecreases, the possibility of fracture increases. Figure 5 shows the variation of J integral according to the inclined crack angle for patched and unpatched crack plate. Two behaviors are observed: The first is where the maximum value of J integral is recorded for crack inclination of 0° corresponding to the pure mode I and the J integral value is null (zero) for an angle of 90° corresponding to the case of a crack parallel to the loading axis (unpatched crack plate). The second behavior is when the, J integral value is maximal for 45°. This last value corresponds to the high shear stresses (patched crack plate). The beneficial effect of the application of the repair technique can be seen. For the best estimate of the advantages of the presence of the composite patch, the variation of the J* as a function of the inclined crack angle is illustrated in Figure 6. The reduction ratio of the J integral for the repaired and unrepaired plate is between of 38 and 87%. This reduction does not have any sense when the crack inclination tends towards 90° because the J integral for repaired and unrepaired tend towards zero. This fact is due the closing of crack under this loading case. One can thus confirm that the use of a composite patch offers an enormous advantage for repaired inclined cracks and consequently increasing the service life. On the other hand, the decrease in repair efficiency for angle of 45 ° is essentially due to the dominance of mode II of rupture, which generates high shear stresses, as shown by Hosseini [13] for an elastic plate. To overcome this drawback, another type of composite patch must be used to improve this technique. 4.2. Effect of the crack orientation on the shape of plastic zone A plastic zone is characterized by a Von Mises stress value higher than 350 N/mm2, i.e. the yield stress value of the considered material. The shape and size of the plastic zone is shown in Figure 7 for unrepaired plate, for β = 0° (pure mode I), 45° and 60° respectively. It can be seen that the size of the plastic zone is significantly affected by the variation of the crack orientation. The size of the plastic zone decreases with the increment of the crack inclination angle β between 0° and 90°). The crack tip plastic zone in mode I present a “butterfly-like” shape. The plastic zone has a symmetrical distribution to the initial crack plane. The results are consistent with those shown by Lu et al. and Gao Xin et al. [23, 24].
To examine the efficiency of the patch, the shape and size of the plastic zone were plotted. It can be observed from Figure 8 that the plastic zone has a non-symmetrical distribution to this crack plane β = 45o and the size of the plastic zone for repaired plate is smaller by 40% than the size in the case of unrepaired plate at the crack tip. This behavior is due to the fact that the shear stress field at the crack tip under the patch becomes smaller and it contributes to minimize the creation of plastic deformation and a plastic zone around the crack tip. In correlation with Figure 5, it has been concluded that the energy approach of the nonlinear mechanics of the fracture based on the integral J affects the plastic zone size for the repaired and unrepaired structure. The reduction in the plastic zone is due to the beneficial effect of the patch on the absorption of the stresses at the crack tip vicinity and that improves the fatigue life of the structure. 4.3. Effect of the material properties of composite patch A three dimensional finite element (FE) analysis was performed to understand further the variation of the J integral through the thickness. In particular, the behavior of the J integral for unrepaired plate and repaired plate with various composite patch configurations was investigated. The details are presented in the following sections. Transversely graded material (TGM) as the patch material is proposed in this section. This material is new and still in the development stage. TGM is an orthotropic composite material whose Young's modulus varies linearly along the thickness [25]. The transversely graded material was used as a sequence of homogeneous layers for 1.5 mm thickness of composite patch for the bonded repair, each having a constant elastic modulus. The overall elastic modulus gradient was used by assigning different modulus values to different layers. Two transversely graded materials with different Young's modulus gradient variations were considered namely TGM1 and TGM2. Figure 9 shows the variation of Elastic modulus profile (E) of both transversely graded materials across the composite thickness where TGM1 varies linearly with the Young's modulus gradient from 200 GPa to 250 GPa and TGM2 varies linearly with the Young's modulus gradient from 300 GPa to 450 GPa. In this paragraph the effects of different parameters such as composite layup and composite material on the J integral is highlighted. Six configurations of patch are used: transversely graded material TGM1 & TGM2 and Boron/epoxy composite having different layup orientations such as [+45]2/[-45]2, [+90]2/ [-45]2, [+90]2/[0]2 and [0]4 (unidirectional). The
comparison of J integral throughout the plate thickness between repaired and unrepaired face is shown in Figure 10. The transversely graded patched model and the balanced patched layup [+90]2/[0]2 have an efficient effect to significantly reduce the value of J integral in both faces of the repaired plate compared to the unidirectional composite [0]4. In addition, in the repaired face a reduction of J integral has been observed in the TGM configuration compared to other configurations, because the transversely graded composite TGM absorbs more energy than the others. The fiber orientation based on the crack direction is important to consider to ensure good absorption of the stresses at the crack tip and the repairing material strength under mixed mode loading as shown by Khodja et al. [7]. Of further importance is material choice and the bonding durability to ensure better mechanical behavior of bonded composite repairs and proposed material models with the use of finite element assessment and the rich background in the literature to provide detailed and reliable information on the stress distribution, failure onset, and crackpath propagation as demonstrated by A. Riccio et al. [26]. To clearly illustrate the previous results in Figure 10, Figure 11 shows the data in the form of bar charts where one can see and compare easily the results of the various configurations of repair. Distinctly, the TGM1 and 2 materials composite (patch) reduce the J integral considerably compared to the other configurations of repairs. This is due to the variation across the thickness that is influenced by the strength of gradation where the gradation in TGM1 is lower than TGM2 due to differences in the Young modulus gradient. TGM2 gives better results, which influences the plastic zone shape and size. 4.4. Peel stress analysis The adhesive is the weak point of reinforcement by composite patch materials. It is a material having very low mechanical shear characteristics and it is known that de-bonding is triggered by the development of the peel stress concentrations (σz) at the overlap ends. The peel stresses distribution and variation in the repaired face and the overlap area along a virtual path line [AB] drawn to plot the results as shown in Figure 12, with different composite patches configurations. Three behaviors have been noted on the peel stresses along the area of interest, starting from the point A towards the point B. The first area covers the repaired face where the patch is bonded and covers the plate and low peel stresses are found in the second area in the middle, peak values of peels stresses are high as a side effect of the bond line between the patch
composite and the adherent. For this reason, tapered patch design is recommended to reduce the edge effect. Finally, the third area of the free edge has low values of peel stresses at the end of bond line and goes towards zero where the undamaged area of the plate is reached. The plots shows similar peel stresses behavior has been noticed with various composite patch configurations with different levels but it is clear that the use of a TGM material generates low peel stresses in the proximity of the patch free edge and consequently decreases the debonding probabilities. This is due to the influence of elastic gradient properties that vary along the thickness compared to the use of the boron /epoxy with different lay-up orientation. The use of TGM is recommended to extend the service life of cracked structural components. This result is also confirmed by Ramji and Srilakshmi [27]. 5. Conclusion The nature of the plastic zone that is formed ahead of a crack tip plays a very important role in the determination of the type of failure that occurs. In this study the shape and size of crack tip plastic zones have been estimated by three finite element analyses in an aluminum 2024-T3 plate repaired and unrepaired under mixed mode loading according to Von Mises yield criteria. The J integral was also calculated at the crack tip. It was found that: x
The reduction ratio of the J integral for the repaired and unrepaired plate is between 38 and 87% depending on the crack type. This reduction does not have any sense when the crack inclination tends towards 90° because the J integral for repaired and unrepaired tend towards zero. This fact is due the closing of crack under this loading case.
x
The plastic zone size and shape are influenced by the crack inclination which decreases with the increment of the crack inclination angle β (β = 0° corresponding to the pure mode I). The shear stress field at the crack tip under the patch composite decreases; it contributes to minimize the creation of plastic deformation and a plastic zone around the crack tip.
x
The transversely graded patched model and the balanced patched layup [+90]2/[0]2 have an efficient effect to significantly reduce the value of the J integral in both faces of repaired plate compared to the unidirectional composite (patch) repaired in these cases.
x
The use of a TGM generates low peel stress in the proximity of the patch and consequently decreases the debonding probability.
x
The energy approach based on the integral J of fracture of nonlinear mechanics affect the plastic zone size for repaired and unrepaired structure. The reduction in the plastic zone is due to the beneficial effect of the patch repair on the absorption of the stresses at the crack tip, which improves the fatigue life of the structure.
x
The variation across the thickness is influenced by the strength of gradation of transversely graded material properties. Optimal gradation with respect to applied load and crack direction must be investigated.
References
[1] Baker Alan A. A proposed approach for certification of bonded composite repair to flightcritical airframe structure, Applied Composite Materials. 18 (2011) 337–69. [2] Ahn JS, Basu PK and Woo KS. Analysis of cracked aluminum plates with one-sided patch repair using p-convergent layered model, Finite Elements in Analysis and Design. 46 (2010) 438–48. [3] Bakuckas John G, Westerman Bud. Fatigue and residual strength performance of bonded repair to metallic fuselage, Structural Integrity: Influence of Efficiency and Green Imperatives. 8 (2011) 735–52. [4] Breitzman TD, Iarve EV, Cook BM, Schoeppner GA, Lipton RP. Optimization of a composite scarf repair patch under tensile loading, Composites Part A: Applied Science and Manufacturing. 40 (2009) 1921–30. [5] Ouinas D, Hebbar A, Bachir Bouiadjra B, Belhouari M, Serier B. Numerical analysis of the stress intensity factors for repaired cracks from a notch with bonded composite semicircular patch, Composites Part B: Engineering (2009) 804–10. [6] A. Riccio, G. Di Felice, F. Scaramuzzino, A. Sellitto. A Practical Tool for the Preliminary Design of Bonded Composite Repairs. Applied Composite Materials. 2014; 21(3):495-509. [7] M. Khodja,H. Fekirini, G. Corderley, S. Govender, Repaired crack in AA7075-T6 structures subjected to biaxial tensile stresses, Insights and Innovations in Structural Engineering, Mechanics and Computation, Taylor & Francis Group, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 CRC Press 2016, 563-566.
[8] Umamaheswar Turaga VRS and Ripudaman Singh.Modeling of a patch repair to a thin cracked sheet, Engineering Fracture Mechanics. 62 (1999) 267–289. [9] Chukwujekwu Okafor A, Navdeep Singh, Enemouh UE, Design, analysis and performance of adhesively bonded composite patch repair of cracked aluminium aircraft Panels, Composite Structures. 71 (2005) 258–270. [10] Marie-Laetitia Pastor, Xavier Balandraud, Michel Grediac, On the fatigue response of aluminium specimens reinforced with carbon epoxy patches, Composite Structures. 83 (2008) 237–246. [11] A.O Randolph, M.F. Clifford, Journal of Reinforced Plastics and Composites. 4 (2002) 311–332. [12] Hosseini-Toudeshky H, Mohammadi B and Daghyani HR. Mixed mode fracture analysis of aluminium repaired panels using composite patches, Composites Science and Technology. 66 (2006) 188–198. [13] Hosseini-Toudeshky H, Mohammadi B and Bakhshandeh S. Mixed-mode fatigue crack growth of thin aluminium panels with single-side repair using experimental and numerical methods, Fatigue & Fracture of Engineering Materials & Structures. 30 (2007) 629–639. [14] Ayatollahi MR and Hashemi R. Mixed mode fracture in an inclined center crack repaired by composite patching, Composite Structures. 81 (2007) 264–273. [15] Ayotollahi MR and Hashemi R. Computation of stress intensity factors (KI, KII) and Tstress for cracks reinforced by composite patching, Composite Structures. 78 (2007) 602–609. [16] Bouiadjra Bachir B, Fekirini H, Belhouari M. SIF for double and single sided composite repair in mode I and mixed mode, Journal of Reinforced Plastics and Composites. 29 (2010) 1463–1477. [17] Ramji M and Srilakshmi R. Finite element modeling of composite patch repair. In: Proceedings of 5th international conference on theoretical, applied computational and experimental mechanics, IIT Kharagpur, India. December 27–29 (2010) 286–288. [18] Duong CH. A unified approach to geometrically nonlinear analysis of tapered bonded joints and doublers, International Journal of Solids and Structures. 43 (2005) 3498–3526.
[19] G.R. Irwin, Fracturing of Metals, ASM, Cleveland, Chio (1949), p 147. [20] H. Tada, P.C. Paris, G.R. Irwin, The stress analysis of cracks handbook, seconded., Del Research, Paris Production, Inc., St. Louis (1985). [21] Deryagin B.V. ET Krotova N.A. Doklady Akademii Nauk SSSR, vol. 61 (1948), pp 849. [22] ABAQUS/CAE Ver 6.9. User’s manual. Hibbitt, Karlsson & Sorensen, Inc (2007). [23] Lu, Y.B., Guan, B.T.,Lei,J.P. Influence of anisotropic material constants on plastic zone of crack tip and fracture characteristic. Wuhan JIAO TONG Sci. Tech. Univ.20 (1996), 273278. [24] Gao Xin, Wang Hangong, Kang Xingwu, Jiang Liangzhou, Analytic solutions to crack tip plastic zone under various loading conditions, European Journal of Mechanics A/Solids. 29 (2010), 738–45. [25] Erdogon F, Fracture mechanics of functionally graded material, J Comp Eng (1995); 5: 753–770. [26] A. Riccio, R. Ricchiuto, F. Di Caprio, A. Sellitto, A. Raimondo. Numerical investigation of constitutive material models on bonded joints in scarf repaired composite laminates. Engineering Fracture Mechanics. 2017; 173:91-106. [27] M Ramji, R Srilakshmi, Design of composite patch reinforcement applied to mixed-mode cracked panel using finite element analysis, Journal of Reinforced Plastics and Composites (2012) 31: 585-595. [28] Carl David Mortimer Liljedahl. Modelling the interfacial degradation in adhesively bonded joints. Submitted for a degree of Doctor of Philosophy in February 2006. University of Surrey Guildford Surrey United Kingdom.
Table 1. Elastic properties of the different materials Properties Longitudinal Young’s modulus E1 (GPa) Transversal Young’s modulus E2 (GPa) Transversal Young’s modulus E3 (GPa) Longitudinal Poisson’s ratio ߥ12 Transversal Poisson’s ratio ߥ13 Transversal Poisson’s ratio ߥ23 Longitudinal shear modulus G12 (GPa) Transversal shear modulus G13 (GPa) Transversal shear modulus G23 (GPa)
Materials Aluminum alloy T3 72 0.33 -
Boron/epoxy
Adhesive (FM-73)
200 25 25 0.21 0.21 0.21 7.2 5.5 5.5
2.55 0.32 -
Fig 1: Geometry of the repaired model (a) Front view (b) side view of asymmetrical patch (c) 3D Finite element model
500
Stress (MPa)
400
300
200
100
0
0,02
0,04
0,06
0,08
0,10
0,12
0,14
0,16
0,18
Strain (%)
Fig 2: Stress–strain curve of aluminum alloy 2024-T3.
50
stress (MPa)
40
30
20
10
0 0
1
2
3
4
5
strain (
Fig 3: Stress–strain curve of FM-73 adhesive [28]
(a)
(b)
Fig 4: Typical mesh model by finite elements of the plate, the patch and in the vicinity of the crack. (a) The global structure and (b) near the crack tip
5
4
without patch with patch
J (N/mm)
3
2
1
0 0
20
40
60
80
crack orientation Eq
Fig 5: J integral vs crack orientation β°
1,0 0,9 0,8
J*
0,7 0,6 0,5 0,4 0,3
0
20
40
60
80
crack orientation Eq
Fig 6: J* vs crack orientation for repaired face
0,6
PZS for crack angle 0° PZS for crack angle 45° PZS for crack angle 60°
0,4
Y(mm)
0,2 0,0 -0,4
-0,3
-0,2
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
-0,2 -0,4
X(mm)
-0,6 crack frond
Fig 7: Plastic zone shape (PZS) of unrepaired plate with different β
0,20 0,15
PZS for repaired plate PZS for unrepaired plate
crack frond
0,10 0,05
y (mm)
0,00 -0,2
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
-0,05 -0,10
x (mm)
-0,15 -0,20 -0,25 -0,30
Fig 8: Plastic zone shape (PZS) of repaired and unrepaired plate at β = 45 °
450
TGM2 TGM1
400
Young's modulus of the patch (GPa)
350 300 250 200 150
100 0,0
0,2
0,4
0,6 0,8 1,0 Thikness of the patch (mm)
1,2
1,4
1,6
Fig 9: Elastic modulus profile of the transversely graded material. (Variation of Young’s modulus (E) along the thickness of the patch)
3,0
2,5
1,5
unpatched face
patched face
J (N/mm)
2,0
[0]4 TGM1 TGM2 Composite [+45] /[-45]2
1,0
Composite [90]/[-45]2 Composite[ 90]/[0]2
0,5
Unreperaid plate
0
1
2
3
4
5
6
7
8
9
10
11
Normalised distance of thickness
Fig 10: Comparison of J integral variation through the thickness for the unrepaired and repaired plate having an inclined crack (2a = 10 mm and β = 45 °) with different configurations of patch
1,0
J integral (N/mm)
0,8
0,6
0,4
0,2
0,0
TGM1
TGM2
[90]2/[0]2
[90]2/[-45]2
[+45]2/[-45]2
[0]4
composite configuration
Fig 11: Comparison of the different configurations of repaired patch (a=5mm with β=45°).
14 plate
12 10
virtual line
Composite
A
8
Peel stress Vz (MPa)
free edge
B
6
[0]4
4
TGM2 TGM1 [90]2/[45]2
2
crack
0 -2 -4 -6 0,0
0,2
0,4
0,6
0,8
1,0
Path line [A-B] (mm)
Fig 12: Variation of peel stress along the path [A-B] of the repaired face