Numerical Analysis of the Influences of Geometrical Deviation on Delamination in Composite Laminates around the Countersunk Hole

During countersunk hole machining, defects like geometrical deviation of the chamfer angle and delamination are easily introduced into the structure. To investigate the influences of geometrical deviation on delamination propagation around the countersunk hole during assembly, a progressive damagemodel (PDM) combining cohesive element was proposed and validated. Numerical analyses were then carried out to study delamination propagation behavior under the influences of geometrical parameters including delamination factor, chamfer angle, and location of delamination. The results show that when delamination appears at the transition area of the countersunk hole, the load causing the delamination evolution is much smaller than other cases.


Introduction
Carbon fiber reinforced plastic (CFRP) possesses the properties of lightweight and high-specific stiffness and strength, all of which make them suitable for a wide range of high responsibility applications in aircraft structures.Although there are many ways to assemble two different CFRP parts, countersunk fasteners provide aerodynamic benefits compared to protruding head ones and are of particular interest for aircraft skin applications [1].However, delamination is easily introduced at the neighborhood of the countersunk hole in the process of machining such as drilling, which decreases the stiffness of structure, leads to the destruction of laminated plates in advance and seriously affects the safety of the structure of the aircraft.Studies on the effects of delamination on strength of composite structure have been carried out for many years.
Finite element method (FEM) is widely employed to predict the mechanical behavior of composite structures with delamination by means of the virtual crack closure technique (VCCT) [2][3][4][5] and cohesive zone model (CZM) [6][7][8].The VCCT is based on the assumption of the energy release rate, an initial defect or crack length, and complex mesh moving technique, while CZM can predict both the onset and the non-self-similar propagation of delamination.In recent years, the cohesive element methods have been used widely which can conveniently and accurately predict the delamination initiation and growth.Turon et al. [9,10] presented an accurate analytical solution for delamination growth under mixed-mode loading using cohesive elements.The significant effects of buckling and delamination growth for various parameters (delamination size and distribution) in slender composite laminate were investigated with FEM based on a cohesive element [11,12].Ataş et al. [13][14][15] conducted numerical analysis to study the clamping force effects on the delamination onset and growth: pinned and bolted (protruding head bolts) composite laminates.One can expect that the compressive normal stresses (σ 33 ) developed under the washer area due to the initial clamping force could suppress the onset of delamination.
McCarthy et al. [16,17] carried out an experimental and numerical investigation into the effect of clearance on the stiffness and strength of single-bolt, single-lap, and carbonepoxy joints.Finger-tight and fully torqued conditions were also investigated.Meanwhile, Chishti et al. [18,19] also performed an experimental and numerical investigation of the effects of bolt torque, clearance, and countersunk height ratio on damage progression and strength of countersunk composite joints.It indicated that the introduction of the countersunk hole roughly halved the bearing stress and caused delamination for some configurations.Increasing the bolt torque was found to increase damage in the countersunk area whilst reducing it close to the shear plane, without affecting the distribution of damage around the hole.Most of them were devoted to investigate that the initial bolt clamping force can suppress the delamination initiation and growth in CFRP under tensile load; few attempts have been made to predict the delamination initiation and propagation in composite laminates during the process of the preloading of bolts, especially when delamination occurs around the countersunk hole.
The aim of this paper is to systematically investigate the factors that affect the growth of delamination around the countersunk hole under the condition of bolt preloading with the progressive damage model (PDM) and cohesive element, which is based on a three-dimensional Hashin-type criterion and bilinear traction-separation law, respectively.

Geometry and Materials
Load conditions of wing skin and beam connection structure are shown in Figure 1.Due to the anisotropic properties of CFRP, composite parts are easily prone to warpage during formation.Consequently, there will always be an interfacial gap between assembly components, especially for large parts.Meanwhile, stress concentration is prone to take place at the countersunk located at the skin.In order to facilitate this study, only wing skin is concerned.The configuration of the structure with a countersunk hole and subjected to preloading is shown in Figure 2.
An orthotropic material system of 20 plies of carbon fiber/ epoxy (IMS194/CYCOM 977-2) with symmetrical stacking sequence [45/90/−45/0/90/0/−45/90/45/−45] S is considered in the analysis of the present study [20].The nominal singlelayer thickness of the laminate is 0.15 mm, and its mechanical properties are given in Table 1.The properties of the cohesive layer are listed in Table 2. Bolt is considered as a titanium single solid and the properties are E = 110 GPa and υ = 0.3.

Failure Theory and Finite Element Model
The failure model employed here includes the progressive failure and delamination.The initiation of the failure was   2 International Journal of Aerospace Engineering determined by three-dimensional Hashin-type criteria and the modified Tserpes' material degradation model [22,23].
The delamination, on the other hand, was modeled using cohesive elements.
3.1.Progressive Failure Model.Generally, the progressive failure analysis of the composite laminate material consists of two major steps [24].The first step is to choose the appropriate failure criteria to determine which kind of failure mode will occur.Although many failure criteria had been reported so far, the Hashin-type criteria have been widely used because it has the ability to distinguish seven failure models of composite laminate and suits for three-dimensional conditions as present in this literature.Therefore, the 3D Hashintype failure criteria are adopted for the failure predictions of composite materials, which is shown in Table 3, where σ ij (i,j = 1, 2, 3) are the scalar components of the stress tensor, and S ij (i,j = 1, 2, 3) are the material strengths in longitude, transverse, and through-the-thickness directions of lamina, respectively.The superscripts T and C denote tension and compression, respectively, and α is a constant parameter related to the shear nonlinear behavior, which is determined experimentally.In any element, when the criterion e > 1 is satisfied, the element fails by the associated failure.The second step is to choose the suitable material degradation rules for a reduction in stiffness of the composite material after the occurrence of a certain type of failure.The commonly used degradation methods are the total discount method, the limit discount method, and the residual property method.Since Tserpes' material degradation model contains seven failure modes and corresponds to the 3D Hashin-type failure criteria, on the basis of the modified Tserpes' material degradation model, the limit discount method of the unidirectional lamina is applied in this research, and the degradation rules are shown in Table 4.The failure criteria and the degradation rules of composite materials are implemented through the ABAQUS user subroutine UMAT.
The flow chart of the UMAT subroutine is given in Figure 3.The simulation procedure stops as soon as either the displacement load is reached or the simulation fails to converge prematurely.In this study, the equal strain formats of the failure criteria are used for simplicity.In addition, in order to improve convergence, the viscous regularization technique of the damage variables is implemented with discretized form.Since the subroutine is not the focus of this study, for details, please refer to [25].

Delamination.
It has been proved that the bilinear traction-separation law can perfectly represent delamination initiation and propagation.So, in this study, the interface damage is modelled using the traction-separation law, which is commonly used to describe the delamination of the composite.Interfacial failure or delamination is assumed to initiate once the following quadratic interactive criterion is satisfied: where the symbol represents the Macaulay bracket operator.t n , t s , and t t are the normal stress and the two shear The properties of the cohesive layer [21].Under mixed-mode loading, the bilinear softening constitutive law was used for delamination propagation and the mixed-mode fracture energy criterion developed by Benzeggagh and Kenane [26] was used here (B-K criterion).If mode III occurs, the criterion can be given by where G I , G II , and G III are the modes I, II, and III strain energy release rates.G IC and G IIC are the critical strain energy release rate (fracture toughness) in modes I and II.η is a material parameter, which can be obtained by the mixmode bending test and η = 2 0 in this study [27].
Fiber failure Calculate the fiber damage variable In-plane matrix failure Calculate the in-plane matrix damage variable Out-of-plane matrix failure   The structure of the composite layer and interface including the embedded delamination defect around the countersunk hole is illustrated in Figure 4.The cohesive layer was created by the offset method and inserted into two laminate plies [20].In order to assure that the results are not dependent upon the element's size, the mesh refinement process is necessary.For the details, please refer to [28].After refinement, the composite plate was modelled with 26,140 elements and the cohesive layer with 940 elements, and 20,916 elements were used to model the bolt.The finite element mesh of the specimen is shown in Figure 4(a).The cohesive element has been positioned in the area (Figure 5(a), zones II and III) around the embedded delamination defect (Figure 5(a), zone I) to predict the delamination evolution.The boundary conditions are shown in Figure 5(b).Surface-to-surface contact element has been placed in the delamination zone to avoid overlaps between elements.Due to large displacements and composite material behavior, the nonlinear stress calculation of the problems presented here was performed using the ABAQUS/ Standard code.

Results and Discussion
4.1.Model Validation.In order to verify the effectiveness of the proposed method of combining the abovementioned progressive damage method and cohesive model, the finite element model was established according to the geometry and material parameters in the literature [29].The countersunk head (CS) specimens used was the HL523 pin with HL97 collar.The CFRP material used was CYCOM970 PWC T300 3 K ST, a plain weave epoxy prepreg with ply thickness of 0.22 mm.A symmetric, nominally quasiisotropic lay-up was used in all specimens.A test rig was designed to determine the pullout loads of fasteners in composite laminates under a quasi-static tensile load condition.The obtained numerical results were compared with the experimental data and numerical prediction from literature [29], as shown in Figure 6.An excellent agreement between numerical prediction and experimental data was found in both stiffness and bearing strength.The proposed method predicted the initial subcritical failure load of 5005 N, similar to the experimental result of 5000 N, with an error less than 0.1%, and the error of bearing failure load obtained from numerical result of 5561 N and experimental data of 5545 N was only 0.3%.Moreover, the present model can reproduce

Delamination Propagation around the Countersunk
Hole.As mentioned above, the structure of the study under the bolt preload can be simulated by applying the pullthrough load.When there is an embedded delamination defect around the countersunk hole in composite laminate, the defect is easy to propagate under the condition of large bolt preloading.Three typical processes of delamination propagation are damage initiation, damage evolution, and structural failure.Figure 7 shows a typical load/displacement curve of a laminate with circular shape delamination defect (R = 6 mm), which has been examined by the similar experimental results in the literature [30].Those processes can be decomposed into four main steps: (i) Step I: no damage, linear elastic behavior (ii) Step II: delamination initiation (see Figure 8 The delamination factor (F d ) can be calculated from the ratio of the maximum diameter (D max ) of the delamination zone to the drill diameter (D 0 ), as given in [30] In order to expatiate on the delamination size factor, four kinds of the composite laminates with different F d (1.0, 1.5, 2.0, and 2.5) were investigated, which were under the condition of the same countersunk angle (100 °) and through-the-thickness position (t/t 0 = 1/2), as shown in Figure 9(a).Figure 10 shows the variation of bearing load as a function of F d for the case of through-the-thickness position t 0 /t = 1/2.The initial subcritical failure load decreases as F d increases from 1.0 to 2.5.Meanwhile, the increase of F d also leads to the structural stiffness decreasing.When F d is increased up to 2.5 (i.e., D max = 15 mm), the stiffness of structure directly declined by 17.75% compared with that of the no defect one.It is easy to understand that the extremely large delamination area results in a decrease in the plate resistance against the bearing behavior.Thus, new damage is produced at a far lower bolt preloading level.
Moreover, the load causing delamination evolution decreases with the increase of F d , which can be seen from Figure 11.When F d is increased up to 2.5, the load declined obviously by 10.26% compared with that of the none embedded one and was very close to the initial subcritical failure load.As explained in the literature [31], the delamination occurs abruptly at the beginning of the test when the load is relatively low (i.e., the lower initial subcritical failure load).
The shapes of the initial damage vary a lot with the change of F d (Figures 11(a)-11(d)).When there is no embedded delamination defect or in small size, the area of the initial damage is larger (Figures 11(a) and 11(b)).In addition, the damage evolves from the nearby hole to the hole boundary for the no embedded defect case, while the propagation path is opposite for the other case.This is due to the appearance of the embedded delamination defect and its size affects the propagation path.

Effect of the Chamfer Angle.
As explained before, we investigated the rib-and wing skin-connecting structure, using the countersunk bolt joints, and the wing skin needs to be drilled by countersunk holes.In the process of drilling, however, it is not only possible to introduce the delamination defect around the countersunk hole but also the deviation of chamfer angle, as shown in Figure 12.As we only focus on the standard 100 °countersunk-head fasteners in this study, which were made of titanium alloy (Ti-6Al-4 V), so the countersunk angle θ c of the composite laminate is 100 °± 2 °(i.e., θ c = 98 °, 99 °, 100 °, 101 °, and 102 °).Figures 13 and 14 show the effects of the countersunk in the laminate plate on the delamination evolution and the initial subcritical failure load under different delamination factors.When the countersunk angle of the laminate is small (θ c < 100 °, Figure 12(a)), the load causing delamination evolution and bearing capacity increase by 59 N and 58 N, respectively, with the reduction of the countersunk angle per 1 °.The reason for this phenomenon is that the bolthead will firstly contact with the upper surface of the plate and then the transition area due to the fact that the bolt angle is larger than that of the countersunk.Thus, the bolt will bear the upper surface of the plate and then press the transition area (see Figure 12      International Journal of Aerospace Engineering when the countersunk angle of the composite plate is large (θ c > 100 °), with per 1 °increasing, the load causing delamination evolution declines by 129 N, and the bearing capacity decreases by 117 N. As the countersunk hole of the plate is larger than that of the countersunk bolt, the bolt directly bears the transition area of the countersunk, leading to delamination evolution which occurs at the transition zone.
No matter how different the countersunk angle is, the damage occurs at almost the same location on condition that the embedded delamination defect locates at the same through-the-thickness position (Figures 13(a)-13(e)), which implies that the countersunk angle only has influence on the load causing delamination evolution.

Effect of the Delamination Location.
In order to describe the position of the embedded delamination defect through the thickness of the plate, the thickness distribution factor (T d ) is defined as the ratio of the distance (t) from the embedded delamination defect to the upper plate surface to the plate thickness (t 0 ), as shown in Figure 9(a).When the plates are drilled, it is easy to introduce delamination defects into the transition area of the countersunk holes.Therefore, we only focus on the delamination appearing in the transition region.
Figures 15 and 16 show the variation of delamination evolution thresholds and the initial subcritical failure loads as a series of T d , namely, T d = 0.05, 0.25, 0.40, 0.45, 0.50, 0.55, 0.60, 0.75, and 0.95, respectively, under the condition of different countersunk angles in the laminated plate.It is easy to notice that the delamination evolution thresholds and the initial subcritical failure loads are influenced by T d remarkably.When there is an embedded delamination at the transition area of the countersunk hole, the delamination evolution threshold decreases to 67% with that of the other cases.Meanwhile, the initial subcritical failure load declines to 42%.The appearance of the delamination in the transition area leads to lower delamination evolution threshold and initial subcritical failure load and is the most dangerous case, which should be avoided during hole drilling.
Meanwhile, it was found that the dangerous transition area is within upper and lower 10% plate thickness from the junction of the countersunk part and straight shank part.The reason is that in the process of preloading of the bolt, the stress concentrates in the transition zone of the countersunk hole, and delamination in this region is easier to propagate.The parameter of T d plays a predominant role in determining the delamination evolution behavior and the composite plate with the delamination defect in the transition zone of the countersunk which has an extremely low-structure bearing capacity.

Conclusions
In this paper, the delamination initiation and evolution in composite laminates are investigated by using FEM based on PDM and the cohesive element method.When there is an embedded delamination around the countersunk hole in composite laminate, the critical load causing delamination growth is much smaller than other cases, and the defect is easy to propagate under the process of larger bolt preloading.The size and through-the-thickness position of delamination and plate countersunk angle have significant effects upon the delamination initiation and propagation behavior of composite laminates.
Generally, as the delamination factor F d increases the load causing delamination evolution decreases.Meanwhile, the increase of F d also leads to the structural stiffness decreasing.With the increase of θ c , the delamination evolution threshold and the initial subcritical failure load decrease.In terms of the position of delamination, when there is an embedded delamination at the transition area of the countersunk hole, the critical load causing delamination evolution decreases to 67% of other cases; however, the initial subcritical failure load declines to 42%.Simultaneously, it was found that the dangerous region of the transition area is within upper and lower 10% plate thickness from the junction of the countersunk part and the straight shank part.
The shape of damage is affected by the delamination factor F d and delamination position T d , instead of the plate countersunk angle θ c .As the embedded delamination in the countersunk part is away from the transition area, only a larger bolt preloading can result in delamination evolution.Meanwhile, the width of the propagation zone is ever decreasing as the embedded delamination through-thethickness positions closer to the transition area.

Figure 1 :
Figure 1: Load condition of wing skin and rib connection structure.

Figure 4 :Figure 5 :
Figure 4: The structure of the composite layer and interface.

4 . 3 .
(a)) (iii) Step III: delamination propagation (see Figures 8(b) and 8(c)) (iv) Step IV: stiffness decrease, with first audible cracks: structural failure (see Figures 8(d) and 8(e)) Effects of Geometrical Deviation on the Delamination 4.3.1.Effect of the Delamination Factor.The delamination factor has been widely used to characterize the level of damage on drilling the countersunk hole, as shown in Figure 9

Figure 10 :
Figure 10: Effect of delamination scale on bearing capacity and structural stiffness.

Figure 11 :
Figure 11: Effect of F d on the delamination initiation, initial subcritical failure load, and the propagation of delaminated area: (a) F d = 1.0,(b) F d = 1.5, (c) F d = 2.0, and (d) F d = 2.5.

Figure 14 :Figure 15 :igure 16 :
Figure 14: Influence of deviation of the countersunk on the initial subcritical failure load under different delamination factors.

Figure 17 :
Figure 17: Delamination evolution with F d = 2.0 and different T d .