Prediction of failure behavior of adhesively bonded CFRP scarf joints using a cohesive zone model

Engineering Fracture Mechanics 228 (2020) 106897

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Prediction of failure behavior of adhesively bonded CFRP scarf joints using a cohesive zone model

T

⁎

Ligang Sun, Ying Tie, Yuliang Hou, Xingxue Lu, Cheng Li

School of Mechanical and Power Engineering, Zhengzhou University, Science Road 100, 450001 Zhengzhou, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Composite Scarf joints Finite element analysis Cohesive zone model Stress distribution

In this article, the tensile behavior of adhesively bonded scarf joints (SAJs) of CFRP laminates with ductile adhesive is investigated by experiments and finite element (FE) simulations. Experimental tests considering joints with six different scarf angles are performed, and threedimensional (3D) FE models are established in ABAQUS to study the failure mechanism. A userdefined cohesive zone model (UCZM) is proposed to predict the failure of ductile adhesive. The load-displacement response and failure modes of SAJs obtained by FE simulations are compared with the experimental results to validate the accuracy and applicability of UCZM. Moreover, the widely used triangular cohesive zone model (TCZM) is also employed to predict the failure. The comparison between experimental and numerical results exhibits that UCZM is able to predict the joint strength and failure displacement more precisely although TCZM also achieves certain accuracy. Furthermore, the influence of adhesive properties and composite stacking sequence on the tensile performance of joints is discussed using UCZM. Finally, the stress distribution within the adhesive layer and its variation along with different scarf angles are analyzed.

1. Introduction In recent years, the application of composite materials is increasingly wider and deeper for their high stiffness- and strength-toweight ratios, high temperature resistance, corrosion resistance, excellent fatigue resistance and other advantages [1–3]. In this context, composite adhesive joints and adhesive repairs are being used in many fields, such as aerospace engineering, civil engineering, automotive industry and so on [4–8]. Compared with mechanical joints, adhesive joints can not only reduce structural weight and manufacturing costs, but also possess longer fatigue life and higher resistance to vibration and corrosion. On this basis, adhesive joints and adhesive repair technology witness rapid development. Unlike single-lap joints, scarf joints (SAJs) are not affected by load eccentricity, which causes the bending of adherends and has an adverse effect on the joint strength. Superior to external patch repairs, scarf patch repairs have certain advantages for repairing surfaces with extensive damage, pronounced curvature and aerodynamic requirements and are capable of restoring the structural strength to the maximum extent. Despite the need for complex operations and related equipments, scarf bonded structures are increasingly being applied in industrial and other applications. The theoretical approach for the analysis of scarf repairs or joints includes analytical method and FE method. Hart-Smith [9] first proposed two-dimensional (2D) models for the stress and strength analysis of adhesively bonded scarf and stepped-lap joints based on continuum mechanics. However, the models were mainly for homogeneous materials, not involving the anisotropy of composites. Based on the modified Hart-Smith model, Ahn et al. [10] regarded the adhesive as an elastic-perfectly plastic material to calculate the failure loads of composite double lap and scarf joints, taking into account the heterogeneous properties of composite laminates and ⁎

Corresponding author. E-mail address: [email protected] (C. Li).

https://doi.org/10.1016/j.engfracmech.2020.106897 Received 16 November 2019; Received in revised form 17 January 2020; Accepted 17 January 2020 Available online 25 January 2020 0013-7944/ © 2020 Elsevier Ltd. All rights reserved.

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repair patches. Harman et al. [11] presented an analytical method to predict the shear stress distributions along the bondline for the sake of optimal design of the SAJ, but the method was not accurate enough to reflect the variation of the stresses in the local ply orientations and the stress peaks adjacent to 0° plies. Compared with analytical method, the FE method is characterized by higher precision and wider application range. It can successfully deal with the problems of heterogeneous materials, arbitrary boundary conditions, complex geometric shapes and so on. Charalambides et al. [12] adopted 2D plane-strain models to predict failure paths and strength of SAJs, in which two material models were used for both composite adherends and adhesive. Three different types of failure were observed, showing good accordance with experimental results. Wang et al. [13] employed elastoplastic analysis to investigate the influence of adhesive plastic yielding, ply stacking sequence and thickness of composite laminates on the stress and strain concentrations in SAJs, then proposed an improved design method of scarf repairs based on the maximum plastic strain criterion. Breitzman et al. [14] performed a 3D nonlinear analysis to predict the failure of composite scarf repairs under tensile loads, and optimized the patch ply orientations to minimize the von Mises stress within the adhesive layer. Pinto et al. [15] introduced a trapezoidal CZM to simulate the damage initiation and propagation in the adhesive, and studied the effects of repair width, scarf angle and over-laminating plies on the tensile behavior of 3D scarf repairs, but the damage of composite laminates was not considered. Leone et al. [16] utilized progressive damage analyses to predict the load–displacement response and failure mechanism of adhesive joints boned with the sandwich structure composite, and the FE results coincided well with those of experiments. Liu et al. [17] applied the modified 3D Hashin damage initiation criteria, Yeh delamination criterion and the damage evolution based on fracture energy dissipation to simulate progressive failure in composite laminates, and applied CZM to simulate the adhesive failure. Moreover, they came up with a modified semi-analytical method (MAM), which was capable of making up the deficiency of Harman’s method and predicting the shear stress distribution of adhesive more precisely. Alves et al. [18] investigated hybrid SAJs bonded with composite and aluminium adherends, considering different scarf angles and adhesives, and verified the accuracy and applicability of CZM in predicting the joint strength. The peel and shear stresses of the adhesive were also obtained and their variations with various scarf angles were discussed. In this paper, the tensile performance of CFRP SAJs with different scarf angles is investigated experimentally and numerically, and the load–displacement response as well as failure modes are analyzed. In the numerical simulations, 3D FE models with the same configurations and boundary conditions of experiments are established and ABAQUS/Explicit solver is used to predict the tensile behavior of joints. 3D Hashin criteria and damage evolution based on fracture energy are implemented by scripting a subroutine VUMAT to predict composite intralaminar failure. Moreover, triangular cohesive zone model (TCZM) is applied to forecast interlaminar delamination, because delamination essentially belongs to brittle failure. Many scholars also employ TCZM to simulate the behavior of adhesive, while TCZM is not very appropriate to evaluate cohesive failure of ductile adhesive. The objective of this study is to offer a user-defined cohesive zone model (UCZM) that better conforms to the behavior of ductile adhesive, and further to make more accurate predictions of the performance of SAJs. To verify the superiority of the new model, cohesive failure of adhesive is performed by TCZM and UCZM, respectively. The numerical results agree well with those obtained from experiments, verifying the models’ effectiveness and UCZM possesses higher precision in comparison with TCZM. The influence of composite ply stacking sequence and adhesive properties on the joint strength is discussed by FE models using UCZM approach. The variations of peel and shear stress distributions in adhesive with scarf angle are presented to facilitate understanding the failure mechanism. The remainder of the paper includes the following contents: Section 2 briefly describes the experimental materials, manufacturing process of specimens and test apparatus; Section 3 presents the material constitutive models and the FE modeling; Section 4 firstly verifies the accuracy and applicability of the FE method, and then discusses the influence of ply stacking sequence and adhesive properties on the joint strength, finally analyzes the stress distributions in adhesive. 2. Experimental work 2.1. Material and properties Made from T300 /QY8911 unidirectional carbon/epoxy prepreg with ply thickness of 0.12 mm, the adherend has a ply stacking sequence of [45/0/-45/90]3S and a nominal thickness of 2.88 mm. From a macroscopic perspective, the laminate is regarded as a homogeneous orthotropic continuum material. The anisotropic properties of off-axis plies, for example ±45° plies, are acquired by coordinate transformation based on those of 0° plies. Because of the nonexistence of composite plasticization in the test, the properties of the elastic orthotropic composite adherends are given in Table 1 [19,20]. Subscripts 1, 2 and 3 signify the longitudinal, inplane transverse and through-thickness directions, respectively. Cohesive properties of CFRP interlaminar interface are presented in Table 2 and obtained from Ref. [21]. The film-type adhesive is LJM-170, which is an intermediate temperature curing thermosetting and toughness-enhanced epoxy resin film with the areal density of 170 g/m2 produced by Weihai Guangwei Composite Corp. in China. The parameters of adhesive require calibrated by experiments for accurate simulation analysis. Bulk tests and Thick Adherend Shear Tests (TAST) have been employed to characterize the adhesive in tension and shear in previous studies. Although the cohesive parameters of thin adhesive layer and the parameters of buck adhesive are different, since thin adhesive layer is constrained between stiff adherends and damage growth occurs under mixed-mode along the predefined path, the cohesive parameters of adhesive were assumed as equal to its bulk quantities as an approximation and good correspondence of experimental and simulation results was observed in Ref. [22,23]. Dumb-bell specimens of bulk adhesive were manufactured by following the method described in Ref. [24]. The tensile tests were carried out by a WDW-300 electronic tensile testing machine with a 300 kN load cell using a displacement rate of 0.5 mm/min at room temperature, and the loads were obtained. In order to determine the tensile curve, a 3D digital image correlation (DIC) 2

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Table 1 Mechanical properties of T300/QY8911 unidirectional ply [19,20]. Property

T300/QY8911

Longitudinal modulus, E1 [GPa ] Transverse modulus, E2 = E3 [GPa ] Poisson's ratio, ν12 = ν13 Poisson's ratio, ν23 Shear modulus, G12 = G13 [GPa ] Shear modulus, G23 [GPa ] Longitudinal tensile strength, XT [MPa ] Longitudinal compressive strength, XC [MPa ] Transverse tensile strength, YT = ZT [MPa ] Transverse compressive strength, YC =ZC [MPa ] Shear strength, S12 = S13 [MPa ] Shear strength, S23 [MPa ]

135 8.8 0.33 0.35 4.47 4.00 1548 1226 55.8 218 89.9 51.2 91.6a

2 Fiber tensile fracture energy, G C ft [kJ m ]

79.9a

2 Fiber compressive fracture energy, G C fc [kJ m ]

Matrix tensile fracture energy,

C Gmt

[kJ

0.22b

m2 ]

1.1b

C Matrix compressive fracture energy, Gmc [kJ m2 ]

Density, ρ a b

1600

[kg/m3 ]

Values from Ref. [20]. Merely estimated values from the literature.

Table 2 Cohesive properties of adherend’s ply interface and the adhesive LJM-170. Property

ply interface

LJM-170

6

Tensile stiffness, Knn [N mm3]

10

Shear stiffness, Kss = Ktt [N mm3] Young's modulus, E [GPa ] Shear modulus, G [GPa ] Tensile yield stress, σny [MPa]

106

Tensile strength, σn0 [MPa]

15

2.20 0.815 16.8 31.9

Tensile failure strain, εn0 [%]

3.25

Shear yield stress, τsy, t [MPa]

7.5

Shear strength, τs0 = τt0 [MPa]

20

21.2

0.5

0.48

1

1.83

1.45

1.8

12

Shear failure strain, εs0, t [%] Toughness in Toughness in η

tension, GnC [kJ m2] shear, GsC = GtC [kJ

m2]

equipment was used to measure the strains. The shear stress–strain curve of the adhesive was obtained from TAST according to previous research in Ref. [25]. Five specimens were prepared for the tensile test as well as the shear test to guarantee the accuracy of the measurement. After getting the tensile and shear stress–strain curves, both the Young’s modulus (E) and shear modulus (G) could be acquired by means of linear fitting to the data in the elastic region with small strains. The tensile and shear yield stress and strength and corresponding strain were also obtained. It should be noted that the curves must be built with real stress and strain values, which are able to be obtained from the nominal values (σ ′ and ε′) by tests through Eqs. (1) and (2). As will be presented in Section 3.2, both curves show obvious nonlinearity.

σ = σ ′ (1 + ε′)

(1)

ε = ln(1 + ε′)

(2)

The Mode I fracture energy, GnC , and the Mode II fracture energy, GsC , were measured by double cantilever beam (DCB) and end notched flexure (ENF) tests, respectively. A simple double compliances method was proposed in Ref. [26] to estimate GnC without measuring the growing crack length during the DCB test. The accuracy of two equivalent crack methods (Corrected Beam Theory with Effective Crack Length and Compliance-Based Beam Method) in measuring GsC was verified by numerical simulations of the ENF test in Ref. [27]. The DCB and ENF specimens were fabricated with the same thickness of adhesive as that of SAJs to avoid the influence of this parameter. Five specimens were prepared for each of the two tests to ensure the reliability of the measured results. The properties of LJM-170 are given in Table 2. 3

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Fig. 1. Configuration of test specimen.

2.2. Specimen configuration and fabrication The configuration and geometric dimension of the SAJ specimen are shown in Fig. 1, including total length (L ), width (W ), endtab length (LT ), scarf angle (θ ), adherend thickness (tP ), end-tab thickness (tT ) and adhesive thickness (tA ). The parameter values were chosen as follows: L = 200 mm, W = 25 mm, LT = 40 mm, tP = 2.88 mm, tT = 1 mm and tA = 0.15 mm, and 6 scarf angles of 3°, 5°, 10°, 15°, 20° and 30° were considered. Scarf angles less than 3° are not considered because the specimen preparation is difficult and such small angles are rarely used in engineering. The composite laminates were cut according to the required specimen dimensions using the HSQ1020S high pressure waterjet cutting machine produced by AoYu CNC Technology Company. The waterjet cutting has the following advantages: high cutting speed, materials saving, low cost, even cuts and no burr. More important, there is no thermal deformation during cutting, which avoids changes of the material physical and chemical characteristics. The mating inclined surfaces of adherends were processed by a CNC milling machine to satisfy the fore-mentioned scarf angles. The bonding surfaces were hand polished with #400-grit sandpaper to obtain adequate surface roughness and then scrubbed clean using a cotton swab dipped in acetone to remove scrap from machining process, dust and other contaminants. The surface preparation ensured strong adhesion between the adherend and the adhesive and avoided interfacial debonding as far as possible. After that, a layer of adhesive film (LJM170) was placed between the mating surfaces. Aluminum end-tabs measuring 40 mm × 25 mm × 1 mm were bonded to the grip areas of SAJs to enable uniform load distribution over the surface of grip section, thus reducing failure in this part. Then all the SAJ specimens were cured at 120 °C and 0.1 MPa for about 2 h as the manufacturer recommended in oven. The adhesive uniformly distributed along the bondline after the curing process with approximate 0.15 mm thick. The exceeding adhesive was removed carefully using a portable electric grinder so as to provide the ideal joint configuration

2.3. Quasi-static testing procedure Uniaxial tensile load was applied to the specimens using the WDW-300 machine by displacement control of 0.5 mm/min at room temperature. During the test, the specimen was in severe vertical state and clamped firmly by the grips to prevent slippage, as displayed in Fig. 2. The DIC equipment was used to obtain the displacement data more accurately. Then the load (P ) data and the simultaneous displacement (δ ) were extracted to draw the P − δ curve. Eight specimens were tested for each group and three specimens with the greatest deviation from the average value of ultimate failure load were removed from the analysis, therefore five results were analyzed for each scarf angle. The experimental results are presented in detail in Section 4.

Fig. 2. The uniaxial tensile test: (a) specimen clamped in the machine for test and DIC equipment; (b) schematic diagram of specimen under tension load. 4

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3. Numerical work The strength and failure modes of SAJs can be obtained through aforementioned experiments. In order to study the failure mechanism and investigate the stress distributions of SAJs during the loading process, FE models were established. This section will begin with descriptions of the failure criteria of composite adherend and two different CZMs of adhesive, followed by the FE modeling process. 3.1. Composite failure criteria There are five basic damage modes in the composite adherend, i.e., fiber tensile and compression damages, matrix tensile and compression damages and delamination damage. Progressive failure model based on continuum damage mechanics includes damage initiation and evolution, and it has been widely applied in the strength analysis of composite materials. Various criteria have been developed to predict the composite failure. Among them, Hashin initiation criteria can describe four failure modes separately and reach high precision, leading to their wide application. For different failure modes, the degradation of material properties in the damaged locations is implemented by introducing respective damage variables. Lapczyk et al. [28] used Hashin criteria to predict the damage initiation of FRP, and introduced a damage evolution law based on fracture energy dissipation to model progressive failure, and addressed problems in the numerical simulation, such as mesh sensitivity and convergence during the damage process. Following our previous studies [29–32], the user-defined subroutine was developed based on coupling theories for the 3D Hashin criteria and damage evolution in order to accurately analyze the composite intralaminar progressive failure. With regard to delamination (the interlaminar damage) in composite laminates, Tabiei and Zhang [33] gave a review about recent development of composite delamination simulation and experimental method, emphatically analyzing the two most common methods (VCCT and CZM) to model delamination. In recent years, the CZM has been widely used to model delamination due to its ability to simulate both onset and propagation of delamination without the existence of an initial crack. Many researchers have employed interfacial decohesion elements based on TCZM to simulate delamination in composites because TCZM is easy to be realized and is an appropriate model for predicting the brittle delamination growth [34–36]. In the current study, zero-thickness interface elements are placed between different plies of the composite laminates. Interface elements are based on TCZM since the brittle delamination occurs in the selected laminates. 3.2. Adhesive failure criteria CZM is also employed to model the failure of adhesive and several CZM law shapes, such as the triangular, exponential and trapezoidal shapes, have been used depending on the adhesive’s performance (brittle or ductile) [15–19]. Some researchers have applied the CZM by a continuum approach that treats the adhesive layer as an entirety and model it using a row of cohesive elements [17,18]. Others have applied the CZM by a local approach which models the adhesive layer as solid elements with elastoplastic behaviors accompanied by cohesive elements placed at the adherend/adhesive interface and inside the adhesive to simulate damage propagation [37,38]. In this section, the cohesive failure of adhesive is simulated by TCZM and UCZM using a continuum approach owing to its simplicity, respectively. Note that UCZM is implemented through VUMAT, while TCZM is readily available in ABAQUS. 3.2.1. Triangular cohesive zone model TCZM assumes initially linear elastic behavior followed by the initiation and linear evolution of damage. The constitutive relation can be defined as:

Kn ε σ ⎤ ⎧ δn ⎫ ⎤⎧ n⎫ ⎡ ⎧ τn ⎫ ⎡ E Ks = ⎢ G ⎥ εs = ⎢ s ⎥ δs ⎨ ⎬ ⎨ τt ⎬ ⎨ ⎬ G ⎦ ⎩ εt ⎭ ⎢ Kt ⎥ ⎩ ⎭ ⎣ ⎦ ⎩ δt ⎭ ⎣

(3)

where σn , τs and τt are the normal traction and two shear tractions, respectively, and εn , εs and εt the corresponding strain components, and δn , δs and δt the corresponding separations. Denoting by T0 the initial thickness of the thin adhesive layer, the relationship between the nominal strains and the separation displacements can be defined by Eq. (4) and the stiffness parameters are obtained by Eq. (5). (4)

δn = T0 εn, δs = T0 εs, δt = T0 εt

Kn =

E G G , Ks = , Kt = T0 T0 T0

(5)

Quadratic nominal stress (QUADS) criterion represented in Eq. (6) is used for damage initiation under mixed-mode loading. denotes Macaulay brackets, signifying that the compressive stress does not cause damage. 2

2

2

⎧ 〈σn 〉 ⎫ + ⎧ τs ⎫ + ⎧ τt ⎫ = 1 0 0 0 ⎨ ⎨ ⎨ ⎭ ⎩ τt ⎬ ⎭ ⎩ τs ⎬ ⎭ ⎩ σn ⎬

(6)

When Eq.(6) is satisfied, the values of σn, τs and τt are σn0, m , τs0, m and τt0, m , which represent the normal and the two shear tractions at damage onset under mixed-mode loading, respectively. The corresponding separations are denoted by δn0, m , δs0, m and δt0, m . 5

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The single-mode relative displacements δn0, δs0 and δt0 at softening onset are:

δn0 =

σn0 τ0 τ0 , δs0 = s , δt0 = t Kn Ks Kt

(7)

For an opening displacement δn greater than zero, the following mode mixity ratios are defined as:

βd =

δshear = δn

δs2 + δt2

βs =

σshear = σn

σs2 + σt2

(8)

δn

(9)

σn τs0

τt0 ,

In the present paper, Ks = Kt and = therefore damage onset under mixed - mode loading is:

δs0

=

δt0 .

Substituting Eqs. (7) and (8) into Eq. (6), the normal displacement at

δs0 δn0

δn0, m =

(δs0 )2

+ (βd δn0 )2

(10)

Camanho et al. [34] introduced an effective displacement, δm , which can be calculated by the normal and the two shear displacements according to:

δm =

〈δn 〉2 + δs2 + δt2 =

2 〈δn 〉2 + δshear

(11)

Substituting Eqs. (8) and (10) into Eq. (11), the effective displacement corresponding to the damage onset,

δm0 =

1 + βd2 (δs0 )2 + (βd δn0 )2

⎧ (δ 0 )2 (1 + β 2) = δ 0 δ 0 n, m s n d ⎨ 0 δshear , δn ⩽ 0 ⎩

δm0 ,

is obtained as:

, δn > 0 (12)

The evolution of damage is defined based on the fracture energy which is equal to the area under the traction-separation curve. The dependence of the fracture energy on the mode mix can be defined based on the B-K fracture criterion, given by: η

Gs + Gt ⎫ = GC GnC + (GsC − GnC ) ⎧ ⎨ ⎭ ⎩ Gn + Gs + Gt ⎬

(13)

The strain energy release rate components are obtained from: f

Gn =

∫0

δn, m

σn dδn =

σn0, m δnf, m 2

f

, Gs =

∫0

δs, m

τs dδs =

Substituting Eqs. (8), (9) and (14) into Eq. (13),

GC =

(

⎧GnC + (GsC − GnC ) βs βd 1 + βs βd ⎨ C G , δ n ⩽ 0 s ⎩

GC

τs0, m δsf, m 2

f

, Gt =

∫0

δs, m

τt dδt =

τt0, m δt ,fm 2

(14)

can be obtained according to:

η

), δ

n

>0 (15)

For TCZM, damage evolution is linear, and the damage variable, D, is defined as [39]:

D=

δmf (δmmax − δm0 ) δmmax (δmf − δm0 )

δmmax

δmf

δmf =

2GC = σm0

(16)

and refer to the maximum value of effective displacement attained during the loading history and the effective where displacement at complete failure, respectively. δmf can be obtained by the following equation:

2GC 〈σn0, m 〉2

+ (τs0, m )2 + (τt0, m )2

(17)

The definitions of the three stress components under the influence of the damage are as follows:

(1 − D) σn, σn ⩾ 0 σn = ⎧ σn, σn < 0 ⎨ ⎩ τs = (1 − D) τs τt = (1 − D) τt

(18)

where σ¯n, τ¯s and τ¯t are the stress components corresponding to the current strains without stiffness degradation. The value of D monotonically increases from 0 to 1 with further loading after damage initiation. D = 0 stands for undamaged state and D = 1 for fully damaged state. 6

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Fig. 3. Experimental and fitted stress–strain curves for adhesive under tension (a) and shear (b).

3.2.2. User-defined cohesive zone model UCZM takes the large plastic strain of ductile adhesive in consideration. The adhesive’s constitutive model can be divided into the following phase: initially linear elasticity, plastic hardening, damage softening and final failure. The stress–strain relationships are expressed by Eqs. (19) and (20):

σn = Eεn, 0⩽ σn ⩽ σny ⎧ ⎪ 0 n1 ⎨ σn = σn0 ⎜⎛1 − e−m1 (εn εn ) ⎟⎞, σny < σn ⩽ σn0 ⎪ ⎝ ⎠ ⎩

(19)

τsy, t

τs, t = Gεs, t , 0⩽ τs, t ⩽ ⎧ ⎪ 0 n2 ⎨ τs, t = τs0, t ⎛⎜1 − e−m2 (εs, t εs, t ) ⎟⎞, τsy, t < τs, t ⩽ τs0, t ⎪ ⎝ ⎠ ⎩

(20)

Here, the newly introduced constants m1, n1, m2 and n2 , which denote hardening parameters, are determined by nonlinear fitting using experimental data. Fig. 3 shows experimental and fitted stress–strain curves for adhesive under tension and shear, respectively. It can be seen that the fitted equations can quite accurately describe the adhesive’s stress–strain relation. With regard to UCZM, QUADS is also used as damage initiation criterion. When Eq.(6) is satisfied, δm0 can be computed by:

δm0 =

〈δn0, m 〉2 + (δs0, m )2 + (δt0, m )2

(21)

The damage evolution is nonlinear and D is expressed as:

D=

max δm 0 m

∫δ

σm dδm GC − G 0

(22)

When adopting the UCZM, the values of Gn, Gs and Gt cannot be calculated by Eq. (14), but Eq. (15) is still used to obtain adhesive. The strain energy at damage initiation, G 0 , is obtained as:

G0 =

∫0

0 δm

σm dδm

GC

for

(23)

The stress components for the softening stage is given by Eq. (18). In general, TCZM and UCZM have both similarities and differences. The traction-separation law shapes of TCZM and UCZM under pure mode and mixed mode are shown in Fig. 4. 3.3. Finite element model 3D FE models were established using ABAQUS® 6.14 to analyze the strength, failure modes and stress distributions of SAJs. To simplify modeling, the shell part was created and meshed in the first place, then the shell elements were extruded in a direction perpendicular to the shell face to construct 3D models, which allowed ply orientation at any angle other than 0° and 90°. Fig. 5 presents the 3D FE model and mesh details. The 8-node reduced integration hexahedral solid elements (C3D8R) together with a small number of 6-node triangular prism elements (C3D6) were selected to model the composite adherend, with each ply consisting of one row of elements in the thickness direction. The 3D, 8-node cohesive elements (COH3D8) with zero thickness were placed between adjacent two plies with different ply orientations to simulate the interlaminar delamination. A single 0.15 mm thick 7

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Fig. 4. Traction-separation law of the triangular CZM (a) and user-defined CZM (b).

Fig. 5. Illustration of the boundary conditions applied in the FE models and mesh details.

row of COH3D8 elements was created to model the adhesive layer by sweep meshing technique with the sweep path along adhesive’s thickness direction. In addition, the area near the free surface was meshed using C3D8R and C3D6 elements for the adhesive. Note that the adherend and adhesive were connected by sharing element nodes, so the interface debonding was not considered. For the purpose of reducing element quantities, and thus shortening the calculation time without the compromise of calculation precision, various mesh densities were applied at different locations of the model. A higher mesh refinement was adopted in the scarf region, and the meshes outside there were created automatically considering bias effect with smaller sized elements close to the scarf region and larger ones close to the ends. The boundary condition at one end of the SAJ limited all degrees of freedom and only longitudinal displacement was allowed at the other end. In order to make it convenient to get the P − δ curve, the displacement was implemented by a kinematic coupling constraint that limited the motion of coupling nodes to the rigid body motion defined by the reference node. The translation U1 and reaction force RF1 of the reference node, which corresponded to δ and P , respectively, were output. Because of complex material nonlinearity, the FE analysis simultaneously considering both composite failure and adhesive failure often takes a long time, or even can't be convergent for the ABAQUS/Standard solver. Though widely used in transient response analysis, the explicit solver has been applied for highly-efficient modelling failure of the composite joints under quasi-static load as its higher ability of solution in strongly nonlinear problem [19,40]. The application of displacement was in the form of smooth step so as to achieve an efficient solution and the mass scaling method was introduced to save computational time. In the post-processing, different failure modes could be demonstrated by corresponding output variables.

4. Results and discussion In this section, the strength and failure modes of SAJs with different scarf angles obtained by experiments and FE simulations, respectively, are made comparison analysis firstly, proving the accuracy of FE models and UCZM is superior to TCZM in modeling the failure of ductile adhesive to some extent. Then, the influence of different composite ply stacking sequences and varied adhesive properties on joint strength is investigated based on the numerical simulation. Finally, the stress distribution within the adhesive 8

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Fig. 6. Comparison between experimental and numerical P − δ curves for SAJs as a function of the scarf angle.

layer and its variation along with different scarf angles is analyzed. 4.1. Validation of FE models The experimental P − δ curves and the numerical ones predicted by TCZM and UCZM for SAJs bonded with 6 scarf angles are presented in Fig. 6. The deviation of experimental curves in each group is acceptable, showing the reliability of experiments. It is observed that the numerical curves are basically consistent with the experimental ones, especially for curves predicted by UCZM. The pre-peak P − δ curves predicted by TCZM are approximately linear, while the experimental curves and the ones predicted by UCZM show initial linearity, followed by descending slope prior to the peak load. The curves predicted by TCZM show faster failure of SAJs because TCZM assumes linear elastic behavior of adhesive before damage initiation without regard to elastic–plastic hardening. For all curves, the value of P does not fall suddenly after it reaches the maximum, but decreases with a certain amount of increase in δ . During experiments, there are no sharp crack sounds, indicating SAJs experience ductile fracture. Fig. 7 gives an example of the P − δ curve predicted by TCZM and UCZM, respectively, for the SAJ boned with θ = 10° . From Fig. 7(a), the curve obtained by TCZM can be divided into three parts. The first part shows P increases linearly with δ up to point b1 that stands for damage initiation in either adherend or adhesive. The second part indicates that the SAJ is still able to withstand greater load until the peak (point c1) accompanied by damage accumulation and the slight nonlinearity is caused by the damage of adhesive and adherend. The remaining part describes a softening behavior, which manifests that the SAJ is incapable of bearing further load due to the rapid propagation of damage. From Fig. 7(b), the curve predicted by UCZM is divided into four parts, and point b2 and c2 signify the beginning of adhesive plastic deformation and damage initiation, respectively. The first part (namely segment a2-b2) shows P increases linearly with δ , and the second nonlinear part (segment b2-c2) shows the decreasing slope of the curve on account of the plasticity of ductile adhesive. In the third part (segment c2-d2), P still increases nonlinearly with δ due to the combined effect of adhesive’s elastic–plastic behavior and joint damage. The last part exhibits that the carrying capacity of the SAJ drops due to damage propagation and the joint is about to fracture. The effects of θ on the ultimate joint failure load (Pm , namely joint strength) and the ultimate joint failure displacement (δ Pm ) of SAJs are presented in Fig. 8, and the percentile deviations (Δ ) between average values of experiments and numerical values are also included. From Fig. 8(a), Pm presents exponential growth with decreasing θ . This is because reducing θ can significantly increase the bonding area and effectively decrease the adhesive stress (discussed in Section 4.3) and then postpone adhesive’s failure, thereby improving joint’s bearing capacity. It shows that FE models adopting UCZM method can accurately predict Pm of SAJs and Δ is 9

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Fig. 7. P − δ curves predicted by triangular CZM (a) and by user-defined CZM (b) for SAJs bonded with.θ = 10°.

Fig. 8. Comparison between experimental and numerical results and percentile deviation for the joints as a function of the scarf angle: (a) ultimate failure load and (b) ultimate failure displacement.

negligible with a maximum of 1.9% for θ = 10° . Whereas, TCZM method consistently underestimates Pm and the general tendency of discrepancy enlarges with increasing θ (Δ = −0.4% for θ = 3° and Δ = −8.9% for θ = 30° ). From Fig. 8(b), δ Pm also increases exponentially with the reduction of θ , showing a similar trend to Pm . The UCZM method has high prediction accuracy, and the maximum Δ is − 2.0% for θ = 15°. However, the accuracy of TCZM approach declines with increasing θ (Δ = −20.7% for θ = 30° ). The predictions of Pm and δ Pm by TCZM are under the average experimental results and this is mainly due to no consideration for the plasticity of the ductile adhesive. Conclusion can be drawn that reducing θ can effectively improve joint’s tensile performance. However, the smaller is the value of θ , the more adherend material is removed during the manufacturing process and the higher is the manufacturing cost. Therefore, the selection of θ should trade off well between production efficiency and joint strength. Fig. 9 displays typical failure modes of SAJs with different values of θ. From Fig. 9(a) and (b), cohesive failure of adhesive and adhesive/adherend interface debonding (i.e. adhesive failure) exist together for θ = 30° and θ = 20° , and the former proportion of interface debonding is larger than the later. It seems that large θ is likely to cause adhesive failure which will weaken the joint strength, and there is a relatively small bonding area in this case, making the joint fragile. Therefore it is not advisable to choose large θ. Fig. 9(c) shows that cohesive failure occurs along the entire bondline for θ = 15° . When θ decreases to 10°, from Fig. 9(d), matrix crack of mid 90° ply and the delamination between 90° ply and adjacent 45° ply are observed, which is because the strength of 90° ply is relatively low along the loading direction. As shown in Fig. 9(e), the range of 90° ply matrix crack and delamination expands, due to the fact that smaller θ signifies bigger bonding area and then reduces the adhesive stress, not conducive to cohesive failure. Moreover, matrix crack and some few fiber pull-out appear in the surface 45° ply located at the feathered tip of adherend. With regard to θ = 3° , matrix crack happens in all 90° plies and some 45° plies and delamination expands further from Fig. 9(f). Of the overall 10

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Fig. 9. Failure modes of specimens with different scarf angles (a) θ = 30° , (b) θ = 20° , (c) θ = 15° , (d) θ = 10° , (e) θ = 5° and (f) θ = 3° .

joints, the conclusion can be drawn that cohesive failure takes dominant, and matrix crack and delamination increase with the decrease of θ , while adhesive failure exists in case of relatively large θ. Taking the SAJ with θ = 3° as an instance, the joint failure is mainly caused by cohesive failure of the adhesive, the matrix crack and delamination of the adherend, accompanied by a small part of fiber breakage. Fig. 10 presents the damage propagation in both adherend and adhesive during the load process simulated by UCZM approach. As shown in Fig. 10(a), the damage coefficient from damage initiation to complete failure is demonstrated using rainbow spectrum. Under 70% Pm , the matrix crack appears in three 90° plies and spreads across the joint’s width direction, and cohesive failure of the adhesive occurs close to the 0° plies and on both sides of the adherend with small damage coefficient, as indicated in Fig. 10(b) and (c). Under 87% Pm , Fig. 10(d) and (e) show the damaged region of composite adherend expands further, accompanied by new matrix crack in ±45° and 90° plies, and cohesive failure also rapidly spreads across the width direction accounting for about 18.8% of the bonded area. Till the load reaches Pm , as depicted in Fig. 10(f) and (g), the matrix crack extends to all of 90° plies and 45° plies associated with continuously expanding adhesive damage whose proportion comes up to 47.5% and increasing damage coefficient, makng the joint unable to withstand a larger tensile load any more. After that, although the load drops down, the damage expansion is still ongoing. When the joint is broken and completely losts its carrying capacity, matrix crack propagation becomes more evident, and 80.7% of the adhesive fails thoroughly, as illustrated in Fig. 11(a) and (b). It is noted that fiber breakage occurs in the surface 45° ply and the next 0° ply from Fig. 11(c), and because the feathered tip has a small thickness, this region is susceptible to damage. Fig. 11(d) exhibits adherend’s interlaminar delamination damage which mainly occurs between 90° and neighboring plies. According to the simulation results, cohesive failure holds dominant position, indicating that the ultimate bearing capacity of the joint is mainly determined by the performance of adhesive. The simulated failure modes are consistent well with those of experiments. 4.2. Joint strength prediction under different design parameters Robustness of the proposed FE models is shown in Section 4.1, and these models can also be applied to investigate the effect of different design parameters, including composite ply stacking sequence and adhesive properties, on the tensile performance of SAJs. 4.2.1. Effect of stacking sequence As shown in Fig. 12, the strength analysis is performed, considering composite adherends with eleven different ply stacking sequences and θ = 3° , 5° , 10°and15° . Adherends with θ = 20°and30° are no longer considered for their quite small joint strength. From Fig. 12(a), it is observed that Pm of SAJs bonded with [0]24, [0/45/-45/0]3S, [0/90]6S and [45/0/-45/90]3S adherends increases rapidly with decreasing θ and the rate of increase gets faster as θ decreases. The absence of a constant value of Pm is due to the high strength of the adherend (i.e., the tensile strength of these laminates is not yet attained for the tested values of θ ) and the failure of these kinds of joints are determined primarily by adhesive’s cohesive failure. With regard to joints with [45/-45]6S and [45/90–45/ 90]3S adherends, Pm exhibits a small amount of increase when θ decreses from 15° to 10°. However, Pm keeps at a constant level or even decreases slightly with continuously decreasing θ. This is because the tensile strength of the laminates is reached and the 11

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Fig. 10. The damage propagation in both adherend and adhesive of SAJ boned with θ = 3° and [45/0/-45/90]3S adherends: (a) the damage coefficient expressed in the form of rainbow spectrum; (b) matrix crack in three 90° plies under 70% Pm ; (c) cohesive failure of the adhesive appearing close to the 0° plies and on both sides of the adherend under 70% Pm ; (d) matrix crack in multiple 90° and ± 45° plies under 87% Pm ; (e) the proportion of damaged adhesive accounting for 18.8% under 87% Pm ; (f) increasing matrix crack in 90° and ± 45° plies under Pm ; (g) damaged adhesive accounting for 47.5% of the bonding area under Pm .

Fig. 11. The failure mode at the moment of joint fracture: (a) matrix crack in 90°, ± 45° and 0° plies; (b) the proportion of damaged adhesive accounting for 80.7%; (c) fiber breakage in the surface 45° ply and the next 0° ply at adherend’s feathered tip; (d) interlaminar delamination mainly occurring between 90° and adjacent plies.

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Fig. 12. Pm for the joints boned with adherends of different ply stacking sequences under varying θ : (a) 0°, ± 45° and 90° plies in different proportions; (b) 0°, ± 45° and 90° plies in same proportion and different stacking sequences.

Fig. 13. The matrix crack of SAJs boned with [45/-45]6S adherends (a) and [45/90–45/90]3S adherends (b).

ultimate failure of these two types of joints is caused by adherend fracture due to the low strength of 45° and 90° plies, as shown in Fig. 13(a) and (b). The strength of the six kinds of joints shows that the higher the proportion of 0° layer is, the stronger the joint becomes, and 90° ply leads to the reduction of strength, because the load carrying capacity of 0° ply is maximum, 45° ply comes next and 90° ply is minimum. The strength of SAJs bonded with adherends of the same proportion of 0°, ±45° and 90° plies but different stacking sequences as a function of θ is presented in Fig. 12(b). The conclusion is that Pm of joints bonded with [0/45/-45/90]3S, [45/ 0/-45/90]3S and [90/-45/0/45]3S adherends have a similar value, and Pm for [03/453/-453/903]S, [453/03/-453/903]S and [903/453/453/03]S are also similar with each other and are much less than the former three joints. The reason is that ply blocking effect exacerbates the non-uniformity of the adhesive stress and then brings down the joint strength. 4.2.2. Effect of mechanical properties of adhesive Previous research has shown that cohesive parameters, such as the maximal strength and fracture toughness, have great influence on the interface failure [41]. So, in this paper, SAJs bonded with [45/0/-45/90]3S adherends and the four aforementioned scarf angles are taken as examples to discuss the influence of adhesive properties, such as cohesive strength (σn0, τs0 andτt0 ) and fracture toughness (GnC , GsC andGtC ) on the joint strength. As shown in Fig. 14(a) and (b), percentile variations of Pm (i.e., (Pm − Pm0) Pm0, Pm0 stands for Pm for the reference value in Table 2) are analyzed by changing either of the two parameters in the normal and shear directions, while keeping the other one invariable. The changing ranges of the two parameters are all from −80% to + 100% compared to original values at the same intervals of 20%. Results indicate that reducing the cohesive strength of adhesive has significant influence on the joint strength notwithstanding the value of θ . Pm decreases approximately in proportion with cohesive strength, resulting in a reduction of Pm around 75% when the cohesive strength decreases by 80%. However, the increase of this parameter above the reference value has different influence on Pm for joints with different scarf angles. When θ = 15° , Pm varies most significantly, and the maximum percentage of growth is 43.8% when the cohesive strength increases by 100%. As for θ = 10° , the variation of Pm is very similar to that of θ = 15° , with the largest percentage increase of Pm reaching 42.2%. With regard to θ = 5°and3°, owing to damage of the adherend itself, the improvement of Pm with increasing cohesive strength declines markedly compared with θ = 15°and10° , and the maximum increasing percentage reaches only 22.4% and 16.5%. When decreasing the adhesive fracture toughness, the maximum percentage reduction of Pm for joints with θ = 3° , 5° , 10°and15° is −25.3%, −19.4%, −8.7% and −6.8%, respectively. The results show that the smaller the scarf angle is, the more obviously the decrease of adhesive fracture toughness effects on Pm . However, increasing the parameter above the reference value has a negligible influence on Pm with the maximum improvement being only 0.65%, 1%, 2.75% and 2%, respectively. In 13

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Fig. 14. Effects of the variation of the cohesive parameters on joint strength: (a) cohesive strength and (b) fracture toughness.

conclusion, the cohesive strength has a higher influence on the joint strength than fracture toughness, and the decrease of the two parameter values has a great impact on Pm , while positive variation has a relatively small impact, especially for the adhesive fracture toughness. 4.3. Stress analysis The stress analysis of the SAJs bonded with [45/0/-45/90]3S adherends and the aforementioned values of θ is performed to better understand failure mechanism. The stress analysis focuses on the normal and shear stress distributions within the adhesive layer. As shown in Fig. 15, a local coordinate system is created to extract the normal stress (or peel stress) and shear stress, with the former (S33 ) perpendicular to the scarf bondline and the latter (S13 ) along the bondline. S33 and S13 stresses as a function of ζ L (relative position along the adhesive bondline) are illustrated in Fig. 16. It should be noted that they are taken from the half width (W/2) of the joint to avoid edge effects and normalized by the respective tensile stress in far field (σx , equal to the ratio between the load and the joint’s cross-sectional area), as written in Eq. (24) .

S33 S33 S S13 = , 13 = σx P (W × T ) σx P (W × T )

(24)

It is observed that S13 and S33 stresses present the wave distribution along the bondline and are closely associated with the ply orientation angle. Both S13 and S33 reach the peak value at the position of 0° plies and drop to the valley at the position of 90° plies. The variation of the stress can clearly reflect the change of stiffness between adherend’s different plies because 0° plies have much higher stiffness than that of 90° plies in the direction of load. The symmetric stress distribution along the bondline is due to the symmetrisity of the ply stacking sequence. The maximum point of S13 and S33 appears at the two outermost 0° plies and the minimum occurs at half the thickness of the adherend. Overall, both S13 and S33 gradually increase with θ and S13 is much larger than S33 in magnitude for small θ . 5. Conclusion The tensile performance of SAJs bonded with [45/0/-45/90]3S adherends and different scarf angles was studied through combination of experiments and FE simulations that adopted TCZM and UCZM to predict the failure of ductile adhesive, respectively. By contrasting the P − δ curves and failure modes with those of experiments, the robustness of FE models was verified and UCZM exhibited a higher accuracy. The influence of scarf angle, ply stacking sequence of composite laminates and mechanical properties of adhesive on the strength of SAJs was analyzed. Stress distributions within the adhesive layer were investigated in order to better

Fig. 15. The global coordinate system of SAJ and the local coordinate system of adhesive layer. 14

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Fig. 16. Normalized S13 and S33 along the bondline for SAJs with different θ .

understand the failure mechanism. According to the experimental and numerical results, the following conclusions can be reached: 1. The pre-peak P − δ curves predicted by TCZM are approximately linear, while the experimental curves and the ones predicted by UCZM show initial linearity, followed by the descending slope prior to the peak load. UCZM can predict both Pm and δ Pm more correctly than TCZM which consistently underestimates Pm and δ Pm due to no consideration for the plasticity of the ductile adhesive. 2. The failure modes of SAJs bonded with [45/0/-45/90]3S adherends include the cohesive failure in adhesive, the matrix crack, interface delamination and adhesive failure at the adherend/adhesive interface, sometimes accompanied by fiber pull-out, and changed with θ . Adhesive failure usually occurs when θ is relatively large, and the matrix crack and delamination are more likely to happen with the decrease of θ , while cohesive failure always takes the dominant. 3. Pm presents an exponential increase with decreasing θ , except for the joints bonded with [45/-45]6S and [45/90–45/90]3S adherends. The proportion of ply in various orientations and ply stacking sequence make a big difference in Pm. The higher the proportion of 0° ply is, the stronger the joint becomes and the effect is reversed for 90° ply. Moreover, the ply blocking of laminates has a negative effect on Pm . Beside the adherend factors, Pm is also closely related with adhesive’s properties. The cohesive strength has a higher influence on Pm than fracture toughness, and the negative variation of the two parameter values has a greater impact on Pm than the positive variation, especially for the adhesive fracture toughness. 4. The stress distribution within adhesive is not uniform and depends on θ and adherend’s layup sequence. Both peel stress and shear stress peak at the position of 0° plies, while their minimum values appear at the position of 90° plies. On the whole, the magnitude of shear stress exceeds peel stress and both of them increase with increasing θ because larger θ leads to smaller bonded area. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Authors acknowledge the sponsorship and support from National Natural Science Foundation of China (No. U1333201 and U1833116). This study is also supported by Key Scientific Research Project of Colleges and Universities in Henan Province (20A460003). References [1] Knops M. Analysis of failure in fiber polymer laminates: The theory of alfred puck. Germany: Spring; 2008. [2] Rajak DK, Pagar DD, Kumar R, Pruncu C. Recent progress of reinforcement materials: a comprehensive overview of composite materials. J Mater Res Technol 2019. https://doi.org/10.1016/j.jmrt.2019.09.068. [3] Rajak DK, Menezes PL, Linul E. Fiber-Reinforced Polymer Composites: Manufacturing, Properties, and Applications. Polymers 2019;11(10):1667–704. https:// doi.org/10.3390/polym11101667. [4] Banea MD, da Silva LFM. Adhesively bonded joints in composite materials: an overview. Proc IME J Mater Des Appl 2009;223:1–18. https://doi.org/10.1243/ 14644207JMDA219. [5] Katnam K, Da Silva L, Young T. Bonded repair of composite aircraft structures:a review of scientific challenges and opportunities. Prog Aerosp Sci 2013;61:26–42. https://doi.org/10.1016/j.paerosci.2013.03.003. [6] Cheng XQ, Yasir B, Rw Hu, Gao YJ, Zhang JK. Study of tensile failure mechanisms in scarf repaired CFRP laminates. Int J Adhes Adhes 2013;41:177–85. https:// doi.org/10.1016/j.ijadhadh.2012.10.015. [7] Budhe S, Banea MD, de Barros S, da Silva LFM. An updated review of adhesively bonded joints in composite materials. Int J Adhes Adhes 2017;72:30–42. https:// doi.org/10.1016/j.ijadhadh.2016.10.010. [8] Cheng X, Du X, Zhang J, Zhang J, Guo X, Bao J. Effects of stacking sequence and rotation angle of patch on low velocity impact performance of scarf repaired laminates. Compos B Engng 2018;133:78–85. https://doi.org/10.1016/j.compositesb.2017.09.020. [9] Hart-Smith LJ. Adhesively-bonded scarf and stepped-lap joints. NASA Technical Report CR 112237. Washington, United States: NASA, January 1973.

15

Engineering Fracture Mechanics 228 (2020) 106897

L. Sun, et al.

[10] Ahn SH, Springer GS. Repair of composite laminatesⅡ: Models. J Compos Mater 1998;32(11):1076–114. https://doi.org/10.1177/002199839803201103. [11] Harman AB, Wang CH. Improved design methods for scarf repairs to highly strained composite aircraft structure. Compos Struct 2006;75:132–44. https://doi. org/10.1016/j.compstruct.2006.04.091. [12] Charalambides MN, Hardouin R, Kinloch AJ. Adhesively bonded repairs to fibre-composite materials II: finite element modelling. Compos Part A-Appl Sci 1998;29(11):1383–96. https://doi.org/10.1016/S1359-835X(98)00061-X. [13] Wang CH, Gunnion AJ. On the design methodology of scarf repairs to composite laminates. Compos Sci Technol 2008;68(1):35–46. https://doi.org/10.1016/j. compscitech.2007.05.045. [14] Breitzman TD, Iarve EV, Cook BM. Optimization of a composite scarf repair patch under tensile loading. Compos Part A-Appl Sci 2009;40(12):1921–30. https:// doi.org/10.1016/j.compositesa.2009.04.033. [15] Pinto AMG, Campilho R, de Moura M. Numerical evaluation of three-dimensional scarf repairs in carbon-epoxy structures. Int J Adhes Adhes 2010;30(5):329–37. https://doi.org/10.1016/j.ijadhadh.2009.11.001. [16] Leone FA, Davila CG, Girolamo D. Progressive damage analysis as a design tool for composite bonded joints. Compos B Engng 2015;77:474–83. https://doi.org/ 10.1016/j.compositesb.2015.03.046. [17] Liu B, Xu F, Feng W, Yan R. Experiment and design methods of composite scarf repair for primary-load bearing structures. Compos Part A-Appl Sci 2016;88:27–38. https://doi.org/10.1016/j.compositesa.2016.05.011. [18] Alves DL, Campilho RDSG, Moreira RDF, Silva FJG, da Silva LFM. Experimental and numerical analysis of hybrid adhesively-bonded scarf joints. Int J Adhes Adhes 2018;83:87–95. https://doi.org/10.1016/j.ijadhadh.2018.05.011. [19] Ye J, Yan Y, Li J, Hong Y. 3D explicit finite element analysis of tensile failure behavior in adhesive-bonded composite single-lap joints. Compos Struct 2018;201:261–75. https://doi.org/10.1016/j.compstruct.2018.05.134. [20] Pinho ST, Robinson P, Iannucci L. Fracture toughness of the tensile and compressive fibre failure modes in laminated composites. Compos Sci Technol 2006;66:2069–79. https://doi.org/10.1016/j.compscitech.2005.12.023. [21] Li J, Yan Y, Zhang T, Liang Z. Experimental and Numerical Study of Adhesively Bonded CFRP Scarf-Lap Joints Subjected to Tensile Loads. J Adhes 2016;92:1–17. https://doi.org/10.1080/00218464.2014.987343. [22] Nunes SLS, Campilho RDSG, da Silva FJG, de Sousa CCRG. Comparative failure assessment of single and double-lap joints with varying adhesive systems. J Adhes 2016;92:610–34. https://doi.org/10.1080/00218464.2015.1103227. [23] Ribeiro TEA, Campilho RDSG. da Silva LFM. Goglio L. Damage analysis of composite–aluminium adhesively-bonded single-lap joints. Compos Struct 2016;136:25–33. https://doi.org/10.1016/j.compstruct.2015.09.054. [24] Aydin MD, Özel A, Temiz Ş. Non-Linear stress and failure analyses of adhesively-bonded joints subjected to a bending moment. J Adhes Sci Technol 2004;18:1589–602. https://doi.org/10.1163/1568561042411286. [25] Banea MD, da Silva LFM. Mechanical Characterization of Flexible Adhesives. J Adhes 2009;85(4):261–85. https://doi.org/10.1080/00218460902881808. [26] Xu W, Guo ZZ. A simple method for determining the mode I interlaminar fracture toughness of composite without measuring the growing crack length. Eng Frac Mech 2018;191:476–85. https://doi.org/10.1016/j.engfracmech.2018.01.014. [27] de Moura MFSF, de Morais AB. Equivalent crack based analyses of ENF and ELS tests. Eng Frac Mech 2008;75:2584–96. https://doi.org/10.1016/j.engfracmech. 2007.03.005. [28] Lapczyk I, Hurtado JA. Progressive damage modeling in fiber-reinforced materials. Compos Part A-Appl Sci 2007;38:2333–41. https://doi.org/10.1016/j. compositesa.2007.01.017. [29] Tie Y, Hou YL, Li C. An insight into the low-velocity impact behavior of patch-repaired CFRP laminates using numerical and experimental approaches. Compos Struct 2018;190:179–88. https://doi.org/10.1016/j.compstruct.2018.01.075. [30] Hou YL, Tie Y, Li C. Low-velocity impact behaviors of repaired CFRP laminates: effect of impact location and external patch configurations. Compos Part B 2019;163:669–80. https://doi.org/10.1016/j.compositesb.2018.12.153. [31] Tie Y, Hou Y, Li C, Meng L, Sapanathan T, Rachik M. Optimization for maximizing the impact-resistance of patch repaired CFRP laminates using a surrogatebased model. Int. J. Mech. Sci. 2020;172:105407https://doi.org/10.1016/j.ijmecsci.2019.105407. [32] Hou Y, Tie Y, Li C, Meng L, Sapanathan T, Rachik M. On the damage mechanism of high-speed ballast impact and compression after impact for CFRP laminates. Compos. Struct. 2019;229:111435https://doi.org/10.1016/j.compstruct.2019.111435. [33] Tabiei A, Zhang WL. Composite Laminate Delamination Simulation and Experiment: A Review of Recent. Development. Appl Mech Rev 2018;70(3)::030801-. https://doi.org/10.1115/1.4040448. [34] Camanho PP, Dávila CG. Mixed-mode decohesion finite elements for the simulation of delamination in composite materials. 2002,;NASA/TM2002–211737:1–37. [35] De Moura MFSF, Goncalves JPM. Cohesive zone model for high-cycle fatigue of composite bonded joints under mixed-mode I+II loading. Eng Frac Mech 2015;140:31–42. https://doi.org/10.1016/j.engfracmech.2015.03.044. [36] Mohammadi B, Shahabi F. On computational modeling of postbuckling behavior of composite laminates containing single and multiple through-the-width delaminations using interface elements with cohesive law. Eng Frac Mech 2016;152:88–104. https://doi.org/10.1016/j.engfracmech.2015.04.005. [37] PardoenaT Ferracin T, Landis CM, Delannay F. Constraint effects in adhesive joint fracture. J Mech Phys Solids 2005;53(9):1951–83. https://doi.org/10.1016/j. jmps.2005.04.009. [38] Sugiman S, Crocombe AD, Aschroft IA. Modelling the static response of unaged adhesively bonded structures. Eng Frac Mech 2013;98:296–314. https://doi.org/ 10.1016/j.engfracmech.2012.10.014. [39] ABAQUS User Manual. Version 6.14. Providence (RI): Dassault Systemes Simulia Corp; 2014. [40] Liu P, Cheng X, Wang S. Numerical analysis of bearing failure in countersunk composite joints using 3D explicit simulation method. Compos Struct 2016;138:30–9. https://doi.org/10.1016/j.compstruct.2015.11.058. [41] Shao Y, Zhao HP, Feng XQ, Gao HJ. Discontinuous crack-bridging model for fracture toughness Analysis of nacre. J Mech Phys Solids 2012;60:1400–19. https:// doi.org/10.1016/j.jmps.2012.04.011.

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