Experimental-numerical analysis of failure of adhesively bonded lap joints under transverse impact and different temperatures

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Experimental-numerical analysis of failure of adhesively bonded lap joints under transverse impact and different temperatures Yibo Li , Yuancheng Yang , Jian Li , Bisheng Wang , Yashi Liao PII: DOI: Reference:

S0734-743X(19)30646-3 https://doi.org/10.1016/j.ijimpeng.2020.103541 IE 103541

To appear in:

International Journal of Impact Engineering

Received date: Revised date: Accepted date:

11 June 2019 13 October 2019 16 February 2020

Please cite this article as: Yibo Li , Yuancheng Yang , Jian Li , Bisheng Wang , Yashi Liao , Experimental-numerical analysis of failure of adhesively bonded lap joints under transverse impact and different temperatures, International Journal of Impact Engineering (2020), doi: https://doi.org/10.1016/j.ijimpeng.2020.103541

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Highlights 

Low-velocity impact investigations were conducted on single lap joints fabricated by 2024-T3 and FM 94K at different temperatures.



DIC (digital image correction) based DCB and ENF tests were conducted to obtain load-displacement curves, and then, R-curves as well as the cohesive laws of the bonded joints were induced through the direct modeling method.



A finite element model is developed successfully to predict the low-velocity response of SLJ.

1

Experimental-numerical analysis of failure of adhesively bonded lap joints under transverse impact and different temperatures Yibo Lia,b, Yuancheng Yanga,*, Jian Lib,c, Bisheng Wangb, Yashi Liaoa a

Light Alloy Research Institute, Central South University, Changsha 410012, China

b

School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China

*Correspondence: E-mail address: [email protected] Abstract: Cohesive failure, energy absorption, and post-load-bearing capacity of the adhesive bonded lap joints were studied through experimental–numerical comparative analysis under four different temperatures (-30℃, room temperature (RT), 50℃, and 80℃) and four different transverse impact (parallel to the normal direction of the bonded surfaces) energies (4, 8, 12, and 16J). A bilinear cohesive zone model was used for simulations, DIC (digital image correction) based DCB and ENF tests were conducted to obtain loaddisplacement curves, and then, R-curves as well as the cohesive laws of the bonded joints were induced through the direct modeling method. The CZM based simulation results showed good agreement with the experimental results on both the impact damage and post-load-bearing capacity of the adhesive joints and showed that both the absorbed energy and maximum post-bearing load did not significantly change with increasing temperature except at 80℃, which is near the glass-transition temperature (Tg). Keyword: bonded joints; transverse impact; post-load-bearing capacity; CZM 1. Introduction Continuous attentions are being paid to reducing the weight and increasing the abilities for energy absorption and load bearings in the aircraft and automotive industries. For this purpose, adhesive bonded structures are receiving increasingly more attention because of their good performances, such as lightweighted, low cost, strong energy absorption capacity and great reduction of the stress concentration and corrosion [1][2]. Mechanical performances, i.e. the strength and energy absorption of the adhesive structures under service conditions at high and low temperatures under salt and spray corrosive environment are attracting more attention of aircraft and automobile manufacturers. Previous studies have investigated the characteristics of bonded joints under impact loading. Asgharifar [3] analyzed and numerically studied the 2

stress wave propagation and stress distribution of bonded joints under impact loading. The influences of the modulus, resin layer thickness, and impact velocity on stress wave propagation were studied. Vaidya [4] numerically simulated low-velocity impacts under in-plane and transverse loads to study single-lap joints. Cohesive failure and crack propagation of the layer always occur at the edges of the adhesive layer under tensile stress. Park [5] and Choudhry [6] focused mainly on the the modes of damage and the damage areas imaged by ultrasonic scanning. Additional, numerical simulation of the hailstone impact was conducted and the result demonstrated that there was a higher peel ans shear stress between the adhesively layer. In practice, single-lap joints are usually subjected to external lateral deflection in the area of the adhesive layer. Sayman [7] evaluated the effect of a low-speed impact on the joint failure response. The tensile test after impact treatment showed that the tensile shear strength of bonded joints significantly decreased. Avendaño and Carbas [8] investigated the effects of temperature on the failure characteristics of bonded joints and found that the strength of joints decreased as the temperature increased. Survana et al. [9] studied the effect of temperature on LVI damage at 30 to 90 ℃. in a previous study, cross-ply carbon fibre/epoxy laminates were subjected to 4.3 J-impact. The damage area exhibited a widely diverse result in relation to cooling laminates, and the results showed that the size of damage area decreased with increasing temperature. However, different adhesions and structures exhibit large differences of mechanical behaviors, and determining the characteristics of the mechanical performances of the related structures as well as the modeling algorithm, parameter identification, and correction of the simulated results are essential. The ability to simulate failure processes is crucial for predicting experimental results. Cohesive zone models (CZM) are a very useful tool for investigating the behavior of bonded joints. Barenblatt [10] proposed the use of the CZM theory to study the problem of quasi-brittle material fracture. A CZM unites elastic stress calculations with classical fracture mechanics such that fracture processes and locations can be effectively modeled. In a recent study, Ribeiro [11] effectively used FEM coupled with CZM to fully describe the failure process and predict the joint strength. Marami et al. [12] employed a finite element model to assess the failure strength of Reduced Graphene Oxide reinforced adhesive bonded joints. Machado [13] and Yuan [14] applied traction-separation laws to model the degradation of elements in order to obtain the relationship between force and displacement under tensile loads. Abir et al. [15] employed CZM to simulate inter-laminar failure using a quadratic traction-separation law for damage initiation. Parameter identifications for the CZM models are essential, to reduce the modelling and identification complexities. Dias [16] and Silva [17] 3

proposed a direct method to obtain the cohesive laws of epoxy bonded joints under different loading modes. After experimental observation of the P-δ curve and crack tip opening displacement ( CTOD), through the so-called equivalent crack length method, the cohesive laws for different failure mode can be obtained. Because of the high precision and ease of use, the Dias method was adopted in this study. The purpose of this paper is to evaluate the damage evolution, energy absorption capacity, as well as the post-load-bearing capacity of adhesive bonded aviation structures under different temperatures. The bonded structure is made of AA2024-T3 aluminum alloys and bonded by the FM 94K epoxy adhesion. Considering the actual load-bearing condition and the frontal collision of the bonded structure, transverse impact failure behavior is studied in this paper. According to the serving environment of the bonded aluminum structures, experimental and CZM model-based simulations were conducted to study the relationship between the damage area, adhesive failure shapes, and energy absorption capacity of the bonding joints. In addition, the post-load-bearing capacity of the impacted joints under different temperatures and transverse impact energies were studied. Four temperatures (-30℃, RT, 50℃ and 80℃) according to maximum servable temperature range of the adhesion, from the -40℃ ~ 92℃ (Tg of the epoxy adhesive) and four transverse impact energies (4 J, 8 J, 12 J and 16 J, considered from the related references [7]) were applied to the joints. 2. Experimental details 2.1. Materials and properties Two 150 mm × 25 mm × 2 mm AA2024-T3 alloy plates (China Southwest aluminum corporation) were bonded using the modified epoxy resin, FM 94K adhesive film (Cytec Industries, USA) in this study. Properties of AA2024-T3 and the FM 94K provided by manufacturer are listed in Table 1 and Table 3, respectively. Table 1. Properties of AA2024-T3. E (MPa)

68000

Young’s modulus

ν (-)

0.34

Poisson’s ratio

ρ (Kg/m3)

2820

Density

σ0 (MPa)

351

Yield tensile strength 4

σult (MPa)

480

Ultimate tensile strength

τ0 (MPa)

264

Yield shear strength

τult (MPa)

283

Ultimate shear strength

α (1/℃)

2.24 x 10-5

Coefficient of thermal expansion

с (KJ/Kg∙ K)

0.88

Specific heat capacity

Table 2. Constants determined for calibration of MRK model for AA2024-T3[22]. 𝑌(𝑀𝑃𝑎)

𝐵0 (𝑀𝑃𝑎)

𝑣(−)

𝑛0 (−)

𝐷2 (−)

𝜉1 (−)

𝜉2 (−)

𝑇𝑚 (𝐾)

351

1000.2

0.07023

0.4805

0.0292

0.001159

0.00845

900

𝜀̇𝑚𝑖𝑛 (𝑠 −1 ) 𝜀̇𝑚𝑎𝑥 (𝑠 −1 ) 10-5

107

𝜃∗ 0.9

Table 3. Properties of adhesive. E(MPa)

48600a

Young’s modulus

ν12 (-)

0.38a

Poisson’s ratio

G (MPa)

35000a

shear modulus

ρ (Kg/m3)

1172a

Density

αL (1/℃)

6.10 x 10-6a

coefficient of thermal expansion

с (KJ/Kg∙ K)

1.7a

Specific heat capacity

GⅠc (N/mm)

1.75b

Mode Ⅰ fracture toughness

GⅡc (N/mm)

3.1b

Mode Ⅱ fracture toughness

Tg (℃)

92

Glass transition temperture a

manufacturer’s data; b tested.

Table 4. Mechanical properties of adhesive under impact conditions. Impact conditions c T (℃)

Properties

Quasi-static a

1 s-1b

100 s-1b

0.36 m/s

0.42 m/s

0.54 m/s

0.62 m/s

Tensile strength, σn (MPa)

35

35

38.5

42

42.27

42.43

42.53

Shear strength, ts (MPa)

27.9

27.9

29.295

30.76

30.87

30.93

30.98

Tensile strength, σn (MPa)

43.8

43.8

48.18

52.55

52.89

53.08

53.22

Shear strength, ts (MPa)

36.5

36.5

38.325

40.184

40.328

40.408

40.464

-30

RT

5

Tensile strength, σn (MPa)

26.8

26.8

29.48

32.3

32.5

32.6

32.6

Shear strength, ts (MPa)

25.7

25.7

26.99

28.4

28.5

28.5

28.6

Tensile strength, σn (MPa)

21.5

21.5

23.65

25.9

26.1

26.2

26.4

Shear strength, ts (MPa)

22.1

22.1

23.205

23.9

24.1

24.1

24.1

50

80

a

tested; b obtained from references[34][35]; c logarithmic extrapolation[33].

2.2 Adhesive characterization tests In order to characterize the tensile strength of the adhesive, the typical specimen geometry used (see Fig.1(a)) was in accordance to EN ISO 527-2 [18]. While ASTM D3165-07 test method [19] is intended for use in metal-to-metal applications, it was used to characterize the shear strength of the adhesive. The stiffness of the aluminium substrate combined with the overlap length obtains an almost pure shear stress in bonded layer. Fig.1(b) shows the specimen geometry. To measure the tensile and shear strength of adhesive at different temperature, the tests were carried at four kinds of temperatures (-30℃, RT, 50℃ and 80℃) and three specimens were performed for each condition. Fig.2 showed the results obtained for all the tests at four different temperatures. Table 3 lists the experimentally measured mechanical properties for the FM94K adhesive under quasi-static condition.

a)

b)

Fig.1. a) Geometry of tensile specimens. b) Geometry of shear specimens.

6

Fig.2. Tensile and shear strength of the FM 94K at different temperatures.

2.3. Specimen fabrication Primarily, aluminum alloy plates were cut into a dimension of 25 × 150 mm, as shown in Fig.3. Then, the surfaces of adherends were prepared as follows: (1) The surfaces of the aluminum alloy plates were roughened with abrasive 320 grit sandpapers, and the aluminum alloy plates were subsequently degreased with anhydrous ethanol; (2) The surfaces of the aluminum alloy plates were anodized, and orthogonal experiments were performed under the following conditions: determined to anodization parameters-15 V direct-current voltage, 10% phosphate solution, 30℃ solution temperature, and 20 min processing time. Before bonding, the parts and adhesive film must be properly assembled; therefore, patterns of FM94K adhesive were cut as required. Finally, the curing temperature was maintained at 120℃ and the curing time was set at 60 min under a pressure of 0.28 MPa. Details of the process have been described in our previous study [20].

25

2

150

substrate

0.2

substrate 25

adhesive 150

Fig.3. Geometry of SLJ specimens (dimensions in mm).

2.4. Impact and tensile test procedure 7

Low-velocity impact tests were performed under the CEAST 9350 Drop Tower Impact System (Instron, USA) equipped with the CEAST DAS 16000 data collection system. The impact test fixture was illustrated in Fig.4. A hemispherical impactor with a diameter of 16.0 mm was used, and a force sensor was installed on the impactor. In this study, the impact energies of 4, 8, 12, and 16 J were applied at various temperatures. More precisely, all specimens were maintained for 10 min under their test temperature, in order to ensure homogeneous temperature of the specimens. Three specimens were tested for each condition. The specimens were placed in a fixture and the impactor was sliding down along a fixed channel (see Fig.5). Before each test, the plumb bob was hung from the drop center to ensure the impactor drop in the same place. The specimens subjected to various transverse impact energies are shown in Fig.6.

Fig.4. The impact test fixture.

Guide Roller

Impactor Plumb bob: determine the drop point before Test

Laser

Specimen

Clamp

Support

8

Fig.5. schematic of the impact test setup.

Fig.6. Specimens with subjected to impact.

The post-load-bearing captivity tests of the impacted joints through the uniaxial tensile method were conducted under the CMT5105 universal testing machine (SUST, China) with a capacity of ±100 KN (see Fig.7). All the tests were performed with a crosshead speed of 2 mm/min, three specimens were tested under each condition, and then the average value are discussed.

Fig.7. SLJ specimens clamped in the tensile fixture.

3. Models of the simulation To better investigate the impact and energy absorption behaviors of the adhesive bonding joints, as well 9

as the post-bearing capacity of partially damaged structures, a series of CZM model-based impact analysis was conducted. Two steps of the simulations were conducted: the former was the explicit dynamic analysis of the low-velocity transverse impact response for the bonding structures, and the latter was the implicit axial quasi-static tensile analysis for the impacted joints. For both models, a bilinear cohesive zone model was used to represent the damage and residual strength of the adhesions. 3.1. Constitutive model for the AA2024-T3 In this paper, the Modified Rusinek-Klepaczko model was used. The MRK constitutive relation is based on the additive disintegration of the Huber-Mises equivalent stress [21][22]. 𝐸(𝑇) ̅(𝜀̅𝑝 , 𝜀̅̇𝑝 , 𝑇) = 𝐸 [𝜎̅𝜇 + 𝜎̅ ∗ (𝜀̅𝑝 , 𝜀̅̇𝑝 , 𝑇)] σ

(1)

0

The multiplicate factor 𝐸(𝑇)⁄𝐸0 defines the Young’s modulus evolution process with temperature: 𝐸 (𝑇) = 𝐸0 {1 −

𝑇 𝑇𝑚

𝑒𝑥𝑝 [𝜃 ∗ (1 −

𝑇𝑚 𝑇

)]} , 𝑇 > 0

(2)

where 𝐸0 , 𝑇𝑚 and 𝜃 ∗ denote the Young’s modulus at 𝑇 = 0𝐾 , the melting temperature and the characteristic homologous temperature, respectively. According to FCC metal such as aluminium alloys, 𝜃 ∗ ≈ 0.9, as mentioned in [22], the athermal stress 𝜎̅𝜇 is assumed independent of plastic strain. This stress component is defined below: 𝜎̅𝜇 = 𝑌

(3)

where 𝑌 is the flow stress at 𝜀̅𝑝 = 0. The thermal stress 𝜎̅ ∗ is the flow stress component defining expressions in regard to rate dependent. Based on the theory of thermodynamics and kinetics of slip[23], the expression can be written as: 𝑝 𝑇 𝜀̇ 𝜎̅ ∗ (𝜀̅𝑝 , 𝜀̅̇𝑝 , 𝑇) = 𝐵( 𝜀̅̇𝑝 , 𝑇) ∙ (𝜀̅𝑃 )𝑛( 𝜀̅̇ ,𝑇) ∙ 〈1 − 𝜉1 (𝑇 ) 𝑙𝑜𝑔 ( 𝑚𝑎𝑥 )〉1⁄𝜉2 𝜀̅̇ 𝑝 𝑚

(4)

where 𝜉1 and 𝜉2 are material constants describing temperature and rate sensitivity of the material, 𝜀̇𝑚𝑎𝑥 is the upper limit of deformation rate. The explicit formulations describing the modulus of plasticity 𝐵(𝜀̇ 𝑝 , 𝑇) and the strain hardening exponent 𝑛(𝜀̇ 𝑝 , 𝑇) are expressed by the following equations: 10

−𝑣 𝑝

𝐵(𝜀̇ , 𝑇) =

𝑇

𝜀̇ 𝐵0 ((𝑇 ) log( 𝑚𝑎𝑥 )) 𝜀̅̇ 𝑝 𝑚

,𝑇 > 0

𝑇

𝜀̇ 𝑝

𝑚

𝑚𝑖𝑛

𝑛(𝜀̇ 𝑝 , 𝑇) = 𝑛0 〈1 − 𝐷2 (𝑇 ) 𝑙𝑜𝑔 (𝜀̇

)〉

(5)

(6)

where 𝐵0 is a material constant, 𝑣 is the proportional to temperature sensitivity, 𝑛0 is the strain hardening exponent at 𝑇 = 0𝐾, 𝐷2 is the material constant and 𝜀̇𝑚𝑖𝑛 is the minimum rate level for application of the model. The material constants of the MRK model corresponding to the AA 2024-T3 was given by RodríguezMartínez et al. [22] and is listed in Table 2. According to test results [22], strain rate (between 0.0001𝑠 −1 𝑎𝑛𝑑 100𝑠 −1 ) has little influence on the properties of aluminum materials under 100℃. 3.2. Cohesive Zone Model and the damage criterial Cohesive zone model (CZM) based on Dugdale-Barenblatt model supposes that there is a small plastic zone at the crack tip. The CZM elements indicate the plastic zone and do not require prefabricated initial crack. Fig.8 reveals the elastic-plastic crack and the stress state of CZM element [24][25]. The CZM elements suffer normal and sliding shear stress, as shown in Fig.8. The CZM elements have three directions (n, s, t) strain as following Eq. (7): 𝜀𝑛 =

𝛿𝑛 𝑇0

𝛿

𝛿

, 𝜀𝑠 = 𝑇𝑠 , 𝜀𝑡 = 𝑇𝑡 0

0

(7)

where, 𝛿𝑚 is the equivalent displacement as Eq. (8), and 𝑇0 shows CZM element thickness. 2 𝛿𝑚 = √𝛿𝑠2 + 𝛿𝑡2 + 〈𝛿𝑛 〉2 = √𝛿𝑠ℎ𝑒𝑎𝑟 + 〈𝛿𝑛 〉2

(8)

The CZM element constitutive relation is shown in Eq. (9). 𝑡𝑛 𝐾𝑛𝑛 𝑡 = { 𝑡𝑠 } = [ 𝐾𝑛𝑠 𝑡𝑡 𝐾𝑛𝑡

𝐾𝑛𝑠 𝐾𝑠𝑠 𝐾𝑠𝑡

𝐾𝑛𝑡 𝜀𝑛 𝐾𝑠𝑡 ] { 𝜀𝑠 } = 𝐾𝜀 𝐾𝑡𝑡 𝜀𝑡

(9)

The CZM element damage failure process includes initiation and propagation of damage crack, as well as debonding. The debonding process is described in Eq. (10)-(13). (1 − 𝐷)𝑡𝑛̅ , 𝑡𝑛̅ ≥ 0 𝑡𝑛 = { ̅ 𝑡𝑛 , 𝑡𝑛̅ ≤ 0(𝑛𝑜 𝑑𝑎𝑚𝑎𝑔𝑒) 11

(10)

𝑡𝑠 = (1 − 𝐷)𝑡𝑠̅

(11)

𝑡𝑡 = (1 − 𝐷)𝑡𝑡̅

(12)

𝑓

𝐷=

𝑚𝑎𝑥 −𝛿 0 ) 𝛿𝑚 (𝛿𝑚 𝑚

(13)

𝑓

𝑚𝑎𝑥 (𝛿 −𝛿 0 ) 𝛿𝑚 𝑚 𝑚

Models using linear, trapezoidal, and exponential softening traction–separation law for the CZM elements are analyzed. The adhesive strength is defined as the starting point of nonlinearity while the critical energy release rate is denoted by the area under the traction–separation curves [23]. Because this test adhesive shows brittle performance, the bilinear traction-separation law was performed in this paper. Fig.9 shows how the cohesive law is defined by an initial stiffness until a maximum strength value (𝑡𝑡0 , 𝑡𝑛0 ), and 𝐶 𝐶 from which it begins to degrade. The degradation is defined by the fracture energy value (𝐺Ⅰ , 𝐺Ⅱ ). Scalar

stiffness degradation (SDEG) factor is used to describe the extent of damage, and its scope is 0 ≤ SDEG ≤ 1. SDEG = 1 denotes that the cohesive element has failed, and it would be deleted to prevent a serious distortion. 0 This study used quadratic nominal stress criterion as described in Eq. (14). 𝛿𝑚 can be calculated from

Eq. (15). 〈𝑡𝑛 〉 2

{

0 𝑡𝑛

0 𝛿𝑚

𝑡

2

2

𝑡

} + { 𝑠0 } + { 0𝑡 } = 1 𝑡 𝑡 𝑠

={

(14)

𝑡

𝛿𝑛0 𝛿𝑠0 √

1+𝛽2 2 0 )2 (𝛿𝑠0 ) +(𝛽𝛿𝑛

0 𝛿𝑠ℎ𝑒𝑎𝑟 ,

, 𝛿𝑛 > 0

(15)

𝛿𝑛 ≤ 0

where 𝛽 indicates the mixed-mode ratio described in Eq. (16). 𝛽=

𝛿𝑠ℎ𝑒𝑎𝑟

(16)

𝛿𝑛

The mixed-mode fracture energy release rate criterion of law [24] was adopted as demonstrated in Eq. (17). 𝛼

𝛼

𝛼

𝐺𝑛

𝐺𝑠

𝐺𝑡

Ⅰ

Ⅱ

Ⅱ

{𝐺 𝐶 } + {𝐺 𝐶 } + {𝐺 𝐶 } = 1

(17)

𝐶 𝐶 where 𝐺Ⅰ is mode Ⅰ critical fracture energy release rate, 𝐺Ⅱ is mode Ⅱ critical fracture energy release rate.

12

𝛼 is a material-specific power law coefficient. It has been assumed that 𝛼 = 1 so that Eq. (17) becomes a 𝑓

linear interaction criterion[24]. Using Eq. (18) to calculate 𝛿𝑚 . 2(1+𝛽2 ) 𝑓 𝛿𝑚

0 𝐾𝛿𝑚

= {

𝛼

[(

1 𝐺𝐶

Ⅰ

𝛽2

𝛼 −1⁄𝛼

) + (𝐺 𝐶 ) ]

2 √(𝛿𝑠𝑓 )

, 𝛿𝑛 > 0

Ⅱ

+

2 (𝛿𝑡𝑓 ) ,

(18) 𝛿𝑛 ≤ 0

Fig.8. Schematic presentation of the cohesive zone concept.

13

Fig.9. Mixed-mode bilinear cohesive law.

3.3. Simulation model descriptions Cohesive zone models are integrated in many finite element software, such as Abaqus, and zerothickness and actual geometrical thickness cohesive elements can be used to identify the cohesive behaviors described above. However, as a typical interface element, only one layer of the cohesive zone element can be inserted along the thickness direction between the bonded surfaces. Research on the cohesive zone elements have shown that even one layer of the cohesive zone elements can give enough precision for the damage and stress evolution simulation[27], Furthermore, simulation results are slightly influenced by mesh density with the use of cohesive element[28]. For the purposes of the impact and strength simulations, COH3D8 elements, which have a thickness of 0.2 mm (the real thickness of the adhesive), were used with bilinear traction-separation law in the models, while the adherends with C3D8R elements had a thickness of 2 mm. Meshes of the adhesives along the length and width directions were much finer than the meshes of adherends to improve the convergence and avoid stress concentration as much as possible. Then, tie constraints were applied between the contact surfaces of the adhesive and the adherends to make the correct connection between the adhesion and the adherends. As shown in Fig.10(a)~(b), low-velocity transverse impact analysis under different temperatures and impact energies were conducted at first. For these models, two ends of the adherends were fixed and the impact load was applied through the definition of certain initial impact velocities in the negative z direction of the hemispherical impactor with the diameter of ϕ16.0 mm. The weight of the impactor was 0.650 kg and thus the initial velocities of the impactors were 2.01 m/s, 2.85 m/s, 3.48 m/s, and 4.02 m/s, which represented impact energies of 4, 8, 12, and 16 J, respectively. Post-load-bearing quasi-static analysis of the impacted joints were performed based on the impacted structure performed in the previous step (Fig.10(a)). Result files (.odb) of the impact analysis were imported firstly (Fig.10(c)), and then one end of the adherends was fixed and the other end of the adherends was pulled by a definition of displacement in the direction of x axis shown in Fig.10(a).

14

a) work-flow and boundary conditions

b) impact analysis model

c) imported deformed model

Fig.10. Schematic representation of the numerical models

Material properties related to the analysis are listed in the Table 1-4. However, for the models, cohesive laws of the joints must be studied, and crucial parameters of the CZM model, such as the critical stress of 𝜎𝑚𝑎𝑥 and the strain energy of 𝐺𝑐 for different fractural modes, must be determined exactly. The parameter 𝜎𝑚𝑎𝑥 of the model can be confirmed by an inverse method [13], but the parameter 𝐺𝑐 , where 𝐺𝐼 for the fractural mode I and 𝐺𝐼𝐼 for the fractural mode II, must be determined from the experimental load– displacement curves (P-δ curves) of the double cantilever beam (DCB) tests and end notched flexure (ENF) tests, respectively. More details on the definition of cohesive laws will be discussed in the next section. 3.4. Cohesive laws of the bonded joints under different fractural modes Two key parameters of the cohesive law, the fracture energy (G) and the peak stress (𝜎), are required to be determined under different test conditions. However, it was found that little influence of the temperature 15

and the strain rate on the fracture energy while tested below glass transition temperature (Tg) of the adhesive, but the stiffness and tensile strength of the adhesive were increasing with the decreasing temperature [29][30]. To be simplify, fracture energy (G) of the cohesive law was tested and induced at room temperature, while the peak stresses under different temperatures were tested. All the tests were conducted by the DIC method. 3.4.1. Algorithm for determining the cohesive laws A direct method for determining the cohesive laws under different modes is based on the following relationship [16][17]: 𝑢

𝐺𝐼 = ∫0 (𝜎)𝑑𝑤

(19)

𝑢

𝐺𝐼𝐼 = ∫0 (𝜏)𝑑𝑢

(20)

The differential form of Eq.(19)–Eq. (20) leads to the cohesive law 𝜎 = 𝑓 (𝑤) and 𝜏 = 𝑓 (𝑢): 𝜎 (𝑤 ) =

𝑑𝐺𝐼

𝜏 (𝑢 ) =

𝑑𝐺𝐼𝐼

(21)

𝑑𝑤

(22)

𝑑𝑢

where, 𝐺𝐼 and 𝐺𝐼𝐼 are the strain energy release rate in mode I and mode II, respectively; 𝜎 and 𝜏 refer to the tension and shear stress and are functions 𝐶𝑇𝑂𝐷𝐼 and 𝐶𝑇𝑂𝐷𝐼𝐼 , respectively. Based on the Timoshenko beam theory, the specimen compliance (𝐶 = 𝛿/𝑃) can be written as: 8𝑎 3 𝐸

𝐵ℎ 3

12𝑎

+ 5𝐵ℎ𝐺

𝐶 = {3𝑎1 3 +2𝐿3 8𝐸1 𝐵ℎ 3

𝑓𝑜𝑟 𝑚𝑜𝑑𝑒𝐼

13

(23)

3𝐿

+ 10𝐵ℎ𝐺

13

𝑓𝑜𝑟 𝑚𝑜𝑑𝑒𝐼𝐼

where 𝐸1 and 𝐺13 are the elastic properties of the specimens. The dimensions of the remaining parameters of the specimen are defined in Fig.11. Considering the initial crack length 𝑎0 and compliance 𝐶0 , the equivalent modulus of each specimen (𝐸𝑓 ) to be used instead of 𝐸1 in Eq.(24) is:

16

12(𝑎0 +ℎ∆) −1 8(𝑎0 +ℎ∆)3

(𝐶0 −

𝐸𝑓 = {

3𝑎03 +2𝐿3 8𝐵ℎ 3

)

5𝐵ℎ𝐺13

(𝐶0 −

3𝐿 10𝐵ℎ𝐺13

)

𝐵ℎ 3 −1

𝑓𝑜𝑟 𝑚𝑜𝑑𝑒 𝐼 (24) 𝑓𝑜𝑟 𝑚𝑜𝑑𝑒 𝐼𝐼

where ∆ reflects the crack length correction by accounting for the root rotation effects and can be calculated by [31] : 𝐸𝑓

Γ

2

Δ = √11𝐺 [3 − 2 (1+Γ) ]

(25)

13

Γ = 1.18

√ 𝐸𝑓 𝐸3

(26)

𝐺13

The equivalent crack 𝑎𝑒 from the current compliance, which takes the nonlinear effects for consideration can be obtained from the iterative solution for Eq.(24) (for mode I only) or directly calculated from(for mode II only): 1

𝐶𝑐

𝑎𝑒 = [ 𝐶

0𝑐

3𝐿

𝑎03

2

𝐶𝑐

3 ]3

+ 3 (𝐶 − 1) 𝐿 0𝑐

(27)

3𝐿

where 𝐶𝑐 = 𝐶 − 10𝐵ℎ𝐺 , 𝐶0𝑐 = 𝐶0 -10𝐵ℎ𝐺 13

13

Then, the R-curve can be obtained from the Irwin–Kies equation, 𝑃 2 𝑑𝐶

𝐺 = 2𝐵 𝑑𝑎

(28)

Finally, the strain energy release rates can be calculated from: 6𝑃 2

2𝑎𝑒2

1

𝐺𝐼 = 𝐵2 ℎ (𝐸

2 𝑓ℎ

+ 5𝐺 ) 13

9𝑃 2 𝑎 2

𝐺𝐼𝐼 = 16𝐵2 ℎ3𝑒𝐸

𝑓

Calculation procedure of the strain energy release rates was illustrated as Fig.12.

17

(29) (30)

a) DCB test

b) ENF test

Fig.11. Schematic representation of the DCB and ENF test Table 5. dimension of the DCB and ENF specimens DCB specimens

ENF specimens

h /mm

t /mm

L/mm

B/mm

h /mm

t /mm

L /mm

B /mm

6

0.2

140

6

6

0.2

70

6

Fig.12. Procedures of the calculation of strain energy rates.

3.4.2. CTOD tests and cohesive laws calculation Digital image correlation (DIC) based double cantilever beam (DCB) tests and end notched flexure (ENF) tests, with the guidance of the GOM ARAMIS system were conducted to obtain the P-δ curves, as 18

well as the 𝐶𝑇𝑂𝐷𝐼 (for mode I through DCB test) and 𝐶𝑇𝑂𝐷𝐼𝐼 (for mode II through ENF test). Table 5 listed the dimensions for DCB and ENF specimens. All of the experiments were performed on the CMT5105 (SUST, China) universal testing machine, with a loading ratio of 2 mm/min according to ISO25217 standard. Testing systems and specimens are shown as Fig.13.

Fig.13. Experimental setup for DCB and ENF tests.

Codes were programed by MATLAB® software to calculate the strain energy release rate under different modes according to the Eq.(29)–Eq.(30). P-δ curves, and the corresponding R-curves of DCB tests and ENF tests were obtained or predicted, as shown in Fig.14 and Fig.15. Finally, the R-curves were used to identify the fracture energy form the plateau (starting with the marking point in Fig.14(b) and Fig.15(b)) corresponding to the self-similar crack propagation. Cohesive laws under different modes were obtained and are listed in Table 3 and Table 4. Fig.16 presents the simulated adhesive fracture process of the DCB and ENF specimens using the determined parameters. P-δ curve prediction results are shown in Fig.14(a) and 19

Fig.15(a). Comparing the simulated P-δ curves and the tested P-δ curves, good agreements were obtained.

a) P-δ curves

b) R-curves

Fig.14. Load–displacement (P-δ) curves and the corresponding R-curves from the DCB tests for mode Ⅰ failure.

a) P-δ curves

b) R-curves

Fig.15. Load–displacement (P-δ) curves and the corresponding R-curves from the ENF tests for mode ⅠI failure.

a) DCB simulation 20

b) ENF simulation Fig.16. The simulated adhesive fracture process of the DCB and ENF specimens.

3.5. Strain rate effect The adhesive demonstrated to be highly strain rate dependent because of its polymeric property. In this case, the properties that were introduced in the model were obtained by using a logarithmic extrapolation from the test rate values at 1 and 100 mm/min. In spite of the simplicity of this calculation, it has been used in previous research works to define the behaviour of adhesive as a function of the strain rate [32][33]. This logarithmic extrapolation expresses Eq. (27): 𝑥 = 𝐴𝑙𝑛(𝜀̇) + 𝐵

(27)

where 𝑥 is the property to be ensured, 𝜀̇ is the strain rate and 𝐴 and 𝐵 are constants determined experimentally. The strain rates associated to the test speeds were deduced form Eq. (28): 𝜀̇ = 𝑣/𝑙0

(28)

where 𝑣 is the test speed and 𝑙0 is the calibrated length of specimens. Shokrieh [34] and Morinière [35] reported strain rate below 1 s-1 scarcely effects the FM94 adhesive mechanical properties, but a strain rate of 100 s-1 augments the tensile strength of monodirectional epoxy resin by 10% when compared to quasi-static conditions. In regard to the transverse behaviour, a 5% increase in strength at s strain rate of 100 s-1 was mentioned in the references. Table 4 lists the mechanical properties (average values) in this reference and the extrapolated properties for impact velocity. 4. Results and Discussion 21

In order to preferably study dynamic responses of the bonded joints subjected to transverse impact, low velocity impact tests were performed under different impact energies and different temperatures. In this study, according to the damage process, absorbed energy, and tensile shear strength, the influence of transverse impact and temperature was studied. In addition, numerical simulations were performed and verified through the experimental results. 4.1. Damage characterization Table 6 shows the damage area, absorbed energy ratio, tensile shear strength, and crater deep of the experimental and simulation results. The damage area of the adherend increased with the increasing impact energies. Relative to the experimental results, the errors of the damage area were 4.2%, 9.6%, 6.5%, and 7%, respectively. The experimental damage extent picture of the front side is depicted in Fig.17(a)(c), and the numerical damage picture is depicted in Fig.17(b)(d). The adherend is observed to exhibit a permanent indention. In addition, by comparing the impact experiment and simulations at different temperatures, it is observed that the numerical model is effectively validated by the experimental results. Because of the sensitivity of the damage area to higher temperature, at 12 J impact energy, the damage area increased from 5.9 × 6.1 mm2 to 6.8 × 7.2 mm2, representing an increment of approximately 26.5% when the temperature rises to 80℃. Table 6. Experimental and simulation results under impact loading at room temperature. Damage area

Tensile shear strength

Crater deep

(KN)

(mm)

Absorbed energy ratio (mm × mm) Experiment- 4J

4.7 x 4.8

0.36

8.049

0.72

Simulation-4J

4.6 x 4.7

0.38

7.814

0.66

Error

4.2%

5.6%

3%

8.3%

Experiment- 8J

5 x 5.2

0.42

7.185

0.88

Simulation-8J

5 x 5.7

0.4

7.052

0.84

Error

9.6%

4.8%

1.95%

4.5%

22

Experiment-12J

5.9 x 6.1

0.54

4.847

0.95

Simulation-12J

5.7 x 5.9

0.51

4.721

0.90

Error

6.5%

5.6%

2.6%

5.2%

Experiment-16J

6.5 x 7

0.62

6.625

1.16

Simulation-16J

6.3 x 6.7

0.57

2.416

1.11

Error

7%

8%

63%

4.3%

a)

b)

c)

d)

Fig.17. Comparison of the impact damage area between experiment and simulation.(a,b,c,d)

The finite element model based on CZM can effectively simulate the failure condition of the bonded 23

joints and predict their performance. More precisely, the finite element simulation reflects the process of fracture failure of the bonded joint structure. The internal damage in single lap joints under impact loading is a significant focus, especially for the interface damage. Fig.18 shows the progressive failure process in the finite element simulation. When the SDEG value is greater than 0, the element begins to undergo damage. As the value of SDEG reaches 1, the elements are at the damage limit and the damaged elements are deleted. From the simulation results, the damage of the adhesive layer can be observed to begin at the impact point, causing the glue layer to detach, which in turn reduces the contact area between the substrate and the glue layer, resulting in a decrease in strength. On the other hand, Fig.19 shows the progressive failure of the 12Jimpact specimens at RT and at 80℃. It is noticed that the damage failure of specimens at 80℃ is higher than that at room temperature. This is due to the degradation of the adhesive. Moreover, the impacts with 12 J at 80℃ and 16 J at RT reveal similar damage failures, and their tensile shear strengths exhibit similar results.

a)

b)

c)

d)

24

Fig.18. Progressive failure of specimens subjected to different impact energies: a) 4J, b) 8J, c) 12J, and d) 16J.

Fig.19. Progressive failure of 12J-impact specimens at RT and 80℃: a) RT-12J; b) 80℃-12J.

The failure of the adhesive on the impact area is experimentally observed (Fig.20), similar to the simulation results shown in Fig.19. Stress distributions of the cohesive layer at room temperature under different impact energies are shown in Fig.21. In addition to the V-shaped stress distribution in the adhesive bonded area, stress concentrations on the edges are also found. Consequently, as shown in Fig.22, cracks were observed to start from the edges of the adhesively bonded joint, thus affecting the strength of the joints. The corresponding crack images under different impact energies were displayed to determine the extent of damage. The crack length clearly increases with the impact energy. In Fig.23 the crack length is plotted against the impact energies. For the impacts at -30℃, the crack length increases gradually, similar to the case for impact at 50℃. However, in the case for impact at 80℃, the crack length for impacts distinctly increases. This is mainly due to the higher stress generated at higher temperatures.

25

Fig.20. The fracture surface of 12J-impact specimen.

a)

b)

c)

d)

Fig.21. The stress distribution in the adhesively area when subjected to impact loading: a) 4J, b) 8J, c) 12J, and d) 16J.

26

Fig.22. damaged crack of joints for the different impact energies at RT.

Fig.23. The crack length at different impact energies and temperatures.

4.2. Absorbed energy Fig.24 shows the changing curve of transmitted energy over time in the impact test. The rebound of the impactor is due to the energy stored in the elastic deformation. The absorbed energy is the inelastic part of the impact; Thus, it is dissipated in the form of heat and damage in the specimens. The failure processes of the bonded joint mainly include matrix deformation, bondline cracking, and interface delamination. Therefore, the primary part of the absorbed energy is used to generate the required damage. 27

Fig.24. The changing curve of transmitted energy over time in impact test.

Fig.25 shows energy-time curves for various impact energies at room temperature. The impact transmitted energy (Ei) is the maximum value of the curve and the absorbed energy is defined as (Eabs). Because high impact energy was imposed, more serious damage occurred. Therefore, absorbed energy was found to increase with increasing impact energy. In particular, the ratio Eabs/Ei was used to indicate the energy absorption capacity of the bonded joints. Then according to the experimental results, 4 J, 8 J, 12 J, and 16 J respectively correspond to ratios of 0.36, 0.42, 0.54, and 0.62. In the comparison, the deviations between the numerical and experimental energy curves are within 7%.

Fig.25. Energy vs time curves for different impact energies.

Fig.26 shows the relationship between energy absorption and impact energies at different temperatures. It is noteworthy that the absorbed energy did not increase significantly with increasing temperature up to 28

50℃, but the absorbed energy obviously increased at 80℃, mainly because the higher temperature is close to Tg.

Fig.26. The energy absorption capacity (Eabs/Ei) at different impact energies and temperatures.

4.3. Post-load-bearing capacity Tensile tests of the adhesively bonded joints of alloy plate specimens to evaluate the post-load-bearing capacity were conducted by subjecting them to various transverse impact energies at different temperatures (-30℃, RT, 50℃, and 80℃). Fig.27 shows force-displacement curves of joints subjected to impact load prior to tensile tests at room temperature. It can be clearly seen that the failure loads and displacements decreased as the impact energies increased. This experimental phenomenon can be attributed to the high peeling stress on the edges of joints induced by transverse impact loading, because of which initial cracks usually appear on the edges. Furthermore, instantaneous impact loading can cause debonding of the adhesive layer region at the interface, resulting in insufficient adhesion. Nevertheless, the adhesively bonded joints subjected to the impact energy of 16 J show comparatively higher post-load-bearing capacity and failure displacements. For specimens subjected to 16 J-impact, the failure displacement and failure load were observed to increase by 38.3% and 31.7% compared to those subjected to 12 J. Comparing the numerical and experimental results, the numerical curves were reasonably close to experimental results for the failure load and displacement, and the differences were roughly controlled below 5%. However, the numerical results showed slightly lower values than the experimental results. This explains 29

why the experimentally identified properties, such as tensile strength and fracture energy properties, are slightly conservative. Secondly, to simplify the numerical simulation, the fracture energy was considered as strain-rate independent. Therefore, the numerical model exhibited slightly lower strength results than those of the experimental test. Furthermore, obvious differences were observed between the experimental and numerical curves of specimens subjected to 16 J-impact. The reason for this experimental result is that adherend plates are stamped during the impact process to create a deeper crater, thereby obtaining a clinching mechanical joint. Table 6 shows the crater deep of specimens subjected to 16J-impact is observed to increase by 61.1 % compared to 4J-impact. Secondly, there was actual contact between the upper and lower adherend plate in the real experiment. However, the impact damage of adhesives was deleted resulting in contact in the adherend plates in the simulation, but the interaction between the upper adherend and the lower adherend was not defined.

Fig.27. Experimental vs Numerical results of SLJs tested at different impact energy conditions.

The post-load-bearing capacity of joint specimens tested at different energies and temperatures are given in Fig.28. The failure loads of specimens subjected to different impact energies did not significantly decrease at -30℃ and 50℃. However, it clearly decreased at 80℃. This is mainly due to the degradation of the epoxy resin adhesive. With increasing temperature the modulus and strength are decrease.

30

Fig.28. The post-load-bearing capacity at different energies and temperatures.

5. Conclusions In the present study, the influence of temperatures and transverse impact on failure responses of singlelap joints were investigated numerically and experimentally. Impact and tensile tests were performed on bonded joints at different impact energies and temperatures. In addition, numerical models were developed and validated using the experimental results. The following main conclusions can be drawn: a) The adhesive bonded area produced a V-shaped stress distribution under transverse impact loading and the stress increased towards the edges. Consequently, cracks were observed to start from the edge of the adhesively bonded joints. The damage area and crack length grew with the increasing impact energies and temperatures. b) The absorbed energy of bonded joints increased with increasing impact energy. In addition, the absorbed energy did not increase significantly with increasing temperature up to 50℃, but it obviously increased at 80℃, mainly because the higher temperature is close to T g. c) Post-load-bearing capacity of the bonded joints decreased with increasing impact energies. In contrast, the failure loads of specimens subjected to impact loading did not significantly decrease at -30℃ and 50℃. However, the failure load of specimens obviously decreased at 80℃. d) Performing DCB and ENF fracture tests and obtaining the cohesive parameters is sufficient to investigate the response of impact loading. The simulations adopted bilinear cohesive zone model. The 31

numerical model could reasonably predict the process of failure of progressive damage of the bonded joints. Overall, the predicted results are in agreement with the experimental results. Acknowledgments This research was supported by National Natural Science Foundation of China (51575535), the Key Technology R&D Program of Liuzhou City (2018BB30501) , Science and Technology Plan of Hunan Province(2013RS4083, 2012RS4076) and the CSC scholarship (201806375038). References [1] Silva LD. Handbook of Adhesion Technology. Springer Berlin Heidelberg 2011. [2] Salih Y, Yiannis A, Robert E J, Daniel S, Douglas J, Feridun D. Characterization of adhesively bonded aluminum plates subjected to shock-wave loading. Int J Impact Eng 2019;127:86-99. [3] Asgharifar M, Kong F, Carlson B, Kovacevic R. Dynamic analysis of adhesively bonded joint under solid projectile impact. Int J Adhes Adhes 2014;50:17-31. [4] Vaidya UK, Gautam ARS, Hosur M, Dutta P. Experimental–numerical studies of transverse impact response of adhesively bonded lap joints in composite structures. Int J Adhes Adhes 2006;26:184-198. [5] Park H, Kim H. Damage resistance of single lap adhesive composite joints by transverse ice impact. Int J Impact Eng 2010;37:177-184. [6] Choudhry RS, Syed F H, S.Li, R.Day. Damage in single lap joints of woven fabric reinforced polymeric composites subjected to transverse impact loading. Int J Impact Eng 2015;80:76-93. [7] Sayman O, Arikan V, Dogan A, Ibrahim FS, Tolga D. Failure analysis of adhesively bonded composite joints under transverse impact and different temperatures. Compos Part B: Eng 2013;54:409-414. [8] Avendaño R, Carbas RJC, Chaves FJP, Costa M, Da Silva LFM, Fernandes AA. Impact Loading of Single Lap Joints of Dissimilar Lightweight Adherends Bonded with a Crash-Resistant Epoxy Adhesive. J Mater Sci Technol 2016;138. [9] Suvarna R, Arumugam V, Bull DJ, Chambers AR, Santulli C. Effect of temperature on low velocity impact damage and postimpact flexural strength of CFRP assessed using ultrasonic C-scan and micro-focus computed tomography. Compos Part B: Eng 2014;66:58-64. [10] Barenblatt GI. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Adv Appl Mech 1962;7:55-129. [11] 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. [12] Marami G, Nazari SA, Faghidian SA, Faghidian SA, Vakili-Tahami F, Etemadi S. Improving the Mechanical Behavior of the Adhesively Bonded Joints Using RGO Additive. Int J Adhes Adhes 2016;70:277-286.

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[30] Banea MD, Da Silva LFM, Campilho RDSG. Mode I fracture toughness of adhesively bonded joints as a function of temperature: Experimental and numerical study. Int J Adhes Adhes 2011;31:273-279. [31] Jorge C, Almudena MM, Xavier José, Guaita M. Determination of the resistance-curve in Eucalyptus globulus through double cantilever beam tests. Mater Struct 2018;51:77-. [32] Avendao R, Carbas RJC, Marques EAS, da Silva LFM, Fernandes AA. Effect of temperature and strain rate on single lap joints with dissimilar lightweight adherends bonded with an acrylic adhesive. Compos Struct 2016. [33] Araújo HAM, Machado JJM, Marques EAS, da Silva LFM. Dynamic behaviour of composite adhesive joints for the automotive industry. Compos Struct 2017;171:549-561. [34] Shokrieh MM, Omidi MJ. Investigating the transverse behavior of Glass–Epoxy composites under intermediate strain rates. Compos Struct 2011;93:690-696. [35] Morinière FD, Alderliesten RC, Benedictus R. Modelling of impact damage and dynamics in fibre-metal laminates-A review. Int J Impact Eng 2014;67:27-38.

Declaration of interests

☐ 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. Yes, we declare that we have no conflicts of interest to this work.

☒The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

no

34