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The effect of angle on the strain of scarf lap joints subjected to tensile loads
Applied Mathematical Modelling 36 (2012) 2858–2867
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The effect of angle on the strain of scarf lap joints subjected to tensile loads Hamit Adin ⇑ Mechanical Engineering, Batman University, 72060 Batman, Turkey
a r t i c l e
i n f o
Article history: Received 14 April 2011 Received in revised form 5 September 2011 Accepted 26 September 2011 Available online 2 October 2011 Keywords: Adhesives Composite material Scarf lap joints (SLJs) Strain Finite Element Method (FEM)
a b s t r a c t In this paper, the mechanical behavior of the scarf lap joints (SLJs) bonded with adhesive (Vinylester Atlac 580) under a tensile load was analyzed. The effects of scarf angle at the interface strain distributions of SLJs were examined. The stress analyses were performed via three dimensional Finite Element Method (3D-FEM). The 3D-FEM code employed was Ansys (12.0). Experimental results were compared with the 3D-FEM results and were found quite reasonable. The results indicated that the maximum values of the normalized ex strain values were determined at h = 60° in all joints. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Adhesive joints have provided attractive solutions to many engineering problems. In recent years, adhesive bonding has been applied successfully in many technological applications, especially in aerospace and automotive industries. The analysis of adhesively bonded joints requires knowledge of the stress–strain behavior of the structural adhesive [1] and there are several test methods for this purpose [2]. Adhesive bonding offers many advantages over classical fastening techniques such as welding, riveting and mechanical fastening. The substantial reduction in weight that can be achieved using adhesive bonding is an important advantage, especially for lightweight structures. Despite these advantages, the use of adhesive bonding is not yet widespread, because adhesively bonded joints lose their strength when exposed to high humidity, aqueous environments and high temperatures [3,4]. A method for making the shear stress uniform along the bond length was presented by Cherry and Harrison [5]. Profile of ideal adherend for making the shear stress uniform was found to be a symmetric taper of the adherend along the bond line. It was also found that in addition to being a function of the adherend thickness, the shear stress was also a function of Young’s modulus of the adherends. Temiz et al. [6] investigated the effect of adhesive layer thickness on a film type adhesive and they demonstrated that as the thickness of adhesive layer increased, the loss in strength increased. Sancaktar and Nirantar [7] reported that the smaller the taper angle of adherend at the ends of lap zone, the higher the strength of joint. Özel et al. [8] observed that there was a significant effect of adherend dimensional in adhesive joint. The stress distributions in scarf adhesive joints under static tensile loadings are analyzed using three dimensional finite element calculations. The effects of adhesive Young’s modulus, adhesive thickness and scarf angle in the adherend at the interface stress distributions are examined. As a results, it is found that the maximum value of the maximum principal stress occur at the edge of the interfaces [9]. The strength of single lap joints and scarf joints between carbon cloth laminated ⇑ Tel.: +90 488 217 3553; fax: +90 488 215 7205. E-mail addresses: [email protected], [email protected], [email protected] 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.09.079
H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
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plastics (CFRP) and carbon steel bonded with epoxy resin was investigated. The stress and strain distributions under tensile loads of the joints were analyzed by applying the elastic finite element method [10]. Solmaz and Turgut [11] investigated that the increasing of both overlap length and taper angle increase the joint strength and, in particular, the highest strength values in all joint geometries are attained by specimens having a taper angle of 15°. In a study, which strength of epoxy adhesively bonded scarf joints of dissimilar adherends was examined on several scarf angles and various bond thicknesses under uniaxial tensile loading. Scarf angle, h = 45°, 60° and 75° were employed. From analytical solutions, stress singularity existed most pronouncedly at the steel/adhesive interface corner of joint having 45–75° scarf angle. The failure surface observations confirmed [12]. Additionally, in another study, strength and fracture toughness of epoxy adhesively bonded scarf joints of dissimilar adherends were examined on several scarf angles various bond thickness under uniaxial tensile loading. Scarf angles, h = 45°, 60° and 75° are employed. From analytical solutions, stress singularity existed most pronouncedly at the steel/adhesive interface corner of joints having 45–75° scarf angle. The strength of scarf joints increased as the bond thickness decreased [13]. Gacoin et al. [14] investigated the influence of singularity created by the internal geometry of the double scarf joint, on the damage evolution to the adhesively bonded joint. The obtained result showed that the better behavior of the adhesively bonded scarf joint when the scarf angle is lower than a = 18°. In another study, the strength of adhesively bonded scarf joints under tensile load was investigated both analytically and experimentally. The strain and stress distributions were analyzed using the elastic finite element method and the strain distributions obtained were compared with experimental results. The effects of scarf length and elastic imbalance on the deformation and strength of the joints were examined [15]. Brink et al. [16] investigated the role of geometry on the mechanical performance of scarf joints in Al-matrix composites reinforced with continuous polycrystalline alumina fibers. Reference composite specimens were produced in the same manner and exhibited tensile strengths of the order of 700 MPa, compared with matrix yield strength, r0, of approximately 100 MPa. The strength of the scarf joints with aspect ratios of 17–48 and angles below 45° was relatively constant, but increased at higher angles, reaching over 485 MPa at 75°. Failure of these joints was typically by debonding at the compositeinterlayer interface. In this paper, the mechanical behavior of scarf lap joints (SLJs) under a tensile load was analyzed, both experimentally and numerically. In order to assess the performance of the adhesive, tensile experiments of the joints with different scarf angle were carried out. The strain analysis in the SLJs was performed via 3-D FEM by considering strain behavior of adhesive and adherend. The joint strengths are estimated using the obtained at the interface strain distributions in elastic deformation range. The 3-D FEM results were also compared with experimental results. 2. Experimental details 2.1. Adhesive used and determination of mechanical properties In the study, as adhesive was chosen Vinylester Atlac 580 produced by Huntsman. Vinylester Atlac 580 is a pre-accelerated thixotropic, high grade bisphenol A vinyl ester urethane resin. It combines exceptional chemical resistance and an outstanding combination of heat resistance and flexibility. Furthermore, Vinylester Atlac 580 has very good handling and curing properties. Temperature of cure and time is 100 °C and 3 h, respectively. Vinylester Atlac 580 is resistant to many aqueous acidic salts and alkaline solutions. Especially against alkaline media and hot water Vinylester Atlac 580 has an outstanding performance [17]. The stress–strain behavior of adhesive are estimated using bulk specimens. For this purpose, in this study were prepared as follows: In order to adjust the thickness of the bulk specimens, a U-shaped spacer frame as seen Fig. 1a, was fixed on the lower plate of a hot press used for curing the adhesives. The height of the frame determines the specimen thickness and height of 2–3 mm is suitable for most tests [18,19]. In this study, the thickness and width of bulk specimens were 2 mm and 15 mm,
Fig. 1. Preparation of the adhesives: (a) hot press, (b) tensile test specimen (dog-bone) (all dimension in mm).
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
Fig. 2. (a) Tensile stress–strain behavior of adhesive and (b) bulk specimens.
respectively. A release agent was applied to bulk the specimens of the hot press, so as to easily separate the cured adhesive from of the hot press. In order to prevent air entrapment between the bulk specimens and for curing adhesives, the pressure was applied to the bulk specimens of hot press, which was 100 °C for Vinylester Atlac 580. Filter curing, bulk specimens were machined to the dimension shown in Fig. 1b [20,21]. A non-contacting device must be used to measure the strains in bulk specimens, if the mechanical properties of the flexible adhesives are determined from the bulk specimens (see Fig. 2b). Although deriving strains from the position of the crosshead is not recommended for measuring a small range of displacement (<100 lm) which is typically required for determining the tensile modulus of materials due to its limited accuracy, these accuracy limitations are of less significance in measuring comparatively large displacements i.e. >1 mm. Therefore, the crosshead position is better suited for measuring failure strains in flexible materials which are often in excess of 50% [22,23]. The stress–strain (r–e) behavior of the adhesives were determined from bulk specimens tested under the specified conditions. The experiments of bulk specimen were performed using video extensometer, Shimadzu (Shimadzu Corporation, Tokyo, Japan) (100 kN) machine at room temperature and relative humidity 50% ± 5. During tensile tests, the crosshead speeds were maintained at 1 mm/min. Four specimens were tested for each analyzed experimental, condition and the average values are shown in Fig. 2a, where it observed that the stress increased in an almost linear manner until failure (R2 = 0.9947). Upon reaching the peak of stress, the stress dropped suddenly and the bulk specimen failured. Hence, adhesive has showed linear elastic behavior. Additionally, it can be seen that the maximum stress of the adhesive is measured as 40.618 MPa. Therefore, because of the maximum stress values of the Vinylester Atlac 580 is more effective in joint. This situation provides an important increase in the performances of the joint. 2.2. The properties of adherend _ _ Fiber reinforced epoxy/glass (Hgw 2372.4 Grade G11 EP GC 203, produced by hand lay up method by Izoreel in Izmir/Turkey) was chosen for the study as adherend. Adherends are carried out reinforced angles (0°/90°) composite laminated plates and have four layers. High quality construction materials are for electric equipment and transformers. It has high retention of flexural strength at elevated temperatures [24]. 2.3. Production of SLJs SLJs shown in table were made of Fiber reinforced epoxy/glass bonded using the resin type adhesive. The dimensions of adherend and adhesive are shown in Table 1. The thickness of adherend and adhesive were chosen as 10 mm and 0.25, Table 1 Geometrical parameters used in experimental and numerical studies. Adherend thickness (mm)
Overlap angle, h (°)
Adherend width, w (mm)
Adhesive thickness, t (mm)
10 10 10 10 10
15 30 45 60 75
6 6 6 6 6
0.25 0.25 0.25 0.25 0.25
H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
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Fig. 3. Configuration of composite material under load: (a) geometry (all dimensions in mm) and (b) mesh details and boundary conditions.
respectively. In addition, the width of adherend was chosen as 6 mm. Four different scarf angles were used. The SLJs were manufactured as four different plates for each condition of angle. They were prepared by hand lay-up with cure room temperature. Before bonding, the SLJs were wiped with acetone and later on abraded with 320 and 400 grid silicon carpide abrasive, which was visible. Then, the adhesive was prepared. It was applied at the joint surface and the adherend were clamped for curing as cure of adhesives were waited for 3 h at 100 °C. 2.4. Experimental method The configuration of the adherend used in the experiments in order to measure the strains in the SLJs is shown in Fig. 3a. _ The material of adherend was chosen as fiber reinforced epoxy/glass (Hgw 2372.4 Grade G11 EP GC 203, produced by Izoreel _ Corporation, Izmir, Turkey) composites. The experiments of SLJS were performed using video extensometer Shimadzu (Shimadzu Corporation, Tokyo, Japan) (100 kN) machine at room temperature and relative humidity 50% ± 5. In addition, during tensile tests, the crosshead speeds were maintained at 1 mm/min. 3. The strain results obtained from 3D-FEM in interface In this step, 3-D FEM was employed in order to analyze the behavior of the SLJs. The obtained stress–strain characteristic of adhesive is shown in Fig. 2a. The geometrical parameters and material properties of adhesive and adherend used in the 3D FEM calculations are given in Fig. 3, Tables 1 and 2, respectively. The 3D-FEM calculations were the ANSYS (Academic Teaching Advanced, Ver. 12.0.1) software [25]. Additionally, the strain analysis of the SLJs was obtained according to vonMises yield criterion. Because, Gali et al. [26] showed that the von-Misses yield criterion was suitable to model the stress–strain behavior of the adhesives used in the joint. By means of this criterion, the strain (eeqv) distributions in the adhesive layer were calculated. Loading, boundary conditions and mesh conditions were presented in Fig. 3. In this study, Solid 46 and Solid 191 elements were used. The element is composed of eight and 20 different nodes with three degrees of freedom. Baylor and Sancaktar
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867 Table 2 Material properties of the adherends and adhesives. Adherend Hgw2372.4 Grade G11EPGC 203
Adhesive Vinylester Atlac580
m Ex (GPa) Ey (GPa) Ez (GPa) Gxy (GPa) Gxz (GPa) Gyz (GPa) r⁄ (MPa)
44 10.5 10.5 3.74 3.74 2.5 320
m
0.32 0.41 0.32
1.4 0.37 0.44246
40.618 ⁄
⁄
E: Young’s modulus, m: Poisson’s ratio, r : ultimate strength, G : Shear modulus.
[27] showed that if the mesh density along the transverse direction of the overlap was greater than 3 elements per mm, then the variation in maximum principal stress and von Misses stress with mesh density would be effectively removed. It was also shown that for an adhesive thickness of 0.2 mm, 25 elements per mm in the peel direction would result in the uncoupling of these stresses with mesh density. The mesh density can effect the strain predictions in the adhesive layer. A smaller element size will generally give a higher strain. For this reason, the size of elements in the mesh was reduced until a stable strain value had been achieved. Eventually, 25 elements through the adhesive thickness (0.25 mm) were used in the models, as shown in Fig. 3b. The adherend thickness (10 mm) was meshed with 100 layers. So, the smallest element sizes were used as 0.01 mm in the adhesives and 0.1 mm in the adherends. The total number of nodes and elements was chosen 246,413 and 162,000, respectively. Since effects of scarf angle were examined, the same element dimension was used in all models as often as practicable (see Table 1 and Fig. 3a). The upper and lower adherends have the same dimensions and materials, and are subjected to static tensile loadings. In the analysis, the strain behavior of adhesive was considered as linear (Fig. 2a). 3.1. Formulation of stresses and strains If scarf joints are submitted to the axial tensile load, stresses and strains inside the adhesive layer of scarf joints are relatively uniform except for the small region at the vicinity of interface corner. In this section, the discussion will be restricted only to the stresses and strains in the central region of adhesive layer in scarf joints. Fig. 3a shows the coordinate system that is typically used to evaluate stresses and strains in the central region of adhesive layer in scarf joints. Theoretically, for scarf joints loaded axially with r stress, normal and shear stresses are given by
rx ¼ r sin2 h
ð1Þ
sxy ¼ r sin h cos h;
ð2Þ
and
respectively. According to these stresses, maximum and minimum principal stresses can be derived as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
r1;3 ¼ rx þ ry ðrx ry Þ2 þ 4s2xy
ð3Þ
and von Misses equivalent stress is obtained as follow:
reqv ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðrx þrz Þðm2 m þ 1Þ 3rx rz g;
ð4Þ
where m is Poisson’s ratio of the adhesive. The adhesive layer was obtained. The shear deformation of the adhesive is described by an elastic stress–strain curve, see Fig. 2a. The shear strain in the adhesive layer is approximated by bond line displacements,
c¼
ue uf ; t
ð5Þ
where ue , uf are end and beginning longitudinal displacement of the adherends, respectively and t is the thickness of adhesive layer. The shear stress–strain relations for the adhesive layer are stated as follow:
s ¼ Gc; r ¼ Ee;
ð6Þ
where G is the elastic shear modulus of the adhesive layer, and e is the strain of adhesive, defined as,
G¼
E ; 2ð1 þ mÞ
where E is Young’s modulus of elasticity of the adhesive layer.
ð7Þ
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
To evaluate the stresses and strains of scarf joint, 3D-FEM linear analysis was performed using ANSYS 12 code. To constitute the adhesive layer in the calculations, the true stress–strain curve was used from the actual uniaxial tensile test data of specimen as shown in Fig. 2a. The adherend and adhesives were used the data of mechanical properties are taken from Tables 1 and 2. 4. Results and discussion 4.1. Experimental results The specimens were carefully and closely observed to understand failure mechanism during the tensile experiments using video extensometer. After the experiments were finished and average failure loads for each four specimen were recorded; finally, failure was occurred at the interface of the joints. As can be clearly seen from Table 3, the lowest failure load was determined at h = 75° for each specimen. For all joints, the experiments revealed a cohesive failure of the SLJs, which justifies the effectiveness of the chosen adhesive and surface preparation method to bond the composite adherends. Fig. 4 shows a cohesive failure of the adhesive layer for the SLJs. Damage initiated cohesively in the adhesive layer at the overlap edges, propagating towards the inner region of the bond up to complete failure. 4.2. 3-D FEM results To find out, the failure loads of the specimens in calculations, ultimate strength of adhesives (r⁄) were used. The r⁄ value of adhesive is measured as 40.618 MPa (see Fig. 2a). The equivalent stress (reqv) and strain (eeqv) were calculated using the von-Misses yield criterion and it was assumed that the failure occured when the equivalent stress and strain calculated at any point of adhesive layer reached the ultimate strength of adhesive. As the maximum von-Mises yielding stress values at adhesive interface approach to these values, then, they are multiplied with the frontal area of specimens (t b = thickness width and failure load = von-Mises stress frontal area) from which the failure loads seen in Table 3 is obtained. As it can be seen from the so-called Table 3, the failure load values of 3-D FEM calculations and the experimentally measured failure loads seem to be quite close. In the last columns of the tables, the convergence ratios are found by dividing the experimental loads ðPE Þ with failure loads (PFEM) found by 3-D FEM calculations. In general, the ratio values were found to be very close to 1. As a result, it can be said that experimentally found values are in consistence with the values of 3-D FEM calculations. When the 3-D FEM stress values eventuate near to the stress values of bulk specimen, the geometry of adhesive are turned on. There have been defined paths through the line A–B at the adhesive interface. After that, the path values are separately found for each specimen and adhesive. The ex, ey, ez, cxy, cxz and eeqv values are selected for each of the specimen and
Table 3 Experimental and calculation failure loads. Scarf angle (°)
15 30 45 60 75
Hgw 2372 Grade G10 EP GC 203 Vinylester Atlac 580 P E
PFEM
PR
2.437 2.428 2.393 2.386 2.338
2.416 2.412 2.369 2.366 2.312
1.009 1.007 1.010 1.008 1.011
P E (kN): Experimental failure load of adhesives; P FEM (kN): 3-D FEM failure load of adhesives; PR = PE/PFEM (experimental load/Finite Element Method load) (kN).
Fig. 4. Cohesive failures on surface of joints.
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867 2.44 EXP
2.43
FEM
2.41
1
2.4
0.99
2.39
0.98
2.38
σ FEM/σ EXP
Failure load (kN)
2.42
2.37 2.36 2.35 2.34
0.97 0.96 0.95 0.94
2.33 0.93
2.32
0.92
2.31 15
30
45 Scarf Angle (degree)
60
15
75
30
45
60
75
Scarf Angle (degree)
(a)
(b)
Fig. 5. (a) Failure loads of experimental and 3D-FEM and (b) ratios of stress experimental and 3D-FEM.
4.0E-04
5.0E-05
3.5E-04
0.0E+00
15º
0
3.0E-04
30º
0.1
0.2
45º
0.3
60º
0.4
0.5
75º
0.6
0.7
0.8
0.9
1
0.8
0.9
1
-5.0E-05
εy
εx
2.5E-04 2.0E-04
-1.0E-04
1.5E-04
-1.5E-04
1.0E-04
-2.0E-04
5.0E-05 15º
30º
45º
60º
-2.5E-04
75º
0.0E+00 0
-5.0E-06
-1.5E-05
0
0.1
0.1 15º
0.2
0.2
0.3
0.3 30º
0.4
0.4 45º
0.5
0.6
0.7
0.8
0.9
1
-3.0E-04
x/L
x/L
(a)
(b)
0.5 60º
0.6
0.7
0.8
0.9
1
2.5E-03
75º
2.0E-03
-2.5E-05
ε eqv
εz
-3.5E-05 -4.5E-05
1.5E-03
1.0E-03 -5.5E-05 -6.5E-05
5.0E-04
-7.5E-05
15º
30º
45º
60º
75º
0.0E+00
0
-8.5E-05 x/L
(c)
0.1
0.2
0.3
0.4
0.5 x/L
0.6
0.7
(d)
Fig. 6. Normal and von-Misses equivalent strain distributions along the line A–B obtained from the adhesive layer: (a) normalized ex strain distributions; (b) normalized ey strain distributions; (c) normalized ez strain distributions and (d) normalized eeqv von-Misses stress distributions.
their results are found. As cyz values are considerably small to be neglected and since it is possible to get six strain values while drawing the paths in the 3-D FEM, cyz values are not taken to the list. Fig. 5a shows 3-D FEM and experimental failure load of joint with respect to the overlap angle of SLJs bonded by Vinylester Atlac 580. The failure loads were minimum and maximum when the scarf angle was around h = 75° and h = 15°,
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867 x/L 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.8
0.9
1
1.1E-03 15º
30º
45º
60º
75º
6.0E-04
1.0E-04
γ xy
-4.0E-04
-9.0E-04
-1.4E-03
-1.9E-03
-2.4E-03
(a) x/L 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1
1,0E-04 15º
30º
45º
60º
75º
5,0E-05 0,0E+00 -5,0E-05
γ xz
-1,0E-04 -1,5E-04 -2,0E-04 -2,5E-04 -3,0E-04 -3,5E-04 -4,0E-04
(b)
Fig. 7. Shear stress distributions along the line A–B obtained from the adhesive layer: (a) normalized cxy strain distributions and (b) normalized cxz strain distributions.
respectively. Fig. 5b shows 3-D FEM and experimental ratios of stress load of joint depending on the scarf angle. Ratio of stress was maximum when the scarf angle was around h = 15°. It is clearly seen from Fig. 5 that the maximum load and stress ratio of scarf joints were increased with the decreasing of scarf angle. So, these results have been consistent with Refs. [12–14]. 4.3. 3-D FEM results and comparison with experiments In this section, maximum stress values of bulk specimens that are obtained from experiments are compared with the maximum stress values of adhesive interface that are obtained from 3-D FEM. Since it is at the interface adhesive where the damage generally occurs in adhesive joints, they are compared with stress values of interface. Additionally, as the rupture stress of specimens is greater than the rupture stress of adhesives, the rupture occurs in the adhesive region. The distributions of strains along the line A–B on the adhesive side obtained from 3D-FEM analyses are presented in Figs. 6 and 7. In Fig. 6, as a result of the 3-D FEM, normal (ex, ey and ez), shear (cxy) and equivalent (eeqv) strain distributions are obtained from the adhesive layers along the line A–B.
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
20 15º
6
30º
45º
60º
75º
18 16
5
σ eqv (MPa)
σ x (MPa)
14 4
3
2
12 10 8 6 4
1 15º
30º
45º
60º
2
75º
0
0 0
0.1
0.2
0.3
0.4
0.5 x/L
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 x/L
0.6
0.7
0.8
0.9
1
Fig. 8. Stress distributions along the line A–B obtained from the adhesive layer: (a) normalized rx stress distributions and (b) normalized reqv stress distributions.
Thanks to Fig. 6a, it was observed that maximum and minimum values of normal strain ex occurred in h = 60° and at h = 15°, respectively, occurring at the edge of the interfaces. Minimum values of normalized (ey) strain are determined in the edges of specimens by means of Fig. 6b. Maximum value of the ey strain was obtained in the middle of x/L and at h = 15° angle. Normal ez strain distributions are shown in Fig. 6c. The values of ez strain are minimum at h = 60°. The strain values of adhesive obtained higher values in the middle of overlap length and maximum values occurred in h = 45°. Fig. 6d exhibited that the minimum values of equivalent strain (eeqv) were obtained in the middle of joints at h = 15° respectively. The maximum equivalent strain were located around at x/L = 1 of the SLJs. The failures initiates at the edges of the interfaces. This result is corresponded to the result shown in Fig. 5a. The equivalent strain of SLJs also increases with increasing scarf angle up to 45°. But decreases after this point. This situation is compatible with another study made by Dan et al. [9]. As shown in Fig. 7a maximum and minimum values of cxy are obtained in the edges of the SLJs at h = 75° and h = 15°, respectively. In generally, when angle of SLJs was increased, the values of cxy strain were increased. As shown in Fig. 7b, minimum and maximum values of cxz are at h = 15° and h = 75° sections of SLJs, respectively. In addition, these values were shown at the edge of the interface on adhesive layer along the line A–B. Also, Fig. 7 indicated that more shear strain is transferred from the end to the center of x/L with increasing of the scarf in adherend. The distributions of normal stress (rx) and equivalent (reqv) along the line A–B on the adhesive layer are presented in Fig. 8. The maximum value of equivalent stress was occurred in the edge of the adhesive layer. The equivalent stress increases with increasing scarf angle up to 45°. However, decreases after from 60o. This situation among others, the maximum stress and equivalent stress are the most widely accepted as the appropriate failure criteria for scarf joints [9,21]. So, for scarf joints h smaller than 45°, the equivalent stress becomes the dominant failure criterion. However, for scarf joints h larger than 45°, the effect of equivalent stress is decreased. When the magnitude of the equivalent stresses is considered, it is clearly that the equivalent stresses have very important influence on the initiation and the propagation of failure at the edges of scarf joint. Consequently, failure initiation is probably occurred on the edges of overlap (the line A–B) at the interface of adhesives. Then, the failure at both free ends promotes to the center of overlap before joining each other. In addition, it is found that the 3-D FEM results of stress are well consistent with the experimental results (see Table 3). 5. Conclusions In this work, at the interface strain distributions in SLJs subjected to static tensile loadings were calculated by the three dimensional Finite Element Method (3D-FEM) and the joint strengths were estimated using the obtained at the interface strain distributions. In addition, the effects of angle were examined at the interface strain distributions. The following results were obtained: The 3-D FEM results were well consistent with the experimental results. The experiments to measure the joints strengths were carried out for verification of the 3D-FEM results. The SLJs strength was estimated using at the interface stress distributions obtained from the 3D-FEM results. In addition, the failure loads were maximum when the scarf angle was around h = 15° in the present scarf adhesive joints. So, it was shown that the angle of scarf affected the performance of the SLJs. The peak strain values of ex occur at the adhesive upper adherend at the interface (along line A–B) and at the point of 60°. Both 3D-FEM stress analyses and experimental results presented that failure occurred around the end zone of the overlap
H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
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due to the effect of shear stresses, while the failure at the edges of the adhesive layer originated from the ey strain in tensile. The effect of shear strain on the failure and strength of the SLJs increases with increasing of SLJs angle. Minimum and maximum values of cxy are at x = 0 and x = L. Furthermore, failure initiation will probably occur at the left and right edges (along the line A–B) of the adhesive interface. The most critical points of failure are ends of scarf as illustrated with the graphics of strain. All cases of the SLJs showed the failures of cohesive starting from the edges. Equivalent strain (eeqv) is transferred from the end to the center of the scarf with increasing of SLJs angle. This is the reason for increase in the strength of joints. In addition, the effect of equivalent strain increases with increasing up to h = 60° angle of SLJs. Minimum and maximum values of eeqv are at h = 15° and h = 60°, respectively.
Acknowledgements _ _ The authors are extremely thankful to Murat Erog˘lu, owner of Izoreel Corporation, Izmir/Turkey, for his discussions and supply of specimens, advice and encouragement. The author would like to express their deep gratitude towards Professor Dr. Erol Sancaktar for his valuable technical suggestions for improvement of the current paper. And also we would like to thank _ to Ihsan Pilatin, the lecturer of English at Batman University, for his preliminary contributions in correcting of grammar faults. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
W.K. Chiu, R. Jones, Int. J. Adhesion Adhesives 12 (1992) 219–225. F. Kadioglu, R.D. Adams, J. Adhesion Sci. Technol. 16 (2002) 179–195. J. Verhoff, K. Ramani, N. Blank, S. Rosenberg, J. Adhesion Sci. Technol. 16 (2002) 373–393. A.J. Kinloch (Ed.), Durability of Structural Adhesives, Elsevier, Amsterdam, 1983. B.W. Cherry, N.L. Harrison, J. Adhesion 2 (1) (1970) 125–130. S ß . Temiz, A. Özel, M.D. Aydın, J. Adhesion Sci. Technol. 19 (8) (2005) 705–718. E. Sancaktar, P. Nirantar, J. Adhesion Sci. Technol. 17 (2003) 655–675. A. Özel, M.D. Aydın, S ß . Temiz, J. Adhesion Sci. Technol. 18 (3) (2004) 313–325. H. Dan, T. Sawa, T. Iwamoto, Y. Hirayama, Int. J. Adhesion Adhesives 30 (2010) 387–392. C. Sato, K. Ikegami, J. Adhesion 39 (1) (1992) 29–41. M.Y. Solmaz, A. Turgut, Math. Comput. Appl. 16 (1) (2011) 159–170. M. Afendi, T. Teramoto, H.B. Bakri, Int. J. Adhesion Adhesives 31 (6) (2011) 402–411. M. Afendi, T. Teramoto, A. Matsudo, Key Eng. Mater. 462–563 (2011) 768–773. A. Gacoin, P. Lestriez, J. Assih, A. Objois, Y. Delmas, Int. J. Adhesion Adhesives 29 (5) (2009) 572–579. K. Ikegami, T. Takeshita, K. Matsuo, T. Sugibayashi, Int. J. Adhesion Adhesives 10 (3) (1990) 199–206. D.D. Brink, C.G. Levi, A.C.F. Cocks, F.A. Leckie, Acta Mater. 45 (7) (1997) 2765–2775. _ßcan, A. Turgut, e-J. New World Sci. Acad. 5 (3) (2010). H. Adin, B. Is ISO 527-2, Plastics – Determination of tensile properties. Part 2: Test conditions for molding and extrusion of plastics, 1993. B.C. Duncan, M.A. Girardi, B.E. Read, The preparation of bulk adhesive samples for mechanical testing, NPL Report DMM (B) 339, 1994. ISO 527-1, Plastics—Determination of tensile properties. Part 1: General principles, 1993. R.D. Adams, V.J. Coppendale, Mallick, H. Al-Hamdan, Int. J. Adhesion Adhesives 12 (1992) 185–190. S ß . Temiz, J. Adhesion Sci. Technol. 20 (14) (2006) 1547–1560. ISO 15166-2, Adhesives-methods of preparing bulk specimens. Part 2: Elevated-temperature-curing one-part systems, 2000. http://www.izoreel.com.tr/En/index.html. ANSYS, The general purpose finite element software, Swanson Analysis Systems, Houston, TX. S. Gali, G. Dolev, O. Ishai, Int. J. Adhesion Adhesives 1 (1981) 135–140. J.S. Baylor, E. Sancaktar, Reliability, Stress Analysis and Failure Prevention Issues in Emerging Technologies and Materials, vol. 87, ASME-DE, 1995, pp. 41–48.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
The effect of angle on the strain of scarf lap joints subjected to tensile loads Hamit Adin ⇑ Mechanical Engineering, Batman University, 72060 Batman, Turkey
a r t i c l e
i n f o
Article history: Received 14 April 2011 Received in revised form 5 September 2011 Accepted 26 September 2011 Available online 2 October 2011 Keywords: Adhesives Composite material Scarf lap joints (SLJs) Strain Finite Element Method (FEM)
a b s t r a c t In this paper, the mechanical behavior of the scarf lap joints (SLJs) bonded with adhesive (Vinylester Atlac 580) under a tensile load was analyzed. The effects of scarf angle at the interface strain distributions of SLJs were examined. The stress analyses were performed via three dimensional Finite Element Method (3D-FEM). The 3D-FEM code employed was Ansys (12.0). Experimental results were compared with the 3D-FEM results and were found quite reasonable. The results indicated that the maximum values of the normalized ex strain values were determined at h = 60° in all joints. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Adhesive joints have provided attractive solutions to many engineering problems. In recent years, adhesive bonding has been applied successfully in many technological applications, especially in aerospace and automotive industries. The analysis of adhesively bonded joints requires knowledge of the stress–strain behavior of the structural adhesive [1] and there are several test methods for this purpose [2]. Adhesive bonding offers many advantages over classical fastening techniques such as welding, riveting and mechanical fastening. The substantial reduction in weight that can be achieved using adhesive bonding is an important advantage, especially for lightweight structures. Despite these advantages, the use of adhesive bonding is not yet widespread, because adhesively bonded joints lose their strength when exposed to high humidity, aqueous environments and high temperatures [3,4]. A method for making the shear stress uniform along the bond length was presented by Cherry and Harrison [5]. Profile of ideal adherend for making the shear stress uniform was found to be a symmetric taper of the adherend along the bond line. It was also found that in addition to being a function of the adherend thickness, the shear stress was also a function of Young’s modulus of the adherends. Temiz et al. [6] investigated the effect of adhesive layer thickness on a film type adhesive and they demonstrated that as the thickness of adhesive layer increased, the loss in strength increased. Sancaktar and Nirantar [7] reported that the smaller the taper angle of adherend at the ends of lap zone, the higher the strength of joint. Özel et al. [8] observed that there was a significant effect of adherend dimensional in adhesive joint. The stress distributions in scarf adhesive joints under static tensile loadings are analyzed using three dimensional finite element calculations. The effects of adhesive Young’s modulus, adhesive thickness and scarf angle in the adherend at the interface stress distributions are examined. As a results, it is found that the maximum value of the maximum principal stress occur at the edge of the interfaces [9]. The strength of single lap joints and scarf joints between carbon cloth laminated ⇑ Tel.: +90 488 217 3553; fax: +90 488 215 7205. E-mail addresses: [email protected], [email protected], [email protected] 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.09.079
H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
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plastics (CFRP) and carbon steel bonded with epoxy resin was investigated. The stress and strain distributions under tensile loads of the joints were analyzed by applying the elastic finite element method [10]. Solmaz and Turgut [11] investigated that the increasing of both overlap length and taper angle increase the joint strength and, in particular, the highest strength values in all joint geometries are attained by specimens having a taper angle of 15°. In a study, which strength of epoxy adhesively bonded scarf joints of dissimilar adherends was examined on several scarf angles and various bond thicknesses under uniaxial tensile loading. Scarf angle, h = 45°, 60° and 75° were employed. From analytical solutions, stress singularity existed most pronouncedly at the steel/adhesive interface corner of joint having 45–75° scarf angle. The failure surface observations confirmed [12]. Additionally, in another study, strength and fracture toughness of epoxy adhesively bonded scarf joints of dissimilar adherends were examined on several scarf angles various bond thickness under uniaxial tensile loading. Scarf angles, h = 45°, 60° and 75° are employed. From analytical solutions, stress singularity existed most pronouncedly at the steel/adhesive interface corner of joints having 45–75° scarf angle. The strength of scarf joints increased as the bond thickness decreased [13]. Gacoin et al. [14] investigated the influence of singularity created by the internal geometry of the double scarf joint, on the damage evolution to the adhesively bonded joint. The obtained result showed that the better behavior of the adhesively bonded scarf joint when the scarf angle is lower than a = 18°. In another study, the strength of adhesively bonded scarf joints under tensile load was investigated both analytically and experimentally. The strain and stress distributions were analyzed using the elastic finite element method and the strain distributions obtained were compared with experimental results. The effects of scarf length and elastic imbalance on the deformation and strength of the joints were examined [15]. Brink et al. [16] investigated the role of geometry on the mechanical performance of scarf joints in Al-matrix composites reinforced with continuous polycrystalline alumina fibers. Reference composite specimens were produced in the same manner and exhibited tensile strengths of the order of 700 MPa, compared with matrix yield strength, r0, of approximately 100 MPa. The strength of the scarf joints with aspect ratios of 17–48 and angles below 45° was relatively constant, but increased at higher angles, reaching over 485 MPa at 75°. Failure of these joints was typically by debonding at the compositeinterlayer interface. In this paper, the mechanical behavior of scarf lap joints (SLJs) under a tensile load was analyzed, both experimentally and numerically. In order to assess the performance of the adhesive, tensile experiments of the joints with different scarf angle were carried out. The strain analysis in the SLJs was performed via 3-D FEM by considering strain behavior of adhesive and adherend. The joint strengths are estimated using the obtained at the interface strain distributions in elastic deformation range. The 3-D FEM results were also compared with experimental results. 2. Experimental details 2.1. Adhesive used and determination of mechanical properties In the study, as adhesive was chosen Vinylester Atlac 580 produced by Huntsman. Vinylester Atlac 580 is a pre-accelerated thixotropic, high grade bisphenol A vinyl ester urethane resin. It combines exceptional chemical resistance and an outstanding combination of heat resistance and flexibility. Furthermore, Vinylester Atlac 580 has very good handling and curing properties. Temperature of cure and time is 100 °C and 3 h, respectively. Vinylester Atlac 580 is resistant to many aqueous acidic salts and alkaline solutions. Especially against alkaline media and hot water Vinylester Atlac 580 has an outstanding performance [17]. The stress–strain behavior of adhesive are estimated using bulk specimens. For this purpose, in this study were prepared as follows: In order to adjust the thickness of the bulk specimens, a U-shaped spacer frame as seen Fig. 1a, was fixed on the lower plate of a hot press used for curing the adhesives. The height of the frame determines the specimen thickness and height of 2–3 mm is suitable for most tests [18,19]. In this study, the thickness and width of bulk specimens were 2 mm and 15 mm,
Fig. 1. Preparation of the adhesives: (a) hot press, (b) tensile test specimen (dog-bone) (all dimension in mm).
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
Fig. 2. (a) Tensile stress–strain behavior of adhesive and (b) bulk specimens.
respectively. A release agent was applied to bulk the specimens of the hot press, so as to easily separate the cured adhesive from of the hot press. In order to prevent air entrapment between the bulk specimens and for curing adhesives, the pressure was applied to the bulk specimens of hot press, which was 100 °C for Vinylester Atlac 580. Filter curing, bulk specimens were machined to the dimension shown in Fig. 1b [20,21]. A non-contacting device must be used to measure the strains in bulk specimens, if the mechanical properties of the flexible adhesives are determined from the bulk specimens (see Fig. 2b). Although deriving strains from the position of the crosshead is not recommended for measuring a small range of displacement (<100 lm) which is typically required for determining the tensile modulus of materials due to its limited accuracy, these accuracy limitations are of less significance in measuring comparatively large displacements i.e. >1 mm. Therefore, the crosshead position is better suited for measuring failure strains in flexible materials which are often in excess of 50% [22,23]. The stress–strain (r–e) behavior of the adhesives were determined from bulk specimens tested under the specified conditions. The experiments of bulk specimen were performed using video extensometer, Shimadzu (Shimadzu Corporation, Tokyo, Japan) (100 kN) machine at room temperature and relative humidity 50% ± 5. During tensile tests, the crosshead speeds were maintained at 1 mm/min. Four specimens were tested for each analyzed experimental, condition and the average values are shown in Fig. 2a, where it observed that the stress increased in an almost linear manner until failure (R2 = 0.9947). Upon reaching the peak of stress, the stress dropped suddenly and the bulk specimen failured. Hence, adhesive has showed linear elastic behavior. Additionally, it can be seen that the maximum stress of the adhesive is measured as 40.618 MPa. Therefore, because of the maximum stress values of the Vinylester Atlac 580 is more effective in joint. This situation provides an important increase in the performances of the joint. 2.2. The properties of adherend _ _ Fiber reinforced epoxy/glass (Hgw 2372.4 Grade G11 EP GC 203, produced by hand lay up method by Izoreel in Izmir/Turkey) was chosen for the study as adherend. Adherends are carried out reinforced angles (0°/90°) composite laminated plates and have four layers. High quality construction materials are for electric equipment and transformers. It has high retention of flexural strength at elevated temperatures [24]. 2.3. Production of SLJs SLJs shown in table were made of Fiber reinforced epoxy/glass bonded using the resin type adhesive. The dimensions of adherend and adhesive are shown in Table 1. The thickness of adherend and adhesive were chosen as 10 mm and 0.25, Table 1 Geometrical parameters used in experimental and numerical studies. Adherend thickness (mm)
Overlap angle, h (°)
Adherend width, w (mm)
Adhesive thickness, t (mm)
10 10 10 10 10
15 30 45 60 75
6 6 6 6 6
0.25 0.25 0.25 0.25 0.25
H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
2861
Fig. 3. Configuration of composite material under load: (a) geometry (all dimensions in mm) and (b) mesh details and boundary conditions.
respectively. In addition, the width of adherend was chosen as 6 mm. Four different scarf angles were used. The SLJs were manufactured as four different plates for each condition of angle. They were prepared by hand lay-up with cure room temperature. Before bonding, the SLJs were wiped with acetone and later on abraded with 320 and 400 grid silicon carpide abrasive, which was visible. Then, the adhesive was prepared. It was applied at the joint surface and the adherend were clamped for curing as cure of adhesives were waited for 3 h at 100 °C. 2.4. Experimental method The configuration of the adherend used in the experiments in order to measure the strains in the SLJs is shown in Fig. 3a. _ The material of adherend was chosen as fiber reinforced epoxy/glass (Hgw 2372.4 Grade G11 EP GC 203, produced by Izoreel _ Corporation, Izmir, Turkey) composites. The experiments of SLJS were performed using video extensometer Shimadzu (Shimadzu Corporation, Tokyo, Japan) (100 kN) machine at room temperature and relative humidity 50% ± 5. In addition, during tensile tests, the crosshead speeds were maintained at 1 mm/min. 3. The strain results obtained from 3D-FEM in interface In this step, 3-D FEM was employed in order to analyze the behavior of the SLJs. The obtained stress–strain characteristic of adhesive is shown in Fig. 2a. The geometrical parameters and material properties of adhesive and adherend used in the 3D FEM calculations are given in Fig. 3, Tables 1 and 2, respectively. The 3D-FEM calculations were the ANSYS (Academic Teaching Advanced, Ver. 12.0.1) software [25]. Additionally, the strain analysis of the SLJs was obtained according to vonMises yield criterion. Because, Gali et al. [26] showed that the von-Misses yield criterion was suitable to model the stress–strain behavior of the adhesives used in the joint. By means of this criterion, the strain (eeqv) distributions in the adhesive layer were calculated. Loading, boundary conditions and mesh conditions were presented in Fig. 3. In this study, Solid 46 and Solid 191 elements were used. The element is composed of eight and 20 different nodes with three degrees of freedom. Baylor and Sancaktar
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867 Table 2 Material properties of the adherends and adhesives. Adherend Hgw2372.4 Grade G11EPGC 203
Adhesive Vinylester Atlac580
m Ex (GPa) Ey (GPa) Ez (GPa) Gxy (GPa) Gxz (GPa) Gyz (GPa) r⁄ (MPa)
44 10.5 10.5 3.74 3.74 2.5 320
m
0.32 0.41 0.32
1.4 0.37 0.44246
40.618 ⁄
⁄
E: Young’s modulus, m: Poisson’s ratio, r : ultimate strength, G : Shear modulus.
[27] showed that if the mesh density along the transverse direction of the overlap was greater than 3 elements per mm, then the variation in maximum principal stress and von Misses stress with mesh density would be effectively removed. It was also shown that for an adhesive thickness of 0.2 mm, 25 elements per mm in the peel direction would result in the uncoupling of these stresses with mesh density. The mesh density can effect the strain predictions in the adhesive layer. A smaller element size will generally give a higher strain. For this reason, the size of elements in the mesh was reduced until a stable strain value had been achieved. Eventually, 25 elements through the adhesive thickness (0.25 mm) were used in the models, as shown in Fig. 3b. The adherend thickness (10 mm) was meshed with 100 layers. So, the smallest element sizes were used as 0.01 mm in the adhesives and 0.1 mm in the adherends. The total number of nodes and elements was chosen 246,413 and 162,000, respectively. Since effects of scarf angle were examined, the same element dimension was used in all models as often as practicable (see Table 1 and Fig. 3a). The upper and lower adherends have the same dimensions and materials, and are subjected to static tensile loadings. In the analysis, the strain behavior of adhesive was considered as linear (Fig. 2a). 3.1. Formulation of stresses and strains If scarf joints are submitted to the axial tensile load, stresses and strains inside the adhesive layer of scarf joints are relatively uniform except for the small region at the vicinity of interface corner. In this section, the discussion will be restricted only to the stresses and strains in the central region of adhesive layer in scarf joints. Fig. 3a shows the coordinate system that is typically used to evaluate stresses and strains in the central region of adhesive layer in scarf joints. Theoretically, for scarf joints loaded axially with r stress, normal and shear stresses are given by
rx ¼ r sin2 h
ð1Þ
sxy ¼ r sin h cos h;
ð2Þ
and
respectively. According to these stresses, maximum and minimum principal stresses can be derived as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
r1;3 ¼ rx þ ry ðrx ry Þ2 þ 4s2xy
ð3Þ
and von Misses equivalent stress is obtained as follow:
reqv ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fðrx þrz Þðm2 m þ 1Þ 3rx rz g;
ð4Þ
where m is Poisson’s ratio of the adhesive. The adhesive layer was obtained. The shear deformation of the adhesive is described by an elastic stress–strain curve, see Fig. 2a. The shear strain in the adhesive layer is approximated by bond line displacements,
c¼
ue uf ; t
ð5Þ
where ue , uf are end and beginning longitudinal displacement of the adherends, respectively and t is the thickness of adhesive layer. The shear stress–strain relations for the adhesive layer are stated as follow:
s ¼ Gc; r ¼ Ee;
ð6Þ
where G is the elastic shear modulus of the adhesive layer, and e is the strain of adhesive, defined as,
G¼
E ; 2ð1 þ mÞ
where E is Young’s modulus of elasticity of the adhesive layer.
ð7Þ
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
To evaluate the stresses and strains of scarf joint, 3D-FEM linear analysis was performed using ANSYS 12 code. To constitute the adhesive layer in the calculations, the true stress–strain curve was used from the actual uniaxial tensile test data of specimen as shown in Fig. 2a. The adherend and adhesives were used the data of mechanical properties are taken from Tables 1 and 2. 4. Results and discussion 4.1. Experimental results The specimens were carefully and closely observed to understand failure mechanism during the tensile experiments using video extensometer. After the experiments were finished and average failure loads for each four specimen were recorded; finally, failure was occurred at the interface of the joints. As can be clearly seen from Table 3, the lowest failure load was determined at h = 75° for each specimen. For all joints, the experiments revealed a cohesive failure of the SLJs, which justifies the effectiveness of the chosen adhesive and surface preparation method to bond the composite adherends. Fig. 4 shows a cohesive failure of the adhesive layer for the SLJs. Damage initiated cohesively in the adhesive layer at the overlap edges, propagating towards the inner region of the bond up to complete failure. 4.2. 3-D FEM results To find out, the failure loads of the specimens in calculations, ultimate strength of adhesives (r⁄) were used. The r⁄ value of adhesive is measured as 40.618 MPa (see Fig. 2a). The equivalent stress (reqv) and strain (eeqv) were calculated using the von-Misses yield criterion and it was assumed that the failure occured when the equivalent stress and strain calculated at any point of adhesive layer reached the ultimate strength of adhesive. As the maximum von-Mises yielding stress values at adhesive interface approach to these values, then, they are multiplied with the frontal area of specimens (t b = thickness width and failure load = von-Mises stress frontal area) from which the failure loads seen in Table 3 is obtained. As it can be seen from the so-called Table 3, the failure load values of 3-D FEM calculations and the experimentally measured failure loads seem to be quite close. In the last columns of the tables, the convergence ratios are found by dividing the experimental loads ðPE Þ with failure loads (PFEM) found by 3-D FEM calculations. In general, the ratio values were found to be very close to 1. As a result, it can be said that experimentally found values are in consistence with the values of 3-D FEM calculations. When the 3-D FEM stress values eventuate near to the stress values of bulk specimen, the geometry of adhesive are turned on. There have been defined paths through the line A–B at the adhesive interface. After that, the path values are separately found for each specimen and adhesive. The ex, ey, ez, cxy, cxz and eeqv values are selected for each of the specimen and
Table 3 Experimental and calculation failure loads. Scarf angle (°)
15 30 45 60 75
Hgw 2372 Grade G10 EP GC 203 Vinylester Atlac 580 P E
PFEM
PR
2.437 2.428 2.393 2.386 2.338
2.416 2.412 2.369 2.366 2.312
1.009 1.007 1.010 1.008 1.011
P E (kN): Experimental failure load of adhesives; P FEM (kN): 3-D FEM failure load of adhesives; PR = PE/PFEM (experimental load/Finite Element Method load) (kN).
Fig. 4. Cohesive failures on surface of joints.
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867 2.44 EXP
2.43
FEM
2.41
1
2.4
0.99
2.39
0.98
2.38
σ FEM/σ EXP
Failure load (kN)
2.42
2.37 2.36 2.35 2.34
0.97 0.96 0.95 0.94
2.33 0.93
2.32
0.92
2.31 15
30
45 Scarf Angle (degree)
60
15
75
30
45
60
75
Scarf Angle (degree)
(a)
(b)
Fig. 5. (a) Failure loads of experimental and 3D-FEM and (b) ratios of stress experimental and 3D-FEM.
4.0E-04
5.0E-05
3.5E-04
0.0E+00
15º
0
3.0E-04
30º
0.1
0.2
45º
0.3
60º
0.4
0.5
75º
0.6
0.7
0.8
0.9
1
0.8
0.9
1
-5.0E-05
εy
εx
2.5E-04 2.0E-04
-1.0E-04
1.5E-04
-1.5E-04
1.0E-04
-2.0E-04
5.0E-05 15º
30º
45º
60º
-2.5E-04
75º
0.0E+00 0
-5.0E-06
-1.5E-05
0
0.1
0.1 15º
0.2
0.2
0.3
0.3 30º
0.4
0.4 45º
0.5
0.6
0.7
0.8
0.9
1
-3.0E-04
x/L
x/L
(a)
(b)
0.5 60º
0.6
0.7
0.8
0.9
1
2.5E-03
75º
2.0E-03
-2.5E-05
ε eqv
εz
-3.5E-05 -4.5E-05
1.5E-03
1.0E-03 -5.5E-05 -6.5E-05
5.0E-04
-7.5E-05
15º
30º
45º
60º
75º
0.0E+00
0
-8.5E-05 x/L
(c)
0.1
0.2
0.3
0.4
0.5 x/L
0.6
0.7
(d)
Fig. 6. Normal and von-Misses equivalent strain distributions along the line A–B obtained from the adhesive layer: (a) normalized ex strain distributions; (b) normalized ey strain distributions; (c) normalized ez strain distributions and (d) normalized eeqv von-Misses stress distributions.
their results are found. As cyz values are considerably small to be neglected and since it is possible to get six strain values while drawing the paths in the 3-D FEM, cyz values are not taken to the list. Fig. 5a shows 3-D FEM and experimental failure load of joint with respect to the overlap angle of SLJs bonded by Vinylester Atlac 580. The failure loads were minimum and maximum when the scarf angle was around h = 75° and h = 15°,
2865
H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867 x/L 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.8
0.9
1
1.1E-03 15º
30º
45º
60º
75º
6.0E-04
1.0E-04
γ xy
-4.0E-04
-9.0E-04
-1.4E-03
-1.9E-03
-2.4E-03
(a) x/L 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1
1,0E-04 15º
30º
45º
60º
75º
5,0E-05 0,0E+00 -5,0E-05
γ xz
-1,0E-04 -1,5E-04 -2,0E-04 -2,5E-04 -3,0E-04 -3,5E-04 -4,0E-04
(b)
Fig. 7. Shear stress distributions along the line A–B obtained from the adhesive layer: (a) normalized cxy strain distributions and (b) normalized cxz strain distributions.
respectively. Fig. 5b shows 3-D FEM and experimental ratios of stress load of joint depending on the scarf angle. Ratio of stress was maximum when the scarf angle was around h = 15°. It is clearly seen from Fig. 5 that the maximum load and stress ratio of scarf joints were increased with the decreasing of scarf angle. So, these results have been consistent with Refs. [12–14]. 4.3. 3-D FEM results and comparison with experiments In this section, maximum stress values of bulk specimens that are obtained from experiments are compared with the maximum stress values of adhesive interface that are obtained from 3-D FEM. Since it is at the interface adhesive where the damage generally occurs in adhesive joints, they are compared with stress values of interface. Additionally, as the rupture stress of specimens is greater than the rupture stress of adhesives, the rupture occurs in the adhesive region. The distributions of strains along the line A–B on the adhesive side obtained from 3D-FEM analyses are presented in Figs. 6 and 7. In Fig. 6, as a result of the 3-D FEM, normal (ex, ey and ez), shear (cxy) and equivalent (eeqv) strain distributions are obtained from the adhesive layers along the line A–B.
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H. Adin / Applied Mathematical Modelling 36 (2012) 2858–2867
20 15º
6
30º
45º
60º
75º
18 16
5
σ eqv (MPa)
σ x (MPa)
14 4
3
2
12 10 8 6 4
1 15º
30º
45º
60º
2
75º
0
0 0
0.1
0.2
0.3
0.4
0.5 x/L
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5 x/L
0.6
0.7
0.8
0.9
1
Fig. 8. Stress distributions along the line A–B obtained from the adhesive layer: (a) normalized rx stress distributions and (b) normalized reqv stress distributions.
Thanks to Fig. 6a, it was observed that maximum and minimum values of normal strain ex occurred in h = 60° and at h = 15°, respectively, occurring at the edge of the interfaces. Minimum values of normalized (ey) strain are determined in the edges of specimens by means of Fig. 6b. Maximum value of the ey strain was obtained in the middle of x/L and at h = 15° angle. Normal ez strain distributions are shown in Fig. 6c. The values of ez strain are minimum at h = 60°. The strain values of adhesive obtained higher values in the middle of overlap length and maximum values occurred in h = 45°. Fig. 6d exhibited that the minimum values of equivalent strain (eeqv) were obtained in the middle of joints at h = 15° respectively. The maximum equivalent strain were located around at x/L = 1 of the SLJs. The failures initiates at the edges of the interfaces. This result is corresponded to the result shown in Fig. 5a. The equivalent strain of SLJs also increases with increasing scarf angle up to 45°. But decreases after this point. This situation is compatible with another study made by Dan et al. [9]. As shown in Fig. 7a maximum and minimum values of cxy are obtained in the edges of the SLJs at h = 75° and h = 15°, respectively. In generally, when angle of SLJs was increased, the values of cxy strain were increased. As shown in Fig. 7b, minimum and maximum values of cxz are at h = 15° and h = 75° sections of SLJs, respectively. In addition, these values were shown at the edge of the interface on adhesive layer along the line A–B. Also, Fig. 7 indicated that more shear strain is transferred from the end to the center of x/L with increasing of the scarf in adherend. The distributions of normal stress (rx) and equivalent (reqv) along the line A–B on the adhesive layer are presented in Fig. 8. The maximum value of equivalent stress was occurred in the edge of the adhesive layer. The equivalent stress increases with increasing scarf angle up to 45°. However, decreases after from 60o. This situation among others, the maximum stress and equivalent stress are the most widely accepted as the appropriate failure criteria for scarf joints [9,21]. So, for scarf joints h smaller than 45°, the equivalent stress becomes the dominant failure criterion. However, for scarf joints h larger than 45°, the effect of equivalent stress is decreased. When the magnitude of the equivalent stresses is considered, it is clearly that the equivalent stresses have very important influence on the initiation and the propagation of failure at the edges of scarf joint. Consequently, failure initiation is probably occurred on the edges of overlap (the line A–B) at the interface of adhesives. Then, the failure at both free ends promotes to the center of overlap before joining each other. In addition, it is found that the 3-D FEM results of stress are well consistent with the experimental results (see Table 3). 5. Conclusions In this work, at the interface strain distributions in SLJs subjected to static tensile loadings were calculated by the three dimensional Finite Element Method (3D-FEM) and the joint strengths were estimated using the obtained at the interface strain distributions. In addition, the effects of angle were examined at the interface strain distributions. The following results were obtained: The 3-D FEM results were well consistent with the experimental results. The experiments to measure the joints strengths were carried out for verification of the 3D-FEM results. The SLJs strength was estimated using at the interface stress distributions obtained from the 3D-FEM results. In addition, the failure loads were maximum when the scarf angle was around h = 15° in the present scarf adhesive joints. So, it was shown that the angle of scarf affected the performance of the SLJs. The peak strain values of ex occur at the adhesive upper adherend at the interface (along line A–B) and at the point of 60°. Both 3D-FEM stress analyses and experimental results presented that failure occurred around the end zone of the overlap
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due to the effect of shear stresses, while the failure at the edges of the adhesive layer originated from the ey strain in tensile. The effect of shear strain on the failure and strength of the SLJs increases with increasing of SLJs angle. Minimum and maximum values of cxy are at x = 0 and x = L. Furthermore, failure initiation will probably occur at the left and right edges (along the line A–B) of the adhesive interface. The most critical points of failure are ends of scarf as illustrated with the graphics of strain. All cases of the SLJs showed the failures of cohesive starting from the edges. Equivalent strain (eeqv) is transferred from the end to the center of the scarf with increasing of SLJs angle. This is the reason for increase in the strength of joints. In addition, the effect of equivalent strain increases with increasing up to h = 60° angle of SLJs. Minimum and maximum values of eeqv are at h = 15° and h = 60°, respectively.
Acknowledgements _ _ The authors are extremely thankful to Murat Erog˘lu, owner of Izoreel Corporation, Izmir/Turkey, for his discussions and supply of specimens, advice and encouragement. The author would like to express their deep gratitude towards Professor Dr. Erol Sancaktar for his valuable technical suggestions for improvement of the current paper. And also we would like to thank _ to Ihsan Pilatin, the lecturer of English at Batman University, for his preliminary contributions in correcting of grammar faults. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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