Single lap joints loaded in tension with high strength steel adherends

International Journal of Adhesion & Adhesives 43 (2013) 81–95

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Single lap joints loaded in tension with high strength steel adherends E.F. Karachalios a, R.D. Adams a,b, Lucas F.M. da Silva c,n a b c

Department of Mechanical Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK Department of Engineering Science, University of Oxford, Parks Road, Oxford, UK Departamento de Engenharia Mecˆ anica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

a r t i c l e i n f o

abstract

Available online 1 February 2013

Single lap joints in many different geometric and material configurations were analysed using finite element analysis and tested in tension. Geometric parameters, such as the overlap length and adherend thickness, together with material parameters such as the adherend and adhesive stress–strain behaviour, were all tested. The mechanisms and modes of failure were observed for different cases, and positions of damage initiation were identified. Failure patterns were related to failure mechanisms. A failure prediction methodology has been proposed and a good correlation was obtained between the experimental and finite element predictions of strength for a variety of joint configurations. The study is presented in two parts. In the first (present paper), high strength steel adherends are considered and in the second paper ductile steel adherends are studied. For high strength steel adherends and a relatively short overlap, failure is dominated by adhesive global yielding. As the overlap gets longer, however, failure is no longer due to global yielding, but due to high local shear strains. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Epoxy/epoxides Steels Lap-shear Finite element stress analysis Fracture Joint design

1. Introduction In the continuum mechanics approach, the maximum values of stress, strain or strain energy, predicted by the finite element analyses, are normally used in the failure criterion and are compared with the corresponding material allowable values [1–3]. However, it is known that these maximum predicted values are usually found very near the singular points of the model (sharp corners, bi-material interfaces), and therefore their magnitude depends strongly on how well the stress field around the singularity is modelled (i.e. mesh refinement). In order to overcome this problem, an approach used by many researchers is to use the same variables (stress, strain or energy) but this time at some arbitrary distance from the singularity, where the stress field is clear of any effects from the singular point [4,5]. The critical distances must be calibrated from the FE results and, as a consequence, the critical values obtained can only be used for similar geometric and material configurations. There is no physical explanation relating the critical distances with experimental observations. Continuum mechanics assumes that the structure and its material are continuous. Defects or two materials with reentrant corners obviously violate such an assumption. Cracks are the most common defects in structures, for which the method

n

Corresponding author. Tel.: þ351 2250 81706; fax: þ351 2250 81445. E-mail address: [email protected] (L.F.M. da Silva).

0143-7496/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijadhadh.2013.01.016

of fracture mechanics has been developed. Fracture mechanics can be used to predict joint strength or residual strength if there is a crack tip or a known and calibrated singularity [6]. Fracture mechanics is more difficult to apply to strength predictions for joints bonded with ductile adhesives since the fracture energy is not independent of the joint geometry [7,8]. It may be seen from the above discussions that it is not easy to use fracture mechanics for predicting the strength of lap joints made with ductile adhesives. Damage mechanics, such as the cohesive zone model (CZM), has been used to model the progressive damage and failure of a pre-defined crack path [9–13]. A CZM simulates the fracture process, extending the concept of continuum mechanics by including a zone of discontinuity modelled by cohesive zones, thus using both strength and energy parameters to characterise the debonding process. However, the cohesive zone models present a limitation, as it is necessary to know in advance the critical zones where damage is likely to occur and to place the cohesive zone elements accordingly. Also, for ductile materials, the shape of the traction–separation law is controversial and may lead to convergence problems. The concept of defining failure by a progressively growing damage zone in the adhesive layer, until a critical limit is reached, seems physically correct since failure of the joints usually occurs after some damage in the adhesive has developed (typically seen as a stress whitening area in the adhesive indicating extensive plastic deformation). It appears that rather than trying to concentrate on the singular points at the ends of the joint, failure

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adhesive due to its weak behaviour. ESP 110 adhesive is a paste adhesive, which curing schedule is 45 min at 150 1C. AV 119 is a rubber-toughened paste epoxy. A curing schedule of 1 h at 120 1C was followed, according to the manufacturer’s instructions. EC 3448 is a rubber modified paste epoxy. This adhesive is regarded as a top of the range aerospace structural adhesive. It cures in 1 h at 120 1C. It is well known that the behaviour of polymeric materials, including adhesives, depends on the type of loading they are subjected to. It was, therefore, decided to carry out tensile and shear tests on the adhesives used. Tensile tests were carried out on dogbone shaped bulk adhesive specimens with the extension measured using clip-on extensometers. The shear tests were carried out using the standard thick adherend shear test (TAST) (ISO 11003) and in-house butt torsion specimens. Although it is known that polymeric materials may be rate sensitive, in this work rate effects were not taken into account since the adhesives are tested at temperatures well below the glass transition temperature Tg, where rate effects have much less influence. In all tests, a monotonic load was applied to failure under displacement control at a constant speed (5 mm/min). The stress–strain behaviour of the adhesives used, both in tension and shear, can be seen in Figs. 1 and 2, respectively.

should be predicted by considering other parts of the overlap too, perhaps the whole overlap in some cases. However, the damage size and location depend on many parameters such as adhesive and adherend plasticity, overlap length, type of loading (tension or bending), adherend thickness and the geometry of the joint in general. These variables have been studied experimentally and numerically for high strength steel (present paper) and low strength steel (low and medium carbon steel) in an accompanying paper [14]. A general failure methodology was developed for single lap joints. Three different cases have been considered: adhesive global yielding, adherend yield, and the maximum strain limit of the adhesive. Experimentally determined critical values of stress (maximum or plateau stress) are generally more consistent than critical values of strain (strain to failure in shear or tension) [15] and should therefore be preferred. However, a criterion based on some critical value of strain is physically meaningful for a material that exhibits large plastic or non-linear deformations, such as are found in several of the adhesives used in the current work.

2. Experimental details 2.1. Materials

2.2. Joint geometry A high carbon steel (AISI O1) heat-treated to its hard condition was used. Standard dogbone shaped specimens and special strain gauges capable of recording the post yield behaviour of the steel were used for the derivation of the stress–strain curve. The yield strength was measured at around 1800 MPa and the steel in the bonded joints is expected to behave elastically throughout the loading regime that the joints would be expected to experience. Four different epoxy adhesives were used throughout this investigation. Three out of four are one-part structural epoxies [ESP 110 from Permabond, AV 119 from Huntsman (formerly Ciba Polymers) EC 3448 from 3 M] and the fourth is a more brittle, two-part epoxy system (MY 753 from Huntsman (formerly Ciba Polymers) with hardener HY 951) which was mainly used for comparison reasons and should not be regarded as a structural

Single lap joints were manufactured individually in a mould. All the specimens were manufactured in accordance with the standard ASTM D 1002. One of the aims of the current investigation was to assess the effect of the overlap length (l) on the strength and failure of a SLJ. Joints with five different overlap lengths were manufactured, ranging from 12.5 mm to 60 mm. End tabs of the same thickness as the adherends were bonded at the ends of the joints as shown in Fig. 3, in order to reduce the eccentricity of the load path. In order to comply with the ASTM standards, the free length, Lf (Fig. 3) was kept constant at 63.5 mm for all the joint configurations. Another parameter affecting the strength of joints is the bondline thickness (t). This is a parameter of great importance,

80 MY 753 ESP 110

70

AV 119 EC 3448

True stress (MPa)

60

50

40

30

20

10

0 0

0.01

0.02

0.03

0.04

0.05

0.06

True strain Fig. 1. Tensile stress–strain curves for the four adhesives tested.

0.07

0.08

0.09

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83

60

50

Shear stress (MPa)

40

30

20 MY 753 ESP 110

10

AV 119 EC 3448 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Shear strain Fig. 2. Comparison of shear stress–strain curves for all the adhesives used (butt torsion test for MY 753, EC 3448 and AV 119, thick adherend shear test for ESP 110).

Fig. 3. Three-dimensional view of a single lap joint with strain gauge positions.

especially for the manufacturing industries, as it is quite difficult to control, especially in an industrial scale manufacturing process. Many researchers have tackled the subject. Results presented by Grant et al. [16] for bond lines up to 3 mm thick have shown that joint strength should increase as the bondline thickness decreases because the bending moment at the overlap end due to load misalignment is considerably increased. Below a certain point of approximately 0.3 mm, no significant changes were noted. Based on the above, it was decided to manufacture joints with a bondline thickness of 0.1 mm. Identifying a failure criterion that applies to a wide range of adhesive thicknesses is more challenging than the other joint geometry parameters. The failure criteria used in the present study might not be applicable to bondlines much thicker than 1 mm but should apply to the bondline thicknesses used in most lap joints made to the ASTM 1002 or similar standards. The majority of the joints were manufactured with a full depth fillet (Fig. 3). Fillets have been shown to reduce significantly the peak stresses at the ends of the joints. The majority of the adherends used were 1.6 mm thick. Some tensile specimens were also manufactured using 1 mm and 2 mm thick adherends, in order to assess the effect of adherend thickness on the strength of the joints.

2.3. Testing All the specimens were tested in a universal testing machine using wedge grips to clamp the specimens, and all the tests were displacement controlled. The maximum load recorded by the machine was taken as the strength of the joint. The jaws separating speed (test speed) was chosen to be 5 mm per minute for all joint tests to represent quasi-static testing conditions. All the tests were conducted at room temperature of approximately 23 1C and at 50% humidity. At least five specimens were tested for each joint configuration. 2.4. Strain gauging Strain gauges were used both in the adherend materials characterisation and in the testing of joints. The standard procedure of sanding and etching the adherend surfaces was followed [17] to ensure perfect bonding of the strain gauges, the location of which is shown in Fig. 3. 2.5. Video camera—microscopes A video camera capable of recording at up to 25 frames per second was used to record different configurations of joints

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while they were tested. In order to visualise the crack formation better, the surfaces of interest were painted white, using a thin layer of paint. Photographs of the broken surfaces were taken using a still camera. Microscopes were used in some cases to observe the failure patterns whenever a more detailed view was needed.

3. Experimental results 3.1. Strength results Typical load–displacement curves from tests can be seen in Fig. 4 for adhesive AV 119. As can be seen, the behaviour of the joints is linear to failure. The results are presented in Table 1. Fig. 5 represents the trend lines of the failure load vs. overlap lap length for the four adhesives. The experimental points were omitted for clarity. The increase in the strength of the joints is almost proportional to the overlap length. No definite plateau is reached for the long overlaps. However, there is a tendency for a non-proportional increase for very long overlaps. It seems that further increase in the strength of the joints could be possible by increasing the overlap length, but a limit should be expected because of the adhesive strength. The strength of the joints is different for the four adhesives used. The MY 753 adhesive gives joints with the lowest strength followed by ESP 110 adhesive, while the more ductile AV 119 and EC 3448 give the strongest

joints. The adhesive ductility appears to be the parameter governing strength. 3.2. Fillet effect Joints without fillets were manufactured for the 25 mm and 40 mm overlaps. The adhesive used was ESP 110. The comparison between the joints with and without fillets for this case can be seen in Table 2. Results from these tests show no differences in strength compared with the filleted joints. Variations only arise due to experimental scatter. Thus, the presence of a full depth spew fillet does not seem to influence the strength of the joints for this case. A similar conclusion was obtained by Grant et al. [16] for joints with thin bondlines and ductile adherends. 3.3. Adherend thickness effect One of the main advantages of adhesive bonding is the ability to join thin sheets of materials, thus producing efficient joints. The efficiency of a joint is usually defined as the ratio of the joint strength divided by the strength of the weakest of the adherends Joint efficiency ¼ ðjoint strength=strength of weakest adherendÞ ð1Þ In a lap joint configuration, due to the eccentricity of the load path, the thicker the adherends the larger will be the bending moment induced at the edges of the joint.

60.0E+3

50.0E+3

12.5 mm overlap 20 mm overlap 25 mm overlap 40 mm overlap 60 mm overlap

Load (N)

40.0E+3

30.0E+3

20.0E+3

10.0E+3

000.0E+0 0.00

1.00

2.00

3.00 4.00 5.00 6.00 Cross head displacement (mm)

7.00

8.00

9.00

Fig. 4. Load–displacement curves for SLJs with AV 119 adhesive and various overlaps.

Table 1 Results from tensile testing of SLJs. Overlap Length (mm)

12.5 20 25 40 60

ESP 110 adhesive

AV 119 adhesive

EC 3448 adhesive

MY753 adhesive

Average failure load (kN)

Standard deviation

Average failure load (kN)

Standard deviation

Average failure load (kN)

Standard deviation

Average failure load (kN)

Standard deviation

14.82 19.01 23.44 32.39 43.26

0.827 0.937 1.05 1.55 1.65

15.01 23.17 30.76 43.09 53.58

0.88 1.96 1.59 2.38 3.05

15.92 25.28 29.58 46.85 55.72

0.98 0.90 2.88 2.25 2.09

8.79 – 13.76 18.80 21.86

1.37 – 3.06 1.43 1.36

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85

60 EC 3448 AV 119 ESP 110

50

MY 753

Failure load (kN)

40

30

20

10

0 0

10

20

30

40

50

60

70

Overlap (mm) Fig. 5. Failure load vs. overlap for the four adhesives.

Table 2 Results from tensile testing of SLJs with and without fillets using ESP 110 adhesive. Overlap length (mm)

25 40

With 451 fillet

Without fillet

Av. fail. load (kN)

St. deviation

Av. fail. load (kN)

St. deviation

23.44 32.39

1.05 1.55

22.67 33.85

1.17 1.10

An investigation into the effect of adherend thickness on the strength of lap joints was carried out. The 25 mm overlap configuration using AV 119 adhesive was chosen. Three different adherend thicknesses were compared: 1, 1.6, and 2 mm. The effect of adherend thickness on the stress distribution on bonded joints has been also investigated theoretically by Crocombe and Adams [18] and Chiu and Jones [19]. They found that increasing the adherend thickness results in a reduction of the peak stresses at the edges of the overlap and a more even distribution of the shear stress in the adhesive layer. The results are presented in Fig. 6. It seems that there is only a small difference between the three different cases. The lowest strength was obtained for the case of the 2 mm thick adherends while the highest strength was obtained by using the 1.6 mm thick adherends; the differences are, however, small. We conclude that there seems to be no significant influence on the strength of the joints from the adherend thickness. 3.4. Failure mechanisms There was no plastic deformation of the adherends and, in all configurations, the adherends behaved as linearly elastic materials. The rotations around the edges of the overlap were small. However, it is useful to investigate the effect of the bending moments around the overlap edges on the strength of the joints for this case, and also to compare the different configurations. Slight differences in the failure were noted for the different adhesives used. These differences mainly depend on how brittle

or ductile is the adhesive. When an adhesive behaves in a brittle manner, then a crack will form and propagate until catastrophic failure occurs. When the adhesive has the ability to deform plastically, then a plastic zone forms ahead of the ‘crack tip’ creating a damaged zone. It has been mentioned by Hunston et al. [20] that tough materials owe their high fracture energies to their ability to generate crack tip deformation zones of significant size prior to fracture. As will be shown in the next paragraphs, and has been also noted by Papini and Fernlund [21], that fracture in modern toughened adhesive joints occurs by the development and propagation of a damage zone, rather than a single sharp crack. Of the four adhesives used in the current work, ESP 110 and MY 753 behaved in a brittle manner while AV 119 and EC 3448 behaved in a more ductile manner. For the case of ESP 110, cracks could be seen forming, while for AV 119 and EC 3448 adhesives, a ‘stress whitened’ damaged area was visible instead of a crack (but there is a crack behind the stress whitened zone). For the remainder of this section, the term ‘damage’ will be sometimes used instead of ‘crack’, as it describes the situation more realistically. Optical observation shows that damage initiates around the embedded adherend corner (under the overlap) at the middle of the joint (in the width direction). The damage then spreads out to the fillet and under the overlap. In the case of ESP 110, cracks appear in the face of the fillet at the middle of the width of the joint while, in the case of AV 119 and EC 3448 the cracks take the form of a stress whitened damaged zone as explained above. Fig. 7 shows some cracks in the case of ESP 110 adhesive, and Fig. 8 is a schematic of the same process for the more ductile adhesives. Damage initiates at the centre of the width of the joint because of the effect of anticlastic bending [22]. As the adherends start to bend elastically, the middle part of the joint experiences through-thickness tensile stresses due to the anticlastic effect, while the edges will go into compression. It is well known that adhesives are stronger in compression than in tension: as a consequence, the middle of the joint is in a more critical condition for failure. Different conditions also apply at different parts of the joint. The middle of the joint is in a condition of plane strain while the

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35

30

Failure load (kN)

25

20

15

10

5

0 0.8

1

1.2

1.4

1.6

2

1.8

2.2

Adherend thickness (mm) Fig. 6. Variation of strength of SLJ in tension vs. adherend thickness: 25 mm overlap, AV 119 adhesive.

Visible crack in the middle of the width of the joint (i)

(ii)

Width of joint

Top adherend (unloaded end)

Bottom adherend (loaded end) Z

X

Y

overlap

overlap Fillet face

Fillet face

Top adherend

Bottom adherend

Z Crack

Y

X

Fig. 7. The 40 mm (i) and 25 mm (ii) overlap joints: detailed view of fillet’s face, ESP 110 adhesive, initiation of cracks in the centre of the width of the joint due to anticlastic bending effect.

edges (sides) approach a condition of plane stress. Theoretical analysis [23] has shown that in the middle of the joints there exist high hydrostatic tensile components of stress while, at the edges high deviatoric components dominate. Consequently, cracks are more likely to develop from the middle part of the joint.

Kinloch and Shaw [7] found that the fracture toughness GIc in plane strain is less than GIc in plane stress conditions. The reason is because the tensile stress at which a material yields is greater in a triaxial stress field (plane strain) than in a biaxial stress field (plane stress) and thus, in the former, a more limited degree

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Top adherend

87

Location of damage initiation

I

III Bottom adherend

Typical final mode of failure

IV

Damage propagation inside the overlap

II

Direction of damage propagation

Bottom adherend

Top adherend's edge ji

"Hackle region" Cusps forming - rough texture Rapid crack growth Catastrophic failure

"Transition zone" Propagation of damage

Initiation of damage Adhesive whitening due to excessive yielding

Fig. 8. Description of failure mechanisms for ductile adhesives.

Adhesive stress whitening

Curved damage front Hackle region

Width of joint

Loaded adherend end

of plasticity develops at the crack tip, resulting in GIc (plane strain)oGIc (plane stress). This also suggests that cracks are more likely to form in the middle of the width of the joints. The damage shown in the fillet face (Figs. 7 and 8) is a result of the damage that initiates near the adherend embedded corner. Further evidence for that was given by noises emitted from the joints before any visible damage could be seen on the fillet face, something that was observed for all four different adhesives. Three fracture regions have been recognised, depending on the surface roughness as observed by optical microscopy [24], but in all cases failure was cohesive. The mirror region is smooth and featureless and normally surrounds the origin of failure and is associated with slow propagation as the incipient damage develops. A smooth, matt region called mist follows as the fracture accelerates. This changes to hackle, a rough-textured surface of rapid damage growth. The speed at which the damage propagates is related to the fracture surface morphology [25]. The smooth regions correspond to the zone where damage (or cracks) is accelerating. The rough region corresponds to catastrophic failure (very fast damage growth—crack bifurcation). Many of the above features were found in the fractured single lap joints we tested in this investigation. For the case of the two ductile adhesives, there is an area very near the edge of the overlap, and around the embedded adherend corner, where ‘stress whitening’ due to excessive deformation of the adhesive occurs (initiation of damage) [26]. This forms a ‘damaged zone’ which spreads inwards through the overlap as the load increases (propagation of damage). Adams et al. [23] and Adams and Pepiatt [27] also showed that cracks initiate at the corner. At the same time, damage spreads into the fillet (stress whitening of the fillet face Fig. 8). The propagation of damage inside the overlap follows a parabolic shape due to the anticlastic bending effect. Damage propagates faster in the middle rather than at the edges of the width of the joint. In the case of ESP 110, this shows up as dark and shiny patterns while, in the case of AV 119 and EC 3448, it shows up as a stress whitened area (Fig. 9). Then follows the rough-textured (hackle) area, which corresponds to catastrophic failure (very rapid damage growth). These three regions are not clearly distinguished from each other, but there is a gradual transition from one region to the other. By observing the fractured surfaces of the joints, it was noted that the stress whitened area around the adherend embedded corner was bigger for longer overlaps. This means that longer overlaps can sustain

Z

Y X

Overlap length

Fig. 9. Failure pattern of a 60 mm overlap with AV 119 adhesive.

more damage at the ends of the joint before catastrophic failure occurs. All joints failed in a catastrophic manner. As already said, noises were emitted in some cases before catastrophic failure, but no cracks were visible (apart from those referred to in Fig. 7) until failure occurred. From results presented in this section, it can be seen that the strongest joints were with EC 3448 adhesive followed by AV 119, ESP 110 and MY 753, and this correlated directly with the strain to failure (both in tension and shear) of the adhesive, although for the case of MY 753 the adhesive low strength should also be taken into account. 3.5. Strain gauging of joints Joints with EC 3448 adhesive were strain gauged for all five overlap configurations. In addition, joints with AV 119 adhesive and the three different adherend thicknesses in the overlap configuration of the 25 mm were also strain gauged. As has already been shown, the adherends remain elastic throughout the loading of the joints. Therefore, the geometric effect of the overlap length on the load vs. bending moment distribution can be assessed for the whole loading regime.

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0.009 12.5 - Exper.

0.008

Inner strain gauges

20 - Exper. 25 - Exper.

0.007

40 - Exper. 60 - Exper.

Longitudinal strain - e11

0.006

12.5 - F.E 20 - F.E

0.005

25 - F.E 40 - F.E

0.004

60 - F.E

0.003

Outer strain gauges

0.002

0.001

0 0

10

20

30

40

50

60

70

-0.001 Load (kN)

Fig. 10. FE prediction—experimental correlation for the strain variation in the adherends with EC 3448 adhesive and various overlap lengths under tensile loading.

0.0016 12.5

Bending strain

0.0014

25

0.0012 0.001

40

0.0008 0.0006

60

0.0004 0.0002 0 0

10

20

30

40

50

60

70

Load (kN) Fig. 11. Variation of bending strain in the adherends with load—EC 3448 adhesive SLJs for various overlap lengths (mm).

Results from experiments for all overlaps and the EC 3448 adhesive are presented in Fig. 10. The amount of difference between the strain from the inner and outer strain gauge is directly related to the bending moment applied at the particular point of the cross section of the adherend under consideration and the average gives the applied tensile strain. It is obvious from Fig. 10 that, for the same axial applied load, joints with longer overlap experience less bending. This is very important in understanding the load and bending moment distribution in single lap joints. This is a purely geometric effect, resulting from changing the length of the overlap, since all the rest of the parameters have been kept the same and there is no adherend plasticity. Fig. 11 shows the bending strain (difference of the strain between the strain readings in Fig. 10 divided by two) as a function of the applied load for the adhesive EC 3448. It is interesting to note the decrease of the bending strain with overlap and also for a given overlap the decrease of the bending strain corresponding to a decrease of the bending moment as the joint rotates and aligns the load. A few experiments have been performed using other adhesives, such as ESP 110 and AV 119, to investigate the effect of the adhesive properties on the moment distribution. Results show that the distribution, for the same overlap, remained the same

whatever adhesive used. The only difference in this case is the failure load, which is different, as already shown in Section 3.1. The bending moment distribution is therefore mainly affected by the adherend properties, and geometric parameters such as the overlap length, and is independent of the adhesive used. The only point where differences can occur is very near the failure load (the very end of the curves). As long as the load is transferred through the joint, the moment distribution will be governed by the length of the overlap and the properties of the adherend. It must be mentioned here that all the adhesives compared were epoxy paste adhesives. If a very different type of adhesive (i.e. a much more compliant such as a polyurethane or acrylic) was used then results could be different. The effect of the adherends thickness (for a given overlap length) on the bending moment distribution and therefore the strength of the joints was investigated. Three different adherend thicknesses were investigated: 1 mm, 1.6 mm and 2 mm. Results are presented in Fig. 12. Comparisons can be made from the three different configurations. As would be expected, the overall stiffness of the joint increases as the thickness of the adherends increases. The difference between the inner and outer strain gauge readings is proportional to the amount of bending at the edges of the overlap, as has already been mentioned. If comparisons between the joints with different adherend thicknesses are made, it can be seen that the thinner the adherend, the less is the bending moment for a given axial applied load. The thicker the adherend, the stiffer is the joint in flexure so the local curvature of the adherend is less. That means the adhesive has to deform less in the critical areas around the overlap edges. Thus, there is less peeling load for the thicker adherends and a more uniform shear stress distribution under the overlap.

4. Finite element analysis 4.1. Details A commercial finite element package (ABAQUS) was used for the numerical analysis using two-dimensional analyses. Both

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89

0.006 1mm - Exper. 1mm - F.E 0.005

1.6mm - Exper. 1.6mm - F.E 2mm - Exper. 2mm - F.E

Strain e11

0.004

0.003

0.002

0.001

0 0

5

10

15

20

25

30

35

-0.001 Load (kN) Fig. 12. FE prediction—experimental correlation for the strain variation in the adherends for 25 mm overlap with AV 119 adhesive and various adherend thicknesses under tensile loading.

geometric and material non-linearities were taken into account. Fully integrated, solid, quadrilateral, 8-noded elements were used throughout the majority of the two-dimensional models. Some 6noded triangular elements were used in areas of mesh refinement and around the fillets. Generalised plane strain formulated elements were used, where the strains in the width direction (ezz) are assumed to be a non-zero constant, effectively modelling the middle of the width of the joints. Yield in polymeric materials, including adhesives, is known to be dependent on both the deviatoric and the hydrostatic stress components of stress. This means that the yield behaviour is different in tension and compression. The yield criterion of Raghava is a pressure-dependent yield criteria defined by the following equation [28,29]: Ss2yt ¼ 3J2 þ ðS1Þsyt I1

Table 3 Values of yield ratio S derived from single element models. Adhesive

S ¼ syc/syt

ESP 110 AV 119 EC 3448 MY753

1.40 1.35 1.55 1.37

ð2Þ

where I1 is the first invariant of the stress tensor and is defined in terms of the principal stresses as: I1 ¼ s1 þ s2 þ s3, and S is the ratio of yield stress in compression to the yield stress in tension corresponding to the same equivalent plastic strain. The term J2 is the second deviatoric stress invariant, J 2 ¼ ð1=6Þ½ðs1 s2 Þ2 þ ðs2 s3 Þ2 þ ðs3 s1 Þ2 , and syt is the yield stress in tension. When a pressure-dependent yield criterion such as the Raghava criterion is used, the yield surface is a paraboloid in the principal stress space. This model requires the elastic parameters such as Young’s modulus E and Poisson’s ratio n, the hardening curve (stress vs. corresponding plastic strain) and the yield parameter S. The Raghava criterion was implemented in ABAQUS using the Exponent Drucker–Prager criterion option. It is assumed that the direction of the plastic flow is the same as the direction of the normal to the yield surface at the point of yield which is called associated flow. In order to calculate the yield parameter S, tensile and compressive yield data are normally needed. In the current work, a standard procedure of using a single element model to verify the behaviour of the model under different loading conditions was followed. Tensile, compressive and shear loading were applied in a single solid element and the response (in the FE

Adherend

Adhesive

Fig. 13. Detailed view of the finite element mesh used in the analyses.

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model) in tension, compression, and shear was obtained. The experimental tensile data were used as input for the material behaviour. A trial and error procedure was followed in order to define the S value for each adhesive material. The value of S was changed until the experimentally measured shear behaviour was matched by the behaviour predicted by FE analysis. The S values used are summarised in Table 3. Appropriate boundary conditions were applied at the ends of the joints to simulate the gripping of the specimens during testing. One end of the joint was fully constrained. The load was applied at the other end as a uniform pressure at the element faces. This end was also constrained in the vertical direction. The constraints cover a length of 25 mm which corresponds with the gripping area of the specimens. Since different models were created for the different overlap configurations, it was decided to keep the mesh consistent around the overlap edges. Three elements were used through the adhesive thickness. Part of the FE mesh used in the analyses can be seen in Fig. 13. The results are presented for two different interfaces (the ‘loaded’ and ‘unloaded’ interfaces) along the overlap length. 4.2. Results Joints with the EC 3448 adhesive at five different overlap configurations were modelled and the results for the predicted variation of longitudinal strain in the adherends from the FE analysis are shown in Fig. 10 along with the experimental results. The strain results calculated from the FE analysis are the average strain over the area that the strain gauges cover (mid-point of the gauge is 9 mm from the overlap edge). Two strain variations are presented for each overlap case (inner and outer side of the adherends). The correlation is very good up to the failure load for all overlap configurations. The FE analysis correctly predicts that, for a given axial applied load, the bending moment is less for joints with longer overlaps. FE results predict that the adhesive behaviour does not alter the bending moment distribution, which is in agreement with what was observed experimentally. Only very minor differences

can be observed for the range of adhesives tested. This is because the bending moment distribution is mainly dependent on the geometry of the joints (overlap length in particular) and of course the adherend properties. The adherend thickness is also another variable that affects the bending moment distribution around the overlap edges. Comparisons between experimental results, for the configuration of 25 mm overlap, AV 119 adhesive and different adherend thicknesses, and the FE predictions can be seen in Fig. 12. As can be seen, the correlation is very good for all cases and the FE results predict that the thinner the adherends the less is the applied bending moment at the edges of the joint for a given applied load. It can be therefore concluded that, for the range of structural adhesives tested and modelled in the current work, the bending moment distribution around the overlap edges is mainly dependent on geometric quantities such as the adherend thickness and the overlap length. 4.3. Failure load prediction In the FE models normally used, it has been noted that there exists a region very near the sharp corner of the unloaded adherend where values of strain reach extremely high values, mainly due to the idealised modelling behaviour and also due to the concentrating effect of the sharp corner. These very high strain values are, in some cases, physically meaningless, in the sense that they are much higher than the adhesive strain limits defined by tensile or shear tests. In reality, such sharp corners do not exist; but, even so, some small volume of the adhesive may be well beyond the critical limits (as defined by uniaxial tests) on the unloaded interface, although not at the high strain levels predicted by the FE analysis. As an example, the variation of the shear strain (e12) along the loaded and unloaded interfaces for the case of 25 mm overlap with the EC 3448 adhesive is presented in Figs. 14 and 15 for different levels of loading during an FE simulation. Only half of the overlap is presented since the variation is symmetric for the other half of the overlap. As can be seen from the graphs, the highest predicted shear strains exist very near the embedded corner of the unloaded adherend. On the

0.00E+00 -4

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2

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6

8

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12

14

-5.00E-02

Shear strain (e12)

-1.00E-01

-1.50E-01

-2.00E-01 Load (kN) -2.50E-01

9.41 17.74 23.90

-3.00E-01

26.63 30.63 33.48

-3.50E-01 Distance along overlap (mm)

Fig. 14. Variation of shear strain along the overlap length for the loaded interface (EC3448 adhesive, 25 mm overlap, 1.6 mm thick adherends).

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91

0.00E+00 -4

-2

0

2

4

6

8

10

12

14

-2.00E-01

Shear strain (e12)

-4.00E-01

-6.00E-01

-8.00E-01

-1.00E+00 Load (kN) 9.41 17.74 23.90 26.63 30.63 33.48

-1.20E+00

-1.40E+00

-1.60E+00 Distance along overlap (mm)

Fig. 15. Variation of shear strain along the overlap length for the unloaded interface (EC3448 adhesive, 25 mm overlap, 1.6 mm thick adherends).

Fig. 16. Shear stress on the loaded interface with and without a crack for an applied load of 17.74 kN (EC3448 adhesive, 25 mm overlap, 1.6 mm thick adherends).

loaded adherend side, the strains peak further inside the overlap and the maximum predicted strains on this side are considerably lower than the high values (caused by the concentrating effects of the embedded corner) on the unloaded adherend side. Similar strain variations can be obtained for the case of peel strains e22 of the same or other adhesives tested in the current work. In a single lap joint configuration, the fact that the adhesive around the unloaded interface corner has exceeded the adhesive strain limit does not necessarily mean that failure of the joint will occur. This is because the load can be transferred by alternative routes further inside the overlap. A model was built where a crack was inserted at the unloaded adherend interface to study the shear stress distribution in case a crack has formed due to excessive strain. The results are presented in Fig. 16 for the case of a lap joint with 25 mm overlap and adhesive EC 3448 for an applied load of

17.74 kN for which, according to Fig. 15, the shear strain at the unloaded interface has already exceeded the shear failure strain. It can be seen that the crack alters the stress distribution and causes a shift of the load transfer further inside the joint. However, the level of stress further inside is approximately equal to that for a joint without a crack, i.e. the crack does not significantly change the level of stress on the loaded adherend side. This is an important point because it means that the high strain levels seen at the unloaded interface are not greatly influencing the strain distribution at the loaded adherend interface. Because more load is transferred further inside the overlap, the shear strains start to build up on the loaded interface and the strains peak further inside the overlap as the load increases. When the strains in the loaded interface exceed the adhesive limiting values, a crack can develop and depending on how the remainder of the overlap is strained,

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Fig. 14 shows the limiting case for the 25 mm long EC 3448 joint. While the peak shear stress on the loaded adherend does not reach the ultimate shear strain of 50%, at the middle of the joint (12.5 mm) the shear strain reaches the level (5%) where the shear stress–strain graph for EC 3448 shows the onset of plasticity. At this point, then, the whole of the adhesive layer is yielding in shear and no more load can be carried. A similar situation occurs with larger overlaps and the predictions for the 40 mm joint (Fig. 18) are at the limit. At 5 mm under the overlap, the shear strain reaches 50% and at 20 mm (middle of the joint) the shear strain reaches the 5% needed for global yield. Thus, the 40 mm overlap is just on the straight line indicating global yield. However, for the 60 mm overlap, a new situation exists (Fig. 19). Here, the adhesive exceeds 50% shear strain about 4.5 mm under the overlap, while the middle of the joint has not reached the 5% limit for global yield. The adhesive will then fail at this point, and no further load can be carried. In effect, we have a crack which will grow until the effective overlap is such that we have global yield. In Fig. 20, if the strength at 60 mm is projected back to the global yield line, and then down to the overlap length, we have an effective overlap length of about 46 mm. In other words, the crack must have grown a distance of ½(60– 46)¼6.5 mm from the end. If we now look at Fig. 9, the zone of stress whitening stretches about 6 mm into the overlap before the fracture mode changes. This correlates well with the 6 mm predicted above. For those adhesives which have lower ductility, these limiting conditions towards the joint ends will be reached at lower loads.

catastrophic failure occurs. If it is a larger overlap, more local damage can be sustained stably than with a shorter overlap. However, there are some cases (depending on the adhesive properties and the geometry) where failure of the joints may occur due to more ‘global’ stresses and strains becoming critical. This is failure due to global yielding [30]. At medium applied loads, some part of the overlap is outside the yield surface but there are still parts in the middle of the overlap that have not yet yielded. The concept of global yielding could be schematically drawn as in Fig. 17. The thick lines, above the idealised stress strain curve, represent the condition of the adhesive along the overlap length at different levels of loading. At medium applied loads, some part of the overlap is outside the yield surface but there are still parts in the middle of the overlap that have not yet yielded (in zone I). With further increase in the applied load, more parts of the overlap yield. At the same time, some stress whitening may occur at the very edge of the overlap (part of the curve is now in region III), but the limiting values of strain in the loaded interface have not yet reached the critical limits. A point is reached when the whole of the overlap has yielded, and there is no part of the overlap in the elastic region (I). An unstable situation is now reached and the joints are about to fail.

Failure due to global yield in the overlap

(II)

σ

Load increasing

(III)

(I)

4.3.1. Definition of failure due to global yielding The global yielding criterion effectively means that the whole of the overlap is responsible for transferring the load. Experimentally, it was proven that such a situation can occur with some of the adhesives tested in the current work. It can be said therefore that, in general, the global yielding failure criterion usually applies in joints with relatively short overlaps or when the adhesive behaviour is ductile and the adherends remain elastic. The load corresponding to the total plastic deformation of the

ε Failure load - failure due to global damage High applied load - variable damage throughout the adhesive layer - all regions (I), (II) &(III) Medium applied load - some of the adhesive in the elastic region (I) and some in the plastic region

Fig. 17. Definition of damage is due to global failure in the adhesive layer.

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14

15

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22

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Shear strain (e12)

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-4.00E-01

-5.00E-01

-6.00E-01 Distance along overlap (mm)

Fig. 18. Variation of shear strain along the overlap length for the loaded interface: EC 3448 adhesive, 40 mm overlap, 1.6 mm thick adherends, tensile loading.

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1

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9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

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9.69 24.17 33.85 38.41 42.04 47.47 52.90 57.96

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Fig. 19. Variation of shear strain along the overlap length for the loaded interface: EC 3448 adhesive, 60 mm overlap, 1.6 mm thick adherends, tensile loading.

Table 4 Critical strains used in the FE strength predictions for the different adhesives. Adhesive

Max. shear strain e12 (%)

Max. tens. strain e22 (%)

EC 3448 AV 119 ESP 110 MY753

50 45 15 20

8 7 3 4

adhesive (global yielding) may also be calculated using the following simple equation: P GY ¼ ty UbUl

ð3Þ

where PGY is the failure load of the adhesive due to global yielding, ty is the yield strength of the adhesive, b is the joint width and l is the overlap length. 4.3.2. Definition of failure due to local strains exceeding limiting values Failure of the joint is predicted when the critical strain (shear strain e12 or peel strain e22) on the loaded adherend interface reaches the strain limits of the adhesive in shear or tension. It is not necessary to know which of the two variables (e12 or e22) would be the critical one, since this will mainly depend on the relative ratio of tension and bending applied in the joint. The limiting values of strain, based on the stress–strain curves of the adhesives, that were used in the failure predictions are summarised in Table 4. It can generally be said that failure due to local strains in the joint exceeding the adhesive strain limits usually applies in cases where failure is controlled by the local strain field. This is usually the case for joints with long overlaps and joints with brittle adhesives. 4.3.3. Strength predictions The predictions are presented in Figs. 20–23, for joints with EC 3448, AV 119, ESP 110 and MY 753, respectively. The graphs contain a straight line corresponding to the application of Eq. (2), i.e. global yielding of the adhesive. The experimental results are also included for comparison purposes.

For joints with the EC 3448 adhesive in tension (Fig. 20), failure due to global yielding occurs for overlaps up to 40 mm. Eq. (3) may be applied up to 40 mm overlap with enough accuracy. The critical variable for longer overlaps (60 mm) is the shear strain e12 (50%). For joints with AV 119 in tension (Fig. 21), the same situation exists. Joints with overlaps up to 25 mm fail due to global yielding and failure in longer overlaps is dominated by shear strains along the loaded interface. Eq. (3) is in this case not as precise for 40 mm as it is in the case of joints with adhesive EC 3448. That is because adhesive AV 119 is slightly less ductile than EC 3448 adhesive. For ESP 110 joints in tension (Fig. 22) failure due to global yielding occurs for joints up to 20 mm overlap. Eq. (3) is useful only up to 20 mm overlaps for this adhesive. The critical variable dominating failure for longer overlaps is again shear strain. A similar situation exists for the MY 753 adhesive in tension (Fig. 23). Failure is due to global yielding for the case of 12.5 mm overlap, but longer overlap joints fail due to high local shear strains. For brittle adhesives such as ESP 110 and MY 750, Eq. (3) is only applicable for short overlaps.

5. Conclusion As can be seen from the above results for structural adhesives and a relatively short overlap in single lap shear that failure is dominated by global yielding. This means that the whole of the overlap is involved in transferring the load. As the overlap gets longer, however, failure is no longer due to global yielding, but due to high local shear strains. This is because a critical overlap length is reached at which some part in the middle of the overlap is still elastic. A crack will therefore propagate until the strain at the middle of the overlap reaches the condition of initial yield strain (about 5% for EC 3448) when failure will then occur. As the adhesive behaviour becomes more brittle, then the conditions at the edges of the joint become more and more important and failure is dominated by the high local shear strains near the ends of the overlap. This is because as the adhesive gets more brittle, it cannot withstand the high shear strains that develop around the

Fig. 20. Experimental—FE strength prediction correlation with EC 3448 adhesive and various overlap lengths under tensile loading.

Fig. 21. Experimental—FE strength prediction correlation with AV 119 adhesive and various overlap lengths under tensile loading.

Fig. 22. Experimental—FE strength prediction correlation with ESP 110 adhesive and various overlap lengths under tensile loading.

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Fig. 23. Experimental—FE strength prediction correlation with MY753 adhesive and various overlap lengths under tensile loading.

ends of the overlap. Plasticity of the adhesive cannot spread under the whole of the overlap, because the critical strain limits at the overlap ends are exceeded first.

Acknowledgements One of the authors, E.F. Karachalios, would like to thank the Needham Cooper Foundation for supporting the research work by means of a scholarship. References [1] Crocombe AD, Adams RD. An elasto-plastic investigation of the peel test. J Adhes 1982;13:241–67. [2] Harris JA, Adams RD. Strength prediction of bonded single lap joints by the non-linear finite element methods. Int J Adhes Adhes 1984;4:65–78. [3] Dorn L, Liu W. The stress state and failure properties of adhesive-bonded plastic/metal joints. Int J Adhes Adhes 1993;13:21–31. [4] Zhao X, Adams RD, da Silva LFM. Single lap joints with rounded adherend corners: Stress and strain analysis. J Adhe Sci Technol 2011;25:819–36. [5] Zhao X, Adams RD, da Silva LFM. Single lap joints with rounded adherend corners: experimental results and strength prediction. J Adhe Sci Technol 2011;25:837–56. [6] Clarke JD, Mcgregor IJ. Ultimate tensile criterion over a zone: a new failure criterion for adhesive joints. J Adhe 1993;42:227–45. [7] Kinloch AJ, Shaw SJ. A fracture mechanics approach to the failure of structural joints. In: Kinloch AJ, editor. Developments in adhesives: 2. London: Applied Science Publishers; 1981. p. 82–124. [8] Hunston DL, Kinloch AJ, Shaw SJ, Wang SS. Characterisation of the fracture behaviour of adhesive joints. In: Mittal KL, editor. Adhesive joints, formation, characteristics and testing. New York: Plenum Press; 1984. p. 789–807. [9] Duan K, Hu X, Mai Y-W. Substrate constraint and adhesive thickness effects on fracture toughness of adhesive joints. J Adhe Sci Technol 2004;18:39–54. [10] Needleman A. A continuum model for void nucleation by inclusion debonding. J Appl Mech 1987;54:525–31. [11] Ungsuwarungsri T, Knauss WG. The role of damage-softened material behavior in the fracture of composites and adhesives. Int J Fract 1987;35: 221–41.

[12] Tvergaard V, Hutchinson JW. The relation between crack growth resistance and fracture process parameters in elastic–plastic solids. J Mech Phys Solids 1992;40:1377–97. [13] Campilho RDSG, Banea MD, Neto JABP, da Silva LFM. Modelling of single-lap joints using cohesive zones models: effect of the cohesive parameters on the output of the simulations. J Adhe 2012;88:513–33. [14] Karachalios EF, Adams RD, da Silva LFM. Single lap joints loaded in tension with ductile steel adherends, Int J Adhes Adhes, http://dx.doi.org/10.1016/j. ijadhadh.2013.01.017, this issue. [15] da Silva LFM, Adams RD. Measurement of the mechanical properties of structural adhesives in tension and shear over a wide range of temperatures. J Adhe Sci Technol 2005;19:109–42. [16] Grant LDR, Adams RD, da Silva LFM. Experimental and numerical analysis of single lap joints for the automotive industry. Int J Adhes Adhes 2009;29: 405–13. ¨ chsner A, Adams RD, editors. Heidelberg: Springer; 2011. [17] da Silva LFM, O [18] Crocombe AD, Adams RD. Influence of spew fillet and other parameters on the stress distribution in the single lap joint. J Adhe 1981;13:141–55. [19] Chiu WK, Jones R. A numerical study of adhesively bonded lap joints. Int J Adhes Adhes 1992;12:219–25. [20] Hunston DL, Kinloch AJ, Wang SS. Micromechanics of fracture in structural adhesive bonds. J Adhe 1989;28:103–14. [21] Papini M, Fernlund G, Spelt JK. The effect of geometry on the fracture of adhesive joints. Int J Adhes Adhes 1994;14:5–13. [22] Adams RD, Davies RGH. Strength of joints involving composites. J Adhe 1996;59:171–82. [23] Adams RD, Comyn J, Wake WC. Structural adhesive joints in engineering. second ed. London: Chapman & Hall; 1997. [24] Purslow D. Matrix fractography of fibre reinforced epoxy composites. Composites 1986;17:289–303. [25] Roulin-Moloney AC. Fractography and failure mechanisms of polymers and composites. London: Elsevier Science Publishers Ltd; 1989. [26] Guild FJ, Kinloch AJ, Taylor AC. Particle cavitation in rubber toughened epoxies: the role of particle size. J Mater Sci 2010;45:3882–94. [27] Adams RD, Peppiatt NA. Stress analysis of adhesive-bonded lap joints. J Strain Anal 1974;9:185–96. [28] Raghava R, Caddell R. A macroscopic yield criterion for crystalline polymers. Int J Mech Sci 1973;15:967–74. [29] Raghava R, Caddell R. The macroscopic yield behaviour of polymers. J Mater Sci 1973;8:225–32. [30] Crocombe AD. Global yielding as a failure criterion for bonded joints. Int J Adhes Adhes 1989;9:145–53.