Closed-form Solutions for Epoxy Adhesively Bonded Dissimilar Joint

8th Asian-Australasian Conference on Composite Materials (ACCM-8)

6 – 8 November 2012 Kuala Lumpur, Malaysia

P-JOI-166

Closed-form Solutions for Epoxy Adhesively Bonded Dissimilar Joint

M. Afendi1*

, A. Abdul Rahman2, Haftirman

1, M.S. Abdul Majid

1, and A. Syayuthi A.R.

1

1 School of Mechatronic Engineering, Universiti Malaysia Perlis,

02600 Pauh Putra, Perlis, Malaysia 2PETRONAS Chemical Fertiliser Kedah Sdn Bhd

Km 3, Jalan Jeniang, 08300 Gurun, Kedah, Malaysia

*[email protected]

ABSTRACT

A two-dimensional adhesively bonded dissimilar single lap joint model is analyzed under

tension. An explicit closed-form solution is formulated by using MATLAB tool for analysis of

shear and peel stresses distribution along the bondline under effect of variation of overlap length,

adherend thickness ratio and adherend Young’s modulus ratio. The solution is formulated based

on analysis of Bo Zhao et al. [2]. The bending moment at the edge joint of the Bo Zhao’s solution

is replaced by the bending moment at the edge joint that have been proposed by X. Zhao et al. [5]

to compare the accuracy of solutions. The least stress intensities in dissimilar joint could be

achieved with a suitable ratio of thickness and Young’s modulus of adherends.

Keywords: adhesive, dissimilar joint, single lap, closed-form, finite element method

INTRODUCTION

Many researchers so far consider the closed-form solutions for similar adhesively bonded

joint and literature for dissimilar adhesively bonded joint is limited. The study of dissimilar

adhesively bonded joint is vital as well. Besides, bonding similar and dissimilar adherends can

bring different effect to the strength of the joint. Comparison of a similar and dissimilar

adhesively bonded joint can allow a designer to choose the most suitable joint for application.

Meanwhile, maximum stress is found at the edge of a joint. The concentration of stress at the edge

of joint could be reduced by altering Young’s modulus of adherends, thickness of adherends and

overlap length of a joint. It is important for a designer to choose the optimum overlap length and

optimum ratio of adherends’ thickness and Young’s modulus for adhesive joint application.

Based on limitation of Volkersen, Goland and Reissner’s classical equation, many

researchers have tried to modify and correct it but most of the researches involve adhesively

bonded similar lap joint [1]. Other researchers have tried to modify the classical equation to study

adhesively bonded dissimilar joint but their research still needs further studies and correction [2-



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4]. Hence, the objective of this study is to formulate a closed-form solution for adhesively bonded

dissimilar single lap joint and to study stress distribution along bondline under effect of variation

of Young’s modulus, thickness of adherends and overlap length.

METHODOLOGY

MATLAB tool is used to formulate closed-form solutions to investigate adhesive shear

stress and peel stress distribution along bondline under effect of variation of overlap length,

adherends’ Young’s modulus and thickness. Results between similar and dissimilar adhesively

bonded joint is compared to develop an optimum single lap joint. Finally, the accuracy of

solutions is discussed by changing bending moment parameter of the single lap joint. Bo Zhao

and co-researchers claimed that Goland and Reissner’s model could predict the stress state in

adhesive layer well but it did not satisfy the zero shear stress condition at the free joint edges [3].

So, they adopted two-dimensional elasticity theory to formulate a solution which could satisfy the

zero shear stress condition at the free edges of joint. They also utilized equations of bending

moment and shear force at the joint edges from Shun Cheng et al. [4]. By using global

equilibrium equations and boundary conditions, they has successfully developed solutions for

shear stress (τm) and peel stress (σm) at the middle of adhesive layer as shown as below:

( ) ( ) (Eq. 1)

( ) ( ) (Eq. 2)

where, (i=1-9) and (j=1-8) are the integration constants whereas

are eight eigenvalues of the characteristic equation. The detailed solving

procedure of Bo Zhao’s solutions can be referred in [2].

Eqs. (1) and (2) are known as full analytical solution for the shear and peel stresses at the

middle layer of adhesive. The solutions are not easily applied due to two factors: the requirement

of during derivation of solutions for unbalanced joints and when the

unbalanced joints become balanced joints, becomes zero. Thus, Bo Zhao has made

some simplification for the solutions and it becomes simplified solutions as shown below:

(Eq. 3)

( ) ( ) (Eq. 4)

RESULTS AND DISCUSSION

A closed-form solution was formulated based on Bo Zhao’s solution [2,3]. The result has

been verified with Bo Zhao’s result. This solution was used to study the effect of overlap length,

adherend thickness and adherend Young’s modulus on the strength of a single lap joint. Figures 1

and 2 illustrate the effect of the overlap length on the adhesive shear and peel stress distribution

along the overlap length for joint with dissimilar adherend. The distribution of the shear stress and



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peel stress acting on the left and right portion of the joint. The peak peel stress is found at the joint

edges but zero shear stress is found at the joint edges. The zero shear stress rapidly rising to peak

value, then decrease to small value as approaching to the middle portion. The adhesive shear and

peel stress at the both joint edges decreases when overlap length is increasing. It is noted that the

joint strength increases as the overlap length is increasing because it provides larger bonded area

to distribute the load.

In Figure 2, the peel stress at the middle portion of the joint is approximately zero for

overlap length = 20 mm, 30 mm and 40 mm except overlap length = 10 mm. For overlap length =

10 mm, the peel stress is far away from zero value because overlap length = 10 mm provides

limited area for load distribution. When larger overlap length is used, adhesive peel stress will

decrease but adherend peel stress will increase. If the adherend peel stress exceeds the yield point

of adherend, the adherend will deform and cracking initiates. This is because an excessive

deformation of adherend cannot be compensated by relatively rigid adhesive layer. However, the

shear stress and peel stress can be decreased by applying larger overlap length but beyond a

certain limit, only little effect it gives.

When larger overlap length is used, the bonding between the adherend and adhesive layer

becomes stronger in which adhesive peel stress will decrease and the adhesive layer will not

easily be broken. Conversely, the adherend tensile stress will increase because the bonding has

become stronger, larger tensile force are required to pull the adherend apart.

Figure 1: Adhesive shear stress distribution at the middle plane along

overlap length for different overlap length

Figure 2: Normalized average peel stress distribution along overlap length for different overlap length

0

10

20

30

40

50

60

-1.5 -1 -0.5 0 0.5 1 1.5

τm

, M

Pa/

mm

x/l

c=10mm

c=20mm

c=30mm

c=40mm

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

σm/p

x/l

c=10mm

c=20mm

c=30mm

c=40mm



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Figures 3 and 4 show the normalized average shear and peel stress distribution along

overlap length for joint with different adherend thickness ratios. When the ratio of is

increasing, the adhesive shear stress and peel stress at the left portion decreases but the adhesive

shear stress and peel stress at the right portion increases. The value of adhesive shear stress at the

right portion is lower than those at the left portion because lower adherend is stiffer and has lower

strain deformation. However, the maximum shear stress for single lap joint with dissimilar

adherend is higher than that in joint with similar adherend. When larger ratio of adherend

thickness is used, the bending moment acting the edge of joints become larger which will cause

the peel stress increasing. The stress distribution can be balanced by adjusting the adherend

thickness. It implies that there should be an optimal ratio of which is approximately 1.5.

Figure 3: Normalized average shear stress distribution along overlap length for various adherend thickness

Figure 4: Normalized average peel stress distribution along overlap length for different adherend thickness

Figures 5 and 6 show the effect of the ratio of Young’s modulus on the distributions of

shear stress and peel stress along overlap length. The peel stress and shear stress at the left portion

increases when the ratio of is increasing but the peel stress and shear stress at the right

portion decreases. The maximum shear stress along adhesive layer for joint with similar adherend

is lower than that for joint with dissimilar adherend. The maximum shear stress is found at the left

portion which has the most probability for cracking initiates. There is a gap for shear stress and

peel stress between left portion and right portion which causes it becomes the unfavorable feature

for adhesive joints but the distribution can be balanced by choosing the suitable ratio of .

0

0.5

1

1.5

2

2.5

-1.5 -1 -0.5 0 0.5 1 1.5

τm/τ

o

x/l

t1/t2=1

t1/t2=2

t1/t2=3

t1/t2=4

-0.1

0

0.1

0.2

0.3

0.4

0.5

-1.5 -1 -0.5 0 0.5 1 1.5

σm/p

x/l

t1/t2=1

t1/t2=2

t1/t2=3

t1/t2=4



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The result of the project was summarized in Table 1. Based on the Table 1, the optimal

ratio of and for single lap joint are 2 and 1.5 since it gives low shear stress and peel

stress compared to other ratio. The closed-solution used in the previous analysis is denoted as

method 1. Shun Cheng’s bending moment equation was adopted in method 1 [4].

The bending moment equation was then replaced by another equation which was

proposed by X. Zhao [5]. The latter closed-form solution is denoted as method 2. These two

solutions were utilized to analyze the strength of dissimilar single lap joint. Table 2 shows that the

adhesive peak shear stress and peel stress at both end of the joint for method 2 is higher than those

of method 1. By comparing Bo Zhao’s finite element analysis (FEA) result, it was found that

method 1 is more accurate than method 2. Hence, the closed-form solution for adhesively bonded

dissimilar joint by using Shun Cheng’s bending moment equation can yield more accurate

analysis on the joint strength.

Figure 5: Normalized average shear stress distribution along overlap length for different Young’s modulus

Figure 6: Normalized average peel stress distribution along overlap length for different Young’s modulus

CONCLUSION

The strength of adhesively bonded single lap joint with dissimilar adherend has been

studied under effect of variation of overlap length, adherend thickness, and adherend Young’s

modulus. These various parameters have significant effect on the peak shear stress and peak peel

stress within the adhesive layer. A closed-form solution has been successfully formulated for

adhesively bonded joint with dissimilar joint and it satisfies the zero shear stress at the free edge

of joints. Larger overlap length should be used to decrease the shear and peel stress in the

0

0.5

1

1.5

2

2.5

-1.5 -1 -0.5 0 0.5 1 1.5

τm/τ

o

x/l

E2/E1=1

E2/E1=2

E2/E1=3

E2/E1=4

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-1.5 -1 -0.5 0 0.5 1 1.5

σm/p

x/l

E2/E1=1

E2/E1=2

E2/E1=3

E2/E1=4



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adhesive layer. Besides, optimal strength of a single lap joint can be achieved by using the

combination of the ratio of E2/E1 = 2 and t1/t2 = 1.5. The method 1 which is using Shun Cheng’s

bending moment equation is more suitable to be used for this analysis if compared to method 2

due to its accuracy. In the future, an experiment can be carried out to compare the accuracy of

closed-form solution because there are lots of assumptions have been made in derivation and this

could deviate from practical result. This closed-form solution would be a great tool for engineers

in order to analyze the strength of a single lap joint since it is simpler, straight-forward and

quicker than FEA.

Table 1: Summary result for the effect of ratio of E2/E1 and t1/t2

left

edge

right

edge

left

edge

right

edge

1 1 1.8073 1.8073 0.2318 0.2318

2 1 2.0663 1.3523 0.2757 0.1597

3 1 2.2016 1.1421 0.2963 0.1252

4 1 2.2844 1.0165 0.3075 0.1040

1 1 1.8078 1.8078 0.2318 0.2318

1 2 1.6332 1.6332 0.2164 0.2164

1 3 1.5245 1.5245 0.2380 0.2380

1 4 1.4127 1.4127 0.2656 0.2656

2 1 2.0663 1.3523 0.2757 0.1597

2 1.5 1.6358 1.5588 0.1573 0.2460

2 2 1.3443 1.6968 0.0878 0.3048

2 3 1.0165 1.7825 0.0334 0.3828

2 4 0.8890 1.7411 0.01708 0.4418

Table 2: Comparison of accuracy for normalized peak shear and peel stresses

along normalized overlap length

Approach Bo Zhao’s FEA solution Method 1 Method 2

Position left edge right edge left edge right edge left edge right edge

Normalized peak

shear stress 2.00 1.60 2.07 1.35 2.45 1.36

Normalized peak

peel stress 0.30 0.21 0.28 0.16 0.37 0.19

REFERENCES

[1] Lucas F.M. da Silva, Paulo J.C. das Neves, R.D. Adams, J.K. Spelt: Analytical models of

adhesively bonded joints - Part I: Literature survey: Int. J. Adhes & Adhes Vol. 29

(2009), pp.319-330

[2] B. Zhao, Z-H. Lu.: A two-dimensional approach of single-lap adhesive bonded joints.

Mech Adv Mater Struct Vol. 16(2) (2009), pp. 130-159

[3] B. Zhao, Z-H. Lu, Y-N. Lu: Closed-form solutions for elastic stress-strain analysis in

unbalanced adhesive single-lap joints considering adherend deformations and bond

thickness: Int. J. Adhes & Adhes Vol. 31 (2011), pp. 434-445

http://www.sciencedirect.com/science/article/pii/S0143749608000730

http://www.sciencedirect.com/science/article/pii/S0143749608000730



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[4] S. Cheng, D. Chen, Y. Shi: Analysis of adhesive-bonded joints with nonidentical

adherends. J. Eng. Mech.-ASCE Vol. 117(3) (1991), pp.605-623

[5] X. Zhao, R.D. Adams, Lucas F.M. da Silva: A new method for the determination of

bending moments in single lap joints. Int. J. Adhes & Adhes Vol. 30 (2010), pp.63-71

Documents

Closed-form Solutions for Epoxy Adhesively Bonded Dissimilar Joint