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Nonlinear stress analysis in adhesivelybonded single-lap jointOnur Sayman a , Adnan Ozel b , Ahmet Pasinli c & Mustafa Ozen aa Faculty of Engineering, Department of Mechanical Engineering ,Dokuz Eylul University , Bornova, Izmir , Turkeyb Faculty of Engineering, Department of Mechanical Engineering ,Ataturk University , Erzurum , Turkeyc Ege Technical Training High School , Ege University , Bornova,Izmir , TurkeyPublished online: 19 Mar 2013.

To cite this article: Onur Sayman , Adnan Ozel , Ahmet Pasinli & Mustafa Ozen (2013) Nonlinearstress analysis in adhesively bonded single-lap joint, Journal of Adhesion Science and Technology,27:21, 2304-2314, DOI: 10.1080/01694243.2013.773696

To link to this article: http://dx.doi.org/10.1080/01694243.2013.773696

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Nonlinear stress analysis in adhesively bonded single-lap joint

Onur Saymana*, Adnan Ozelb, Ahmet Pasinlic and Mustafa Ozena

aFaculty of Engineering, Department of Mechanical Engineering, Dokuz Eylul University, Bornova,Izmir, Turkey; bFaculty of Engineering, Department of Mechanical Engineering, Ataturk University,Erzurum, Turkey; cEge Technical Training High School, Ege University, Bornova, Izmir, Turkey

(Received 14 October 2011; final version received 9 June 2012; accepted 13 January 2013)

In this investigation, an analytical elastic–plastic solution was proposed for a single-lapjoint. A ductile adhesive joint material was used as the bond material. FM-73 was utilizedin the study. The influence of the bending moment was neglected in the solution. The solu-tion was modified for shear stresses. The analytical solution was compared with the FEMsolution. An ANSYS 10.0 solution was employed in the numerical solution. Both solutionswere compared with each other. These two solutions produced close agreements.

Keywords: elastic-plastic solution; FEM solution; single-lap joint; adhesive joint

1. Introduction

Adhesively bonded joints are preferred in several industries due to lower cost and weight incomparison with the conventional methods such as riveting, bolting and welding. They enablestructures with easy and simple joints.

Kaye and Heller [1] developed an optimal design of free-form bounded repairs and lap-joints, with the aim of achieving reduced adhesive stresses. In that study, significant improve-ments over conventional designs are obtained, as assessed by the reduction in peak adhesivestresses. Krishna et al. [2] studied the finite element analysis of composite bonded single lap-joint under axial tensile force. In this study, a special six nodded isoparametric element isdeveloped for the adhesive layer and the adhesive layer is assumed to be relatively thin andbehaves elastically as simple tension-compression springs and shear springs connecting theadherends. Apalak and Gunes [3] investigated three-dimensional elastic stress state of anadhesively bonded single lap joint with functionally graded adherends in tension. The adher-ends compose of a functionally gradient layer between a pure ceramic (Al2O3) layer and puremetal (Ni) layer. Kadioglu et al. [4] presented a very ductile adhesive and a structural epoxy.Four-point bending test was performed using a single-lap joint configuration. Finite elementanalysis was also used to investigate the stress distribution in the configuration. Kang et al.[5] tested three types of adhesives for the ability of bond carbon fiber reinforced plastic(CFRP) composites developed for cryogenic use and aluminum alloy (Al 6061-T6) for liningthe tank using double-lap joint specimens. In addition, ABAQUS was used for analyzing theexperimental results. Kaye and Heller [6] proposed a numerical evaluation of the stress con-centration for a single-layer patch, a multi-layer equivalent-stiffness patch with uniform taper

*Corresponding author. Email: onur.sayman@deu.edu.tr

Journal of Adhesion Science and Technology, 2013Vol. 27, No. 21, 2304–2314, http://dx.doi.org/10.1080/01694243.2013.773696

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stepping. Xiao et al. [7] proposed an analytical solution for the in-plane stiffness response ofadhesively bonded double-lap shear joints.

Zou et al. [8] presented analytical solutions for adhesively bonded balanced compositeand metallic joints. The classical laminate plate theory and adhesive interface constitutivemodel are employed for this deduction. Both theoretical and numerical studies of the bal-anced joints are conducted to reveal the adhesive peel and shear stresses. Kadioglu [9]introduced a different aspect of adhesive joints under environmental conditions. He con-cluded that the load transfer capacity of adhesive joints can increase under environmentalconditions such as temperature and humidity. Kadioglu et al. [10] investigated beneficialeffects of temperature increase on the adhesive joints in the single-lap joint configuration;four point bending test was employed for experimental and numerical examination. Ribeiroet al. [11] performed a numerical stress analysis in carbon-fiber-reinforced epoxy compos-ites using a finite element method. They used an epoxy adhesive layer in the study. Her[12] presented a simplified one-dimensional model based on the classical elasticity theoryfor adhesively bonded single-lap and double-lap joints. He compared the results withnumerical solutions.

Chataigner et al. [13] presented an analytical procedure based on the well-known shearlag theory of Volkersen. They studied non-linear failure analysis of a doubl-lap-bondedjoint. Malvade et al. [14] investigated simulation of nonlinear mechanical behaviors ofadhesively bonded double-lap shear joints for variable extension rates and temperaturesusing the implicit ABAQUS solver. Kadioglu and Adams [15] studied nonlinear analysisof a ductile adhesive in a single-lap joint under tensile loading. They compared their solu-tions with the finite element method. For this purpose, they used a very ductile adhesivewith high strain to failure.

Markolefas and Papathanassiou [16] developed a shear-lag model in order to evaluatestress redistributions in double-lap joints under axial lap-shear cyclic loading. In their solu-tion, the adherend materials represent linear elastic behavior; however; the material of theadhesive layer exhibits the elastic-perfectly plastic shear stress–strain relation.

Edlund et al. [17] proposed a model for an adhesively bonded joint with elastic-plasticadherends and a softening adhesive. In the solution, a finite element formulation and a numer-ical treatment of the constitutive law were discussed.

In this study, the adhesive was chosen as a high ductile material, as FM-73. The adherentlayers were elastic and brittle. Single-lap joint was loaded by traction forces. Due to ductility,an elastic–plastic stress analysis was carried out in the adhesive. In addition, finite elementsolution was carried out for comparing the analytical and numerical results.

2. Problem statement

Some structures are bonded by an adhesive due to simplicity of bonding and transferringthe forces easily. In this study, an elastic–plastic stress analysis is carried out in an adhe-sively-bonded single-lap joint. The adhesive material was chosen ductile in order to obtainplastic deformations. Loading of the single-lap joint is eccentric. The eccentric loading pro-duces bending stresses. In order to neglect the bending stresses, the thicknesses of the lami-nates and the bonding material are assumed to be small. As a result of this, the bendingstresses are neglected in comparison with the traction stresses. Her [12] proposed someassumptions:

(a) The shear stress in the adhesive layer does not vary through the thickness.

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(b) The longitudinal stresses in the adherents do not vary through the thickness.(c) The adherent layers are linear elastic.

3. Elastic solution

A single–lap joint is represented in Figure 1.Static equilibrium can be written by using the free-body-diagram of an infinitesimal

element.[12] Static equation produces the following equations as:

dTo

dxþ s ¼ 0 ð1Þ

dTi

dx� s ¼ 0 ð2Þ

where To and Ti are the longitudinal tension per unit width in the upper and lower adherents,respectively. τ is the shear stress in the adhesive layer.

d2To

dx2� k2eTo ¼ � GP

Eitigð3Þ

where

k2e ¼G

g1

Eitiþ 1

Eoto

� �ð4Þ

The integration constants can be found by using the boundary conditions. The boundary con-ditions can be written as

To ¼ 0; x ¼ lTo ¼ P; x ¼ �l

ð5Þ

Using boundary conditions and Ti=P�To gives the elastic solution,[12]

To ¼ P �1

2

sinhðkexÞsinhðkelÞ þ

Eiti � Eoto2ðEiti þ EotoÞ

coshðkexÞcoshðkelÞ þ

EotoEiti þ Eoto

� �ð6Þ

Ti ¼ P 1þ 1

2

sinhðkexÞsinhðkelÞ �

Eiti � Eoto2ðEiı ti þ EotoÞ

coshðkexÞcoshðkelÞ �

EotoEiti þ Eoto

� �ð7Þ

Figure 1. Single-lap joint.

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s ¼ Pke2

coshðkexÞsinhðkelÞ � Eiti � Eoto

Eiti þ Eoto

sinhðkexÞcoshðkelÞ

� �ð8Þ

where uo, ɛo, To, to are the displacement, unit strain, longitudinal force and thickness of theupper layer, respectively. Similarly, ui, ɛi, Ti, ti are the symbols of the lower layer.

4. Elastic–plastic solution

In this solution, only the adhesive layer presents plastic deformations. When plastic deforma-tions occurs, the total strain increment is equal to the summation of elastic and plastic strainincrements as

dexy ¼ deexy þ depxy ð9Þ

The total strain can be expressed in terms of displacement as in Figure 2,Distortion angle is written as,

exy ¼ c2¼ ui � uo

2gð10Þ

Equation (10) can be written in the incremental form as

dexy ¼ dui � duo2g

¼ ds2G

þ sdk ð11Þ

where dλ is given in Prandtl–Reuss equations. Dividing both sides by dx gives the followingequation,

dexydx

¼ 1

2G

dsdx

þ sdkdx

ð12Þ

dexydx

¼ 1

2G

dsdx

þ depxydx

ð13Þ

The total strain increment can be written as

dexydx

¼ 1

2gTi

tiEi� To

toEo

� �ð14Þ

If the Ludwik equation is obtained from the shear stress–shear strain relations in the bilin-ear form, the equation can be expressed as,

Figure 2. Distortional angle.

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s ¼ so þ Kc p ð15Þ

Derivation of the above equation produces as

ds ¼ Kdc p ð16Þ

depxy can be expressed in terms of the equivalent strain, dγ p as

dcp ¼ffiffiffi6

p

3depxy ð17Þ

Substituting in the relation produces as following,

depxydx

¼ 3ffiffiffi6

p dcp

dx¼ 3ffiffiffi

6p ds

Kdxð18Þ

Now, the Equation (12) becomes as,

dexydx

¼ dsdx

1

2Gþ 3

Kffiffiffi6

p� �

ð19Þ

by using Equation (14) and after some mathematical manipulations, the following relation isfound as,

dsdx

¼ cTi

Eiti� To

Eoto

� �ð20Þ

where c is written instead of

1=g1

Gþ

ffiffiffi6

p

K

� �; ð21Þ

then the equation becomes as,

d2To

dx2¼ �ds

dx¼ �c

Ti

Eiti� To

Eoto

� �ð22Þ

writing Ti=P� To produces the following relation as,

d2To

dx2� k2pTo ¼ � cP

Eitið23Þ

where,

k2p ¼ c1

Eitiþ 1

Eoto

� �ð24Þ

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The integration constants can be obtained from the boundary conditions. The longitudinal ten-sions, Ti and To in the upper and lower adherents and shear stress τ in the adhesive ply canbe expressed as,

To ¼ P� sinhðkpxÞ2 sinhðkplÞ þ Eiti � Eoto

2ðEiti þ EotoÞcoshðkpxÞcoshðkplÞ þ

EotoEiti þ Eoto

� �ð25Þ

Ti ¼ P 1þ sinhðkpxÞ2 sinhðkplÞ �

Eiti � Eoto2ðEiti þ EotoÞ

coshðkpxÞcoshðkplÞ �

EotoEiti þ Eoto

�������� ð26Þ

s ¼ Pkp2

coshðkpxÞsinhðkplÞ � Eiti � Eoto

Eiti þ Eoto

sinhðkpxÞcoshðkplÞ

�������� ð27Þ

It is seen that a similar relation is found between the elastic and elastic–plastic solutions.When the shear stress (τ) attains a value higher than the shear yield stress (τy), these aboveequations are employed in the solution.

5. Results and discussion

In this study, composite layers have been bonded by FM-73 adhesive. The adhesive presentsductile material properties. τ� γ diagram of FM-73 is shown in Figure 3. As seen that it is avery high ductile adhesive. The true shear stress–strain values were used in constructing thediagram.

The mechanical properties of the adhesive are given in the Table 1. As seen the yieldpoint of τ is τy= 17.0MPa and plasticity constants are K = 40.4MPa, n = 0.596.

Mechanical properties of the glass-epoxy composite layers are given in Table 2.The joint is shown in Figure 4. Where 2 l is 25mm and ti= to were chosen as 1.8, 3.6 and

2.5mm. The magnitude of P is chosen as 450N for all the layers, which produces the plasticdeformations in the adhesive. In addition, the finite element analysis was performed in theANSYS 10 solution. Solid 182 of four node elements were utilized for the numerical solu-tion. Symbolic meshing of the single–lap joint is shown in Figure 5. Roller supports were puton the right layer in order to omit the influence of the bending moment. As a result of this,

Figure 3. τ� γ diagram of FM-73.

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the finite element solution performs a good distribution of the shear stress through the adhe-sive. This numerical solution provides two-dimensional stress analysis. Besides, the analyticalsolution produces one-dimensional stress analysis.

Table 1. Mechanical properties of the adhesive.

E (MPa) υ K (MPa) n τy (MPa)

5000 0.35 40.4 0.596 17.0

Table 2. Mechanical properties of the composite layers.

E1 (GPa) E2 (GPa) G12 (GPa) υ12 Xt (MPa) Yt (MPa) Xc (MPa) Yc (MPa) S (MPa)

33.0 10.4 4.9 0.27 840 82 380 94 62

Figure 4. Single-lap joint.

Figure 5. Mesh model of the joint.

Figure 6. Shear stress through the adhesive for t= 1.8mm.

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The shear stress (τ) in the adhesive for to = ti= 1.8mm is shown in Figure 6. As shown inthe Figure, a close result is obtained between the analytical and numerical solutions, espe-cially at the ends of the adhesive.

The shear stress variation throughout the adhesive for t= 2.5mm is shown in Figure 7. Asseen, a good agreement was obtained between these solutions. The large stresses agree at theends of the adhesive. Both methods produce close results.

The shear stress variation throughout the adhesive length for t= 3.6mm is shown inFigure 8. It is again found that there is a good agreement between two these solutions.Especially, large stresses are in good agreement.

Figure 7. Shear stress through the adhesive for t= 2.5mm.

Figure 8. Shear stress through the adhesive for t= 3.6mm.

Figure 9. Residual stress τ through the adhesive for t= 1.8mm.

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The strength of the adhesive is weaker than the strength of the composite layers. For thisreason, the adhesive can be strengthened by using the plastic deformations. When plasticdeformations occur, the tensile force is released slowly, residual shear stresses occur in theadhesive. Since the adhesive is ductile, residual stresses can be obtained in the adhesive only.The distribution of the residual stress τ for t= 1.8mm is shown in Figure 9. As seen in thisFigure, the adhesive becomes stronger at the ends of the adhesive, due to the residual stres-ses. It can be loaded more than the virgin material with safety.

The residual stresses for t= 3.6 and 2.5mm are shown in Figures 10 and 11. As seen, thestrength of adhesive increases at the ends of the layers. So, the adhesive enables more tensileforces than the virgin material.

Figure 10. Residual stress τ through the adhesive for t = 3.6mm.

Figure 11. Residual stress τ through the adhesive for t = 2.5mm.

Figure 12. Peel stress through the adhesive for t= 1.8mm.

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The peel stress for the elastic–plastic case along the adhesive was investigated by someresearchers (Markolefas and Papathanassiou [16] and Hart-Smith [18]). The peel stress (σ)variation throughout the adhesive length for t= 1.8mm is shown in Figure 12. In the numeri-cal solution, the adhesive is assumed as an ideal plastic and as a result of this dτ/dx= 0. Thesolution was performed by using Newton–Cotes numerical integration. Obtained solution wascompared with the finite element solution. As seen in Figure 12, both the numerical and finiteelement solutions produce a close result. It is seen that, magnitude of the peel stress is thehighest at the ends of the adhesive.

6. Conclusion

An elastic–plastic stress analysis was carried out in a single-lap joint. The adhesive was cho-sen very ductile as, FM-73. An analytical solution was proposed in the solution for the duc-tile adhesive. Finite element solution was carried out for comparing the results.

(1) Elastic–plastic analysis can be proposed under some assumptions.(2) Both analytical and numerical methods produce close results in the joints.(3) They both show good agreement at the ends of the adhesive.(4) The ductile adhesive can be strengthened by using the residual stresses.

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