Composites: Part B 43 (2012) 204–209

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Composites: Part B

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Elasto-plastic stress analysis in an adhesively bonded single-lap joint

O. Sayman ⇑Dokuz Eylul University, Engineering Faculty, Department of Mechanical Engineering, Bornova, Izmir, Turkey

a r t i c l e i n f o

Article history:Received 22 March 2011Received in revised form 23 June 2011Accepted 9 August 2011Available online 17 August 2011

Keywords:C. Analytical modelingC. Finite element analysisE. Joints/joiningElasto-plasticity

1359-8368/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compositesb.2011.08.006

⇑ Tel.: +90 232 38831 38/203; fax: +90 232 38878E-mail address: onur.sayman@deu.edu.tr

a b s t r a c t

In this study, an analytical elasto-plastic stress analysis was proposed to determine the shear stress in aductile adhesively bonded single-lap joint. DP460 adhesive was employed in the present study. The von-Mises criterion was used for checking the yield condition of the adhesive material. In the solution, theshear stress was assumed to be constant across the thickness of the adhesive. Bending moment wasneglected in the solution. Analytical results were checked by using the finite element analysis. ANSYS10 was employed in the numerical solution. Analytical and numerical solutions are found to be in a goodagreement.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Adhesively bonded joints have been used in several industries.Adhesive bonding usually enables structures to have lower costand weight in comparison with the conventional methods suchas, welding, riveting and bolting. For instance, a bolting joint hasmany component parts such as a bolt, a nut, a washer and adher-ents. However an adhesive bonding has fewer component parts.Adhesive bonding can also offer an improvement in quality,strength to weight ratio due to the small unit mass of the adhesivein comparison with the conventional joint materials, and an in-crease in work productivity. This method is especially suitable tothe joining of thin materials. To ensure the safety of adhesivelybonded lap joints in structures, it is necessary to analyze the stressdistributions on the joints.

Her [1] produced an elastic analytical formulation for the adhe-sively-bonded joints considered as single-lap joints. The stress dis-tributions in the single lap and double lap joints for the elastic casewere proposed by Her. He showed that both analytical and finiteelement method results agree well in adhesive layer. Kadioglu[2] examined the increase of load transfer capacity of adhesivejoints under environmental conditions such as temperature in-crease and humidity. He found that the adhesive joints load trans-fer capacity can be increased under environmental conditions suchas temperature increase and humidity with the condition that ade-quate surface preparation of adherends to be used is provided.Kadioglu et al. [3] examined a very ductile adhesive and a struc-

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68.

tural epoxy under a four point bending loading. Results werechecked by using finite element analysis. Some investigations canbe found in papers [4–7]. Xiao et al. [8] presented an analyticalsolution for the in-plane stiffness response of adhesively bondeddouble-lap shear joints. The analytical solution was verified withexperimental results. Zou et al. [9] presented analytical solutionsfor adhesively bonded balanced composite and metallic joints. Inthat solution, the classical lamination plate theory and adhesiveinterface constitutive model are employed for that deduction. Boththeoretical and numerical studies of the balanced joints were con-ducted to reveal the adhesive peel and shear stresses.

da Silva and Adams [10] investigated joint strengths for adhe-sively bonded joints over a wide temperature range. In that study,the mixed modulus concept described by Hart-Smith is taken asthe starting point of the analysis. A numerical analysis was carriedout using finite element models to determine stress distribution intitanium/titanium and titanium/composite double lap joints. Theyobtained the shear and peel stresses. At 200 �C, the peel stresses inSupreme 10 HT were found to be very close to zero. da Silva andAdams [11] also carried out studies to decrease the transversestresses in composites and so to increase the joint strength partic-ularly at low temperatures.

Kang et al. [12] tested three types of adhesive bonding CFRPcomposites for cryogenic use and aluminum alloy (Al 6061-T6)for lining the tank using double – lap joint specimens. Besides,ABAQUS was used for the purpose of analyzing the experimentalresults. Krishna et al. [13] investigated composite bonded singlelap joints under axial tensile loading by using the finite elementanalysis. He pointed out that, the peel and shear stress for thevarious adhesive thicknesses are found out to be maximum at

Fig. 2. Free body diagrams.

O. Sayman / Composites: Part B 43 (2012) 204–209 205

the ends of the overlap and the distribution is symmetric with re-spect to the centre of the joint. Kadioglu and Adams [14] investi-gated a single-lap joint under axial loading using non-linearanalysis. For this purpose, a very ductile adhesive was used byusing finite element analysis. They found out that ductile adhesiveshave potential uses in structural applications where considerablyhigh stress concentrations are to be expected. Malvade et al. [15]studied the simulation of non-linear mechanical behaviors of adhe-sively bonded double – lap shear joints using ABAQUS solver. Theyemployed a shell-solid model, in which 4-node shell and 8-nodesolid elements were used for representing substrates and adhesive.These elements produced good correlation with test force-exten-sion behaviors. Chataigner et al. [16] proposed an analytical proce-dure based on the shear lag theory of Volkersen for a double-lapjoint.

In this study, DP460 adhesive was used as a ductile material. Ananalytical elasto-plastic solution based on Her’s elastic solution [1]was proposed in the stress analysis. The analytical solution waschecked by using the finite element analysis. A close result was ob-tained in the mentioned solutions. It is concluded that, the strengthof the adhesively bonded single-lap joint can be increased by usingthe residual stresses.

2. Problem statement

Elastic–plastic stress analysis of adhesively-bonded single-lapjoints is carried out analytically and numerically. The length ofthe adhesive bonding is taken long, in order to satisfy a character-istic stress distribution. In a characteristic stress distribution, shearstresses are high at the free ends and small at or around the centerof the adhesive [1]. An adhesive material was chosen for satisfyingthe plastic deformations. In this solution, loading of the single-lapjoint is eccentric. The eccentric loading produces bending stresses.In order to neglect the bending stresses, the thicknesses of the lam-inates and the bonding material are assumed to be small. As a re-sult of this, the bending stresses are neglected in comparison withthe tensile or peeling stresses. Assumptions in the investigation areas follows, Her [1];

a. The shear stresses in the adhesive layer do not vary throughthe thickness.

b. The longitudinal stresses in the adherents do not varythrough the thickness.

c. The adherent layers are linearly elastic.

3. Elastic solution

A single-lap joint is represented in Fig. 1. In this solution, thenonuniformity of the stress distribution within the width isneglected.

Static equilibrium can be written by using the free-body-dia-gram of an infinitesimal element in the over zone as shown inFig. 2.

Static equation produces the following equations as [1],

Pl lt0

y

η

Fig. 1. Single-

dT0

dxþ s ¼ 0 ð1Þ

dTi

dx� s ¼ 0 ð2Þ

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

The longitudinal displacement and strain relations can be foundas,

eo ¼duo

dx¼ ro

Eo¼ To

Eotoð3Þ

ei ¼dui

dx¼ ri

Ei¼ Ti

Eitið4Þ

where uo, eo, To, to are the displacement, unit strain, longitudinalforce, and thickness of the upper layer, respectively. Similarly,ui, ei, Ti, ti are the symbols of the lower layer.

The shear strain angle can be written as, as shown in Fig. 3.

c ¼ ui � uo

gð5Þ

The shear stress is,

s ¼ Gac ¼Ga

gðui � uoÞ ð6Þ

where Ga is the shear modulus of the adhesive material. Derivationof both sides of Eq. (6) gives the following equation as,

dsdx¼ Ga

gdui

dx� duo

dx

� �ð7Þ

By using Eqs. (1)–(7) the following equation can be produced as,

d2To

dx2 � k2e To ¼ �

GaPEitig

ð8Þ

where

k2e ¼

Ga

g1

Eitiþ 1

Eoto

� �ð9Þ

P ti

x

lap joint.

iu

0uγ η

Fig. 3. Distortion angle.

206 O. Sayman / Composites: Part B 43 (2012) 204–209

The integration constants can be found by using the boundaryconditions. The boundary conditions can be written as,

To ¼ 0; x ¼ l

To ¼ P; x ¼ �lð10Þ

Using boundary conditions and Ti = P � To gives the elastic solu-tion [1],

To ¼ P �12

sinhðkexÞsinhðkelÞ þ

Eiti � Eoto

2ðEiti þ EotoÞcoshðkexÞcoshðkelÞ þ

Eoto

Eiti þ Eoto

� �ð11Þ

Ti ¼ P 1þ 12

sinhðkexÞsinhðkelÞ �

Eiti � Eoto

2ðEiı ti þ EotoÞcoshðkexÞcoshðkelÞ �

Eoto

Eiti þ Eoto

� �ð12Þ

s ¼ Pke

2coshðkexÞsinhðkelÞ �

Eiti � Eoto

Eiti þ Eoto

sinhðkexÞcoshðkelÞ

� �ð13Þ

4. Elasto-plastic solution

In the plastic regions, the total strain increment is equal to thesummation of elastic and plastic strain increments as,

dexy ¼ deexy þ dep

xy ð14Þ

The total strain can be expressed in terms of displacement as,

exy ¼c2¼ ui � u0

2gð15Þ

Eq. (14) can be written in the incremental form as,

dexy ¼dui � duo

2g¼ ds

2Gaþ sdk ð16Þ

where dk is given in Prandtl–Reuss equations. Dividing both sidesby dx gives the following equation,

dexy

dx¼ 1

2Ga

dsdxþ sdk

dxð17Þ

From the equivalent strain formula depxy can be written as,

depxy ¼

ffiffiffi3p

2dep ð18Þ

If the Ludwik equation is obtained from the stress–strainrelations in exponential form, the equation can be expressed as,

r ¼ ro þ Kenp ð19Þ

where K and n are the plasticity constants. If n is used in thesolution, the analytical solution may not be solved. Therefore,Ludwik equation is written in the bilinear form as,

r ¼ r0 þ Kep ð20Þ

Derivation of the above equation produces,

dr ¼ Kdep ð21Þ

If theffiffiffi3p

ds is written instead of dr from the von-Mises criterion,the plastic strain increment is found as,

dep ¼ drK¼

ffiffiffi3p

dsK

ð22Þ

and

dePxy ¼

ffiffiffi3p

2deP ¼

ffiffiffi3p

2

ffiffiffi3p

dsK

!¼ 3ds

2Kð23Þ

Now, the Eq. (17) becomes as,

dexy

dx¼ 1

2Ga

dsdxþ 3

2Kdsdx

ð24Þ

by using Eq. (24) and after some mathematical manipulations thefollowing relation is found as,

dsdx

12Gaþ 3

2K

� �¼ dexy

dx¼ 1

2gdui

dx� duo

dx

� �ð25Þ

If c is written instead of 1=g 1Gaþ 3

K

� �, the equation becomes,

dsdx¼ c

dui

dx� duo

dx

� �¼ c

Ti

Eiti� To

Eoto

� �ð26Þ

then

d2To

dx2 ¼ �dsdx¼ �c

Ti

Eiti� To

Eoto

� �ð27Þ

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

d2To

dx2 � k2pTo ¼ �

cPEiti

ð28Þ

where

k2p ¼ c

1Eitiþ 1

Eoto

� �ð29Þ

The integration constants can be obtained from the boundaryconditions. The longitudinal tensions, Ti and To in the upper andlower adherents and shear stress s in the adhesive ply can be writ-ten as,

To ¼ P� sinhðkpxÞ2 sinhðkplÞ þ

Eiti � Eoto

2ðEiti þ EotoÞcoshðkpxÞcoshðkplÞ þ

Eoto

Eiti þ Eoto

� �ð30Þ

Ti ¼ P 1þ sinhðkpxÞ2 sinhðkplÞ �

Eiti � Eoto

2ðEiti þ EotoÞcoshðkpxÞcoshðkplÞ �

Eoto

Eiti þ Eoto

� �ð31Þ

s ¼ Pkp

2coshðkpxÞsinhðkplÞ �

Eiti � Eoto

Eiti þ Eoto

sinhðkpxÞcoshðkplÞ

� �ð32Þ

It is seen that a similar relation is found between the elastic andelasto-plastic solutions.

5. Results and discussions

In this study, composite layers have been bonded by usingDP460 adhesive. The adhesive presents ductile material properties.Mechanical properties of the composite layers are given in Table 1.

Glass fiber epoxy composite were manufactured by using thevacuum assisted resin infusion process. The study is solved for 0�orientation. Mechanical properties of the adhesive are given inTable 2. As seen in this table, the yield point of the adhesive isry = 26 MPa and plasticity constants K and n are 81.50 MPa and0.625, respectively.

The joint is shown in Fig. 4, where 2l is 25 mm and ti, t0 werechosen as 1.9, 2.4 and 3.8 mm. P is chosen as 380 N, whichproduces the plastic deformations.

Table 1Mechanical properties of the composite layers.

E1

(GPa)E2

(GPa)G12

(GPa)t12 Xt

(MPa)Yt

(MPa)Xc

(MPa)Yc

(MPa)S(MPa)

31.0 10.5 4.8 0.27 821 78 377 62 64

Table 2Mechanical properties of the adhesive.

E (MPa) t K (MPa) n ry (MPa)

5000 0.35 81.50 0.625 26.0

t0

ti

η=0,13 mm

100 mm

100 mm

25 mm

Fig. 4. Dimensions of single-lap joint.

O. Sayman / Composites: Part B 43 (2012) 204–209 207

In addition, the finite element analysis was performed in theANSYS 10 solution. Solid 182 four nodes element was used forthe numerical solution. The symbolic representation of the finiteelement mesh of the single-lap joint is shown in Fig. 5.

As seen in this figure, roller supports were put in the secondlayer for omitting the influence of the bending moment. As a resultof this, the finite element solution has produced a good distribu-tion of the shear stress through the adhesive.

A free mesh generation was carried out in the mesh, as shown inFig 6. This numerical solution provides two-dimensional stressanalysis. However, the analytical solution provides only onedimensional stress analysis.

The shear stress (s) in the adhesive for to = ti = 1.9 mm is shown inFig. 7. As shown in Fig. 7, a good agreement is obtained between boththe analytical and finite element solutions, especially at the ends ofthe adhesive. As seen that the magnitude of the shear stresses arefound to be highest at the ends of the adhesive. It is the smallest

Fig. 5. Mesh mode

A

(a)Fig. 6. (a) Mesh generation of single lap joint for t =

around the center of the adhesive. The number of finite elementswas 2044, in this solution. The size of the finite element was chosen0.40 mm in the adherents and 0.025 mm in the adhesive.

The shear stress variation throughout the adhesive fort = 2.4 mm is shown in Fig. 8. As seen, a good agreement was ob-tained between both solutions. The large stresses agree at the endsof the adhesive. It is also seen that the magnitude of the shearstresses at the ends of the adhesive are smaller than those inFig. 7. It can be concluded that when the thicknesses of the adher-ents increase under ti = t0 condition, the magnitude of the shearstresses at the ends of the adhesive becomes smaller. The numberof elements was 2484 in this solution. The size of the finite elementwas chosen 0.40 mm in the adherents and 0.025 mm in theadhesive.

The shear stress variation throughout the adhesive length of t= 3.8 mm is shown in Fig. 9. It is again found that there is a goodagreement between two these solutions. The number of finite ele-ments was 3571. In the performed solutions, the dimensions of thefinite elements were chosen to be the same as 0.40 mm in theadherents and 0.025 mm in the adhesive. It is again seen that themagnitude of the shear stresses are obtained to be the highest atthe ends of the adhesive. However, the magnitude of the shearstresses is smaller than those are found for ti = t0 = 1.9 and2.4 mm. When ti = t0 increase the absolute value of the shear stressalong the adhesive changes gradually.

As seen in all figures, the highest shear stress was obtained inthe thinnest layer, as t0 = ti = 1.9 mm. When the thickness of thelayers are increased, the highest shear stress, in the overlap jointdecreases, as observed in t = 2.4 and t = 3.8 mm layers.

The strength of the adhesive is weaker than the strength of thelayers. For this reason, the adhesive can be strengthened by usingthe plastic deformations. After creating plastic deformations, sub-sequently residual shear stresses develop in the adhesive. Sincethe adhesive is ductile, the residual stresses can be obtained inthe adhesive only. The distribution of the residual stress s fort = 1.9 mm is shown in Fig. 10. As seen in this figure, the adhesivebecomes stronger at the ends of the adhesive, due to the residualstresses. Since the magnitude of the shear stress is the highest atthe ends of the adhesive, the residual stress causes an increase inthe strength of the adhesive.

The residual shear stress for t = 2.4 and 3.8 mm are shown inFigs. 11 and 12, respectively. As seen in these figures, the magni-

P

l of the joint.

B

(b)1.9 mm and (b) the detail of mesh of adhesive.

-21

-19

-17

-15

-13

-11

0 5 10 15 20 25x (mm)

τ (M

Pa)

finiteanalytic

Fig. 7. Shear stress through the adhesive for t = 1.9 mm.

-21

-19

-17

-15

-13

-11

0 5 10 15 20 25x (mm)

τ (M

Pa)

finiteanalytic

Fig. 8. Shear stress through the adhesive for t = 2.4 mm.

-21

-19

-17

-15

-13

-11

0 5 10 15 20 25x (mm)

τ (M

Pa)

finiteanalytic

Fig. 9. Shear stress through the adhesive for t = 3.8 mm.

-15

-5

5

15

25

35

45

55

0 5 10 15 20 25

x (mm)

Res

idua

l Str

ess

Fig. 10. Residual stress s (MPa) through the adhesive for t = 1.9 mm.

-15

-5

5

15

25

35

45

55

0 5 10 15 20 25

x (mm)

Res

idua

l Str

ess

(MPa

)

Fig. 11. Residual stress s through the adhesive for t = 2.4 mm.

-10

0

10

20

30

0 5 10 15 20 25

x (mm)

Res

idua

l Str

ess

(MPa

)

Fig. 12. Residual stress s through the adhesive for t = 3.8 mm.

Table 3Shear stress at A and B points at different number of elements.

Element size Number ofelements

Thickness ofadherends(mm)

sxy (MPa)

Inadherends(mm)

Inadhesive(mm)

Point A Point B

0.40 0.025 2044 1.9 �20.38 �21.342484 2.4 �18.95 �19.873571 3.8 �17.04 �17.32

0.20 0.010 8301 1.9 �20.56 �22.7710,025 2.4 �18.92 �21.2415,654 3.8 �17.01 �18.74

0.10 0.005 33,100 1.9 �20.32 �23.2939,859 2.4 �18.98 �21.6463,637 3.8 �17.12 �19.64

-25

-20

-15

-10

-5

00 10 20 30

x (mm)

τ(M

Pa) finite

analytic

Fig. 13. Shear stress through the adhesive for L = 30 mm and t = 1.9 mm.

208 O. Sayman / Composites: Part B 43 (2012) 204–209

tudes of the shear stresses at the free ends for t = ti = 2.4 and3.8 mm are small in comparison with t = ti = 1.9 mm. They can beloaded greater than the virgin material in safety. The increases ofthe shear strengths of the joints are 43%, 28% and 19% for thick-nesses as 1.9, 2.4 and 3.8 mm, respectively. The highest increaseis obtained in the thinnest adherends.

The highest shear stresses at A and B for different number of fi-nite elements are given in Table 3, where A and B are the ends of

overlap. As seen in this Table, when the number of finite elementsis increased, the highest shear stresses at A and B increase gradu-ally, or not. Besides, when the thickness of adherends is increased,the highest shear stresses decreases at A and B.

O. Sayman / Composites: Part B 43 (2012) 204–209 209

The shear stress distribution through the adhesive forL = 30 mm and to = ti = 1.9 mm is shown in Fig. 13. It is seen that fi-nite element solution produces larger shear stresses at A and Bthan those for L = 25 mm.

6. Conclusion

An elasto-plastic stress analysis was carried out in this study.An analytical formulation was obtained in the solution. Finite ele-ment solution can be compared with the analytical solution.

(a) Elasto-plastic analysis can be employed in the ductileadhesive.

(b) Finite element solution can be compared with the analyticalsolution.

(c) They both produce good agreement at the ends of theadhesive.

(d) The ductile adhesive can be strengthened by using the resid-ual stresses.

(e) When the number of finite elements is increased, the highestshear stress increases gradually, or not.

(f) The proposed method presents a closed form solution forobtaining shear stresses in the elasto-plastic analysis. How-ever, this method is suitable for one dimensional stress anal-ysis. The peel stresses cannot be obtained by using thismethod.

(g) The strength of the joints increases 43%, 28% and 19% for thethicknesses as 1.9, 2.4 and 3.8 mm by using the residualstresses in the adhesive, respectively.

(h) The increase of the shear strength of the joint was obtainedto be the highest in the thinnest adherents as t = t0 = 1.9 mm.When the thicknesses of the adherents are increased, themagnitude of the shear stress at the free ends decreases

(i) Ductile adhesives have potential uses in adhesive joints andsuch adhesives may have some advantages over traditionalstructural adhesives.

(j) The applied tensile force at the free ends is limited by theultimate shear strength of the adhesive.

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