Research ArticleA Comparative Evaluation of Numerical and AnalyticalSolutions to the Biadhesive Single-Lap Joint

Halil Oumlzer1 and Oumlzkan Oumlz2

1 Mechanical Engineering Department Faculty of Mechanical Engineering Yıldız Technical University YıldızBesiktas 34349 Istanbul Turkey

2Machine Education Division Faculty of Technical Education Karabuk University 78050 Karabuk Turkey

Correspondence should be addressed to Halil Ozer hozeryildizedutr

Received 23 December 2013 Accepted 15 January 2014 Published 10 March 2014

Academic Editor Evangelos J Sapountzakis

Copyright copy 2014 H Ozer and O Oz This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper attempts to address the detailed verification of Zhaorsquos analytical solution including the moment effect with the two- andthree-dimensional finite element results Zhao compared the analytical results with only the 2D FEA results and used the constantbond-length ratio for the biadhesive bondline In this study overlap surfaces of the adherends and the adhesives were modelledusing surface-to-surface contact elements Both analytical and numerical analyses were performed using four different biadhesivebondline configurations The 3D FEA results reveal the existence of complex stress state at the overlap ends However the generalresults show that analytical and numerical results were in a good agreement

1 Introduction

Structural adhesives have been used extensively in the spaceaviation automotive and naval industries Single-lap jointsare the most widely used adhesive joints and have beeninvestigated by many researchers [1ndash8] Techniques reduc-ing peel and shear stress concentrations are tapering theadherend forming an adhesive fillet changing the lap jointgeometry and so forth However these techniques can havesome disadvantages For example tapering the adherenddamage fiber structure of the fiber-reinforced compositesand forming an adhesive fillet are quite difficult when lowviscosity adhesives are used

An alternative technique is to use a combination of stiffand flexible adhesives along the overlap region The stiffadhesive should be located in themiddle and flexible adhesiveat the ends Different names for this type of joint are usedin the literature such as mixed-adhesive biadhesive andhybrid-adhesive joints The joints bonded with biadhesivetransfer the stresses from the ends towards the centre of theoverlap more than the joints bonded with a monoadhesivealone Therefore high stress concentrations at the ends canbe reduced by using this technique

Biadhesive joints have been studied in a limited numberof papers in the literature Raphaelrsquos early paper [9] relatedto the biadhesive joint showed the possible benefits of usinga mixed-modulus bondline He neglected the peel stresseffect in his model Srinivas [10] investigated the applicationof combined flexible and stiff adhesives in the bondlineHis model has an ability to include dissimilar as well ascomposite adherends Pires et al [11] investigated a biadhesivejoint with aluminium adherends using both experimentaland numerical (finite element) techniques They proved thatjoint strength can be optimized by choosing appropriatejoint geometry and material Fitton and Broughton [12]used a linear elastic 2D finite element method (FEM) tocompare hybrid and monoadhesive bondlines They showedthat significant strength improvement can be obtained ifjoint failure stresses are considerably less than the shearstrength of the adhesive Three-dimensional finite elementanalyses of hybrid-adhesive joints under cleavage and tensileload were carried out by Kong et al [13] They showed thatmaximum stresses along the bondline can be decreased byusing appropriate bond-length ratios Also they emphasizedthat it is necessary to take into account the change ofloading modes when optimizing the stress distributions of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 852872 16 pageshttpdxdoiorg1011552014852872

2 Mathematical Problems in Engineering

the biadhesive joint Kumar [14] investigated the effects offunctionally graded bondlines on the stress components intubular joints He suggested controlling the modulus of theadhesive spatially for optimization of the peel and shearstrengths Variable flexibility and strength along the overlaplength were described as an ideal adhesive joint by da Silvaand Lopes [15] They concluded that if the ductile adhesivehas a joint strength lower than that of the brittle adhesivea mixed-adhesive joint with both adhesives gives a jointstrength higher than the joint strength of the adhesives usedindividually This synergetic effect can be explained by theshear stress distribution of the adhesive at failure The effectof the biadhesive bondline on the stress distribution of weld-bonded joints was studied by You et al [16] They showedthat the load bearing capacity of the weld-bonded joints maybe increased by transferring some parts of stress from theadhesive layer to the weld nugget Da Silva and Adams [17]studied titaniumtitanium and titaniumcomposite double-lap joints formed using a hybrid-adhesive bondline Theyshowed that the suitable combination of two adhesives givesa better performance over a wide temperature range (minus55∘Cto 200∘C) than a high temperature adhesive alone for a jointwith dissimilar adherends Temiz [18] studied the effect ofa hybrid-adhesive bondline on the strength of double strapjoints subjected to external bending moments He concludedthat stress concentration at the overlap ends decreases byapplying the flexible adhesive towards the ends of the overlapin bonded joints By using the flexible adhesive in biadhesivejoints the strains do not increase significantly when com-pared with increase in predicted failure load This indicatesthat the stiffer adjacent adhesive has the constraining effecton the strain in the flexible adhesive Das Neves et al [19]developed an analytical model for hybrid-adhesive single anddouble lap joints subjected to low and high temperaturesThey compared the solutions of the analytical model with afinite element analysis and observed only small differencesclose to the overlap ends where the maximum adhesive shearand peel stresses occurred Pires et al [20] discussed thefailure mechanism of biadhesive joints Their results showedan increase in shear strength of the biadhesive-bonded jointscompared with those in which monoadhesives were usedover the full length of the bondline The increase in theapparent lap-shear strength was predicted through finiteelement model Kumar and Pandey [21] performed the two-dimensional and three-dimensional nonlinear (geometricand material) finite element analyses of adhesively bondedsingle-lap joints having modulus-graded bondline undermonotonic loading conditions The adhesives were modelledas an elastoplastic multilinear material while the substrateswere regarded as both linear elastic and bilinear elastoplasticmaterial They observed that the static strength was higherfor joints with biadhesive bondlines compared to those withmonoadhesive bondlines Effects of load level and bondlinethickness on stress distribution in the biadhesive bondlinewere also studied

Carbas et al [22] developed a functionally modifiedadhesive in order to have mechanical properties that varygradually along the overlap allowing a more uniform stressdistribution along the overlap and to reduce the stress

concentrations at the ends of the overlap Grading wasachieved by induction heating giving a graded cure of theadhesive Analytical analyses were performed to predict thefailure load of the joints with graded cure and isothermalcureThe functionally graded joint was found to have a higherjoint strength compared to the cases where the adhesive wascured uniformly at low temperature or at high temperatureThe simple analytical analysis proposed by the authors wasshown to be a valid tool to predict the maximum failure loadof the functionally graded joint Carbas et al [23] studied afunctionally modified adhesive Simple analytical model tostudy the performance of the functionally graded joints wasdevelopedThe differential equation of this model was solvedby a power series Finite element analysis was performed tovalidate the analytical model developed The joints with theadhesive properties functionally modified along the overlapshowed a high strength when compared with the joints withhomogeneous adhesive properties along the overlap Bothshear stress distributions of the bond line were found tohave a similar behavior comparing the analytical analysisby Power series expansion with the numerical analysis bya FE analysis Bavi et al [24] optimized the geometry ofthe overlap in mixed adhesive single- and double-lap jointsusing a modified version of Bees and Genetic Algorithms(BA and GA) Four and five optimization variables wereconsidered within bi- and triadhesive joint configurationsrespectively Eventually the efficiencies of the two employedalgorithms namely modified BA and GA were comparedwith each other Most optimal joint configurations weredescribed by a long adhesive bond length and thick layers ofadherends Comparing a modified version of BA (MBA) andGA it was observed that the first algorithm has a significantrobustness producing a 100 success rate in all consideredcases They concluded that the proposed MBA proved tobe a very suitable candidate for these types of engineeringproblems

Many closed-form solutions are available in the openliterature The early analytical model was developed byVolkersen [25] However Volkersen method known as theshear-lag model neglected the rotation of the joint A lot ofimprovements were made over the following seventy yearsincluding the addition of the rotation of the adherendsadhesive plasticity and the adherend shear deformation[26 27] Recent contribution has been made by Zhao Heproposed some closed-form solutions to evaluate the stresscomponents along the adhesive bondline and then extendedthese solutions to the biadhesive bondline by taking intoaccount the bending effect [28]

In this study after reviewing the current literature onthe existing analytical models for the biadhesive single-lapjoints some expressions related to stress components forthe single-lap joint with biadhesive bondlines were derivedby following the same steps as Zhaorsquos solutions [28] Defor-mations due to the bending effect which causes the jointto rotate were included in the formulations A MAPLEprogram was written Prepared MAPLE program employsthese expressions to calculate stress componentsThe validityof the analytical results was assessed by comparing the 2Dand 3D FEA results for the mono- and biadhesive bondlines

Mathematical Problems in Engineering 3

The analytical model was based on the plane elasticity andsome restrictive assumptions but the state of stress in thejoint has the three-dimensional nature Especially the adhe-sive peel stress is sensitive to the three-dimensional effectsTherefore the three-dimensional FEM model can providemore accurate predictions for comparing the results withoutintroducing any simplification at the modelling procedureIn this study the 3D FEM model is based on the surface-to-surface contact elements Aluminum adherends were bondedwith stiff and flexible adhesivesThe stiff adhesive was locatedin themiddle and the flexible adhesive was located at the endsof the overlap The overlap length the adhesive thicknessthe adherend thickness and modulus of stiff and flexibleadhesives were kept constant Both analytical and numericalanalyses were performed using four different bond-lengthratios for the biadhesive joint The effect of biadhesiveand monoadhesive bondlines on the peel stress (transversenormal stress) and shear stress distributions was investigatedThe numerical analyses were performed using the Ansysfinite element code The results of both FEM analyses andanalytical solutions were compared

It must be stated that this paper aims especially tocompare the analytical solutions with the 3D FEA solutionsand to show the three-dimensional nature of the state of stressin the joint Zhao [28] compared stress results predicted bythe closed-form expressions only with the 2D FEA resultsThe bond-length ratio for the biadhesive bondline was keptconstant in his study

2 Analytical Evaluation of Stress Componentsfor Biadhesive Joint

The biadhesive bondline can be modeled as three individ-ual regions according to their shear modulus components(Figure 1) The stiff adhesive was located in the middle andflexible adhesive at the ends of the overlap Two adherendswith the thicknesses of 120575

1and 120575

2were bonded with an

adhesive layer with a thickness of 1205753 where 120575

4= 120575

3 The

regions I and III at the overlap ends are the left and rightflexible adhesive regions respectively The region II at thecentre of the overlap is the stiff adhesive region The twoends of the adherends are simply supported and right endis subjected to an axial load 119865 (As seen in Figure 1 the 119909-axispasses through the midplane of the adhesive layer)

In Figure 1 119897119891is the flexible adhesive length and 119897

119904is the

stiff adhesive length where 119897119891= (119897 minus 119904) and 119897

119904= 2119904 The upper

and the lower adherends are denoted by the subscripts 1 and 2respectivelyTheflexible and stiff adhesives are denoted by thesubscripts 3 and 4 respectivelyThen 119864

119894and119866

119894(119894 = 1 2 3 4)

are the modulus of elasticity and shear modulus of the fourindividual components The total overlap length is 2ℓ Thejoint width perpendicular to the (119909 119910)-plane is 119887119881

0and119872

0

are the shear force and bendingmoment acting on the ends ofthe upper and lower adherends respectively (Figure 1(b)) Adifferential section 119889119909 can be cut out from the overlap regionof the biadhesive joint as shown in Figure 2

In Figure 2 1205903119910

is the peel stress at the upper and lowerinterfaces of the adhesive 119865

119894119909 119881119894and 119872

119894(119894 = 1 2) are

the tensile forces shear forces and bending moments relatedto the upper and lower adherends respectively

The distributions of the longitudinal shear stresses anddisplacements of adherends 1 and 2 are illustrated in Figure 3A linear shear stress and strain distributions through thethickness of the adherends is assumed Two local coordinatesystems 119910

1and 119910

2 were introduced where 119910

1and 119910

2are

the distances from the top of the upper and lower adherendsrespectively Free surface stress conditions are considered

The detailed derivation of the stress components forthe bi-adhesive joint along the bondline was given in theAppendix For the three regions (I II and III) shown inFigure 1 the three sets of expressions are given as follows

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) + 119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(1a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(1b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+ 119861

11cosh (120596

1119909) sin (120596

1119909)

+ 119861

12cosh (120596

1119909) cos (120596

1119909)

(1c)

The abbreviations used in the expressions above weregiven in theAppendixThemoment effect is introduced in theformulation (see the Appendix)The superscripts I II and IIIdenote the relevant stress components of the three individualregions (see Figure 1) 120590

1119909is the longitudinal normal stress

of the adherends 120590119894119910and 120591119894119909(119894 = 3 4) are the peel and shear

stresses along the adhesive midplane Set of (1a) and (1c) arerelated to the right and left flexible adhesive regions Set of(1b) is related to the stiff adhesive region the central regionNormal shear and peel stresses can then be evaluated by

4 Mathematical Problems in Engineering

1

L

L

E1G1

ll

s s

E3G3 E4G4

y

xE2G2 2 F

12057511205752

b

Overlap widthcross-section

1205753 1205754I II III

(a)

E1G1

l l

y

x

E2G2

F

F

1205751

1205752

12057531205754

M0M0

V0

V0

lf lfls

E3G3 E4G4 o

(b)

Figure 1 Biadhesive single lap joint under a tensile load (a) geometric and material parameters (b) force equilibrium free-body diagram

dx

M1 M1 + dM1

F1xF1x + dF1x

V1 V1 + dV1

1205911x

1205913xbdx

1205913xbdx

1205913xbdx

1205913xbdx

1205903ybdx

1205903ybdx

1205903ybdx

1205903ybdx

1205912x

M2 + dM2

V2 + dV2

F2x + dF2x

M2

F2x

V2

Figure 2 Free-body force equilibrium diagram

using the appropriate expressions for each regionThe resultsof analytical solutions were compared with both the results of2D and 3D FEM analyses

3 Finite Element Model

Geometry dimensions and boundary conditions of thebiadhesive single-lap joint are shown in Figure 4

1205911x = 0

1205751

1205911x = 1205913x

y1

u1t

u1

u1b

(a)

1205752

1205912x = 1205913x

1205912x = 0

y2

u2t

u2

u2b

(b)

Figure 3 Longitudinal shear stress and displacement distributionsthrough the thickness of the adherends (a) upper adherend (b)lower adherend

Finite Element Analysis based on the surface to surfacecontact model was performed by using commercial FEAsoftwareANSYSThe adherendswere constrained at the endspreventing excessive deflection of the joint and simulating thegripping in the testing machine The effect of the grippinglength on the stress components was studied to validate thestress analysis results Convergence tests were carried outby using the gripping lengths of 10 20 and 30mm It wasseen that the gripping length does not have a considerableeffect on the stress results [29 see page 118] Finally it wasconcluded that the 20mm optimum length was enough andsufficient to prevent excessive deflection and allow the free

Mathematical Problems in Engineering 5

b =25

mm

1205750 = 15mm

1205750 = 15mm

875mm 875mmlf lfls

lt = 125mm

Flexible adhesiveStiff adhesive

1205753 1205754 = 025mm

(a)

x

y

F

L = 675mm

(b)

Fz

(c)

Figure 4 Biadhesive single-lap joint (a) geometry and dimensions (b) 2D and 3D FEM model boundary conditions (front view) (c) 3DFEMmodel boundary conditions (top view)

Table 1 Material properties of adherends and adhesives

Adherend(Aluminium alloy 7075)

Flexible Adhesive(Terokal 5045)

Stiff Adhesive(Hysol EA 9313)

Modulus of elasticity (GPa)119864

1

= 119864

2

= 71700 119864

3

= 0437 119864

4

= 2274

Modulus of shear (GPa)119866

1

= 119866

2

= 26955 119866

3

= 0158 119866

4

= 0836

Poissonrsquos ratio 033 038 036Shear strength (MPa) 152 20 276Elongation at break () 10 113 8

rotation of the joint for the problem considered (Figure 4(b))Thus overlap regionwas not influenced by the boundary con-ditions Due to the longitudinal symmetry half-symmetryboundary condition was used in order to reduce the solutiontime for the three-dimensional analysis (Figure 4(c)) [30]

Aluminium alloy 7075 was used as identical adherendsHysol EA 9313 and Terokal 5045 epoxy adhesives producedby Henkel were used as stiff and flexible adhesives respec-tively Material properties of adherends and adhesives aregiven in Table 1 [31]

The thicknesses of the adherends and adhesives are15mm and 025mm respectively The total overlap length

119897

119905 2119897119891+ 119897

119904 was taken as 125mm where 119897

119891is the length of

the flexible adhesive and 119897119904is the length of the stiff adhesive

The bond-length ratios of the adhesive bondline varied as 120585 =119897

119891119897

119904= 02 04 07 13 Therefore four different biadhesive

bondline configurations were investigated A static force of36 kN was applied to the right side of the lower adherend(Loading was chosen as the same in our recent paper [31])The finite element models of the joint for the 2D and 3D FEMare shown in the same figure (Figure 5)

The finite element contact model recognizes possiblecontact pairs by the presence of specific contact elementsThe overlap surfaces of the adherends and the adhesives were

6 Mathematical Problems in Engineering

(b)

(a)

x

y

z

x

y

Figure 5 Finite element model of the joint (a) two-dimensional (b) three-dimensional

modelled using surface-to-surface contact for the 2D and 3Dfinite element models Surface-to-surface contact model usesGauss integration points as a default which generally providemore accurate results than the nodal detection scheme whichuses the nodes themselves as the integration pointsThis typeof contact model transmits contact pressure between Gausspoints and not force between nodes

In 2D FEA model contact pairs were created using con-tact element ldquoCONTA172rdquo and target element ldquoTARGE169rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA172 elements were generated on thebottom and upper surfaces of adhesive CONTA172 is a2D 3-node higher order parabolic element TARGE169elements were used on the overlap surfaces of the adherendsTARGE169 is used to represent various 2D ldquotargetrdquo surfacesfor the associated contact element The 2D model wasdiscretized using the 8-node plane stress elements (Plane 183)

In 3D FEA model contact pairs were created using con-tact element ldquoCONTA174rdquo and target element ldquoTARGE170rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA174 elements were generated on thebottom and upper surfaces of adhesive CONTA174 is a3D 8-node higher order quadrilateral element TARGE170elements were used on the overlap surfaces of the adherendsTARGE170 is used to represent various 3D ldquotargetrdquo surfacesfor the associated contact element (More detailed infor-mation on the 3D contact modelling of biadhesive joint isavailable in our recent paper [31]) Adherends and adhesiveswere meshed with solid hexahedral elements (solid95) Thiselement is defined by twenty nodes having three degrees offreedom per node Three elements were used through thethickness of the adherends and the bondline (Figure 5(b))

The mesh density affects the stress values in the adhesivelayer The size of the elements in the mesh was reduced untila stable stress value was achieved Five different meshingschemes were investigated to optimize the number and sizeof the elements The sizes of the elements were graded forthe finer mesh in the critical regions (at the edges of overlapand contact interfaces of adhesives) Finally the element sizenear the stress concentration and contact interface were set

Target element

Contact element

Adherend

Adhesive

Contact pair

Figure 6 Surface-to-surface contact pair

to 003mm and 004mm respectively which is fine enoughto describe the severe stress variation near the critical regionsThenumber of elements varied for FEMmodels ofmono- andbiadhesive joints However the mesh size was kept constantin all models This was important in order to compare theresults between the mono- and biadhesive joints and also tocompare results of biadhesive joints among themselves

The two-dimensional and three-dimensional finite ele-ment analyses were performed The results were used tocompare the analytically predicted adhesive stresses Wepresent the FEA results and their comparisons with theanalytical results in the next section

4 Results and Discussion

The adhesive shear and peel stress distributions in themidplane of the bondline were obtained both analyticallyand numericallyThis section presents a comparison betweenthe numerical and analytical results The distributions of thestress components were normalized with respect to averageshear stress (120591avg) and distributions of the same type of stresscomponents were given in the same figure In addition bar

Mathematical Problems in Engineering 7

Contact interfaces

Flexible adhesive Stiff adhesive Flexible adhesive

Midplane

Adherend

Adherend

Overlap edge Overlap edge

Figure 7 Overlap region of the biadhesive single-lap joint

graphs were used to illustrate a visual comparison among thestress results of analytical model 2D FEA and 3D FEA

Some comparisons among the peak stress values werecarried out on the basis of the percent error Percentage erroris defined as follows

Error =Numeric minus Analytic

Numerictimes 100

(2)

where word ldquoNumericrdquo denotes 2D and 3D FEA resultsPercent errors between analytical and numerical results forthe biadhesive case were tabulated in tables while the errordiscussions were presented in a text for monoadhesive case

In the bar graphs and tables peak values of the stress com-ponents obtained through analytical model were comparedwith those obtained through 2D FEA and 3D FEA Howeverthere are two peak stress locations for the biadhesive bond-line one just at the overlap edges and the other at contactinterfaces of adhesives Therefore in the case of biadhesivejoint two peak stresses were considered for the comparison(Figure 7) In 3D FEA shear and peel stresses peak at themidwidth of the adhesive midplane for both of mono- andbiadhesive bondlines Therefore stress distributions plottedat themidwidth along the adhesive midplane Peak values arealso related to the midwidth of the adhesive midplane in 3DFEA

Figure 8 shows the shear stress distributions of themono-and biadhesive joints obtained from analytical and numerical(FEA) solutions It can be seen from Figure 8 that the shearstress distributions in the monoadhesive joint is not uniformThis is because the applied load causes bending momentswhich lead to peel stresses especially at the overlap edgesHowever peel stress component is more dominant especiallyover the mono-stiff adhesive Therefore the shear stressdistribution for the mono-flexible adhesive is more uniformand there is a very small stress concentration at the overlapedges In the monoadhesive joint the higher level of stressregion exists at the overlap edges (Figure 8)

In the biadhesive bondline as can be seen in Figure 8higher shear stress occurs at the contact interfaces of theadhesives and the lower shear stress region exists at theoverlap edges Peak stress decreases at the overlap edges andincreases at the contact interfaces With the increase in the 120585ratio the values of peak shear stress are decreasing althoughthere is no great reduction in the peak shear stress for ratiohigher than 07

The shear stress distributions from the analytical solutionwere plotted in Figure 8(c)This shows a good agreementwithnumerical ones although some differences can be seen at theoverlap edges Peak shear stresses of the numerical solutionsoccur very near the edge of the overlap region while those ofthe analytical results were always at the overlap edges Thatis because the numerical model tries to model the zero stressstate at traction-free end surfaces However in practice thereis an adhesive spew fillet and the shear stress does not go tozero at the bondline ends

The normalized peak shear stress values at the overlapedges and the contact interfaces are showed in Figure 9 as avertical bar chart

From Figure 9 it can be seen that the normalized peakshear stress values for analytical and the numerical modelshad close correlation for the majority of mono- and biad-hesive joints The maximum shear stresses occurred at theoverlap edges for monoadhesive joints and at the contactinterfaces for biadhesive joints

According to Figure 9 the shear peak stress which occursat the overlap edges increased with increasing 120585 ratio Fur-thermore increasing rate in the shear peak stress at theedges was observed to be a parallel changing rate withthe decreasing rate in the shear peak stress at the contactinterfaces

Figures 9(a) and 9(b) show that analytical results are ingood agreement with 2D FEA results rather than 3D FEAresults for both the mono- and biadhesive joint casesThe 3DFEA solution gives slightly higher shear peak stress valuesthan those from the analytical and 2D FEA solutions

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the biadhesive joint Kumar [14] investigated the effects offunctionally graded bondlines on the stress components intubular joints He suggested controlling the modulus of theadhesive spatially for optimization of the peel and shearstrengths Variable flexibility and strength along the overlaplength were described as an ideal adhesive joint by da Silvaand Lopes [15] They concluded that if the ductile adhesivehas a joint strength lower than that of the brittle adhesivea mixed-adhesive joint with both adhesives gives a jointstrength higher than the joint strength of the adhesives usedindividually This synergetic effect can be explained by theshear stress distribution of the adhesive at failure The effectof the biadhesive bondline on the stress distribution of weld-bonded joints was studied by You et al [16] They showedthat the load bearing capacity of the weld-bonded joints maybe increased by transferring some parts of stress from theadhesive layer to the weld nugget Da Silva and Adams [17]studied titaniumtitanium and titaniumcomposite double-lap joints formed using a hybrid-adhesive bondline Theyshowed that the suitable combination of two adhesives givesa better performance over a wide temperature range (minus55∘Cto 200∘C) than a high temperature adhesive alone for a jointwith dissimilar adherends Temiz [18] studied the effect ofa hybrid-adhesive bondline on the strength of double strapjoints subjected to external bending moments He concludedthat stress concentration at the overlap ends decreases byapplying the flexible adhesive towards the ends of the overlapin bonded joints By using the flexible adhesive in biadhesivejoints the strains do not increase significantly when com-pared with increase in predicted failure load This indicatesthat the stiffer adjacent adhesive has the constraining effecton the strain in the flexible adhesive Das Neves et al [19]developed an analytical model for hybrid-adhesive single anddouble lap joints subjected to low and high temperaturesThey compared the solutions of the analytical model with afinite element analysis and observed only small differencesclose to the overlap ends where the maximum adhesive shearand peel stresses occurred Pires et al [20] discussed thefailure mechanism of biadhesive joints Their results showedan increase in shear strength of the biadhesive-bonded jointscompared with those in which monoadhesives were usedover the full length of the bondline The increase in theapparent lap-shear strength was predicted through finiteelement model Kumar and Pandey [21] performed the two-dimensional and three-dimensional nonlinear (geometricand material) finite element analyses of adhesively bondedsingle-lap joints having modulus-graded bondline undermonotonic loading conditions The adhesives were modelledas an elastoplastic multilinear material while the substrateswere regarded as both linear elastic and bilinear elastoplasticmaterial They observed that the static strength was higherfor joints with biadhesive bondlines compared to those withmonoadhesive bondlines Effects of load level and bondlinethickness on stress distribution in the biadhesive bondlinewere also studied

Carbas et al [22] developed a functionally modifiedadhesive in order to have mechanical properties that varygradually along the overlap allowing a more uniform stressdistribution along the overlap and to reduce the stress

concentrations at the ends of the overlap Grading wasachieved by induction heating giving a graded cure of theadhesive Analytical analyses were performed to predict thefailure load of the joints with graded cure and isothermalcureThe functionally graded joint was found to have a higherjoint strength compared to the cases where the adhesive wascured uniformly at low temperature or at high temperatureThe simple analytical analysis proposed by the authors wasshown to be a valid tool to predict the maximum failure loadof the functionally graded joint Carbas et al [23] studied afunctionally modified adhesive Simple analytical model tostudy the performance of the functionally graded joints wasdevelopedThe differential equation of this model was solvedby a power series Finite element analysis was performed tovalidate the analytical model developed The joints with theadhesive properties functionally modified along the overlapshowed a high strength when compared with the joints withhomogeneous adhesive properties along the overlap Bothshear stress distributions of the bond line were found tohave a similar behavior comparing the analytical analysisby Power series expansion with the numerical analysis bya FE analysis Bavi et al [24] optimized the geometry ofthe overlap in mixed adhesive single- and double-lap jointsusing a modified version of Bees and Genetic Algorithms(BA and GA) Four and five optimization variables wereconsidered within bi- and triadhesive joint configurationsrespectively Eventually the efficiencies of the two employedalgorithms namely modified BA and GA were comparedwith each other Most optimal joint configurations weredescribed by a long adhesive bond length and thick layers ofadherends Comparing a modified version of BA (MBA) andGA it was observed that the first algorithm has a significantrobustness producing a 100 success rate in all consideredcases They concluded that the proposed MBA proved tobe a very suitable candidate for these types of engineeringproblems

Many closed-form solutions are available in the openliterature The early analytical model was developed byVolkersen [25] However Volkersen method known as theshear-lag model neglected the rotation of the joint A lot ofimprovements were made over the following seventy yearsincluding the addition of the rotation of the adherendsadhesive plasticity and the adherend shear deformation[26 27] Recent contribution has been made by Zhao Heproposed some closed-form solutions to evaluate the stresscomponents along the adhesive bondline and then extendedthese solutions to the biadhesive bondline by taking intoaccount the bending effect [28]

In this study after reviewing the current literature onthe existing analytical models for the biadhesive single-lapjoints some expressions related to stress components forthe single-lap joint with biadhesive bondlines were derivedby following the same steps as Zhaorsquos solutions [28] Defor-mations due to the bending effect which causes the jointto rotate were included in the formulations A MAPLEprogram was written Prepared MAPLE program employsthese expressions to calculate stress componentsThe validityof the analytical results was assessed by comparing the 2Dand 3D FEA results for the mono- and biadhesive bondlines

Mathematical Problems in Engineering 3

The analytical model was based on the plane elasticity andsome restrictive assumptions but the state of stress in thejoint has the three-dimensional nature Especially the adhe-sive peel stress is sensitive to the three-dimensional effectsTherefore the three-dimensional FEM model can providemore accurate predictions for comparing the results withoutintroducing any simplification at the modelling procedureIn this study the 3D FEM model is based on the surface-to-surface contact elements Aluminum adherends were bondedwith stiff and flexible adhesivesThe stiff adhesive was locatedin themiddle and the flexible adhesive was located at the endsof the overlap The overlap length the adhesive thicknessthe adherend thickness and modulus of stiff and flexibleadhesives were kept constant Both analytical and numericalanalyses were performed using four different bond-lengthratios for the biadhesive joint The effect of biadhesiveand monoadhesive bondlines on the peel stress (transversenormal stress) and shear stress distributions was investigatedThe numerical analyses were performed using the Ansysfinite element code The results of both FEM analyses andanalytical solutions were compared

It must be stated that this paper aims especially tocompare the analytical solutions with the 3D FEA solutionsand to show the three-dimensional nature of the state of stressin the joint Zhao [28] compared stress results predicted bythe closed-form expressions only with the 2D FEA resultsThe bond-length ratio for the biadhesive bondline was keptconstant in his study

2 Analytical Evaluation of Stress Componentsfor Biadhesive Joint

The biadhesive bondline can be modeled as three individ-ual regions according to their shear modulus components(Figure 1) The stiff adhesive was located in the middle andflexible adhesive at the ends of the overlap Two adherendswith the thicknesses of 120575

1and 120575

2were bonded with an

adhesive layer with a thickness of 1205753 where 120575

4= 120575

3 The

regions I and III at the overlap ends are the left and rightflexible adhesive regions respectively The region II at thecentre of the overlap is the stiff adhesive region The twoends of the adherends are simply supported and right endis subjected to an axial load 119865 (As seen in Figure 1 the 119909-axispasses through the midplane of the adhesive layer)

In Figure 1 119897119891is the flexible adhesive length and 119897

119904is the

stiff adhesive length where 119897119891= (119897 minus 119904) and 119897

119904= 2119904 The upper

and the lower adherends are denoted by the subscripts 1 and 2respectivelyTheflexible and stiff adhesives are denoted by thesubscripts 3 and 4 respectivelyThen 119864

119894and119866

119894(119894 = 1 2 3 4)

are the modulus of elasticity and shear modulus of the fourindividual components The total overlap length is 2ℓ Thejoint width perpendicular to the (119909 119910)-plane is 119887119881

0and119872

0

are the shear force and bendingmoment acting on the ends ofthe upper and lower adherends respectively (Figure 1(b)) Adifferential section 119889119909 can be cut out from the overlap regionof the biadhesive joint as shown in Figure 2

In Figure 2 1205903119910

is the peel stress at the upper and lowerinterfaces of the adhesive 119865

119894119909 119881119894and 119872

119894(119894 = 1 2) are

the tensile forces shear forces and bending moments relatedto the upper and lower adherends respectively

The distributions of the longitudinal shear stresses anddisplacements of adherends 1 and 2 are illustrated in Figure 3A linear shear stress and strain distributions through thethickness of the adherends is assumed Two local coordinatesystems 119910

1and 119910

2 were introduced where 119910

1and 119910

2are

the distances from the top of the upper and lower adherendsrespectively Free surface stress conditions are considered

The detailed derivation of the stress components forthe bi-adhesive joint along the bondline was given in theAppendix For the three regions (I II and III) shown inFigure 1 the three sets of expressions are given as follows

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) + 119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(1a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(1b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+ 119861

11cosh (120596

1119909) sin (120596

1119909)

+ 119861

12cosh (120596

1119909) cos (120596

1119909)

(1c)

The abbreviations used in the expressions above weregiven in theAppendixThemoment effect is introduced in theformulation (see the Appendix)The superscripts I II and IIIdenote the relevant stress components of the three individualregions (see Figure 1) 120590

1119909is the longitudinal normal stress

of the adherends 120590119894119910and 120591119894119909(119894 = 3 4) are the peel and shear

stresses along the adhesive midplane Set of (1a) and (1c) arerelated to the right and left flexible adhesive regions Set of(1b) is related to the stiff adhesive region the central regionNormal shear and peel stresses can then be evaluated by

4 Mathematical Problems in Engineering

1

L

L

E1G1

ll

s s

E3G3 E4G4

y

xE2G2 2 F

12057511205752

b

Overlap widthcross-section

1205753 1205754I II III

(a)

E1G1

l l

y

x

E2G2

F

F

1205751

1205752

12057531205754

M0M0

V0

V0

lf lfls

E3G3 E4G4 o

(b)

Figure 1 Biadhesive single lap joint under a tensile load (a) geometric and material parameters (b) force equilibrium free-body diagram

dx

M1 M1 + dM1

F1xF1x + dF1x

V1 V1 + dV1

1205911x

1205913xbdx

1205913xbdx

1205913xbdx

1205913xbdx

1205903ybdx

1205903ybdx

1205903ybdx

1205903ybdx

1205912x

M2 + dM2

V2 + dV2

F2x + dF2x

M2

F2x

V2

Figure 2 Free-body force equilibrium diagram

using the appropriate expressions for each regionThe resultsof analytical solutions were compared with both the results of2D and 3D FEM analyses

3 Finite Element Model

Geometry dimensions and boundary conditions of thebiadhesive single-lap joint are shown in Figure 4

1205911x = 0

1205751

1205911x = 1205913x

y1

u1t

u1

u1b

(a)

1205752

1205912x = 1205913x

1205912x = 0

y2

u2t

u2

u2b

(b)

Figure 3 Longitudinal shear stress and displacement distributionsthrough the thickness of the adherends (a) upper adherend (b)lower adherend

Finite Element Analysis based on the surface to surfacecontact model was performed by using commercial FEAsoftwareANSYSThe adherendswere constrained at the endspreventing excessive deflection of the joint and simulating thegripping in the testing machine The effect of the grippinglength on the stress components was studied to validate thestress analysis results Convergence tests were carried outby using the gripping lengths of 10 20 and 30mm It wasseen that the gripping length does not have a considerableeffect on the stress results [29 see page 118] Finally it wasconcluded that the 20mm optimum length was enough andsufficient to prevent excessive deflection and allow the free

Mathematical Problems in Engineering 5

b =25

mm

1205750 = 15mm

1205750 = 15mm

875mm 875mmlf lfls

lt = 125mm

Flexible adhesiveStiff adhesive

1205753 1205754 = 025mm

(a)

x

y

F

L = 675mm

(b)

Fz

(c)

Figure 4 Biadhesive single-lap joint (a) geometry and dimensions (b) 2D and 3D FEM model boundary conditions (front view) (c) 3DFEMmodel boundary conditions (top view)

Table 1 Material properties of adherends and adhesives

Adherend(Aluminium alloy 7075)

Flexible Adhesive(Terokal 5045)

Stiff Adhesive(Hysol EA 9313)

Modulus of elasticity (GPa)119864

1

= 119864

2

= 71700 119864

3

= 0437 119864

4

= 2274

Modulus of shear (GPa)119866

1

= 119866

2

= 26955 119866

3

= 0158 119866

4

= 0836

Poissonrsquos ratio 033 038 036Shear strength (MPa) 152 20 276Elongation at break () 10 113 8

rotation of the joint for the problem considered (Figure 4(b))Thus overlap regionwas not influenced by the boundary con-ditions Due to the longitudinal symmetry half-symmetryboundary condition was used in order to reduce the solutiontime for the three-dimensional analysis (Figure 4(c)) [30]

Aluminium alloy 7075 was used as identical adherendsHysol EA 9313 and Terokal 5045 epoxy adhesives producedby Henkel were used as stiff and flexible adhesives respec-tively Material properties of adherends and adhesives aregiven in Table 1 [31]

The thicknesses of the adherends and adhesives are15mm and 025mm respectively The total overlap length

119897

119905 2119897119891+ 119897

119904 was taken as 125mm where 119897

119891is the length of

the flexible adhesive and 119897119904is the length of the stiff adhesive

The bond-length ratios of the adhesive bondline varied as 120585 =119897

119891119897

119904= 02 04 07 13 Therefore four different biadhesive

bondline configurations were investigated A static force of36 kN was applied to the right side of the lower adherend(Loading was chosen as the same in our recent paper [31])The finite element models of the joint for the 2D and 3D FEMare shown in the same figure (Figure 5)

The finite element contact model recognizes possiblecontact pairs by the presence of specific contact elementsThe overlap surfaces of the adherends and the adhesives were

6 Mathematical Problems in Engineering

(b)

(a)

x

y

z

x

y

Figure 5 Finite element model of the joint (a) two-dimensional (b) three-dimensional

modelled using surface-to-surface contact for the 2D and 3Dfinite element models Surface-to-surface contact model usesGauss integration points as a default which generally providemore accurate results than the nodal detection scheme whichuses the nodes themselves as the integration pointsThis typeof contact model transmits contact pressure between Gausspoints and not force between nodes

In 2D FEA model contact pairs were created using con-tact element ldquoCONTA172rdquo and target element ldquoTARGE169rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA172 elements were generated on thebottom and upper surfaces of adhesive CONTA172 is a2D 3-node higher order parabolic element TARGE169elements were used on the overlap surfaces of the adherendsTARGE169 is used to represent various 2D ldquotargetrdquo surfacesfor the associated contact element The 2D model wasdiscretized using the 8-node plane stress elements (Plane 183)

In 3D FEA model contact pairs were created using con-tact element ldquoCONTA174rdquo and target element ldquoTARGE170rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA174 elements were generated on thebottom and upper surfaces of adhesive CONTA174 is a3D 8-node higher order quadrilateral element TARGE170elements were used on the overlap surfaces of the adherendsTARGE170 is used to represent various 3D ldquotargetrdquo surfacesfor the associated contact element (More detailed infor-mation on the 3D contact modelling of biadhesive joint isavailable in our recent paper [31]) Adherends and adhesiveswere meshed with solid hexahedral elements (solid95) Thiselement is defined by twenty nodes having three degrees offreedom per node Three elements were used through thethickness of the adherends and the bondline (Figure 5(b))

The mesh density affects the stress values in the adhesivelayer The size of the elements in the mesh was reduced untila stable stress value was achieved Five different meshingschemes were investigated to optimize the number and sizeof the elements The sizes of the elements were graded forthe finer mesh in the critical regions (at the edges of overlapand contact interfaces of adhesives) Finally the element sizenear the stress concentration and contact interface were set

Target element

Contact element

Adherend

Adhesive

Contact pair

Figure 6 Surface-to-surface contact pair

to 003mm and 004mm respectively which is fine enoughto describe the severe stress variation near the critical regionsThenumber of elements varied for FEMmodels ofmono- andbiadhesive joints However the mesh size was kept constantin all models This was important in order to compare theresults between the mono- and biadhesive joints and also tocompare results of biadhesive joints among themselves

The two-dimensional and three-dimensional finite ele-ment analyses were performed The results were used tocompare the analytically predicted adhesive stresses Wepresent the FEA results and their comparisons with theanalytical results in the next section

4 Results and Discussion

The adhesive shear and peel stress distributions in themidplane of the bondline were obtained both analyticallyand numericallyThis section presents a comparison betweenthe numerical and analytical results The distributions of thestress components were normalized with respect to averageshear stress (120591avg) and distributions of the same type of stresscomponents were given in the same figure In addition bar

Mathematical Problems in Engineering 7

Contact interfaces

Flexible adhesive Stiff adhesive Flexible adhesive

Midplane

Adherend

Adherend

Overlap edge Overlap edge

Figure 7 Overlap region of the biadhesive single-lap joint

graphs were used to illustrate a visual comparison among thestress results of analytical model 2D FEA and 3D FEA

Some comparisons among the peak stress values werecarried out on the basis of the percent error Percentage erroris defined as follows

Error =Numeric minus Analytic

Numerictimes 100

(2)

where word ldquoNumericrdquo denotes 2D and 3D FEA resultsPercent errors between analytical and numerical results forthe biadhesive case were tabulated in tables while the errordiscussions were presented in a text for monoadhesive case

In the bar graphs and tables peak values of the stress com-ponents obtained through analytical model were comparedwith those obtained through 2D FEA and 3D FEA Howeverthere are two peak stress locations for the biadhesive bond-line one just at the overlap edges and the other at contactinterfaces of adhesives Therefore in the case of biadhesivejoint two peak stresses were considered for the comparison(Figure 7) In 3D FEA shear and peel stresses peak at themidwidth of the adhesive midplane for both of mono- andbiadhesive bondlines Therefore stress distributions plottedat themidwidth along the adhesive midplane Peak values arealso related to the midwidth of the adhesive midplane in 3DFEA

Figure 8 shows the shear stress distributions of themono-and biadhesive joints obtained from analytical and numerical(FEA) solutions It can be seen from Figure 8 that the shearstress distributions in the monoadhesive joint is not uniformThis is because the applied load causes bending momentswhich lead to peel stresses especially at the overlap edgesHowever peel stress component is more dominant especiallyover the mono-stiff adhesive Therefore the shear stressdistribution for the mono-flexible adhesive is more uniformand there is a very small stress concentration at the overlapedges In the monoadhesive joint the higher level of stressregion exists at the overlap edges (Figure 8)

In the biadhesive bondline as can be seen in Figure 8higher shear stress occurs at the contact interfaces of theadhesives and the lower shear stress region exists at theoverlap edges Peak stress decreases at the overlap edges andincreases at the contact interfaces With the increase in the 120585ratio the values of peak shear stress are decreasing althoughthere is no great reduction in the peak shear stress for ratiohigher than 07

The shear stress distributions from the analytical solutionwere plotted in Figure 8(c)This shows a good agreementwithnumerical ones although some differences can be seen at theoverlap edges Peak shear stresses of the numerical solutionsoccur very near the edge of the overlap region while those ofthe analytical results were always at the overlap edges Thatis because the numerical model tries to model the zero stressstate at traction-free end surfaces However in practice thereis an adhesive spew fillet and the shear stress does not go tozero at the bondline ends

The normalized peak shear stress values at the overlapedges and the contact interfaces are showed in Figure 9 as avertical bar chart

From Figure 9 it can be seen that the normalized peakshear stress values for analytical and the numerical modelshad close correlation for the majority of mono- and biad-hesive joints The maximum shear stresses occurred at theoverlap edges for monoadhesive joints and at the contactinterfaces for biadhesive joints

According to Figure 9 the shear peak stress which occursat the overlap edges increased with increasing 120585 ratio Fur-thermore increasing rate in the shear peak stress at theedges was observed to be a parallel changing rate withthe decreasing rate in the shear peak stress at the contactinterfaces

Figures 9(a) and 9(b) show that analytical results are ingood agreement with 2D FEA results rather than 3D FEAresults for both the mono- and biadhesive joint casesThe 3DFEA solution gives slightly higher shear peak stress valuesthan those from the analytical and 2D FEA solutions

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

The analytical model was based on the plane elasticity andsome restrictive assumptions but the state of stress in thejoint has the three-dimensional nature Especially the adhe-sive peel stress is sensitive to the three-dimensional effectsTherefore the three-dimensional FEM model can providemore accurate predictions for comparing the results withoutintroducing any simplification at the modelling procedureIn this study the 3D FEM model is based on the surface-to-surface contact elements Aluminum adherends were bondedwith stiff and flexible adhesivesThe stiff adhesive was locatedin themiddle and the flexible adhesive was located at the endsof the overlap The overlap length the adhesive thicknessthe adherend thickness and modulus of stiff and flexibleadhesives were kept constant Both analytical and numericalanalyses were performed using four different bond-lengthratios for the biadhesive joint The effect of biadhesiveand monoadhesive bondlines on the peel stress (transversenormal stress) and shear stress distributions was investigatedThe numerical analyses were performed using the Ansysfinite element code The results of both FEM analyses andanalytical solutions were compared

It must be stated that this paper aims especially tocompare the analytical solutions with the 3D FEA solutionsand to show the three-dimensional nature of the state of stressin the joint Zhao [28] compared stress results predicted bythe closed-form expressions only with the 2D FEA resultsThe bond-length ratio for the biadhesive bondline was keptconstant in his study

2 Analytical Evaluation of Stress Componentsfor Biadhesive Joint

The biadhesive bondline can be modeled as three individ-ual regions according to their shear modulus components(Figure 1) The stiff adhesive was located in the middle andflexible adhesive at the ends of the overlap Two adherendswith the thicknesses of 120575

1and 120575

2were bonded with an

adhesive layer with a thickness of 1205753 where 120575

4= 120575

3 The

regions I and III at the overlap ends are the left and rightflexible adhesive regions respectively The region II at thecentre of the overlap is the stiff adhesive region The twoends of the adherends are simply supported and right endis subjected to an axial load 119865 (As seen in Figure 1 the 119909-axispasses through the midplane of the adhesive layer)

In Figure 1 119897119891is the flexible adhesive length and 119897

119904is the

stiff adhesive length where 119897119891= (119897 minus 119904) and 119897

119904= 2119904 The upper

and the lower adherends are denoted by the subscripts 1 and 2respectivelyTheflexible and stiff adhesives are denoted by thesubscripts 3 and 4 respectivelyThen 119864

119894and119866

119894(119894 = 1 2 3 4)

are the modulus of elasticity and shear modulus of the fourindividual components The total overlap length is 2ℓ Thejoint width perpendicular to the (119909 119910)-plane is 119887119881

0and119872

0

are the shear force and bendingmoment acting on the ends ofthe upper and lower adherends respectively (Figure 1(b)) Adifferential section 119889119909 can be cut out from the overlap regionof the biadhesive joint as shown in Figure 2

In Figure 2 1205903119910

is the peel stress at the upper and lowerinterfaces of the adhesive 119865

119894119909 119881119894and 119872

119894(119894 = 1 2) are

the tensile forces shear forces and bending moments relatedto the upper and lower adherends respectively

The distributions of the longitudinal shear stresses anddisplacements of adherends 1 and 2 are illustrated in Figure 3A linear shear stress and strain distributions through thethickness of the adherends is assumed Two local coordinatesystems 119910

1and 119910

2 were introduced where 119910

1and 119910

2are

the distances from the top of the upper and lower adherendsrespectively Free surface stress conditions are considered

The detailed derivation of the stress components forthe bi-adhesive joint along the bondline was given in theAppendix For the three regions (I II and III) shown inFigure 1 the three sets of expressions are given as follows

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) + 119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(1a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(1b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+ 119861

11cosh (120596

1119909) sin (120596

1119909)

+ 119861

12cosh (120596

1119909) cos (120596

1119909)

(1c)

The abbreviations used in the expressions above weregiven in theAppendixThemoment effect is introduced in theformulation (see the Appendix)The superscripts I II and IIIdenote the relevant stress components of the three individualregions (see Figure 1) 120590

1119909is the longitudinal normal stress

of the adherends 120590119894119910and 120591119894119909(119894 = 3 4) are the peel and shear

stresses along the adhesive midplane Set of (1a) and (1c) arerelated to the right and left flexible adhesive regions Set of(1b) is related to the stiff adhesive region the central regionNormal shear and peel stresses can then be evaluated by

4 Mathematical Problems in Engineering

1

L

L

E1G1

ll

s s

E3G3 E4G4

y

xE2G2 2 F

12057511205752

b

Overlap widthcross-section

1205753 1205754I II III

(a)

E1G1

l l

y

x

E2G2

F

F

1205751

1205752

12057531205754

M0M0

V0

V0

lf lfls

E3G3 E4G4 o

(b)

Figure 1 Biadhesive single lap joint under a tensile load (a) geometric and material parameters (b) force equilibrium free-body diagram

dx

M1 M1 + dM1

F1xF1x + dF1x

V1 V1 + dV1

1205911x

1205913xbdx

1205913xbdx

1205913xbdx

1205913xbdx

1205903ybdx

1205903ybdx

1205903ybdx

1205903ybdx

1205912x

M2 + dM2

V2 + dV2

F2x + dF2x

M2

F2x

V2

Figure 2 Free-body force equilibrium diagram

using the appropriate expressions for each regionThe resultsof analytical solutions were compared with both the results of2D and 3D FEM analyses

3 Finite Element Model

Geometry dimensions and boundary conditions of thebiadhesive single-lap joint are shown in Figure 4

1205911x = 0

1205751

1205911x = 1205913x

y1

u1t

u1

u1b

(a)

1205752

1205912x = 1205913x

1205912x = 0

y2

u2t

u2

u2b

(b)

Figure 3 Longitudinal shear stress and displacement distributionsthrough the thickness of the adherends (a) upper adherend (b)lower adherend

Finite Element Analysis based on the surface to surfacecontact model was performed by using commercial FEAsoftwareANSYSThe adherendswere constrained at the endspreventing excessive deflection of the joint and simulating thegripping in the testing machine The effect of the grippinglength on the stress components was studied to validate thestress analysis results Convergence tests were carried outby using the gripping lengths of 10 20 and 30mm It wasseen that the gripping length does not have a considerableeffect on the stress results [29 see page 118] Finally it wasconcluded that the 20mm optimum length was enough andsufficient to prevent excessive deflection and allow the free

Mathematical Problems in Engineering 5

b =25

mm

1205750 = 15mm

1205750 = 15mm

875mm 875mmlf lfls

lt = 125mm

Flexible adhesiveStiff adhesive

1205753 1205754 = 025mm

(a)

x

y

F

L = 675mm

(b)

Fz

(c)

Figure 4 Biadhesive single-lap joint (a) geometry and dimensions (b) 2D and 3D FEM model boundary conditions (front view) (c) 3DFEMmodel boundary conditions (top view)

Table 1 Material properties of adherends and adhesives

Adherend(Aluminium alloy 7075)

Flexible Adhesive(Terokal 5045)

Stiff Adhesive(Hysol EA 9313)

Modulus of elasticity (GPa)119864

1

= 119864

2

= 71700 119864

3

= 0437 119864

4

= 2274

Modulus of shear (GPa)119866

1

= 119866

2

= 26955 119866

3

= 0158 119866

4

= 0836

Poissonrsquos ratio 033 038 036Shear strength (MPa) 152 20 276Elongation at break () 10 113 8

rotation of the joint for the problem considered (Figure 4(b))Thus overlap regionwas not influenced by the boundary con-ditions Due to the longitudinal symmetry half-symmetryboundary condition was used in order to reduce the solutiontime for the three-dimensional analysis (Figure 4(c)) [30]

Aluminium alloy 7075 was used as identical adherendsHysol EA 9313 and Terokal 5045 epoxy adhesives producedby Henkel were used as stiff and flexible adhesives respec-tively Material properties of adherends and adhesives aregiven in Table 1 [31]

The thicknesses of the adherends and adhesives are15mm and 025mm respectively The total overlap length

119897

119905 2119897119891+ 119897

119904 was taken as 125mm where 119897

119891is the length of

the flexible adhesive and 119897119904is the length of the stiff adhesive

The bond-length ratios of the adhesive bondline varied as 120585 =119897

119891119897

119904= 02 04 07 13 Therefore four different biadhesive

bondline configurations were investigated A static force of36 kN was applied to the right side of the lower adherend(Loading was chosen as the same in our recent paper [31])The finite element models of the joint for the 2D and 3D FEMare shown in the same figure (Figure 5)

The finite element contact model recognizes possiblecontact pairs by the presence of specific contact elementsThe overlap surfaces of the adherends and the adhesives were

6 Mathematical Problems in Engineering

(b)

(a)

x

y

z

x

y

Figure 5 Finite element model of the joint (a) two-dimensional (b) three-dimensional

modelled using surface-to-surface contact for the 2D and 3Dfinite element models Surface-to-surface contact model usesGauss integration points as a default which generally providemore accurate results than the nodal detection scheme whichuses the nodes themselves as the integration pointsThis typeof contact model transmits contact pressure between Gausspoints and not force between nodes

In 2D FEA model contact pairs were created using con-tact element ldquoCONTA172rdquo and target element ldquoTARGE169rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA172 elements were generated on thebottom and upper surfaces of adhesive CONTA172 is a2D 3-node higher order parabolic element TARGE169elements were used on the overlap surfaces of the adherendsTARGE169 is used to represent various 2D ldquotargetrdquo surfacesfor the associated contact element The 2D model wasdiscretized using the 8-node plane stress elements (Plane 183)

In 3D FEA model contact pairs were created using con-tact element ldquoCONTA174rdquo and target element ldquoTARGE170rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA174 elements were generated on thebottom and upper surfaces of adhesive CONTA174 is a3D 8-node higher order quadrilateral element TARGE170elements were used on the overlap surfaces of the adherendsTARGE170 is used to represent various 3D ldquotargetrdquo surfacesfor the associated contact element (More detailed infor-mation on the 3D contact modelling of biadhesive joint isavailable in our recent paper [31]) Adherends and adhesiveswere meshed with solid hexahedral elements (solid95) Thiselement is defined by twenty nodes having three degrees offreedom per node Three elements were used through thethickness of the adherends and the bondline (Figure 5(b))

The mesh density affects the stress values in the adhesivelayer The size of the elements in the mesh was reduced untila stable stress value was achieved Five different meshingschemes were investigated to optimize the number and sizeof the elements The sizes of the elements were graded forthe finer mesh in the critical regions (at the edges of overlapand contact interfaces of adhesives) Finally the element sizenear the stress concentration and contact interface were set

Target element

Contact element

Adherend

Adhesive

Contact pair

Figure 6 Surface-to-surface contact pair

to 003mm and 004mm respectively which is fine enoughto describe the severe stress variation near the critical regionsThenumber of elements varied for FEMmodels ofmono- andbiadhesive joints However the mesh size was kept constantin all models This was important in order to compare theresults between the mono- and biadhesive joints and also tocompare results of biadhesive joints among themselves

The two-dimensional and three-dimensional finite ele-ment analyses were performed The results were used tocompare the analytically predicted adhesive stresses Wepresent the FEA results and their comparisons with theanalytical results in the next section

4 Results and Discussion

The adhesive shear and peel stress distributions in themidplane of the bondline were obtained both analyticallyand numericallyThis section presents a comparison betweenthe numerical and analytical results The distributions of thestress components were normalized with respect to averageshear stress (120591avg) and distributions of the same type of stresscomponents were given in the same figure In addition bar

Mathematical Problems in Engineering 7

Contact interfaces

Flexible adhesive Stiff adhesive Flexible adhesive

Midplane

Adherend

Adherend

Overlap edge Overlap edge

Figure 7 Overlap region of the biadhesive single-lap joint

graphs were used to illustrate a visual comparison among thestress results of analytical model 2D FEA and 3D FEA

Some comparisons among the peak stress values werecarried out on the basis of the percent error Percentage erroris defined as follows

Error =Numeric minus Analytic

Numerictimes 100

(2)

where word ldquoNumericrdquo denotes 2D and 3D FEA resultsPercent errors between analytical and numerical results forthe biadhesive case were tabulated in tables while the errordiscussions were presented in a text for monoadhesive case

In the bar graphs and tables peak values of the stress com-ponents obtained through analytical model were comparedwith those obtained through 2D FEA and 3D FEA Howeverthere are two peak stress locations for the biadhesive bond-line one just at the overlap edges and the other at contactinterfaces of adhesives Therefore in the case of biadhesivejoint two peak stresses were considered for the comparison(Figure 7) In 3D FEA shear and peel stresses peak at themidwidth of the adhesive midplane for both of mono- andbiadhesive bondlines Therefore stress distributions plottedat themidwidth along the adhesive midplane Peak values arealso related to the midwidth of the adhesive midplane in 3DFEA

Figure 8 shows the shear stress distributions of themono-and biadhesive joints obtained from analytical and numerical(FEA) solutions It can be seen from Figure 8 that the shearstress distributions in the monoadhesive joint is not uniformThis is because the applied load causes bending momentswhich lead to peel stresses especially at the overlap edgesHowever peel stress component is more dominant especiallyover the mono-stiff adhesive Therefore the shear stressdistribution for the mono-flexible adhesive is more uniformand there is a very small stress concentration at the overlapedges In the monoadhesive joint the higher level of stressregion exists at the overlap edges (Figure 8)

In the biadhesive bondline as can be seen in Figure 8higher shear stress occurs at the contact interfaces of theadhesives and the lower shear stress region exists at theoverlap edges Peak stress decreases at the overlap edges andincreases at the contact interfaces With the increase in the 120585ratio the values of peak shear stress are decreasing althoughthere is no great reduction in the peak shear stress for ratiohigher than 07

The shear stress distributions from the analytical solutionwere plotted in Figure 8(c)This shows a good agreementwithnumerical ones although some differences can be seen at theoverlap edges Peak shear stresses of the numerical solutionsoccur very near the edge of the overlap region while those ofthe analytical results were always at the overlap edges Thatis because the numerical model tries to model the zero stressstate at traction-free end surfaces However in practice thereis an adhesive spew fillet and the shear stress does not go tozero at the bondline ends

The normalized peak shear stress values at the overlapedges and the contact interfaces are showed in Figure 9 as avertical bar chart

From Figure 9 it can be seen that the normalized peakshear stress values for analytical and the numerical modelshad close correlation for the majority of mono- and biad-hesive joints The maximum shear stresses occurred at theoverlap edges for monoadhesive joints and at the contactinterfaces for biadhesive joints

According to Figure 9 the shear peak stress which occursat the overlap edges increased with increasing 120585 ratio Fur-thermore increasing rate in the shear peak stress at theedges was observed to be a parallel changing rate withthe decreasing rate in the shear peak stress at the contactinterfaces

Figures 9(a) and 9(b) show that analytical results are ingood agreement with 2D FEA results rather than 3D FEAresults for both the mono- and biadhesive joint casesThe 3DFEA solution gives slightly higher shear peak stress valuesthan those from the analytical and 2D FEA solutions

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

1

L

L

E1G1

ll

s s

E3G3 E4G4

y

xE2G2 2 F

12057511205752

b

Overlap widthcross-section

1205753 1205754I II III

(a)

E1G1

l l

y

x

E2G2

F

F

1205751

1205752

12057531205754

M0M0

V0

V0

lf lfls

E3G3 E4G4 o

(b)

Figure 1 Biadhesive single lap joint under a tensile load (a) geometric and material parameters (b) force equilibrium free-body diagram

dx

M1 M1 + dM1

F1xF1x + dF1x

V1 V1 + dV1

1205911x

1205913xbdx

1205913xbdx

1205913xbdx

1205913xbdx

1205903ybdx

1205903ybdx

1205903ybdx

1205903ybdx

1205912x

M2 + dM2

V2 + dV2

F2x + dF2x

M2

F2x

V2

Figure 2 Free-body force equilibrium diagram

using the appropriate expressions for each regionThe resultsof analytical solutions were compared with both the results of2D and 3D FEM analyses

3 Finite Element Model

Geometry dimensions and boundary conditions of thebiadhesive single-lap joint are shown in Figure 4

1205911x = 0

1205751

1205911x = 1205913x

y1

u1t

u1

u1b

(a)

1205752

1205912x = 1205913x

1205912x = 0

y2

u2t

u2

u2b

(b)

Figure 3 Longitudinal shear stress and displacement distributionsthrough the thickness of the adherends (a) upper adherend (b)lower adherend

Finite Element Analysis based on the surface to surfacecontact model was performed by using commercial FEAsoftwareANSYSThe adherendswere constrained at the endspreventing excessive deflection of the joint and simulating thegripping in the testing machine The effect of the grippinglength on the stress components was studied to validate thestress analysis results Convergence tests were carried outby using the gripping lengths of 10 20 and 30mm It wasseen that the gripping length does not have a considerableeffect on the stress results [29 see page 118] Finally it wasconcluded that the 20mm optimum length was enough andsufficient to prevent excessive deflection and allow the free

Mathematical Problems in Engineering 5

b =25

mm

1205750 = 15mm

1205750 = 15mm

875mm 875mmlf lfls

lt = 125mm

Flexible adhesiveStiff adhesive

1205753 1205754 = 025mm

(a)

x

y

F

L = 675mm

(b)

Fz

(c)

Figure 4 Biadhesive single-lap joint (a) geometry and dimensions (b) 2D and 3D FEM model boundary conditions (front view) (c) 3DFEMmodel boundary conditions (top view)

Table 1 Material properties of adherends and adhesives

Adherend(Aluminium alloy 7075)

Flexible Adhesive(Terokal 5045)

Stiff Adhesive(Hysol EA 9313)

Modulus of elasticity (GPa)119864

1

= 119864

2

= 71700 119864

3

= 0437 119864

4

= 2274

Modulus of shear (GPa)119866

1

= 119866

2

= 26955 119866

3

= 0158 119866

4

= 0836

Poissonrsquos ratio 033 038 036Shear strength (MPa) 152 20 276Elongation at break () 10 113 8

rotation of the joint for the problem considered (Figure 4(b))Thus overlap regionwas not influenced by the boundary con-ditions Due to the longitudinal symmetry half-symmetryboundary condition was used in order to reduce the solutiontime for the three-dimensional analysis (Figure 4(c)) [30]

Aluminium alloy 7075 was used as identical adherendsHysol EA 9313 and Terokal 5045 epoxy adhesives producedby Henkel were used as stiff and flexible adhesives respec-tively Material properties of adherends and adhesives aregiven in Table 1 [31]

The thicknesses of the adherends and adhesives are15mm and 025mm respectively The total overlap length

119897

119905 2119897119891+ 119897

119904 was taken as 125mm where 119897

119891is the length of

the flexible adhesive and 119897119904is the length of the stiff adhesive

The bond-length ratios of the adhesive bondline varied as 120585 =119897

119891119897

119904= 02 04 07 13 Therefore four different biadhesive

bondline configurations were investigated A static force of36 kN was applied to the right side of the lower adherend(Loading was chosen as the same in our recent paper [31])The finite element models of the joint for the 2D and 3D FEMare shown in the same figure (Figure 5)

The finite element contact model recognizes possiblecontact pairs by the presence of specific contact elementsThe overlap surfaces of the adherends and the adhesives were

6 Mathematical Problems in Engineering

(b)

(a)

x

y

z

x

y

Figure 5 Finite element model of the joint (a) two-dimensional (b) three-dimensional

modelled using surface-to-surface contact for the 2D and 3Dfinite element models Surface-to-surface contact model usesGauss integration points as a default which generally providemore accurate results than the nodal detection scheme whichuses the nodes themselves as the integration pointsThis typeof contact model transmits contact pressure between Gausspoints and not force between nodes

In 2D FEA model contact pairs were created using con-tact element ldquoCONTA172rdquo and target element ldquoTARGE169rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA172 elements were generated on thebottom and upper surfaces of adhesive CONTA172 is a2D 3-node higher order parabolic element TARGE169elements were used on the overlap surfaces of the adherendsTARGE169 is used to represent various 2D ldquotargetrdquo surfacesfor the associated contact element The 2D model wasdiscretized using the 8-node plane stress elements (Plane 183)

In 3D FEA model contact pairs were created using con-tact element ldquoCONTA174rdquo and target element ldquoTARGE170rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA174 elements were generated on thebottom and upper surfaces of adhesive CONTA174 is a3D 8-node higher order quadrilateral element TARGE170elements were used on the overlap surfaces of the adherendsTARGE170 is used to represent various 3D ldquotargetrdquo surfacesfor the associated contact element (More detailed infor-mation on the 3D contact modelling of biadhesive joint isavailable in our recent paper [31]) Adherends and adhesiveswere meshed with solid hexahedral elements (solid95) Thiselement is defined by twenty nodes having three degrees offreedom per node Three elements were used through thethickness of the adherends and the bondline (Figure 5(b))

The mesh density affects the stress values in the adhesivelayer The size of the elements in the mesh was reduced untila stable stress value was achieved Five different meshingschemes were investigated to optimize the number and sizeof the elements The sizes of the elements were graded forthe finer mesh in the critical regions (at the edges of overlapand contact interfaces of adhesives) Finally the element sizenear the stress concentration and contact interface were set

Target element

Contact element

Adherend

Adhesive

Contact pair

Figure 6 Surface-to-surface contact pair

to 003mm and 004mm respectively which is fine enoughto describe the severe stress variation near the critical regionsThenumber of elements varied for FEMmodels ofmono- andbiadhesive joints However the mesh size was kept constantin all models This was important in order to compare theresults between the mono- and biadhesive joints and also tocompare results of biadhesive joints among themselves

The two-dimensional and three-dimensional finite ele-ment analyses were performed The results were used tocompare the analytically predicted adhesive stresses Wepresent the FEA results and their comparisons with theanalytical results in the next section

4 Results and Discussion

The adhesive shear and peel stress distributions in themidplane of the bondline were obtained both analyticallyand numericallyThis section presents a comparison betweenthe numerical and analytical results The distributions of thestress components were normalized with respect to averageshear stress (120591avg) and distributions of the same type of stresscomponents were given in the same figure In addition bar

Mathematical Problems in Engineering 7

Contact interfaces

Flexible adhesive Stiff adhesive Flexible adhesive

Midplane

Adherend

Adherend

Overlap edge Overlap edge

Figure 7 Overlap region of the biadhesive single-lap joint

graphs were used to illustrate a visual comparison among thestress results of analytical model 2D FEA and 3D FEA

Some comparisons among the peak stress values werecarried out on the basis of the percent error Percentage erroris defined as follows

Error =Numeric minus Analytic

Numerictimes 100

(2)

where word ldquoNumericrdquo denotes 2D and 3D FEA resultsPercent errors between analytical and numerical results forthe biadhesive case were tabulated in tables while the errordiscussions were presented in a text for monoadhesive case

In the bar graphs and tables peak values of the stress com-ponents obtained through analytical model were comparedwith those obtained through 2D FEA and 3D FEA Howeverthere are two peak stress locations for the biadhesive bond-line one just at the overlap edges and the other at contactinterfaces of adhesives Therefore in the case of biadhesivejoint two peak stresses were considered for the comparison(Figure 7) In 3D FEA shear and peel stresses peak at themidwidth of the adhesive midplane for both of mono- andbiadhesive bondlines Therefore stress distributions plottedat themidwidth along the adhesive midplane Peak values arealso related to the midwidth of the adhesive midplane in 3DFEA

Figure 8 shows the shear stress distributions of themono-and biadhesive joints obtained from analytical and numerical(FEA) solutions It can be seen from Figure 8 that the shearstress distributions in the monoadhesive joint is not uniformThis is because the applied load causes bending momentswhich lead to peel stresses especially at the overlap edgesHowever peel stress component is more dominant especiallyover the mono-stiff adhesive Therefore the shear stressdistribution for the mono-flexible adhesive is more uniformand there is a very small stress concentration at the overlapedges In the monoadhesive joint the higher level of stressregion exists at the overlap edges (Figure 8)

In the biadhesive bondline as can be seen in Figure 8higher shear stress occurs at the contact interfaces of theadhesives and the lower shear stress region exists at theoverlap edges Peak stress decreases at the overlap edges andincreases at the contact interfaces With the increase in the 120585ratio the values of peak shear stress are decreasing althoughthere is no great reduction in the peak shear stress for ratiohigher than 07

The shear stress distributions from the analytical solutionwere plotted in Figure 8(c)This shows a good agreementwithnumerical ones although some differences can be seen at theoverlap edges Peak shear stresses of the numerical solutionsoccur very near the edge of the overlap region while those ofthe analytical results were always at the overlap edges Thatis because the numerical model tries to model the zero stressstate at traction-free end surfaces However in practice thereis an adhesive spew fillet and the shear stress does not go tozero at the bondline ends

The normalized peak shear stress values at the overlapedges and the contact interfaces are showed in Figure 9 as avertical bar chart

From Figure 9 it can be seen that the normalized peakshear stress values for analytical and the numerical modelshad close correlation for the majority of mono- and biad-hesive joints The maximum shear stresses occurred at theoverlap edges for monoadhesive joints and at the contactinterfaces for biadhesive joints

According to Figure 9 the shear peak stress which occursat the overlap edges increased with increasing 120585 ratio Fur-thermore increasing rate in the shear peak stress at theedges was observed to be a parallel changing rate withthe decreasing rate in the shear peak stress at the contactinterfaces

Figures 9(a) and 9(b) show that analytical results are ingood agreement with 2D FEA results rather than 3D FEAresults for both the mono- and biadhesive joint casesThe 3DFEA solution gives slightly higher shear peak stress valuesthan those from the analytical and 2D FEA solutions

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

b =25

mm

1205750 = 15mm

1205750 = 15mm

875mm 875mmlf lfls

lt = 125mm

Flexible adhesiveStiff adhesive

1205753 1205754 = 025mm

(a)

x

y

F

L = 675mm

(b)

Fz

(c)

Figure 4 Biadhesive single-lap joint (a) geometry and dimensions (b) 2D and 3D FEM model boundary conditions (front view) (c) 3DFEMmodel boundary conditions (top view)

Table 1 Material properties of adherends and adhesives

Adherend(Aluminium alloy 7075)

Flexible Adhesive(Terokal 5045)

Stiff Adhesive(Hysol EA 9313)

Modulus of elasticity (GPa)119864

1

= 119864

2

= 71700 119864

3

= 0437 119864

4

= 2274

Modulus of shear (GPa)119866

1

= 119866

2

= 26955 119866

3

= 0158 119866

4

= 0836

Poissonrsquos ratio 033 038 036Shear strength (MPa) 152 20 276Elongation at break () 10 113 8

rotation of the joint for the problem considered (Figure 4(b))Thus overlap regionwas not influenced by the boundary con-ditions Due to the longitudinal symmetry half-symmetryboundary condition was used in order to reduce the solutiontime for the three-dimensional analysis (Figure 4(c)) [30]

Aluminium alloy 7075 was used as identical adherendsHysol EA 9313 and Terokal 5045 epoxy adhesives producedby Henkel were used as stiff and flexible adhesives respec-tively Material properties of adherends and adhesives aregiven in Table 1 [31]

The thicknesses of the adherends and adhesives are15mm and 025mm respectively The total overlap length

119897

119905 2119897119891+ 119897

119904 was taken as 125mm where 119897

119891is the length of

the flexible adhesive and 119897119904is the length of the stiff adhesive

The bond-length ratios of the adhesive bondline varied as 120585 =119897

119891119897

119904= 02 04 07 13 Therefore four different biadhesive

bondline configurations were investigated A static force of36 kN was applied to the right side of the lower adherend(Loading was chosen as the same in our recent paper [31])The finite element models of the joint for the 2D and 3D FEMare shown in the same figure (Figure 5)

The finite element contact model recognizes possiblecontact pairs by the presence of specific contact elementsThe overlap surfaces of the adherends and the adhesives were

6 Mathematical Problems in Engineering

(b)

(a)

x

y

z

x

y

Figure 5 Finite element model of the joint (a) two-dimensional (b) three-dimensional

modelled using surface-to-surface contact for the 2D and 3Dfinite element models Surface-to-surface contact model usesGauss integration points as a default which generally providemore accurate results than the nodal detection scheme whichuses the nodes themselves as the integration pointsThis typeof contact model transmits contact pressure between Gausspoints and not force between nodes

In 2D FEA model contact pairs were created using con-tact element ldquoCONTA172rdquo and target element ldquoTARGE169rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA172 elements were generated on thebottom and upper surfaces of adhesive CONTA172 is a2D 3-node higher order parabolic element TARGE169elements were used on the overlap surfaces of the adherendsTARGE169 is used to represent various 2D ldquotargetrdquo surfacesfor the associated contact element The 2D model wasdiscretized using the 8-node plane stress elements (Plane 183)

In 3D FEA model contact pairs were created using con-tact element ldquoCONTA174rdquo and target element ldquoTARGE170rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA174 elements were generated on thebottom and upper surfaces of adhesive CONTA174 is a3D 8-node higher order quadrilateral element TARGE170elements were used on the overlap surfaces of the adherendsTARGE170 is used to represent various 3D ldquotargetrdquo surfacesfor the associated contact element (More detailed infor-mation on the 3D contact modelling of biadhesive joint isavailable in our recent paper [31]) Adherends and adhesiveswere meshed with solid hexahedral elements (solid95) Thiselement is defined by twenty nodes having three degrees offreedom per node Three elements were used through thethickness of the adherends and the bondline (Figure 5(b))

The mesh density affects the stress values in the adhesivelayer The size of the elements in the mesh was reduced untila stable stress value was achieved Five different meshingschemes were investigated to optimize the number and sizeof the elements The sizes of the elements were graded forthe finer mesh in the critical regions (at the edges of overlapand contact interfaces of adhesives) Finally the element sizenear the stress concentration and contact interface were set

Target element

Contact element

Adherend

Adhesive

Contact pair

Figure 6 Surface-to-surface contact pair

to 003mm and 004mm respectively which is fine enoughto describe the severe stress variation near the critical regionsThenumber of elements varied for FEMmodels ofmono- andbiadhesive joints However the mesh size was kept constantin all models This was important in order to compare theresults between the mono- and biadhesive joints and also tocompare results of biadhesive joints among themselves

The two-dimensional and three-dimensional finite ele-ment analyses were performed The results were used tocompare the analytically predicted adhesive stresses Wepresent the FEA results and their comparisons with theanalytical results in the next section

4 Results and Discussion

The adhesive shear and peel stress distributions in themidplane of the bondline were obtained both analyticallyand numericallyThis section presents a comparison betweenthe numerical and analytical results The distributions of thestress components were normalized with respect to averageshear stress (120591avg) and distributions of the same type of stresscomponents were given in the same figure In addition bar

Mathematical Problems in Engineering 7

Contact interfaces

Flexible adhesive Stiff adhesive Flexible adhesive

Midplane

Adherend

Adherend

Overlap edge Overlap edge

Figure 7 Overlap region of the biadhesive single-lap joint

graphs were used to illustrate a visual comparison among thestress results of analytical model 2D FEA and 3D FEA

Some comparisons among the peak stress values werecarried out on the basis of the percent error Percentage erroris defined as follows

Error =Numeric minus Analytic

Numerictimes 100

(2)

where word ldquoNumericrdquo denotes 2D and 3D FEA resultsPercent errors between analytical and numerical results forthe biadhesive case were tabulated in tables while the errordiscussions were presented in a text for monoadhesive case

In the bar graphs and tables peak values of the stress com-ponents obtained through analytical model were comparedwith those obtained through 2D FEA and 3D FEA Howeverthere are two peak stress locations for the biadhesive bond-line one just at the overlap edges and the other at contactinterfaces of adhesives Therefore in the case of biadhesivejoint two peak stresses were considered for the comparison(Figure 7) In 3D FEA shear and peel stresses peak at themidwidth of the adhesive midplane for both of mono- andbiadhesive bondlines Therefore stress distributions plottedat themidwidth along the adhesive midplane Peak values arealso related to the midwidth of the adhesive midplane in 3DFEA

Figure 8 shows the shear stress distributions of themono-and biadhesive joints obtained from analytical and numerical(FEA) solutions It can be seen from Figure 8 that the shearstress distributions in the monoadhesive joint is not uniformThis is because the applied load causes bending momentswhich lead to peel stresses especially at the overlap edgesHowever peel stress component is more dominant especiallyover the mono-stiff adhesive Therefore the shear stressdistribution for the mono-flexible adhesive is more uniformand there is a very small stress concentration at the overlapedges In the monoadhesive joint the higher level of stressregion exists at the overlap edges (Figure 8)

In the biadhesive bondline as can be seen in Figure 8higher shear stress occurs at the contact interfaces of theadhesives and the lower shear stress region exists at theoverlap edges Peak stress decreases at the overlap edges andincreases at the contact interfaces With the increase in the 120585ratio the values of peak shear stress are decreasing althoughthere is no great reduction in the peak shear stress for ratiohigher than 07

The shear stress distributions from the analytical solutionwere plotted in Figure 8(c)This shows a good agreementwithnumerical ones although some differences can be seen at theoverlap edges Peak shear stresses of the numerical solutionsoccur very near the edge of the overlap region while those ofthe analytical results were always at the overlap edges Thatis because the numerical model tries to model the zero stressstate at traction-free end surfaces However in practice thereis an adhesive spew fillet and the shear stress does not go tozero at the bondline ends

The normalized peak shear stress values at the overlapedges and the contact interfaces are showed in Figure 9 as avertical bar chart

From Figure 9 it can be seen that the normalized peakshear stress values for analytical and the numerical modelshad close correlation for the majority of mono- and biad-hesive joints The maximum shear stresses occurred at theoverlap edges for monoadhesive joints and at the contactinterfaces for biadhesive joints

According to Figure 9 the shear peak stress which occursat the overlap edges increased with increasing 120585 ratio Fur-thermore increasing rate in the shear peak stress at theedges was observed to be a parallel changing rate withthe decreasing rate in the shear peak stress at the contactinterfaces

Figures 9(a) and 9(b) show that analytical results are ingood agreement with 2D FEA results rather than 3D FEAresults for both the mono- and biadhesive joint casesThe 3DFEA solution gives slightly higher shear peak stress valuesthan those from the analytical and 2D FEA solutions

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

(b)

(a)

x

y

z

x

y

Figure 5 Finite element model of the joint (a) two-dimensional (b) three-dimensional

modelled using surface-to-surface contact for the 2D and 3Dfinite element models Surface-to-surface contact model usesGauss integration points as a default which generally providemore accurate results than the nodal detection scheme whichuses the nodes themselves as the integration pointsThis typeof contact model transmits contact pressure between Gausspoints and not force between nodes

In 2D FEA model contact pairs were created using con-tact element ldquoCONTA172rdquo and target element ldquoTARGE169rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA172 elements were generated on thebottom and upper surfaces of adhesive CONTA172 is a2D 3-node higher order parabolic element TARGE169elements were used on the overlap surfaces of the adherendsTARGE169 is used to represent various 2D ldquotargetrdquo surfacesfor the associated contact element The 2D model wasdiscretized using the 8-node plane stress elements (Plane 183)

In 3D FEA model contact pairs were created using con-tact element ldquoCONTA174rdquo and target element ldquoTARGE170rdquobetween the adherends and adhesive overlap surfaces(Figure 6) CONTA174 elements were generated on thebottom and upper surfaces of adhesive CONTA174 is a3D 8-node higher order quadrilateral element TARGE170elements were used on the overlap surfaces of the adherendsTARGE170 is used to represent various 3D ldquotargetrdquo surfacesfor the associated contact element (More detailed infor-mation on the 3D contact modelling of biadhesive joint isavailable in our recent paper [31]) Adherends and adhesiveswere meshed with solid hexahedral elements (solid95) Thiselement is defined by twenty nodes having three degrees offreedom per node Three elements were used through thethickness of the adherends and the bondline (Figure 5(b))

The mesh density affects the stress values in the adhesivelayer The size of the elements in the mesh was reduced untila stable stress value was achieved Five different meshingschemes were investigated to optimize the number and sizeof the elements The sizes of the elements were graded forthe finer mesh in the critical regions (at the edges of overlapand contact interfaces of adhesives) Finally the element sizenear the stress concentration and contact interface were set

Target element

Contact element

Adherend

Adhesive

Contact pair

Figure 6 Surface-to-surface contact pair

to 003mm and 004mm respectively which is fine enoughto describe the severe stress variation near the critical regionsThenumber of elements varied for FEMmodels ofmono- andbiadhesive joints However the mesh size was kept constantin all models This was important in order to compare theresults between the mono- and biadhesive joints and also tocompare results of biadhesive joints among themselves

The two-dimensional and three-dimensional finite ele-ment analyses were performed The results were used tocompare the analytically predicted adhesive stresses Wepresent the FEA results and their comparisons with theanalytical results in the next section

4 Results and Discussion

The adhesive shear and peel stress distributions in themidplane of the bondline were obtained both analyticallyand numericallyThis section presents a comparison betweenthe numerical and analytical results The distributions of thestress components were normalized with respect to averageshear stress (120591avg) and distributions of the same type of stresscomponents were given in the same figure In addition bar

Mathematical Problems in Engineering 7

Contact interfaces

Flexible adhesive Stiff adhesive Flexible adhesive

Midplane

Adherend

Adherend

Overlap edge Overlap edge

Figure 7 Overlap region of the biadhesive single-lap joint

graphs were used to illustrate a visual comparison among thestress results of analytical model 2D FEA and 3D FEA

Some comparisons among the peak stress values werecarried out on the basis of the percent error Percentage erroris defined as follows

Error =Numeric minus Analytic

Numerictimes 100

(2)

where word ldquoNumericrdquo denotes 2D and 3D FEA resultsPercent errors between analytical and numerical results forthe biadhesive case were tabulated in tables while the errordiscussions were presented in a text for monoadhesive case

In the bar graphs and tables peak values of the stress com-ponents obtained through analytical model were comparedwith those obtained through 2D FEA and 3D FEA Howeverthere are two peak stress locations for the biadhesive bond-line one just at the overlap edges and the other at contactinterfaces of adhesives Therefore in the case of biadhesivejoint two peak stresses were considered for the comparison(Figure 7) In 3D FEA shear and peel stresses peak at themidwidth of the adhesive midplane for both of mono- andbiadhesive bondlines Therefore stress distributions plottedat themidwidth along the adhesive midplane Peak values arealso related to the midwidth of the adhesive midplane in 3DFEA

Figure 8 shows the shear stress distributions of themono-and biadhesive joints obtained from analytical and numerical(FEA) solutions It can be seen from Figure 8 that the shearstress distributions in the monoadhesive joint is not uniformThis is because the applied load causes bending momentswhich lead to peel stresses especially at the overlap edgesHowever peel stress component is more dominant especiallyover the mono-stiff adhesive Therefore the shear stressdistribution for the mono-flexible adhesive is more uniformand there is a very small stress concentration at the overlapedges In the monoadhesive joint the higher level of stressregion exists at the overlap edges (Figure 8)

In the biadhesive bondline as can be seen in Figure 8higher shear stress occurs at the contact interfaces of theadhesives and the lower shear stress region exists at theoverlap edges Peak stress decreases at the overlap edges andincreases at the contact interfaces With the increase in the 120585ratio the values of peak shear stress are decreasing althoughthere is no great reduction in the peak shear stress for ratiohigher than 07

The shear stress distributions from the analytical solutionwere plotted in Figure 8(c)This shows a good agreementwithnumerical ones although some differences can be seen at theoverlap edges Peak shear stresses of the numerical solutionsoccur very near the edge of the overlap region while those ofthe analytical results were always at the overlap edges Thatis because the numerical model tries to model the zero stressstate at traction-free end surfaces However in practice thereis an adhesive spew fillet and the shear stress does not go tozero at the bondline ends

The normalized peak shear stress values at the overlapedges and the contact interfaces are showed in Figure 9 as avertical bar chart

From Figure 9 it can be seen that the normalized peakshear stress values for analytical and the numerical modelshad close correlation for the majority of mono- and biad-hesive joints The maximum shear stresses occurred at theoverlap edges for monoadhesive joints and at the contactinterfaces for biadhesive joints

According to Figure 9 the shear peak stress which occursat the overlap edges increased with increasing 120585 ratio Fur-thermore increasing rate in the shear peak stress at theedges was observed to be a parallel changing rate withthe decreasing rate in the shear peak stress at the contactinterfaces

Figures 9(a) and 9(b) show that analytical results are ingood agreement with 2D FEA results rather than 3D FEAresults for both the mono- and biadhesive joint casesThe 3DFEA solution gives slightly higher shear peak stress valuesthan those from the analytical and 2D FEA solutions

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Contact interfaces

Flexible adhesive Stiff adhesive Flexible adhesive

Midplane

Adherend

Adherend

Overlap edge Overlap edge

Figure 7 Overlap region of the biadhesive single-lap joint

graphs were used to illustrate a visual comparison among thestress results of analytical model 2D FEA and 3D FEA

Some comparisons among the peak stress values werecarried out on the basis of the percent error Percentage erroris defined as follows

Error =Numeric minus Analytic

Numerictimes 100

(2)

where word ldquoNumericrdquo denotes 2D and 3D FEA resultsPercent errors between analytical and numerical results forthe biadhesive case were tabulated in tables while the errordiscussions were presented in a text for monoadhesive case

In the bar graphs and tables peak values of the stress com-ponents obtained through analytical model were comparedwith those obtained through 2D FEA and 3D FEA Howeverthere are two peak stress locations for the biadhesive bond-line one just at the overlap edges and the other at contactinterfaces of adhesives Therefore in the case of biadhesivejoint two peak stresses were considered for the comparison(Figure 7) In 3D FEA shear and peel stresses peak at themidwidth of the adhesive midplane for both of mono- andbiadhesive bondlines Therefore stress distributions plottedat themidwidth along the adhesive midplane Peak values arealso related to the midwidth of the adhesive midplane in 3DFEA

Figure 8 shows the shear stress distributions of themono-and biadhesive joints obtained from analytical and numerical(FEA) solutions It can be seen from Figure 8 that the shearstress distributions in the monoadhesive joint is not uniformThis is because the applied load causes bending momentswhich lead to peel stresses especially at the overlap edgesHowever peel stress component is more dominant especiallyover the mono-stiff adhesive Therefore the shear stressdistribution for the mono-flexible adhesive is more uniformand there is a very small stress concentration at the overlapedges In the monoadhesive joint the higher level of stressregion exists at the overlap edges (Figure 8)

In the biadhesive bondline as can be seen in Figure 8higher shear stress occurs at the contact interfaces of theadhesives and the lower shear stress region exists at theoverlap edges Peak stress decreases at the overlap edges andincreases at the contact interfaces With the increase in the 120585ratio the values of peak shear stress are decreasing althoughthere is no great reduction in the peak shear stress for ratiohigher than 07

The shear stress distributions from the analytical solutionwere plotted in Figure 8(c)This shows a good agreementwithnumerical ones although some differences can be seen at theoverlap edges Peak shear stresses of the numerical solutionsoccur very near the edge of the overlap region while those ofthe analytical results were always at the overlap edges Thatis because the numerical model tries to model the zero stressstate at traction-free end surfaces However in practice thereis an adhesive spew fillet and the shear stress does not go tozero at the bondline ends

The normalized peak shear stress values at the overlapedges and the contact interfaces are showed in Figure 9 as avertical bar chart

From Figure 9 it can be seen that the normalized peakshear stress values for analytical and the numerical modelshad close correlation for the majority of mono- and biad-hesive joints The maximum shear stresses occurred at theoverlap edges for monoadhesive joints and at the contactinterfaces for biadhesive joints

According to Figure 9 the shear peak stress which occursat the overlap edges increased with increasing 120585 ratio Fur-thermore increasing rate in the shear peak stress at theedges was observed to be a parallel changing rate withthe decreasing rate in the shear peak stress at the contactinterfaces

Figures 9(a) and 9(b) show that analytical results are ingood agreement with 2D FEA results rather than 3D FEAresults for both the mono- and biadhesive joint casesThe 3DFEA solution gives slightly higher shear peak stress valuesthan those from the analytical and 2D FEA solutions

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(a)

0 2 4 6

005

115

225

335

minus6 minus4 minus2minus1

minus05

Bondline length x (mm)

120591 xy120591

av

(b)

0 2 4 60

051

152

253

35

Stiff adhesiveFlexible adhesive

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

minus6 minus4 minus2

120591 xy120591

av

(c)

Figure 8 (a) Normalized shear stress distributions obtained from 2D FEA (b) Normalized shear stress distributions at midwidth obtainedfrom 3D FEA (c) Normalized shear stress distributions obtained from analytical model

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(a)

005

115

225

335

2D FEA3D FEA

Analytic

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

120591 xy120591

av

(b)

Figure 9 (a) Normalized peak shear stress values at the overlap edges (b) Normalized peak shear stress values at the contact interfaces

As stated before percent error comparisons for monoad-hesive joint will be given in a text After doing some calcu-lations by data used to plot Figure 9 it was concluded thatfor mono-stiff adhesive joint the analytical model predictshigher peak shear stress than the 2D FEA with an error of atmost 20 while it predicts a lower peak shear stress than 3D

FEA with an error of at most 380 However for the mono-flexible adhesive bondline analytical model predicts lowerpeak shear stress than both the 2D FEA and 3D FEA witherrors of 140 and 594 respectively For more detailedcomparison about the biadhesive joint percent errors werecalculated by using (2) and tabulated in Table 2

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

19195

values at the contact interfaces

2

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(a)

0 2 4 6

012345

2205

21215

2222523

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

minus6 minus4 minus2

minus1

Bondline length x (mm)

120590y120591

av

(b)

0 2 4 6

012345

Normalized peak peel stress

Normalized peak peel stressvalues at the overlap interfaces

values at the contact interfaces

218

21621421221208206

Stiff adhesiveFlexible adhesive

minus6 minus4 minus2

minus1

Bondline length x (mm)

120585 = 13

120585 = 07120585 = 04120585 = 02

120590y120591

av

(c)

Figure 10 (a) Normalized peel stress distributions obtained from 2D FEA (b) Normalized peel stress distributions at the midwidth obtainedfrom 3D FEA (c) Normalized peel stress distributions obtained from analytical model

Table 2 Percent error between analytical and numerical results forthe peak shear stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus214 393120585 = 04 minus362 302120585 = 07 minus411 206120585 = 13 minus257 119

3-D FEA120585 = 02 398 662120585 = 04 4 579120585 = 07 484 508120585 = 13 732 460

Peak shear stress values were given at the adhesive contactinterface and overlap edge for the four bond-length ratiosAs can be seen from Table 2 the analytical results agreewell with the numerical ones It can be calculated form thedata in Table 2 that at the overlap edges peak shear stressespredicted by the analytical model agree well with the 2D FEAresults with a maximum error of about 393 compared tothat of 662 for the 3D FEA results The analytical results

have a high level of agreement with the 2D FEA results at thecontact interfaces of adhesives The maximum percent erroris minus 411 If 3D FEA results were considered there wasa lower level of agreement between analytical and 3D FEAresults with a maximum error of 732

Figure 10 shows the peel stress distributions of the mono-and biadhesive joints obtained from analytical and numericalsolutions It is clear from Figure 10 that the peel stress peaksat the overlap edges for both mono- and biadhesive jointsThe biadhesive bondline gives lower peak peel stresses at theoverlap edges However the peak peel stress in the flexiblepart of the biadhesive bondline is approximately withinthe same order of magnitude for both mono- flexible andbiadhesive joints

From Figure 10 it must be noted that at the overlapedges changing the 120585 ratio has a moderating effect on thevalues of the peel peak stresses The peel peak stress in thebiadhesive bondline was lower than that of the mono-stiffadhesive bondline At the contact interfaces of adhesives itcan be seen fromFigure 10 that the presence of a little amountof flexible adhesive can cause a secondary peak in the peelstress distribution in the biadhesive joint

Figure 11 shows a comparison as a vertical bar chart of thenormalized peel peak stresses obtained from analytical andnumerical solutions

It can be seen from Figure 11 that the analytical andnumerical peel peak stresses have close correlation for

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

005

115

225

335

445

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

120590y120591

av

(a)

0

12

3

4

5

Stiff

adhe

sive

Flex

ible

adhe

sive

120585=13

120585=07

120585=04

120585=02

2D FEA3D FEA

Analytic

minus1

120590y120591

av

(b)

Figure 11 (a) Normalized peak peel stress values at the overlap edges (b) Normalized peak peel stress values at the contact interfaces

the majority of mono- and biadhesive joints especially atthe overlap edges The peak values occurred at the overlapedges for both the mono- and biadhesive joints Figure 11(a)shows that at the mono- and biadhesive overlap edgesthe analytical results are in a good agreement with the 3DFEA results For mono-stiff adhesive joint the analyticalmodel predicts higher peel peak stress than both the 2DFEA and 3D FEA results with an error of minus1362 andminus148 respectivelyHowever for themono-flexible adhesivebondline analytical model predicts higher peel peak stressthan the 2D FEA with an error of minus855 while it predictslower peel peak stress than 3D FEA results with an error of440 Apparent difference about the results for the mono-stiff adhesive joint is that analytical results show a higherpeak peel stress compared to the peak values of the 3D FEAresults (Figure 11(a))The largest difference in the peak valuesoccurred at the contact interfaces of adhesives when the120585 ratio is set to 02 This difference is not very importantbecause in order to design an adhesively bonded joint thevalues of the maximum peak stresses are more importantthan the exact stress distribution [32]

For more detailed comparison about the biadhesive jointpeak peel stress values were tabulated in Table 3 Table 3compares the analytical results with numerical results for peelpeak stresses of biadhesive joint

As seen in Table 3 a good correlation was found betweenthe analytical and numerical results at the overlap edges Themaximum error between the analytical and 2D FEA resultsis minus912 However if 3D FEA results are considered thereis a higher level of agreement between analytical and 3DFEA results with a maximum error of 7 The maximumerror in the peak values occurred at the contact interfacesof adhesives The magnitude of the error was maximum forthe bond-length ratio of 02 For this ratio analytical modelpredicts higher peel peak stress than both the 2D FEA withan error of minus11146 and 3-D FEA results with an error ofminus40886

Table 3 Percent error between analytical and numerical results forthe peel peak stresses

Percent error ()At the contact interfaces At the overlap edges

2-D FEA120585 = 02 minus11146 minus803120585 = 04 1141 minus912120585 = 07 minus1057 minus818120585 = 13 minus2457 minus794

3-D FEA120585 = 02 minus40886 7120585 = 04 1087 516120585 = 07 minus828 578120585 = 13 minus4885 598

5 Conclusions

Some expressions were derived related to stress componentsfor the single lap joint with biadhesive bondlines by followingthe same steps as Zhaorsquos solutions However Zhao comparedstress results predicted by the closed-form expressions onlywith the 2D FEA results Moreover the bond-length ratio forthe biadhesive layer stayed constant in his study

In this study the analytical formulations were presentedin a detailed format (including further details) and givenin the Appendix A MAPLE program was written PreparedMAPLE program employs these expressions to calculatestress components Some figures were plotted using the datacalculated with MAPLE The closed-form solution accountsthe bending effect It is capable of determining the distribu-tion of the stress components only in the middle of bondline(It must be noted that stress expressions are only a function of119909 As seen in Figure 4 the119909-axis passes through themidplaneof the adhesive layer)

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

The validity of the analytical results was assessed by com-paring the analytical results with the 2D and 3D FEA resultsEspecially the adhesive peel stress is sensitive to the three-di-mensional effects Therefore the three-dimensional FEMmodel can provide more accurate predictions for comparingthe results In this study the 2D and 3D FEM models werebased on the surface-to-surface contact elements Analyseswere performed for four different bond-length ratios both an-alytically and numerically In addition a comparison of peakstress values was carried out on the basis of percentage error

The following conclusions can bemade about the analysisresults

(1) The analytical model gives similar stress distributionsto those of numerical models for all the stress compo-nents

(2) The peak stress comparison shows that analyticalresults are generally in a good agreement with thenumerical ones The analytical predictions are morecompatible with 3D FEA predictions for the peel peakstresses in the case of the biadhesive bondline Asstated above the adhesive peel stress is sensitive tothe three-dimensional effectsThus as to its peel peakstress predictions it can be said that the analyticalmodel has somewhat more ability to model the 3Deffects than 2D FEM model However for the shearpeak stresses the analytical model gives closer resultsto those of the 2D (plane stress) FEA These com-parisons are valid for both the mono- and biadhesivejoints Zhao [28] reported in his paper that the peakshear stresses at the overlap edges predicted by 2DFEAwere lower than those predicted by the analyticalsolution Our study offers opposite case This can beattributed to the differences in FEMmodels andmeshconcentrations In our FEM model the surface-to-surface contact model was used In addition therewas also a difference in the predicted peak locationsZhao concluded that shear peak stresses of the 2DFEA results occurred at about one bondline thicknessaway from the ends while those of the analyticalresults were always at the ends of the overlap regionHowever in this study peak shear stress occurs at apoint closer to the ends approximately 70 of thebondline thickness

(3) The analytical and numerical results show that appro-priate 120585 ratio must be used to effectively reduce theconcentration of peel and shear stresses in the joint

(4) Choosing improper bond-length ratio using insuf-ficient amount of flexible adhesive in the biadhesivebondline such as 120585 = 02 in our model caused asecondary peak in the peel stress distribution becauseof the change of peel stresses from compressive totensile Therefore selecting the appropriate bond-length ratio is crucial for biadhesive joints

(5) The results show a measurable decrease in the stresscomponents of the biadhesive joints compared withthose in whichmonoadhesives were used over the fulllength of the bondline

Appendix

The detailed procedure for deriving stress componentsexpressions for the biadhesive single-lap joint was presentedbelow and given by Zhao [28] Firstly we begin with thefree-body diagram of the flexible adhesive layer given inFigure 2 By referring to Figure 2 the differential equationsof equilibrium equations can be obtained as

119889119865

1119909

119889119909

+ 120591

3119909119887 = 0

119889119865

2119909

119889119909

minus 120591

3119909119887 = 0

(A1)

119889119881

1

119889119909

minus 120590

3119910119887 = 0

119889119881

2

119889119909

+ 120590

3119910119887 = 0

(A2)

119889119872

1

119889119909

minus 119881

1minus 120591

3119909119887

120575

1

2

= 0

119889119872

2

119889119909

minus 119881

2minus 120591

3119909119887

120575

2

2

= 0

(A3)

Based upon the assumptions (see Figure 3) one can write thefollowing stress-strain relationships

120590

1119909= 119864

1120576

1119909= 119864

1

119889119906

1

119889119909

120590

2119909= 119864

2120576

2119909= 119864

2

119889119906

2

119889119909

120590

3119910= 119864

3120576

3119910

(A4a)

120574

1119909=

120591

1119909

119866

1

120574

2119909=

120591

2119909

119866

2

120574

3119909=

120591

3119909

119866

3

(A4b)

From the equilibrium of 119909 forces in Figure 2 we have

119865 = 119865

1119909+ 119865

2119909 (A5)

The stress distributions through the thickness of adherendsare assumed to behave linearly (Figure 3) Then the shearstresses acting on upper and lower adherends can beexpressed as

120591

1119909= 120591

3119909

119910

1

120575

1

0 le 119910

1le 120575

1 (A6a)

120591

2119909= 120591

3119909(1 minus

119910

2

120575

2

) 0 le 119910

2le 120575

2 (A6b)

where 1199101and 119910

2are the distances from the top of adherend 1

and adherend 2 respectively Substituting (A6a) and (A6b)into (A4b) we obtain the shear strains of adherends asfollows

120574

1119909=

120591

3119909

120575

1

119910

1

119866

1

120574

2119909=

120591

3119909

119866

1

(1 minus

119910

2

120575

2

) (A7)

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Longitudinal displacements of adherend 1 and 2 due to thelongitudinal forces are given as follows

119906

119879

1

(119909 119910

1) = 119906

119879

1119905

+ int

1199101

0

120574

1119909119889119910

1= 119906

119879

1119905

+

120591

3119909119910

2

1

2119866

1120575

1

(A8a)

119906

119879

2

(119909 119910

2) = 119906

119879

2119905

+ int

1199102

0

120574

2119909119889119910

2= 119906

119879

2119905

+

120591

3119909

119866

2

(119910

2minus

119910

2

2

2120575

2

)

(A8b)

where subscripts 1119905 and 2119905 correspond to the top surfacesof adherend 1 and adherend 2 respectively Superscript 119879was used for emphasizing the longitudinal force effectswhich serves to distinguish it from the moment effects Themoment effect was also introduced in the later steps offormulation As an example we can try to write an expressionfor the displacement at the interface between adherend 1 andadhesive by using (A8a) as follows

119906

119879

1119887

= 119906

119879

1

(119910

1= 120575

1) = 119906

119879

1119905

+

120591

3119909120575

1

2119866

1

(A9)

where subscript 1119887 corresponds to the top surface ofadherend 1 Using (A9) we can rewrite (A8a) as

119906

119879

1

(119909 119910

1) = 119906

119879

1119887

minus

120591

3119909120575

1

2119866

1

+

120591

3119909119910

2

1

2119866

1120575

1

(A10)

Taking (A8a) and (A8b) into account longitudinal forcecomponents can be written as

119865

1119909= int

1205751

0

120590

119879

1119909

(119910

1) 119887119889119910

1

= 119864

1int

1205751

0

119889119906

119879

1

119889119909

(119910

1) 119887119889119910

1

= 119864

1119887120575

1[

119889119906

119879

1119887

119889119909

minus

120575

1

3119866

1

119889120591

3119909

119889119909

]

(A11a)

119865

2119909= int

1205752

0

120590

119879

2119909

(119910

2) 119887119889119910

2

= 119864

2int

1205752

0

119889119906

119879

2

119889119909

(119910

2) 119887119889119910

2

= 119864

2119887120575

2[

119889119906

119879

2119905

119889119909

+

120575

2

3119866

2

119889120591

3119909

119889119909

]

(A11b)

From Figure 3 shear strain in the low modulus adhesive is

120574

3119909=

119906

2119905minus 119906

1119887

120575

3

(A12)

Substituting (A12) into (A4b) and then differentiating it withrespect to 119909 we obtain

120575

3

119866

3

119889120591

3119909

119889119909

= [

119889119906

2119905

119889119909

minus

119889119906

1119887

119889119909

] (A13)

To get the displacements induced by the bending momentwe can rearrange 119906

1119887and 119906

2119905displacements in terms of the

tensile and moment effects in the following form

119906

1119887= 119906

119879

1119887

minus 119906

119872

1119887

119906

2119905= 119906

119879

2119905

+ 119906

119872

2119905

(A14)

Longitudinal strains may now be written in terms of bendingmoments by using classical beam theory as

119889119906

119872

1

119889119909

=

119872

1120575

1

119864

1119868

12

=

6119872

1

119864

1119887120575

2

1

119889119906

119872

2

119889119909

=

119872

2120575

2

119864

2119868

22

=

6119872

2

119864

2119887120575

2

2

(A15)

Differentiating (A14) with respect to119909 and substituting it and(A15) into (A13) we have

120575

3

119866

3

120591

3119909

119889119909

=

119889119906

119879

2119905

119889119909

minus

119889119906

119879

1119887

119889119909

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A16)

Substituting (A15) into (A11a) and (A11b) and after simpli-fying the expression we get

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889120591

3119909

119889119909

=

119865

2119909

119864

2120575

2119887

minus

119865

1119909

119864

1119887120575

1

+

6119872

1

119864

1119887120575

2

1

+

6119872

2

119864

2119887120575

2

2

(A17)

Differentiating (A17) with respect to 119909 and combining thisexpression with (A1) and (A3) we obtain

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

120591

3119909

119889119909

2

minus 4(

1

119864

2120575

2

+

1

119864

1120575

1

) 120591

3119909

=

6119881

1

119864

1119887120575

2

1

+

6119881

2

119864

2119887120575

2

2

(A18)

Rearranging (A18) with (A1) (A5) and (A17) yields

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

2

119865

1119909

119889119909

2

minus (

1

119864

2120575

2

+

1

119864

1120575

1

)119865

1119909

+

6119872

1

119864

1120575

2

1

+

6119872

2

119864

2120575

2

2

+

119865

119864

2120575

2

= 0

(A19)

Differentiating (A19) twice with respect to 119909 and recalling(A1) (A2) and (A3) and rearranging give us

[

120575

3

119866

3

+

120575

1

3119866

1

+

120575

2

3119866

2

]

119889

4

119865

1119909

119889119909

2

minus 4(

1

119864

1120575

1

+

1

119864

2120575

2

)

119889

2

119865

1119909

119889119909

2

+ 6119887(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)120590

3119910= 0

(A20)

Curvatures of adherends 1 and 2 are

119889

2]1

119889119909

2

= minus

119872

1

119864

1119868

1

119889

2]2

119889119909

2

= minus

119872

2

119864

2119868

2

(A21)

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

where V1

and V2

are the transverse displacements ofadherends 119868

1and 1198682are the second moments of areas for

adherends 1 and 2 and defined as 1198681= 119887120575

3

1

12 and 1198682=

119887120575

3

2

12 The interfacial peel stress-strain relationship may bewritten in terms of the transverse deflections of adherends as

120590

3119910

119864

3

= 120576

3119910=

]1minus ]2

120575

3

997904rArr

120575

3

119864

3

120590

3119910= ]1minus ]2 (A22)

Differentiating the second expression of (A22) with respectto 119909 yields

120575

3

119864

3

119889120590

3119910

119889119909

=

119889]1

119889119909

minus

119889]2

119889119909

= 120579

1minus 120579

2

(A23)

where 1205791and 1205792are the relevant slopes of adherends Differ-

entiating (A23) with respect to 119909 and introducing (A21) leadto

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

=

12119872

2

119864

2119887120575

3

2

minus

12119872

1

119864

1119887120575

3

1

(A24)

Differentiating (A24) with respect to 119909 and rearranging yield

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

119889119872

2

119889119909

12

119864

2119887120575

3

2

minus

119889119872

1

119889119909

12

119864

1119887120575

3

1

(A25)

Substituting (A3) into (A25) we obtain

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

=

12119881

2

119864

2119887120575

3

2

minus

12119881

1

119864

1119887120575

3

1

+ 6120591

3119909(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A26)

Differentiating (A26) with respect to 119909 and using (A1) and(A2) give us

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

=

119889119881

2

119889119909

12

119864

2119887120575

3

2

minus

119889119881

1

119889119909

12

119864

1119887120575

3

1

+ 6

119889120591

3119909

119889119909

(

1

119864

2120575

2

2

minus

1

119864

1120575

2

1

)

(A27)

Substituting (A1) and (A2) into (A27) we have

120575

3

119864

3

119889

4

120590

3119910

119889119909

4

+ (

12

119864

2120575

3

2

+

12

119864

1120575

3

1

)120590

3119910

minus

6

119887

(

1

119864

1120575

2

1

minus

1

119864

2120575

2

2

)

119889

2

119865

1119909

119889119909

= 0

(A28)

When the same geometric characteristics and material prop-erties are used for the adherends that is identical adherendsthat allows the straightforward stress analysis of the biadhe-sive joint That is

119866

1= 119866

2= 119866 119864

1= 119864

2= 119864 120575

1= 120575

2= 120575

0 (A29)

Rearranging (A20) gives us the below differential equation

[

120575

3

119866

3

+

2120575

0

3119866

]

119889

4

119865

1119909

119889119909

2

minus

8

119864120575

0

119889

2

119865

1119909

119889119909

2

= 0(A30)

Rearranging this equation gives

119889

4

119865

1119909

119889119909

4

minus 120582

2

1

119889

2

119865

1119909

119889119909

4

= 0

(A31)

where 12058221

is

120582

2

1

=

4120572

120578 + 1205733

(A32)

Here 120572 = 21198641205750 120578 = 120575

3119866

3 and 120573 = 2120575

0119866 By treating

the stiff adhesive layer as the flexible adhesive layer then 12058222

becomes

120582

2

2

=

4120572

120578

1015840

+ 1205733

(A33)

where 120572 = 21198641205750 120578 = 120575

3119866

4and 120573 = 2120575

0119866 Rearranging

(A28) by considering (A29) leads to differential equationgiven below

119889

4

120590

3119910

119889119909

4

+ 4120596

4

1

120590

3119910= 0

(A34)

where 12059641

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

3 By

treating the stiff adhesive layer as the flexible adhesive layerthen1205964

1

becomes12059642

= 3120601120594 Here 120601 = 211986412057530

and 120594 = 1205753119864

4

The governing equations (A31) and (A34) are the forth-orderordinary differential equationswith constant coefficientsThegeneral solution of (A31) is

119865

1119909= 119860

1015840 sinh (1205821119909) + 119861

1015840 cosh (1205821119909) + 119862

1015840

119909 + 119863

1015840

(A35a)

where 1198601015840 1198611015840 1198621015840 and 1198631015840 are constants of integration Thenthe longitudinal normal stress can be written as

120590

1119909=

119865

1119909

119887120575

0

= 119860

10158401015840 sinh (1205821119909) + 119861

10158401015840 cosh (1205821119909) + 119862

10158401015840

119909 + 119863

10158401015840

(A35b)

where 11986010158401015840 11986110158401015840 11986210158401015840 and 11986310158401015840 are also constants of integrationThe general solution of (A34) is

120590

3119910= 119860 sinh (120596

1119909) sin (120596

1119909) + 119861 cosh (120596

1119909) sin (120596

1119909)

+ 119862 sinh (1205961119909) cos (120596

1119909) + 119863 cosh (120596

1119909) cos (120596

1119909)

(A35c)

Using the first expression of (A1) and the derivative expres-sion of (A35a) with respect to 119909 the shear stress at theinterface then becomes

120591

3119909= minus120575

1(119860

10158401015840

120582

1cosh (119909) + 11986110158401015840120582

1sinh (119909) + 11986210158401015840) (A35d)

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

Rearranging (A35b) (A35c) and (A35d) for the threeregions (I II and III) shown in Figure 1 we have the threesets of expressions below

Region I (minus119897 le 119909 le minus119904)

120590

120484

1119909

= 119860

1sinh (120582

1119909) + 119860

2cosh (120582

1119909) + 119860

3119909 + 119860

4

120591

120484

3119909

= minus120575

0(119860

1120582

1cosh (120582

1119909) + 119860

2120582

1sinh (120582

1119909) sinh+119860

3)

120590

120484

3119910

= 119861

1sinh (120596

1119909) sin (120596

1119909) + 119861

2sinh (120596

1119909) cos (120596

1119909)

+ 119861

3cosh (120596

1119909) sin (120596

1119909) + 119861

4cosh (120596

1119909) cos (120596

1119909)

(A36a)

Region II (minus119904 le 119909 le 119904)

120590

120484120484

1119909

= 119860

5sinh (120582

2119909) + 119860

6cosh (120582

2119909) + 119860

7119909 + 119860

8

120591

120484120484

4119909

= minus120575

0(119860

5120582

2cosh (120582

2119909) + 119860

6120582

2sinh (120582

2119909) + 119860

7)

120590

120484120484

4119910

= 119861

5sinh (120596

2119909) sin (120596

2119909) + 119861

6sinh (120596

2119909) cos (120596

2119909)

+ 119861

7cosh (120596

2119909) sin (120596

2119909) + 119861

8cosh (120596

2119909) cos (120596

2119909)

(A36b)

Region III (119904 le 119909 le 119897)

120590

120484120484120484

1119909

= 119860

9sinh (120582

1119909) + 119860

10cosh (120582

1119909) + 119860

11119909 + 119860

12

120591

120484120484120484

3119909

= minus120575

0(119860

9120582

1cosh (120582

1119909) + 119860

10120582

1sinh (120582

1119909) + 119860

11)

120590

120484120484120484

3119910

= 119861

9sinh (120596

1119909) sin (120596

1119909) + 119861

10sinh (120596

1119909) cos (120596

1119909)

+119861

11csch (120596

1119909) sin (120596

1119909)+119861

12cosh (120596

1119909) cos (120596

1119909)

(A36c)

Set of (A36a) and (A36c) is related to the right and leftflexible adhesive region Set of (A36b) is related to thestiff adhesive region the central region Normal shear andpeel stresses are can be evaluated by using the appropriateexpressions for each region

Boundary Conditions (BCs) Continuity conditions of shearstrain normal strain and slope at the interface betweenregions I and II lead to

120591

120484

3119909

119866

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120591

120484120484

4119909

119866

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37a)

120590

120484

3119910

119864

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120590

120484120484

4119910

119864

4

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37b)

120575

3

119864

3

119889120590

120484

3119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889120590

120484120484

4119910

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A37c)

Thenormal stress boundary conditions at the interfaces of theadhesives are

120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

= 120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119904

(A37d)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

= 120590

120484120484

1119909

1003816

1003816

1003816

1003816119909=minus119904

(A37e)

By using (A17) (A24) and (A26) the boundary conditionsat the interface between regions I and II may be written as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

120484

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

= (

120575

3

119866

4

+

2120575

0

3119866

)

119889120591

120484120484

4119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38a)

120575

3

119864

3

119889

2

120590

120484

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

2

120590

120484120484

4119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38b)

120575

3

119864

3

119889

3

120590

120484

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

=

120575

3

119864

4

119889

3

120590

120484120484

4119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119904

(A38c)

The boundary conditions at 119909 = plusmn 119897 for the momentsshear forces and longitudinal forces are

119872

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119872

0

119881

1

1003816

1003816

1003816

1003816119909=minus119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

= 119865

119872

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119881

2

1003816

1003816

1003816

1003816119909=minus119897

= 0

119865

2119909

1003816

1003816

1003816

1003816119909=minus119897

= 0

119872

1

1003816

1003816

1003816

1003816119909=119897

= 0

119881

1

1003816

1003816

1003816

1003816119909=119897

= 0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 0

119872

2

1003816

1003816

1003816

1003816119909=119897

= 119872

0

119881

2

1003816

1003816

1003816

1003816119909=119897

= minus119881

0

119865

1119909

1003816

1003816

1003816

1003816119909=119897

= 119865

(A39)

Referring to Figure 1(b) and considering (A39) the bound-ary conditions at119909 = minus119897 can bewritten by using (A17) (A18)(A24) and (A25) as

(

120575

3

119866

3

+

2120575

0

3119866

)

119889120591

3119909

119889119909

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

= minus(

119865

119864119887120575

0

+

6119872

0

119864119887120575

2

0

) (A40a)

(

120575

3

119866

3

+

2120575

0

3119866

)

119889

2

120591

3119909

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

minus

8

119864120575

0

120591

3119909

1003816

1003816

1003816

1003816119909=minus119897

= minus

6119881

0

119864119887120575

2

0

(A40b)

120575

3

119864

3

119889

2

120590

3119910

119889119909

2

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119872

0

119864119887120575

3

0

(A40c)

120575

3

119864

3

119889

3

120590

3119910

119889119909

3

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=minus119897

=

12119881

0

119864119887120575

3

0

(A40d)

The other two boundary conditions are

120590

120484120484120484

1119909

1003816

1003816

1003816

1003816119909=119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=119897

119887120575

0

= 0(A41a)

120590

120484

1119909

1003816

1003816

1003816

1003816119909=minus119897

=

119865

1119909

1003816

1003816

1003816

1003816119909=minus119897

119887120575

1

=

119865

119887120575

0

(A41b)

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

In (A36a) (A36b) and (A36c) there are a total of 24unknown coefficients corresponding to the three regionsHowever we constructed only 14 BCs above Ten boundaryconditions are needed for the problem at hand We wrote theboundary conditions above for the left group of flexible-stiffadhesive layer (regions I and II) From 119910-axis symmetry wecan again write additional new boundary conditions for theright group of stiff-flexible adhesive layer (regions II and III)Then the 24 unknown coefficients can be obtained from theseBCs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to reviewers for their valuablecomments on this paper

References

[1] R D Adams andN A Peppiatt ldquoEffect of poissonrsquos ratio strainsin adherends on stresses of an idealized lap jointrdquoThe Journal ofStrain Analysis for Engineering Design vol 8 no 2 pp 134ndash1391973

[2] L J Hart-Smith ldquoAdhesive-bonded single-lap jointsrdquo NASAReport CR-112236 NASA Washington DC USA 1973

[3] R D Adams and N A Peppiatt ldquoStress analysis of adhesive-bonded lap jointsrdquoThe Journal of Strain Analysis for EngineeringDesign vol 9 no 3 pp 185ndash196 1974

[4] D J Allman ldquoA theory for elastic stresses in adhesive bondedlap jointsrdquo The Quarterly Journal of Mechanics amp AppliedMathematics vol 30 no 4 pp 415ndash436 1977

[5] D A Bigwood and A D Crocombe ldquoElastic analysis andengineering design formulae for bonded jointsrdquo InternationalJournal of Adhesion amp Adhesives vol 9 no 4 pp 229ndash242 1989

[6] M Y Tsai DWOplinger and JMorton ldquoImproved theoreticalsolutions for adhesive lap jointsrdquo International Journal of Solidsand Structures vol 35 no 12 pp 1163ndash1185 1998

[7] B Zhao and Z-H Lu ldquoA two-dimensional approach of single-lap adhesive bonded jointsrdquo Mechanics of Advanced Materialsand Structures vol 16 no 2 pp 130ndash159 2009

[8] B Zhao Z-H Lu and Y-N Lu ldquoClosed-form solutions forelastic stressstrain analysis in unbalanced adhesive single-lapjoints considering adherend deformations and bond thicknessrdquoInternational Journal of Adhesion amp Adhesives vol 31 no 6 pp434ndash445 2011

[9] C Raphael ldquoVariable-adhesive bonded jointsrdquo Applied PolymerSymposia vol 3 pp 99ndash108 1966

[10] S Srinivas ldquoAnalysis of bonded jointsrdquo NASA Report TN D-7855 NASA Washington DC USA 1975

[11] I Pires L Quintino J F Durodola and A Beevers ldquoPerfor-mance of bi-adhesive bonded aluminium lap jointsrdquo Interna-tional Journal of Adhesion amp Adhesives vol 23 no 3 pp 215ndash223 2003

[12] M D Fitton and J G Broughton ldquoVariable modulus adhesivesan approach to optimised joint performancerdquo International

Journal of Adhesion amp Adhesives vol 25 no 4 pp 329ndash3362005

[13] F-R KongM You and X-L Zheng ldquoThree-dimensional finiteelement analysis of the stress distribution in bi-adhesive bondedjointsrdquoThe Journal of Adhesion vol 84 no 2 pp 105ndash124 2008

[14] S Kumar ldquoAnalysis of tubular adhesive joints with a func-tionally modulus graded bondline subjected to axial loadsrdquoInternational Journal of Adhesion amp Adhesives vol 29 no 8 pp785ndash795 2009

[15] L FM da Silva andMCQ Lopes ldquoJoint strength optimizationby the mixed-adhesive techniquerdquo International Journal ofAdhesion amp Adhesives vol 29 no 5 pp 509ndash514 2009

[16] M You J Yan X Zheng D Zhu J Li and J Hu ldquoThenumerical analysis on theweld-bonded joints with bi-adhesiverdquoin Proceedings of the 2nd International Conference on ComputerModeling and Simulation (ICCMS rsquo10) pp 25ndash28 Sanya ChinaJanuary 2010

[17] L F M da Silva and R D Adams ldquoAdhesive joints at high andlow temperatures using similar and dissimilar adherends anddual adhesivesrdquo International Journal of Adhesion amp Adhesivesvol 27 no 3 pp 216ndash226 2007

[18] S Temiz ldquoApplication of bi-adhesive in double-strap jointssubjected to bending momentrdquo Journal of Adhesion Science andTechnology vol 20 no 14 pp 1547ndash1560 2006

[19] P J C das Neves L FM da Silva and R D Adams ldquoAnalysis ofmixed adhesive bonded joints part I theoretical formulationrdquoJournal of Adhesion Science and Technology vol 23 no 1 pp1ndash34 2009

[20] I Pires L Quintino andRMMiranda ldquoNumerical simulationof mono- and bi-adhesive aluminium lap jointsrdquo Journal ofAdhesion Science and Technology vol 20 no 1 pp 19ndash36 2006

[21] S Kumar and P C Pandey ldquoBehaviour of Bi-adhesive jointsrdquoJournal of Adhesion Science and Technology vol 24 no 7 pp1251ndash1281 2010

[22] R J C Carbas L F M da Silva and G W CritchlowldquoAdhesively bonded functionally graded joints by inductionheatingrdquo International Journal of Adhesion amp Adhesives vol 48pp 110ndash118 2014

[23] R J C Carbas L F M da Silva M L Madureira and G WCritchlow ldquoModelling of functionally graded adhesive jointsrdquoThe Journal of Adhesion 2013

[24] O Bavi N Bavi and M Shishesaz ldquoGeometrical optimizationof the overlap in mixed adhesive lap jointsrdquo The Journal ofAdhesion vol 89 no 12 pp 948ndash972 2013

[25] O Volkersen ldquoDie niektraftverteilung in zugbeanspruchtenmitkonstanten laschenquerschrittenrdquo Luftfahrtforschung vol 15pp 41ndash47 1938

[26] L F M da Silva P J C das Neves R D Adams and J KSpelt ldquoAnalytical models of adhesively bonded jointsmdashpart Iliterature surveyrdquo International Journal of AdhesionampAdhesivesvol 29 no 3 pp 319ndash330 2009

[27] L F M da Silva P J C das Neves R D Adams A Wangand J K Spelt ldquoAnalytical models of adhesively bonded jointsmdashpart II comparative studyrdquo International Journal of Adhesion ampAdhesives vol 29 no 3 pp 331ndash341 2009

[28] B Zhao ldquoTheoretical stressmodel ofmixed-modulus single-lapadhesive bonded joints considering the bending effectrdquoChineseJournal of Mechanical Engineering vol 44 no 10 pp 129ndash1372008

[29] D A Dillard and A V Pocius Adhesion Science andEngineering-I The Mechanics of Adhesion Elsevier ScienceAmsterdam The Netherlands 2002

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Mathematical Problems in Engineering

[30] J PM GoncalvesM F S F deMoura and PM S T de CastroldquoA three-dimensional finite element model for stress analysis ofadhesive jointsrdquo International Journal of Adhesion amp Adhesivesvol 22 no 5 pp 357ndash365 2002

[31] H Ozer and O Oz ldquoThree dimensional finite element analysisof bi-adhesively bonded double lap jointrdquo International Journalof Adhesion amp Adhesives vol 37 pp 50ndash55 2012

[32] G P Zou K Shahin and F Taheri ldquoAn analytical solution forthe analysis of symmetric composite adhesively bonded jointsrdquoComposite Structures vol 65 no 3-4 pp 499ndash510 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Complex AnalysisJournal of

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OptimizationJournal of

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International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of