Research Article An Analytical Model for Predicting the...

Research ArticleAn Analytical Model for Predicting the Stress Distributionswithin Single-Lap Adhesively Bonded Beams

Xiaocong He and Yuqi Wang

Innovative Manufacturing Research Centre Kunming University of Science and Technology Kunming 650500 China

Correspondence should be addressed to Xiaocong He hhxxccyahoocouk

Received 13 November 2013 Accepted 23 December 2013 Published 11 February 2014

Academic Editor Filippo Berto

Copyright copy 2014 X He and Y Wang 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

An analytical model for predicting the stress distributions within single-lap adhesively bonded beams under tension is presentedin this paper By combining the governing equations of each adherend with the joint kinematics the overall system of governingequations can be obtained Both the adherends and the adhesive are assumed to be under plane strain condition With suitableboundary conditions the stress distribution of the adhesive in the longitudinal direction is determined

1 Introduction

The use of adhesively bonded joint which uses metalliccomposite and ceramic materials is of great interest tomany industrial sectors including aerospace automotivemarine machine tools package and appliance industriesThis widespread use of adhesive joints is due to ease ofapplication time and cost savings high corrosion and fatigueresistance crack retardance and good damping characteris-tics [1ndash3]

Any joint occurring in practice is designed to carry agiven set of loads Most of the adherends are loaded intension The subsequent loads on the adhesive are then afunction of the geometry of the joint Under most operatingloads and environmental conditions the adherends behave ina linearly elastic manner However the adhesive may exhibitviscoelastic or nonlinear behaviour The exact analyticalsolution to the problem of stress distributions in the bondedarea is complex The existing analytical studies are thereforebased on certain simplifying assumptions with regard to themodelling of the adhesive and adherends

In Volkersenrsquos shear-lag analysis it was assumed that theadhesive deforms only in shear while the adherend deformedonly in tension [4] The consequences of the rotation ofthe adherends were first taken into account by Goland andReissner [5] The authors derived equations to evaluate the

shearing and normal stresses in the bond layer as wellas those in the jointed plates assuming that the peel andshear stresses were constants across the adhesive thicknessIn Cornellrsquos work [6] a variation and extension of Golandand Reissnerrsquos method were presented for determining thestresses in adhesive lap joints The author assumed that thetwo lap joint plates act like simple beam and the more elasticadhesive layer is an infinite number of shear and tensionsprings Differential equations were set up which describe thetransfer of the load in one beam through the springs to theother beam From the solution of these differential equationsa fairly complete analysis of the stresses in the lap joint wasobtained

Ojalvo and Eidinoff [7] presented results of an analyticalinvestigation on the influence of bond thickness upon thestress distribution in single adhesive lap joints The workextended the basic approach for bonded joints originallyintroduced by Goland and Reissner through the use of amore complete shear-straindisplacement equation for theadhesive layer The work uncovers several interesting phe-nomena without adding any significant complication to theanalysis Delale et al [8] analyzed a general plane strainproblemof adhesively bonded structureswhich consist of twodifferent orthotropic adherends Both the transverse shearstress effects in the adherends and the in-plane normal strainin the adhesive were taken into account The solution was

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2014 Article ID 346379 5 pageshttpdxdoiorg1011552014346379

2 Advances in Materials Science and Engineering

z w

x u

Figure 1 Coordinate system

obtained by assuming linear stress-strain relations for theadhesive The peak values of the shear as well as those ofthe normal stress in the adhesive were found to be at theedges of the overlap region Rossettos et al [9] establishedthe governing equations for a step lap joint with a voidusing a modified shear-lag model where the adhesive canhave extensional as well as shear deformations The modelconsiders quadratic axial deformation across the adhesivethickness

The objective of this work is to present an analyticalmodel for predicting the stress distributions within a single-lap adhesively bonded beam under tension This model is adevelopment of the shear-lag analysismodel of Volkersen [4]The derivations are similar to those of Goland and Reissner[5] but the geometry employed is different By combiningthe governing equations of each adherend with the jointkinematics the overall system of governing equations can beobtained Both the adherends and the adhesive are assumedto be under plane strain condition and the adhesive stressesare assumed to be uniform across the thicknessWith suitableboundary conditions the stress distribution of the adhesive inthe longitudinal direction is determined

2 Formulation of the Problem

Figure 1 shows the coordinate system used in this study 119909and 119911 are coordinates while 119906 and119908 represent displacementsA single-lap-jointed beam is shown in Figure 2 The jointlength is 2119888 The two adherends are considered to be of equalthickness 119905 and equal length (119897 + 2119888) Two points 119886 and 119887 arethe points at the centre of two free edge of the single-lap-jointed beam The adhesive thickness is 120578 The joint width isassumed to be large compared with the adherend thicknessIf the system is now loaded by tensile forces 119879 per unit ofadherend width at the points 119886 and 119887 the line of action ofthe forces will be 119886119900119887

For studying the system deformation two coordinatesystems (119909

1 1199111) and (119909

2 1199112) are introduced in Figure 2

The first system is used to analyze the behavior of the left-hand adherend under load The coordinate 119909

1has its origin

at the point 119886 extends along the longitudinal axis of the

adherend and is positive to the right1199081represents transverse

deformation of the adherend from the unloaded conditionand is positive upwards A similar definition is taken for(1199092 1199112) with reference to the joint

If 1198721is equal to the bending moment in the adherend

at station 1199091and 119872

2is equal to the moment in the joint at

station 1199092 each per unit of width then

1198721= 119879 [tan120572

1198991199091minus 1199081] (0 le 119909

1le 119897)

1198722= 119879 [tan120572

119899(119897 + 119909

2) minus 1199082minus

119905 + 120578

2

] (0 le 1199092le 119888)

(1)

where

tan120572119899=

(119905 + 120578) 2

119897 + 119888

(2)

As 120578 ≪ 119905 and 119905 ≪ 119897 (1) then reduce to the simplified formsby using Taylor series

1198721= 119879 [120572

1198991199091minus 1199081] (0 le 119909

1le 119897)

1198722= 119879 [120572

119899(119897 + 119909


119905

2

] (0 le 1199092le 119888)

(3)

As for deformations 1199081and 119908

2 we have

11988921199081

1198891199092

1

= minus

1198721

1198631

11988921199082

1198891199092

2

= minus

1198722

1198632

(4)

where 1198631and 119863

2are the flexural rigidities of the adherend

and joint respectivelyFrom (3) and (4) we have

11988921199081

1198891199092

1

= minus

119879

1198631

[1205721198991199091minus 1199081] (0 le 119909

1le 119897) (5)

11988921199082

1198891199092

2

= minus

119879

1198632

[120572119899(119897 + 119909


119905

2

] (0 le 1199092le 119888) (6)

From the four boundary conditions

at 1199091= 0 119908

1= 0

at 1199091= 119897 119909

2= 0 119908

1= 1199082

at 1199091= 119897 119909

2= 0

1198891199081

1198891199091

=

1198891199082

1198891199092

at 1199092= 119888 119908

2= 0

(7)

The two preceding equations (5) and (6) have solutions of theform

1199081= 1198601cosh 119887

11199091+ 1198611sinh 119887

11199091+ 1205721198991199091

(0 le 1199091le 119897)

1199082= 1198602cosh 119887

21199092+ 1198612sinh 119887

21199092+ 120572119899(119897 + 119909

2minus

119905

2

)

(0 le 1199092le 119888)

(8)

Advances in Materials Science and Engineering 3

Upper adherend

Lower adherend

Adhesive

T

Tb

l lc c

o

z2

x2120578

t

z1

x1a

Figure 2 A single-lap-jointed beam

where [5]

1198601=

minus (1199052) sinh (1198872119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198602=

(1199052) (11988721198871) cosh (119887

1119897) sinh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198611=

minus (1199052) cosh (1198872119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198612=

(1199052) (11988721198871) cosh (119887

1119897) cosh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198872

1=

119879

1198631

1198872

2=

119879

1198632

(9)

3 Stress Distributions in the Adhesive Layer

Figure 3 shows elements of the upper and lower adherends119872 119881 and 119879 are the bending moments vertical shearand axial tension in the adherends The subscripts 119906 and119897 designate quantities pertaining to the upper and loweradherend respectively 120590

0and 1205910are the transverse normal

stress and the shear stress respectivelyThe conditions of moment equilibrium for the elements

of the adherends are

119889119872119906

119889119909

minus 119881119906+ 1205910

119905

2

= 0

119889119872119897

119889119909

minus 119881119897+ 1205910

119905

2

= 0

(10)

The conditions of horizontal force equilibrium are

119889119879119906

119889119909

minus 1205910= 0

119889119879119897

119889119909

+ 1205910= 0

(11)

The conditions of vertical force equilibrium for theelements of the adherends are

119889119881119906

119889119909

minus 1205900= 0

119889119881119897

119889119909

+ 1205900= 0

(12)

The transverse deflections of the upper and loweradherends are denoted by ]

119906and ]

119897 respectively both

measured positively upwardThen the thin plate theory gives

1198892V119906

1198891199092= minus

119872119906

1198631

1198892V119897

1198891199092= minus

119872119897

1198631

with 1198631=

1198641199053

12 (1 minus ]2)

(13)

where 119864 and ] are Youngrsquos modulus and Poissonrsquos ratio ofadherends respectively 119863

1refers to the flexural rigidity of

the adherendsThe longitudinal displacements of the adherend at the

adherend boundaries adjacent to the adhesive are denoted by119906119906and 119906

119897 Then from the stress-strain relations

119889119906119906

119889119909

=

1

119864

(

119879119906

119905

minus 6

119872119906

1199052)

119889119906119897

119889119909

=

1

119864

(

119879119897

119905

+ 6

119872119897

1199052)

(14)

Let 119864119886119889

and 119866119886119889

refer to Youngrsquos and shear modulirespectively of the adhesive material then

1205910

119866119886119889

=

119906119906minus 119906119897

120578

1205900

119864119886119889

=

V119906minus V119897

120578

(15)

Combining equations from (10) to (14) and differentiatingequation (15) we have

11988931205910

1198891199093minus

8119866119886119889

119864119905120578

1198891205910

119889119909

= 0 (16)

11988941205900

1198891199094+

24 (1 minus ]2) 119864119886119889

1198641199053120578

1205900= 0 (17)


Tu

Mu Vu

dx

Vu + (dVudx)dx

Tu + (dTudx)dx

Mu + (dMudx)dx

1205910dx

1205900dx

dx

Vl + (dVldx)dx

Tl + (dTldx)dx

Ml + (dMldx)dx

Tl

Ml Vl

1205910dx

1205900dx

Figure 3 Free body diagram of joint with adhesive layer

The boundary conditions can be written as

at 119909 = 119888 119872119906= 119879119906= 119881119906= 0

119872119897= 1198720 119881

119897= 1198810 119879

119897= 119879

at 119909 = minus119888 119872119897= 119879119897= 119881119897= 0

119872119906= minus119872

0 119881

119906= 1198810 119879

119906= 119879

(18)

From (8) and (9) we have

1198720= (1198721)1199091=119897= minus1198631(

11988921199081

1198891199092

1

)

1199091=119897

= 119896

119879119905

2

1198810= (

1198891198721

1198891199091

)

1199091=119897

= 119896119879[3 (1 minus ]2)119879

119864119905

]

12

119896 =

sinh (1198871119897) cosh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

11198872) cosh (119887

1119897) sinh (119887

2119888)

(19)

On the basis of the differential equations (16) and (17) andthe boundary conditions the distributions of 120590

0and 1205910in the

adhesive are obtained in the form

1205900=

1199011199052

Δ1198882[(11987721205822 119896

2

+ 1205821198961015840 cosh 120582 cos 120582) cosh 120582119909

119888

cos 120582119909119888

+(11987711205822 119896

2

+ 1205821198961015840 sinh 120582 sin 120582) sinh 120582119909

119888

sin 120582119909119888

]

1205910= minus

119901119905

8119888

[

120573119888

119905

(1 + 3119896)

cosh (120573119909119905)sinh (120573119888119905)

+ 3 (1 minus 119896)]

(20)

where

120582 = 120574

119888

119905

1205744= 6

119864119886119889

119864

119905

120578

1205732= 8

119866119886119889

119864

119905

120578

1198771= cosh 120582 sin 120582 + sinh 120582 cos 120582

1198772= sinh 120582 cos 120582 minus cosh 120582 sin 120582

Δ =

1

2

(sinh 2120582 + sin 2120582) 119896 =

21198720

1199011199052 1198961015840=

1198810119888

1199011199052

1198871= 2[

3119901 (1 minus ]2)

1198641199052

]

12

1198872=

1198871

2radic2

(21)

If the materials and the thickness of the adherends aredifferent then (17) may become a 7th order differentialequation It is obvious that the analytical model is verycomplex and only the stress distributions of the adhesivein the longitudinal direction are determined In additionthe adhesive stresses actually are not uniform across thethickness In other words the simplifications have restrictedthe results Usually when we consider different boundaryconditions by a closed-form analysis the limitation is howtractable a realisticmathematicalmodel is within an algebraicsolution To overcome this limitation the finite elementanalysis (FEA) technique can be employed

4 Summary

An analytical model was presented for predicting the stressdistributions within a single-lap adhesively bonded beamunder tension By combining the governing equations of eachadherend with the joint kinematics the overall system ofgoverning equations can be obtained Both the adherends andthe adhesive are assumed to be under plane strain conditionand the adhesive stresses are assumed to be uniform acrossthe thickness With suitable boundary conditions the stress


distribution of the adhesive in the longitudinal direction isdetermined

Conflict of Interests

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

Acknowledgment

This study is partially supported by the Special Program ofthe Ministry of Science and Technology China (Grant no2012ZX04012-031)

References

[1] X He ldquoA review of finite element analysis of adhesively bondedjointsrdquo International Journal of Adhesion and Adhesives vol 31no 4 pp 248ndash264 2011

[2] X He F Gu and A Ball ldquoFatigue behaviour of fastening jointsof sheet materials and finite element analysisrdquo Advances inMechanical Engineering vol 2013 Article ID 658219 9 pages2013

[3] X He ldquoFinite element analysis of torsional free vibration ofadhesively bonded single-lap jointsrdquo International Journal ofAdhesion and Adhesives vol 48 pp 59ndash66 2014

[4] O Volkersen ldquoDie Nietkraftverteilung in ZugbeanspruchtenNietverbindungen mit Konstanten LaschenquerschnittenrdquoLuftfarhtforschung vol 15 pp 41ndash47 1938

[5] M Goland and E Reissner ldquoStresses in cemented jointsrdquoJournal of Applied Mechanics vol 11 pp A17ndashA27 1944

[6] R W Cornell ldquoDetermination of stresses in cemented lapjointsrdquo Journal of Applied Mechanics vol 20 pp 355ndash364 1953

[7] I U Ojalvo and H L Eidinoff ldquoBond thickness effects uponstresses in single-lap adhesive jointsrdquo AIAA Journal vol 16 no3 pp 204ndash211 1978

[8] F Delale F Erdogan andMN Aydinoglu ldquoStress in adhesivelybonded joints a closed-form solutionrdquo Journal of CompositeMaterials vol 15 no 3 pp 249ndash271 1981

[9] J N Rossettos P Lin and H Nayeb-Hashemi ldquoComparison ofthe effects of debonds and voids in adhesive jointsrdquo Journal ofEngineering Materials and Technology vol 116 no 4 pp 533ndash538 1994

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Journal ofNanomaterials


z w

x u

Figure 1 Coordinate system

obtained by assuming linear stress-strain relations for theadhesive The peak values of the shear as well as those ofthe normal stress in the adhesive were found to be at theedges of the overlap region Rossettos et al [9] establishedthe governing equations for a step lap joint with a voidusing a modified shear-lag model where the adhesive canhave extensional as well as shear deformations The modelconsiders quadratic axial deformation across the adhesivethickness

The objective of this work is to present an analyticalmodel for predicting the stress distributions within a single-lap adhesively bonded beam under tension This model is adevelopment of the shear-lag analysismodel of Volkersen [4]The derivations are similar to those of Goland and Reissner[5] but the geometry employed is different By combiningthe governing equations of each adherend with the jointkinematics the overall system of governing equations can beobtained Both the adherends and the adhesive are assumedto be under plane strain condition and the adhesive stressesare assumed to be uniform across the thicknessWith suitableboundary conditions the stress distribution of the adhesive inthe longitudinal direction is determined

2 Formulation of the Problem

Figure 1 shows the coordinate system used in this study 119909and 119911 are coordinates while 119906 and119908 represent displacementsA single-lap-jointed beam is shown in Figure 2 The jointlength is 2119888 The two adherends are considered to be of equalthickness 119905 and equal length (119897 + 2119888) Two points 119886 and 119887 arethe points at the centre of two free edge of the single-lap-jointed beam The adhesive thickness is 120578 The joint width isassumed to be large compared with the adherend thicknessIf the system is now loaded by tensile forces 119879 per unit ofadherend width at the points 119886 and 119887 the line of action ofthe forces will be 119886119900119887

For studying the system deformation two coordinatesystems (119909

1 1199111) and (119909

2 1199112) are introduced in Figure 2

The first system is used to analyze the behavior of the left-hand adherend under load The coordinate 119909

1has its origin

at the point 119886 extends along the longitudinal axis of the

adherend and is positive to the right1199081represents transverse

deformation of the adherend from the unloaded conditionand is positive upwards A similar definition is taken for(1199092 1199112) with reference to the joint

If 1198721is equal to the bending moment in the adherend

at station 1199091and 119872

2is equal to the moment in the joint at

station 1199092 each per unit of width then

1198721= 119879 [tan120572

1198991199091minus 1199081] (0 le 119909

1le 119897)

1198722= 119879 [tan120572

119899(119897 + 119909


119905 + 120578

2

] (0 le 1199092le 119888)

(1)

where

tan120572119899=

(119905 + 120578) 2

119897 + 119888

(2)

As 120578 ≪ 119905 and 119905 ≪ 119897 (1) then reduce to the simplified formsby using Taylor series

1198721= 119879 [120572

1198991199091minus 1199081] (0 le 119909

1le 119897)

1198722= 119879 [120572

119899(119897 + 119909


119905

2

] (0 le 1199092le 119888)

(3)

As for deformations 1199081and 119908

2 we have

11988921199081

1198891199092

1

= minus

1198721

1198631

11988921199082

1198891199092

2

= minus

1198722

1198632

(4)

where 1198631and 119863

2are the flexural rigidities of the adherend

and joint respectivelyFrom (3) and (4) we have

11988921199081

1198891199092

1

= minus

119879

1198631

[1205721198991199091minus 1199081] (0 le 119909

1le 119897) (5)

11988921199082

1198891199092

2

= minus

119879

1198632

[120572119899(119897 + 119909


119905

2

] (0 le 1199092le 119888) (6)

From the four boundary conditions

at 1199091= 0 119908

1= 0

at 1199091= 119897 119909

2= 0 119908

1= 1199082

at 1199091= 119897 119909

2= 0

1198891199081

1198891199091

=

1198891199082

1198891199092

at 1199092= 119888 119908

2= 0

(7)

The two preceding equations (5) and (6) have solutions of theform

1199081= 1198601cosh 119887

11199091+ 1198611sinh 119887

11199091+ 1205721198991199091

(0 le 1199091le 119897)

1199082= 1198602cosh 119887

21199092+ 1198612sinh 119887

21199092+ 120572119899(119897 + 119909

2minus

119905

2

)

(0 le 1199092le 119888)

(8)


Upper adherend

Lower adherend

Adhesive

T

Tb

l lc c

o

z2

x2120578

t

z1

x1a


where [5]

1198601=

minus (1199052) sinh (1198872119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198602=

(1199052) (11988721198871) cosh (119887

1119897) sinh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198611=

minus (1199052) cosh (1198872119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198612=

(1199052) (11988721198871) cosh (119887

1119897) cosh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198872

1=

119879

1198631

1198872

2=

119879

1198632

(9)






119889119872119906

119889119909

minus 119881119906+ 1205910

119905

2

= 0

119889119872119897

119889119909

minus 119881119897+ 1205910

119905

2

= 0

(10)


119889119879119906

119889119909

minus 1205910= 0

119889119879119897

119889119909

+ 1205910= 0

(11)


119889119881119906

119889119909

minus 1205900= 0

119889119881119897

119889119909

+ 1205900= 0

(12)


119906and ]



1198892V119906

1198891199092= minus

119872119906

1198631

1198892V119897

1198891199092= minus

119872119897

1198631

with 1198631=

1198641199053

12 (1 minus ]2)

(13)






119889119906119906

119889119909

=

1

119864

(

119879119906

119905

minus 6

119872119906

1199052)

119889119906119897

119889119909

=

1

119864

(

119879119897

119905

+ 6

119872119897

1199052)

(14)

Let 119864119886119889

and 119866119886119889


1205910

119866119886119889

=

119906119906minus 119906119897

120578

1205900

119864119886119889

=

V119906minus V119897

120578

(15)


11988931205910

1198891199093minus

8119866119886119889

119864119905120578

1198891205910

119889119909

= 0 (16)

11988941205900

1198891199094+

24 (1 minus ]2) 119864119886119889

1198641199053120578

1205900= 0 (17)


Tu

Mu Vu

dx

Vu + (dVudx)dx

Tu + (dTudx)dx

Mu + (dMudx)dx

1205910dx

1205900dx

dx

Vl + (dVldx)dx

Tl + (dTldx)dx

Ml + (dMldx)dx

Tl

Ml Vl

1205910dx

1205900dx



at 119909 = 119888 119872119906= 119879119906= 119881119906= 0

119872119897= 1198720 119881

119897= 1198810 119879

119897= 119879

at 119909 = minus119888 119872119897= 119879119897= 119881119897= 0

119872119906= minus119872

0 119881

119906= 1198810 119879

119906= 119879

(18)


1198720= (1198721)1199091=119897= minus1198631(

11988921199081

1198891199092

1

)

1199091=119897

= 119896

119879119905

2

1198810= (

1198891198721

1198891199091

)

1199091=119897

= 119896119879[3 (1 minus ]2)119879

119864119905

]

12

119896 =

sinh (1198871119897) cosh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

11198872) cosh (119887

1119897) sinh (119887

2119888)

(19)


0and 1205910in the


1205900=

1199011199052

Δ1198882[(11987721205822 119896

2

+ 1205821198961015840 cosh 120582 cos 120582) cosh 120582119909

119888

cos 120582119909119888

+(11987711205822 119896

2

+ 1205821198961015840 sinh 120582 sin 120582) sinh 120582119909

119888

sin 120582119909119888

]

1205910= minus

119901119905

8119888

[

120573119888

119905

(1 + 3119896)

cosh (120573119909119905)sinh (120573119888119905)

+ 3 (1 minus 119896)]

(20)

where

120582 = 120574

119888

119905

1205744= 6

119864119886119889

119864

119905

120578

1205732= 8

119866119886119889

119864

119905

120578

1198771= cosh 120582 sin 120582 + sinh 120582 cos 120582


Δ =

1

2

(sinh 2120582 + sin 2120582) 119896 =

21198720

1199011199052 1198961015840=

1198810119888

1199011199052

1198871= 2[

3119901 (1 minus ]2)

1198641199052

]

12

1198872=

1198871

2radic2

(21)


4 Summary






Acknowledgment


References

















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Upper adherend

Lower adherend

Adhesive

T

Tb

l lc c

o

z2

x2120578

t

z1

x1a


where [5]

1198601=

minus (1199052) sinh (1198872119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198602=

(1199052) (11988721198871) cosh (119887

1119897) sinh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198611=

minus (1199052) cosh (1198872119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198612=

(1199052) (11988721198871) cosh (119887

1119897) cosh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

21198871) sinh (119887

1119897) cosh (119887

2119888)

1198872

1=

119879

1198631

1198872

2=

119879

1198632

(9)






119889119872119906

119889119909

minus 119881119906+ 1205910

119905

2

= 0

119889119872119897

119889119909

minus 119881119897+ 1205910

119905

2

= 0

(10)


119889119879119906

119889119909

minus 1205910= 0

119889119879119897

119889119909

+ 1205910= 0

(11)


119889119881119906

119889119909

minus 1205900= 0

119889119881119897

119889119909

+ 1205900= 0

(12)


119906and ]



1198892V119906

1198891199092= minus

119872119906

1198631

1198892V119897

1198891199092= minus

119872119897

1198631

with 1198631=

1198641199053

12 (1 minus ]2)

(13)






119889119906119906

119889119909

=

1

119864

(

119879119906

119905

minus 6

119872119906

1199052)

119889119906119897

119889119909

=

1

119864

(

119879119897

119905

+ 6

119872119897

1199052)

(14)

Let 119864119886119889

and 119866119886119889


1205910

119866119886119889

=

119906119906minus 119906119897

120578

1205900

119864119886119889

=

V119906minus V119897

120578

(15)


11988931205910

1198891199093minus

8119866119886119889

119864119905120578

1198891205910

119889119909

= 0 (16)

11988941205900

1198891199094+

24 (1 minus ]2) 119864119886119889

1198641199053120578

1205900= 0 (17)


Tu

Mu Vu

dx

Vu + (dVudx)dx

Tu + (dTudx)dx

Mu + (dMudx)dx

1205910dx

1205900dx

dx

Vl + (dVldx)dx

Tl + (dTldx)dx

Ml + (dMldx)dx

Tl

Ml Vl

1205910dx

1205900dx



at 119909 = 119888 119872119906= 119879119906= 119881119906= 0

119872119897= 1198720 119881

119897= 1198810 119879

119897= 119879

at 119909 = minus119888 119872119897= 119879119897= 119881119897= 0

119872119906= minus119872

0 119881

119906= 1198810 119879

119906= 119879

(18)


1198720= (1198721)1199091=119897= minus1198631(

11988921199081

1198891199092

1

)

1199091=119897

= 119896

119879119905

2

1198810= (

1198891198721

1198891199091

)

1199091=119897

= 119896119879[3 (1 minus ]2)119879

119864119905

]

12

119896 =

sinh (1198871119897) cosh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

11198872) cosh (119887

1119897) sinh (119887

2119888)

(19)


0and 1205910in the


1205900=

1199011199052

Δ1198882[(11987721205822 119896

2

+ 1205821198961015840 cosh 120582 cos 120582) cosh 120582119909

119888

cos 120582119909119888

+(11987711205822 119896

2

+ 1205821198961015840 sinh 120582 sin 120582) sinh 120582119909

119888

sin 120582119909119888

]

1205910= minus

119901119905

8119888

[

120573119888

119905

(1 + 3119896)

cosh (120573119909119905)sinh (120573119888119905)

+ 3 (1 minus 119896)]

(20)

where

120582 = 120574

119888

119905

1205744= 6

119864119886119889

119864

119905

120578

1205732= 8

119866119886119889

119864

119905

120578

1198771= cosh 120582 sin 120582 + sinh 120582 cos 120582


Δ =

1

2

(sinh 2120582 + sin 2120582) 119896 =

21198720

1199011199052 1198961015840=

1198810119888

1199011199052

1198871= 2[

3119901 (1 minus ]2)

1198641199052

]

12

1198872=

1198871

2radic2

(21)


4 Summary






Acknowledgment


References

















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Tu

Mu Vu

dx

Vu + (dVudx)dx

Tu + (dTudx)dx

Mu + (dMudx)dx

1205910dx

1205900dx

dx

Vl + (dVldx)dx

Tl + (dTldx)dx

Ml + (dMldx)dx

Tl

Ml Vl

1205910dx

1205900dx



at 119909 = 119888 119872119906= 119879119906= 119881119906= 0

119872119897= 1198720 119881

119897= 1198810 119879

119897= 119879

at 119909 = minus119888 119872119897= 119879119897= 119881119897= 0

119872119906= minus119872

0 119881

119906= 1198810 119879

119906= 119879

(18)


1198720= (1198721)1199091=119897= minus1198631(

11988921199081

1198891199092

1

)

1199091=119897

= 119896

119879119905

2

1198810= (

1198891198721

1198891199091

)

1199091=119897

= 119896119879[3 (1 minus ]2)119879

119864119905

]

12

119896 =

sinh (1198871119897) cosh (119887

2119888)

sinh (1198871119897) cosh (119887

2119888) + (119887

11198872) cosh (119887

1119897) sinh (119887

2119888)

(19)


0and 1205910in the


1205900=

1199011199052

Δ1198882[(11987721205822 119896

2

+ 1205821198961015840 cosh 120582 cos 120582) cosh 120582119909

119888

cos 120582119909119888

+(11987711205822 119896

2

+ 1205821198961015840 sinh 120582 sin 120582) sinh 120582119909

119888

sin 120582119909119888

]

1205910= minus

119901119905

8119888

[

120573119888

119905

(1 + 3119896)

cosh (120573119909119905)sinh (120573119888119905)

+ 3 (1 minus 119896)]

(20)

where

120582 = 120574

119888

119905

1205744= 6

119864119886119889

119864

119905

120578

1205732= 8

119866119886119889

119864

119905

120578

1198771= cosh 120582 sin 120582 + sinh 120582 cos 120582


Δ =

1

2

(sinh 2120582 + sin 2120582) 119896 =

21198720

1199011199052 1198961015840=

1198810119888

1199011199052

1198871= 2[

3119901 (1 minus ]2)

1198641199052

]

12

1198872=

1198871

2radic2

(21)


4 Summary






Acknowledgment


References

















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







Acknowledgment


References

















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Research Article An Analytical Model for Predicting the...

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