Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads

ARTICLE IN PRESS International Journal of Adhesion & Adhesives 29 (2009) 785–795

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Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads S. Kumar  Solid Mechanics and Materials Engineering Group, Department of Engineering Science, University of Oxford, Parks road, Oxford OX1 3PJ, UK

a r t i c l e in fo

abstract

Article history: Accepted 12 June 2009 Available online 11 August 2009

The strength and lifetime of adhesively bonded joints can be significantly improved by reducing the stress concentration at the ends of overlap and distributing the stresses uniformly over the entire bondline. The ideal way of achieving this is by employing a modulus graded bondline adhesive. This study presents a theoretical framework for the stress analysis of adhesively bonded tubular lap joint based on a variational principle which minimizes the complementary energy of the bonded system. The joint consists of similar or dissimilar adherends and a functionally modulus graded bondline (FMGB) adhesive. The varying modulus of the adhesive along the bondlength is expressed by suitable functions which are smooth and continuous. The axisymmetric elastic analysis reveals that the peel and shear stress peaks in the FMGB are much smaller and the stress distribution is more uniform along its length than those of mono-modulus bondline (MMB) adhesive joints under the same axial tensile load. A parametric evaluation has been conducted by varying the material and geometric properties of the joint in order to study their effect on stress distribution in the bondline. Furthermore, the results suggest that the peel and shear strengths can be optimized by spatially controlling the modulus of the adhesive. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Functionally graded bondline Adhesive joint Stress distribution Stress analysis Variational method

1. Introduction Adhesively bonded joints are widely used in the aerospace and automotive industries for joining dissimilar materials since they provide more uniform load transfer over the bonded area. Nevertheless, stress distribution in the bondline adhesive is non-uniform. The stress analysis of lap joints has undergone continuous development and refinement for more than seven decades. Volkersen [1] analysed the stress distribution in a singlelap joint geometry under the load due to stretching of the adherends, while ignoring the tearing stresses at the free ends. Goland and Reissner [2] modelled bonded joints by using beam elements for the adherends, considering shear and normal transverse deformations of adhesive. Hart-Smith [3] included elastic–plastic adhesive behaviour. All these studies neglected shear deformations of adherend. Finite element methods have been used increasingly since the 1970s to analyse lap joints. Wooley and Carver [4] conducted a geometrically linear finite element analysis on bonded lap joints to predict stress concentration factors. Adams and Peppiatt [5] and Crocombe and Adams [6] performed two-dimensional finite element analysis of single-lap joints with a spew fillet. Adams and Peppiatt [7] used axisymmetric quadratic isoparametric finite elements to carry out a stress analysis of tubular lap joints

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subjected to axial and torsional loads. Nagaraja and Alwar [8] identified a decrease of the peel and shear stresses in the adhesive of a single lap joint based on finite element analyses with a viscoelastic material model. Harris and Adams [9] predicted the strength of a single-lap joint with a spew fillet by considering elastic and elastic–plastic material properties of the adherend and adhesive through a nonlinear finite element stress analysis. Pickett and Hollaway [10] performed theoretical and finite element studies on lap joints with elastic-perfectly plastic adhesive. Oplinger [11] developed a layered beam analysis, which considered the large deflection of the joint overlap. Tsai and Morton [12] evaluated the single-lap joint analytically and compared with nonlinear finite element analysis results. Deb et al. [13] recently studied the mechanical behaviour of adhesively bonded joints at different extension rates and temperatures. Several researchers have proposed two-dimensional analytical solutions for cylindrical bonded joints, which were focused on the joint overlap, to ensure that the stress free boundary conditions would be satisfied at the free end. For instance, Allman [14] used a minimum strain energy, with given bending, stretching and shearing at the end of the overlap and assuming that the longitudinal normal stress was zero, the shear stress constant, and the transverse normal stress was linearly distributed across the thickness of the adhesive. Shi and Cheng [15] developed approximate closed form solutions for cylindrical single-lap joints employing the variational principle of complementary energy with similar boundary conditions and assumptions to those of Allman. Nemes et al. [16] proceeded further, employing the same

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Nomenclature E1 , n1 E2 , n2 E, n

Young’s modulus and Poison’s ratio of the inner tube Young’s modulus and Poison’s ratio of the outer tube Young’s modulus and Poison’s ratio of the MMB adhesive El1 , Et1 , ntl1 , G1 transversely isotropic outer adherend’s properties E2l , E2t , ntl2 , G2 transversely isotropic inner adherend’s properties Ef 1 , Ef 2 , Ef 3 , Ef 4 , Ef 5 modulus functions of the FMGB adhesive maximum and minimum value of Young’s modulus of Em , E0 the FMGB adhesive EðzÞ modulus function of the FMGB adhesive a, b inner and outer radii of the inner tube c, d inner and outer radii of the outer tube thickness of inner and outer tubes t1 , t2

methodology as that of Shi and Cheng [15] to provide a statically determinate elastic solution for the same cylindrical lap-joint by omitting radial stress component in the joint. Numerous techniques have been used to reduce the stress concentrations at the ends of the overlap of single lap joints and hence to improve structural capability [17–19]. These include modifying the adherend geometry [20–22], the adhesive geometry [23] and the spew geometry [24,25]. These studies mostly concentrated on geometrical aspect of constituent components of joint to maximize joint performance. However, in a few cases, change of geometry is restricted by complexities involved in production/fabrication besides the cost. A few investigators altered the material of the adhesive globally to reduce the stress concentration. They studied the effect of the shear modulus of adhesive on the shear stress distribution in the bondline and showed that it has a significant effect [17,26]. Sadek [27] has tried to achieve higher lap-shear strength by introducing high modulus adhesives in the bondline. However, in this case, adhesives are prone to interfacial/cohesive brittle failure due to the high peel stresses they produce. Even if we employ a flexible adhesive in the bondline, the stress distribution in the bonded area would be non-uniform [27]. On the other hand, changing the material property of the adherend would not be possible because the adherend material is selected based on the functional requirement of the structural members to be joined. A study of the stress distribution in the adhesive of a single-lap joint with controlled and spatially varying adherend material modulus in the overlap region was the first attempt to locally alter both the geometric and material properties of a composite adherend [28]. Recently, Pires et al. [29] and Fitton and Broughton [30] evaluated performance of bi-adhesive single lap joints both experimentally and theoretically with a brittle adhesive in the middle portion of the bondline and a ductile adhesive at the ends of the overlap. Kumar and Pandey [31] examined the performance bi-adhesive joints by performing geometrical and material nonlinear finite element analyses. The performance of adhesive joints with dual adhesive at high and low temperatures using similar and dissimilar adherends was studied by Da Silva and Adams [32]. These investigators considered only one step change in the property over the bondline. Therefore, in this investigation, a multi-step variation of the modulus of the adhesive along its length has been considered so as to reduce the shear and peel stress concentrations at the ends of the overlap. The objective of the problem is to formulate the adhesive stresses in terms of geometrical and material properties of the adherends and adhesive.

t P L r, y, z

thickness of the adhesive layer axial tensile load bondlength of the joint radial, circumferential and axial coordinates of the tubular system axial edge stresses in the inner and outer tubes of the f1 , f2 jointed portion ð1Þ ð1Þ ð1Þ sð1Þ stress components in the inner tube rr , syy , szz , trz ð2Þ ð2Þ ð2Þ sð2Þ , s , s , t stress components in the outer tube rr zz rz yy srr , syy , szz , trz stress components in the adhesive O1 , O2 , O3 complementary energy in the inner tube, outer tube and adhesive O complementary energy of the joint max tmax maximum shear and peel stress in the FMGB rz , syy adhesive xsh , xpe shear and peel stress concentration factors

2. Axisymmetric formulation Consider two tubes of different materials and dimensions as shown in Fig. 1a. The two tubes are lap-jointed by a FMGB adhesive. The joint is subjected to an axial tensile load P. Fig. 1b shows the coordinate system with coordinates r and z and the edge stresses (f1 and f2 ) of the bonded portion whose length is L. The aim of the problem is to determine the stress distribution in the adhesive layer when using an adhesive where properties vary along the length of the joint. The following assumptions have been adopted to analyse this statically determinate system.

 The radial stresses in all the three domains are neglected  



ð2Þ ðsð1Þ rr ¼ srr ¼ srr ¼ 0Þ. Axisymmetric condition implies that the following shear stresses are zero i.e try ¼ 0; tzy ¼ 0 in all three domains. For a thin adhesive, the difference between the two shearing stresses acting on the outer surface of the adhesive trz ðc; zÞ and that on the inner surface of the adhesive trz ðb; zÞ is very small and, hence, the longitudinal stress szz in the adhesive may be neglected as compared with shearing stress trz . The longitudinal stress in the inner and outer tubes is a ð1Þ function of the axial coordinate z only i.e. sð1Þ zz ¼ szz ðzÞ; ð2Þ ð2Þ szz ¼ szz ðzÞ.

Therefore, the non-zero stress components in the bonded system are: ð1Þ ð1Þ  Inner tube: tð1Þ rz ðr; zÞ, syy ðr; zÞ, szz ðzÞ.  Adhesive: trz ðr; zÞ, syy ðr; zÞ. ð2Þ ð2Þ  Outer tube: tð2Þ rz ðr; zÞ, syy ðr; zÞ, szz ðzÞ.

Incorporating the aforementioned assumptions, the differential equations of equilibrium are reduced to the following. The stress field in the axisymmetric system should satisfy the equations of equilibrium, the traction boundary conditions at z ¼ 0; z ¼ L, and the conditions of stress continuity across the dividing surfaces ðr ¼ b; r ¼ cÞ @trz 1  syy ¼ 0 @z r

ð1Þ

@trz @szz 1 þ þ trz ¼ 0 @z @z r

ð2Þ

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Fig. 1. (a) Adhesively bonded tubular joint. (b) Coordinate system (r, y, z) and edge stresses on jointed portion.

Using tð1Þ rz given by Eq. (5) in equilibrium equation (1), we can get the tangential stress in the inner tube which is given by

sð1Þ yy ðr; zÞ ¼

ðr 2  a2 Þ d2 sð1Þ zz 2 2 dz

ð6Þ

Similarly considering equilibrium of the elemental length dz of the inner tube and adhesive together as depicted in Fig. 2b, we can express trz as

trz ðr; zÞ ¼

ðb2  a2 Þ dsð1Þ zz dz 2r

ð7Þ

Again, using trz in equilibrium equation (1), the circumferential stress syy in the adhesive is obtained as

syy ðr; zÞ ¼

ðb2  a2 Þ d2 sð1Þ zz 2 2 dz

ð8Þ

Note that the circumferential stress in the adhesive is independent of r since we assumed that szz is negligible. Considering equilibrium of elemental length dz of the outer tube as shown in Fig. 2c, the shear stress in the outer tube can be given as a function of the gradient of longitudinal stress in the outer tube ðr 2  d2 Þ dsð2Þ zz dz 2r

Fig. 2. (a) Equilibrium of the inner tube. (b) Equilibrium of the inner tube and adhesive. (c) Equilibrium of the outer tube.

tð2Þ rz ðr; zÞ ¼

The equilibrium of the assembly gives the relationship between f1 and f2 as

Applying the shear stress continuity condition at the adhesive– adherend outer interface (tð2Þ rz at r ¼ c is equal to trz at r ¼ c), we can relate the longitudinal stress gradients of both tubes

2 2 ð2Þ 2 2 f1 ðb2  a2 Þ ¼ f2 ðd2  c2 Þ ¼ sð1Þ zz ðb  a Þ þ szz ðd  c Þ

ð3Þ

Therefore, the longitudinal stress in tube 2 can be expressed by

sð2Þ zz ¼ f2 þ

ðb2  a2 Þ ð1Þ s ðc2  d2 Þ zz

ð9Þ

ð4Þ

dsð2Þ ðb2  a2 Þ dsð1Þ zz zz ¼ 2 dz ðc  d2 Þ dz

ð10Þ

Using the above in Eq. (9), we get ðr 2  d2 Þ ðb2  a2 Þ dsð1Þ zz 2r ðc2  d2 Þ dz

2.1. Stress fields in the adherends and adhesive

tð2Þ rz ðr; zÞ ¼

Considering equilibrium of an elemental length dz of the inner tube as shown in Fig. 2a, the shear stress tð1Þ rz in the inner tube can be expressed by

Now, either continuity of circumferential stress condition or equilibrium equations can be used to obtain the tangential stress in the outer tube as

tð1Þ rz ðr; zÞ ¼

ðr 2  a2 Þ dsð1Þ zz dz 2r

ð5Þ

sð2Þ yy ðr; zÞ ¼

ðr 2  d2 Þ ðb2  a2 Þ d2 sð1Þ zz 2 ðc2  d2 Þ dz2

ð11Þ

ð12Þ

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1.1

interfaces. Thus, the stress components in the inner tube (tð1Þ rz ðr; zÞ,

1

ð1Þ

syy ðr; zÞ), in the adhesive trz ðr; zÞ, syy ðr; zÞ and in the outer tube ð2Þ ð2Þ tð2Þ rz ðr; zÞ, syy ðr; zÞ, szz ðzÞ are expressed in terms of a single unknown stress function sð1Þ zz ðzÞ. Now the statically determinate problem is solved applying the traction boundary conditions prescribed at the ends of overlap. The boundary conditions are

sð1Þ zz ð0Þ ¼ f1 ; trz ðr; 0Þ ¼ 0;

sð1Þ zz ðLÞ ¼ 0

ð13Þ

trz ðr; LÞ ¼ 0; r 2 ½b; c

ð14Þ

2.2. Functionally modulus graded bondline (FMGB) adhesive A few researchers have considered a single step variation of bondline modulus and justified the studies as showing a better joint performance. In this work, the bondline adhesive considered has a multi-step variation of modulus along the bondlength as shown in Fig. 3. The multi-step variation of modulus in the tubular joint can be obtained by applying a number of rings of adhesive of different moduli in the bondline. The stiff ones are applied in the middle portion of the bondline while the flexible ones are applied at the overlap end zones. As the lengths of individual slices tend to zero, the multi-modulus bondline exactly represents the smoothly varying modulus function. The smoothly varying modulus function given by Ef 2 is shown in Fig. 3. The modulus function is approximated such that

Normalised modulus of the adhesive (Ef/Em)

It is clear from the abovementioned equations that both shear and circumferential stresses are continuous across the adherend–adhesive

0.9 0.8

Ef1

0.7

Ef2

0.6

Ef4

Ef3 Ef5

0.5 0.4 0.3 0.2 0.1 0

Ef ðzÞ dzC2E0 L0 þ 2E1 L1 þ    þ 2Em1 Lm1 þ Em Lm

1

Table 1 Geometric and material properties of adhesive and adherends. Material

L 0

0.4 0.6 0.8 Normalised bond length (η=z/L)

Fig. 4. Young’s modulus of adhesive as a function of bondlength.

Item

Z

0.2

E (GPa)

n

75 75 2.7

0.3 10 0.3 – 0.35 –

a (mm)

b (mm)

c (mm)

d (mm)

f2 (MPa)

11 – 11

– 11.2 11.2

– 12.2 –

– 1000 –

ð15Þ

The various modulus profiles examined in the analysis are given below in normalized form and are shown in Fig. 4. These modulus

Tube 1 AU 4G Tube 2 AU 4G Adhesive Araldite AV119

3000 60

FMGB MMB

2500 50 2000 40

1500

τrz [MPa]

Young’s moduluss of adhesive layer [MPa]

L L Em−1 m−1 Lm m−1 Em−1 Em

Ef2: Quadratic modulus function

L2 E2

Multi−modulus adhesive

E2 L2

1000 E1 L1

L1 E 1

30

20

10

500 L0 E0

E0 L0

0

60

−10

0 0

10

20 30 40 Bond Length [mm]

50

Fig. 3. Representation of multi-modulus bondline adhesive as a functionally modulus graded bondline adhesive.

0

10

20 30 40 Bond Length (L) [mm]

50

Fig. 5. Shear stress distribution at the midplane of adhesive layer.

60

ARTICLE IN PRESS S. Kumar / International Journal of Adhesion & Adhesives 29 (2009) 785–795

functions are arbitrarily chosen:

FMGB MMB

300

Ef 1

200

"    # Ef 1 Em z 1 2 ¼ ¼ exp 4ln  Em E0 L 2

100 Ef 2 ¼

ð16Þ

  Ef 2 ðE0  Em Þ z2 z E0 ¼4  þ Em Em Em L L

ð17Þ

0

Ef 3 ¼

−100

"     # Ef 3 ðEm  E0 Þ z 1 4 z 1 2 þ1 ¼8 2    Em Em L 2 L 2

−200 Ef 4 ¼

−300 0

10

20 30 40 Bond Length (L) [mm]

50

   # z 1 6 z 1 4 þ1 þ   L 2 L 2

80 τrz [MPa]

100

FMGB MMB

50

60 FMGB MMB

40 20

0

0 0

2

4 6 L [mm]

8

10

0

60

5

10 L [mm]

15

20

60

50

FMGB MMB

40 τrz [MPa]

40 30 20

FMGB MMB

10

20 0 −20

0 0

10

20

30

0

10

L [mm] 60

20 30 L [mm]

40

50

60 FMGB MMB

FMGB MMB

40 τrz [MPa]

40 20 0

20 0

−20

−20 0

20

40 60 L [mm]

ð19Þ

ð20Þ

100

150

τrz [MPa]

"

Ef 5 ¼ Em

200

τrz [MPa]

Ef 4 64 ðE0  Em Þ ¼ Em 5 Em

ð18Þ

60

Fig. 6. Peel stress distribution at the midplane of adhesive layer.

τrz [MPa]

σθθ [MPa]

789

80

100

0

50

100 L [mm]

150

Fig. 7. Shear stress distribution at the midplane of adhesive as a function of bondlength.

200

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S. Kumar / International Journal of Adhesion & Adhesives 29 (2009) 785–795

1000

400 FMGB MMB

0 −500

0 −200

−1000

−400 0

2

4 6 L [mm]

8

10

0

400

5

15

20

FMGB MMB

σθθ [MPa]

200

0 −200

0 −200

−400

−400 0

10

20

30

0

10

L [mm] 400

20 30 L [mm]

40

50

400 FMGB MMB

FMGB MMB

200 σθθ [MPa]

200 σθθ [MPa]

10 L [mm]

400 FMGB MMB

200 σθθ [MPa]

FMGB MMB

200 σθθ [MPa]

σθθ [MPa]

500

0 −200

0 −200

−400

−400 0

20

40 60 L [mm]

80

100

0

50

100 L [mm]

150

200

Fig. 8. Peel stress distribution in the adhesive as a function of bondlength.

3. Variational method

60

O ¼ O1 þ O2 þ O3

ð21Þ

Ef1 Ef2 Ef3 Ef4 Ef5

50

40

τrz [MPa]

The variational method can be based either on an assumed infinitesimal displacement field in conjunction with the principle of minimum potential energy or on an assumed small stress variation associated with the complementary energy theorem [33]. In the current work, the second route has been pursued, following the analysis developed by Shi and Cheng [15] for tubular-lap joints and extending it to bonded systems with a FMGB adhesive. The problem can be defined as obtaining a solution for sð1Þ zz by minimizing the complementary energy of the bonded system where the stresses for the adherends and FMGB adhesive have been defined in terms of a single stress function sð1Þ zz . The admissible stress states are those which satisfy continuum differential equations of equilibrium, stress boundary conditions, stress-free end conditions of the joint and stress continuity at the adherend–adhesive interfaces. Once sð1Þ zz has been obtained, then all the stress components in the adhesive can be obtained. A true lower bound solution can be obtained in this way. The complementary energy of the joint comprising transversely isotropic adherends and a functionally modulus graded isotropic adhesive can be given by O, where

30

20

10

0

−10 0

10

20 30 Bond Length (L) [mm]

40

50

Fig. 9. Shear stress distribution at the midplane of adhesive for different modulus functions.

ARTICLE IN PRESS S. Kumar / International Journal of Adhesion & Adhesives 29 (2009) 785–795

O1 is the complementary energy of the inner tube, O2 is the complementary energy of the outer tube and O3 is the complementary energy of the adhesive. O1 , O2 and O3 are given

Ef1 Ef2 Ef3 Ef4 Ef5

300

200

791

by

O1 ¼ p

Z

L

Z

0

b

"

sð1Þ zz 2 El1

a

þ

sð1Þ yy 2

þ

sð2Þ yy 2

Et1



# 2ntl1 ð1Þ ð1Þ tð1Þ 2 szz syy þ rz r dr dz Et1 G1

ð22Þ



# 2ntl2 ð2Þ ð2Þ tð2Þ 2 szz syy þ rz r dr dz Et2 G2

ð23Þ

σθθ [MPa]

100

O2 ¼ p

Z

L

Z

0

0

O3 ¼ p

−100

Z

d

"

El2

c

L

Z

0

sð2Þ zz 2

c b

Et2

1 ½s2 þ 2ð1 þ nÞt2rz r dr dz EðzÞ yy

ð24Þ

For an isotropic system, O1 and O2 become

−200

O1 ¼ p

Z

L

Z

0

b

1 ð1Þ ð1Þ ð1Þ ð1Þ ½s 2 þ sð1Þ yy 2  2n1 szz syy þ 2ð1 þ n1 Þtrz 2r dr dz E1 zz ð25Þ

d

1 ð2Þ ð2Þ ð2Þ ð2Þ ½s 2 þ sð2Þ yy 2  2n2 szz syy þ 2ð1 þ n2 Þtrz 2r dr dz E2 zz ð26Þ

a

−300 0

10

20 30 Bond length (L) [mm]

40

50

O2 ¼ p

Z

L 0

Fig. 10. Peel stress distribution in the adhesive for different modulus function profiles.

Z c

E1/E2=0.5

E1/E2=1.0 60

80 FMGB MMB τrz [MPa]

τrz [MPa]

60 40 20

FMGB MMB

40 20 0

0 −20

−20 0

20

40 60 L [mm]

80

100

0

20

E1/E2=1.5

100

80

100

80

100

E1/E2=2.5

FMGB MMB

40

FMGB MMB

60 τrz [MPa]

τrz [MPa]

80

80

60

20 0

40 20 0 −20

−20 0

20

40 60 L [mm]

80

100

0

20

E1/E2=2.8

40 60 L [mm] E1/E2=3.0

80

80 FMGB MMB

FMGB MMB

60 τrz [MPa]

60 τrz [MPa]

40 60 L [mm]

40 20

40 20 0

0

−20

−20 0

20

40 60 L [mm]

80

100

0

20

40 60 L [mm]

Fig. 11. Shear stress distribution at the midplane of adhesive as a function of stiffness mismatch.

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ð1Þ ð2Þ ð2Þ ð2Þ Introducing expressions for stresses (sð1Þ yy , trz , szz , syy , trz , syy and trz ) in O1 , O2 and O3 and integrating the resulting expressions over the radius r, the energy functional for an isotropic bonded system becomes

O¼p

Z

L 0

0

L

ð1Þ 2

Þ  dz þ p

Z

L

0

½b5 szz00

ð1Þ

þ b6 sð1Þ zz þ b7  dz

ð28Þ

    d @j d2 @j þ ¼0 ð1Þ 2 dz @szz0 ð1Þ dz @szz00

3

dz

þ ðb3  b4 ðzÞ þ b200 ðzÞÞ

d2 sð1Þ zz 2

dz

dsð1Þ b6 zz þ b1 sð1Þ zz þ dz 2 ð30Þ

ð29Þ

4. Results and discussion Initially, analysis was performed considering similar isotopic adherends whose geometrical and mechanical properties are given in Table 1 (aluminium adherends and AV119 adhesive properties were taken from Nemes et al. [16]) with a FMGB adhesive of modulus function Ef 2 (Em ¼ 2700 MPa, E0 ¼ 280 MPa and L ¼ 60 mm). The results were then compared with a MMB adhesive joint under the same load. The shear and peel stress intensities both at interface and as well as at the midplane of adhesive are much smaller and their distribution along the bondline is more uniform than those of a MMB adhesive joint. Figs. 5 and 6 show the shear and peel stress distribution at the midplane of the bondline, respectively. Note that the shear stress distribution is symmetric and the peel stress distribution is antisymmetric. A very small deviation from the symmetric

E1/E2=1.0

E1/E2=0.5 400

600 FMGB MMB σθθ [MPa]

σθθ [MPa]

400 200 0 −200 −400

FMGB MMB

200 0 −200 −400

0

20

40 60 L [mm]

80

100

0

20

E1/E2=1.5

40 60 L [mm]

80

100

80

100

80

100

E1/E2=2.5 200

400 200

σθθ [MPa]

σθθ [MPa]

0 −200

0 −200

FMGB MMB

FMGB MMB −400

−400 0

20

40 60 L [mm]

80

100

0

20

E1/E2=2.8

40 60 L [mm] E1/E2=3.0

200

200 σθθ [MPa]

@szz

þ b40 ðzÞ

d3 sð1Þ zz

ð27Þ

0 ð1Þ 00 ð1Þ jðsð1Þ ; zÞ dz zz ; szz ; szz

σθθ [MPa]

 ð1Þ

dz

þ 2b20 ðzÞ

The above differential equation can be solved with the stress boundary conditions given by Eqs. (13) and (14).

We now need the differential equation satisfied by the function sð1Þ zz which minimizes the above functional. Performing variational calculus on the above functional yields @j

4

¼0

where the constant coefficients b1 , b3 , b5 , b6 , b7 and the variable coefficients b2 ðzÞ and b4 ðzÞ depend on geometrical and material properties and the loading conditions of the bonded joint. Here, 2 0 ð1Þ szz00 ð1Þ ¼ d2 sð1Þ ¼ dsð1Þ zz =dz , szz zz =dz. The above functional can be expressed as a function of j Z

d4 sð1Þ zz

00 ð1Þ 2 ½b1 sð1Þ Þ þ b3 szz00 ð1Þ sð1Þ zz 2 þ b2 ðzÞðszz zz

þb4 ðzÞðszz0

O¼

b2 ðzÞ

0 −200

0 −200

FMGB MMB

FMGB MMB −400

−400 0

20

40 60 L [mm]

80

100

0

20

40 60 L [mm]

Fig. 12. Peel stress distribution at the midplane of adhesive as a function of stiffness mismatch.

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distribution of shear stress and anti-symmetric distribution of peel stress arises due to the geometric stiffness mismatch between the two tubes as although they are of same thickness, they necessarily have different diameters. If modulus function Ef 5 is employed (i.e. MMB), stress distribution in the adhesive is in line with the stress distribution predicted by Shi and Cheng [15] and Nemes et al. [16]. The shear and peel stress concentration factors ðxsh Þ and ðxpe Þ, respectively, are given by

xsh ¼

2prm Ltmax rz P

ð31Þ

xpe ¼

2prm Lsmax yy P

ð32Þ

where tmax and smax rz yy are the maximum shear and peel stresses in the bondline, respectively, and rm ¼ ðb þ cÞ=2 is the mean radius of the adhesive. xsh reduces from 1.53 to 1.28, while xpe reduces from 9.10 to 3.03 by replacing a MMB adhesive with a FMGB adhesive. The peak peel stress in the FMGB adhesive appears close to the overlap ends, while it appears exactly at the overlap ends in a MMB adhesive joint. This is because the stiffness jump in FMGB joint is gradual than in MMB joint. 4.1. Effect of bondlength (L) Stress analyses have been carried out by varying the bondlength from 10 to 200 mm, and adopting the quadratic modulus

4.2. Effect of modulus function Different modulus function profiles have been examined to reduce the stress peaks and gradients in the FMGB adhesive. The shear and peel stress distributions for different modulus functions are shown in Figs. 9 and 10, respectively. The shear stress intensity is less for modulus function Ef 3 while the peel stress intensity is

ta=0.1mm

80

FMGB MMB

FMGB MMB

60 τrz [MPa]

60 τrz [MPa]

function profile Ef 2 for the adhesive in order to study the effect of bondlength on stress distribution. Figs. 7 and 8 show the shear and peel stress distributions at the midplane of adhesive for selected values of bondlength. At small bondlengths, the shear stress distribution in both FMGB and MMB adhesives are parabolic, with stress peaks at mid-bondlength. Both shear and peel stress peaks decrease and the distribution becomes more uniform with increase of bondlength. Beyond a certain bondlength, the increase of bondlength does not reduce the shear stress appreciably. On the other hand, an increase of bondlength reduces the peel stress upto a certain bondlength and increases thereafter. Therefore, the bondlength at which the peel stress starts to increase with increase of bondlength is considered to be an optimum bondlength. The optimum bondlength in this case is L ¼ 100 mm. Both shear stress and peel stress peaks move towards the overlap ends with an increase of bondlength. The shear stress distribution in the FMGB is more severe than in the MMB joint for small bondlengths (Lp20 mm). However, the stress distribution does not change in the MMB joint after L ¼ 50 mm, for the variables used here.

ta=0.05mm

80

40 20 0

40 20 0

−20

−20 0

20

40 60 L [mm]

80

100

0

20

ta=0.2mm

80

100

80

100

80

100

ta=0.3mm

FMGB MMB

40

τrz [MPa]

τrz [MPa]

40 60 L [mm]

60

60

20

FMGB MMB

40 20 0

0

−20

−20 0

20

40 60 L [mm]

80

100

0

20

ta=0.4mm 40

40 60 L [mm] ta=0.5mm

40

FMGB MMB τrz [MPa]

τrz [MPa]

793

20 0

FMGB MMB

20 0 −20

−20 0

20

40 60 L [mm]

80

100

0

20

40 60 L [mm]

Fig. 13. Shear stress distribution at the midplane of adhesive as a function of bondline thickness.

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S. Kumar / International Journal of Adhesion & Adhesives 29 (2009) 785–795

ta=0.1mm

ta=0.05mm 1000

FMGB MMB

400 σθθ [MPa]

σθθ [MPa]

FMGB MMB 500 0

200 0 −200

−500

−400 0

20

40 60 L [mm]

80

100

0

20

ta=0.2mm

40 60 L [mm]

80

100

80

100

80

100

ta=0.3mm

FMGB MMB

200

FMGB MMB

200 σθθ [MPa]

σθθ [MPa]

300

0 −200

100 0 −100 −200

0

20

40 60 L [mm]

80

100

0

20

ta=0.4mm

σθθ [MPa]

σθθ [MPa]

ta=0.5mm 200

FMGB MMB

200

40 60 L [mm]

100 0 −100 −200

FMGB MMB

100 0 −100 −200

0

20

40 60 L [mm]

80

100

0

20

40 60 L [mm]

Fig. 14. Peel stress distribution at the midplane of adhesive as a function of bondline thickness.

less for modulus function Ef 2 . If we choose a stiff MMB adhesive to have maximum shear strength, it will fail due to high peel stresses. Unlike the MMB adhesive, the modulus function of the FMGB adhesive can be so tailored simultaneously to achieve both shear and peel strengths.

4.3. Effect of stiffness mismatch Figs. 11 and 12 show the shear and peel stress distribution at the midplane of adhesive as a function of stiffness mismatch between two tubes. Note that the shear stress distribution looses its symmetry and peel stress distribution looses its anti-symmetry about mid-bondlength when E1 aE2 . The stress distribution is compared with the MMB adhesive and found that the stress distribution in FMGB adhesive is much less than MMB adhesive.

4.4. Effect of adhesive thickness Analyses were performed varying the thickness of the adhesive while keeping the thickness of the two tubes the same ðt1 ¼ t2 Þ. The influence of adhesive thickness variation on the shear and peel stresses and their distribution is shown in Figs. 13 and 14, respectively. As the thickness of the adhesive increases, both shear and peel stress peaks reduce and their distribution becomes more uniform. The stress distribution in the FMGB is compared with that of MMB adhesive.

5. Conclusions A simple analytical framework has been provided to study stress intensity and its distribution in a FMGB adhesive joint based on a variational method, which minimizes the complimentary energy of the bonded system. It has been observed that the shear and peel stress concentrations at the overlap ends in the FMGB adhesive joints are much less than those of MMB adhesive joints under the same axial load. Reduced shear and peel stress concentrations provide improved joint strength and lifetime. Analysis also indicated that an optimized joint performance can be achieved by grading the modulus of the bondline adhesive. It has been observed through parametric evaluation that shear and peel stress peaks and its gradients in the bondline can be significantly reduced by selectively perturbing the geometrical and material properties of the bonded system. References [1] Volkersen O. Luftfahrtforschung 1938;15:41–7. [2] Goland M, Reissner E. The stresses in cemented joints. J Appl Mech Trans ASME 1944;66:A17–27. [3] Hart-Smith LJ. An engineer’s viewpoint on design and analysis of aircraft structural joints. J Aerosp Eng GDD IMechE 1995;209:105–29. [4] Wooley GR, Carver DR. Stress concentration factors for bonded lap joints. J Aircraft 1971;8:817–20. [5] Adams RD, Peppiatt NA. Stress analysis of adhesively bonded lap joints. J strain Anal 1974;9:185–96. [6] Crocombe AD, Adams RD. Influence of spew fillet and other parameters on the stress distribution in the single-lap joint. J Adhesion 1981;13:141–55.

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[7] Adams RD, Peppiatt NA. Stress analysis of adhesively bonded tubular lap joint. J Adhesion 1977;9(1):1–18. [8] Nagaraja YR, Alwar RS. Viscoelastic analysis of an adhesive-bonded plane lap joint. Comput Struct 1980;11:621–7. [9] Harris JA, Adams RD. Strength prediction of bonded single lap joints by nonlinear finite element methods. Int J Adhesion Adhesives 1984;4:65–78. [10] Pickett AK, Hollaway L. The analysis of elastic–plastic adhesive stress in bonded lap joints in FRP structures. Compos Struct 1985;4:135–60. [11] Oplinger DW. A layered beam theory for single lap joints. Army Materials Technology Laboratory Report, MTL TR91-23; 1991. [12] Tsai MY, Morton J. On analytical and numerical solutions to the single-lap joint. Center for Adhesive and Sealant Science, VPI, Report CASS/ESM-92-8; 1993. [13] Deb A, et al. An experimental and analytical study of mechanical behaviour of adhesively bonded joints for variable extension rates and temperatures. Int J Adhesion Adhesives 2007;28:1–15. [14] Allman DJ. A theory for elastic stresses in adhesive bonded lap joints. Q J Mech Appl Math 1977;30:415–36. [15] Shi YP, Cheng S. Analysis of adhesive-bonded cylindrical lap joints subjected to axial load. J Eng Mech 1993;119(3):584–602. [16] Nemes O, et al. Contribution to the study of cylindrical adhesive joining. Int J Adhesion Adhesives 2006;26:474–80. [17] Her SC. Stress analysis of adhesively bonded lap joints. Compos Struct 1999;47:673–8. [18] Apalak MK. Geometrically nonlinear analysis of adhesively bonded corner joints. J Adhesion Sci Technol 1999;13(11):A17–27. [19] Penado FE. Analysis of singular regions in bonded joints. Int J Fract 2000;105(1):1–25. [20] Erdogan F, Ratwani M. Stress distribution in bonded joints. J Compos Mater 1971;5:378–793.

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