Open access peer-reviewed chapter

# A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

By Sontipee Aimmanee

Submitted: July 13th 2017Reviewed: November 7th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.72288

## Abstract

### Keywords

• coupler joint
• lap joint
• elasticity
• finite-segment method

## 1. Introduction

Structures usually need to have joints connecting each part together due to the limitation of manufacturing, transportation, and installation. These structures are generally vulnerable at the joints because of the stress concentrations from material discontinuity. There are many types of joint, such as mechanical joints, welding joints, and adhesive-bonded joints. Over the other kinds of joints, adhesive-bonded joints have advantages due to less stress concentration, higher capability of joining dissimilar materials, lighter weight, and better corrosion resistance. Nonetheless, stress concentration in the adhesive layer is still existing. The adhesive stress distribution is locally nonuniform and always highest at the edges of the bonding region. Thus, in order to use an adhesive-bonded joint safely, it is important to predict the developed adhesive stress accurately. A good analysis also provides the understanding of the joint behavior, yielding a design for improving the joint performance by decreasing the joint stress concentration.

There are lots of literature dealing with the stresses in lap joints and coupler joints between two tubular adherends. Among several types of adhesive-bonded joint, cylindrical or tubular joints subjected to axial, torsional, and external as well as internal pressure loads have been one of the main focus of mechanics of adhesion research for a half century due to their popular usage in many engineering applications. The analytical modeling and finite-element analysis are the two popular approaches for predicting the stresses developed in adhesive and adherends. In order to recognize the progress in this field, some examples of important work in mathematical modeling are given below.

## 2. Elasticity theory of a laminated cylindrical structure

A bonded-coupler joint is illustrated in Figure 1(a). The joint consists of two inner tubes (adherend part 1), a coupler of length 2 L (or adherend part 2), and an adhesive layer. The cylindrical coordinates xθrdepicted in Figure 2 are used to describe the joint geometry. Because of symmetry about the cross-sectional plane in the middle, only half of the coupler joint is demonstrated in Figure 1(b). Either half is equivalent to a single tubular lap joint and applicable for modeling and analysis. The inner tubes are considered to be made of an ordinary material, such as an isotropic metal or a more sophisticated material, namely, orthotropic material or laminated composite. On the contrary the coupler is proposed to be fabricated from a symmetric-balanced laminated composite with variable fiber orientation in the x direction.

For the sake of generality, this section discusses the elasticity theory of a laminated cylindrical tube [18]. A sketch of a general open-ended, cylindrical, laminated N-layer tube subjected to uniform loads is shown in Figure 2. Each layer is made of a unidirectional fiber-reinforced composite material. The principal material coordinates (1, 2, 3), whose axes are mutually orthogonal, are defined along the fiber orientation, tangent, and normal to the tube surface, respectively. The layers in the tube are perfectly bonded between each other. Evidently, this considered laminated cylinder can be simply degenerated into a single isotropic or orthotropic tube by letting N = 1 and employing the related elastic properties. For the tube with axisymmetric geometry and circumferentially independent material properties under a uniform load, the strain–displacement relations in the kth layer in the cylindrical coordinates are

εxk=ukx,εθk=wkr,εrk=wkrγθrk=vkrvkr,γxrk=ukr,γk=vkxE1

where εand γdenote normal and shear strains, respectively. u, v, and w are displacements in axial, tangential, and radial directions, respectively. Superscript (k) indicates that the corresponding quantities are in the kth layer.

According to the prescribed loading conditions and constant fiber orientation, the normal stresses, σ, and the shear stresses, τ, are independent of x and θ. The equilibrium equations in the kth layer along the r-, θ-, x-directions are reduced to ordinary differential equations with respect to r, respectively, as

σrkr+1rσrkσθk=0E2
τθrkr+2rτθrk=0E3
τxrkr+1rτxrk=0E4

The stresses and strains in the kth layer in xθrcoordinates expressed in Eqs. (1)(4) can be transformed to those in the principal material coordinates (1, 2, 3) as follows:

σ123k=Tkσxθrk,ε123k=TkεxθrkE5

where σ123kand ε123kare tensorial stress and tensorial strain components, respectively. Tkis transformation matrix of the kth layer as shown in Eq. (6), in which mk=cosØkand nk=sinØk. Økis fiber angle of the kth layer as shown in Figure 2.

Tk=mk2nk20002mknknk2mk20002mknk001000000mknk0000nkmk0mknkmknk000mk2nk2E6

The constitutive relation in the kth layer in the cylindrical coordinates can be written as

σxθrk=C¯kεengxθrkE7

In the above, C¯kis the transformed stiffness matrix, and εengxθrkis engineering strain components in the global cylindrical coordinate system. The transformed stiffness matrix C¯kcan be evaluated as

C¯k=Tk1CkRTkR1E8

where Ck, as shown in Eq. (9), is the stiffness matrix in the principle material coordinate system in the kth layer.

Ck=1/E1kυ12k/E1kυ31k/E3k000υ21k/E2k1/E2kυ32k/E3k000υ13k/E1kυ23k/E2k1/E3k0000001/G23k0000001/G13k0000001/G12k1E9

Ekand Gkare Young’s modulus and shear modulus, respectively. Ris the Reuter’s matrix, which is defined as

R=100000010000001000000200000020000002E10

With the strains defined in Eq. (1), three out of six equations of the compatibility in the cylindrical coordinates described in [19] are automatically satisfied. Solving the equilibrium equations in Eqs. (2)(4) and using the strain–displacement relations in Eq. (1), the constitutive relation in Eq. (7), the remaining three compatibility equations, as well as the displacement continuity between each layer yield the displacement expressions in the kth layer of the laminated tube as illustrated in Eqs. (11)(13):

ukx=εx0xE11
vkxr=γ0xrE12
wkr=A1krλk+A2krλk+Γkεx0r+Ωkγ0r2E13

In the above, εx0and γ0are axial strain and angle of twist per unit length constants, respectively. A1kand A2kare the integration constants in the kth layer. λk, Γk,and Ωkare described in Eq. (14) in terms of components of the transformed stiffness matrix in the kth layer:

λk=C¯22kC¯33k,Γk=C¯12kC¯13kC¯33kC¯22k,Ωk=C¯26k2C¯36k4C¯33kC¯22kE14

Note that when a layer is made of 0o fiber orientation, the stiffness coefficients C¯16k, C¯26k, and C¯36kare zero and C¯12k=C¯13kas well as C¯22k=C¯33k. As such, Eq. (13) is degenerated to become wkr=A1krk+A2kr1k[13].

For an N-layer laminated tube, there are 2 N + 2 unknown integration constants to be evaluated. Therefore, 2 N + 2 equations are required to solve for the constants. The first four equations written in Eqs. (15)(18) are obtained from two equations of the force equilibrium with the external loads and two equations from the surface traction boundary conditions. Note that F in Eq. (15) is an axial force, T in Eq. (16) an applied torque, piin Eq. (17) a normal traction or internal pressure on the inner surface, and poin Eq. (18) a normal traction or external pressure on the outer surface. The remaining 2 N-2 equations can be obtained from N-1 continuity conditions of the interfacial radial stresses, σr, and N-1 continuity conditions of the interfacial radial displacements, w, as shown in Eqs. (19) and (20), respectively:

R0RN2πσxrdr=2πk=1NRk1Rkσxkrdr=FE15
R0RN2πτr2dr=2πk=1NRk1Rkτkr2dr=TE16
σr1R0=piE17
σrNRN=p0E18
σrkRk=σrk+1Rk(k=1,2,3,…,N-1)E19
wkRk=wk+1Rk(k=1,2,3,…,N-1)E20

Eqs. (15)(20) give the system of algebraic equations written in matrix form as

k11k12k13k21k22k23k31k41k32k42k33k43k1,2N+2k2,2N+2k3,2N+2k4,2N+2k2N+2,1k2N+2,2k2N+2,3k2N+2,2N+2FTpipo000=ϵx0γ0A11A21A1NA2NE21

where kijij=1232N+2are the coefficients obtained from the equations above. By solving Eq. (21), the constants ϵx0,γ0,and A11,A21,,A1N,A2Ncan be obtained. Subsequently, all displacements, strains, and stresses are calculated by using Eqs. (11)(13), (1), and (7), respectively.

## 3. Formulation of an equivalent lap joint model

### 3.1. Derivation of governing equations

In order to derive the governing equations, let us initially consider the torque transmission through a coupler joint. The applied torque T is assumed to distribute only in the adherend part 1 and adherend part 2 as denoted as T1and T2, respectively. Hence, the applied torque T can be written as

T=T1+T2E22

To determine the variation of the T2along the bonding length, the adherend 2 is divided into elements with an infinitesimal length dx. The equilibrium between the resultant torque in the element and the adhesive hoop shear stress can be expressed as follows:

12πR2i2dT2xdx=τθra=GaγθraE23

In Eq. (23), Gais shear modulus of adhesive. By considering the deformation of an adhesive element on a cross-sectional plane in the overlap region of the perfectly bonded joint, the kinematic condition in the adhesive can be written as

γθra=v2iv1otaE24

and its derivative with respect to x is.

Combining Eqs. (23) and (25) yields the first governing equation:

d2T2xdx2=2πR2i2Gataγ2iγ1oE26

Next, consider equilibrium of resultant axial force. When the joint is subjected to tension or compression loads, the resultant axial force in the adherend 1, F1,and in adherend 2, F2,are produced at any given cross section in the overlap region, similar to Eq. (22). The force equilibrium is

F=F1+F2E27

The variation of the F2along the length can be examined by considering an infinitesimal elements in adherend part 2 with the differential length dx. The equilibrium between resultant axial force in the element and the adhesive longitudinal shear stress τxracan consequently be expressed as follows:

12πR2idF2xdx=τxra=GaγxraE28

By considering compatibility of the joint, it can be shown that.

Combining Eqs. (28) and (29) yields the axial force governing equation:

d2F2xdx2=2πR2iGataεx2iεx1oE30

Next, interacting through the adhesive thickness, the resultant normal traction acting on the outer surface of adherend 1, p1o, and that exerting on the inner surface of adherend 2, p2i,are generated. Under the assumption of thin adhesive layer, the resultant normal tractions p1oand p2iare related to each other as

p1o=p2iE31

Lastly, instead of directly equating adhesive radial normal stress to normal traction in (31), σracan be more accurately determined by the equilibrium equation in cylindrical coordinates of the adhesive layer showing in Eq. (32):

σrr+1rσrσθ+1rτθrθ+τxrx=0E32

With the conditions of axisymmetry, the equilibrium equation is reduced to

1R2iσraσθa+τxrax=0E33

According to the study conducted in [6], σθais observed to have the same distribution as σraso they are legitimately regarded as being proportional to each other via adhesive normal stress ratio α. Their relation can be mathematically expressed in Eq. (34):

σθa=ασraE34

As a consequence, the equilibrium equation in Eq. (32) can then be written as

σra=12π1αd2F2xdx2E35

### 3.2. Implementation of elasticity theory for adherends

First, further modification of the torque governing equation of Eq. (26) is performed, adherend in-plane shear strains γ1oand γ2imust be expanded in terms of the internal resultant loads. It can be seen that they are equal to γ01R1oand γ02R2i, respectively, where γ01and γ02are denoted for γ0of adherend parts 1 and 2. Utilizing Eq. (21) yields the relations:

R1ok211F1+k221T1+k231p1i+k241p1o=γ,F1oF1+γ,T1oT1+γ,pi1op1i+γ,po1op1o=γ1oE36
R2ik212F2+k222T2+k232p2i+k242p2o=γ,F2iF2+γ,T2iT2+γ,pi2ip2i+γ,po2ip2o=γ2iE37

where quantities k2X1and k2X2, where X=1,2,3, and 4 are the first four elements compliances in the second row of matrix kin Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1 and 2, respectively. γ,F1o, γ,T1o, γ,pi1o, γ,po1o, γ,F2i, γ,T2i, γ,pi2i, and γ,po2iare in-plane shear strains per unit load on the outer interfacial surface of adherend part 1 and inner interfacial surface of adherend part 2, respectively.

By substituting Eqs. (36) and (37) into the governing equation Eq. (26) and utilizing Eqs. (22), (27), and (31), the governing equation in term of T2becomes

d2T2xdx2=KFF2x+KTT2x+Kpp2ix+KCE38

where the parameters KF, KT, Kp,and KCare.

KF=2πR2i2Gataγ,F2i+γ,F1o,KT=2πR2i2Gataγ,T2i+γ,T1o,Kp=2πR2i2Gataγ,pi2iγ,po1o,KC=2πR2i2Gataγ,po2ipoγ,pi1opiγ,F1oFγ,T1oTE39

Accompanying with the boundary conditions of Eq. (40), which are implied that torque in adherend part 2 is zero at x = 0 and fully transmitted at x = L, Eq. (38) is well-defined for solving resultant torque in the adherend part 2,T2:

T20=0,T2L=TE40

Second, analogous to Eqs. (36) and (37), adherend in-plane normal strains εx1oand εx2imust also be written in terms of the internal resultant loads. Again, using the expression in Eq. (21), the following expressions are obtained:

k111F1+k121T1+k131p1i+k141p1o=εx,F1oF1+εx,T1oT1+εx,pi1op1i+εx,po1op1o=εx1oE41
k112F2+k122T2+k132p2i+k142p2o=εx,F2iF2+εx,T2iT2+εx,pi2ip2i+εx,po2ip1o=εx2iE42

In the above, k1X1and k1X2, where X=1,2,3,and 4 are the first four elements in the first row of matrix kin Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1 and 2, respectively. εx,F1o, εx,T1o, εx,pi1o, εx,po1oεx,F2i, εx,T2i, εx,pi2i, and εx,po2iare newly denoted to indicate the physical meaning of the parameters. They represent the in-plane normal strains due to unit load on the adhesive-interfacial surface in the adherends. The unit load quantities are distinguished by after-comma subscripts F, T, or p. Quantities εx1oand εx2iare εx0of adherends 1 and 2, respectively.

Combining the governing equations Eqs. (30), (41), and (42), as well as the load equilibriums in Eqs. (22), (27), and (31), yields a new form of the axial force governing equation:

d2F2xdx2=kFF2x+kTT2x+kpp2ix+kCE43

where the parameters kF, kT, kp, and kCare.

kF=2πR2iGataεx,F2i+εx,F1o,kT=2πR2iGataεx,T2i+εx,T1okp=2πR2iGataεx,pi2iεx,po1o,kC=2πR2iGataεx,po2ipoεx,pi1opiεx,F1oFεx,T1oTE44

To specify the boundary conditions of Eq. (43), one can consider the disappearance of F2at x = 0. This is because the left end surfaces of the adherend are normal traction-free. The right end at x = L on the other hand must take the full axial load F if there exists the application of external axial load. Thus, in the mathematical form, these boundary conditions are as follows:

F20=0,F2L=FE45

Finally, it should be noted that the occurrence of p2iin Eq. (43) is closely related with the existence of F2because the tensile or compressive loading can induce the peeling traction. Therefore, p2iin the equation is considered as unknown. However, it is possible to find the approximated relation between the two variables by letting T2and p2ibe zero; F2then can be evaluated and expressed in Eq. (46):

F2x=a0ekFx+b0ekFxkCkFE46

where a0and b0are integration constants.

Reinstating the resultant normal traction p2iand substituting Eq. (46) into Eq. (43), it is found that p2ican be simply estimated as

p2ia1ekFx+b1ekFxE47

in which, a1and b1are unknown parameters. In order to determine these two parameters, two more boundary conditions are required from zero longitudinal shear stress τxrain the adhesive layer at left and right ends as shown in Eq. (48):

dF2dxx=0=0dF2dxx=L=0E48

Up to this point, the unified formulation of an analysis of adhesive-bonded coupler joint has been developed. The model can be universally used to determine the stresses in the adhesive layer for any particular load case previously mentioned. To elaborate the applicability of the model for each loading condition, i.e., torsion, axial, or external and internal pressure, the pertinent details are given below:

• For torsional load, the secondary variables, namely, F2and p2i,are initially neglected in Eq. (38). In addition, F=p1i=p2o=0. Consequently, T2and τθracan be evaluated. Subsequently, F2, τxra, and σraand can be recovered and computed by employing the full form of Eqs. (43) and (47), (28), and (35), respectively.

• For axial load, the secondary variable T2is initially neglected. Additionally, T=p1i=p2o=0.The primary variables F2, τxra, and σraare solved by using the governing equation, Eqs. (43), (47), (28), and (35). T2and τθrais later calculated from the full form of Eqs. (38) and (23), respectively.

• For external and internal pressure, the secondary variable T2is firstly omitted. In this case T=F=0.If only external pressure is present, p1i=0, whereas if only internal pressure exists, p2o=0. The first variables F2, τxra, and σraare solved by using the governing equation, Eqs. (43), (47), (28), and (35). T2and τθracan be later recovered the same way as those for the axial load.

## 4. Results

E1(GPa)1.30200.00128.00
E2, E3(GPa)1.30200.0010.00
G12, G13(GPa)0.4676.9050.00
G23(GPa)0.4676.9050.00
υ12, υ130.410.300.28
υ230.410.300.47

### Table 1.

Mechanical properties of materials in the principle material coordinate [20].

In the case of torsional loading, the joints are assumed to have a torque of 1 N.m as an input without loss of generality. Also, the adhesive mean shear stress τmain Eq. (49) is utilized to normalize the induced adhesive hoop shear stress in the coupler joint:

τma=T2πR2i2LE49

The resultant torque of the adherend part 2 and normalized adhesive hoop shear stress can be calculated and plotted in Figures 4 and 5, respectively. It can be noticed that the joints considered develop the nonconstant slopes in Figure 4 with relatively high torque gradients at both ends. This is equivalent to the peak adhesive hoop shear stresses at x = 0 and L in Figure 5. Note that the torque and stress distributions for Ø2= 0°and 30°are identical to those for 90°and 60°, respectively, so they cannot be clearly seen. In addition, since the fiber orientation of 45°is the most suitable angle to withstand the in-plane shear loads, the coupler with Ø2= 45°provides the lowest magnitude of τθraas expected.

Stress distributions in the composite coupler are illustrated in Figure 6 for the case of Ø2= 30o. The normal stress in the fiber direction σ11in Figure 6(a) is the dominant stress component compared to those in the other directions. The radial normal stress σ33illustrated in Figure 6(b) is relatively small at x = 0 mm and noticeably larger at x = 40 mm. In addition, σ33at adhesive-coupler interface r = 10.1 mm or σrais also minimal because of being the secondary effect. Note that the number of segments used to calculate the stresses in Figure 6 is 40. Finally, the figure shows that the developed model is capable of capturing the variation of these two stress components through the coupler thickness thanks to the advantage of the elasticity theory.

When the coupler joints are subjected to an axial loading, a tension force with the magnitude of 1 N is used in calculation. For this particular case, the adhesive mean shear stress τmain Eq. (50) is adopted to normalize the induced longitudinal shear stress and radial normal stress in the adhesive. Same as above, the normalized stresses can be utilized to indicate the distribution intensity of load transfer within the joints:

τma=F2πR2iLE50

Figures 7 and 8 show the effect of fiber orientation on the distributions of F2and τxraalong the overlap region, respectively. Observation in Figure 8 reveals that by adjusting fiber orientation, the composite coupler can generate mostly uniform load transmission in the central bonding region. The internal forces F2of Figure 7 in that region concomitantly reveal linear relationships with the spatial coordinate x/L. The optimum fiber angle Ø2is about 30°, which provides the lowest maximum τxra/τmaof 1.2. Figure 9 shows the radial normal stress in the adhesive. Generally speaking, the smaller peak of τxra/τma, the lower magnitude of σra/τma.

Figure 10 shows the normal stresses σ11and σ33of adherend part 2 in the principal material coordinate system, when the fiber orientation is equal to 30o. It can be seen from the figure that under the application of the axial force, σ11is vanished at the left of the bonding region due to the traction-free surface, while σ33is disappeared on the outermost area of the coupler. The stress in the fiber direction σ11steadily attains the same maximum value along the bond length in all laminae of composite coupler. The radial normal stress σ33, which is induced from the resultant axial force, is highest at the adhesive-coupler interface.

Lastly, for the case of pressure loads, 1 MPa internal pressure is exerted inside the adherend part 1, but no external pressure is present on the outer surface of the adherend part 2. The adhesive longitudinal shear stress and adhesive radial normal stress can then be normalized by the internal pressure to form the dimensionless variables.

Figures 11 and 12 show the effect of fiber orientation on the distributions of F2and τxraalong the overlap region, respectively. Figure 11 indicates that peak values of F2are generated in the central region of the composite couplers, but their values are null at both ends. The longitudinal shear stresses in adhesive τxrain Figure 12 illustrate the antisymmetric characteristic along the bond length. It can be seen that the optimum fiber angle Ø2is 90°. This fiber orientation delivers the lowest maximum τxra/piof 0.6. Figure 13 shows the radial normal stress in the adhesive. Interestingly, the values of σraare reduced by four to five times compared to the internal pressure applied over the whole range of the fiber angles considered.

The normal stresses σ11and σ33of adherend part 2 in the principal material coordinate system, when the fiber orientation are equal to 30°, are displayed in Figure 14. It can be noticed that under the application of the uniform internal pressure with 1 MPa magnitude, σ11is maximum at the mid-length of bonding region, whereas σ33is peak at x = 0 and 40 mm on the adhesive-adherend interface.

## 5. Conclusions

A unified mathematical model for predicting the joint stresses of the adhesive-bonded tubular-coupler joints or the equivalent bonded-lap joints under several types of load is formulated. The inner and outer adherends can be considered as an isotropic material, orthotropic material, or a laminated composite, whose fiber angle is constant along the tube axis. They are modeled as three-dimensional body and satisfied the equilibrium, kinematic, and constitutive equations in theory of elasticity. The adhesive is only treated to be a very thin isotropic elastic material with relative low modulus, and thus, merely three out-of-plane stress components are present. The finite-segment method is developed to compute adherend stresses in each small portion of the coupler. The analytical results obtained indicate the viability of the model for many joint conditions and configurations. The model can be used conveniently in the preliminary process of the joint design, which is usually critical in huge, complex, or integrated structures.

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Sontipee Aimmanee (December 20th 2017). A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints, Applied Adhesive Bonding in Science and Technology, Halil Özer, IntechOpen, DOI: 10.5772/intechopen.72288. Available from:

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