**Meet the editor**

Halil Özer is currently a full professor at the Department of Mechanical Engineering of the Yıldız Technical University in Istanbul and the committee member of PhD proficiency in Mechanical Engineering. He completed his postdoctoral training at the Eindhoven University of Technology in the Netherlands, in the area of damage mechanics. He is the author and coauthor of scientific

publications, especially in the field of the bi-adhesive joint, the self-healed adhesive joint, and cohesive zone modeling. He supervised PhD/MSc students mainly on adhesive joints.

Contents

**Preface VII**

Halil Özer

**Section 1 Structural Adhesive Bonding 1**

**Coupler Joints 11** Sontipee Aimmanee

**Generation 33** Yusuke Okabe

**Section 2 Wood Adhesive Bonding 47**

**Indonesia 73**

Chapter 4 **Green Binders for Wood Adhesives 49**

Khabbaz and Eva Malmström

Chapter 1 **Introductory Chapter: Structural Adhesive Bonded Joints 3**

Chapter 3 **Development and Application of Low-Temperature Curable**

Chapter 5 **A Review of Isocyanate Wood Adhesive: A Case Study in**

Arif Nuryawan and Eka Mulya Alamsyah

**Section 3 Adhesive Bonding in Medical Applications 91**

Chapter 6 **Silicone Adhesives in Medical Applications 93**

Nartker and Xavier Thomas

**Isotropic Conductive Adhesive Toward to Fabrication in IoT**

Emelie Norström, Deniz Demircan, Linda Fogelström, Farideh

Gerald K. Schalau II, Alexis Bobenrieth, Robert O. Huber, Linda S.

Chapter 2 **A Unified Analysis of Adhesive-Bonded Cylindrical**

## Contents

#### **Preface XI**


#### Chapter 7 **Adhesives: Applications and Recent Advances 119** Elena Dinte and Bianca Sylvester

Preface

help of numerical techniques.

metal-to-wood, and rubber-to-skin).

obtain a simple stretchable bonding system.

Indonesia, are studied.

Adhesives produced from the traditional tools have been used for thousands of years. To‐ day, adhesives are used extensively in aerospace and industrial and medical applications. In addition to the development in adhesives, modern computers have also made it easier to evaluate the performance of the adhesive joints by giving the results in seconds with the

The subject of the book is multidisciplinary in nature as it deals with adhesives drawn from the disciplines of chemical, mechanical, medical, biological, and other sciences. Advantages of using adhesive bonding can be summarized as follows: leading to a significant decrease in stress concentrations, providing more uniform stress distributions along the overlap length, providing savings in weight and cost, eliminating any cuts/holes in the joint, and so on. Adhesive bonding is often an appropriate choice for joining similar/dissimilar substrates (various substrate combinations, e.g., metal-to-metal, wood-to-wood, metal-to-composite,

This book is divided into three sections: "Structural Adhesive Bonding," "Wood Adhesive Bonding," and "Adhesive Bonding in Medical Applications." Each section presents the ap‐

In the first section, some applications of structural bonding techniques are presented. It is known that stress concentrations developed at the bonded joints reduce their failure loads and hence joint strengths. In the first chapter, the role of the adhesive layer on stress distri‐ butions is discussed by using some applications of bi-adhesive and modulus-graded joints. In the second chapter, a unified mathematical model for predicting the joint stresses of the adhesive-bonded tubular-coupler joints under several types of load is formulated. In the third chapter, possible benefits of using STPE in the area of flexible electronics are studied to

In the second section, the fourth chapter summarizes some of the most recent scientific liter‐ ature regarding the development of green adhesives. It was reported that fundamental re‐ search is still required in order to have a strong bonding between the adhesive and substrates. In the fifth chapter, two types of isocyanate wood adhesives, commonly used in

The third section presents a comprehensive review on some medical applications of the ad‐ hesive bonding. The sixth chapter reviews the silicone-based adhesive technologies, applica‐ tions, and characterization, emphasizing those self-adhesive materials often used in skincontact applications including transdermal drug delivery and wound care device attachment. The last chapter presents a brief history of adhesive use. In addition, the chapter

plications of the adhesive bonding in various different disciplines.

## Preface

Chapter 7 **Adhesives: Applications and Recent Advances 119**

Elena Dinte and Bianca Sylvester

**VI** Contents

Adhesives produced from the traditional tools have been used for thousands of years. To‐ day, adhesives are used extensively in aerospace and industrial and medical applications. In addition to the development in adhesives, modern computers have also made it easier to evaluate the performance of the adhesive joints by giving the results in seconds with the help of numerical techniques.

The subject of the book is multidisciplinary in nature as it deals with adhesives drawn from the disciplines of chemical, mechanical, medical, biological, and other sciences. Advantages of using adhesive bonding can be summarized as follows: leading to a significant decrease in stress concentrations, providing more uniform stress distributions along the overlap length, providing savings in weight and cost, eliminating any cuts/holes in the joint, and so on. Adhesive bonding is often an appropriate choice for joining similar/dissimilar substrates (various substrate combinations, e.g., metal-to-metal, wood-to-wood, metal-to-composite, metal-to-wood, and rubber-to-skin).

This book is divided into three sections: "Structural Adhesive Bonding," "Wood Adhesive Bonding," and "Adhesive Bonding in Medical Applications." Each section presents the ap‐ plications of the adhesive bonding in various different disciplines.

In the first section, some applications of structural bonding techniques are presented. It is known that stress concentrations developed at the bonded joints reduce their failure loads and hence joint strengths. In the first chapter, the role of the adhesive layer on stress distri‐ butions is discussed by using some applications of bi-adhesive and modulus-graded joints. In the second chapter, a unified mathematical model for predicting the joint stresses of the adhesive-bonded tubular-coupler joints under several types of load is formulated. In the third chapter, possible benefits of using STPE in the area of flexible electronics are studied to obtain a simple stretchable bonding system.

In the second section, the fourth chapter summarizes some of the most recent scientific liter‐ ature regarding the development of green adhesives. It was reported that fundamental re‐ search is still required in order to have a strong bonding between the adhesive and substrates. In the fifth chapter, two types of isocyanate wood adhesives, commonly used in Indonesia, are studied.

The third section presents a comprehensive review on some medical applications of the ad‐ hesive bonding. The sixth chapter reviews the silicone-based adhesive technologies, applica‐ tions, and characterization, emphasizing those self-adhesive materials often used in skincontact applications including transdermal drug delivery and wound care device attachment. The last chapter presents a brief history of adhesive use. In addition, the chapter provides important review materials about the new generation of adhesives, based on mod‐ ern technologies such as nanotechnology, polymers, and biomimetic adhesives.

This book brings together scientists and provides the reader with a comprehensive overview of some recent developments in the field of adhesive bonding with the contributions of inter‐ nationally recognized authors. This book provides an important review on the adhesive bond‐ ing practices. I would like to express my gratitude to all the authors who contributed to this book. I hope that the book published in open access will help researchers to benefit from it.

> **Halil Özer** Yıldız Technical University Faculty of Mechanical Engineering Mechanical Engineering Department Istanbul, Turkey

**Section 1**

**Structural Adhesive Bonding**

**Structural Adhesive Bonding**

provides important review materials about the new generation of adhesives, based on mod‐

This book brings together scientists and provides the reader with a comprehensive overview of some recent developments in the field of adhesive bonding with the contributions of inter‐ nationally recognized authors. This book provides an important review on the adhesive bond‐ ing practices. I would like to express my gratitude to all the authors who contributed to this book. I hope that the book published in open access will help researchers to benefit from it.

**Halil Özer**

Istanbul, Turkey

Yıldız Technical University

Faculty of Mechanical Engineering Mechanical Engineering Department

ern technologies such as nanotechnology, polymers, and biomimetic adhesives.

VIII Preface

**Chapter 1**

**Provisional chapter**

**Introductory Chapter: Structural Adhesive Bonded**

**Introductory Chapter: Structural Adhesive Bonded** 

DOI: 10.5772/intechopen.74229

Adhesives produced from the traditional tools have been used for thousands of years. The advantages of using adhesively bonding techniques instead of classical mechanical fasteners can be listed as joining similar/dissimilar materials, significantly reducing the stress concentrations, providing more-uniform stress distributions along the overlap length, savings in weight and cost, eliminating any cuts/holes in the joint, etc. Adhesive bonding is often an appropriate choice for joining similar/dissimilar substrates (various substrate combinations, e.g., metal-to-metal, metal-to-composite, metal-to-rubber, metal-to-glass, metal-towood, etc.). Subject of adhesive bonding is also multidisciplinary in nature since it deals with adhesives drawn from the disciplines of chemical, mechanical, medical and medicine, biological, and other sciences. Adhesives have therefore become a key research area because of their potential applications. Today, adhesives are used extensively in aerospace, industrial, and medical applications. Three basic types of adhesively bonded joints used commonly are

Choosing an appropriate joining technique is important to have strong joints. Single lap-joint (SLJ) is a simple joint type that allows for joining two adherends easily (**Figure 1a**). The slope of the scarf is the main factor determining the stresses developed on the inclined section of scarf joints (**Figure 1b**). Butt joint is another simple joining technique, and can have some disadvantages due to the small overlap area (**Figure 1c**). Each lap-joint type has therefore comparative advantages and disadvantages over the others. It is therefore important to choose the appropriate lap-joint type considering the application purposes. In addition to joint types, mechanical properties of adhesive/adherend materials, overlap length, thicknesses of adhe-

sive/adherend, etc., affect stresses developed in the joint and hence the joint strength.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74229

**Joints**

**Joints**

Halil Özer

**1. Introduction**

shown in **Figure 1**.

Halil Özer

#### **Introductory Chapter: Structural Adhesive Bonded Joints Introductory Chapter: Structural Adhesive Bonded Joints**

DOI: 10.5772/intechopen.74229

Halil Özer Halil Özer

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74229

## **1. Introduction**

Adhesives produced from the traditional tools have been used for thousands of years. The advantages of using adhesively bonding techniques instead of classical mechanical fasteners can be listed as joining similar/dissimilar materials, significantly reducing the stress concentrations, providing more-uniform stress distributions along the overlap length, savings in weight and cost, eliminating any cuts/holes in the joint, etc. Adhesive bonding is often an appropriate choice for joining similar/dissimilar substrates (various substrate combinations, e.g., metal-to-metal, metal-to-composite, metal-to-rubber, metal-to-glass, metal-towood, etc.). Subject of adhesive bonding is also multidisciplinary in nature since it deals with adhesives drawn from the disciplines of chemical, mechanical, medical and medicine, biological, and other sciences. Adhesives have therefore become a key research area because of their potential applications. Today, adhesives are used extensively in aerospace, industrial, and medical applications. Three basic types of adhesively bonded joints used commonly are shown in **Figure 1**.

Choosing an appropriate joining technique is important to have strong joints. Single lap-joint (SLJ) is a simple joint type that allows for joining two adherends easily (**Figure 1a**). The slope of the scarf is the main factor determining the stresses developed on the inclined section of scarf joints (**Figure 1b**). Butt joint is another simple joining technique, and can have some disadvantages due to the small overlap area (**Figure 1c**). Each lap-joint type has therefore comparative advantages and disadvantages over the others. It is therefore important to choose the appropriate lap-joint type considering the application purposes. In addition to joint types, mechanical properties of adhesive/adherend materials, overlap length, thicknesses of adhesive/adherend, etc., affect stresses developed in the joint and hence the joint strength.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

**Figure 1.** Three basic types of adhesively bonded joints.

This study aims to review some advancements in applications of the structural adhesively bonded joints. SLJ, the simplest form of adhesive joints, was introduced first and its characteristic behaviors of stress distributions along the bondline were given and discussed. In the adhesively bonded joint applications, reducing the stress concentrations and maximizing the failure load are the important issues to be solved. Many different techniques have been proposed to reduce stress concentrations. Grading the adhesive band has recently come to the fore among the reported remedies to overcome the problems faced by lap-joint applications. Two important applications of grading the adhesive band are using the bi-adhesive and modulus-graded bondlines in lap joints. The bi-adhesive bondline consists of three individual regions by a combination of stiff and flexible adhesives along the bondline, in which the flexible adhesives locate at the bondline ends and the stiff adhesive locates in the middle of the overlap. Second remedy is to use the modulus-graded bondline, in which bondline is graded functionally along the overlap length. This study aims to discuss the role of the adhesive layer on shear stress distributions and review some advancements in applications of the bi-adhesive and modulus-graded joints. Each joining technique was discussed briefly and compared with the joints bonded with a mono-adhesive alone.

bonded with a mono-adhesive alone are widely used joints, however, weak in local stress concentrations at the overlap ends. High stress regions at the overlap ends reduce their failure loads and hence joint strengths. When adhesively bonded joint is subjected to tensile load, load is transferred mainly by shear stress developed in the adhesive layer. The first analytical model for assessing the adhesive shear stresses of bonded joints was developed by Volkersen [9]. His model is also called shear-lag model, which neglects the bending moment. The adhesive shear

> tt ‐ <sup>t</sup> \_\_\_\_b tt <sup>+</sup> tb)(

\_\_\_\_\_\_\_\_\_ \_\_\_\_ Ga Ett ta (1+ t \_t

thicknesses, respectively. *b* and *l* are the bond width and bond length, respectively. E, Ga

to reduce the stress concentrations of SLJ and improve its failure loads [10–12].

F are the adherend modulus, adhesive shear modulus, and applied force, respectively. The x-axis passes through the mid-plane of the adhesive layer. As seen from **Figure 2**, high peel and shear stress concentrations develop at the overlap ends. There have been many attempts

\_\_\_\_\_\_\_\_ ω *l*

and tb

<sup>2</sup> ) sinh(<sup>ω</sup> x) \_\_\_\_\_\_\_\_\_\_\_\_\_\_

cosh(<sup>ω</sup> *<sup>l</sup>*/2) (1)

are the top and bottom adherend

, and

tb) (2)

Introductory Chapter: Structural Adhesive Bonded Joints http://dx.doi.org/10.5772/intechopen.74229 5

stress distribution is defined in the Volkersen model as follows:

**Figure 2.** Deformed shape of an SLJ and stress distributions along its bondline length.

Fω 2b

cosh(ω x) \_\_\_\_\_\_\_\_\_\_\_\_\_\_ sinh(<sup>ω</sup> *<sup>l</sup>*/2) <sup>+</sup> (

τ = \_\_\_

ω= <sup>√</sup>

In which, ω is the characteristic shear-lag distance. t<sup>t</sup>

where

#### **2. Structural adhesive joints**

#### **2.1. Mono-adhesive bondline**

Single lap joints have been studied by many authors [1–8]. As seen in **Figure 2**, load eccentricity results in developing the bending moments in an SLJ subjected to an axial loading. **Figure 2** shows the characteristic behavior of the peel and shear stress distributions along the bondline length.

It is seen from **Figure 2** that both shear and peel stresses become peak at overlap ends. However, higher peel stresses develop at the overlap ends due to bending moment effect. Single lap joints

**Figure 2.** Deformed shape of an SLJ and stress distributions along its bondline length.

bonded with a mono-adhesive alone are widely used joints, however, weak in local stress concentrations at the overlap ends. High stress regions at the overlap ends reduce their failure loads and hence joint strengths. When adhesively bonded joint is subjected to tensile load, load is transferred mainly by shear stress developed in the adhesive layer. The first analytical model for assessing the adhesive shear stresses of bonded joints was developed by Volkersen [9]. His model is also called shear-lag model, which neglects the bending moment. The adhesive shear stress distribution is defined in the Volkersen model as follows:

$$\pi = \frac{\text{F}\omega}{2\text{b}} \frac{\cosh(\omega \,\text{x})}{\sinh(\omega \, l/2)} + \left(\frac{\text{t}\_{\text{i}} \,\text{-t}\_{\text{b}}}{\text{t}\_{\text{i}} \,\text{+t}\_{\text{b}}}\right) \left(\frac{\omega \, l}{2}\right) \frac{\sinh(\omega \,\text{x})}{\cosh(\omega \, l/2)}\tag{1}$$

where

This study aims to review some advancements in applications of the structural adhesively bonded joints. SLJ, the simplest form of adhesive joints, was introduced first and its characteristic behaviors of stress distributions along the bondline were given and discussed. In the adhesively bonded joint applications, reducing the stress concentrations and maximizing the failure load are the important issues to be solved. Many different techniques have been proposed to reduce stress concentrations. Grading the adhesive band has recently come to the fore among the reported remedies to overcome the problems faced by lap-joint applications. Two important applications of grading the adhesive band are using the bi-adhesive and modulus-graded bondlines in lap joints. The bi-adhesive bondline consists of three individual regions by a combination of stiff and flexible adhesives along the bondline, in which the flexible adhesives locate at the bondline ends and the stiff adhesive locates in the middle of the overlap. Second remedy is to use the modulus-graded bondline, in which bondline is graded functionally along the overlap length. This study aims to discuss the role of the adhesive layer on shear stress distributions and review some advancements in applications of the bi-adhesive and modulus-graded joints. Each joining technique was discussed briefly and

Single lap joints have been studied by many authors [1–8]. As seen in **Figure 2**, load eccentricity results in developing the bending moments in an SLJ subjected to an axial loading. **Figure 2** shows the characteristic behavior of the peel and shear stress distributions along the

It is seen from **Figure 2** that both shear and peel stresses become peak at overlap ends. However, higher peel stresses develop at the overlap ends due to bending moment effect. Single lap joints

compared with the joints bonded with a mono-adhesive alone.

**2. Structural adhesive joints**

**Figure 1.** Three basic types of adhesively bonded joints.

4 Applied Adhesive Bonding in Science and Technology

**2.1. Mono-adhesive bondline**

bondline length.

$$
\omega = \sqrt{\frac{G\_s}{\mathcal{E}t\_r} \left(1 + \frac{t\_i}{t\_p}\right)}\tag{2}
$$

In which, ω is the characteristic shear-lag distance. t<sup>t</sup> and tb are the top and bottom adherend thicknesses, respectively. *b* and *l* are the bond width and bond length, respectively. E, Ga , and F are the adherend modulus, adhesive shear modulus, and applied force, respectively. The x-axis passes through the mid-plane of the adhesive layer. As seen from **Figure 2**, high peel and shear stress concentrations develop at the overlap ends. There have been many attempts to reduce the stress concentrations of SLJ and improve its failure loads [10–12].

#### **2.2. Modulus-graded bondline**

One of the important remedies to overcome some deficiencies (i.e., stress concentration, decreasing in joint strength, etc.) arising in lap joints is grading the adhesive properties along the bondline. The earliest study on grading the modulus of an adhesive along the overlap length was performed by Raphael [14]. He splitted the adhesive bondline into finite number of discrete parts (**Figure 3**).

His model is based on the shear-lag concept of Volkersen, and therefore neglects the peel stress effect [13]. The work was undertaken as part of a program to design and test bonded rocket motor cases. The aim was to obtain the highest possible joint strength for a simple overlap. However, Raphael did not report any experimental work nor, indeed, quantify the possible benefits of a variable modulus bondline [14].

The stiff adhesive should be located in the middle and flexible adhesive at the ends of the bondline. The earliest study on the bi-adhesive joints was performed by Raphael [13]. Özer and Öz [24, 25] performed numerical studies to investigate the state of stress in the bi-adhesive bondline. In their other study [26], they also performed an experimental study to assess the effect of a bi-adhesive bondline on the failure load of both mono- and bi-adhesive SLJs. Their results were discussed briefly in **Figure 6**. **Figure 6** shows the characteristic behaviors of shear stress distributions along the mid-plane of the mono- and bi-adhesive layers. For comparison purposes, the shear stress distributions for the monoflexible and mono-stiff adhesives were also given for mono-adhesive SLJs. As can be seen from **Figure 6**, shear stress distribution for the mono-flexible adhesive is more uniform than that of the mono-stiff adhesive and there is a lower stress concentration at the overlap edges. As a result, it is seen that the shear stress concentrations occurred at the overlap

Introductory Chapter: Structural Adhesive Bonded Joints http://dx.doi.org/10.5772/intechopen.74229 7

However, in the bi-adhesive bondline, as can be seen in **Figure 6**, the position of the maximum shear stress moves to a new position between adhesives (i.e., to the ends of the stiff adhesive in the middle). As reported above, the maximum shear stresses becomes peak at the overlap edges for mono-adhesive joints, however, it becomes peak at the contact interfaces for bi-

**Figure 6.** Characteristic behavior of shear stress distributions along mono- and bi-adhesive bondlines.

edges for mono-adhesive joints.

**Figure 5.** Bi-adhesive single lap-joint.

adhesive joints.

**Figure 4** shows an SLJ bonded with the modulus-graded bondline. Recently, Carbas et al. [15] developed a simple analytical model to study the performance of the functionally graded joints.

In addition to numerical study, Carbas and others [16] also performed experimental study and used induction heating system to have a graded cure and joints with the adhesive gradually modified along the overlap. The induction system was set to allow the induction heating at the overlap ends and the induction cooling in the middle. They also performed analytical analyses to predict the failure load of the joints with graded cure and isothermal cure.

Modulus-graded joints have been studied in a limited number of papers in the literature and still open for numerical/analytical/experimental studies. The reader may refer to the following articles for current applications [17–23].

#### **2.3. Bi-adhesive bondline**

Bi-adhesive joint is an alternative stress-management technique for adhesively bonded lap joints. Its bondline includes a combination of stiff and flexible adhesives (**Figure 5**). This joint type including two types of adhesives in the overlap region is called as bi-adhesive, hybridadhesive, and mixed-adhesive joints in the literature.

**Figure 4.** An SLJ bonded with the modulus-graded bondline.

**Figure 5.** Bi-adhesive single lap-joint.

**2.2. Modulus-graded bondline**

6 Applied Adhesive Bonding in Science and Technology

of discrete parts (**Figure 3**).

possible benefits of a variable modulus bondline [14].

ing articles for current applications [17–23].

adhesive, and mixed-adhesive joints in the literature.

**Figure 4.** An SLJ bonded with the modulus-graded bondline.

**Figure 3.** Raphael's modulus-graded bondline.

**2.3. Bi-adhesive bondline**

One of the important remedies to overcome some deficiencies (i.e., stress concentration, decreasing in joint strength, etc.) arising in lap joints is grading the adhesive properties along the bondline. The earliest study on grading the modulus of an adhesive along the overlap length was performed by Raphael [14]. He splitted the adhesive bondline into finite number

His model is based on the shear-lag concept of Volkersen, and therefore neglects the peel stress effect [13]. The work was undertaken as part of a program to design and test bonded rocket motor cases. The aim was to obtain the highest possible joint strength for a simple overlap. However, Raphael did not report any experimental work nor, indeed, quantify the

**Figure 4** shows an SLJ bonded with the modulus-graded bondline. Recently, Carbas et al. [15] developed a simple analytical model to study the performance of the functionally graded joints. In addition to numerical study, Carbas and others [16] also performed experimental study and used induction heating system to have a graded cure and joints with the adhesive gradually modified along the overlap. The induction system was set to allow the induction heating at the overlap ends and the induction cooling in the middle. They also performed analytical

analyses to predict the failure load of the joints with graded cure and isothermal cure.

Modulus-graded joints have been studied in a limited number of papers in the literature and still open for numerical/analytical/experimental studies. The reader may refer to the follow-

Bi-adhesive joint is an alternative stress-management technique for adhesively bonded lap joints. Its bondline includes a combination of stiff and flexible adhesives (**Figure 5**). This joint type including two types of adhesives in the overlap region is called as bi-adhesive, hybridThe stiff adhesive should be located in the middle and flexible adhesive at the ends of the bondline. The earliest study on the bi-adhesive joints was performed by Raphael [13]. Özer and Öz [24, 25] performed numerical studies to investigate the state of stress in the bi-adhesive bondline. In their other study [26], they also performed an experimental study to assess the effect of a bi-adhesive bondline on the failure load of both mono- and bi-adhesive SLJs. Their results were discussed briefly in **Figure 6**. **Figure 6** shows the characteristic behaviors of shear stress distributions along the mid-plane of the mono- and bi-adhesive layers. For comparison purposes, the shear stress distributions for the monoflexible and mono-stiff adhesives were also given for mono-adhesive SLJs. As can be seen from **Figure 6**, shear stress distribution for the mono-flexible adhesive is more uniform than that of the mono-stiff adhesive and there is a lower stress concentration at the overlap edges. As a result, it is seen that the shear stress concentrations occurred at the overlap edges for mono-adhesive joints.

However, in the bi-adhesive bondline, as can be seen in **Figure 6**, the position of the maximum shear stress moves to a new position between adhesives (i.e., to the ends of the stiff adhesive in the middle). As reported above, the maximum shear stresses becomes peak at the overlap edges for mono-adhesive joints, however, it becomes peak at the contact interfaces for biadhesive joints.

**Figure 6.** Characteristic behavior of shear stress distributions along mono- and bi-adhesive bondlines.

Therefore, it is seen that peak shear stress decreases at the overlap edges and increases at the contact interfaces (i.e., at the ends of the stiff adhesive) in the bi-adhesive bondline. It can be concluded that stiff adhesive in the middle contributes its high shear-strength-capacity to the bi-adhesive joint. Therefore, high stress concentrations at the bondline ends can be reduced by using bi-adhesive bondline. However, it is important to select the appropriate adhesive type for the bi-adhesive bondline. In addition, amounts of the stiff/flexible adhesives used in the bi-adhesive bondline also affect the shear stress values.

[5] Bigwood DA, Crocombe AD. Elastic analysis and engineering design formulae for bonded joints. International Journal of Adhesion and Adhesives. 1989;**9**:229-242

Introductory Chapter: Structural Adhesive Bonded Joints http://dx.doi.org/10.5772/intechopen.74229 9

[6] Tsai MY, Oplinger DW, Morton J. Improved theoretical solutions for adhesive lap joints.

[7] Zhao B, Lu Z-H. A two-dimensional approach of single-lap adhesive bonded joints.

[8] Zhao B, Lu Z-H, Lu Y-N. Closed-form solutions for elastic stress–strain analysis in unbalanced adhesive single-lap joints considering adherend deformations and bond

[9] Volkersen O. Rivet strength distribution in tensile-stressed rivet joints with constant

[10] Hua Y, Gu L, Trogdon M. Three-dimensional modeling of carbon/epoxy to titanium single-lap joints with variable adhesive recess length. International Journal of Adhesion

[11] Belingardi G, Goglio L, Tarditi A. Investigating the effect of spew and chamfer size on the stresses in metal/plastics adhesive joints. International Journal of Adhesion and

[12] Çalık A, Yıldırım S. Effect of adherend recessing on bi-adhesively bonded single-lap

[13] Raphael C. Variable-adhesive bonded joints. Journal of Applied Polymer Science:

[14] Broughton JG, Fitton MD. Science of mixed-adhesive joints. In: da Silva L, Pirondi A, Öchsner A, editors. Hybrid Adhesive Joints. Berlin, Heidelberg: Springer; 2011. pp. 257-281

[15] Carbas RJC, da Silva LFM, Madureira ML, Critchlow GW. Modelling of functionally

[16] Carbas RJC, da Silva LFM, Critchlow GW. Adhesively bonded functionally graded joints by induction heating. International Journal of Adhesion and Adhesives. 2014;**48**:110-118

[17] Stein N, Mardani H, Becker W. An efficient analysis model for functionally graded adhesive single lap joints. International Journal of Adhesion and Adhesives. 2016;**70**:117-125

[18] Stein N, Felger J, Becker W. Analytical models for functionally graded adhesive single lap joints: A comparative study. International Journal of Adhesion and Adhesives.

[19] Nimje SV, Panigrahi SK. Strain energy release rate based damage analysis of functionally graded adhesively bonded tubular lap joint of laminated FRP composites. The Journal of

[20] Stein N, Rosendahl PL, Becker W. Homogenization of mechanical and thermal stresses in functionally graded adhesive joints. Composites Part B: Engineering. 2017;**111**:279-293

graded adhesive joints. The Journal of Adhesion. 2014;**90**:698-716

thickness. International Journal of Adhesion and Adhesives. 2011;**31**:434-445

International Journal of Solids and Structures. 1998;**35**:1163-1185

Mechanics of Advanced Materials and Structures. 2009;**16**:130-159

cross-section. Luftfahrorschung. 1938;**15**:41-47

joints with spew fillet. Sadhana. 2017;**42**:317-325

Applied Polymer Symposium. 1965;**3**:99-108

and Adhesives. 2012;**38**:25-30

Adhesives. 2002;**22**:273-282

2017;**76**:70-82

Adhesion. 2017;**93**:389-411

There are a limited number of publications in the open literature about the bi-adhesive joints. The reader may refer to the following articles for current applications of the joints bonded with bi-adhesive bondline [27–33].

#### **3. Conclusion**

It is known that high stress concentrations develop at the overlap ends of the adhesively bonded joints. Grading the adhesive band has recently come to the fore among the reported remedies to overcome the problems faced by lap-joint applications. In this study, the role of the adhesive layer on stress distributions was reviewed. Joining techniques using the bi-adhesive and modulus-graded bondlines were discussed briefly and compared with the joints bonded with a mono-adhesive alone. It is seen that high stress concentrations at the ends can be reduced by using these techniques. It is therefore concluded that stress concentration and joint strength can be optimized by using modulus-graded and bi-adhesive bondlines in the lap joints.

## **Author details**

#### Halil Özer

Address all correspondence to: hozer@yildiz.edu.tr

Mechanical Engineering Department, Yıldız Technical University, Istanbul, Turkey

#### **References**


[5] Bigwood DA, Crocombe AD. Elastic analysis and engineering design formulae for bonded joints. International Journal of Adhesion and Adhesives. 1989;**9**:229-242

Therefore, it is seen that peak shear stress decreases at the overlap edges and increases at the contact interfaces (i.e., at the ends of the stiff adhesive) in the bi-adhesive bondline. It can be concluded that stiff adhesive in the middle contributes its high shear-strength-capacity to the bi-adhesive joint. Therefore, high stress concentrations at the bondline ends can be reduced by using bi-adhesive bondline. However, it is important to select the appropriate adhesive type for the bi-adhesive bondline. In addition, amounts of the stiff/flexible adhesives used in

There are a limited number of publications in the open literature about the bi-adhesive joints. The reader may refer to the following articles for current applications of the joints bonded

It is known that high stress concentrations develop at the overlap ends of the adhesively bonded joints. Grading the adhesive band has recently come to the fore among the reported remedies to overcome the problems faced by lap-joint applications. In this study, the role of the adhesive layer on stress distributions was reviewed. Joining techniques using the bi-adhesive and modulus-graded bondlines were discussed briefly and compared with the joints bonded with a mono-adhesive alone. It is seen that high stress concentrations at the ends can be reduced by using these techniques. It is therefore concluded that stress concentration and joint strength can be optimized by using modulus-graded and bi-adhesive bondlines in the lap joints.

Mechanical Engineering Department, Yıldız Technical University, Istanbul, Turkey

[2] Hart-Smith LJ. Adhesive-bonded single-lap joints. 1973. NASA CR-112236

Journal of Mechanics and Applied Mathematics. 1977;**30**:415-436

Strain Analysis for Engineering Design. 1974;**9**:185-196

[1] Adams RD, Peppiatt NA. Effect of Poisson's ratio strains in adherends on stresses of an idealized lap joint. The Journal of Strain Analysis for Engineering Design. 1973;**8**:134-139

[3] Adams RD, Peppiatt NA. Stress analysis of adhesive-bonded lap joints. The Journal of

[4] Allman DJ. A theory for elastic stresses in adhesive bonded lap joints. The Quarterly

the bi-adhesive bondline also affect the shear stress values.

Address all correspondence to: hozer@yildiz.edu.tr

with bi-adhesive bondline [27–33].

8 Applied Adhesive Bonding in Science and Technology

**3. Conclusion**

**Author details**

Halil Özer

**References**


[21] Khan MA, Kumar S. Interfacial stresses in single-side composite patch repairs with material tailored bondline. Mechanics of Advanced Materials and Structures. 2018;**25**:304-318 **Chapter 2**

Provisional chapter

**A Unified Analysis of Adhesive-Bonded Cylindrical**

DOI: 10.5772/intechopen.72288

A Unified Analysis of Adhesive-Bonded Cylindrical

In the past years, many studies have been conducted on behaviors of adhesive tubular joints subjected to various loading conditions, such as torsion, axial, and internal and external pressure. However, the previous models are conceptually distinct, since they were developed to analyze only for each type of load. Mostly, homogeneous isotropic or orthotropic material were considered and thin-walled joint structures were examined. Therefore, the aim of this chapter is to present for the first time a generalized mathematical formulation and modeling of adhesive-bonded cylindrical coupler joints taking into account all loading scenarios. The inner and outer adherends can be made of isotropic, orthotropic, or laminated composite materials, and they are modeled as three-dimensional elastic body, so adherends with any thickness can be analyzed. Assumptions of an axisymmetric joint with linearly elastic adherends and adhesive materials are employed. Thin adhesive layer is considered so that only the out-of-plane adhesive stresses are concerned, and they are treated to be uniform through its thickness. Using elasticity theory and the newly developed finite-segmented method, stress distributions in both adherends and adhesive can be evaluated. Calculation examples of laminated composite joints are given. This model provides the unified analysis of adhesive-bonded cylindri-

Keywords: adhesive, coupler joint, lap joint, elasticity, finite-segment method

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,

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Coupler Joints**

Coupler Joints

Sontipee Aimmanee

Sontipee Aimmanee

Abstract

cal coupler joints.

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72288


Provisional chapter

## **A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints** A Unified Analysis of Adhesive-Bonded Cylindrical

DOI: 10.5772/intechopen.72288

Sontipee Aimmanee

Coupler Joints

[21] Khan MA, Kumar S. Interfacial stresses in single-side composite patch repairs with material tailored bondline. Mechanics of Advanced Materials and Structures. 2018;**25**:304-318

[22] Stapleton SE, Weimer J, Spengler J. Design of functionally graded joints using a polyurethane-based adhesive with varying amounts of acrylate. International Journal of

[23] Kumar S, Wardle BL, Arif MF. Strength and performance enhancement of bonded joints by spatial tailoring of adhesive compliance via 3D printing. ACS Applied Materials and

[24] Özer H, Öz Ö. Three dimensional finite element analysis of bi-adhesively bonded double lap joint. International Journal of Adhesion and Adhesives. 2012;**37**:50-55

[25] Özer H, Öz Ö. A comparative evaluation of numerical and analytical solutions to the biadhesive single-lap joint. Mathematical Problems in Engineering. 2014;**2014**:852872

[26] Öz Ö, Özer H. An experimental investigation on the failure loads of the mono and biadhesive joints. Journal of Adhesion Science and Technology. 2017;**31**:2251-2270

[27] das Neves PJC, da Silva LFM, Adams RD. Analysis of mixed adhesive bonded joints. Part I: Theoretical formulation. Journal of Adhesion Science and Technology. 2009;**23**:1-34 [28] Yousefsani SA, Tahani M. Relief of edge effects in bi-adhesive composite joints.

[29] Breto R, Chiminelli A, Lizaranzu M, Rodríguez R. Study of the singular term in mixed adhesive joints. International Journal of Adhesion and Adhesives. 2017;**76**:11-16

[30] Temiz S. Application of bi-adhesive in double-strap joints subjected to bending moment.

[31] Marques EAS, Campilho RDSG, da Silva LFM. Geometrical study of mixed adhesive joints for high-temperature applications. Journal of Adhesion Science and Technology.

[32] Akpinar S, Aydin MD, Özel A. A study on 3-D stress distributions in the bi-adhesively

[33] Chiminelli A, Breto R, Izquierdo S, Bergamasco L, Duvivier E, Lizaranzu M. Analysis of mixed adhesive joints considering the compaction process. International Journal of

bonded T-joints. Applied Mathematical Modelling. 2013;**37**:10220-10230

Adhesion and Adhesives. 2017;**76**:38-46

Composites Part B: Engineering. 2017;**108**:153-163

Adhesion and Adhesives. 2017;**76**:3-10

Journal of Adhesion Science and Technology. 2006;**20**:1547-1560

Interfaces. 2017;**9**:884-891

10 Applied Adhesive Bonding in Science and Technology

2016;**30**:691-707

Additional information is available at the end of the chapter Sontipee Aimmanee

http://dx.doi.org/10.5772/intechopen.72288 Additional information is available at the end of the chapter

#### Abstract

In the past years, many studies have been conducted on behaviors of adhesive tubular joints subjected to various loading conditions, such as torsion, axial, and internal and external pressure. However, the previous models are conceptually distinct, since they were developed to analyze only for each type of load. Mostly, homogeneous isotropic or orthotropic material were considered and thin-walled joint structures were examined. Therefore, the aim of this chapter is to present for the first time a generalized mathematical formulation and modeling of adhesive-bonded cylindrical coupler joints taking into account all loading scenarios. The inner and outer adherends can be made of isotropic, orthotropic, or laminated composite materials, and they are modeled as three-dimensional elastic body, so adherends with any thickness can be analyzed. Assumptions of an axisymmetric joint with linearly elastic adherends and adhesive materials are employed. Thin adhesive layer is considered so that only the out-of-plane adhesive stresses are concerned, and they are treated to be uniform through its thickness. Using elasticity theory and the newly developed finite-segmented method, stress distributions in both adherends and adhesive can be evaluated. Calculation examples of laminated composite joints are given. This model provides the unified analysis of adhesive-bonded cylindrical coupler joints.

Keywords: adhesive, 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,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

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.

three-dimensional stress analysis of a bonded tubular-coupler joint subjected to torsion. Their purpose was to investigate all of the adherend and adhesive stress components without the assumption of through-thickness constant stresses across the adhesive layer. Oh [11] performed an analysis of the bonded tubular lap joint of laminated tubes with softening adhesive's stiffness properties under torsion using an elasticity model. Oh concluded that the load capacity in the linear analysis can be quite underestimated when compared to the nonlinear modeling. Spaggiari and Dragoni [12] investigated the joint studied in the Kumar's work in [7], but the joint is subjected to torsion instead. They developed the closed-form function of the adhesive shear modulus in order to minimize adhesive shear stress over the bonding region and addressed the limitation of shear modulus and thicknesses ratio for joint manufacturing with functionally graded adhesives. Recently, Aimmanee and Hongpimolmas [13] formulated a mathematical model of an adhesive-bonded tubular joint with a variablestiffness composite coupler. The optimal variable fiber orientation in the coupler was deter-

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

http://dx.doi.org/10.5772/intechopen.72288

13

Stresses in cylindrical joints under pressure have also been studied, even though investigation of this type of load was comparatively scarce compared to the above two loadings. Terekhova and Skoryi [14] provided a close-form solution for the stresses in tubular lap joints under external and internal pressures and axial forces. Their model neglected the effect of adherend bending. Baishya et al. [15] conducted research in individual and combined effect of internal pressure and torsional loading on stress and failure characteristics of tubular single lap joints made of composite materials. The onset of different joint fracture modes was investigated in their work. Strength analysis of adhesive joints of riser pipes in deep sea environment loadings was performed by Zhang et al. [16] External pressure, internal pressure, tension, torsion, and bending were examined to understand singular stress fields existing around end of the interface. Apalak [17] investigated elastic stresses in the adhesive layer and tubes of an adhesively bonded tubular joint with functionally graded tubes subjected to an internal pressure. Finite-element method was used to model the tubes having gradient layer between a ceramic layer and a metal layer.

According to the former analytical research work presented in the literature, the problems can be mathematically complicated even though the joint is made of simple conventional isotropic adherends. In addition, the previous models usually are distinct for each type of load, since they were developed to analyze only for a specific loading case. Therefore, this chapter aims to present a mathematical modeling of adhesive-bonded cylindrical coupler joints taking into account all loading scenarios, i.e., torsion, axial, and pressure loadings. The inner and outer adherends can be made of isotropic, orthotropic, or laminated composite materials, and they are modeled as three-dimensional elastic body, so a thick or solid cylinder adherend can also be analyzed. Stresses in adhesive layer and adherends can be evaluated by newly developed finitesegment method. The unified formulation of the model will be discussed in the next section.

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

2. Elasticity theory of a laminated cylindrical structure

mined to minimize the adhesive hoop shear stress.

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.

For axial loads, the early investigation was conducted by Lubkin and Reissner [1] who analyzed the tubular lap joints. Axial shear stresses and radial normal stresses in the adhesive layer were predicted under the assumption of approximating the layer as an infinite number of tensile and shear springs. However, disappearance of axial shear stress on the free surfaces at the ends of the adhesive was not considered. Adam and Peppiatt [2] performed axisymmetric finite-element analysis (FEA) of tubular lap joints subjected to stretching and twisting loads. The effects of an adhesive filet and partial tapering adherends on stress distribution were also reported. Using a minimum strain energy, Allman [3] proposed two-dimensional analytical solution for lap joints that ensure the traction-free boundary condition. Bending, stretching, and shearing of the adherends and shearing and tearing of the adhesive layer were taken into account. Shi and Cheng [4] formulated closed-form solutions for tubular lap joints utilizing the variational principle of complementary energy. Boundary conditions and assumptions of Allman were adopted in their model development. Nemes et al. [5] further developed the stress analysis of adhesive in a cylindrical assembly of two tubes. Variational method of the potential energy was also employed. Nonetheless, Nemes et al. neglected radial stress component in the joint. Kumar [6] presented a theoretical framework for the stress analysis of shafttube adhesive joints subjected to tensile loads. The joint assembly was considered to consist of similar or dissimilar isotropic or orthotropic adherends. The principle of minimum complementary energy and a stress function approach were used to establish the governing equations in order to determine the stress state in each constituent. To reduce the stress concentration, Kumar [7] also studied the use of functionally graded adhesive in a tubular lap joint with an isotropic adherend under tension. In his model, the adhesive was divided into annular rings to take into account the gradient property of shear modulus.

Regarding the case of the tubular joint subjected to torsion, Volkersen [8] provided a closedform solution for circumferential shear stresses at the interface of tubular lap joint exerted by a torque. Pugno and Surace [9] investigated the analysis of the joint subjected to torsion. They utilized the common function of resultant torques in adherends and achieved the uniform adhesive hoop shear stress by tapering adherend surfaces. Xu and Li [10] investigated the full three-dimensional stress analysis of a bonded tubular-coupler joint subjected to torsion. Their purpose was to investigate all of the adherend and adhesive stress components without the assumption of through-thickness constant stresses across the adhesive layer. Oh [11] performed an analysis of the bonded tubular lap joint of laminated tubes with softening adhesive's stiffness properties under torsion using an elasticity model. Oh concluded that the load capacity in the linear analysis can be quite underestimated when compared to the nonlinear modeling. Spaggiari and Dragoni [12] investigated the joint studied in the Kumar's work in [7], but the joint is subjected to torsion instead. They developed the closed-form function of the adhesive shear modulus in order to minimize adhesive shear stress over the bonding region and addressed the limitation of shear modulus and thicknesses ratio for joint manufacturing with functionally graded adhesives. Recently, Aimmanee and Hongpimolmas [13] formulated a mathematical model of an adhesive-bonded tubular joint with a variablestiffness composite coupler. The optimal variable fiber orientation in the coupler was determined to minimize the adhesive hoop shear stress.

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

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

For axial loads, the early investigation was conducted by Lubkin and Reissner [1] who analyzed the tubular lap joints. Axial shear stresses and radial normal stresses in the adhesive layer were predicted under the assumption of approximating the layer as an infinite number of tensile and shear springs. However, disappearance of axial shear stress on the free surfaces at the ends of the adhesive was not considered. Adam and Peppiatt [2] performed axisymmetric finite-element analysis (FEA) of tubular lap joints subjected to stretching and twisting loads. The effects of an adhesive filet and partial tapering adherends on stress distribution were also reported. Using a minimum strain energy, Allman [3] proposed two-dimensional analytical solution for lap joints that ensure the traction-free boundary condition. Bending, stretching, and shearing of the adherends and shearing and tearing of the adhesive layer were taken into account. Shi and Cheng [4] formulated closed-form solutions for tubular lap joints utilizing the variational principle of complementary energy. Boundary conditions and assumptions of Allman were adopted in their model development. Nemes et al. [5] further developed the stress analysis of adhesive in a cylindrical assembly of two tubes. Variational method of the potential energy was also employed. Nonetheless, Nemes et al. neglected radial stress component in the joint. Kumar [6] presented a theoretical framework for the stress analysis of shafttube adhesive joints subjected to tensile loads. The joint assembly was considered to consist of similar or dissimilar isotropic or orthotropic adherends. The principle of minimum complementary energy and a stress function approach were used to establish the governing equations in order to determine the stress state in each constituent. To reduce the stress concentration, Kumar [7] also studied the use of functionally graded adhesive in a tubular lap joint with an isotropic adherend under tension. In his model, the adhesive was divided into annular rings to

Regarding the case of the tubular joint subjected to torsion, Volkersen [8] provided a closedform solution for circumferential shear stresses at the interface of tubular lap joint exerted by a torque. Pugno and Surace [9] investigated the analysis of the joint subjected to torsion. They utilized the common function of resultant torques in adherends and achieved the uniform adhesive hoop shear stress by tapering adherend surfaces. Xu and Li [10] investigated the full

concentration.

modeling are given below.

12 Applied Adhesive Bonding in Science and Technology

take into account the gradient property of shear modulus.

Stresses in cylindrical joints under pressure have also been studied, even though investigation of this type of load was comparatively scarce compared to the above two loadings. Terekhova and Skoryi [14] provided a close-form solution for the stresses in tubular lap joints under external and internal pressures and axial forces. Their model neglected the effect of adherend bending. Baishya et al. [15] conducted research in individual and combined effect of internal pressure and torsional loading on stress and failure characteristics of tubular single lap joints made of composite materials. The onset of different joint fracture modes was investigated in their work. Strength analysis of adhesive joints of riser pipes in deep sea environment loadings was performed by Zhang et al. [16] External pressure, internal pressure, tension, torsion, and bending were examined to understand singular stress fields existing around end of the interface. Apalak [17] investigated elastic stresses in the adhesive layer and tubes of an adhesively bonded tubular joint with functionally graded tubes subjected to an internal pressure. Finite-element method was used to model the tubes having gradient layer between a ceramic layer and a metal layer.

According to the former analytical research work presented in the literature, the problems can be mathematically complicated even though the joint is made of simple conventional isotropic adherends. In addition, the previous models usually are distinct for each type of load, since they were developed to analyze only for a specific loading case. Therefore, this chapter aims to present a mathematical modeling of adhesive-bonded cylindrical coupler joints taking into account all loading scenarios, i.e., torsion, axial, and pressure loadings. The inner and outer adherends can be made of isotropic, orthotropic, or laminated composite materials, and they are modeled as three-dimensional elastic body, so a thick or solid cylinder adherend can also be analyzed. Stresses in adhesive layer and adherends can be evaluated by newly developed finitesegment method. The unified formulation of the model will be discussed in the next section.

#### 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 ð Þ x; θ;r depicted 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.

axisymmetric geometry and circumferentially independent material properties under a uni-

xr <sup>¼</sup> <sup>∂</sup>uð Þ<sup>k</sup>

where ε and γ denote normal and shear strains, respectively. u, v, and w are displacements in axial, tangential, and radial directions, respectively. Superscript (k) indicates that the

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

th layer along the r-, θ-, x-directions are reduced to ordinary differential equations with

<sup>x</sup>θ<sup>r</sup>, f g<sup>ε</sup> ð Þ<sup>k</sup>

0 0 10 0 0 0 00 <sup>m</sup>ð Þ<sup>k</sup> �nð Þ<sup>k</sup> <sup>0</sup> 0 00 nð Þ<sup>k</sup> mð Þ<sup>k</sup> 0 �mð Þ<sup>k</sup> <sup>n</sup>ð Þ<sup>k</sup> <sup>m</sup>ð Þ<sup>k</sup> <sup>n</sup>ð Þ<sup>k</sup> 00 0 <sup>m</sup>ð Þ<sup>k</sup> � �<sup>2</sup> � <sup>n</sup>ð Þ<sup>k</sup> � �<sup>2</sup>

<sup>r</sup> , <sup>ε</sup>ð Þ<sup>k</sup>

<sup>∂</sup><sup>r</sup> , <sup>γ</sup>ð Þ<sup>k</sup>

ð Þk <sup>θ</sup> <sup>¼</sup> <sup>w</sup>ð Þ<sup>k</sup>

<sup>r</sup> , <sup>γ</sup>ð Þ<sup>k</sup>

th layer.

∂σð Þ<sup>k</sup> r ∂r þ 1 r σð Þ<sup>k</sup> <sup>r</sup> � <sup>σ</sup>ð Þ<sup>k</sup> θ

> ∂τ ð Þk θr ∂r þ 2 r τ ð Þk

> ∂τ ð Þk xr ∂r þ 1 r τð Þ<sup>k</sup>

transformed to those in the principal material coordinates (1, 2, 3) as follows:

f g<sup>σ</sup> ð Þ<sup>k</sup>

<sup>x</sup>θ<sup>r</sup> ¼ C � �ð Þ<sup>k</sup>

<sup>123</sup> <sup>¼</sup> ½ � <sup>T</sup> ð Þ<sup>k</sup> f g<sup>σ</sup> ð Þ<sup>k</sup>

th layer in the cylindrical coordinates are

http://dx.doi.org/10.5772/intechopen.72288

<sup>r</sup> <sup>¼</sup> <sup>∂</sup>wð Þ<sup>k</sup> ∂r

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

<sup>x</sup><sup>θ</sup> <sup>¼</sup> <sup>∂</sup>vð Þ<sup>k</sup> ∂x

� � <sup>¼</sup> <sup>0</sup> (2)

<sup>θ</sup><sup>r</sup> ¼ 0 (3)

xr ¼ 0 (4)

<sup>x</sup>θ<sup>r</sup> (5)

<sup>x</sup>θ<sup>r</sup> (7)

(6)

th layer in ð Þ <sup>x</sup>; <sup>θ</sup>;<sup>r</sup> coordinates expressed in Eqs. (1)–(4) can be

<sup>123</sup> <sup>¼</sup> ½ � <sup>T</sup> ð Þ<sup>k</sup> f g<sup>ε</sup> ð Þ<sup>k</sup>

th layer as shown in Eq. (6), in which <sup>m</sup>ð Þ<sup>k</sup> <sup>¼</sup> cos <sup>Ø</sup>ð Þ<sup>k</sup> and

<sup>123</sup> are tensorial stress and tensorial strain components, respectively. ½ � <sup>T</sup> ð Þ<sup>k</sup>

th layer as shown in Figure 2.

nð Þ<sup>k</sup> � �<sup>2</sup> 00 0 2mð Þ<sup>k</sup> nð Þ<sup>k</sup>

<sup>m</sup>ð Þ<sup>k</sup> � �<sup>2</sup> 00 0 �2mð Þ<sup>k</sup> <sup>n</sup>ð Þ<sup>k</sup>

<sup>ε</sup>eng f gð Þ<sup>k</sup>

th layer in the cylindrical coordinates can be written as

(1)

15

form load, the strain–displacement relations in the k

<sup>∂</sup><sup>x</sup> , <sup>ε</sup>

<sup>∂</sup><sup>r</sup> � <sup>v</sup>ð Þ<sup>k</sup>

ε ð Þk <sup>x</sup> <sup>¼</sup> <sup>∂</sup>uð Þ<sup>k</sup>

γð Þ<sup>k</sup> <sup>θ</sup><sup>r</sup> <sup>¼</sup> <sup>∂</sup>vð Þ<sup>k</sup>

corresponding quantities are in the k

respect to r, respectively, as

The stresses and strains in the k

<sup>123</sup> and f g<sup>ε</sup> ð Þ<sup>k</sup>

is transformation matrix of the k

½ � <sup>T</sup> ð Þ<sup>k</sup> <sup>¼</sup>

The constitutive relation in the k

<sup>n</sup>ð Þ<sup>k</sup> <sup>¼</sup> sinØð Þ<sup>k</sup> . <sup>Ø</sup>ð Þ<sup>k</sup> is fiber angle of the <sup>k</sup>

where f g<sup>σ</sup> ð Þ<sup>k</sup>

f g<sup>σ</sup> ð Þ<sup>k</sup>

mð Þ<sup>k</sup> � �<sup>2</sup>

nð Þ<sup>k</sup> � �<sup>2</sup>

the k

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

Figure 1. Schematic of a bonded-coupler joint: (a) full model and (b) half model or tubular lap joint model.

Figure 2. A laminated tube and the defined coordinate systems.

axisymmetric geometry and circumferentially independent material properties under a uniform load, the strain–displacement relations in the k th layer in the cylindrical coordinates are

cylindrical coordinates ð Þ x; θ;r depicted 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

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

Figure 1. Schematic of a bonded-coupler joint: (a) full model and (b) half model or tubular lap joint model.

Figure 2. A laminated tube and the defined coordinate systems.

the x direction.

14 Applied Adhesive Bonding in Science and Technology

$$\begin{aligned} \varepsilon\_x^{(k)} &= \frac{\partial u^{(k)}}{\partial x}, & \varepsilon\_\theta^{(k)} &= \frac{\varpi^{(k)}}{r}, & \varepsilon\_r^{(k)} &= \frac{\partial \varpi^{(k)}}{\partial r} \\ \gamma\_{\partial r}^{(k)} &= \frac{\partial \upsilon^{(k)}}{\partial r} - \frac{\upsilon^{(k)}}{r}, & \gamma\_{xr}^{(k)} &= \frac{\partial u^{(k)}}{\partial r}, & \gamma\_{x\theta}^{(k)} &= \frac{\partial \upsilon^{(k)}}{\partial x} \end{aligned} \tag{1}$$

where ε and γ denote 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 k th 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 k th layer along the r-, θ-, x-directions are reduced to ordinary differential equations with respect to r, respectively, as

$$\frac{\partial \sigma\_r^{(k)}}{\partial r} + \frac{1}{r} \left( \sigma\_r^{(k)} - \sigma\_\theta^{(k)} \right) = 0 \tag{2}$$

$$\frac{\partial \tau\_{\partial r}^{(k)}}{\partial r} + \frac{2}{r} \tau\_{\partial r}^{(k)} = 0 \tag{3}$$

$$\frac{\partial \tau\_{xr}^{(k)}}{\partial r} + \frac{1}{r} \tau\_{xr}^{(k)} = 0 \tag{4}$$

The stresses and strains in the k th layer in ð Þ <sup>x</sup>; <sup>θ</sup>;<sup>r</sup> coordinates expressed in Eqs. (1)–(4) can be transformed to those in the principal material coordinates (1, 2, 3) as follows:

$$\{\{\sigma\}\_{123}^{(k)} = [T]^{(k)} \{\sigma\}\_{x\theta r'}^{(k)} \qquad \qquad \{\varepsilon\}\_{123}^{(k)} = [T]^{(k)} \{\varepsilon\}\_{x\theta r}^{(k)} \tag{5}$$

where f g<sup>σ</sup> ð Þ<sup>k</sup> <sup>123</sup> and f g<sup>ε</sup> ð Þ<sup>k</sup> <sup>123</sup> are tensorial stress and tensorial strain components, respectively. ½ � <sup>T</sup> ð Þ<sup>k</sup> is transformation matrix of the k th layer as shown in Eq. (6), in which <sup>m</sup>ð Þ<sup>k</sup> <sup>¼</sup> cos <sup>Ø</sup>ð Þ<sup>k</sup> and <sup>n</sup>ð Þ<sup>k</sup> <sup>¼</sup> sinØð Þ<sup>k</sup> . <sup>Ø</sup>ð Þ<sup>k</sup> is fiber angle of the <sup>k</sup> th layer as shown in Figure 2.

$$[T]^{(k)} = \begin{bmatrix} \left(m^{(k)}\right)^2 & \left(n^{(k)}\right)^2 & 0 & 0 & 0 & 2m^{(k)}n^{(k)} \\ \left(n^{(k)}\right)^2 & \left(m^{(k)}\right)^2 & 0 & 0 & 0 & -2m^{(k)}n^{(k)} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & m^{(k)} & -n^{(k)} & 0 \\ 0 & 0 & 0 & n^{(k)} & m^{(k)} & 0 \\ -m^{(k)}n^{(k)} & m^{(k)}n^{(k)} & 0 & 0 & 0 & \left(m^{(k)}\right)^2 - \left(n^{(k)}\right)^2 \end{bmatrix} \tag{6}$$

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

$$\{\sigma\}\_{x\theta r}^{(k)} = \left[\overline{\mathbb{C}}\right]^{(k)} \{\epsilon^{\epsilon\eta g}\}\_{x\theta r}^{(k)}\tag{7}$$

In the above, C � �ð Þ<sup>k</sup> is the transformed stiffness matrix, and <sup>ε</sup>eng f gð Þ<sup>k</sup> <sup>x</sup>θ<sup>r</sup> is engineering strain components in the global cylindrical coordinate system. The transformed stiffness matrix C � �ð Þ<sup>k</sup> can be evaluated as

$$\left[\overline{\mathbb{C}}\right]^{(k)} = \left\{ [T]^{(k)} \right\}^{-1} [\mathbb{C}\ ]^{(k)} [\mathbb{R}\ ] [T]^{(k)} [\mathbb{R}\ ]^{-1} \tag{8}$$

<sup>λ</sup>ð Þ <sup>k</sup> <sup>¼</sup>

<sup>1</sup> <sup>r</sup>ð Þ<sup>k</sup> <sup>þ</sup> <sup>A</sup>ð Þ<sup>k</sup>

Cð Þ<sup>k</sup>

<sup>w</sup>ð Þ<sup>k</sup> ð Þ¼ <sup>r</sup> <sup>A</sup>ð Þ<sup>k</sup>

<sup>36</sup> are zero and <sup>C</sup>ð Þ<sup>k</sup>

ffiffiffiffiffiffiffiffi Cð Þ <sup>k</sup> 22 Cð Þ <sup>k</sup> 33

<sup>12</sup> <sup>¼</sup> <sup>C</sup>ð Þ<sup>k</sup>

<sup>2</sup> <sup>r</sup>�1ð Þ<sup>k</sup> [13].

ðRN R0

ðRN R0

σð Þ<sup>k</sup>

k<sup>11</sup> k<sup>12</sup> k<sup>13</sup> k<sup>21</sup> k<sup>22</sup> k<sup>23</sup>

k<sup>32</sup>

k<sup>33</sup>

k<sup>43</sup>

k<sup>42</sup>

k<sup>31</sup>

k<sup>41</sup>

2πτ<sup>x</sup>θr 2 dr ¼ 2π

vuut , <sup>Γ</sup>ð Þ <sup>k</sup> <sup>¼</sup> <sup>C</sup>ð Þ <sup>k</sup>

<sup>13</sup> as well as <sup>C</sup>ð Þ<sup>k</sup>

<sup>12</sup> � <sup>C</sup>ð Þ <sup>k</sup> 13

, <sup>Ω</sup>ð Þ <sup>k</sup> <sup>¼</sup> <sup>C</sup>ð Þ <sup>k</sup>

<sup>26</sup> � 2Cð Þ <sup>k</sup> 36

http://dx.doi.org/10.5772/intechopen.72288

(14)

17

<sup>26</sup> , and

<sup>16</sup> , <sup>C</sup>ð Þ<sup>k</sup>

!

<sup>33</sup> . As such, Eq. (13) is degenerated to become

<sup>x</sup> rdr ¼ F (15)

<sup>r</sup> ð Þ¼ R<sup>0</sup> pi (17)

<sup>r</sup> ð Þ¼ RN p<sup>0</sup> (18)

<sup>r</sup> ð Þ¼ Rk <sup>σ</sup>ð Þ <sup>k</sup>þ<sup>1</sup> <sup>r</sup> ðÞ ð Rk <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, …; <sup>N</sup>�1<sup>Þ</sup> (19)

<sup>w</sup>ð Þ<sup>k</sup> ð Þ¼ Rk <sup>w</sup>ð Þ <sup>k</sup>þ<sup>1</sup> ðÞ ð <sup>R</sup><sup>k</sup> <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, …; <sup>N</sup>�1<sup>Þ</sup> (20)

8

>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>:

F T pi po 0 0 ⋮

9

>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>;

¼

E0 x γ0 xθ Að Þ<sup>1</sup> 1 Að Þ<sup>1</sup> 2 ⋮ Að Þ <sup>N</sup> 1 Að Þ <sup>N</sup> 2

9

>>>>>>>>>>>>>>>=

(21)

>>>>>>>>>>>>>>>;

8

>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>:

0

dr ¼ T (16)

4Cð Þ <sup>k</sup> <sup>33</sup> � <sup>C</sup>ð Þ <sup>k</sup> 22

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

!

Cð Þ <sup>k</sup> <sup>33</sup> � <sup>C</sup>ð Þ <sup>k</sup> 22

Note that when a layer is made of 0o fiber orientation, the stiffness coefficients Cð Þ<sup>k</sup>

interfacial radial displacements, w, as shown in Eqs. (19) and (20), respectively:

2πσxrdr ¼ 2π

<sup>22</sup> <sup>¼</sup> <sup>C</sup>ð Þ<sup>k</sup>

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, pi in Eq. (17) a normal traction or internal pressure on the inner surface, and po in 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

> X<sup>N</sup> k¼1 ðRk Rk�<sup>1</sup> σð Þ<sup>k</sup>

X<sup>N</sup> k¼1 ðRk Rk�<sup>1</sup> τ ð Þk <sup>x</sup><sup>θ</sup> r 2

σð Þ<sup>1</sup>

σð Þ <sup>N</sup>

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

⋯

k1, <sup>2</sup>Nþ<sup>2</sup> k2, <sup>2</sup>Nþ<sup>2</sup> k3, <sup>2</sup>Nþ<sup>2</sup>

k4, <sup>2</sup>Nþ<sup>2</sup>

⋯ … …

⋮ ⋱⋮

k2Nþ2,<sup>1</sup> k2Nþ2, <sup>2</sup> k2Nþ2,<sup>3</sup> ⋯ k2Nþ2, <sup>2</sup>Nþ<sup>2</sup>

where ½ � <sup>C</sup> ð Þ<sup>k</sup> , as shown in Eq. (9), is the stiffness matrix in the principle material coordinate system in the k th layer.

$$\begin{aligned} [\mathbb{C}\ ]^{(k)} &= \begin{bmatrix} 1/E\_1^{(k)} & -\nu\_{12}^{(k)}/E\_1^{(k)} & -\nu\_{31}^{(k)}/E\_3^{(k)} & 0 & 0 & 0\\ -\nu\_{21}^{(k)}/E\_2^{(k)} & 1/E\_2^{(k)} & -\nu\_{32}^{(k)}/E\_3^{(k)} & 0 & 0 & 0\\ -\nu\_{13}^{(k)}/E\_1^{(k)} & -\nu\_{23}^{(k)}/E\_2^{(k)} & 1/E\_3^{(k)} & 0 & 0 & 0\\ 0 & 0 & 0 & 1/G\_{23}^{(k)} & 0 & 0\\ 0 & 0 & 0 & 0 & 1/G\_{13}^{(k)} & 0\\ 0 & 0 & 0 & 0 & 0 & 1/G\_{12}^{(k)} \end{bmatrix}^{-1} \end{aligned} \tag{9}$$

<sup>E</sup>ð Þ<sup>k</sup> and <sup>G</sup>ð Þ<sup>k</sup> are Young's modulus and shear modulus, respectively. ½ � <sup>R</sup> is the Reuter's matrix, which is defined as

$$\begin{aligned} [R\ ] = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 \end{bmatrix} \tag{10} \end{aligned} \tag{11}$$

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 k th layer of the laminated tube as illustrated in Eqs. (11)–(13):

$$
\mu^{(k)}(\mathbf{x}) = \varepsilon\_x^0 \mathbf{x} \tag{11}
$$

$$
\sigma^{(k)}(\mathbf{x}, r) = \boldsymbol{\gamma}\_{\mathbf{x}\boldsymbol{\theta}}^{0} \mathbf{x} r \tag{12}
$$

$$\sigma^{(k)}(r) = A\_1^{(k)} r^{\lambda^{(k)}} + A\_2^{(k)} r^{-\lambda^{(k)}} + \Gamma^{(k)} \varepsilon\_x^0 r + \Omega^{(k)} \gamma\_{x\theta}^0 r^2 \tag{13}$$

In the above, ε<sup>0</sup> <sup>x</sup> and γ<sup>0</sup> <sup>x</sup><sup>θ</sup> are axial strain and angle of twist per unit length constants, respectively. Að Þ<sup>k</sup> <sup>1</sup> and <sup>A</sup>ð Þ<sup>k</sup> <sup>2</sup> are the integration constants in the k th layer. λð Þ<sup>k</sup> , Γð Þ<sup>k</sup> , and Ωð Þ<sup>k</sup> are described in Eq. (14) in terms of components of the transformed stiffness matrix in the k th layer: A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints http://dx.doi.org/10.5772/intechopen.72288 17

$$
\lambda^{(\mathbf{k})} = \sqrt{\frac{\overline{\mathbf{C}}\_{22}^{(\mathbf{k})}}{\overline{\mathbf{C}}\_{33}^{(\mathbf{k})'}}} \quad \Gamma^{(\mathbf{k})} = \begin{pmatrix} \overline{\mathbf{C}}\_{12}^{(\mathbf{k})} - \overline{\mathbf{C}}\_{13}^{(\mathbf{k})} \\ \overline{\mathbf{C}}\_{33}^{(\mathbf{k})} - \overline{\mathbf{C}}\_{22}^{(\mathbf{k})} \end{pmatrix}, \quad \Omega^{(\mathbf{k})} = \begin{pmatrix} \overline{\mathbf{C}}\_{26}^{(\mathbf{k})} - 2\overline{\mathbf{C}}\_{36}^{(\mathbf{k})} \\ 4\overline{\mathbf{C}}\_{33}^{(\mathbf{k})} - \overline{\mathbf{C}}\_{22}^{(\mathbf{k})} \end{pmatrix} \tag{14}
$$

Note that when a layer is made of 0o fiber orientation, the stiffness coefficients Cð Þ<sup>k</sup> <sup>16</sup> , <sup>C</sup>ð Þ<sup>k</sup> <sup>26</sup> , and Cð Þ<sup>k</sup> <sup>36</sup> are zero and <sup>C</sup>ð Þ<sup>k</sup> <sup>12</sup> <sup>¼</sup> <sup>C</sup>ð Þ<sup>k</sup> <sup>13</sup> as well as <sup>C</sup>ð Þ<sup>k</sup> <sup>22</sup> <sup>¼</sup> <sup>C</sup>ð Þ<sup>k</sup> <sup>33</sup> . As such, Eq. (13) is degenerated to become <sup>w</sup>ð Þ<sup>k</sup> ð Þ¼ <sup>r</sup> <sup>A</sup>ð Þ<sup>k</sup> <sup>1</sup> <sup>r</sup>ð Þ<sup>k</sup> <sup>þ</sup> <sup>A</sup>ð Þ<sup>k</sup> <sup>2</sup> <sup>r</sup>�1ð Þ<sup>k</sup> [13].

In the above, C

system in the k

� �ð Þ<sup>k</sup> can be evaluated as

th layer.

16 Applied Adhesive Bonding in Science and Technology

nated tube as illustrated in Eqs. (11)–(13):

<sup>x</sup> and γ<sup>0</sup>

<sup>1</sup> and <sup>A</sup>ð Þ<sup>k</sup>

<sup>w</sup>ð Þ<sup>k</sup> ð Þ¼ <sup>r</sup> <sup>A</sup>ð Þ<sup>k</sup>

<sup>1</sup> r λð Þk

½ � <sup>C</sup> ð Þ<sup>k</sup> <sup>¼</sup>

which is defined as

In the above, ε<sup>0</sup>

tively. Að Þ<sup>k</sup>

C

� �ð Þ<sup>k</sup> is the transformed stiffness matrix, and <sup>ε</sup>eng f gð Þ<sup>k</sup>

C

<sup>1</sup> �υð Þ<sup>k</sup>

<sup>2</sup> <sup>1</sup>=Eð Þ<sup>k</sup>

½ �¼ R

continuity between each layer yield the displacement expressions in the k

<sup>1</sup> �υð Þ<sup>k</sup>

1=Eð Þ<sup>k</sup>

�υð Þ<sup>k</sup> <sup>21</sup> <sup>=</sup>Eð Þ<sup>k</sup>

�υð Þ<sup>k</sup> <sup>13</sup> <sup>=</sup>Eð Þ<sup>k</sup>

� �ð Þ<sup>k</sup> <sup>¼</sup> ½ � <sup>T</sup> ð Þ<sup>k</sup> n o�<sup>1</sup>

<sup>12</sup> <sup>=</sup>Eð Þ<sup>k</sup>

<sup>23</sup> <sup>=</sup>Eð Þ<sup>k</sup>

components in the global cylindrical coordinate system. The transformed stiffness matrix

where ½ � <sup>C</sup> ð Þ<sup>k</sup> , as shown in Eq. (9), is the stiffness matrix in the principle material coordinate

<sup>31</sup> <sup>=</sup>Eð Þ<sup>k</sup>

<sup>32</sup> <sup>=</sup>Eð Þ<sup>k</sup>

0 0 0 0 01=Gð Þ<sup>k</sup>

<sup>1</sup> �υð Þ<sup>k</sup>

<sup>2</sup> <sup>1</sup>=Eð Þ<sup>k</sup>

0 0 0 01=Gð Þ<sup>k</sup>

<sup>E</sup>ð Þ<sup>k</sup> and <sup>G</sup>ð Þ<sup>k</sup> are Young's modulus and shear modulus, respectively. ½ � <sup>R</sup> is the Reuter's matrix,

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

<sup>u</sup>ð Þ<sup>k</sup> ð Þ¼ <sup>x</sup> <sup>ε</sup><sup>0</sup>

<sup>v</sup>ð Þ<sup>k</sup> ð Þ¼ <sup>x</sup>;<sup>r</sup> <sup>γ</sup><sup>0</sup>

<sup>þ</sup> <sup>Γ</sup>ð Þ<sup>k</sup> <sup>ε</sup><sup>0</sup>

<sup>x</sup><sup>θ</sup> are axial strain and angle of twist per unit length constants, respec-

<sup>x</sup><sup>r</sup> <sup>þ</sup> <sup>Ω</sup>ð Þ<sup>k</sup> <sup>γ</sup><sup>0</sup>

<sup>þ</sup> <sup>A</sup>ð Þ<sup>k</sup> <sup>2</sup> r �λð Þ<sup>k</sup>

described in Eq. (14) in terms of components of the transformed stiffness matrix in the k

<sup>2</sup> are the integration constants in the k

<sup>2</sup> �υð Þ<sup>k</sup>

0 0 01=Gð Þ<sup>k</sup>

<sup>x</sup>θ<sup>r</sup> is engineering strain

�1

(9)

(10)

th layer of the lami-

<sup>2</sup> (13)

th layer:

th layer. λð Þ<sup>k</sup> , Γð Þ<sup>k</sup> , and Ωð Þ<sup>k</sup> are

½ � <sup>C</sup> ð Þ<sup>k</sup> ½ � <sup>R</sup> ½ � <sup>T</sup> ð Þ<sup>k</sup> ½ � <sup>R</sup> �<sup>1</sup> (8)

<sup>3</sup> 000

<sup>3</sup> 000

<sup>23</sup> 0 0

<sup>13</sup> 0

<sup>x</sup>x (11)

<sup>x</sup>θxr (12)

xθr

12

<sup>3</sup> 000

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, pi in Eq. (17) a normal traction or internal pressure on the inner surface, and po in 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:

$$\int\_{R\_0}^{R\_N} 2\pi \sigma\_x r dr = 2\pi \sum\_{k=1}^N \int\_{R\_{k-1}}^{R\_k} \sigma\_x^{(k)} r dr = F \tag{15}$$

$$\int\_{R\_{\emptyset}}^{R\_{\aleph}} 2\pi \tau\_{x\emptyset} r^2 dr = 2\pi \sum\_{k=1}^{N} \int\_{R\_{k-1}}^{R\_k} \tau\_{x\emptyset}^{(k)} r^2 dr = T \tag{16}$$

$$
\sigma\_r^{(1)}(\mathcal{R}\_0) = \mathcal{p}\_i \tag{17}
$$

$$
\sigma\_r^{(N)}(R\_N) = p\_0 \tag{18}
$$

$$
\sigma\_r^{(k)}(\mathcal{R}\_k) = \sigma\_r^{(k+1)}(\mathcal{R}\_k) \qquad (k = 1, 2, 3, \dots, N - 1) \tag{19}
$$

$$w^{(k)}(R\_k) = w^{(k+1)}(R\_k) \qquad (k = 1, 2, 3, \dots, N-1) \tag{20}$$

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

$$
\begin{bmatrix}
\begin{array}{cccc}
k\_{11} & k\_{12} & k\_{13} & \dots & k\_{1,2N+2} \\
& k\_{21} & k\_{22} & k\_{23} & \dots & k\_{2,2N+2} \\
& k\_{31} & k\_{32} & k\_{33} & \dots & k\_{3,2N+2} \\
& & & & \\
& \ddots & \ddots & \ddots & \ddots & \\
& & \vdots & & \ddots & \vdots \\
& & & & \ddots & \vdots \\
& & & & & \ddots & \ddots \\
& & & & & & \ddots
\end{bmatrix}
\end{bmatrix}
\begin{Bmatrix}
F \\
T \\
p\_i \\
p\_o \\
\vdots \\
p\_o \\
\vdots \\
p\_1 \\
\vdots \\
p\_N
\end{Bmatrix} = \begin{Bmatrix}
\epsilon\_x^0 \\
\epsilon\_x^0 \\
\vdots \\
A\_1^{(1)} \\
\vdots \\
A\_2^{(N)} \\
A\_3^{(N)} \\
\vdots \\
A\_2^{(N)}
\end{Bmatrix} \tag{21}
$$

where kij ð Þ i; j ¼ 1; 2; 3;…; 2N þ 2 are the coefficients obtained from the equations above. By solving Eq. (21), the constants E<sup>0</sup> <sup>x</sup>, γ<sup>0</sup> <sup>x</sup>θ, and <sup>A</sup>ð Þ<sup>1</sup> <sup>1</sup> , Að Þ<sup>1</sup> <sup>2</sup> , …, Að Þ <sup>N</sup> <sup>1</sup> , Að Þ <sup>N</sup> <sup>2</sup> can 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

All geometric parameters of a perfectly bonded tubular lap joint are shown in Figure 1(b). The adhesive is assumed to be isotropic and linearly elastic. The adhesive thickness ta is considered to be very thin compared to the adherend thicknesses, and thus, the outer radius of part 1 R1<sup>o</sup> is approximately the same as the inner radius of part 2, R2i. In addition, there are only three outof-plane stress components mainly contributed in the adhesive: hoop shear stress τ<sup>a</sup> <sup>θ</sup><sup>r</sup>, longitudinal shear stress τ<sup>a</sup> xr, and radial normal stress σ<sup>a</sup> <sup>r</sup>. These stresses in the adhesive are treated to be uniform through the adhesive thickness. Applied torque T; applied axial force F; internal pressure exerted on the inner surface of adherend part 1, pi or written specifically as p1<sup>i</sup> ; and external pressure exerted on the outer surface of adherend part 2, po or p2o, are all included in the following formulation.

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 T<sup>1</sup> and T2, respectively. Hence, the applied torque T can be written as

$$T = T\_1 + T\_2 \tag{22}$$

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

d2 T2ð Þx dx<sup>2</sup> <sup>¼</sup> <sup>2</sup>πR<sup>2</sup>

axial force in the element and the adhesive longitudinal shear stress τ<sup>a</sup>

1 2πR2<sup>i</sup>

xr <sup>¼</sup> u2i � u1o ta

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

d2 F2ð Þx

∂σ<sup>r</sup> ∂r þ 1 r

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

γa

dF2ð Þx dx <sup>¼</sup> <sup>τ</sup><sup>a</sup>

dx<sup>2</sup> <sup>¼</sup> <sup>2</sup>πR2iGa ta

equilibrium is

expressed as follows:

p2<sup>i</sup> are related to each other as

the adhesive layer showing in Eq. (32):

2i Ga ta

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

The variation of the F<sup>2</sup> along the length can be examined by considering an infinitesimal elements in adherend part 2 with the differential length dx. The equilibrium between resultant

> or dγ<sup>a</sup> xr dx <sup>¼</sup> <sup>ε</sup>2i

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, p2<sup>i</sup>

generated. Under the assumption of thin adhesive layer, the resultant normal tractions p1<sup>o</sup> and

Lastly, instead of directly equating adhesive radial normal stress to normal traction in (31), σ<sup>a</sup>

can be more accurately determined by the equilibrium equation in cylindrical coordinates of

1 r ∂τθ<sup>r</sup> ∂θ þ ∂τxr

ð Þþ σ<sup>r</sup> � σθ

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

γ2i <sup>x</sup><sup>θ</sup> � <sup>γ</sup><sup>1</sup><sup>o</sup> xθ

xr <sup>¼</sup> Ga γa

> ε2i <sup>x</sup> � <sup>ε</sup><sup>1</sup><sup>o</sup> x

<sup>x</sup> � <sup>ε</sup>1o x ta

(26)

http://dx.doi.org/10.5772/intechopen.72288

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

xr can consequently be

(29)

19

, are

r

xr (28)

(30)

p1<sup>o</sup> ¼ p2<sup>i</sup> (31)

<sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup> (32)

F ¼ F<sup>1</sup> þ F<sup>2</sup> (27)

To determine the variation of the T<sup>2</sup> along 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:

$$\frac{1}{2\pi R\_{2i}^2} \frac{dT\_2(\mathbf{x})}{d\mathbf{x}} = \pi\_{\theta r}^{\boldsymbol{a}} = G^{\boldsymbol{a}} \gamma\_{\theta r}^{\boldsymbol{a}} \tag{23}$$

In Eq. (23), Ga is 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

$$\gamma\_{\theta r}^{a} = \frac{\upsilon\_{2i} - \upsilon\_{1o}}{t\_a} \tag{24}$$

and its derivative with respect to x is.

$$\frac{d\gamma^{a}\_{\partial r}}{d\mathbf{x}} = \frac{\gamma^{2i}\_{x\partial} - \gamma^{1o}\_{x\partial}}{t\_a} \tag{25}$$

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

where kij ð Þ i; j ¼ 1; 2; 3;…; 2N þ 2 are the coefficients obtained from the equations above. By

quently, all displacements, strains, and stresses are calculated by using Eqs. (11)–(13), (1), and

All geometric parameters of a perfectly bonded tubular lap joint are shown in Figure 1(b). The adhesive is assumed to be isotropic and linearly elastic. The adhesive thickness ta is considered to be very thin compared to the adherend thicknesses, and thus, the outer radius of part 1 R1<sup>o</sup> is approximately the same as the inner radius of part 2, R2i. In addition, there are only three out-

be uniform through the adhesive thickness. Applied torque T; applied axial force F; internal pressure exerted on the inner surface of adherend part 1, pi or written specifically as p1<sup>i</sup>

external pressure exerted on the outer surface of adherend part 2, po or p2o, are all included in

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 T<sup>1</sup> and T2, respectively. Hence, the applied torque T

To determine the variation of the T<sup>2</sup> along the bonding length, the adherend 2 is divided into elements with an infinitesimal length dx. The equilibrium between the resultant torque in the

In Eq. (23), Ga is 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

> <sup>θ</sup><sup>r</sup> <sup>¼</sup> <sup>v</sup>2<sup>i</sup> � <sup>v</sup>1<sup>o</sup> ta

> > <sup>x</sup><sup>θ</sup> � <sup>γ</sup><sup>1</sup><sup>o</sup> xθ ta

<sup>θ</sup><sup>r</sup> <sup>¼</sup> <sup>G</sup><sup>a</sup> γa

dT2ð Þx dx <sup>¼</sup> <sup>τ</sup><sup>a</sup>

γa

dγ<sup>a</sup> θr dx <sup>¼</sup> <sup>γ</sup><sup>2</sup><sup>i</sup>

of-plane stress components mainly contributed in the adhesive: hoop shear stress τ<sup>a</sup>

xr, and radial normal stress σ<sup>a</sup>

element and the adhesive hoop shear stress can be expressed as follows: 1 2πR<sup>2</sup> 2i

kinematic condition in the adhesive can be written as

and its derivative with respect to x is.

<sup>1</sup> , Að Þ<sup>1</sup>

<sup>2</sup> , …, Að Þ <sup>N</sup>

<sup>1</sup> , Að Þ <sup>N</sup>

<sup>r</sup>. These stresses in the adhesive are treated to

T ¼ T<sup>1</sup> þ T<sup>2</sup> (22)

<sup>θ</sup><sup>r</sup> (23)

<sup>2</sup> can be obtained. Subse-

<sup>θ</sup><sup>r</sup>, longitu-

; and

(24)

(25)

<sup>x</sup>θ, and <sup>A</sup>ð Þ<sup>1</sup>

<sup>x</sup>, γ<sup>0</sup>

3. Formulation of an equivalent lap joint model

solving Eq. (21), the constants E<sup>0</sup>

18 Applied Adhesive Bonding in Science and Technology

3.1. Derivation of governing equations

(7), respectively.

dinal shear stress τ<sup>a</sup>

can be written as

the following formulation.

$$\frac{d^2T\_2(\mathbf{x})}{d\mathbf{x}^2} = \frac{2\pi R\_{2i}^2 G^a}{t\_a} \left(\boldsymbol{\gamma}\_{x\boldsymbol{\theta}}^{2i} - \boldsymbol{\gamma}\_{x\boldsymbol{\theta}}^{1o}\right) \tag{26}$$

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

$$F = F\_1 + F\_2 \tag{27}$$

The variation of the F<sup>2</sup> along 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 τ<sup>a</sup> xr can consequently be expressed as follows:

$$\frac{1}{2\pi R\_{2i}} \frac{dF\_2(\mathbf{x})}{d\mathbf{x}} = \boldsymbol{\tau}\_{\mathbf{x}r}^a = \mathbf{G}^a \boldsymbol{\gamma}\_{\mathbf{x}r}^a \tag{28}$$

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

$$\gamma\_{\rm xr}^{\rm a} = \frac{\mathbf{u}\_{2\rm i} - \mathbf{u}\_{1o}}{\mathbf{t}\_{\rm a}} \text{ or } \frac{\mathbf{d}\gamma\_{\rm xr}^{\rm a}}{\mathbf{dx}} = \frac{\varepsilon\_{\rm x}^{2\rm i} - \varepsilon\_{\rm x}^{1o}}{\mathbf{t}\_{\rm a}} \tag{29}$$

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

$$\frac{d^2 F\_2(\mathbf{x})}{d\mathbf{x}^2} = \frac{2\pi R\_{2i} G^t}{t\_d} \left(\varepsilon\_\mathbf{x}^{2i} - \varepsilon\_\mathbf{x}^{1o}\right) \tag{30}$$

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, p2<sup>i</sup> , are generated. Under the assumption of thin adhesive layer, the resultant normal tractions p1<sup>o</sup> and p2<sup>i</sup> are related to each other as

$$p\_{1o} = p\_{2i} \tag{31}$$

Lastly, instead of directly equating adhesive radial normal stress to normal traction in (31), σ<sup>a</sup> r can be more accurately determined by the equilibrium equation in cylindrical coordinates of the adhesive layer showing in Eq. (32):

$$\frac{\partial \sigma\_r}{\partial r} + \frac{1}{r}(\sigma\_r - \sigma\_\theta) + \frac{1}{r}\frac{\partial \tau\_{\partial r}}{\partial \theta} + \frac{\partial \tau\_{\text{ar}}}{\partial \mathbf{x}} = \mathbf{0} \tag{32}$$

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

$$\frac{1}{R\_{2i}} \left( \sigma\_r^a - \sigma\_\theta^a \right) + \frac{\partial \tau\_{xr}^a}{\partial x} = 0 \tag{33}$$

By substituting Eqs. (36) and (37) into the governing equation Eq. (26) and utilizing Eqs. (22),

2i Ga ta

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

be written in terms of the internal resultant loads. Again, using the expression in Eq. (21), the

matrix ½ � k in Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1 and 2, respectively.

of the parameters. They represent the in-plane normal strains due to unit load on the adhesiveinterfacial surface in the adherends. The unit load quantities are distinguished by after-comma

Combining the governing equations Eqs. (30), (41), and (42), as well as the load equilibriums in

ta

<sup>x</sup> are ε<sup>0</sup>

dx<sup>2</sup> <sup>¼</sup> kFF2ð Þþ <sup>x</sup> kTT2ð Þþ <sup>x</sup> kpp2<sup>i</sup>

Eqs. (22), (27), and (31), yields a new form of the axial force governing equation:

x,FF<sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>1</sup><sup>o</sup>

x,FF<sup>2</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup><sup>i</sup>

2i Ga ta

γ2i

γ2i <sup>x</sup>θ,T <sup>þ</sup> <sup>γ</sup><sup>1</sup><sup>o</sup> xθ,T ,

T2ð Þ¼ 0 0, T2ð Þ¼ L T (40)

x,TT<sup>1</sup> <sup>þ</sup> <sup>ε</sup><sup>1</sup><sup>o</sup>

x,TT<sup>2</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup><sup>i</sup>

<sup>1</sup>X, where X ¼ 1, 2, 3, and 4 are the first four elements in the first row of

ta

ε2i

<sup>x</sup>θ, pipi � <sup>γ</sup><sup>1</sup><sup>o</sup>

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

x, pip1<sup>i</sup> <sup>þ</sup> <sup>ε</sup><sup>1</sup><sup>o</sup>

x, pip2<sup>i</sup> <sup>þ</sup> <sup>ε</sup><sup>2</sup><sup>i</sup>

x, po are newly denoted to indicate the physical meaning

<sup>x</sup> of adherends 1 and 2, respectively.

ε2i x,T <sup>þ</sup> <sup>ε</sup><sup>1</sup><sup>o</sup> x,T 

x, pipi � <sup>ε</sup><sup>1</sup><sup>o</sup>

x,popo � <sup>ε</sup><sup>1</sup><sup>o</sup>

<sup>x</sup>θ,popo � <sup>γ</sup><sup>1</sup><sup>o</sup>

ð Þþ x KC (38)

http://dx.doi.org/10.5772/intechopen.72288

<sup>x</sup>θ,F<sup>F</sup> � <sup>γ</sup><sup>1</sup><sup>o</sup>

<sup>x</sup> and ε<sup>2</sup><sup>i</sup>

x, pop1<sup>o</sup> <sup>¼</sup> <sup>ε</sup><sup>1</sup><sup>o</sup>

x, pop1<sup>o</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup><sup>i</sup>

ð Þþ x kC (43)

x,F<sup>F</sup> � <sup>ε</sup><sup>1</sup><sup>o</sup>

x,TT

(44)

<sup>x</sup>θ,TT

<sup>x</sup> must also

<sup>x</sup> (41)

<sup>x</sup> (42)

(39)

21

dx<sup>2</sup> <sup>¼</sup> KFF2ð Þþ <sup>x</sup> KTT2ð Þþ <sup>x</sup> Kpp2<sup>i</sup>

(27), and (31), the governing equation in term of T<sup>2</sup> becomes

γ2i <sup>x</sup>θ,F <sup>þ</sup> <sup>γ</sup><sup>1</sup><sup>o</sup> xθ,F , KT <sup>¼</sup> <sup>2</sup>πR<sup>2</sup>

<sup>x</sup>θ,po , KC <sup>¼</sup> <sup>2</sup>πR<sup>2</sup>

<sup>13</sup> <sup>p</sup>1<sup>i</sup> <sup>þ</sup> <sup>k</sup>

2 <sup>13</sup> <sup>p</sup>2<sup>i</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

d2 F2ð Þx

kF <sup>¼</sup> <sup>2</sup>πR2iG<sup>a</sup> ta

> ε2i x,pi � <sup>ε</sup><sup>1</sup><sup>o</sup>

where the parameters kF, kT, kp, and kC are.

Second, analogous to Eqs. (36) and (37), adherend in-plane normal strains ε<sup>1</sup><sup>o</sup>

1 <sup>14</sup> <sup>p</sup>1<sup>o</sup> <sup>¼</sup> <sup>ε</sup><sup>1</sup><sup>o</sup>

x, pi, and ε<sup>2</sup><sup>i</sup>

<sup>x</sup> and ε<sup>2</sup><sup>i</sup>

ε2i x,F <sup>þ</sup> <sup>ε</sup><sup>1</sup><sup>o</sup> x,F , kT <sup>¼</sup> <sup>2</sup>πR2iG<sup>a</sup>

x, po , kC <sup>¼</sup> <sup>2</sup>πR2iG<sup>a</sup>

<sup>14</sup> <sup>p</sup>2<sup>o</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup><sup>i</sup>

d2 T2ð Þx

2i Ga ta

solving resultant torque in the adherend part 2,T2:

where the parameters KF, KT, Kp, and KC are.

KF <sup>¼</sup> <sup>2</sup>πR<sup>2</sup>

γ2i <sup>x</sup>θ, pi � <sup>γ</sup><sup>1</sup><sup>o</sup>

following expressions are obtained:

1 <sup>12</sup> <sup>T</sup><sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>1</sup>

2 <sup>12</sup> <sup>T</sup><sup>2</sup> <sup>þ</sup> <sup>k</sup>

<sup>1</sup><sup>X</sup> and k 2

x, pi, ε<sup>1</sup><sup>o</sup> x,po ε<sup>2</sup><sup>i</sup> x,F, ε<sup>2</sup><sup>i</sup> x,T, ε<sup>2</sup><sup>i</sup>

subscripts F, T, or p. Quantities ε<sup>1</sup><sup>o</sup>

kp <sup>¼</sup> <sup>2</sup>πR2iG<sup>a</sup> ta

Kp <sup>¼</sup> <sup>2</sup>πR<sup>2</sup>

k 1 <sup>11</sup> <sup>F</sup><sup>1</sup> <sup>þ</sup> <sup>k</sup>

k2 <sup>11</sup> <sup>F</sup><sup>2</sup> <sup>þ</sup> <sup>k</sup>

In the above, k<sup>1</sup>

ε1o x,F, ε<sup>1</sup><sup>o</sup> x,T, ε<sup>1</sup><sup>o</sup>

2i Ga ta

According to the study conducted in [6], σ<sup>a</sup> <sup>θ</sup> is observed to have the same distribution as σ<sup>a</sup> <sup>r</sup> so 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):

$$
\sigma^a\_\partial = a \sigma^a\_r \tag{34}
$$

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

$$
\sigma\_r^a = -\frac{1}{2\pi(1-a)} \frac{d^2 F\_2(\mathbf{x})}{d\mathbf{x}^2} \tag{35}
$$

#### 3.2. Implementation of elasticity theory for adherends

The two governing equations Eqs. (26) and (30) have already been formulated to determine resultant loads in adherend part 2 of an adhesive-bonded-coupler joint. The resultant loads in adherend part 1 can be then calculated easily by using Eqs. (22) and (27) after all internal loads in adherend part 2 are evaluated. However, related through Eq. (21), the two equations are coupled and need to be solved altogether. To aptly deal with this complicated condition, the problem is separated into primary and secondary effects. When the joint is subjected to torsion, the hoop shear stress in the adhesive τ<sup>a</sup> <sup>θ</sup><sup>r</sup> is primary and dominant compared to the other adhesive stresses as discussed in [8, 10, 13], whereas in the case of the joint being under an application of longitudinal force, or external and internal pressure, the adhesive longitudinal shear stress τ<sup>a</sup> xr and adhesive radial normal stress σ<sup>a</sup> <sup>r</sup> are comparatively crucial [20]. By neglecting the secondary stress components and the corresponding resultant internal loads in the early calculation stage, the problem is then uncoupled and can be readily solved for the primary variables. The initially excluded stress components are later recovered by using the obtained solutions in the coupled set of governing equations.

First, further modification of the torque governing equation of Eq. (26) is performed, adherend in-plane shear strains γ<sup>1</sup><sup>o</sup> <sup>x</sup><sup>θ</sup> and γ<sup>2</sup><sup>i</sup> <sup>x</sup><sup>θ</sup> must be expanded in terms of the internal resultant loads. It can be seen that they are equal to γ<sup>01</sup> <sup>x</sup>θR1<sup>o</sup> and γ<sup>02</sup> <sup>x</sup>θR2i, respectively, where γ<sup>01</sup> <sup>x</sup><sup>θ</sup> and γ<sup>02</sup> <sup>x</sup><sup>θ</sup> are denoted for γ<sup>0</sup> <sup>x</sup><sup>θ</sup> of adherend parts 1 and 2. Utilizing Eq. (21) yields the relations:

$$R\_{1o} \left( k\_{21}^1 F\_1 + k\_{22}^1 T\_1 + k\_{23}^1 p\_{1i} + k\_{24}^1 p\_{1o} \right) = \gamma\_{x\theta, F}^{1o} F\_1 + \gamma\_{x\theta, T}^{1o} T\_1 + \gamma\_{x\theta, p}^{1o} p\_{1i} + \gamma\_{x\theta, p}^{1o} p\_{1o} = \gamma\_{x\theta}^{1o} \tag{36}$$

$$R\_{2i}(k\_{21}^2 F\_2 + k\_{22}^2 T\_2 + k\_{23}^2 p\_{2i} + k\_{24}^2 p\_{2o}) = \gamma\_{x0,F}^{2i} F\_2 + \gamma\_{x0,T}^{2i} T\_2 + \gamma\_{x0,p0}^{2i} p\_{2i} + \gamma\_{x0,p0}^{2i} p\_{2o} = \gamma\_{x0}^{2i} \tag{37}$$

where quantities k 1 <sup>2</sup><sup>X</sup> and k 2 <sup>2</sup>X, where X ¼ 1, 2, 3, and 4 are the first four elements compliances in the second row of matrix ½ � k in Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1 and 2, respectively. γ<sup>1</sup><sup>o</sup> <sup>x</sup>θ,F, γ<sup>1</sup><sup>o</sup> <sup>x</sup>θ,T, γ<sup>1</sup><sup>o</sup> <sup>x</sup>θ,pi, γ<sup>1</sup><sup>o</sup> <sup>x</sup>θ, po, γ<sup>2</sup><sup>i</sup> <sup>x</sup>θ,F, γ<sup>2</sup><sup>i</sup> <sup>x</sup>θ,T, γ<sup>2</sup><sup>i</sup> <sup>x</sup>θ, pi, and γ<sup>2</sup><sup>i</sup> <sup>x</sup>θ, po are 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 T<sup>2</sup> becomes

$$\frac{d^2T\_2(\mathbf{x})}{d\mathbf{x}^2} = K\_F T\_2(\mathbf{x}) + K\_T T\_2(\mathbf{x}) + K\_p p\_{2i}(\mathbf{x}) + K\_{\mathbb{C}} \tag{38}$$

where the parameters KF, KT, Kp, and KC are.

1 R2<sup>i</sup> σa <sup>r</sup> � <sup>σ</sup><sup>a</sup> θ <sup>þ</sup>

ratio α. Their relation can be mathematically expressed in Eq. (34):

According to the study conducted in [6], σ<sup>a</sup>

20 Applied Adhesive Bonding in Science and Technology

∂τ<sup>a</sup> xr

they are legitimately regarded as being proportional to each other via adhesive normal stress

2πð Þ 1 � α

The two governing equations Eqs. (26) and (30) have already been formulated to determine resultant loads in adherend part 2 of an adhesive-bonded-coupler joint. The resultant loads in adherend part 1 can be then calculated easily by using Eqs. (22) and (27) after all internal loads in adherend part 2 are evaluated. However, related through Eq. (21), the two equations are coupled and need to be solved altogether. To aptly deal with this complicated condition, the problem is separated into primary and secondary effects. When the joint is subjected to torsion,

adhesive stresses as discussed in [8, 10, 13], whereas in the case of the joint being under an application of longitudinal force, or external and internal pressure, the adhesive longitudinal

neglecting the secondary stress components and the corresponding resultant internal loads in the early calculation stage, the problem is then uncoupled and can be readily solved for the primary variables. The initially excluded stress components are later recovered by using the

First, further modification of the torque governing equation of Eq. (26) is performed, adherend

<sup>x</sup>θR1<sup>o</sup> and γ<sup>02</sup>

<sup>x</sup><sup>θ</sup> of adherend parts 1 and 2. Utilizing Eq. (21) yields the relations:

<sup>x</sup>θ,FF<sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>1</sup><sup>o</sup>

<sup>x</sup>θ,FF<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup><sup>i</sup>

in the second row of matrix ½ � k in Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1

strains per unit load on the outer interfacial surface of adherend part 1 and inner interfacial

<sup>x</sup>θ,F, γ<sup>2</sup><sup>i</sup>

<sup>x</sup>θ, po, γ<sup>2</sup><sup>i</sup>

d2 F2ð Þx

σa <sup>θ</sup> <sup>¼</sup> ασ<sup>a</sup>

<sup>r</sup> ¼ � <sup>1</sup>

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

σa

xr and adhesive radial normal stress σ<sup>a</sup>

obtained solutions in the coupled set of governing equations.

<sup>x</sup><sup>θ</sup> and γ<sup>2</sup><sup>i</sup>

<sup>¼</sup> <sup>γ</sup><sup>1</sup><sup>o</sup>

<sup>¼</sup> <sup>γ</sup><sup>2</sup><sup>i</sup>

<sup>x</sup>θ,F, γ<sup>1</sup><sup>o</sup>

<sup>23</sup>p2<sup>i</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

<sup>x</sup>θ,T, γ<sup>1</sup><sup>o</sup>

<sup>24</sup>p2<sup>o</sup>

<sup>x</sup>θ,pi, γ<sup>1</sup><sup>o</sup>

3.2. Implementation of elasticity theory for adherends

the hoop shear stress in the adhesive τ<sup>a</sup>

can be seen that they are equal to γ<sup>01</sup>

1 <sup>2</sup><sup>X</sup> and k 2

surface of adherend part 2, respectively.

shear stress τ<sup>a</sup>

denoted for γ<sup>0</sup>

R1<sup>o</sup> k 1 <sup>21</sup>F<sup>1</sup> þ k 1 <sup>22</sup>T<sup>1</sup> þ k 1 <sup>23</sup>p1<sup>i</sup> þ k 1 <sup>24</sup>p1<sup>o</sup>

R2<sup>i</sup> k<sup>2</sup>

where quantities k

and 2, respectively. γ<sup>1</sup><sup>o</sup>

in-plane shear strains γ<sup>1</sup><sup>o</sup>

<sup>21</sup>F<sup>2</sup> þ k 2 <sup>22</sup>T<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>0</sup> (33)

<sup>r</sup> (34)

dx<sup>2</sup> (35)

<sup>r</sup> are comparatively crucial [20]. By

<sup>x</sup><sup>θ</sup> and γ<sup>02</sup>

<sup>x</sup>θ,pop1<sup>o</sup> <sup>¼</sup> <sup>γ</sup><sup>1</sup><sup>o</sup>

<sup>x</sup>θ,pop2<sup>o</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup><sup>i</sup>

<sup>x</sup>θ, po are in-plane shear

<sup>x</sup><sup>θ</sup> are

<sup>x</sup><sup>θ</sup> (36)

<sup>x</sup><sup>θ</sup> (37)

<sup>r</sup> so

<sup>θ</sup> is observed to have the same distribution as σ<sup>a</sup>

<sup>θ</sup><sup>r</sup> is primary and dominant compared to the other

<sup>x</sup><sup>θ</sup> must be expanded in terms of the internal resultant loads. It

<sup>x</sup>θ,TT<sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>1</sup><sup>o</sup>

<sup>x</sup>θ,TT<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup><sup>i</sup>

<sup>x</sup>θ,T, γ<sup>2</sup><sup>i</sup>

<sup>2</sup>X, where X ¼ 1, 2, 3, and 4 are the first four elements compliances

<sup>x</sup>θR2i, respectively, where γ<sup>01</sup>

<sup>x</sup>θ, pip1<sup>i</sup> <sup>þ</sup> <sup>γ</sup><sup>1</sup><sup>o</sup>

<sup>x</sup>θ,pip2<sup>i</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup><sup>i</sup>

<sup>x</sup>θ, pi, and γ<sup>2</sup><sup>i</sup>

$$\begin{aligned} \mathcal{K}\_{\mathcal{F}} &= \frac{2\pi R\_{2i}^{2} \mathcal{G}^{d}}{t\_{\rm d}} \left( \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\mathcal{F}}^{2i} + \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\mathcal{F}}^{1o} \right), \mathcal{K}\_{\mathcal{T}} = \frac{2\pi R\_{2i}^{2} \mathcal{G}^{d}}{t\_{\rm d}} \left( \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\mathcal{T}}^{2i} + \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\mathcal{T}}^{1o} \right), \\\ \mathcal{K}\_{\mathcal{p}} &= \frac{2\pi R\_{2i}^{2} \mathcal{G}^{d}}{t\_{\rm d}} \left( \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\boldsymbol{\mu}}^{2i} - \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\boldsymbol{\mu}}^{1o} \right), \qquad \mathcal{K}\_{\mathcal{C}} = \frac{2\pi R\_{2i}^{2} \mathcal{G}^{d}}{t\_{\rm d}} \left( \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\boldsymbol{\mu}}^{2i} \boldsymbol{p}\_{o} - \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\boldsymbol{\mu}}^{1o} \boldsymbol{p}\_{i} - \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\boldsymbol{\mu}}^{1o} \boldsymbol{F} - \boldsymbol{\gamma}\_{x\boldsymbol{\theta},\boldsymbol{\Gamma}}^{1o} \boldsymbol{T} \right) \end{aligned} \tag{39}$$

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:

$$T\_2(0) = 0, \qquad T\_2(L) = T \tag{40}$$

Second, analogous to Eqs. (36) and (37), adherend in-plane normal strains ε<sup>1</sup><sup>o</sup> <sup>x</sup> and ε<sup>2</sup><sup>i</sup> <sup>x</sup> must also be written in terms of the internal resultant loads. Again, using the expression in Eq. (21), the following expressions are obtained:

$$\varepsilon\_1(k\_{11}^1)F\_1 + \left(k\_{12}^1\right)T\_1 + \left(k\_{13}^1\right)p\_{1i} + \left(k\_{14}^1\right)p\_{1o} = \varepsilon\_{x,F}^{1o}F\_1 + \varepsilon\_{x,T}^{1o}T\_1 + \varepsilon\_{x,p}^{1o}p\_{1i} + \varepsilon\_{x,pv}^{1o}p\_{1o} = \varepsilon\_x^{1o} \tag{41}$$

$$(\mathbf{k}\_{11}^2)\mathbf{F}\_2 + (\mathbf{k}\_{12}^2)T\_2 + (\mathbf{k}\_{13}^2)p\_{2\mathbf{i}} + (\mathbf{k}\_{14}^2)p\_{2\mathbf{o}} = \boldsymbol{\varepsilon}\_{\mathbf{x},\mathbf{F}}^{2\mathbf{i}}\mathbf{F}\_2 + \boldsymbol{\varepsilon}\_{\mathbf{x},\mathbf{T}}^{2\mathbf{i}}T\_2 + \boldsymbol{\varepsilon}\_{\mathbf{x},\mathbf{p}}^{2\mathbf{i}}p\_{2\mathbf{i}} + \boldsymbol{\varepsilon}\_{\mathbf{x},\mathbf{p}}^{2\mathbf{i}}p\_{1\mathbf{o}} = \boldsymbol{\varepsilon}\_{\mathbf{x}}^{2\mathbf{i}}\tag{42}$$

In the above, k<sup>1</sup> <sup>1</sup><sup>X</sup> and k 2 <sup>1</sup>X, where X ¼ 1, 2, 3, and 4 are the first four elements in the first row of matrix ½ � k in Eq. (21). Superscripts 1 and 2 are defined for adherend parts 1 and 2, respectively. ε1o x,F, ε<sup>1</sup><sup>o</sup> x,T, ε<sup>1</sup><sup>o</sup> x, pi, ε<sup>1</sup><sup>o</sup> x,po ε<sup>2</sup><sup>i</sup> x,F, ε<sup>2</sup><sup>i</sup> x,T, ε<sup>2</sup><sup>i</sup> x, pi, and ε<sup>2</sup><sup>i</sup> x, po are newly denoted to indicate the physical meaning of the parameters. They represent the in-plane normal strains due to unit load on the adhesiveinterfacial surface in the adherends. The unit load quantities are distinguished by after-comma subscripts F, T, or p. Quantities ε<sup>1</sup><sup>o</sup> <sup>x</sup> and ε<sup>2</sup><sup>i</sup> <sup>x</sup> are ε<sup>0</sup> <sup>x</sup> of 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:

$$\frac{d^2F\_2(\mathbf{x})}{d\mathbf{x}^2} = k\_\mathsf{F}F\_2(\mathbf{x}) + k\_\mathsf{T}T\_2(\mathbf{x}) + k\_\mathsf{P}p\_{2\mathbf{i}}(\mathbf{x}) + k\_\mathsf{C} \tag{43}$$

where the parameters kF, kT, kp, and kC are.

$$k\_{\rm F} = \frac{2\pi R\_{2i}G^{a}}{t\_{\rm a}} \left(\varepsilon\_{\rm x,F}^{2i} + \varepsilon\_{\rm x,F}^{1o}\right),\\k\_{\rm T} = \frac{2\pi R\_{2i}G^{a}}{t\_{\rm a}} \left(\varepsilon\_{\rm x,T}^{2i} + \varepsilon\_{\rm x,T}^{1o}\right)$$

$$k\_{p} = \frac{2\pi R\_{2i}G^{a}}{t\_{\rm a}} \left(\varepsilon\_{\rm x,pi}^{2i} - \varepsilon\_{\rm x,po}^{1o}\right), \quad k\_{\rm C} = \frac{2\pi R\_{2i}G^{a}}{t\_{\rm a}} \left(\varepsilon\_{\rm x,po}^{2i}p\_{o} - \varepsilon\_{\rm x,pi}^{1o}p\_{i} - \varepsilon\_{\rm x,F}^{1o}F - \varepsilon\_{\rm x,T}^{1o}T\right)$$

To specify the boundary conditions of Eq. (43), one can consider the disappearance of F<sup>2</sup> at 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:

$$F\_2(0) = 0, \qquad F\_2(L) = F \tag{45}$$

Finally, it should be noted that the occurrence of p2<sup>i</sup> in Eq. (43) is closely related with the existence of F<sup>2</sup> because the tensile or compressive loading can induce the peeling traction. Therefore, p2<sup>i</sup> in the equation is considered as unknown. However, it is possible to find the approximated relation between the two variables by letting T<sup>2</sup> and p2<sup>i</sup> be zero; F<sup>2</sup> then can be evaluated and expressed in Eq. (46):

$$F\_2(\mathbf{x}) = a\_0 e^{\sqrt{k\_F \mathbf{x}}} + b\_0 e^{-\sqrt{k\_F \mathbf{x}}} - \frac{k\_\mathbb{C}}{k\_F} \tag{46}$$

where a<sup>0</sup> and b<sup>0</sup> are integration constants.

Reinstating the resultant normal traction p2<sup>i</sup> and substituting Eq. (46) into Eq. (43), it is found that p2<sup>i</sup> can be simply estimated as

$$p\_{2i} \approx a\_1 e^{\sqrt{k\_{\rm f}} \mathbf{x}} + b\_1 e^{-\sqrt{k\_{\rm f}} \mathbf{x}} \tag{47}$$

• For external and internal pressure, the secondary variable T<sup>2</sup> is firstly omitted. In this case T ¼ F ¼ 0: If only external pressure is present, p1<sup>i</sup> ¼ 0, whereas if only internal pressure

As previously discussed, all resultant loads in adherend parts 1 and 2 can be obtained as functions of x-coordinate. Equipped with the elasticity solution discussed in Section 2, the stress analysis in the adherends can be performed by utilizing a technique so-called finitesegment method (FSM) developed in [13]. The joint in the overlap region is divided into n numbers of segment as illustrated in Figure 3 to take care of the axial variation of the resultant loads. The corresponding resultant loads of each segment are then approximated to be constant, and thus, application of the theory delineated in the previous section is valid. For a composite coupler, the fiber angle in the adherend part 2 is denoted as Ø2. Note that if the stresses in adherend part 2 are evaluated, the internal pressure pi in Eq. (21) or equivalently p2<sup>i</sup>

results. Likewise, the procedure must be done for p1<sup>o</sup> if adherend part 1 is considered. When all of the resultant loads in Eq. (21) in each segment are completely determined, stresses in both

Presented in this section are some sample computational results of the model developed. The numerical calculation is performed by using software MATHEMATICA™. The validation of

xr, and σ<sup>a</sup>

<sup>r</sup> are solved by using the governing

<sup>θ</sup><sup>r</sup> can be later recovered the same way as

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<sup>r</sup> obtained from Eq. (35) for more accurate

exists, <sup>p</sup>2<sup>o</sup> <sup>¼</sup> 0. The first variables <sup>F</sup>2, <sup>τ</sup><sup>a</sup>

should be substituted by the radial normal stress σ<sup>a</sup>

adherends can be finally computed and analyzed.

4. Results

those for the axial load.

Figure 3. A segment at a certain axial position x.

equation, Eqs. (43), (47), (28), and (35). T<sup>2</sup> and τ<sup>a</sup>

3.3. Finite segment solution for evaluating adherend stresses

in which, a<sup>1</sup> and b<sup>1</sup> are unknown parameters. In order to determine these two parameters, two more boundary conditions are required from zero longitudinal shear stress τ<sup>a</sup> xr in the adhesive layer at left and right ends as shown in Eq. (48):

$$\frac{dF\_2}{d\mathbf{x}}\Big|\_{\mathbf{x}=0} = 0, \qquad \frac{dF\_2}{d\mathbf{x}}\Big|\_{\mathbf{x}=L} = \mathbf{0} \tag{48}$$

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:


Figure 3. A segment at a certain axial position x.

To specify the boundary conditions of Eq. (43), one can consider the disappearance of F<sup>2</sup> at 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

Finally, it should be noted that the occurrence of p2<sup>i</sup> in Eq. (43) is closely related with the existence of F<sup>2</sup> because the tensile or compressive loading can induce the peeling traction. Therefore, p2<sup>i</sup> in the equation is considered as unknown. However, it is possible to find the approximated relation between the two variables by letting T<sup>2</sup> and p2<sup>i</sup> be zero; F<sup>2</sup> then can be

> ffiffiffiffiffi kFx <sup>p</sup>

Reinstating the resultant normal traction p2<sup>i</sup> and substituting Eq. (46) into Eq. (43), it is found

ffiffiffiffiffi kFx <sup>p</sup>

in which, a<sup>1</sup> and b<sup>1</sup> are unknown parameters. In order to determine these two parameters, two

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

• For axial load, the secondary variable T<sup>2</sup> is initially neglected. Additionally, T ¼ p1<sup>i</sup> ¼

xr, and σ<sup>a</sup>

dx <sup>x</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>; dF<sup>2</sup>

þ b1e

dx

� � � � x¼L

<sup>r</sup> and can be recovered and computed by employing the full form of

þ b0e

� ffiffiffiffiffi kFx <sup>p</sup> � kC kF

� ffiffiffiffiffi kFx <sup>p</sup>

F2ð Þ¼ x a0e

p2<sup>i</sup> ≈ a1e

more boundary conditions are required from zero longitudinal shear stress τ<sup>a</sup>

dF<sup>2</sup>

• For torsional load, the secondary variables, namely, F<sup>2</sup> and p2<sup>i</sup>

Eq. (38). In addition, <sup>F</sup> <sup>¼</sup> <sup>p</sup>1<sup>i</sup> <sup>¼</sup> <sup>p</sup>2<sup>o</sup> <sup>¼</sup> 0. Consequently, <sup>T</sup><sup>2</sup> and <sup>τ</sup><sup>a</sup>

� � � �

F2ð Þ¼ 0 0, F2ð Þ¼ L F (45)

(46)

(47)

xr in the adhesive

¼ 0 (48)

, are initially neglected in

<sup>θ</sup><sup>r</sup> can be evaluated. Subse-

<sup>r</sup> are solved by using the governing equation,

<sup>θ</sup><sup>r</sup> is later calculated from the full form of Eqs. (38)

follows:

evaluated and expressed in Eq. (46):

22 Applied Adhesive Bonding in Science and Technology

where a<sup>0</sup> and b<sup>0</sup> are integration constants.

layer at left and right ends as shown in Eq. (48):

that p2<sup>i</sup> can be simply estimated as

pertinent details are given below:

and (23), respectively.

xr, and σ<sup>a</sup>

<sup>p</sup>2<sup>o</sup> <sup>¼</sup> <sup>0</sup>: The primary variables <sup>F</sup>2, <sup>τ</sup><sup>a</sup>

Eqs. (43), (47), (28), and (35). T<sup>2</sup> and τ<sup>a</sup>

Eqs. (43) and (47), (28), and (35), respectively.

quently, F2, τ<sup>a</sup>

• For external and internal pressure, the secondary variable T<sup>2</sup> is firstly omitted. In this case T ¼ F ¼ 0: If only external pressure is present, p1<sup>i</sup> ¼ 0, whereas if only internal pressure exists, <sup>p</sup>2<sup>o</sup> <sup>¼</sup> 0. The first variables <sup>F</sup>2, <sup>τ</sup><sup>a</sup> xr, and σ<sup>a</sup> <sup>r</sup> are solved by using the governing equation, Eqs. (43), (47), (28), and (35). T<sup>2</sup> and τ<sup>a</sup> <sup>θ</sup><sup>r</sup> can be later recovered the same way as those for the axial load.

#### 3.3. Finite segment solution for evaluating adherend stresses

As previously discussed, all resultant loads in adherend parts 1 and 2 can be obtained as functions of x-coordinate. Equipped with the elasticity solution discussed in Section 2, the stress analysis in the adherends can be performed by utilizing a technique so-called finitesegment method (FSM) developed in [13]. The joint in the overlap region is divided into n numbers of segment as illustrated in Figure 3 to take care of the axial variation of the resultant loads. The corresponding resultant loads of each segment are then approximated to be constant, and thus, application of the theory delineated in the previous section is valid. For a composite coupler, the fiber angle in the adherend part 2 is denoted as Ø2. Note that if the stresses in adherend part 2 are evaluated, the internal pressure pi in Eq. (21) or equivalently p2<sup>i</sup> should be substituted by the radial normal stress σ<sup>a</sup> <sup>r</sup> obtained from Eq. (35) for more accurate results. Likewise, the procedure must be done for p1<sup>o</sup> if adherend part 1 is considered. When all of the resultant loads in Eq. (21) in each segment are completely determined, stresses in both adherends can be finally computed and analyzed.

#### 4. Results

Presented in this section are some sample computational results of the model developed. The numerical calculation is performed by using software MATHEMATICA™. The validation of the model is not given herein, since it has already been shown in [13] and [20]. Adhesivebonded tubular joints with isotropic inner adherend and symmetric-balanced four-layer stacking sequence ½ � �Ø<sup>2</sup> <sup>s</sup> couplers are selected for consideration. The reference joint geometry is given with parameters R<sup>10</sup> ¼ 10, t<sup>1</sup> ¼ 5, t<sup>2</sup> ¼ 5, ta ¼ 0:1, and L ¼ 40 mm. The adherend part 1 is made of steel, whereas the adherend part 2 is fabricated from carbon fiber-reinforced plastic. Epoxy is used as the adhesive material. The material properties of the joints are listed in Table 1. Adhesive normal stress ratio, α, is set to be 20 because it has shown to provide accurate predictions of adhesive radial stresses [20]. In the following computational results, the adhesive stresses in the coupler joint are normalized by the average applied stress in each loading case, because the dimensionless stresses are readily exploited to identify the level of load distribution intensity in the joint.

#### 4.1. Torsional loading

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 τ<sup>a</sup> <sup>m</sup> in Eq. (49) is utilized to normalize the induced adhesive hoop shear stress in the coupler joint:

$$
\tau\_m^a = \frac{T}{2\pi R\_{2i}^2 L} \tag{49}
$$

coupler interface r = 10.1 mm or σ<sup>a</sup>

4.2. Axial loading

<sup>r</sup> is also minimal because of being the secondary effect. Note

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<sup>m</sup> in Eq. (50)

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 τ<sup>a</sup>

Figure 5. Adhesive hoop shear stress distribution of composite coupler joint subjected to torsion.

Figure 4. Torque distribution in adherend part 2 of composite coupler joint subjected to torsion.

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 Ø<sup>2</sup> = 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 Ø<sup>2</sup> = 45� provides the lowest magnitude of τ<sup>a</sup> <sup>θ</sup><sup>r</sup> as expected.

Stress distributions in the composite coupler are illustrated in Figure 6 for the case of Ø<sup>2</sup> = 30o . The normal stress in the fiber direction σ<sup>11</sup> in Figure 6(a) is the dominant stress component compared to those in the other directions. The radial normal stress σ<sup>33</sup> illustrated in Figure 6(b) is relatively small at x = 0 mm and noticeably larger at x = 40 mm. In addition, σ<sup>33</sup> at adhesive-


Table 1. Mechanical properties of materials in the principle material coordinate [20].

Figure 4. Torque distribution in adherend part 2 of composite coupler joint subjected to torsion.

Figure 5. Adhesive hoop shear stress distribution of composite coupler joint subjected to torsion.

coupler interface r = 10.1 mm or σ<sup>a</sup> <sup>r</sup> is 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.

#### 4.2. Axial loading

the model is not given herein, since it has already been shown in [13] and [20]. Adhesivebonded tubular joints with isotropic inner adherend and symmetric-balanced four-layer stacking sequence ½ � �Ø<sup>2</sup> <sup>s</sup> couplers are selected for consideration. The reference joint geometry is given with parameters R<sup>10</sup> ¼ 10, t<sup>1</sup> ¼ 5, t<sup>2</sup> ¼ 5, ta ¼ 0:1, and L ¼ 40 mm. The adherend part 1 is made of steel, whereas the adherend part 2 is fabricated from carbon fiber-reinforced plastic. Epoxy is used as the adhesive material. The material properties of the joints are listed in Table 1. Adhesive normal stress ratio, α, is set to be 20 because it has shown to provide accurate predictions of adhesive radial stresses [20]. In the following computational results, the adhesive stresses in the coupler joint are normalized by the average applied stress in each loading case, because the dimensionless stresses are readily exploited to identify the level of

In the case of torsional loading, the joints are assumed to have a torque of 1 N.m as an input

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

Stress distributions in the composite coupler are illustrated in Figure 6 for the case of Ø<sup>2</sup> = 30o

The normal stress in the fiber direction σ<sup>11</sup> in Figure 6(a) is the dominant stress component compared to those in the other directions. The radial normal stress σ<sup>33</sup> illustrated in Figure 6(b) is relatively small at x = 0 mm and noticeably larger at x = 40 mm. In addition, σ<sup>33</sup> at adhesive-

Properties Epoxy (adhesive) Steel (adherend 1) Carbon/epoxy (adherend 2)

E<sup>1</sup> (GPa) 1.30 200.00 128.00 E2, E<sup>3</sup> (GPa) 1.30 200.00 10.00 G12, G<sup>13</sup> (GPa) 0.46 76.90 50.00 G<sup>23</sup> (GPa) 0.46 76.90 50.00 υ12, υ<sup>13</sup> 0.41 0.30 0.28 υ<sup>23</sup> 0.41 0.30 0.47

Table 1. Mechanical properties of materials in the principle material coordinate [20].

, respectively, so they cannot be clearly seen. In addition, since the fiber

is the most suitable angle to withstand the in-plane shear loads, the coupler

<sup>θ</sup><sup>r</sup> as expected.

τa <sup>m</sup> <sup>¼</sup> <sup>T</sup> 2πR<sup>2</sup> 2i <sup>m</sup> in Eq. (49) is utilized to

are identical to those

.

<sup>L</sup> (49)

and 30�

without loss of generality. Also, the adhesive mean shear stress τ<sup>a</sup>

Figure 5. Note that the torque and stress distributions for Ø<sup>2</sup> = 0�

provides the lowest magnitude of τ<sup>a</sup>

normalize the induced adhesive hoop shear stress in the coupler joint:

load distribution intensity in the joint.

24 Applied Adhesive Bonding in Science and Technology

4.1. Torsional loading

for 90�

and 60�

orientation of 45�

with Ø<sup>2</sup> = 45�

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 τ<sup>a</sup> <sup>m</sup> in Eq. (50)

Figure 6. Distribution of stresses in composite coupler subjected to torsion.

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:

$$
\tau\_m^a = \frac{F}{2\pi R\_{2i}L} \tag{50}
$$

Figure 10 shows the normal stresses σ<sup>11</sup> and σ<sup>33</sup> of adherend part 2 in the principal material

under the application of the axial force, σ<sup>11</sup> is vanished at the left of the bonding region due to the traction-free surface, while σ<sup>33</sup> is disappeared on the outermost area of the coupler. The stress in the fiber direction σ<sup>11</sup> steadily attains the same maximum value along the bond length in all laminae of composite coupler. The radial normal stress σ33, which is induced

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

. It can be seen from the figure that

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coordinate system, when the fiber orientation is equal to 30o

4.3. Pressure loading

from the resultant axial force, is highest at the adhesive-coupler interface.

Figure 9. Interfacial radial stress distribution of composite coupler joint subjected to tension.

Figure 8. Adhesive axial shear stress distribution of composite coupler joint subjected to tension.

Figures 7 and 8 show the effect of fiber orientation on the distributions of F<sup>2</sup> and τ<sup>a</sup> xr along 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 F<sup>2</sup> of Figure 7 in that region concomitantly reveal linear relationships with the spatial coordinate x/L. The optimum fiber angle Ø<sup>2</sup> is about 30�, which provides the lowest maximum τ<sup>a</sup> xr=τ<sup>a</sup> <sup>m</sup> of 1.2. Figure 9 shows the radial normal stress in the adhesive. Generally speaking, the smaller peak of τ<sup>a</sup> xr=τ<sup>a</sup> <sup>m</sup>, the lower magnitude of σ<sup>a</sup> r=τ<sup>a</sup> m.

Figure 7. Force distribution in adherend part 2 of composite coupler joint subjected to tension.

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints http://dx.doi.org/10.5772/intechopen.72288 27

Figure 8. Adhesive axial shear stress distribution of composite coupler joint subjected to tension.

Figure 9. Interfacial radial stress distribution of composite coupler joint subjected to tension.

Figure 10 shows the normal stresses σ<sup>11</sup> and σ<sup>33</sup> of 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, σ<sup>11</sup> is vanished at the left of the bonding region due to the traction-free surface, while σ<sup>33</sup> is disappeared on the outermost area of the coupler. The stress in the fiber direction σ<sup>11</sup> steadily 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.

#### 4.3. Pressure loading

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

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 F<sup>2</sup> of Figure 7 in that region concomitantly reveal linear relationships with the spatial coordinate x/L. The optimum fiber angle Ø<sup>2</sup> is about 30�, which provides

xr=τ<sup>a</sup>

Figure 7. Force distribution in adherend part 2 of composite coupler joint subjected to tension.

<sup>2</sup>πR2iL (50)

r=τ<sup>a</sup> m.

<sup>m</sup> of 1.2. Figure 9 shows the radial normal stress in the adhesive.

<sup>m</sup>, the lower magnitude of σ<sup>a</sup>

xr along the

τa <sup>m</sup> <sup>¼</sup> <sup>F</sup>

Figures 7 and 8 show the effect of fiber orientation on the distributions of F<sup>2</sup> and τ<sup>a</sup>

intensity of load transfer within the joints:

26 Applied Adhesive Bonding in Science and Technology

Figure 6. Distribution of stresses in composite coupler subjected to torsion.

xr=τ<sup>a</sup>

Generally speaking, the smaller peak of τ<sup>a</sup>

the lowest maximum τ<sup>a</sup>

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

Figure 10. Distribution of stresses in composite coupler subjected to tension.

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 F<sup>2</sup> and τ<sup>a</sup> xr along the overlap region, respectively. Figure 11 indicates that peak values of F<sup>2</sup> are generated in the central region of the composite couplers, but their values are null at both ends. The longitudinal shear stresses in adhesive τ<sup>a</sup> xr in Figure 12 illustrate the antisymmetric characteristic along the bond length. It can be seen that the optimum fiber angle Ø<sup>2</sup> is 90. This fiber orientation delivers the lowest maximum τ<sup>a</sup> xr=pi of 0.6. Figure 13 shows the radial normal stress in the adhesive. Interestingly, the values of σ<sup>a</sup> <sup>r</sup> are reduced by four to five times compared to the internal pressure applied over the whole range of the fiber angles considered.

The normal stresses σ<sup>11</sup> and σ<sup>33</sup> of 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, σ<sup>11</sup> is

Figure 13. Interfacial radial stress distribution of composite coupler joint subjected to internal pressure.

Figure 12. Adhesive axial shear stress distribution of composite coupler joint subjected to internal pressure.

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

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29

Figure 14. Distribution of stresses in composite coupler subjected to internal pressure.

Figure 11. Force distribution in adherend part 2 of composite coupler joint subjected to internal pressure.

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints http://dx.doi.org/10.5772/intechopen.72288 29

Figure 12. Adhesive axial shear stress distribution of composite coupler joint subjected to internal pressure.

adhesive longitudinal shear stress and adhesive radial normal stress can then be normalized

the overlap region, respectively. Figure 11 indicates that peak values of F<sup>2</sup> are generated in the central region of the composite couplers, but their values are null at both ends. The longitudi-

the bond length. It can be seen that the optimum fiber angle Ø<sup>2</sup> is 90. This fiber orientation

The normal stresses σ<sup>11</sup> and σ<sup>33</sup> of 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, σ<sup>11</sup> is

xr in Figure 12 illustrate the antisymmetric characteristic along

xr=pi of 0.6. Figure 13 shows the radial normal stress in the

<sup>r</sup> are reduced by four to five times compared to the

xr along

Figures 11 and 12 show the effect of fiber orientation on the distributions of F<sup>2</sup> and τ<sup>a</sup>

internal pressure applied over the whole range of the fiber angles considered.

Figure 11. Force distribution in adherend part 2 of composite coupler joint subjected to internal pressure.

by the internal pressure to form the dimensionless variables.

Figure 10. Distribution of stresses in composite coupler subjected to tension.

nal shear stresses in adhesive τ<sup>a</sup>

delivers the lowest maximum τ<sup>a</sup>

adhesive. Interestingly, the values of σ<sup>a</sup>

28 Applied Adhesive Bonding in Science and Technology

Figure 13. Interfacial radial stress distribution of composite coupler joint subjected to internal pressure.

Figure 14. Distribution of stresses in composite coupler subjected to internal pressure.

maximum at the mid-length of bonding region, whereas σ<sup>33</sup> is peak at x = 0 and 40 mm on the adhesive-adherend interface.

[6] Kumar S, Khan MA. An elastic solution for adhesive stresses in multi-material cylindrical

A Unified Analysis of Adhesive-Bonded Cylindrical Coupler Joints

http://dx.doi.org/10.5772/intechopen.72288

31

[7] Kumar S. Analysis of tubular adhesive joints with a functionally modulus graded bondline subjected to axial loads. International Journal of Adhesion and Adhesives. 2009;29:785-795

[8] Volkersen O. Recheraches sur la theorie des assemblages colles. Construction Metallique.

[9] Pugno N, Surace G. Tubular bonded joint under torsion: Theoretical analysis and optimization for uniform torsional strength. Journal of Strain Analysis. 2001;36(1):17-24

[10] Xu W, Li G. Finite difference three-dimensional solution of stresses in adhesively bonded composite tubular joint subjected to torsion. International Journal of Adhesion and

[11] Nonlinear OJH. Analysis of adhesive bonded tubular single-lap joints for composites in

[12] Spaggiari A, Dragoni E. Regularization of torsional stresses in tubular lap bonded joints by means of functionally graded adhesives. International Journal of Adhesion and Adhe-

[13] Aimmanee S, Hongpimolmas P. Stress analysis of adhesive-bonded tubular-coupler joints with optimum variable-stiffness composite adherend under torsion. Composite Structures.

[14] Terekhova LP, Skoryi IA. Stresses in bonded joints of thin cylindrical shells. Strength of

[15] Baishya N, Das RR, Panigrahi SK. Failure analysis of adhesively bonded tubular joints of laminated FRP composites subjected to combined internal pressure and torsional load-

[16] Zhang Y, Qin TY, Noda, NA, Duan ML. Strength analysis of adhesive joints of riser in

[17] Apalak MK. Stress analysis of an adhesively bonded functionally graded tubular single lap joint subjected to an internal pressure. Science and Engineering of Composite Mate-

[18] Herakovich CT. Mechanics of Fibrous Composites. USA: John Wiley& Sons, Inc; 1998

[19] Carlucci D, Payne N, Mehmedagic I. Small Strain Compatibility Conditions of an Elastic Solid in Cylindrical Coordinates. New Jersey: U.S. Army ARDEC; 2013. pp. 1-12

[20] Aimmanee S, Hongpimolmas P, Ruangjirakit K. Simplified analytical model for adhesive-bonded tubular joints with isotropic and composite adherends subjected to

torsion. Composites Science and Technology. 2007;67:1320-1329

ing. Journal of Adhesion Science and Technology. 2017;31(19-20)

deep sea environment loadings. Applied Adhesion Science. 2013;1:1-9

tension. International Journal of Adhesion and Adhesive (submitted)

joints. International Journal of Adhesion and Adhesives. 2016;64:142-152

1965;4:3-13

Adhesives. 2010;30:191-199

Materials. 1972;4(10):1271-1274

rials. 2006;13(3):183-211

sives. 2014;53:23-28

2017;164:76-89

#### 5. Conclusions

A unified mathematical model for predicting the joint stresses of the adhesive-bonded tubularcoupler 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.

#### Author details

Sontipee Aimmanee

Address all correspondence to: sontipee.aim@kmutt.ac.th

Advanced Materials and Structures Laboratory (AMASS), Department of Mechanical Engineering, Faculty of Engineering, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok, Thailand

#### References


[6] Kumar S, Khan MA. An elastic solution for adhesive stresses in multi-material cylindrical joints. International Journal of Adhesion and Adhesives. 2016;64:142-152

maximum at the mid-length of bonding region, whereas σ<sup>33</sup> is peak at x = 0 and 40 mm on the

A unified mathematical model for predicting the joint stresses of the adhesive-bonded tubularcoupler 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.

Advanced Materials and Structures Laboratory (AMASS), Department of Mechanical Engineering, Faculty of Engineering, King Mongkut's University of Technology Thonburi

[1] Lubkin JL, Reissner E. Stress distribution and design for adhesive lap joints between

[2] Adams R, Peppiatt N. Stress analysis of adhesive-bonded tubular lap joints. Journal of

[3] Allman DJ. A theory for elastic stresses in adhesive bonded lap joints. The Quarterly

[4] Shi YP, Cheng S. Analysis of adhesive-bonded cylindrical lap joints subjected to axial

[5] Nemes O, Lachaud F, Mojtabi A. Contribution to the study of cylindrical adhesive

joining. International Journal of Adhesion and Adhesives. 2006;26:474-448

Address all correspondence to: sontipee.aim@kmutt.ac.th

circular tubes. Transactions of ASME. 1956;78:1213-1221

Journal of Mechanics and Applied Mathematics. 1977;30:415-436

load. Journal of Engineering Mechanics. 1993;119(3):584-602

(KMUTT), Bangmod, Bangkok, Thailand

Adhesion. 1977;9:1-18

adhesive-adherend interface.

30 Applied Adhesive Bonding in Science and Technology

5. Conclusions

Author details

Sontipee Aimmanee

References


**Chapter 3**

**Provisional chapter**

**Development and Application of Low-Temperature**

Flexible electronics is the expected technology in the future, and the bonding material may also require flexibility. Silyl terminated poly-ether (STPE) is a promising material that has both flexibility and low-temperature curability. In combination with tri-block polymer-based stretchable conductive paste and artificially formed fillet formed by elastic resin, it can build simple stretchable bonding system. It will be a prominent technol-

**Keywords:** isotropic conductive adhesive, low-temperature bonding, flexible devices,

Internet of Things (IoT) means all things are connected to the Internet showing the possibility of changing our lives. With the smartphone market becoming steady once, the explosive popularization of smartphones pushed for the miniaturization of electronic components and the spread of wireless Internet. In response to that, the wearable device approached a more practical device. Then, the role of electronics is becoming different from that of the previous one. For example, the reduction of medical expenses by health-care monitoring, efficient use of energy, assistance of workers and disabled people, and so on is said to be an important social task that electronics can solve. It is a great opportunity for printed electronics to join with the flow of giving an electrical function to various things indicated by IoT and create a light and soft device by printing. Attempts to design interfaces between the Internet world and real society have begun, as sensors and other electronic devices are incorporated all over our lives. In the trend,

ogy to satisfy the characteristics required by the near future devices.

**Development and Application of Low-Temperature** 

DOI: 10.5772/intechopen.72662

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Curable Isotropic Conductive Adhesive Toward to**

**Curable Isotropic Conductive Adhesive Toward to**

**Fabrication in IoT Generation**

**Fabrication in IoT Generation**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72662

stretchable devices, IoT

Yusuke Okabe

**Abstract**

**1. Introduction**

Yusuke Okabe

**Provisional chapter**

#### **Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive Toward to Fabrication in IoT Generation Curable Isotropic Conductive Adhesive Toward to Fabrication in IoT Generation**

**Development and Application of Low-Temperature** 

DOI: 10.5772/intechopen.72662

Yusuke Okabe Yusuke Okabe

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.72662

#### **Abstract**

Flexible electronics is the expected technology in the future, and the bonding material may also require flexibility. Silyl terminated poly-ether (STPE) is a promising material that has both flexibility and low-temperature curability. In combination with tri-block polymer-based stretchable conductive paste and artificially formed fillet formed by elastic resin, it can build simple stretchable bonding system. It will be a prominent technology to satisfy the characteristics required by the near future devices.

**Keywords:** isotropic conductive adhesive, low-temperature bonding, flexible devices, stretchable devices, IoT

#### **1. Introduction**

Internet of Things (IoT) means all things are connected to the Internet showing the possibility of changing our lives. With the smartphone market becoming steady once, the explosive popularization of smartphones pushed for the miniaturization of electronic components and the spread of wireless Internet. In response to that, the wearable device approached a more practical device. Then, the role of electronics is becoming different from that of the previous one. For example, the reduction of medical expenses by health-care monitoring, efficient use of energy, assistance of workers and disabled people, and so on is said to be an important social task that electronics can solve. It is a great opportunity for printed electronics to join with the flow of giving an electrical function to various things indicated by IoT and create a light and soft device by printing. Attempts to design interfaces between the Internet world and real society have begun, as sensors and other electronic devices are incorporated all over our lives. In the trend,

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

IoT gives "anywhere anything" according to electric function indicated by IoT, base material has expanded from conventional rigid PCB to heat-sensitive materials, such as plastic, paper, and cloth, which can be formed by flexible circuits. In addition, those materials can be cured by low temperature, which is a major topic and important element technology. In fact, flexible circuits have been created by various materials and have achieved great results. To date, the softening of the wiring materials is largely developed by graphene in [1], carbon nanotubes in [2], silver nanowires in [3], conductive polymers in [4], or metal dispersed elastomers in [5–7]. With the advent of these technologies, flexible electronics can be said to have undergone a dramatic evolution. However, it has been rarely reported about the bonding materials, which coexist with flexibility and low-temperature curability. Furthermore, at present, it is difficult to create ICs, memories, communication modules, and so on by printing; therefore, it is necessary for the next generation devices to use hard parts and flexible substrates together. Additionally, in the connection between flexible substrates and hard components such as a wearable device, the connecting part will bear a large strain (**Figure 1**). As **Figure 2** shows, in Ref. [8], elongation that can occur on human body is around 30%. It is difficult to say that the bonding materials satisfy the requirement of imparting an electrical function to various substrates. In this situation, isotropic conductive adhesives (ICAs) have low-temperature curability and flexibility can be of great potential in creating next generation devices (**Figure 3**).

As the types of base materials are expanding, the most important basic property is to ensure

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive…

http://dx.doi.org/10.5772/intechopen.72662

35

Silyl terminated poly-ether (STPE), as shown in **Figure 4**, is flexible and strong material to repeated strain, such as heat cycle. In present, there are many achievements to sealant for construction and adhesive for electronic components. Conventional conductive adhesives are mainly thermosetting system; the heat energy required for curing itself has disturbed the applications of heat-sensitive substrates and components. The cross-linkable silyl group can react by moisture in the air, even at room temperature, and it is suitable to implement thermally sensitive components. That is, it is possible to assemble the electronic component on the film as well as paper, fabrics, or various materials. We have developed a conductive adhesive having low-temperature curableness and flexibility by distributing the silver microsized to STPE. Here, we describe the characteristics of new isotropic conductive adhesive based on STPE, which has low-temperature curableness and flexibility and its

First, we describe the design and characteristics of new conductive adhesives. Next, we describe the low-temperature curability, flexibility, and stretchable conductive paste for flex-

adhesion to various base materials, and the adhesion technology is meaningful.

**Figure 3.** A comparison of this work with conventional conductive materials.

**2. Characteristics of the STPE based conductive adhesive**

ible devices, and finally, we introduce their application.

application.

**Figure 4.** Chemical structure of STPE.

**Figure 1.** Image of flexible hybrid electronics device in future.


**Figure 2.** Elongation of each part occurring in the human body.

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive… http://dx.doi.org/10.5772/intechopen.72662 35

**Figure 3.** A comparison of this work with conventional conductive materials.

**Figure 4.** Chemical structure of STPE.

IoT gives "anywhere anything" according to electric function indicated by IoT, base material has expanded from conventional rigid PCB to heat-sensitive materials, such as plastic, paper, and cloth, which can be formed by flexible circuits. In addition, those materials can be cured by low temperature, which is a major topic and important element technology. In fact, flexible circuits have been created by various materials and have achieved great results. To date, the softening of the wiring materials is largely developed by graphene in [1], carbon nanotubes in [2], silver nanowires in [3], conductive polymers in [4], or metal dispersed elastomers in [5–7]. With the advent of these technologies, flexible electronics can be said to have undergone a dramatic evolution. However, it has been rarely reported about the bonding materials, which coexist with flexibility and low-temperature curability. Furthermore, at present, it is difficult to create ICs, memories, communication modules, and so on by printing; therefore, it is necessary for the next generation devices to use hard parts and flexible substrates together. Additionally, in the connection between flexible substrates and hard components such as a wearable device, the connecting part will bear a large strain (**Figure 1**). As **Figure 2** shows, in Ref. [8], elongation that can occur on human body is around 30%. It is difficult to say that the bonding materials satisfy the requirement of imparting an electrical function to various substrates. In this situation, isotropic conductive adhesives (ICAs) have low-temperature curability and flexibility can

be of great potential in creating next generation devices (**Figure 3**).

34 Applied Adhesive Bonding in Science and Technology

**Figure 2.** Elongation of each part occurring in the human body.

**Figure 1.** Image of flexible hybrid electronics device in future.

As the types of base materials are expanding, the most important basic property is to ensure adhesion to various base materials, and the adhesion technology is meaningful.

Silyl terminated poly-ether (STPE), as shown in **Figure 4**, is flexible and strong material to repeated strain, such as heat cycle. In present, there are many achievements to sealant for construction and adhesive for electronic components. Conventional conductive adhesives are mainly thermosetting system; the heat energy required for curing itself has disturbed the applications of heat-sensitive substrates and components. The cross-linkable silyl group can react by moisture in the air, even at room temperature, and it is suitable to implement thermally sensitive components. That is, it is possible to assemble the electronic component on the film as well as paper, fabrics, or various materials. We have developed a conductive adhesive having low-temperature curableness and flexibility by distributing the silver microsized to STPE. Here, we describe the characteristics of new isotropic conductive adhesive based on STPE, which has low-temperature curableness and flexibility and its application.

#### **2. Characteristics of the STPE based conductive adhesive**

First, we describe the design and characteristics of new conductive adhesives. Next, we describe the low-temperature curability, flexibility, and stretchable conductive paste for flexible devices, and finally, we introduce their application.

#### **2.1. Experimental section**

ICAs were fabricated by uniformly dispersing microsized silvers as described later. Uniform dispersion was achieved by a high-speed blender operated at 2000 rpm for 3 min under vacuum conditions (ARV-310, THINKY Company). Epoxy-based ICAs were formulated by Epicote 828 (Mitsubishi Chemical Co., LTD.) and 2-ethyl-4-imidazole (Tokyo Chemical Industry Co., LTD.) as curing agents. ICAs were mask-printed onto glass plate. STPE-based ICAs were cured at 80°C for 2 h, and epoxy-based ICAs were cured at 120°C for 1 h. Their dimensions were 80, 100, and 0.5 mm in width, length, and thickness, respectively. Volume resistivity was measured by MCP-T360 (Mitsubishi Analytec Co., LTD.).

density is important in expressing conductivity. Also, urethane bond in the polymer backbone showed less resistivity. Hard segment derived from a hydrogen bond due to a urethane bond plays strengthening of the interaction and the conductive path between the filler. Otherwise, BPA-based conductive adhesive has higher conductivity when using a single shape. Previously, there are few cases mentioning the TAP density in the formulation of the conductive adhesive, and it has been discussed in the shape of fillers. However, these results show the selectivity of charac-

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive…

http://dx.doi.org/10.5772/intechopen.72662

37

Curing behavior of STPE-based conductive adhesive is shown in **Figure 6**. Curing proceeds even at room temperature, the conductivity increased with time. Further, by heating to

**) 3.5 1.0 1.5**

**Base resin Silver flake Aggregated silver Spherical silver**

 STPE (ether) 300 200 – STPE (ether) 300 – 200 STPE (ether) 300 100 100 STPE (ether) 200 300 – STPE (ether) 500 – – STPE (urethane) 300 200 – BPA 500 – – BPA 300 200 – BPA – 500 –

teristic silver fillers to develop the conductivity to flexible binder as STPE (**Table 1**).

**2.3. Curing behavior**

**Figure 5.** Volume resistivity of designed ICAs shown in **Table 1**.

**Table 1.** Formulation of ICAs, silver weight ratio (per 100 resin).

**Entry TAP density of filler (g/cm<sup>3</sup>**

Adhesion strength is tested by die shear tester (Dage 4000) at a shear head speed of 500 μm/s. Epoxy-based ICA (XA-874, FjikuraKasei Co., LTD.) was used as comparison. ICAs were maskprinted onto copper plate on printed circuit board and mounted on 3216-sized chip resistor. Printed thickness was 0.1 mm. The high temperature and high humidity tests were carried out at 85°C and 85% relative humidity (RH), respectively. Heat cycle test was performed at the temperature range between −40 and 105°C. Exposure time was 30 min.

Electrical stability is tested on ISO-16525. ICAs were mask-printed onto copper electrodes, in which dimensions were 2, 4, 8 mm in width, length, interval, respectively, on printed circuit board. Dimensions of ICAs were 4, 100, and 0.1 mm in width, length, and thickness, respectively.

Stretchable conductive paste was fabricated by uniformly dispersing microsized silver flakes. Uniform dispersion was achieved by a high-speed blender operated at 2000 rpm for 3 min under vacuum conditions (ARV-310, THINKY Company). Conductive pastes were mask-printed onto various substrates and dried at 100°C for 30 min. Their dimensions were 80, 100, and 0.5 mm in width, length, and thickness, respectively. Cross cut test was performed based on ISO-2649.

Bending resistance test of conductive paste was performed on IEC62715 by DLDMLH-4U (YUASA SYSTEM Co., LTD.). Conductive pastes were mask-printed onto PET film (Lumirror S10, TORAY Co., LTD.) and dried at 100°C for 30 min. Their dimensions were 5, 100, and 0.2 mm in width, length, and thickness, respectively. To investigate the resistance change in real time, copper and lead were connected to both ends, and the resistance was measured in current of 100 mA by the four-terminal method.

Dynamic boding resistance test was similar to the abovementioned test. Copper foiled polyimide was used as circuit. XA-874 and solder paste (FLF01-BZ(L), Matsuo HANDA Co., LTD.) were used as comparison. Artificially formed fillet was formed by SuperXG No.777 (CEMEDINE Co., LTD.).

In stretched resistance test, conductive pastes were mask-printed onto TPU (Platilon VPT9122, Covestro Japan Co., LTD.) and dried at 100°C for 30 min. Their dimensions were 5, 100, and 0.2 mm in width, length, and thickness, respectively.

#### **2.2. Design of STPE-based conductive adhesive**

**Figure 5** shows formulation and volume resistivity of conductive adhesive. In case of using STPE as base binder, it is understood that combining silver fillers having different shapes and TAP density is important in expressing conductivity. Also, urethane bond in the polymer backbone showed less resistivity. Hard segment derived from a hydrogen bond due to a urethane bond plays strengthening of the interaction and the conductive path between the filler. Otherwise, BPA-based conductive adhesive has higher conductivity when using a single shape. Previously, there are few cases mentioning the TAP density in the formulation of the conductive adhesive, and it has been discussed in the shape of fillers. However, these results show the selectivity of characteristic silver fillers to develop the conductivity to flexible binder as STPE (**Table 1**).

#### **2.3. Curing behavior**

**2.1. Experimental section**

36 Applied Adhesive Bonding in Science and Technology

ICAs were fabricated by uniformly dispersing microsized silvers as described later. Uniform dispersion was achieved by a high-speed blender operated at 2000 rpm for 3 min under vacuum conditions (ARV-310, THINKY Company). Epoxy-based ICAs were formulated by Epicote 828 (Mitsubishi Chemical Co., LTD.) and 2-ethyl-4-imidazole (Tokyo Chemical Industry Co., LTD.) as curing agents. ICAs were mask-printed onto glass plate. STPE-based ICAs were cured at 80°C for 2 h, and epoxy-based ICAs were cured at 120°C for 1 h. Their dimensions were 80, 100, and 0.5 mm in width, length, and thickness, respectively. Volume

Adhesion strength is tested by die shear tester (Dage 4000) at a shear head speed of 500 μm/s. Epoxy-based ICA (XA-874, FjikuraKasei Co., LTD.) was used as comparison. ICAs were maskprinted onto copper plate on printed circuit board and mounted on 3216-sized chip resistor. Printed thickness was 0.1 mm. The high temperature and high humidity tests were carried out at 85°C and 85% relative humidity (RH), respectively. Heat cycle test was performed at the

Electrical stability is tested on ISO-16525. ICAs were mask-printed onto copper electrodes, in which dimensions were 2, 4, 8 mm in width, length, interval, respectively, on printed circuit board. Dimensions of ICAs were 4, 100, and 0.1 mm in width, length, and thickness, respectively.

Stretchable conductive paste was fabricated by uniformly dispersing microsized silver flakes. Uniform dispersion was achieved by a high-speed blender operated at 2000 rpm for 3 min under vacuum conditions (ARV-310, THINKY Company). Conductive pastes were mask-printed onto various substrates and dried at 100°C for 30 min. Their dimensions were 80, 100, and 0.5 mm in width, length, and thickness, respectively. Cross cut test was performed based on ISO-2649.

Bending resistance test of conductive paste was performed on IEC62715 by DLDMLH-4U (YUASA SYSTEM Co., LTD.). Conductive pastes were mask-printed onto PET film (Lumirror S10, TORAY Co., LTD.) and dried at 100°C for 30 min. Their dimensions were 5, 100, and 0.2 mm in width, length, and thickness, respectively. To investigate the resistance change in real time, copper and lead were connected to both ends, and the resistance was measured in

Dynamic boding resistance test was similar to the abovementioned test. Copper foiled polyimide was used as circuit. XA-874 and solder paste (FLF01-BZ(L), Matsuo HANDA Co., LTD.) were used as comparison. Artificially formed fillet was formed by SuperXG No.777 (CEMEDINE

In stretched resistance test, conductive pastes were mask-printed onto TPU (Platilon VPT9122, Covestro Japan Co., LTD.) and dried at 100°C for 30 min. Their dimensions were 5, 100, and

**Figure 5** shows formulation and volume resistivity of conductive adhesive. In case of using STPE as base binder, it is understood that combining silver fillers having different shapes and TAP

resistivity was measured by MCP-T360 (Mitsubishi Analytec Co., LTD.).

temperature range between −40 and 105°C. Exposure time was 30 min.

current of 100 mA by the four-terminal method.

0.2 mm in width, length, and thickness, respectively.

**2.2. Design of STPE-based conductive adhesive**

Co., LTD.).

Curing behavior of STPE-based conductive adhesive is shown in **Figure 6**. Curing proceeds even at room temperature, the conductivity increased with time. Further, by heating to

**Figure 5.** Volume resistivity of designed ICAs shown in **Table 1**.


**Table 1.** Formulation of ICAs, silver weight ratio (per 100 resin).

50 or 80°C, curing is accelerated and conductivity increases quickly. Conventional epoxybased adhesives are often unable to exert their performance unless they adhere to the recommended curing conditions. In other words, the curing conditions themselves may cause trouble in bonding process. With STPE-based conductive adhesive, even if heating is stopped halfway, the reaction proceeds, so that, a flexible production process can be constructed. Also, die shear strength increased with time, curing is accelerated and die shear strength increased quickly. On the other hand, the expression of adhesion strength does not match with the conductivity; this is because the adhesion is rate-limiting between electrode and adhesive interface.

#### **2.4. Adhesion durability**

**Figure 7** shows adhesion durability at high temperature and high humidity, and heat cycle (−40–105°C) compared with thermosetting epoxy resin-based electrical conductive adhesive that is conventionally used. Initial adhesion strength of the STPE-based ICA is smaller than the epoxy resin system, since STPE has a lower elastic modulus than conventional epoxy resin. For this reason, it is better to consider a device having a slightly different design philosophy such as a flexible device than a simple replacement for solder. In particular, STPE-based ICA exhibits excellent bonding strength retention under heat cycle environment. On the other

hand, the adhesion strength of epoxy-based ICA has decreased in every cycle. This phenomenon is induced by internal stress generated at the bonding interface when the temperature change occurs across the glass transition point in Refs. [9–11]. In this situation, rigid base resin cannot reduce the internal stress at bonding interface. STPE has low modulus and flexibility that induce stress relaxation characteristics, and it leads to ensure long-term reliability.

**Figure 8.** Conductor resistance of STPE-based ICA (red) and epoxy-based ICA (blue) at 85°Cand 85% RH on tin plated-

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive…

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39

copper electrode. SEM images show sectional view of PCB in the test of STPE-based ICA after 2000 h.

The excellent characteristics of STPE-based ICA are shown in **Figure 8**. Galvanic corrosion between tin electrode and silver fillers in a high-temperature and high-humidity environment has become a long-standing problem of the epoxy-based ICAs. Therefore, in order to put the conductive adhesive into practical use, an increase in parts' cost has become a problem due to the use of a gold electrode or the like. Corrosion of the tin electrodes is said to be accelerated by the chloride ion in epoxy resin in Ref. [12]. In contrast, STPE is not containing chloride ion, and STPE-based ICA does not occur corrosion on tin electrode. A slight increase is observed in the conductor resistance after 1000 h, and it is found that it was generated by Kirkendall void between tin plating layer and copper used as electrode from

As stated earlier, securing adhesion to enlargement of applied base material is an important factor, and adhesion technology is a key point. Conventional conductive paste does not have

**2.5. Electrical stability**

SEM observation.

**3. Characteristics of conductive paste**

**3.1. Design of flexible/stretchable conductive paste**

**Figure 6.** Curing behavior of STPE-based ICA. Left: curing behavior of volume resistivity. Right: curing behavior of adhesion strength.

**Figure 7.** Adhesion strength at durability test. Left: 85°C and 85% RH. Right: heat cycle (−40–105°C).

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive… http://dx.doi.org/10.5772/intechopen.72662 39

**Figure 8.** Conductor resistance of STPE-based ICA (red) and epoxy-based ICA (blue) at 85°Cand 85% RH on tin platedcopper electrode. SEM images show sectional view of PCB in the test of STPE-based ICA after 2000 h.

hand, the adhesion strength of epoxy-based ICA has decreased in every cycle. This phenomenon is induced by internal stress generated at the bonding interface when the temperature change occurs across the glass transition point in Refs. [9–11]. In this situation, rigid base resin cannot reduce the internal stress at bonding interface. STPE has low modulus and flexibility that induce stress relaxation characteristics, and it leads to ensure long-term reliability.

#### **2.5. Electrical stability**

50 or 80°C, curing is accelerated and conductivity increases quickly. Conventional epoxybased adhesives are often unable to exert their performance unless they adhere to the recommended curing conditions. In other words, the curing conditions themselves may cause trouble in bonding process. With STPE-based conductive adhesive, even if heating is stopped halfway, the reaction proceeds, so that, a flexible production process can be constructed. Also, die shear strength increased with time, curing is accelerated and die shear strength increased quickly. On the other hand, the expression of adhesion strength does not match with the conductivity; this is because the adhesion is rate-limiting between electrode and

**Figure 7** shows adhesion durability at high temperature and high humidity, and heat cycle (−40–105°C) compared with thermosetting epoxy resin-based electrical conductive adhesive that is conventionally used. Initial adhesion strength of the STPE-based ICA is smaller than the epoxy resin system, since STPE has a lower elastic modulus than conventional epoxy resin. For this reason, it is better to consider a device having a slightly different design philosophy such as a flexible device than a simple replacement for solder. In particular, STPE-based ICA exhibits excellent bonding strength retention under heat cycle environment. On the other

**Figure 6.** Curing behavior of STPE-based ICA. Left: curing behavior of volume resistivity. Right: curing behavior of

**Figure 7.** Adhesion strength at durability test. Left: 85°C and 85% RH. Right: heat cycle (−40–105°C).

adhesive interface.

adhesion strength.

**2.4. Adhesion durability**

38 Applied Adhesive Bonding in Science and Technology

The excellent characteristics of STPE-based ICA are shown in **Figure 8**. Galvanic corrosion between tin electrode and silver fillers in a high-temperature and high-humidity environment has become a long-standing problem of the epoxy-based ICAs. Therefore, in order to put the conductive adhesive into practical use, an increase in parts' cost has become a problem due to the use of a gold electrode or the like. Corrosion of the tin electrodes is said to be accelerated by the chloride ion in epoxy resin in Ref. [12]. In contrast, STPE is not containing chloride ion, and STPE-based ICA does not occur corrosion on tin electrode. A slight increase is observed in the conductor resistance after 1000 h, and it is found that it was generated by Kirkendall void between tin plating layer and copper used as electrode from SEM observation.

#### **3. Characteristics of conductive paste**

#### **3.1. Design of flexible/stretchable conductive paste**

As stated earlier, securing adhesion to enlargement of applied base material is an important factor, and adhesion technology is a key point. Conventional conductive paste does not have

**Figure 9.** Schematic image of tri-block elastomer and pre-cured silane coupling agents.


**Table 2.** Test results of block polymer-based conductive paste.

extensive adhesion property. Therefore, we have to choose conductive paste according to substrates. To solve it, we choose tri-block elastomer as binder and pre-cured silane coupling agents (**Figure 9**).

Adhesion test results by cross cut test is shown in **Table 2**. Hydrocarbon polymers such as SIS and SEBS show high conductivity, but they do not exhibit extensive adhesion property. On the other hand, it was found that the acrylic polymer has good adhesion to various substrates and is excellent in balance with conductivity. In subsequent experiments, the conductive paste based on an acrylic polymer is used.

#### **3.2. Dynamic durability of conductive paste**

**Figure 10** shows real time resistance change due to bending on PET film. The resistance change due to bending is smaller than that of conventional epoxy-based flexible conductive paste. In addition, the resistance fluctuation when bending once is also small, which is considered to be based on polymer with hard segment and soft segment coexisting.

**Figure 11** shows the resistance change at elongation on TPU. For example, the strain accruing on the human body is about 50% at the maximum. It is thought that the hysteresis of resistance is small, and it functions as strain sensor, which is capable of detecting elongation of about 50%. Also, by utilizing high adhesion and flexibility, it is possible to form circuits for various substrates.

**4. Application for flexible/stretchable bonding system**

5 mm, length: 100 mm, and thickness: 30 μm.

**Figure 10.** Conductor resistance change of acrylic elastomer-based conductive paste in bending test.

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41

As mentioned earlier, STPE-based ICA does not have strong bonding strength as conventional materials. Therefore, in order to take advantage of this material, it is necessary to construct reinforcing structure. Solder or thermosetting conductive adhesive forms a fillet at the

**Figure 11.** Conductor resistance change of acrylic elastomer-based conductive paste at elongation. Circuit size width:

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive… http://dx.doi.org/10.5772/intechopen.72662 41

**Figure 10.** Conductor resistance change of acrylic elastomer-based conductive paste in bending test.

extensive adhesion property. Therefore, we have to choose conductive paste according to substrates. To solve it, we choose tri-block elastomer as binder and pre-cured silane coupling

**Base polymer Acrylic SIS SEBS** Volume resistivity 3.50E − 04 8.50E − 05 7.50E − 05 Adhesion to PET 100/100 70/100 75/100 Adhesion to PEN 100/100 60/100 70/100 Adhesion to TPU 100/100 50/100 70/100 Adhesion to excimer laser treated SR 100/100 0/100 0/100

**Figure 9.** Schematic image of tri-block elastomer and pre-cured silane coupling agents.

Adhesion test results by cross cut test is shown in **Table 2**. Hydrocarbon polymers such as SIS and SEBS show high conductivity, but they do not exhibit extensive adhesion property. On the other hand, it was found that the acrylic polymer has good adhesion to various substrates and is excellent in balance with conductivity. In subsequent experiments, the conduc-

**Figure 10** shows real time resistance change due to bending on PET film. The resistance change due to bending is smaller than that of conventional epoxy-based flexible conductive paste. In addition, the resistance fluctuation when bending once is also small, which is consid-

**Figure 11** shows the resistance change at elongation on TPU. For example, the strain accruing on the human body is about 50% at the maximum. It is thought that the hysteresis of resistance is small, and it functions as strain sensor, which is capable of detecting elongation of about 50%. Also, by utilizing high adhesion and flexibility, it is possible to form circuits for

ered to be based on polymer with hard segment and soft segment coexisting.

agents (**Figure 9**).

various substrates.

tive paste based on an acrylic polymer is used.

**Table 2.** Test results of block polymer-based conductive paste.

40 Applied Adhesive Bonding in Science and Technology

**3.2. Dynamic durability of conductive paste**

**Figure 11.** Conductor resistance change of acrylic elastomer-based conductive paste at elongation. Circuit size width: 5 mm, length: 100 mm, and thickness: 30 μm.

#### **4. Application for flexible/stretchable bonding system**

As mentioned earlier, STPE-based ICA does not have strong bonding strength as conventional materials. Therefore, in order to take advantage of this material, it is necessary to construct reinforcing structure. Solder or thermosetting conductive adhesive forms a fillet at the time of curing, and it becomes a reinforcing layer against dynamic strain. On the other hand, since STPE-based ICA is a low-temperature curable, it is difficult to form fillets. **Figure 12** shows bending resistance of mounting on FPC by STPE-based ICA and conventional materials and the effect of artificially formed fillet. Although the bonding resistance of solder is stable, bonding resistance of STPE-based ICA does not form a fillet, which has a considerable resistance variation. Even in the case of a thermosetting epoxy-based ICA, a slight resistance

change is observed. On the other hand, fillet formed STPE-based ICA get bending resistance

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive…

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43

**Figure 14.** Pictures of implementation of LED on stretchable substrate. Left: stretched. Right: crumpled.

Detailed structure is shown in **Figure 13**. This structure can also be applied to elastic substrates. Previously, there have been various reports on mounting of rigid parts on stretchable base materials, but it is mainly focused on design of base substrates and how to use rigid bonding materials [13, 14]. By combining STPE-based ICA and flexible reinforcing material, it is not necessary to design a complicated base material and it is possible to simply mount on a flexible base material such as **Figure 14**. In **Figure 14**, a wiring is drawn on a stretchable substrate with a stretchable conductor, and LED chip is mounted with STPE-based ICA and reinforced by elastic resin. Even without constructing a complicated mounting structure, disconnection is not observed, and LED chip continued to light up during repetition of

Characteristics of the conductive adhesive having both low-temperature curability and flexibility, which is based on STPE shows a great advantage compared with conventional conductive adhesive. In addition, it showed dynamic durability rivaled to conventional solder connection. Namely, the bonding system which is almost constructed by elastic resin and highly reliable is a promising technology for electronics in the next generation. Fabrication in IoT and wearable devices still has room for improvement. STPE-based ICA is developed as one solution for the future electronics society and we expect that innovation will happen.

Address all correspondence to: yusuke.okabe@cemedine.co.jp

Research and Development Division, CEMEDINE Co., LTD., Japan

similar to solder connection.

expansion.

**5. Conclusion**

**Author details**

Yusuke Okabe

**Figure 12.** Bonding resistance change of various materials on FPC in bending test.

**Figure 13.** Images of fillet forming at bonding process and detailed image of reinforced STPE bonding structure.

Development and Application of Low-Temperature Curable Isotropic Conductive Adhesive… http://dx.doi.org/10.5772/intechopen.72662 43

**Figure 14.** Pictures of implementation of LED on stretchable substrate. Left: stretched. Right: crumpled.

change is observed. On the other hand, fillet formed STPE-based ICA get bending resistance similar to solder connection.

Detailed structure is shown in **Figure 13**. This structure can also be applied to elastic substrates. Previously, there have been various reports on mounting of rigid parts on stretchable base materials, but it is mainly focused on design of base substrates and how to use rigid bonding materials [13, 14]. By combining STPE-based ICA and flexible reinforcing material, it is not necessary to design a complicated base material and it is possible to simply mount on a flexible base material such as **Figure 14**. In **Figure 14**, a wiring is drawn on a stretchable substrate with a stretchable conductor, and LED chip is mounted with STPE-based ICA and reinforced by elastic resin. Even without constructing a complicated mounting structure, disconnection is not observed, and LED chip continued to light up during repetition of expansion.

#### **5. Conclusion**

time of curing, and it becomes a reinforcing layer against dynamic strain. On the other hand, since STPE-based ICA is a low-temperature curable, it is difficult to form fillets. **Figure 12** shows bending resistance of mounting on FPC by STPE-based ICA and conventional materials and the effect of artificially formed fillet. Although the bonding resistance of solder is stable, bonding resistance of STPE-based ICA does not form a fillet, which has a considerable resistance variation. Even in the case of a thermosetting epoxy-based ICA, a slight resistance

**Figure 12.** Bonding resistance change of various materials on FPC in bending test.

42 Applied Adhesive Bonding in Science and Technology

**Figure 13.** Images of fillet forming at bonding process and detailed image of reinforced STPE bonding structure.

Characteristics of the conductive adhesive having both low-temperature curability and flexibility, which is based on STPE shows a great advantage compared with conventional conductive adhesive. In addition, it showed dynamic durability rivaled to conventional solder connection. Namely, the bonding system which is almost constructed by elastic resin and highly reliable is a promising technology for electronics in the next generation. Fabrication in IoT and wearable devices still has room for improvement. STPE-based ICA is developed as one solution for the future electronics society and we expect that innovation will happen.

#### **Author details**

Yusuke Okabe

Address all correspondence to: yusuke.okabe@cemedine.co.jp

Research and Development Division, CEMEDINE Co., LTD., Japan

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**Section 2**

**Wood Adhesive Bonding**
