Progress in Adhesion and Adhesives, Volume 7 E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
The current book contains eight commissioned chapters and cover topics including stress distribution and design analysis of adhesively bonded tubular composite joints; durability of structural adhesive joints; mechanical surface treatment of adherends for adhesive bonding; surface modification of polymer materials by excimer UV light; corona discharge treatment of materials to enhance adhesion; adhesion activation of aramid fibers; dual-cured hydrogels for bioadhesives and biomedical applications; and non-adhesive SLIPS-like surfaces.
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Veröffentlichungsjahr: 2023
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Preface
Cover
Table of Contents
Series Page
Title Page
Copyright Page
Preface
1 Stress Distribution and Design Analysis of Adhesively Bonded Tubular Composite Joints: A Review
1.1 Introduction
1.2 A Brief Review of Stress Analysis in Tubular Composite Joints
1.3 Governing Equations Based on Linear Elasticity
1.4 Nonlinear Analysis and Stress Distribution in Tubular Composite Joints
1.5 Failure Analysis of Adhesive Layer
1.6 Summary
Acknowledgment
References
2 Durability of Structural Adhesive Joints: Factors Affecting Durability, Durability Assessment and Ways to Improve Durability
Abbreviations
2.1 Introduction
2.2 Factors Affecting Durability
2.3 Durability Assessment
2.4 Methods to Improve Durability
2.5 Summary
References
3 Mechanical Surface Treatment of Adherends for Adhesive Bonding
3.1 Introduction
3.2 Characteristics of Mechanical Surface Treatment Methods
3.3 Types of Abrasive Blasting Operations
3.4 Influence of Mechanical Treatment on the Strength of Adhesive Joints
3.5 Summary
References
4 Surface Modification of Polymer Materials by Excimer 172 nm UV Light: A Review
4.1 Introduction
4.2 Wettability Measurements by Conventional Sessile Drop Technique
4.3 Preference for the Wilhelmy Technique in Wettability Analyses 176
4.4 UV Lithography Technique for Preparation of Mosaic Wettability Pattern
4.5 Chemical and Topographical Changes on Polymer Surfaces Due to UV Treatment
4.6 Determination of Surface Free Energy by Contact Angle Measurements
4.7 Effect of UV Treatment on Particle Adhesion
4.8 Improvement in Textile Performance by UV Treatment
4.9 Summary and Prospects
Acknowledgements
References
5 Corona Discharge Treatment for Surface Modification and Adhesion Improvement
5.1 Introduction
5.2 Historical Development of Corona Treatment Technique and Various Set-Ups Available
5.3 Factors Affecting the Outcome of Corona Treatment
5.4 Effects Produced by Corona Treatment
5.5 Surface Analysis of Corona-Treated Materials
5.6 Summary
References
6 Adhesion Activation of Aramid Fibers for Industrial Use
6.1 Introduction
6.2 Adhesion Between Aramid Fibers and Rubber
6.3 Adhesion Between Aramid Fibers and Other Matrices
6.4 Effect of Processing Oil on Adhesion
6.5 Plasma Activation of Aramid Fibers
6.6 Short-Cut Fibers
6.7 Summary and Prospects
Acknowledgement
References
7 Dual-Cured Hydrogels for Bioadhesives and Various Biomedical Applications
List of Abbreviations
7.1 Introduction
7.2 Discussion
7.3 Summary
References
8 Non-Adhesive SLIPS-Like Surfaces: Fabrication and Applications
List of Abbreviations
8.1 Introduction
8.2 Role of Contact Angle Hysteresis in Repelling Liquids
8.3 Non-Adhesive SLIPS-Like Surfaces
8.4 Applications
8.5 Summary and Outlook
Acknowledgments
References
Index
Also of Interest
End User License Agreement
List of Tables
Chapter 1
Table 1.1 Material properties used in [41] for comparison of results.
Table 1.2 Geometric and mechanical properties of the joint model [51].
Table 1.3 Geometric and mechanical properties of the adherends and adhesive la...
Table 1.4 The effect of adhesive elastic modulus E on the peak shear and peel ...
Table 1.5 The effect of overlap length L on the peak shear and peel stresses a...
Table 1.6 Geometric and mechanical properties of the constituents used in [40]...
Table 1.7 Cohesive zone material properties [31].
Table 1.8 Mechanical properties of the adhesive layer and adherends [31].
Chapter 2
Table 2.1 Dielectric experiments yielded maximal relative permittivity and t...
Table 2.2 Dielectric experiments yielded data on bound and free water with res...
Table 2.3 Mechanical, thermal, and adhesive properties of different adhesives ...
Table 2.4 The effect of different surface treatments and surface roughness on ...
Table 2.5 W
A
values for different interfaces, negative value signifies degrada...
Chapter 3
Table 3.1 Profilograms and surface roughness parameters of steel sheet adheren...
Table 3.2 Profilograms and surface roughness parameters of steel sheet adheren...
Table 3.3 Shear strength of bonded joints with various surface treatments of a...
Table 3.4 Profilograms and surface views of the steel sheet surface after shot...
Chapter 4
Table 4.1 Surface free energy components of untreated and UV-treated polymer f...
Table 4.2 Number of particles adhering to the polymer substrate in liquid medi...
Chapter 5
Table 5.1 Atomic concentration (%) for C1s, O1s and Al2p, and O/C-ratio and O/...
Table 5.2 Surface roughness data (
R
a
and
S
a
values) of untreated and coronatre...
Chapter 6
Table 6.1 XPS results of selected fiber bundle and cord samples.
Chapter 7
Table 7.1 Summary of free radical and coordination dual-crosslinked systems.
Table 7.2 Summary of free radical and condensation dual-crosslinked systems.
Table 7.3 Summary of coordination and condensation dual-crosslinked systems.
Table 7.4 Summary of properties and their relevance for each application.
List of Illustrations
Chapter 1
Figure 1.1 (a) Schematic diagram of an adhesively bonded lap joint and (b) its...
Figure 1.2 Schematic diagrams of an adhesively bonded (a) socket joint with (b...
Figure 1.3 Sectional views of other possible adhesively bonded tubular joints.
Figure 1.4 Schematic view of the bonded region in a tubular single lap joint w...
Figure 1.5 Free-body diagrams of differential elements of the two tubes in con...
Figure 1.6 Geometric dimensions of a defect-free lap joint assembly used in [4...
Figure 1.7 The effects of tube 1 inner radius R
2
and bond length L on the peak...
Figure 1.8 Shear stress distribution in the adhesive layer along the bond leng...
Figure 1.9 Schematic diagram of the adhesively bonded socket joint model used ...
Figure 1.10 The effect of axial load on variations of the interfacial shear st...
Figure 1.11 The effect of stacking sequence of the layers on the variations of...
Figure 1.12 The equivalent linear elastic and bilinear cohesive zone models us...
Figure 1.13 Schematic diagrams of a circular void and debond and their finite ...
Figure 1.14 The cross-sectional view and arrangement of the tubes in a tubular...
Figure 1.15 The cross-sectional view and arrangement of the tubes in a tubular...
Figure 1.16 The effect of a central void size on shear stress distribution in ...
Figure 1.17 Distribution of peel stresses along the bond length in (a) defect-...
Figure 1.18 The effect of central debonds with different sizes on the shear (a...
Figure 1.19 Typical taper/taper joint arrangement (dimensions not to scale) [4...
Figure 1.20 Dimensions (in mm) and geometric view of the aluminum to aluminum ...
Figure 1.21 Shear stress distribution in the adhesive layer based on finite el...
Figure 1.22 Simplified schematic diagrams of the end joint (a) no chamfer, (b)...
Figure 1.23 Sectional view, load arrangement, and geometric dimensions of the ...
Chapter 2
Figure 2.1 Types of stresses in an adhesive joint [11].
Figure 2.2 Arrhenius plot of diffusion coefficient (D) and ▫ - Low molecular w...
Figure 2.3 Weight uptake for various stoichiometric ratios (DGEBA:TETA) [16].
Figure 2.4 Dielectric permittivity graphs for different stoichiometric ratios ...
Figure 2.5 Type I bound water [27].
Figure 2.6 Type II bound water [27].
Figure 2.7 Water retention plots for various water-saturated epoxy systems [27...
Figure 2.8 Water retention plot of Fiberite 934 epoxy at 90 °C [27].
Figure 2.9 Failure stress variation in epoxy adhesive containing CaCO
3
nanofil...
Figure 2.10 Failure stress variation in epoxy adhesive with Al nanoparticles a...
Figure 2.11 Different epoxy adhesives’ adhesive strengths in a water environme...
Figure 2.12 Different epoxy adhesives’ adhesive strengths in a water environme...
Figure 2.13 Morphology of FPL-treated Al surface [47].
Figure 2.14 Morphology of PAA-treated Al surface [47].
Figure 2.15 Schematic of hydration resulting in failure of specimen in wedge t...
Figure 2.16 The ternary surface behaviour (A1PO
4
, A1
2
O
3
and H
2
O) diagram of th...
Figure 2.17 Adhesion strengths of pre-treated aluminium samples and percentage...
Figure 2.18 (a) Crack length versus exposure time plots for variously pre-trea...
Figure 2.19 Variation in lap shear strength of aluminium alloys after discrete...
Figure 2.20 Surface roughness values for various blasting pressures. Annealed ...
Figure 2.21 Surface roughness values for various blasting pressures. Annealed ...
Figure 2.22 Variation in adhesive strength of CFRP/steel joints for different ...
Figure 2.23 Schematic of oxygen diffusion from the oxide into Ti surface at hi...
Figure 2.24 Coefficient of thermal expansion (CTE) for various materials [9].
Figure 2.25 Effect of different loadings (0%, 5%, 10% and 20%) on double lap j...
Figure 2.26 TAST specimen and extensometers for shear deformation measurement ...
Figure 2.27 Screw jack test setup for creep measurement [88].
Figure 2.28 Wedge test specimen [88].
Figure 2.29 Variation in pull-off adhesion strength of epoxy with different na...
Figure 2.30 Mode I fracture toughness of nanocomposites and pure epoxy resin (...
Figure 2.31 Young’s modulus and tensile strength of pure and composite epoxies...
Figure 2.32 SEM images of debonding, cavitation, crack bridging and crack defl...
Figure 2.33 Tensile properties of alumina nanoparticles reinforced epoxy (a) s...
Figure 2.34 Effect of alumina nanoparticle (ANP) concentration on contact angl...
Figure 2.35 Durability of silica nanoparticles reinforced joints under cryogen...
Figure 2.36 Average failure loads of single lap joints reinforced by (a) Al
2
O
3
Figure 2.37 Effect of WC nanoparticles concentration on tensile strength of ep...
Figure 2.38 Effect of WC nanoparticles concentration on flexural strength of e...
Figure 2.39 Lap shear strength of NEAT epoxy and PEEK strengthened epoxy at va...
Figure 2.40 Fracture toughness of PEEK toughened epoxy at various loading perc...
Chapter 3
Figure 3.1 Examples of mechanical treatment methods.
Figure 3.2 Abrasive blasting methods [13,62].
Figure 3.3 Shear strength of steel bonded joint versus surface treatment with ...
Figure 3.4 Shear strength of steel bonded joint versus surface treatment with ...
Figure 3.5 Shear strength of bonded joint versus surface treatment with differ...
Figure 3.6 Isometric views of the steel surface after: a) sandblasting with gl...
Figure 3.7 Shear strength of aluminium alloys bonded joints versus variants of...
Figure 3.8 Shear strength of the adhesive joints versus surface treatment of a...
Figure 3.9 Surface topography versus surface treatment method: (a) after anodi...
Chapter 4
Figure 4.1 Changes in contact angle with time after placing a water drop on po...
Figure 4.2 Advancing contact angle of water on polymer film as a function of U...
Figure 4.3 Advancing contact angle of water on UV-exposed polymer film as a fu...
Figure 4.4 Schematic drawing of apparatus and procedure for wetting force meas...
Figure 4.5 Typical weight recordings and advancing and receding contact angles...
Figure 4.6 Advancing and receding contact angles of water on PET film as a fun...
Figure 4.7 Typical weight recordings for UV-treated PI film covered with a Ni-...
Figure 4.8 Typical weight recordings for UV-treated PET film covered with a Ni...
Figure 4.9 Surface atomic concentrations of carbon, nitrogen and oxygen on pol...
Figure 4.10 AFM images of untreated and UV-treated PET and PEN films. R
rms
, R
a
Figure 4.11 Relationship between the particle adhesion, n
a
, and the calculated...
Figure 4.12 Typical weight recordings and advancing and receding contact angle...
Figure 4.13 Surface reflectance spectrum in the visible region and total color...
Figure 4.14 Images of water spreading 30 s after placing a water drop (upper f...
Figure 4.15 The ratio of absorption coefficient, K, to scattering coefficient,...
Figure 4.16 Removal efficiency of stearic acid from untreated and UV-treated P...
Figure 4.17 Images of untreated and UV-treated PET fabric after dyeing with th...
Chapter 5
Figure 5.1 Illustration of a capacitor in which the corona discharge is formed...
Figure 5.2 Schematic of a DBD corona discharge treater configuration.
Figure 5.3 A corona treater system - (a) Electrical cabinet (power supply), (b...
Figure 5.4 Contact angle measurements with deionized water at a contact time o...
Figure 5.5 Contact angle measurements with deionized water at a contact time o...
Figure 5.6 Contact angle measurements with deionized water at a contact time o...
Figure 5.7 Contact angle measurements with deionized water at a contact time o...
Figure 5.8 SFE data (OWRK) of untreated and corona-treated aluminum foil ( γ
s
t
...
Figure 5.9 SFE data assessed 24 h after corona treatment (OWRK approach) of un...
Figure 5.10 SFE data assessed 24 h after corona treatment (OWRK approach) of u...
Figure 5.11 SFE data assessed 24 h after corona treatment (OWRK approach) of u...
Figure 5.12 SFE data assessed 24 h after corona treatment (methods of Chibowsk...
Figure 5.13 Deconvoluted high resolution C1s XPS spectra (intensity expressed ...
Figure 5.14 AFM-images (3D views) of BOPP film - (a) 0 Wmin/m
2
, (b) 30 Wmin/m
2
Figure 5.15 Adhesion (peel strength) data for untreated and corona-treated alu...
Figure 5.16 Adhesion (peel strength) data for untreated and corona-treated BOP...
Chapter 6
Figure 6.1 Two different process options for aramid fibers to reach sufficient...
Figure 6.2 Typical SPAF adhesion levels of aramid cord in Trelleborg NR-SBR ru...
Figure 6.3 Overlay of the XPS C1s signals of a fiber without finish, with a st...
Figure 6.4 Cross section of a fiber cord dipped in RFL, showing its typical re...
Figure 6.5 Schematic representation of the adhesion mechanism between aramid f...
Figure 6.6 BPO adhesion of three different Twaron
®
samples to an epoxy ...
Figure 6.7 TPO adhesion of AA fibers as percentage of the adhesion of fibers w...
Figure 6.8 Overlay of the XPS C1s signals of the (solvent extracted) samples, ...
Figure 6.9 Chlorine percentage (left Y-axis) and SPAF rubber adhesion (right Y...
Figure 6.10 Chlorine percentage (left Y-axis) and apparent shear strength to a...
Figure 6.11 Typical SEM images of untreated fibers (left hand side) and plasma...
Figure 6.12 Adhesion results of plasma activated fiber bundles, in comparison ...
Figure 6.13 Adhesion results of plasma activated cords, in comparison to refer...
Figure 6.14 Modulus-elongation curves of an EPDM compound without reinforcemen...
Chapter 7
Figure 7.1 Calcium and methacrylate crosslinked alginate [41]. Reprinted, Copy...
Figure 7.2 Schematic illustration of the preparation of single- and dual-cross...
Figure 7.3 Schematic illustration of applying dual-crosslinked hydrogel for wo...
Figure 7.4 Preparation of CMC-based IPN hydrogels. (a) CMC deprotonated in NaO...
Figure 7.5 PEG modification and schematic illustration of the fabrication of d...
Figure 7.6 Dual-crosslinking mechanism of anthracene functionalized chitosan-b...
Figure 7.7 Schematic diagram for preparation of dual-crosslinked and modified ...
Figure 7.8 Schematic synthesis of gelatin-dopamine conjugated system [94]. Rep...
Figure 7.9 Schematic synthesis of dual-crosslinked system. Gelatin-dopamine ra...
Figure 7.10 Modification of Alginate and schematic illustration of dual-crossl...
Figure 7.11 (a) Modification of chitosan (b) oxidation of dextran (c) schemati...
Figure 7.12 Modification of hyaluronic acid with furan-dopamine hydrochloride ...
Figure 7.13 Modification of hyaluronic acid with furan-phenyl boronic acid [98...
Figure 7.14 Synthesis steps of modified chitosan and scheme of dual-crosslinke...
Figure 7.15 The plausible cross-linked mechanism of hydrogels under different ...
Figure 7.16 Preparation of crosslinked hydrogel [100]. Reprinted, Copyright (2...
Figure 7.17 Schematic preparation of dual-crosslinked system [100]. Reprinted,...
Figure 7.18 Dual-network hydrogel preparation. (I) linear PEG carrying NHS est...
Figure 7.19 (I) First curing coordination: chemical structure of sodium algina...
Figure 7.20 Description of fabrication process of photopatterned dual-cured hy...
Figure 7.21 Schematic 3D printing process of hydrogels similar to fused filame...
Chapter 8
Figure 8.1 Schematics of non-wettable surfaces: (a) Textured superhydrophobic ...
Figure 8.2 Schematic representations of a water droplet on (a) homogeneous fla...
Figure 8.3 (A) Liquid residue coverage on the microposts of the SHS. The dashe...
Figure 8.4 Structure of the tris(trimethylsiloxy)silylethyldimethylchlorosilan...
Figure 8.5 Schematic showing the relation between the alkyl chain length and t...
Figure 8.6 Plot of ice adhesion strength with force probe velocity for (a) PFD...
Figure 8.7 (A) Fluorescent images of biomass growth of
P. aeruginosa
on PDMS, ...
Figure 8.8 (a) Schematic diagram of different grafted surfaces: LPDMS-coated s...
Figure 8.9 (A) (a-b) Heat flux and heat transfer coefficient comparison of SLI...
Figure 8.10 (A) (a) Ice adhesion strength for various surfaces in comparison t...
Figure 8.11 (A) (a) CA and SA of water droplet observed at various temperature...
Guide
Cover Page
Table of Contents
Series Page
Title Page
Copyright Page
Preface
Begin Reading
Index
Also of Interest
End User License Agreement
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Scrivener Publishing100 Cummings Center, Suite 541JBeverly, MA 01915-6106
Adhesion and Adhesives: Fundamental and Applied Aspects
The topics to be covered include, but not limited to, basic and theoretical aspects of adhesion; modeling of adhesion phenomena; mechanisms of adhesion; surface and interfacial analysis and characterization; unraveling of events at interfaces; characterization of interphases; adhesion of thin lms and coatings; adhesion aspects in reinforced composites; formation, characterization and durability of adhesive joints; surface preparation methods; polymer surface modi cation; biological adhesion; particle adhesion; adhesion of metallized plastics; adhesion of diamond-like lms; adhesion promoters; contact angle, wettability and adhesion; superhydrophobicity and superhydrophilicity. With regards to adhesives, the Series will include, but not limited to, green adhesives; novel and high-performance adhesives; and medical adhesive applications.
Series Editor: Dr. K.L. MittalEmail: [email protected]
Publishers at ScrivenerMartin Scrivener ([email protected])Phillip Carmical ([email protected])
Progress in Adhesion and AdhesivesVolume 7
Edited by
K.L. Mittal
This edition first published 2024 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA© 2024 Scrivener Publishing LLCFor more information about Scrivener publications please visit www.scrivenerpublishing.com.
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Library of Congress Cataloging-in-Publication Data
ISBN 978-1-394-19810-8
Cover image: Pixabay.ComCover design by Russell Richardson
Preface
The present book constitutes Volume 7 in the book series “Progress in Adhesion and Adhesives”. This book series was conceived in 2014 as an annual publication and the premier volume made its debut in 2015 and Volume 6 was published in 2021. Apropos, we did not label the premier volume as Volume 1 as at that time we did not have the crystal ball to see the future of this book series. It should be recorded for posterity that Volumes 1-6 were based on review articles first published in the journal Reviews of Adhesion and Adhesives (RAA), of which I was the editor. As of July 2022, I am no longer associated with RAA. After a short hiatus, the publisher and I agreed to restore publishing the series with curated original chapters from senior and respected scientists in adhesion and adhesives.
If comments from readers and colleagues are a barometer for the success of a book, then these six volumes have served their intended purpose, i.e. to provide state-of-the-knowledge reviews on many and varied topics within the broad purview of Adhesion and Adhesives. These six volumes have been warmly received by the community and this has vindicated the publication of this book series. We look forward to keeping the series going and volume 8 (2024) is already in the advanced planning stage.
The current book (Volume 7) contains eight chapters that have not been published. The topics covered include stress distribution and design analysis of adhesively bonded tubular composite joints; durability of structural adhesive joints; mechanical surface treatment of adherends for adhesive bonding; surface modification of polymer materials by excimer UV light; corona discharge treatment of materials to enhance adhesion; adhesion activation of aramid fibers; dual-cured hydrogels for bioadhesives and biomedical applications; and non-adhesive SLIPS-like surfaces.
The chapters in this book were all commissioned (invited); as a result, these are written by world-renowned researchers actively involved in research. This book is profusely illustrated and copiously referenced.
This book (Volume 7), like its predecessors, should appeal to and be of interest to adhesionists, adhesive technologists, polymer scientists, materials scientists as well as those involved/interested in adhesive bonding, packaging, printing, modification of polymer surfaces, biomedical applications, and non-adhesive and omniphobic surfaces.
Now comes the pleasant task of thanking those who were instrumental in materializing this book. First and foremost, my thanks go to the authors for their keen interest, sustained enthusiasm, unwavering cooperation, and sharing their valuable research experience in the form of written accounts (which essentially provided the grist for this book), without which this book would not have seen the light of day. Also, Martin Scrivener (publisher) should be thanked for conceiving the idea of publishing this book series and for his steadfast interest and whole-hearted support for this book.
Kash MittalHopewell Jct., NY, USAE-mail: [email protected] 2023
1Stress Distribution and Design Analysis of Adhesively Bonded Tubular Composite Joints: A Review
Mohammad Shishesaz
Engineering School, Department of Mechanical Engineering, The Shahid Chamran University of Ahvaz, Ahvaz, Iran
Abstract
This review is concerned with the stress distribution in adhesively bonded composite and laminated composite tubular joints under external loads. It includes the design aspects of an adhesively bonded joint including the effects of mechanical and geometric properties of the adhesive layer, type of the adhesively bonded joint, adherend properties, and the shapes of a few adhesively bonded joints on the stress distribution in the adhesive layer of different tubular joints. Furthermore, the effects of torsional and axial loads on the induced stresses in the adhesive layer are investigated. Additionally, local damages in terms of voids or debonds in the adhesive layer, as well as delamination in the composite adherends are introduced to determine their adverse effect in terms of adhesive shear and peel stresses on the joint performance. Moreover, the use of hybrid joints is surveyed to explore the benefits of combined mechanical and bonded joints. In contrast to the linear analysis, the nonlinear analysis is also introduced to seek the effect of adhesive nonlinear behavior on the induced stresses, affecting the overall joint performance.
Keywords: Tubular joint, adhesively bonded, lap joint, composite tube, peel stress, shear stress, design aspects, failure analysis
1.1 Introduction
Adhesives have been widely used in a variety of applications ranging from aerospace to marine, medical, dental, construction, automotive, and electronics for joining parts together [1, 2]. One major concern in the use of adhesively bonded joints is the magnitude of peak stresses that develop in the adhesive layer. Depending on the type and magnitude of the applied load, excessive shear and/or peel stresses can develop in the adhesive layer causing failure of the joint. For this reason, different remedies have been proposed to optimize the adhesive performance [3, 4]. However, limited work is available on tubular structural joints (single lap/socket metallic or composite tubular structures) [5]. Many authors have tried to explore the adhesive behavior in tubular joints, and have given a better estimate of the stress magnitudes that develop in the joint under different loading conditions. In a very early study, McCarvill and Bell [6] developed a modified tubular butt assembly to test a modified tubular epoxy-aluminum butt joint that was subjected to torsion. Later, Alwar and Nagaraja [7] used the finite element method (FEM) to include the viscoelastic behavior of the adhesive layer (in a tubular joint under an axial load) and determined the long-term redistribution of stresses in this layer. In a different study, Adams and Peppiatt [8] applied the FEM to investigate the stress distribution in an adhesively bonded tubular joint under axial and torsional loads. Among other factors, the effects of adhesive fillet and partial tapering of the adherends on the stress distribution were investigated in their work.
In general, tubular parts can be joined by different types of adhesives. One major concern about these joints is the magnitude of peak stress that develops in the adhesive layer due to the type of the external load applied to the joint. It has been shown that an excessive load in the adhesive layer can result in excessive peel or shear stress leading to joint failure. In the past decades, many investigations have been performed on different types of plated and tubular adhesive joints [9-14], both analytically and experimentally. Many authors have tried to present easier or modified methods to give better estimates of the magnitudes of the peak stresses that may develop in the adhesive layer under different loading conditions [15, 16]. Thus, for a safe design, the engineers must have a complete knowledge of stress distribution in all constituents forming the joint. In a later study, Oh [17] performed a nonlinear analysis on the adhesively bonded tubular single-lap joints, to obtain the constituent relations under a torsional load. Results showed that the peak shear stress developed at the joint end is less than the value obtained through linear analysis. He further concluded that the torque transmissibility of the joint increases with the joint overlap length.
In a related study, Hosseinzadeh and Taheri [18] studied the torque capacity of a tubular lap joint for load transmission, using large strain finite element analysis. The considered joint was composed of a metal pipe bonded to a composite tube. Both geometric and material nonlinearities were considered in their analysis. Results of the study showed that the peak shear stress in the joint cannot be lowered significantly by increasing the bond length. Esmaeel and Taheri [19] performed further investigation on the adhesive stress distribution in a tubular bonded composite joint with a delaminated adherend. They used the finite element method to examine the effects of location and dimensions of the delaminated region on the resulting stresses. According to their findings, the dimensions and locations of the delaminated region had a major impact on the deduced stresses, magnitudes of which were reduced as the delamination width was decreased.
In the study performed by Xu and Li [20], the finite difference method was used to obtain the three-dimensional solution of stresses developed in an adhesively bonded composite tubular joint subjected to torsion. In their analysis, the variation of stress along the adhesive thickness was considered. Results showed that for tubular joints in torsion, the peel stress in the adhesive layer can be neglected provided the pipes are isotropic and/ or orthotropic. In another study performed by Das and Pradhan [21], failure analysis was performed on the adhesively bonded composite pipe joints. They used Tsai-Wu theory to determine an appropriate length for the joint. They also identified regions prone to initiation of failure or damage. Spaggiari and Dragoni [22] managed to regulate the torsional stresses in a tubular lap joint using functionally graded adhesives, properties of which were considered to vary along the axial direction. Furthermore, they solved the constitutive relations to obtain constant stress along the joint and to reduce the stress concentration at the joint ends. The analyses on delamination damage of laminated bonded tubular single lap joints made of fiber-reinforced polymer composite, as well as failure analysis of the bonded composite pipes were performed in [23] and [24].
Nimje and Panigrahi [25] investigated the effect of the functionally graded adhesive layer on the failure of socket joints, in a laminated fiber reinforced plastic (FRP) composite tube. In a related study, Aimmanee and Hongpimolmas [26] used elasticity theory to analyze the behavior of an adhesively bonded tubular-coupler joint, with an optimum variable stiffness composite coupler, subjected to pure torsion. In their study, the fiber angles were assumed to vary along the coupling length. However, to obtain a solution, the coupling length was divided into small segments in which the fiber angle was assumed to be constant.
Using the differential quadrature method (DQM), Mohieddin Ghomshei and Shahi [27] studied the effects of torsional and hygrothermal loads on stress distribution in a tubular single-lap adhesively bonded composite joint. In this analysis, the effects of several parameters such as adhesive thickness, overlap length, temperature change, and relative humidity on the joint behavior were studied. Moreover, their results were compared with those of finite element findings and the previously published literature. Using isotropic pipes, Shishesaz and Tehrani [28, 29] investigated the effects of a circular void or debond on the overall stress distribution in an adhesively bonded tubular joint subjected to a torsional or axial load. They obtained a semi-analytical solution for the stress distribution in the adhesive layer and compared the results with those of their finite element model. According to their findings, for small central defects away from the ends, it is hardly discernible from the magnitudes of the peak interfacial shear stresses whether the joint contains a void or debond.
In the study performed by Yousefi, et al. [30, 31], the behavior of a hybrid tubular lap joint utilizing rivets/adhesive was studied to determine the joint’s mechanical behavior under a torsional load. In this study, the behaviors of aluminum to aluminum and aluminum to composite joints were investigated to determine the effects of joint type, post-cured temperature, and outer tube material and its thickness on the overall strength and torque capacity of the joint. Results showed that for joining composite to aluminum, application of a hybrid joint setup improves the torque capacity of the joint by 39.5% and 276.9%, compared with the sole mechanical and adhesively bonded joints, respectively. Dantas and coworkers [32, 33] investigated the behavior of a flexible tubular metal-polymer adhesive joint designed to withstand large torsional deflections. To assess the influence of manufacturing procedures on joint performance, three different layouts were considered. The results of this study showed that the joint layout had a major influence on its mechanical performance. Additionally, the torsion test results identified the appropriate overlap length in the joint.
1.2 A Brief Review of Stress Analysis in Tubular Composite Joints
Adhesively bonded tubular composite joints can be classified as follows:
Adhesively bonded metal to metal pipes (tubes) with different properties.
Adhesively bonded metal to non-metal pipes (tubes) (or non-metal to non-metals with different properties).
Adhesively bonded metal to laminated composite pipes (tubes).
Two adhesively bonded laminated composite pipes (tubes).
In these cases, the two pipes (tubes) can be joined by an adhesive layer forming a lap joint (see Fig. 1.1) or using a socket of any shape combined with a layer of adhesive as shown in Fig. 1.2. However, as shown in Fig. 1.3, other shapes may be used to transfer the mechanical and/or thermal loads through the joints.
Figure 1.1 (a) Schematic diagram of an adhesively bonded lap joint and (b) its sectional view.
Figure 1.2 Schematic diagrams of an adhesively bonded (a) socket joint with (b-d) pipe and adhesive arrangements.
These joints may experience axial loads, as well as torsional or bending moments (or their combinations). Depending on the applied loads, typical stresses are developed in the adhesive layer that, if not controlled, may result in different types of adhesion failures [21, 34, 35]. These failures can be caused by progression of stresses beyond the adhesive strength values, due to a bond separation at the adhesive-adherend interface, in the coupling region [36]. Furthermore, an initial crack or defect may cause failure if the adhesive layer is subjected to a load beyond its endurance limit. In such cases, different approaches are adopted to predict the bond strength, as well as how to strengthen the joint and control the development of stresses in the adhesive layer [37, 38]. Although in most cases, the adhesive behavior is assumed to be linear elastic, yet, mostly due to its polymeric nature, a nonlinear behavior can also be considered for its response to the applied loads [17, 18].
One of the major factors influencing the stress distribution in the adhesive layer of a composite joint with laminated adherends (tubes) is the layer arrangement used for each adherend. A few investigations have focused on this issue and how delamination between the adherend layers can influence the overall stresses developed in the adhesive layer, which in turn affect the joint behavior [39, 40] and/or joint failure. Furthermore, for pipes in torsion, the improper stacking sequence of the adherend layers bonded by a defective adhesive layer can increase the torsional stresses, leading to a deterioration of the adhesive bond. This issue has been addressed in [41] where the adhesive layer suffers from a circular void or debond.
An extensive literature survey reveals that many factors can influence the behavior of the adhesive layer, a few of which were outlined above. For this reason, the next sections briefly provide the basic introduction and equations with which these stresses are calculated and their influences on the overall performance of the adhesive layer (or joint behavior) in the adhesively bonded laminated composite tubes.
Figure 1.3 Sectional views of other possible adhesively bonded tubular joints.
Fiber-reinforced polymer (FRP) composite pipes have been widely used in various demanding structural applications such as the automotive and aerospace industries. Some of the main barriers to the extensive use of these pipes (tubes) have been due to insufficient knowledge on stress distribution, design procedures, proper standards, and their relevant performance. Extensive investigations show that FRPs offer many advantages over traditional materials such as steel, in their use for the construction of lightweight structural components, in many industries. Moreover, corrosion resistance, dimensional tolerance, strength, abrasion resistance, and many other factors are additional advantages that make them suitable for their use in the construction of pipes usually joined by an adhesive layer. However, the cost of basic constituents, quality assurance, and testing, as well as the anisotropic properties (that are a function of layup sequence and fiber orientation) are considered to be some of their main drawbacks. For this reason, to eliminate some of these barriers to manage the use of the tubular composite joints, a full knowledge of stress distribution in the adherends and adhesive layer joining the pipes (tubes) is necessary.
According to many investigations, for some laminated composite joints in torsion, the shear stress is considered as the main stress influencing the performance of the adhesive layer [14, 17, 25-27, 37, 39, 41]. However, the bond length and other mechanical and geometrical properties of the adherends and adhesive layer have effects on this stress component [9, 28, 29, 35, 42]. The above-named factors are considered as the major factors influencing the adhesive joint failure. Consequently, for a safe design, a full understanding of the stress distribution within the adhesive layer is necessary [43].
Previous investigations show that for tubular joints with isotropic adherends in torsion, the radial and shear stress components (σr and τrz) are zero. However, for adherends with laminated composites, to satisfy the equilibrium equations and compatibility conditions in the overlap region, these stress components must be included in the analysis [9]. Furthermore, in tubular joints under axial load, the development of stress concentration in the adhesive layer can arise by three different mechanisms: (a) differential straining, (b) end effects, and (c) bending introduced by the non-collinearity of the overlapping tubes. However, for torsional loads, the bending effects and hence the peel stresses are absent and only the end effects and differential straining need to be considered, provided there is no defect in the adherends [40, 43, 44].
An early study by Lubkin and Reissner [45] as well as many other recent studies on the tubular lap joints [29, 39, 40, 41, 46] have shown that under a tensile axial load or a bending moment, an induced shear stress component τrz and the normal stress component σr across the thickness of the adhesive layer are present and occur due to adherend bending. However, for the free surfaces at the ends of the adhesive layer, τrz must be zero. Consequently, a high shear stress gradient exists near the two ends of the joint, magnitude of which depends on the geometric and mechanical properties of the adherends and the adhesive layer. At this location, the interfacial shear stresses increase from zero on the free surface of the adhesive layer to some maximum value over a very short distance along the bond length [28, 29, 39, 40]. As stated before, because of the stress equilibrium considerations this high shear stress gradient is associated with a normal stress gradient across the thickness of the adhesive layer. For this reason, many researchers [e.g. 39-41] have focused on the tubular single lap and socket joints (Figs. 1.1 and 1.2).
1.3 Governing Equations Based on Linear Elasticity
To understand the effects of geometric and mechanical properties of the adhesives and adherends on the stress distribution in the adhesive layer (via the solution of governing equations), first, the required assumptions based on the load type and joint geometry must be introduced. For example, considering the tubular single lap joint shown in Fig. 1.4, the necessary assumptions are defined as follows.
1.3.1 Typical Assumptions in a Tubular Lap Joint Under Torsion
The allowable assumptions for the derivation of governing equations depend on the load type, configuration of the joint, and mechanical properties of the joint constituents. Based on the linear elasticity, for a typical joint under torsional loading (as shown in Fig. 1.4) it can be assumed that:
Only the circumferential displacements in the tubes and adhesive layer need to be considered. In other words, the axial and radial components of the displacements are considered to be zero.
The circumferential components of the displacements in the tubes and the adhesive layer can be assumed to be a function of axial (
x
) and radial (
r
) directions.
For a composite adherend, the stacking sequence in the adherend can be assumed to be either monoclinic [
17
], orthotropic or quasi-isotropic [
41
], transversely isotropic [
47
], or isotropic. Consequently, the main shear stress produced in the adhesive layer is the torsional shear stress (
τ
rθ
), and hence one can neglect the effects of other stress components in the model ([
14
,
17
]).
Figure 1.4 Schematic view of the bonded region in a tubular single lap joint with two composite adherends [41].
However, for a joint under axial force one cannot neglect the peel stress σr in the adhesive layer, as opposed to a tubular joint under torsion, due to the bending moment created in the joint.
Now, using a three-dimensional free-body diagram of the joint (see Fig. 1.5) and using the foregoing assumptions, the governing equilibrium equations for the stress distribution in the joint can be deduced, as will be shown next. For this purpose, τ1 and τ2 designate the interfacial shear stresses resulting from (τrθ)i (i =1, 2) at the lower and upper interfaces of the adhesive layer, respectively. Moreover, denoting the (Nxθ)i, (Qθ), (Mxθ)i, and hi (i =1, 2) as the components of the resultant shear force, transverse shear force, twisting moment, and pipe thicknesses, respectively, application of the force equilibrium equations gives the governing equilibrium equations in the θ-direction as:
It is worth noting that the origin of the local coordinate axes (xi, zi, θi, i = 1, 2) selected in each component (adhesive layer and adherends) is located at the mid-surface of the corresponding component (see Fig. 1.4).
However, if the circumferential components of the displacement in each adherend are assumed to be a function of x and θ, then one can write:
Figure 1.5 Free-body diagrams of differential elements of the two tubes in contact with the adhesive layer [41].
In Eq. (1.2b), V0i(x), Φθi(x), are the mid-plane displacement and rotation in the pipes, respectively, while zi is the local distance from the mid-plane of either pipe.
The normal components of the strain tensor in cylindrical coordinates are:
Additionally, the shear strain components are:
In these equations, . Moreover, the constitutive relations for a composite laminate with n layers and fiber angles βk in each layer (k is the layer number) are . In this equation, the matrices associated with the kth layer are:
and,
For the elements of the reduced stiffened matrix , in terms of the elements of stiffness matrix Q(k) and engineering constants, one can refer to [48].
Now, substituting Eqs. (1.2a) and (1.2b) in the strain-displacement relations we have:
On substituting Eqs. (1.4a), (1.4b), and (1.4c) into the constitutive relations , one can obtain the stresses in each layer of the composite pipes. However, for a single lap joint in torsion, only the force and moment components affecting the shear stress in the adhesive layer can be calculated. These components could be different for other types of loadings. For this purpose, substituting Eqs. (1.4a), (1.4b), and (1.4c) in stress-strain relations and assuming that each pipe is constructed of different layers, one can obtain the resultant forces and moments (in each pipe) as:
In Eqs. (1.5a)-(1.5c), (Nxθ), (Qθ)i, and(Mxθ)i are in-plane and transverse components of the shear loads as well as the twisting moments, respectively. Furthermore, KS represents the shear correction factor that can be assumed to be equal to 5/6 (see [49] and [50]), n is the total number of layers, and k is the layer number. These equations are recast in terms of derivatives of pipes’ mid-plane displacements and rotations as:
The elements Aij, Bij, and Dij represent the elements of extensional, coupling, and stiffness matrices A, B, and D, respectively, values of which (along with the values of Eij and Fij) are given in Eqs. (1.7a)-(1.7e).
Although in some cases, due to the nature of the problem, constant shear stress is considered in the adhesive layer across its thickness, yet in this case to obtain the variations in the shear stress along the adhesive thickness, a second-order displacement vadh(x, r) is considered for the circumferential displacement such that:
In Eq. (1.8)v0 (x), v1 (x), and v2 (x) correspond to the mid-plane displacement, as well as the top and bottom surfaces of the adhesive layer, respectively. Using continuity in the displacements at the tube-adhesive interfaces, it can be shown that the expressions for coefficients η0(r), η1(r), and η2(r) are:
The parameters appearing in Eqs. (1.9a)-(1.9c) are shown in Figs 1.4 and 1.6 where:
In cases in which the adhesive thickness is considered to be constant, one can approximate the magnitude of adhesive shear stress at any section, by Eq. (1.11).
where (v1 − v2) is the relative circumferential displacement of the composite tubes at their interfaces with the adhesive layer. G and tadh are the shear modulus and thickness of the adhesive layer, respectively. However, since (v1 − v2) is a function of both the shear moduli and polar moments of inertia of the tubes (adherends), then it is expected that the adhesive layer experiences a nonsymmetric shear stress distribution as the load is transferred from one tube to another.
Figure 1.6 Geometric dimensions of a defect-free lap joint assembly used in [41].
On using the expression for the adhesive displacement, it can be shown that the expression for variation in the twisting moment in the adhesive layer is given by [41]:
Now, on the proper application of equilibrium equations (forces and moments), one can show that the following equations will determine the complete displacement field, and hence the shear stress distribution in the adhesive layer provided proper boundary conditions are applied.
In this case, for a defect-free tubular joint under pure torsion, the following boundary conditions can be applied. It is worth mentioning that the expressions for S11 to S33 are given in [41].
1.3.2 Stress Distribution in a Defect-Free Tubular Lap Joint Under Torsion
The effects of geometric and mechanical properties of the adhesive and adherends for a tubular lap joint under torsion were fully investigated in [41]. Here, the semi-analytical results were obtained for a single lap joint and supported by the finite element solution. The inner pipe (tube 1) was made of steel and the outer pipe (tube 2) was assumed to be an orthotropic composite pipe with the stacking sequence of [±15]10 as well as other arrangements for a pipe with quasi-isotropic lamination setup. The properties of each tube and the adhesive layer are given in Table 1.1.
Results in [41] showed that the difference between the torsional stiffness of the ±15o laminated outer tube and the inner steel tube produces an unsymmetric shear stress distribution at each adhesive-adherend interface. Furthermore, as the length of the overlap region reduces in size, the effect of stress concentration at one end appears to affect the other, and hence increasing the peak shear stresses that occur at either end of the overlap region. However, there is a so-called “effective length” for which the whole bond length remains active (shear stresses greater than zero) without causing a considerable rise in (τrθ)max. Consequently, to better control the peak shear stresses in the adhesive layer, it is crucial to select the right bond length to properly transfer the applied load.
Table 1.1 Material properties used in [41] for comparison of results.
Geometric and material properties
Outer pipe (pipe 2) [±15
o
]
10
Inner pipe (pipe 1) (steel)
Adhesive layer, EPCO 9923 Epoxy
Inner pipe radius (mm)
15.0
11.9
14.9
Outer pipe radius (mm)
18.0
14.9
15.0
Length (mm)
60.0 (total)
60.0 (total)
20.0 (bonded length)
Elastic modulus
E
3
, GPa
10.0
200.0
1.3
Elastic modulus
E
2
, GPa
10.0
200.0
1.3
Elastic modulus
E
1
, GPa
128.0
200.0
1.3
Shear modulus
G
23
, GPa
3.58
76.92
0.46
Shear modulus
G
13
, GPa
4.49
76.92
0.46
Shear modulus
G
12
, GPa
4.49
76.92
0.46
Poisson’s ratio
v
23
0.47
0.3
0.41
Poisson’s ratio
v
13
0.28
0.3
0.41
Poisson’s ratio
v
12
0.28
0.3
0.41
The simultaneous effects of the inner tube (tube1) torsional stiffness (in terms of a change in tube 1 radius) and the bond length L on the peak shear stress τ1max developed at adhesive-inner tube interface are shown in Fig. 1.7. Here, the adhesive thickness was assumed to be 0.1 mm, while the initial values of 11.9 mm and 12.9 mm were selected for R1 and R2 (see Fig. 1.6).
In Fig. 1.7, the radii of tube 2 were calculated in millimeters according to the formulae R3 = R2 + 0.1 and R4 = R3 + 3. According to this figure, for any fixed bond length, the increase in the inner radius of tube 1 decreases the shear stress concentration in the adhesive layer. Based on the coordinates of points A, B, C, and D in this figure, holding the bond length equal to 70 mm and increasing the inner tube 1 radius from 12.9 to 14.9 mm reduces the peak shear stress by 14.5%. The decrease in the peak shear stress in the adhesive layer for a similar increase in R2 at the short bond length L = 8 mm is 17.9%. Interpolating the results in this figure, the increase in the tube 2 inner radius by 0.2 mm (from 12.9 to 13.1 mm) reduces the peak shear stress only by 0.9% at L =70 mm and by 2.0% at L = 8 mm.
Figure 1.7 The effects of tube 1 inner radius R2 and bond length L on the peak shear stress τ1 max developed in the adhesive layer [41].
Further studies in [41] also indicated that increasing the elastic modulus of the adhesive layer results in an increase in the peak shear stresses at the two ends of the upper and lower adhesive-adherend interfaces (for more explanation see [41]). Furthermore, to investigate the effect of adhesive thickness tadh on the shear stress distribution in the adhesive layer, the inside and outside radii of the inner tube were kept constant and equal to R1 = 11.9 mm and R2 = 14.9 mm (see Fig. 1.6) respectively. However, to increase the adhesive thickness, the inside and outside radii of the outer tube were increased such that its total thickness remained constant and equal to 3 mm. Results indicated that the increase in the adhesive thickness by 0.2 mm reduces the maximum shear stress by about 40%. As explained before, the 0.2 mm increase in the wall thickness of the inner tube decreases the peak shear stress by 2.0%, provided the bond length is short and equal to 8 mm. However, although increasing the adhesive thickness by 0.2 mm (from 0.1 mm to 0.3 mm) increases tube 2 radius by the same amount (resulting in higher polar moment of inertia), yet the peak shear stress is reduced by only 40 %. This means that the peak shear stress in the adhesive layer is more susceptible to the adhesive thickness rather than to a change in tubes radii.
To investigate the effects of lamination setup on the peak stresses in the adhesive layer, Fig. 1.8 was generated to investigate the behavior of orthotropic and quasi-isotropic lamination setups. The mechanical and geometric properties given in Table 1.1 were used for the inner tube, as well as for the adhesive layer. The lamination setup with a total number of 20 layers was selected only for pipe 2. Examination of the results shown in Fig. 1.8 will allow one to decide on the possibility of replacing the steel tubes with laminated composite pipes. In this figure, two types of layer arrangements, namely orthotropic and quasi-isotropic laminations were used for the outer pipe.
Figure 1.8 Shear stress distribution in the adhesive layer along the bond length for various lamination setups in pipe 2 [41].
In the first setup, tube 2 was composed of a series of orthotropic laminates with lamination codes [±15]10, [±30]10, [± 455]s, [± 60]10, and [902/02]10. However, in the second setup, the quasi-isotropic laminations of [04/904/-454/454], [04/364/724/-724/-364], and [02/362/722/-722/-362]s, were used. For the orthotropic arrangement of [902/02]10 the maximum peak shear stress produced in the adhesive layer is 74.5 MPa, while the [±45]10 setup produces a value of 23 MPa. The rise in the peak shear stress between the two arrangements is about 224%. However, among the quasi-isotropic arrangement of layers shown in Fig. 1.8, the [05/905/-455/455] arrangement produces the minimum peak shear stress at both interfaces of the adhesive layer, compared with [02/362/722/-722/-362]s setup. Selecting the former stacking sequence reduces the peak shear stress at the right end by 4.8%. Comparison of the results in Fig. 1.8 indicates that the best lamination setup for pipe 2 to transfer the torsional load (via the joint) with the least peak shear stress in the adhesive layer belongs to the orthotropic pipe with a stacking sequence of [±45]10 for the layers. This will produce the minimum peak shear stress in the adhesive layer. Based on this argument, compared with the quasi-isotropic setup of [05/905/-455/455] the selection of an orthotropic tube with the [±45]10 layer arrangement further reduces the minimum peak stress (on the right side) by an additional 13.5%. Additional investigation in [41] has shown that it is quite possible to replace the steel-composite tubular joints with laminated carbon-epoxy composite-composite tubular joints with almost similar shear stress in the adhesive layer, yet reducing the structural weight by almost 80%. Further studies on stress distribution in tubular joints can be found in [9, 13, 17, 18, 20, 26, 28, 29, 33, 39].
The effects of delamination on the stress distribution in the adhesive layer of a lap joint as well as a socket joint were fully investigated in [40]. The results obtained were based on an extensive finite element solution. In the single lap joint, adherend 1 (or the inner pipe) was made of composite material, while adherend 2 (or the outer pipe) was considered to be either aluminum or a laminated composite pipe. In the socket joint, the two adherends were identical and both adherends and the socket were made of composite materials. Furthermore, any material nonlinearity in the adhesive layer was neglected.
1.3.3 Stress Distribution in Defect-Free Joints Under Bending Moment
To obtain the effect of bending moment on the adhesive stresses, Yang et al. [51] used a generalized model to generate three types of permanent composite pipe joints. The three components, namely the pipes, the coupling, and the adhesive used to generate these models were selected as: (1) for the adhesively bonded socket joint, the two pipes were joined according to Fig. 1.9 using a socket and an adhesive layer; (2) in the butt-strap joint model, the two pipes were joined by a fiber-reinforced coupling using resin matrix as the adhesive layer; and (3) in the heat-activated coupling joint the epoxy resin was used as the adhesive to join the two pipes using epoxy prepreg as the coupling. The properties of these constituents are given in Table 1.2. Furthermore, using separation of variables, a solution was obtained for variations of shear and peel stresses in a tubular joint under bending. The original system of partial differential equations was reduced to the ordinary differential equations, solutions of which were obtained using Maple V. The first-order laminated anisotropic plate theory was used to develop the models introduced before.
Based on the analytical and finite element results, values of about 3.7 MPa for the peel stress and 2.3 MPa for the peak shear stress were obtained at the two ends of the bonded region. Furthermore, there were no discernible differences in the adhesive peel and shear stress distributions between the two solution methods.
Figure 1.9 Schematic diagram of the adhesively bonded socket joint model used in [51].
Table 1.2 Geometric and mechanical properties of the joint model [51].
Property
Coupling (E-Glass/Derakane 470, 50% glass content by weight)
Pipe (E-Glass/Derakane 470, 50% glass content by weight)
Epoxy adhesive
E
xx
(GPa)
25.2
25.2
0.96
Eyy
(GPa)
