DOT/FAA/AR-05/12 

Office of Aviation Research 

Washington, D.C. 20591

Methods of Analysis and Failure 

Predictions for Adhesively Bonded 

Joints of Uniform and Variable 

Bondline Thickness 

May 2005 

Final Report 

This document is available to the U.S. public 

through the National Technical Information 

Service (NTIS), Springfield, Virginia 22161. 

U.S. Department of Transportation 

Federal Aviation Administration 
NOTICE 

This document is disseminated under the sponsorship of the U.S. 

Department of Transportation in the interest of information exchange. The 

United States Government assumes no liability for the contents or use 

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because they are considered essential to the objective of this report. This 

document does not constitute FAA certification policy. Consult your local 

FAA aircraft certification office as to its use. 

This report is available at the Federal Aviation Administration William J. 

Hughes Technical Center's Full-Text Technical Reports page: 

actlibrary.tc.faa.gov in Adobe Acrobat portable document format (PDF). 
 Technical Report Documentation Page 

1. Report No.

DOT/FAA/AR-05/12

2. Government Accession No. 3. Recipient's Catalog No.

5. Report Date 

May 2005

 4. Title and Subtitle 

METHODS OF ANALYSIS AND FAILURE PREDICTIONS FOR ADHESIVELY 

BONDED JOINTS OF UNIFORM AND VARIABLE BONDLINE THICKNESS 6. Performing Organization Code

7. Author(s)

Yuqiao Zhu and Keith Kedward 

8. Performing Organization Report No. 

10. Work Unit No. (TRAIS) 

 

9. Performing Organization Name and Address

Department of Mechanical & Environmental Engineering 

University of California 

Santa Barbara, CA 93106-5070

11. Contract or Grant No.

 

12. Sponsoring Agency Name and Address

U.S. Department of Transportation 

Federal Aviation Administration

13. Type of Report and Period Covered

 Final Report

Office of Aviation Research 

Washington, DC 20591

14. Sponsoring Agency Code

 ACE-120

15. Supplementary Notes 

The FAA William J. Hughes Technical Center COTR was Peter Shyprykevich.

16. Abstract 

Adhesively bonded joints under tensile lap shear loading were analyzed using the finite element method and closed-form 

solutions. Predictions of the stress distribution and failure prediction are compared with experimental failure load data. 

Case studies were performed that addressed the finite element meshing strategies of adhesively bonded joints such as h- and pmethods, mesh density around the overlap regions, element types. Comparisons were made with available closed-form solutions. 

Titanium single lap joints were analyzed using a linear analysis and the effects of bondline thickness and fillet were investigated. 

Parametric studies showed that the maxima strength of the adhesively bonded single lap joint increased with decreasing adhesive

thickness. The proposed use of varying adhesive layer thickness to reduce the stress singularity was investigated by profiling the 

adherends’ thickness quadratically or linearly to reduce or eliminate the shear stress concentration at the ends. Preliminary 

analysis on the effect of variable thickness along the overlap direction showed that maximum stress occurred at the end of least 

thickness. Further analysis on profiling the adherend thickness to reduce the stresses at the ends is recommended. 

To support and validate the analysis, single lap joints were tested with uniform and variable bondline thickness. The specimen

used titanium adherends and 3M’s DP460 adhesive. 

Finally, nonlinear analyses of titanium single lap joints, taking into consideration the ductility of the adhesive were performed, 

and the predicted failure load was shown to be 10% less than the test failure load. The cause of this lower prediction is discussed, 

and future work is suggested. 

17. Key Words 

Adhesively bonded joint, Adaptive mesh, Finite element 

method, Bonding strength, Stress singularity, Variable 

bondline thickness 

18. Distribution Statement 

This document is available to the public through the National 

Technical Information Service (NTIS) Springfield, Virginia 

22161.

19. Security Classif. (of this report) 

 Unclassified

20. Security Classif. (of this page) 

 Unclassified

21. No. of Pages 

 61 

22. Price

Form DOT F1700.7 (8-72) Reproduction of completed page authorized
ACKNOWLEDGEMENTS 

The authors would like to thank Federal Aviation Administration for funding of the project. The 

advice of using true stress-strain curve for the adhesive from Professor Glenn Beltz is greatly 

appreciated. Contributions to the experimental effort by senior undergraduate students, Kris 

Benson and Steven Kiefer, are also gratefully acknowledged. 

iii/iv
TABLE OF CONTENTS 

Page 

EXECUTIVE SUMMARY xi 

1. INTRODUCTION AND FOCUS OF RESEARCH 1-1 

2. LITERATURE REVIEW 2-1 

2.1 Stress Analysis 2-1 

2.2 Failure Predictions 2-2 

2.3 The Effect of Variations in Bondline Thickness 2-3 

3. ANALYSIS APPROACHES AND CASE STUDIES 3-1 

3.1 Stress Distribution in Double Lap Joint Configuration 3-1 

3.1.1 Volkersen 3-1 

3.1.2 Linear Finite Element Analysis 3-1 

3.1.3 Correlation Studies of Predicted Stress Distributions 3-8 

3.2 Stress Distribution in Single Lap Joint Configuration 3-12 

3.2.1 Goland and Reissner 3-12 

3.2.2 Linear Finite Element Analysis With Aluminum Adherends 

(ASTM D 1002-99) 3-12 

3.2.3 Comparisons Between Single Lap and Double Lap Predictions 3-16 

3.3 Linear and Nonlinear Analysis of Titanium Single Lap Joint 3-19 

3.3.1 Linear Analysis 3-19 

3.3.2 Nonlinear Analysis 3-26 

3.4 Approaches for Failure Prediction 3-31 

3.4.1 Maximum Stress/Strain 3-31 

3.4.2 Maximum Stress/Strain Including Nonlinearity of Adhesive 3-32 

3.4.3 Yielding/Nonlinearity (Development and Yield Zones) 3-33 

4. CORRELATION WITH EXPERIMENTS 4-1 

4.1 Single Lap Joint 4-1 

4.1.1 Aluminum/Titanium Adherends 4-1 

v
4.1.2 Composite Adherends 4-5 

4.2 Double Lap Joint 4-6 

4.3 Bondline Thickness Variability 4-6 

5. CONCLUSIONS 5-1 

6. REFERENCES 6-1 

vi
LIST OF FIGURES 

Figures Page 

3-1 A Typical Double Lap Joint 3-2 

3-2 Finite Element Model Using Plane82 3-3 

3-3 Detail Meshes Around the Termination Region of the Adhesive 3-3 

3-4 Peel Stress Contour Using Plane82 3-4 

3-5 Shear Stress Contour Using Plane82 3-4 

3-6 Shear and Peel Stress Distribution Along the Adhesive Centerline Using Plane82 3-5 

3-7 Shear Stress Contours (Local Adhesive Termination Region) Using Plane82 3-5 

3-8 Shear Stress Contours (Local Adhesive Termination Region) Using the p-Method 3-6 

3-9 Peel and Shear Stress Distribution Along the Adhesive Centerline Using the 

p-Method 3-7 

3-10 p-Level Used at Point A (Shear Stress vs Polynomial Level) 3-7 

3-11 Comparisons of Shear and Peel Stresses Along the Centerline of the Adhesive 3-9 

3-12 Three Different Paths in the Adhesive Layer 3-10 

3-13 Comparisons of Shear and Peel Stresses Along Three Different Paths 3-11 

3-14 A Single Lap Joint 3-13 

3-15 A Supported Single Lap Configuration 3-13 

3-16 A Single Lap Joint With Alignment Tabs 3-14 

3-17 Finite Element Models for ASTM D 1002-99 Single Lap Joint 3-14 

3-18 Shear Stress Contours of ASTM D 1002-99 Single Lap Joint 3-15 

3-19 Peel Stress Contours of ASTM D 1002-99 Single Lap Joint 3-15 

3-20 Shear and Peel Stress Distribution Along Centerline of the Adhesive for Single 

Lap Joint 3-16 

3-21 Finite Element Models for the Aluminum Double Lap Joint 3-17 

3-22 Shear Stress Contours of the Aluminum Double Lap Joint 3-17 

vii
3-23 Peel Stress Contours of the Aluminum Double Lap Joint 3-18 

3-24 Shear and Peel Stress Distribution Along Centerline of the Adhesive for the 

Aluminum Double Lap Joint 3-18 

3-25 Stress Singularity Points in a Single Lap Joint 3-20 

3-26 A 0.01-inch Titanium Single Lap Joint With a 45° Fillet 3-22 

3-27 Peel Stress Contour of 0.01-inch Titanium Single Lap Joint With a 45° Fillet 3-23 

3-28 Shear Stress Contour of 0.01-inch Titanium Single Lap Joint With a 45° Fillet 3-23 

3-29 A Lap Joint Configuration With Constant Shear Stress in the Adhesive Layer 3-24 

3-30 A Single Lap Joint With Variance in the Direction of Overlap Length 3-25 

3-31 Shear Stress Distribution Along the Centerline of the Adhesive 3-25 

3-32 True Stress-True Strain Curve for Adhesive DP460 3-26 

3-33 Deformed Shape of the Single Lap Joint of 0.01 inch Thickness 3-27 

3-34 Von Mises Stress Contour of a 0.01-inch-Thick Joint at P = 2.5 lb (τ = 5 psi) 3-27 

3-35 Von Mises Stress Contours of a 0.01-inch-Thick Joint at P = 908 lb 

(τ = 1816 psi) 3-28 

3-36 Von Mises Stress Contours of 0.01-inch-Thick Joint at P = 1545 lb 

(τ = 3090 psi) 3-28 

3-37 Von Mises Stress Contours of a 0.01-inch-Thick Joint at P = 1870 lb 

(τ = 3740 psi) 3-29 

3-38 Von Mises Stress Contours of Variable Bondline Thickness Joint at P = 22 lb 

(τ = 44 psi) 3-29 

3-39 Von Mises Stress Contours of Variable Bondline Thickness Joint at P = 1122 lb 

(τ = 2244 psi) 3-30 

3-40 Von Mises Stress Contours of Variable Bondline Thickness Joint at P = 1865 lb 

(τ = 3730 psi) 3-30 

4-1 Experimental Results for Single Lap Joint Tests 4-1 

4-2 Experimental Results for Coupons With Adhesive Brushed Into Bond Area 4-2 

4-3 Experimental Results for Titanium Single Lap Joint Tests of Brushed and 

Nonbrushed Coupons 4-2

viii
4-4 Experimental Results for Titanium Single Lap Joint Tests, Comparing Ambient 

Temperature vs Low Temperature Exposure 4-3 

4-5 Experimental Results for Titanium Single Lap Joint Tests, Comparing Ambient 

Condition vs High Humidity Exposure 4-3 

4-6 Experimental Results for Titanium Single Lap Joint Tests, Comparing Ambient 

Condition vs Water Soaked Exposure 4-4 

4-7 Experimental Results for Titanium Single Lap Joint Tests, Comparing Ambient 

Condition vs Humidity Exposure Prior to Bonding 4-4 

4-8 Experimental Results for Titanium Single Lap Joint Tests, Comparing Acetone 

Blotting vs Dry Nitrogen Rinsing as Pretreatment 4-5 

4-9 Experimental Results for Single Lap Joints With Composite Adherends 4-5 

4-10 Experimental Data for Double Lap Coupons 4-6 

4-11 Bond Thickness Variation Along the Lap Length and Across the Lap Width 4-6 

4-12 Experimental Results for Titanium Single Lap Joint Tests, Effect of Linear 

Variations in Bond Thickness 4-8 

LIST OF TABLES 

Tables Page 

3-1 Peak Peel and Shear Stress Comparison (for adhesive Ea = 0.6 msi, νa = 0.4) 3-19 

3-2 Peak Peel and Shear Stress Comparison (for adhesive Ea = 0.312 msi, νa = 0.4) 3-19 

3-3 Stresses Along Centerline of the Adhesive for Titanium Single Lap Joint, psi 3-21 

3-4 Stresses at the Singularity Point 1 for Titanium Single Lap Joint, psi 3-21 

3-5 Stresses at Point A for Titanium Single Lap Joint, psi 3-21 

3-6 Failure Load Prediction for Titanium Single Lap Joint, lb 3-27 

4-1 Geometry Data of all the Bondline Cases Shown in Figure 4-12 4-7 

ix/x
EXECUTIVE SUMMARY

Adhesively bonded joints under tensile lap shear loading were analyzed using the finite element 

method and closed-form solutions. Predictions of the stress distribution and failure prediction 

are compared with experimental failure load data. 

Case studies were performed that addressed the finite element meshing strategies of adhesively 

bonded joints such as h- and p- methods, mesh density around the overlap regions, element 

types. Comparisons were made with available closed-form solutions. 

Titanium single lap joints were analyzed using a linear analysis and the effects of bondline 

thickness and fillet were investigated. Parametric studies showed that the maxima strength of the 

adhesively bonded single lap joint increased with decreasing adhesive thickness. The proposed 

use of varying adhesive layer thickness to reduce the stress singularity was investigated by 

profiling the adherends’ thickness quadratically or linearly to reduce or eliminate the shear stress 

concentration at the ends. Preliminary analysis on the effect of variable thickness along the 

overlap direction showed that maximum stress occurred at the end of least thickness. Further 

analysis on profiling the adherend thickness to reduce the stresses at the ends is recommended. 

To support and validate the analysis, single lap joints were tested with uniform and variable 

bondline thickness. The specimen used titanium adherends and 3M’s DP460 adhesive. 

Finally, nonlinear analyses of titanium single lap joints, taking into consideration the ductility of 

the adhesive were performed, and the predicted failure load was shown to be 10% less than the 

test failure load. The cause of this lower prediction is discussed, and future work is suggested. 

xi/xii
1. INTRODUCTION AND FOCUS OF RESEARCH. 

Adhesively bonded joints are widely used in the aerospace and automotive industries; evidence 

of interface failures have been observed in many cases. However, the failure mechanism is not 

well understood and considerable effort has been devoted to testing, theoretical prediction, and 

numerical analysis to effectively address this issue. 

Adhesively bonded joints can provide an efficient method of joining that would be more 

extensively used if reliable methods of analysis and failure prediction were available. 

Improvements in the technology that controls bonding fabrication, such as reliable 

nondestructive inspection, would also advance the field. In this report, the focus is placed on 

methods of analysis and failure prediction and the effects of variations in the thickness of the 

adhesive bondline. 

Well-established closed-form expressions, e.g., Volkersen [1] and Goland and Reissner [2], and 

adaptations thereof, were first used to obtain stress distributions in the adhesive bondlines of 

conventional single- and double lap configurations. The assumptions upon which such methods 

are based were reviewed, and predictions were evaluated by comparisons with linear finite 

element analyses that were developed for the identical configuration. 

One recognized difficulty associated with strength prediction for joints involving variations in 

bondline thickness is addressed. Subsequently, both nonlinear adhesive characteristic and 

fracture mechanics methods were considered in an attempt to resolve this difficulty for a 

relatively ductile adhesive system. 

1-1/1-2
2. LITERATURE REVIEW. 

2.1 STRESS ANALYSIS. 

Adhesively bonded joints were intended to transfer load from one adherend to another simply by 

a shearing mechanism alone. However, due to load eccentricities, transversely normal stress 

(also called peel stress) exists in the adherend and adhesive, which could be the major 

contributor that causes failure in composite adherends with typically low interlaminar strength. 

The analytical prediction of the stress distribution in lap joints has been studied by many 

researchers. 

In 1938, Volkersen [1] first proposed a simple shear lag model for mechanical joints with many 

fasteners, and later on, this model was adopted for adhesively bonded lap joints with the 

assumption that the adherends are in tension and the adhesive is in shear only and both stresses 

are constant across the thickness. However, the Volkersen solution does not reflect the effects of 

the adherend bending and shear deformations, which are potentially significant for composite 

adherends with a low shear and transverse moduli and strength. In 1944, Goland and Reissner 

[2] (hereafter referred to as the GR solution) took into consideration the effects of the adherend 

bending and the peel stress, as well as the shear stress, in the adhesive layer in a single lap joint. 

Subsequent efforts by Oplinger [3] suggested the corrections to the GR solution by using a 

layered beam theory instead of the classical homogeneous beam model for single lap joints. 

The corrections to the shear lag model, or Volkersen solution, include works by Hart-Smith [4 

and 5] Tsai, Oplinger, and Morton [6]. Hart-Smith [4 and 5] modified the shear lag model to 

include the adhesive plasticity. Tsai, Oplinger, and Morton [6] (hereafter referred to as the TOM 

solution) provided a correction to the shear lag model with the assumption that the shear stress is 

linear through the adherends. All the models predict that maximum shear stress occurs at the 

free ends, which is not strictly possible for a free surface condition. 

In many cases, an exact solution is not available due to the complex nature of the stress state, 

complicated geometry, and dissimilar bonded materials. Assumptions can be made to simplify 

the problem, such as neglecting the stress variation across the adhesive thickness and linear 

variation in shear stress through adherends. 

With the development of modern computational tools, some numerical methods, e.g., the finite 

difference method used by H. Kim and K. Kedward [7] and the finite element method used by 

many researchers, e.g., Penado and Dropek [8] and Tessler, Dambach, and Oplinger [9], have 

been used to analyze adhesively bonded joints. Compared with finite difference methods, finite 

element methods can be more conveniently applied to almost any geometric shape under loading 

and is more versatile for the numerical simulation of adhesively bonded joints in general. Due to 

the relatively thin adhesive layer, finite element modeling, considering the element type and the 

mesh density used, has to be carefully designed to obtain a reasonable solution. Penado and 

Dropek [8] indicated several important issues to consider when applying finite element analysis 

(FEA). Tessler, Dambach, and Oplinger [9] presented an adaptive mesh refinement method in 

bonded joints and showed close approximation of the traction free conditions. Another 

significant advantage of FEA is the ability to account for the stress variation through the 

2-1
thickness. In addition, the geometrical and material nonlinearities can be included, providing 

that the discretization, boundary conditions, and the loading are applied properly. 

2.2 FAILURE PREDICTIONS. 

Stress analysis is an important part of the design process. It is also important to predict fracture 

load and thereby enable safe design concepts. Inevitably, analysis and prediction in conjunction 

with carefully selected experimental work is necessary. An appropriate failure mechanism 

should be suggested to guide and better understand the design of adhesively bonded joints. From 

the literature, there are basically two approaches for predicting the strength of joints: strength of 

materials and fracture mechanics. 

Strength of materials approaches typically suggest that when certain stress and strain 

components, or an equivalent stress component (e.g., Von Mises equivalent stress), surpass its 

allowable counterpart, dependent on the material property, failure will occur. 

Using the strength of materials approach, closed-form solutions predict that adhesive joints show 

improved strength with increasing bondline thickness, this is observed by using average stresses 

along the centerline from linear FEA. However, this is contrary to experimental results that 

strength decreases with increasing bondline thickness for thick bondline (Tomblin, et al. [10]). 

Gleich, Tooren and Beukers [11] first pointed out that this apparent contradiction emanates from 

the wrong interpretation of the stresses from the FEA. The peak shear and peel stress occurring 

at the interface of the bonded region increase with increasing adhesive thickness, and average 

adhesive stresses are shown to decrease with increasing bondline thickness. Harris and Adams 

[12] used a nonlinear finite element technique to predict the mode of failure and failure load for 

several single lap joints with aluminum alloy adherends and found that different criteria are 

needed for different adhesive systems. 

As for the fracture mechanics approach, researchers have investigated the use of a generalized 

stress-intensity factor, analogous to the stress-intensity factor in classical fracture mechanics, to 

predict fracture initiation for bonded joints. Sometimes this stress singularity approach is 

referred as the fracture mechanics approach with no initial crack. The intent of this approach is 

to successfully implement a predictive design tool. 

Stress singularity order (some authors call this strength of the singularity) at bimaterial interfaces 

has been studied by many authors [13-19]. Based on numerical and elasticity solutions, Penado 

[15] studied the stress singularity orders at bimaterial interfaces and concluded that the direction 

of crack propagation in lap joints with fillets was different from the cases without fillets based on 

a maximum stress failure criterion. Groth [17] suggested a fracture initiation criterion at the 

interface corners for bonded structure; it was assumed that initiation of fracture occurs when the 

generalized stress-intensity factor reaches its critical value. Akisanya and Meng [19] 

investigated the validity and limitations of the fracture initiation criterion in a butt joint with a 

thin elastic-plastic adhesive layer between two elastic adherends. The plastic zone size was 

compared with the extent of the stress singularity to determine the condition for the fracture 

initiation criterion to be valid. 

2-2
2.3 THE EFFECT OF VARIATIONS IN BONDLINE THICKNESS. 

Using the fracture mechanics approach, it is implied that reducing the singularity factor would 

improve the strength of the joints. For example, Adams, et al. [20] proposed the use of varying 

adhesive layer thickness to reduce the stress singularity. This could be realized by profiling the 

adherends thickness quadratically or linearly to reduce or eliminate the shear stress concentration 

at the ends. Another way to improve the strength of the joints could be done by forming an 

adhesive fillet region around the free end of the adhesive-adherend interfaces, this effect was 

studied by Apalak and Davies [21], Adam and Harris [22], and Tsai and Morton [23]. It was 

found that by rounding the corner of the adhesive and the adherends, the strength of the joint was 

improved. Tsai and Morton [23] also stressed that apart from the fillet effect, nonlinear 

deformation also plays a part in the adhesive stress concentration. 

In this report, the finite element modeling strategy is studied in detail in section 3. This strategy 

is applied to the linear analyses of aluminum and titanium lap joints with some parametric 

studies including the effects of bondline thickness and the fillet effect. Preliminary work on the 

effects of variable bondline thickness was also studied. Nonlinear analyses were performed for 

two different adhesive thicknesses (0.01 and 0.03 inch). The ANSYS general purpose finite 

element software was used to analyze adhesively bonded titanium joints with adhesive thickness 

varying from 0.0085 to 0.0225 inch. The results are compared with the experimental 

observations, and future work on the prediction of joint failure is suggested. 

2-3/2-4
3. ANALYSIS APPROACHES AND CASE STUDIES. 

3.1 STRESS DISTRIBUTION IN DOUBLE LAP JOINT CONFIGURATION. 

3.1.1 Volkersen. 

Volkersen [1] first proposed a simple shear lag model for load transfer from one adherend to 

another by a simple shearing mechanism alone. In the model, the adherends are assumed in 

tension and the adhesive is in shear only, and both are constant across the thickness. The 

important relationships are given by 

⎥

⎦

⎤ ⎢

⎣

⎡

+

− = − )cosh(

)sinh(

2

2

)sinh(

)cosh(

4 )(c

x

tEtE

tEtE

c

P x

x

ii oo

ii oo

λ

λ

λ

λ λ

τ (3-1) 

where 

) 21 ( 2

a oo ii

a

tEtEt

G λ = +

In these expressions, the subscripts o, i, and a denote the respective components relative to the 

outer adherend, inner adherend, and adhesive. The parameter P denotes the loading applied at 

the end of the inner adherend, and the parameter c is half the length of the adhesive. The origin 

of the x coordinate is in the middle of the adhesive, see figure 3-1. 

However, the Volkersen solution does not reflect the effect of the adherend bending and shear 

deformations, which are potentially significant for composite adherends with a low shear and 

transverse moduli and strength. The TOM solution [6] provided a correction to the shear lag 

model with the assumption that the shear stress is linear through the adherends. As a result, the 

λ2 is replaced by 

1

22

36

1

−

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+= +

o

o

i

i

a

a

G

t

G

t

t

G λβ (3-2) 

3.1.2 Linear Finite Element Analysis. 

In this section, a typical balanced double lap joint will be used for the case study. The overlap 

length is 1 inch, the thickness of the outer adherend, inner adherend, and adhesive are 0.05, 0.05 

and 0.005 inch, respectively. Geometrical details are indicated in figure 3-1. The linear-elastic 

material properties are as follows: 

Epoxy Adhesive: Young’s modulus Ea = 0.6 msi, Poisson’s ratio νa = 0.429 

Aluminum Adherend: E = 10.4 msi, ν = 0.33 

3-1
FIGURE 3-1. A TYPICAL DOUBLE LAP JOINT 

The theory of stress analysis using the finite element method is well described in many finite 

element texts, e.g., Zienkiewicz and Taylor [24], and it will not be repeated here. 

For this double lap joint, several different modeling schemes with regard to the element type and 

mesh techniques, assuming plane strain conditions, are discussed below. 

3.1.2.1 h-Method. 

Due to the symmetry of the loads and structure, only one-half of the joint was considered. Eightnode isoparametric elements were used to discretize the whole joint; figure 3-2 shows the finite 

element model where the eight-node isoparametric element (ANSYS plane82) is used. The mesh 

density is biased with a ratio of 12 near the ends of the adhesive due to the high stress 

concentration in that region. Details of this local region are shown in figure 3-3. There are eight 

elements and five elements through the thickness of adherends and adhesive, respectively. There 

are a total of 3,406 elements and 10,625 nodes, the number of degrees of freedom is 20,900. The 

mesh used here is considered to be suitably refined for the present evaluation. 

3-2
FIGURE 3-2. FINITE ELEMENT MODEL USING PLANE82 

FIGURE 3-3. DETAIL MESHES AROUND THE TERMINATION REGION 

OF THE ADHESIVE 

In ANSYS, the following loads and constraints are applied. 

Displacement: v = 0 (along the bottom edge)

 u = 0 (along the left edge)

Pressure: p = -1000 psi (at the right edge) 

3-3
Analysis results show that adhesive peel stresses are the largest among all the stress components, 

and the shear and peel stresses of the adhesive are the most important components. Figures 3-4 

and 3-5 portray the total peel and shear stress distribution, whereas figure 3-6 illustrates the 

stresses along the adhesive centerline. Shear stress contours for the local adhesive termination 

region using plane82 are shown in figure 3-7. These graphs show that maximum peel and shear 

stress along the centerline occur at a very small distance from the free edge. Due to the stress 

singularity, peak stresses occur at the joint corners, shown in figures 3-4 and 3-5. For the current 

meshing density, maximum peel stress and shear stress are 1.350 and 0.594 times the applied 

axial stress, respectively. 

FIGURE 3-4. PEEL STRESS CONTOUR USING PLANE82 

FIGURE 3-5. SHEAR STRESS CONTOUR USING PLANE82 

3-4
FIGURE 3-6. SHEAR AND PEEL STRESS DISTRIBUTION ALONG THE ADHESIVE 

CENTERLINE USING PLANE82 

FIGURE 3-7. SHEAR STRESS CONTOURS (LOCAL ADHESIVE TERMINATION 

REGION) USING PLANE82 

3-5
3.1.2.2 p-Method. 

The p-method, in conjunction with the eight-node elements (plane145), is used in this section. 

The total number of the elements and nodes are the same as in the h-method. Local convergence 

criteria at specified locations in the model are used, and it is defined as the tolerance for 

convergence specifications as 1%, based on shear stress τxy at a distance of 0.0015 inch from the 

free edge along the adhesive centerline (point A in figure 3-8). 

Peak stresses occurs at the joint corners, and peak peel stress and shear stress are 2.592 and 1.028 

of the applied axial stress. The stress distribution along the adhesive centerline is shown in 

figure 3-9. Detail shear stress contours around the adhesive termination are shown in figure 3-8, 

where peak shear stress occurs at the left corner of the adhesive. Compared with conventional hmethod, the advantage of using p-method includes the ability of adaptive refinement to obtain 

good results to a required accuracy, and the error estimate can be made locally and globally. The 

p-level, which refers to the polynomial level used at the local point A, is shown in figure 3-10. 

A

Only the adhesive layer is shown here. 

FIGURE 3-8. SHEAR STRESS CONTOURS (LOCAL ADHESIVE TERMINATION 

REGION) USING THE p-METHOD 

3-6
FIGURE 3-9. PEEL AND SHEAR STRESS DISTRIBUTION ALONG THE ADHESIVE 

CENTERLINE USING THE p-METHOD 

FIGURE 3-10. p-LEVEL USED AT POINT A (SHEAR STRESS VS POLYNOMIAL LEVEL) 

3-7
3.1.2.3 Mixed Element Modeling Using Spring Elements for Adhesive and Eight-Node 

Elements for Adherends. 

According to Loss and Kedward [25], the adhesive can be modeled by a pair of springs. In 

ANSYS, this can be realized by using eight-node elements plane82 for the adherend and a couple 

of spring elements for modeling the shearing and peeling behavior of the adhesive. The stiffness 

coefficients of the peel and shear springs can be calculated by equations 3-3 and 3-4: 

( ) el a eff

peel

a

A E

k

t = (3-3) 

el a

shear

a

A G k

t = (3-4) 

where 

Ae l = ×l b

Ea and na are adhesive Young’s modulus and Poisson’s ratio 

l, b, ta, Ga, (Ea)eff are in-plane distance between nodes, width of the joint being modeled, 

adhesive thickness, elastic adhesive shear modulus, and effective elastic modulus of 

adhesive, respectively. For a triaxial stress state, (Ea)eff is defined below. 

2

(1 ) ( ) (1 2 )

a a

a eff

a a

E E ν

ν ν

− = − −

For uniform meshing along the overlap direction of the adhesive, stiffness coefficients for the 

peel and shear springs at the very ends of the overlap region is one-half the stiffness of other 

springs due to representation of one-half the element area. 

3.1.3 Correlation Studies of Predicted Stress Distributions. 

3.1.3.1 Comparison With Available Closed-Form Solution. 

Figure 3-11 (a)-(f) shows the stress comparison of the spring element and plane eight-node 

element results along the centerline of the adhesive with available closed-form solutions. 

The classical Volkersen solution overestimates the maximum shear stress, and the FEA results 

are closer to the TOM solution (shown in figure 3-11 (d)-(f)). This is because Volkersen 

assumed a one-dimensional model with only shear deformation in the adhesive layer, the effect 

of the adherend shear being ignored. The TOM solution corrected for adherend shear 

deformation by approximating a linear shear stress through the adherends and predicts that 

maximum shear stress of 272.6 psi occurs at the ends of the lap joint overlap. However, the 

maximum shear stress by using eight-node elements occurs at a small distance from the free 

edge. Near the left edge, the maximum shear stress is 245.8 psi, and near the right edge, it is 

3-8
202.9 psi. Figure 3-11 (d)-(f) shows the comparison of peel stress in the adhesive by using the 

spring elements and plane82 elements. The peel stress prediction using spring elements is higher 

than that obtained by using plane82 elements, and maximum stress occurs at the free edge 

instead of at a small distance away from the free edge. Near the left end, the peel stress is in 

tension and near the right end, the peel stress is in compression. 

(a) (d) 

(b) (e) 

(c) (f) 

FIGURE 3-11. COMPARISONS OF SHEAR AND PEEL STRESSES ALONG THE 

CENTERLINE OF THE ADHESIVE 

3-9
As far as the element type used, for the same meshing density, a high-order element is more 

accurate than lower-order element. For the eight-node element analysis, the stress varies within 

the element and the free surface condition is satisfied. For the mixed element modeling, the 

results are reasonable approximations noting that this is only a relatively coarse mesh (502 spring 

elements). A finer mesh will involve more manual work in element generation, and it is not 

convenient to implement. But this mixed element modeling could be very useful for evaluating 

closed-form predictions when the lap joint is simplified as a beam on an elastic foundation 

considering axial and transverse effects, and it will help understand the mechanism of the 

adhesive deformation. Furthermore, the idea of using spring elements to model the adhesive 

behavior can also be used in the cohesive zone models for investigating interface fracture [26-

28]. 

3.1.3.2 Comparison Along Three Different Paths in the Adhesive Layer. 

Figure 3-12 shows three different paths in the adhesive layer, where AB, CD, and EF are 

interface between the outer adherend and the adhesive, centerline of the adhesive, and interface 

between the inner adherend and the adhesive, respectively. Figure 3-13 (a)-(f) shows the stress 

comparison along these paths. Considerable stress variation exists near bondline terminations, 

where points A, B, C, and D are stress singularity points due to material and geometrical 

discontinuity. Apart from these regions, stresses can be considered to be similar along these 

three different paths. 

FIGURE 3-12. THREE DIFFERENT PATHS IN THE ADHESIVE LAYER 

(AB, CD, AND EF) 

3-10
(a) (d) 

(b) (e) 

(c) (f) 

FIGURE 3-13. COMPARISONS OF SHEAR AND PEEL STRESSES ALONG THREE 

DIFFERENT PATHS (AB, CD, AND EF SHOWN IN FIGURE 3-12) 

3-11
3.2 STRESS DISTRIBUTION IN SINGLE LAP JOINT CONFIGURATION. 

3.2.1 Goland and Reissner. 

The GR [2] solution is the most well known classical treatment of single lap joints. This model 

assumes that the transverse normal strain and shear strain in the adherends are negligibly small 

compared with those strains in the adhesive layer. The deformation of the adherends is due to 

the adherend bending effect, whereas the adhesive layer is analogous to a system of shear and 

peel springs positioned between the two adherends. 

The following equations are predictions for shear and peel stress in single lap joints with 

relatively flexible adhesive layers: 

]sinsinh)sinsinh 2 (coscosh)coscosh 2 [( 8 )(

)]1(3

2

sinh

2

)2( cosh

)31( 2[ 4 )(

2 ' ''

1

2 ' ''

2 2

2

σ λλλλλλλλλλλλ

β

β

β τ

k k k R k R

l

pt x

k

t

l

t

lx

k

t

l

l

pt x

+ ++

Δ

− =

−+

−

+ − =

 (3-5) 

where 

λλλλ λλλλ

ν

ν

λλ

λ λλ

ν

β

ν

α

cossinh cossinh,sincosh sincosh

)1(3

2 ',

2

)1(3

2

tanh221

1

2sin2sinh

2 , 6

2

1(

4 , 1(

1 2

2

2

' 4

)

2

)

2

= + = −

− =

− +

=

=Δ +

− = =

+ = + =

R R

E

p

t

kl k

E

p

t

l

k

l

lx

Et

tE

t

l

Et

tE

Ett

E

a

a

aa

a

aa

a

3.2.2 Linear Finite Element Analysis With Aluminum Adherends (ASTM D 1002-99). 

A single lap joint (ASTM D 1002-99) is considered (shown in figure 3-14). The lap length is 

0.5 inch, the thickness of the outer adherend, inner adherend, and adhesive are 0.064, 0.064, and 

0.01 inch, respectively. The grip region is 1 inch at each end, and the total length of the joint is 

7.5 inches. 

The material for the adherend is aluminum, with Young’s modulus E = 10.4 msi, and Poisson’s 

ratio ν = 0.33. The material properties of the adhesive are Ea = 0.6 msi and νa = 0.4. 

Testing of single lap joints shows that the joints could be loaded by in-line axial tensile grip 

loading or by a supported single lap configuration, as shown in figure 3-15. In FEA analysis, for 

in-line axial tensile grip loading, there are two loading steps. The first loading case is called 

3-12
alignment which consists of a uniform displacement 0.074 inch along the transverse direction in 

one grip region. The second loading case is called tensile loading, which is loading of a uniform 

stress or displacement; for this linear elastic case, a uniform tensile stress of 1000 psi is applied. 

Since this is a linear elastic analysis, these two load cases are applied simultaneously. For 

supported single lap configuration or specimen with alignment tabs (figure 3-16), only the 

second load step is needed. 

FIGURE 3-14. A SINGLE LAP JOINT (ASTM D 1002-99) 

compression spring to support

weight of lower grip against pin

hand tighten

grip with wrench

pin grip to

test machine

FIGURE 3-15. A SUPPORTED SINGLE LAP CONFIGURATION 

3-13
FIGURE 3-16. A SINGLE LAP JOINT WITH ALIGNMENT TABS (FROM ADAM, 

COMYN, AND WAKE [29]) 

Stress analyses of aluminum and composite single lap joints show that stresses are a few percent 

larger for in-line axial tensile grip loading than for the supported single lap configuration 

loading. In the interest of conservatism, all the following analyses of single lap joints are loaded 

by in-line axial tensile grip. 

Using the strategy in section 3.1 of the FEA modeling and analysis, the following finite element 

model for the single lap joint (figure 3-17) is used, and the results show center symmetric 

characteristic. Among all the stress components, peel stress has the highest value (shown in 

figures 3-18 and figure 3-19). Figure 3-20 shows the shear and peel stress distribution along the 

centerline of the adhesive. It can be seen that with a sufficiently fine mesh, the free surface 

condition is satisfied. 

FIGURE 3-17. FINITE ELEMENT MODELS FOR ASTM D 1002-99 SINGLE LAP JOINT 

3-14
FIGURE 3-18. SHEAR STRESS CONTOURS OF ASTM D 1002-99 SINGLE LAP JOINT 

FIGURE 3-19. PEEL STRESS CONTOURS OF ASTM D 1002-99 SINGLE LAP JOINT 

3-15
FIGURE 3-20. SHEAR AND PEEL STRESS DISTRIBUTION ALONG CENTERLINE OF 

THE ADHESIVE FOR SINGLE LAP JOINT 

3.2.3 Comparisons Between Single Lap and Double Lap Predictions. 

For comparison, a double lap joint with the same dimensions and materials, as specified in 

ASTM D 1002-99, was also analyzed. For this double lap joint, there was no grip alignment 

required, and only a tensile load of 1000 psi was applied at the end. There was also a change in 

boundary condition because of the symmetric constraints that are characteristic of a double lap 

joint (figure 3-21). Again, shear stress contours, peel stress contours and shear and peel stress 

distribution along centerline of the adhesive are shown in figures 3-22 to 3-24, respectively. The 

shear and peel stress along the centerline of the adhesive for double lap joint are less than half of 

those for single lap joint under the same uniform tensile loading. Also, note that there is no 

symmetry in the stress distribution along the adhesive centerline for a double lap joint, as shown 

in the case for a single lap joint. Comparison of FEA and available closed-form solutions are 

listed in tables 3-1 and 3-2 for two different sets of adhesive properties. 

3-16
FIGURE 3-21. FINITE ELEMENT MODELS FOR THE ALUMINUM DOUBLE LAP JOINT 

(Same dimensions and material as specified in ASTM D 1002-99 single lap joint.) 

FIGURE 3-22. SHEAR STRESS CONTOURS OF THE ALUMINUM DOUBLE LAP JOINT 

3-17
FIGURE 3-23. PEEL STRESS CONTOURS OF THE ALUMINUM DOUBLE LAP JOINT 

FIGURE 3-24. SHEAR AND PEEL STRESS DISTRIBUTION ALONG CENTERLINE OF 

THE ADHESIVE FOR THE ALUMINUM DOUBLE LAP JOINT 

3-18
TABLE 3-1. PEAK PEEL AND SHEAR STRESS COMPARISON 

(FOR ADHESIVE Ea = 0.6 MSI, νa = 0.4) 

FEA GR Solution Volkersen 

TOM 

Solution 

Peel Shear Peel Shear Shear Shear 

Single lap joint 749 498.8 678.8 481.7 NA NA 

Double lap joint 206.6 217.9 NA* NA 266 244 

* Notation: NA means not applicable or not available 

 Unit of measure = psi 

TABLE 3-2. PEAK PEEL AND SHEAR STRESS COMPARISON 

(FOR ADHESIVE Ea = 0.312 MSI, νa = 0.4) 

FEA GR Solution Volkersen 

TOM 

Solution 

Peel Shear Peel Shear Shear Shear 

Single lap joint 576.7 378.3 491 352.2 NA NA 

Double lap joint 154.6 184.2 NA NA 206.9 199 

* Notation: NA means not applicable or not available 

 Unit of measure = psi 

The average shear stress τ (defined as the applied load over the bonded area) is the same for the 

double lap and single lap configurations. In other words, the load supported by single lap joint is 

one-half that supported by the double lap joint. However, the peak stresses are higher for a 

single lap joint than that for a double lap joint, as shown in tables 3-1 and 3-2. 

3.3 LINEAR AND NONLINEAR ANALYSIS OF TITANIUM SINGLE LAP JOINT. 

A single lap joint with Titanium adherends with both linear and nonlinear adhesive properties is 

now considered. The adherend material is changed to titanium and the thickness is 0.035 inch 

(refer to figure 3-14). 

The properties of the titanium adherends are: Young’s modulus E = 16.8 msi, and Poisson’s 

ratio ν = 0.34. The material for the adhesive is DP460, with Ea = 0.312 msi and νa = 0.4. The 

engineering stress-strain curve was supplied by A. Pocius from the 3M company [30]. 

3.3.1 Linear Analysis. 

Stress singularity occurs at joint corners, and the peel and shear stresses in the adhesive around 

the joint corners can be expressed as 

 (3-6) λ σ − = ijij rQ

where σ ij are the stresses, are the generalized stress-intensity factors, r is the distance from 

the point of singularity, and λ is the order of the stress singularity. 

Qij

3-19
According to Bogy [13], the stress singularity orders (termed by some authors as strength of the 

singularity) for the titanium single lap joint corners can be obtained by solving for a complex 

equation. It turns out to be λ1=0.356 and λ2=0.329 for points 1 and 2 (shown in figure 3-25), 

respectively. The stronger singularity order occurs at the two points (the left lower corner and 

the right upper corner of the interface between the adherends and the adhesive, shown in 

figure 3-25), and these two points are believed to be the place where fracture is most likely to 

initiate. 

FIGURE 3-25. STRESS SINGULARITY POINTS IN A SINGLE LAP JOINT 

3.3.1.1 Bondline Thickness Effect. 

For linear analysis, p-method meshing strategy was used, and the convergence criterion was 

defined as 2% based on shear stress τxy at point A (0.0006 inch to the right of point 1 in 

figure 3-25). Some key results on the adhesive thickness effects are shown in tables 3-3 to 3-5. 

• Along the centerline of the adhesive, maximum Sy, Sxy, S1 (maximum principal stress) 

and SEQV (Von Mises equivalent stress) are shown to decrease with increasing adhesive 

thickness (table 3-3). 

• At the singularity point 1, Sy, Sxy, S1, and Von Mises equivalent stress increase with 

increasing adhesive thickness (table 3-4). 

• At point A (figure 3-25), Sy, Sxy, S1, and Von Mises equivalent stress increase with 

increasing adhesive thickness (table 3-5). 

3-20
TABLE 3-3. STRESSES ALONG CENTERLINE OF THE ADHESIVE FOR TITANIUM 

SINGLE LAP JOINT, psi 

Thickness Sx Sy Sz Sxy S1 S2 S3 SEQV 

0.01 158.8 348.5 182.33 231.87 483.32 182.33 -2.735 451.3 

0.02 102.27 266.20 127.09 187.35 365.4 127.09 -0.317 359.33 

0.03 81.76 240.74 107.88 172.13 322.13 107.88 -0.1805 326.17 

0.04 71.129 227.62 97.897 163.23 296.49 97.897 0.001 305.88 

TABLE 3-4. STRESSES AT THE SINGULARITY POINT 1 FOR TITANIUM 

SINGLE LAP JOINT, psi 

Thickness Sx Sy Sz Sxy S1 S2 S3 SEQV 

0.01 2562.8 3698.7 2504.6 1278.5 4529.7 2504.6 1731.8 2502.7 

0.02 2830.1 4076.2 2762.5 1417.0 5001.1 2762.5 1905.2 2768.6 

0.03 3173.5 4564.8 3095.3 1590.8 5605.4 3095.3 2132.9 3105.3 

0.04 3492.6 5021.5 3405.7 1751.4 6168.1 3405.7 2346.1 3417.7 

TABLE 3-5. STRESSES AT POINT A FOR TITANIUM SINGLE LAP JOINT, psi 

Thickness Sx Sy Sz Sxy S1 S2 S3 SEQV 

0.01 705.63 914.2 647.93 331.57 1157.5 647.93 462.33 623.45 

0.02 796.27 1027.3 729.43 355.34 1285.4 729.43 538.15 672.37 

0.03 899.86 1157 822.61 398.26 1447 822.61 609.82 753.38 

0.04 994.08 1276.1 908.08 439.37 1596.5 908.08 673.65 830.86 

The adhesive used here is an isotropic material. If its ductility is neglected and the maximum 

equivalent Von Mises stress along the centerline of the adhesive is used, it yields the predicted 

strength of 506 and 700 lb for 0.01 and 0.03 inch thickness, respectively. The conclusion could 

be drawn that the thicker the adhesive, the stronger the joint will be. 

If the equivalent Von Mises stress at point A is used, the predicted strength will be 366 lb for 

0.01 inch thickness and 303 lb for 0.03 inch thickness. This suggests that the thicker the 

adhesive, the weaker the joint will be. It is noteworthy that the stress predicted at the centerline 

indicates an increasing stress with decreasing bondline thickness. 

Based on the application of fracture mechanics, e.g., the fracture initiation criterion proposed by 

Groth [17], the strength of the adhesively bonded joint can be evaluated quantitatively. Note that 

the stress singularity order is independent of the global geometry or load, and it only depends on 

the local geometry and material properties. Therefore, the stress singularity order for the joint 

with a 0.01-inch adhesive thickness is the same as that for a 0.03 inch thickness, that is 

λ = λ )03.0()01.0( . 

3-21
From table 3-4, )03.0()01.0( σ ij < σ ij

Using equation 3-6, the following can be obtained 

 ) 03.0()01.0( Qij < Qij (3-7) 

where the 0.01 and 0.03 inside the parentheses denote the adhesive thickness of the 

corresponding components. 

Using equation 3-7, for the same critical generalized stress-intensity factor Qcrit, the thicker 

adhesive layer of 0.03 inch thickness would be weaker. When considering ductile adhesive 

systems, the stress singularity extent as well as the plastic zone size needs to be investigated. 

3.3.1.2 Influence of the Adhesive Fillet. 

In an actual joint, there is always some form of fillet existing. To study the effect on the strength 

of the joints, a titanium single lap joint with a 0.01-inch adhesive thickness and 45° fillet (0.01 

by 0.01 inch) is considered (shown in figure 3-26), and the same mesh density along the overlap 

is used as the case without any fillets. The peak peel and shear stresses occur at point 2, which 

are 671.9 and 352.5 psi, respectively (figures 3-27 and 3-28, where only the adhesive is shown). 

While for the case without fillet, the peak stresses are 3698.7 and 1278.5 psi. 

FIGURE 3-26. A 0.01-inch TITANIUM SINGLE LAP JOINT WITH A 45° FILLET 

3-22
FIGURE 3-27. PEEL STRESS CONTOUR OF 0.01-inch TITANIUM SINGLE LAP JOINT 

WITH A 45° FILLET 

FIGURE 3-28. SHEAR STRESS CONTOUR OF 0.01-inch TITANIUM SINGLE LAP JOINT 

WITH A 45° FILLET 

3-23
3.3.1.3 Variable Bondline Thickness Along Overlap Length and Width. 

In practice, bondline thickness varies due to imperfections in assembly and material tolerances. 

A lap shear joint with constant shear stress due to parabolic adhesive thickness (shown in 

figure 3-29) is derived [31]. The main assumptions are shear stress in the adhesive is constant 

over the thickness, shear deformation in the adherends is neglected, and tensile stress in the 

adherends is uniform over the thickness. 

FIGURE 3-29. A LAP JOINT CONFIGURATION WITH CONSTANT SHEAR STRESS IN 

THE ADHESIVE LAYER 

(Based on constant shear stress through adhesive layer and neglecting singularities) 

The variation in thickness of the adhesive layer along the overlap length is given by: 

2

x

Et

G

t a

a = (3-8) 

where Ga is shear modulus of the adhesive, E is Young’s modulus of the adherends, and t is the 

thickness of the adherends. 

It is impracticable to profile the adhesive or the adherends quadratically, this can be achieved 

approximately by profiling the adherend linearly suggested by Adams, et al. [20]. Some possible 

configurations include scarf joints and linearly tapered adherends. 

To analyze the linear variance effect, a preliminary analysis on the variable adhesive thickness 

from 0.0085 to 0.0255 inch along overlap length is considered shown in figure 3-30. A planestrain condition is assumed in the finite element model, and the peel and shear stress distribution 

along centerline is shown in figure 3-31. It can be seen that maximum stress occurs at the end of 

smallest thickness. Further analysis on profiling the adherend thickness to reduce the stresses at 

the ends could be pursued if this is deemed practicable and controllable. 

3-24
As for the variable thickness along the width direction, a plane-strain model with a uniform 

adhesive thickness would be unacceptable. Therefore, a three-dimensional finite element model 

would be needed. 

FIGURE 3-30. A SINGLE LAP JOINT WITH VARIANCE IN THE DIRECTION OF 

OVERLAP LENGTH 

FIGURE 3-31. SHEAR STRESS DISTRIBUTION ALONG THE CENTERLINE 

OF THE ADHESIVE 

3-25
3.3.2 Nonlinear Analysis. 

For the nonlinear analysis, the tensile true stress-true strain curve in figure 3-32 is used. The 

load is applied gradually until the joint fails unstably due to large area plastic deformation. 

Studies have been made to decide whether load control or displacement control should be used 

for this analysis. It was found that there is only 0.5% difference in the maximum simulation load 

in load control and in displacement control. 

FIGURE 3-32. TRUE STRESS-TRUE STRAIN CURVE FOR ADHESIVE DP460 

Analyses for 0.01- and 0.03-inch-thick adhesives yield maximum failure loads of 1870 and 1856 

lb. Both are less than the test failure load of the titanium single lap joint, which varies between 

2000 to 2200 lb for a range of bondline thickness from 0.01 to 0.035 inch (see table 3-6). 

Obviously, these predictions are much higher than the predictions from linear elastic analysis. 

The postprocessing results were the deformed shape of the single lap joint (figure 3-33, which is 

center symmetric) and the Von Mises stress contours, at the termination region of the adhesive 

under different load levels (figure 3-34 to 3-37). Von Mises stress contours show the initial 

yielding starts at the joint corners (stronger singular point), and develops around the joint ends in 

a certain angle through the thickness of the adhesive until the whole adhesive yields, except a 

very small region around the weaker singularity point and the free edge. 

3-26
TABLE 3-6. FAILURE LOAD PREDICTION FOR TITANIUM SINGLE LAP JOINT, lb 

Thickness Linear Analysis 

(inch) Centerline Point A (Figure 3-25) 

Nonlinear 

Analysis Experiments 

0.01 506 366 1870 

0.03 700 303 1856 

2000-2200 

0.017 (variable thickness from 0.0085 to 0.0255 inch) 1865 2150 

Unit of measure = lb

FIGURE 3-33. DEFORMED SHAPE OF THE SINGLE LAP JOINT OF 

0.01 inch THICKNESS 

FIGURE 3-34. VON MISES STRESS CONTOUR OF A 0.01-inch-THICK JOINT AT 

P = 2.5 lb (τ = 5 psi) 

3-27
FIGURE 3-35. VON MISES STRESS CONTOURS OF A 0.01-inch-THICK JOINT AT 

P = 908 lb (τ = 1816 psi) 

FIGURE 3-36. VON MISES STRESS CONTOURS OF 0.01-inch-THICK JOINT AT 

P = 1545 lb (τ = 3090 psi) 

3-28
FIGURE 3-37. VON MISES STRESS CONTOURS OF A 0.01-inch-THICK JOINT AT 

P = 1870 lb (τ = 3740 psi) 

Nonlinear analysis of variable bondline thickness from 0.0085 to 0.0255 inch (ta = 017.0 inch) 

shows in table 3-6 that the failure load is 1865 lb., while the test failure load is around 2150 lb. 

Figures 3-38 to 3-40 show the Von Mises stress contours at different load, where a center 

symmetric characteristic no longer exists and the yield starts from the left corner. With 

increasing load, the plastic zone increases until most of the adhesive region has yielded. 

FIGURE 3-38. VON MISES STRESS CONTOURS OF VARIABLE BONDLINE 

THICKNESS JOINT AT P = 22 lb (τ = 44 psi) 

3-29
FIGURE 3-39. VON MISES STRESS CONTOURS OF VARIABLE BONDLINE 

THICKNESS JOINT AT P = 1122 lb (τ = 2244 psi) 

FIGURE 3-40. VON MISES STRESS CONTOURS OF VARIABLE BONDLINE 

THICKNESS JOINT AT P = 1865 lb (τ = 3730 psi) 

3-30
Both analysis and test show that the particular variable thickness chosen for this study has no 

effect on the original uniform configuration. 

It was assumed that the joint was well bonded and the adhesive was representative of the 

physical condition, the failure load predicted by nonlinear FEA was lower by ≈ 10% than the test 

failure load. A possible rationale may be one or more of the following: 

• Lower yield strength of the adhesive properties in FEA simulation. Adhesive material 

properties come from the test results of the adhesive bulk material, in which case, the 

possibility of more extensive microdefects increases. From the simple Weibull statistics 

viewpoint, the strength of a small volume of adhesive will be higher. 

• Spew fillet effects. There is no fillet included in the FEA analysis, whereas in the actual 

specimen some form of fillet geometry is typically present. It is expected that with such a 

spew fillet, the maximum predicted load may be slightly higher as noted for the previous 

linear analyses, section 3.3.1.2. 

• A different yield criterion may apply. Typically, a polymer material has a higher yield 

stress in compression than in tension, and unlike the case of the metals, its yielding 

behavior is pressure dependent [12, 29, 32, and 33]. The following modified Von Mises 

yield criterion is considered to be appropriate for modeling adhesive yielding: 

2 3( ) σvm − σ σ σσ yc − = yt p yc yt (3-9) 

Where Von Mises stress: 2 2

12 23 31 vm [( ) ( ) ( ) ]/ 2 2 σ = − +− +− σσ σσ σσ

 

Hydrostatic stress: 123 p =− + + ( ) σ σ σ / 3

and σyc, and σyt are yield stress in compression and in tension, respectively. For the adhesive 

used, the ratio yc

yt

σ

σ is assumed to be 1.3 [12, 29, and 32]. 

Using the Von Mises yield criterion for the single lap joint studied, the left corner of the adhesive 

(point 1 in figure 3-25) yields when the equivalent stress reaches the yield stress of the adhesive. 

However, a modified Von Mises yield criterion (3-9) will not predict yielding until the 

hydrostatic stress reaches a certain level. As a result, the yielding process is delayed. Currently, 

ABAQUS is being used to advance this investigation. 

3.4 APPROACHES FOR FAILURE PREDICTION. 

3.4.1 Maximum Stress/Strain. 

Various methods for failure prediction of bonded joints have been used for many years and are 

typically based on a maximum shear stress or maximum shear strain criteria. The most common 

and simplistic of such methods adopt a linear shear lag idealization, e.g., Volkersen [1]. One 

approach was to use test data based on a given representative single lap or double lap joint 

3-31
configuration for the same adherend and adhesive materials and the same adhesive thickness as 

those intended for the design of a given structural joint. It was also recommended that the same 

surface preparation and processing procedures are adopted for the coupon specimens. From this 

database, the average shear stress at failure was obtained and the implied peak shear stress at the 

extremities of the adhesive bondline was computed from the shear lag model. This value of peak 

shear stress was then used as a design allowable value that was then applied to the actual joint 

design. Hence, a prediction for the average shear stress at failure can be estimated for the 

hardware design. 

Clearly the above simplified methodology represents a first level analysis and is limited 

primarily by the assumed linear shear stress-strain behavior of the adhesive system and the 

absence of any consideration of the peel stress developed at the joint extremities. It was also 

assumed that the shear stress distribution is constant at all stations through the thickness and that 

the peak shear stress occurs at the free ends of the adhesive lap. 

For a balanced double lap joint, the following useful relationships are assumed to apply: 

{ })cosh())((cosh( )sinh(2

)( xl x

l

x l λ λ

λ

λ

τ

τ = +− (3-10) 

where τ(x) is the local shear stress developed at distance x from one end of the lap. 

Ett

G

a

2 a 2 λ =

wl

p τ =

This equation gives the same results as equation (3-1), and the maximum shear stress along the 

centerline of the 0.01-inch-thick adhesive for the aluminum double lap joint is 206.9 psi, while 

TOM solution gives 199 psi and FEA shows the maximum shear stress of 184.2 psi (table 3-2). 

3.4.2 Maximum Stress/Strain Including Nonlinearity of Adhesive. 

A common approach introduced by Hart-Smith [5] comprises an elastic-plastic model for the 

adhesive layer. The three-parameter elastic-plastic model, Ga, τp and γmax, requires experimental 

shear stress-strain data for the adhesive for which the basis is the shear strain energy at failure. 

Hence, the area under the shear stress-strain curve is equated for the idealized elastic perfectly 

plastic model and the experimentally obtained curve. This enables the determination of the 

following three parameters: (1) Elastic shear modulus, Ga; (2) shear stress at yield, τp, 

sometimes termed cutoff value; and (3) ultimate shear strain γmax (elastic plus plastic shear strain 

γe+γp). 

The key relationship representing the design guidance is given by: 

max 2 a P = τ l (3-11) 

3-32
where 

Pmax is the failure load per unit width 

τa is the average shear stress at failure 

l is the overlap length 

The parameter λ for the elastic-plastic case can be expressed: 

2

p

a tEt

τ

λ

γ = (3-12) 

And for practical design guidance, the following lap length is recommended [5] 

2

2 R F

p

P l

τ λ

⎡ ⎤ = ⎢ ⎥ + × S

⎢ ⎥ ⎣ ⎦

 (3-13) 

where SF is the factor of safety 

Finally, the peel stress developed at the ends of the adhesive layer is approximated by a 

relationship obtained from a simple beam-on-elastic foundation model, assuming that the shear 

stress induced yield level had been reached for a finite yielded zone to be established. This 

relationship is given by: 

1

' 2 4 3 (1 ) z a

peel p

a

E t

E t

ν

σ τ

⎡ − ⎤ = ⎢ ⎥ ⎣ ⎦ (3-14) 

where 

Ez

’

 is the effective normal elastic modulus of the adhesive in the peel mode assuming inplane constraints are imposed by the stiffer and thicker adherends. 

νa is the Poisson’s ratio of the adhesive material 

t, ta are the thicknesses of the adherends and adhesive layer, respectively 

E is the in-plane elastic modulus of the adherends 

More detailed models, notably finite element-based in most cases, have been used to a significant 

degree in more recent years. However, the general approach for maximum stress/strain criteria is 

often adopted by introducing the same fidelity of discretization in the model of the basic test 

coupon configuration as for the model of the structural hardware that is being designed. 

3.4.3 Yielding/Nonlinearity (Development and Yield Zones). 

As mentioned in reference 34, lap shear tests give only two pieces of information, the shear 

strength (actually the average shear stress at failure) and the failure mode. Only a quantitative 

3-33
assessment of the level of adhesion can be obtained. In FEA modeling, perfect bonding is 

typically assumed for each analysis. 

For adhesively bonded joints, there are three failure modes, depending on the adherend material 

and the bonding quality: (1) cohesive failure in the adherend, (2) cohesive failure in the 

adhesive, and (3) interfacial failure at the interface between one of the adherends and the 

adhesive. A perfectly bonded joint with strong metallic adherends will fail by cohesive failure in 

the adhesive, while with composite adherends, it usually fails by cohesive failure in the 

adherends due to its low transverse strength. Interfacial failure occurs at the interface between 

one of the adherends and the adhesive, and usually reveals a poor surface preparation. In such 

instances, the joints may fail partly by cohesive failure and partly by interfacial failure. 

In section 4, a series of experiments on titanium single lap joint test results are described, the 

adhesive system for these tests is DP460, a relatively ductile system assumed to be an isotropic 

material. For such a ductile adhesive system, the Von Mises failure criterion is generally 

applicable. When considering a joint with composite adherends, a perfect bonded joint will fail 

by cohesive failure in the adherends, the Von Mises stress criterion is not generally applicable to 

such, nonisotropic, materials. 

From the literature, stresses at the termination of the interfaces between the adherend and 

adhesive of single lap joints are singular in a linear elastic analysis due to the dissimilar bonded 

materials; consequently maximum stress/strain criteria cannot be interpreted for the purpose of 

failure prediction. In FEA, the stresses at the singularity point get larger as the element meshing 

is refined. 

Linear analyses neglecting the ductility of the adhesive predict the strength of 506 and 700 lb for 

0.01- and 0.03-inch-thick joints, respectively, if equivalent Von Mises stress along the centerline 

of the adhesive is employed (table 3-6). While the equivalent Von Mises stress at the point next 

to the singular point is used, failure is predicted at 366 lb for 0.01-inch-thick joint and 303 lb for 

0.03-inch-thick joint. In both cases, the prediction was much lower than that of nonlinear 

analysis and test results. It was noted that when stress along the centerline of the adhesive is 

used, it predicts increasing strength with increasing adhesive thickness. This is consistent with 

the closed form solutions, but this is contrary to the experimental observations and nonlinear 

analysis. However, if the stresses along the interface around the singular point are used, the 

prediction would be that the strength slightly decreases with increasing adhesive thickness. 

In this report, as the adhesive used in the test and analysis is relatively ductile, with extensive 

yielding, strength prediction by linear elastic analysis is not applicable. Nonlinear analyses of 

0.01-inch-thick, 0.03-inch-thick, and variable thicknesses from 0.0085- to 0.0255-inch-thick 

adhesive titanium single lap joints with the ductile adhesive DP460 show that failure loads are 

1870 lb, 1856 lb, and 1865 lb, respectively. All are less than the test failure load 2000-2200 lb, 

and possible reasons are listed, see section 3.3.3. As suggested there, a different modeling of the 

adhesive (i.e., with fillet, modified Von Mises) might be selected for predicting the strength of a 

joint. 

In searching for a workable failure criterion for bonded joints, more extensive studies on testing, 

postfracture study, and modeling will be necessary in the future. 

3-34
4. CORRELATION WITH EXPERIMENTS. 

The following section reports the apparent shear strength results of the single lap joint specimens 

prepared and tested in accordance with the established procedures. Variations of these 

procedures were studied in the interest of drawing comparisons. Once data for a control group 

was established, tests were conducted for coupons with altered bond geometry, environmental 

exposure, testing environment, and surface preparation methods. 

This single lap test provided data for the coupon’s apparent shear strength that does not generally 

reflect the true shear strength of the adhesive. Analytical and theoretical predictions were 

obtained for similar materials with linear-elastic properties as described in section 3. From the 

analyses, it was evident that the normal (peel) stress along the adhesive-adherend interface, at the 

edges of the overlap, can dominate coupon failure. Consequently, the results obtained from this 

series of tests are unique to this configuration and should be used only for comparative studies. 

4.1 SINGLE LAP JOINT.

4.1.1 Aluminum/Titanium Adherends. 

A basis for comparison was established by testing coupons with constant bond thickness. 

Figure 4-1 presents data for nonbrushed coupons, bonded with DP460. The apparent shear 

strength of the coupons is plotted against the average bond thickness. Thirty-one coupons were 

tested to obtain the results displayed in this chart. Coupons were gathered (see legend) into 

groups according to their average bond thickness, at approximately 0.005-inch intervals, 

beginning at 0.005 inch. The data points represent average values of bond thickness and the 

corresponding apparent shear strength for each group. 

Adhesive: DP460 (unbrushed) Cured at 180oF for 1 hour 

Adherends: Titanium Tested at room temperature 

FIGURE 4-1. EXPERIMENTAL RESULTS FOR SINGLE LAP JOINT TESTS 

4-1
The next group of coupons was composed of the same adhesive and adherends (DP460 and 

titanium). However, these coupons differ from those presented in figure 4-1, in that the adhesive 

was brushed into the bond area. Eleven coupons were tested and are displayed as individual data 

points in figure 4-2. 

Adhesive: DP460 (brushed) Cured at 180oF for 1 hour 

Adherends: Titanium Tested at room temperature 

FIGURE 4-2. EXPERIMENTAL RESULTS FOR COUPONS WITH ADHESIVE 

BRUSHED INTO BOND AREA 

To emphasize the effect of brushing adhesive into a bond surface, figure 4-3 presents a 

combination of the preceding two figures. The only effect of brushing appears to be lower 

scatter strengths for the thin bondline specimens. 

Adhesive: DP460 Cured at 180o F for 1 hour 

 Adherends: Titanium Tested at Room Temperature 

FIGURE 4-3. EXPERIMENTAL RESULTS FOR TITANIUM SINGLE LAP JOINT TESTS 

OF BRUSHED AND NONBRUSHED COUPONS 

4-2
Data comparing the cooled coupons with coupons tested at room temperature is displayed in 

figure 4-4. Coupons were gathered into three groups according to their average bond thickness 

(see legend). The data points represent average values of bond thickness and apparent shear 

strength for each group. 

Adhesive: DP460 (nonbrushed) 

Adherends: Titanium Cured at 180oF for 1 hour 

FIGURE 4-4. EXPERIMENTAL RESULTS FOR TITANIUM SINGLE LAP JOINT TESTS, 

COMPARING AMBIENT TEMPERATURE VS LOW TEMPERATURE EXPOSURE 

Several tests were conducted to assess the environmental durability of DP460. Figure 4-5 

displays data for coupons that were enclosed in a humidity chamber after curing. Humidity 

ranged from 80% to 100%. Adhesive was not brushed into the bond area. Data for nonexposed 

coupons is presented for comparison. 

Adhesive: DP460 (nonbrushed) Cured at 180oF for 1 hour 

Adherends: Titanium Tested at Room Temperature 

FIGURE 4-5. EXPERIMENTAL RESULTS FOR TITANIUM SINGLE LAP JOINT TESTS, 

COMPARING AMBIENT CONDITION VS HIGH HUMIDITY EXPOSURE 

4-3
Figure 4-6 contains data for several coupons that were submersed in water after curing. Data 

points represent individual coupons. 

Adhesive: DP460 (brushed) Cured at 180o F for 1 hour 

Adherends: Titanium Tested at Room Temperature 

FIGURE 4-6. EXPERIMENTAL RESULTS FOR TITANIUM SINGLE LAP JOINT TESTS, 

COMPARING AMBIENT CONDITION VS WATER SOAKED EXPOSURE 

Tests were conducted to investigate the effect of humidity exposure during the bonding process. 

After being gritblasted, coupons were stored in a humidity chamber for 24 hours. Humidity 

ranged from 90% to 100%. Coupons were bonded within 5 minutes of the time at which they 

were removed from humidity chamber. Data is presented in figure 4-7, where the points 

represent individual coupon data. 

Adhesive: DP460 (brushed) Cured at 180oF for 1 hour 

 Adherends: Titanium Tested at Room Temperature 

FIGURE 4-7. EXPERIMENTAL RESULTS FOR TITANIUM SINGLE LAP JOINT TESTS, 

COMPARING AMBIENT CONDITION VS HUMIDITY EXPOSURE PRIOR TO BONDING 

4-4
Typically, a bond surface is rinsed after being gritblasted. Several methods exist to remove the 

remaining grit media. Figure 4-8 compares coupons blotted with acetone to ones that were 

rinsed with dry nitrogen air at 100 psi. Data points represent individual coupons. 

Adhesive: DP460 (brushed) Cured at 180oF for 1 hour 

Adherends: Titanium Tested at Room Temperature 

FIGURE 4-8. EXPERIMENTAL RESULTS FOR TITANIUM SINGLE LAP JOINT 

TESTS, COMPARING ACETONE BLOTTING VS DRY NITROGEN 

RINSING AS PRETREATMENT 

4.1.2 Composite Adherends. 

Tests were conducted on single lap coupons with composite adherends. The limited database 

reflects concern regarding the relevance of observed failure modes. Failures tended to initiate in 

an adherend rather than the adhesive. Figure 4-9 displays data for coupons with composite 

adherends. The data points represent average values of bond thickness and apparent shear 

strength for each group. 

Adhesive: DP460 Cured at 180oF for 1 hour 

Adherends: Composite Tested at Room Temperature 

FIGURE 4-9. EXPERIMENTAL RESULTS FOR SINGLE LAP JOINTS WITH 

COMPOSITE ADHERENDS 

4-5
4.2 DOUBLE LAP JOINT. 

Double lap coupons were constructed by bonding together the adherends of similar single lap 

coupons. Data for double lap coupons with composite and titanium adherends is presented in 

figure 4-10. Data points represent individual coupons. The titanium-composite lap joints were 

lower in strength because of additional peel stresses and fabrication difficulties. 

Adhesive: DP460 (brushed) Cured at 180oF for 1 hour 

Adherends: see legend Tested at Room Temperature 

FIGURE 4-10. EXPERIMENTAL DATA FOR DOUBLE LAP COUPONS 

4.3 BONDLINE THICKNESS VARIABILITY. 

Tests were conducted to assess the effects of bonds that were variable along the lap length and 

the width (see figure 4-11). Data for brushed coupons with variable and nonvariable bond 

thickness is presented in table 4-1. 

FIGURE 4-11. BOND THICKNESS VARIATION ALONG THE LAP LENGTH AND 

ACROSS THE LAP WIDTH 

4-6
If bondline thickness is linearly changing from t1 (defined as minimum thickness, the least 

thickness of the bondline thickness) to t2 (defined as maximum thickness, the maximum 

thickness of the bondline thickness), then the average thickness ta will be 

2

21 tt

ta

+ = (4-1) 

TABLE 4-1. GEOMETRY DATA OF ALL THE BONDLINE CASES SHOWN IN 

FIGURE 4-12 

Case No. 

Average 

thickness ta

(inch) 

Change of 

thickness Δt

(inch) 

Minimum 

thickness t1 

(inch) 

Maximum 

thickness t2 

(inch) 

Average 

Failure 

stress τa

(psi) 

1 0.003 0 0.003 0.003 3864 

2 0.004 0 0.004 0.004 4219 

3 0.005 0 0.005 0.005 4138 

4 0.012 0 0.012 0.012 4273 

5 0.013 0 0.013 0.013 4510 

6 0.013 0 0.013 0.013 4188 

7 0.015 0 0.015 0.015 4494 

8 0.017 0 0.017 0.017 4265 

9 0.019 0 0.019 0.019 4318 

10 0.022 0 0.022 0.022 4119 

11 0.029 0 0.029 0.029 3992 

12 0.012 0.017 0.0035 0.0205 3734 

13 0.017 0.017 0.0085 0.0255 4300 

14 0.023 0.017 0.0145 0.0315 4209 

15 0.012 0.020 0.002 0.022 4305 

16 0.015 0.018 0.006 0.024 4246 

17 0.017 0.018 0.008 0.026 4160 

18 0.022 0.019 0.0134 0.0324 4200 

Since this variance is linearly along the lap length, the average thickness occurs at the midpoint 

along the length, as shown in the first graph of figure 4-11. This is also true for thickness 

variance across the lap width, where average thickness occurs at the midpoint along the width 

direction, shown in the second graph of figure 4-11. 

If the change of thickness from one end to the other is denoted as Δt (equal to t2-t1) and the 

average bondline thickness ta is shown (as x axis of figure 4-12), then the minimum thickness t1

and maximum t2 can be calculated by 

2 1

t

tt a

Δ −= (4-2) 

4-7
2 2

t

tt a

Δ

+= (4-3) 

Adhesive: DP460 (brushed) Cured at 180oF for 1 hour 

 Adherends: Titanium Tested at Room Temperature 

FIGURE 4-12. EXPERIMENTAL RESULTS FOR TITANIUM SINGLE LAP JOINT TESTS, 

EFFECT OF LINEAR VARIATIONS IN BOND THICKNESS 

In figure 4-12, the label nonvariable bond thickness represents that the bondline thickness is 

uniform in both lap length and width directions. Variable along width (or length) means that the 

bondline is uniform in lap length (or width) direction, but not along width (or length) direction, 

and the change of thickness Δt from the minimum thickness t1 to maximum thickness t2 is 

indicated inside the parentheses. The x axis is average bondline thickness ta, and y axis is the 

average failure stress τa defined as the failure load P divided by the bonding area A (in all the 

cases, A = 0.5 in2

). Therefore, the failure load P is one half of the average failure stress τa

(shown in table 4-1). 

Table 4-1 shows the geometry data of all the bondline cases shown in figure 4-12. Cases 1 to 11 

are cases of uniform bondline thickness brushed, cases 12 to 14 are cases of variable bondline 

thickness along length, and change of thickness is 0.017 inch for these three cases. Cases 15 

through 18 are cases of variable bondline thickness along width, and change of thickness for 

each case is indicated in the third column of table 4-1. 

4-8
5. CONCLUSIONS. 

Numerical and analytical methods for adhesively bonded joints were reviewed, applied, 

developed, and evaluated for one specific paste adhesive system, 3M’s DP460, which exhibits 

ductile characteristics. Commencing with simple shear lag analysis, e.g., Volkersen theory, 

progressively more complex formulations were considered for both stress analysis and failure 

predictions. Finite element methods, singularities, and plasticity treatments were applied for 

prediction of failure loads that were compared to experimentally determined values for both 

single and double lap configurations. The following items summarize the outcome of the 

research. 

1. Predictions based on linear analysis, numerical or analytical, tend to dramatically 

overestimate shear and peel stress levels and, consequently, greatly underestimate the 

failure load levels. Of course this assumes that appropriate surface preparation and 

assembly procedures have been used, resulting in mainly cohesive failure modes. Also, 

linear elastic analyses of single and double lap joints show that, for the same applied 

tensile load, the peak stresses at the critical end of a double lap joint are much lower than 

that in a single lap joint. 

2. For similar reasons, to those stated in item 1, the predicted trends for the effect of 

bondline thickness are significant and misleading, especially for an adhesive with the 

ductility exhibited by the DP460 system. 

3. Nonlinear material simulations approximating adhesive shear and peel stress-strain 

characteristics, introduced via finite element analyses, produce a more realistic and much 

closer predictions of failure load when compared with experiments. Since polymer 

yielding is sensitive to hydrostatic pressure, a modified Von Mises yield criterion is 

recommended for future studies. 

4. Approaches based on the nonlinear characteristics identified in item 3 provide a rationale 

for the essentially constant failure loads with adhesive bondline thickness over the range 

of 0.005 through 0.035 inch that was indicated by experiment. It is highly possible that 

the experimentally indicated tolerance to linear variations of bondline thickness (within a 

given bondline) can also be based on the same rationale. 

5. In single lap shear tests, the indication of high peel stresses appear to drive an initial 

adhesive failure near the ends of the lap. However, the fracture mechanics approaches 

attempted herein have not demonstrated that mode I fracture data enables an adequate 

failure prediction method. 

6. For bonded joint stress analysis purposes, finite element analysis (FEA) has been found 

to be a good choice as long as the mesh, boundary condition, and the loading are applied 

properly. For FEA modeling using ANSYS, an adequately refined mesh and eight-node 

element (plane82) are recommended. 

7. A limited evaluation of spew fillet effects was studied, although no fillet was included in 

the nonlinear FEA analyses. In most practical scenarios, there is some evidence of a fillet 

5-1
at the joint ends, and it is expected that with a spew fillet, the strength of adhesively 

bonded joints will be somewhat enhanced in accordance with other researcher’s results. 

Furthermore, the yielding of adhesive is pressure dependent, and modified Von Mises 

yield criterion is more appropriate for modeling adhesive yield behavior. ABAQUS has 

been adopted for some ongoing studies. 

5-2
6. REFERENCES. 

1. Volkersen, O., Die Niektraftverteilung in Zugbeanspruchten mit Konstanten 

Laschenquerschritten. Luftfahrtforschung 15, 1938, pp. 41-47. 

2. M. Goland and E. Reissner, “The Stresses in Cemented Joints,” Journal of Applied 

Mechanics 11, 1944, pp. A17-A27. 

3. D.W. Oplinger, “A Layered Beam Theory for Single Lap Joints,” Army Materials 

Technology Laboratory Report, MTL TR91-23, 1991. 

4. L.J. Hart-Smith, “Adhesive-Bonded Single Lap Joints,” NASA-Langley Contract Report 

NASA-CR-112236 (1973). 

5. L.J. Hart-Smith, L. J., “Adhesive-Bonded Double Lap Joints,” NASA-Langley Contract 

Report NASA-CR-112235 (1973). 

6. M.Y. Tsai, D.W. Oplinger, and J. Morton, “Improved Theoretical Solutions for Adhesive 

Lap Joints,” Int. J. Solids Structures, Vol. 35, No. 13, 1998, pp.1163-1185. 

7. H. Kim and K. Kedward, “Stress Analysis of Adhesively-Bonded Joints Under In-Plane 

Shear Loading,” J. Adhesion, Vol. 76, 2001, pp.1-36. 

8. F.E. Penado and R.K. Dropek, “Numerical Design and Analysis,” Engineered Materials 

Handbook, Volume 3 “Adhesives & Sealants”, ASM International, 1990. 

9. A. Tessler, M.L. Dambach, and D.W. Oplinger, “Efficient Adaptive Mesh Refinement 

Modeling of Adhesive Joints,” Presented at the Workshop on Bonded Joints and 

Assemblies in Aircraft, ASTM/ASC, Texas A&M, September 2000. 

10. J. Tomblin, P. Harter, W. Seneviratne, and C. Yang, “Characterization of Bondline 

Thickness Effects in Adhesive Joints,” Journal of Composites Technology & Research, 

JCTRER, Vol. 24, No. 2, April 2002, pp. 80–92. 

11. M. Gleich, M. Tooren, and A. Beukers, “Analysis of Bondline Thickness Effects on 

Failure Load in Adhesively Bonded Structures, 32nd International SAMPE Technical 

Conference, November 5-9, 2000. 

12. J.A. Harris and R.D. Adams, “Strength Prediction of Bonded Single Lap Joints by 

Nonlinear Finite Element Methods,” International Journal of Adhesion and Adhesives, 

Volume 4, Issue 2, pp. 65-78, April 1984. 

13. D.B. Bogy, “Two Edge-Bonded Elastic Wedges of Different Materials and Wedge 

Angles Under Surface Tractions,” Journal of Applied Mechanics; 38:377-86, 1971. 

14. D.B. Bogy and K.C.Wang, “Stress Singularities at Interface Corners in Bonded 

Dissimilar Isotropic Materials,” International Journal of Solids and Structures, 7, 1971, 

pp. 993-1005. 

6-1
15. F.E. Penado, “Analysis of Singular Regions in Bonded Joints, International Journal of 

Fracture, 105: 1-25, 2000. 

16. Jin-Quan Xu, Yi-Hua Liu, and Xiao-Gui Wang, “Numerical Methods for the 

Determination of Multiple Stress Singularities and Related Stress Intensity Coefficients,” 

Engineering Fracture Mechanics, Vol. 63, No. 6, August 1999, pp. 775-90.

17. H.L. Groth, “Stress Singularities and Fracture at Interface Corners in Bonded Joints,” 

International Journal of Adhesion and Adhesives, Vol 8, No.2, April 1988, p 55-121. 

18. V.L. Hein, and F. Erdogan, “Stress Singularities in a Two Material Wedge,” 

International Journal of Fracture Mechanics, 7, 1971, pp. 317-330. 

19. A.R. Akisanya and C.S. Meng, “Initiation of Fracture at the Interface Corner of BiMaterial Joints,” Journal of the Mechanics and Physics of Solids, 51, 2003 pp. 27-46. 

20. R.D. Adams, S.H. Chambers, P.J.A. Del Strother, and N.A. Peppiatt, “Rubber Model for 

Adhesive Lap Joints,” J. Strain Anal., 8, 1973, pp. 52-57. 

21. M.K. Apalak and R. Davies, “Analysis and Design of Adhesively Bonded Corner Joints: 

Fillet Effect,” International Journal of Adhesion and Adhesives, Volume 14, Issue 3, July 

1994, pp. 163-174. 

22. R.D. Adams and J.A. Harris, “The Influence of Local Geometry on the Strength of 

Adhesive Joints,” International Journal of Adhesion and Adhesives, Volume 7, Issue 2, 

April 1987, pp. 69-80. 

23. M.Y. Tsai and J. Morton, “The Effect of a Spew Fillet on Adhesive Stress Distributions 

in Laminated Composite Single-Lap-Joints,” Composite Structures, 32, 1995, pp. 123-

131. 

24. O.C. Zienkiewicz and R.L. Taylor, “The Finite Element Method,” ButterworthHeinemann, Ch.15, 2000. 

25. K.R. Loss and K.T. Kedward, “Modeling and Analysis of Peel and Shear Stresses in 

Adhesively Bonded Joints,” Proceedings, 25th AIAA SDM Conference, Palm Springs, 

CA, May 1984. 

26. Q.D. Yang and M.D. Thouless, “Mixed-Mode Fracture Analyses of PlasticallyDeforming Adhesive Joints,” International Journal of Fracture, Vol. 110, No. 2, 2001, 

pp.175-87. 

27. V. Tvergaard and J.W. Hutchinson, “On the Toughness of Ductile Adhesive Joints,” 

Journal of the Mechanics & Physics of Solids, Vol. 44, No. 5, May 1996, pp. 789-800. 

28. J.W. Hutchinson and Z. Suo, “Mixed Mode Cracking in Layered Materials,” Advances in 

Applied Mechanics, Vol. 29, 1992, pp.63-191. 

6-2
29. R.D. Adams, J. Comyn, and the late W.C. Wake, “Structural Adhesive Joints in 

Engineering,” Chapman & Hall, 1997. 

30. A. Posius, Private communication, 3M Company, June 2002. 

31. A. Rothwell, “Adhesive Lap Joint in Tension-Variable Adhesive Thickness for Uniform 

Shear Stress,” Delft University, The Netherlands, 1995. 

32. M.N. Charalambides, A.J. Kinloch, and F.L. Matthews, “Adhesively-Bonded Repairs to 

Fiber-composite Materials II: Finite Element Modeling, Composite Part A 29A,” 1998, 

pp.1383-1396. 

33. N.G. McCrum, C.P. Buckley, and C.B. Bucknall, “Principles of Polymer Engineering,” 

Oxford University Press, 1997. 

34. J. Bardis and K.T. Kedward, “Effects of Surface Preparation on the Long-Term 

Durability of Composite Adhesive Bonds,” FAA report DOT/FAA/AR-01/8, April 2001. 

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