On non-linear thermal stresses in an adhesively bonded single lap joint

On non-linear thermal stresses in an adhesivelybonded single lap joint

M. Kemal Apalak *, Recep Gunes

Department of Mechanical Engineering, Erciyes University, Muhendislik Fakultesi, 38039 Kayseri, Turkey

Received 7 June 2000; accepted 23 July 2001

Abstract

Since adhesive joints consist of adhesive and adherends with different mechanical and thermal properties, the ad-

hesive and adherends present different stress and strain states under thermal loads due to the thermal–mechanical

mismatches. Thermal strains result in serious stresses even though the adhesive joints are not restrained. In this study,

the thermal stress analysis of an adhesively bonded single lap joint (SLJ) was carried out considering the large dis-

placement effects. In the thermal analysis, the outer surfaces of the SLJ are assumed to be subjected to air flows with

different temperature and velocity. The final temperature distribution in the adhesive joint was used to compute thermal

strains. Later, the geometrical non-linear stress analysis of the SLJ was carried out for four adherend edge conditions

using the incremental FEM. Thermal strain concentrations were observed inside the adhesive fillets around the free ends

of the adhesive layer. The top and bottom surfaces of the adherends also experienced high thermal stresses. The detailed

analysis showed that the most critical adhesive regions were the free ends of the adhesive–adherend interfaces. It was

observed that thermal loads caused serious stress and strain concentrations in joint members as well as the structural

loads (Structural adhesive joints in engineering. London: Elsevier Applied Science; 1984). In order to reduce the peak

stresses at the critical adhesive and adherend regions increasing the overlap length was not beneficial for all adherend

edge conditions. � 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Epoxy adhesive; Steel adherends; Single lap joint; Thermal analysis; Geometrical non-linearity; Stress analysis; ANSYS

1. Introduction

The adhesive joints are widely used as a structural

element in the automotive and in aerospace applications.

Therefore, a large number of analytical and numerical

studies has been done so far. Generally, the stress and

deformation states of a single lap joint (SLJ) and its

derivatives subject to a tensile load have been analysed

[1]. In fact, the adhesive joints experience not only me-

chanical loads but also thermal loads. Since the adhesive

joints consist of materials with different mechanical and

thermal properties, the thermal strains in the joint

members may cause serious stresses. Some studies con-

sidering thermal loads in adhesive joints are available:

Lee and Lee [2] presented a method for predicting an

optimum design of adhesive bonded tubular lap joint

based on failure modes of the adhesive layer considering

the residual thermal stresses induced by fabrication.

Reedy and Guess [3] investigated the effect of the re-

sidual stresses in an adhesive butt joint generated by

cooling on the joint strength. They showed that the in-

fluence of the residual stresses on the butt joint strength

could be much smaller than would be predicted by a

linear analysis. Thus, the peak adhesive stresses in the

yield zone at the interface corner could decay signifi-

cantly when given sufficient time. Ioka et al. [4] studied

thermal residual stress distributions on the interface and

around the intersections of the interface and the free

surfaces of bonded dissimilar materials using the

boundary element method. They found that the thermal

stress singularity disappears for certain range of wedge

angles of a pair of materials.

Computers and Structures 80 (2002) 85–98

www.elsevier.com/locate/compstruc

*Corresponding author.

E-mail address: [email protected] (M.K. Apalak).

0045-7949/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0045 -7949 (01 )00139 -0

Anifantis et al. [5] developed an analytical approach

for the solution of the generalized plane strain steady-

state thermoelastic problem and investigated thermal

stress and strain concentrations in unidirectional fibre-

reinforced composites considering the concept of an

inhomogeneous interphase between the fibre and matrix

phases. They showed that the stresses exhibited stronger

discontinuities than the strains, exhibiting a high sensi-

tivity to temperature changes. In the study the location

of peak stress was independent of the degree of adhesion

developed between fibre and matrix while strains are

slightly dependent. They also mentioned that inhomo-

geneous temperature distributions or transient thermal

loads might increase these phenomena drastically.

Kim and coworkers [6,7] investigated the stresses

occurring in an adhesively bonded tubular SLJ consid-

ering the non-linear adhesive properties and thermal

residual stresses due to the fabrication and presented a

failure model. Cho et al. [8] studied the effect of curing

temperature on the adhesion strength of polyamidei-

mide/copper joints and showed the adhesion strength

decreased as the thermal stress increased with the in-

crease of both curing temperature and time. Nakano

et al. [9] carried out the thermal stress analysis of an

adhesive butt joint containing circular holes and rigid

fillers in its adhesive layer and subject to a non-uniform

temperature field. They showed that the size and loca-

tion of the circular holes and rigid fillers had evident

effect on the thermal stresses at the interface and at the

hole and filler peripheries.

Humfeld and Dillard [10] investigated thermal cy-

cling effects on the residual stresses in viscoelastic

polymeric materials bonded to stiff elastic substrates.

The residual stresses in the elastic–viscoelastic bi-mate-

rial system incrementally shifted over time when sub-

jected to thermal cycling, and damaging tensile axial and

peel stresses developed over time due to viscoelastic re-

sponse to thermal stresses induced in the polymeric

layer.

Abedian and Szyszkowski [11] presented a numerical

study in which effects of surface geometry of composites

on the thermal stress distribution are investigated. They

determined that the stress state in the composite at and

near the free surface is very sensitive to the geometric

features of the surface: thus, an ideally flat free surface

resulted in large stress concentrations whereas these

stresses decrease substantially in case the difference be-

tween the fibre and the matrix heights is filled with the

matrix material. Nagakawa et al. [12] investigated ther-

mal stress distributions an in adhesive butt joint con-

taining some hole defects under uniform temperature

changes. They showed that thermal stresses around hole

defects located near the centre of the adhesive were

larger than those around the hole defects located near

the free surface of the adhesive. Katsua et al. [13] carried

out transient thermal stress analysis of an adhesive

butt joint. They assumed the upper and lower end sur-

faces of the joint are in different temperatures at a

certain instant, and analysed the effects of thermal ex-

pansion coefficient and Young’s modulus ratios of the

adhesive and adherends on the transient thermal stress

distribution.

In general the present studies related to thermal stress

and strain distributions in adhesive joints assume that all

joint members had a uniform temperature distribution,

or the temperature distribution along the geometrical

boundaries of the joint was prescribed, and the con-

vective heat transfer was ignored. In addition, the

transient temperature distribution was not considered.

In this study, the thermal stress analysis of a SLJ was

carried out assuming that the outer surfaces of the ad-

hesive joint are subject to fluid flows in different tem-

perature and velocity, and that the adherends have

different edge conditions. The heat transfer throughout

the adhesive joint takes place by conduction whereas it

occurs by convection from the joint surfaces to the flu-

ids, or the fluids to the joint surfaces. The convective

heat transfer between the fluid and adherend surfaces

depends on heat transfer coefficient. These thermal

boundary conditions make the heat transfer problem of

the adhesive joint complex. Since the thermal strain

distributions in the adhesive SLJ are dependent on the

temperature distributions, first the thermal analysis and

later the geometrically non-linear stress analysis of the

Nomenclature

hm heat transfer coefficient, W/m2 �CU air velocity, m/s

Deqv equivalent diameter, m

v kinematic viscosity, m2/s

k0air thermal conductivity of the air, W/m �C

k thermal conductivity of the air, kcal/mh �CL plate length, m

Nu Nusselt number

Re Reynolds number

Pr Prandtl number

cp specific heat, kcal/kg �Cl dynamic viscosity, kg/ms

Ta air temperature, �CTw plate temperatures, �CTf average temperature, �C

86 M.K. Apalak, R. Gunes / Computers and Structures 80 (2002) 85–98

adhesive joint were carried out based on the small

strain–large displacement theory.

2. Joint configuration and finite element model

The SLJ consists of an epoxy adhesive layer and two

steel plates. Material properties are given in Table 1. The

main dimensions of the SLJ are shown in Fig. 1. An

adhesive thickness t2 of 0.2 mm, upper and lower plate

thickness t1 and t3 of 2 mm, upper and lower plate

lengths ‘‘l’’ of 30 mm, plate widthW of 50 mm were kept

constant through the analyses. The overlap length a of 5

mm was used for the first analysis. In order to obtain a

suitable bonding the adhesive layer is pressed between

two adherends, as a result, some amount of adhesive is

squeezed out and accumulated around the adhesive free

ends, called the adhesive fillet. For the simplicity of the

analysis, the shape of this adhesive accumulation was

assumed to be a triangle [1,11]. The height and length of

the adhesive fillet ft were taken as twice the adhesive

thickness (ft ¼ 0:4 mm).

The previous studies on the stress and deformation of

the SLJs have shown that the peak adhesive stress and

strains occur at the adhesive vicinities around the cor-

ners of the adherends [1]. In case the adherend corners

were sharp, the stress singularities occur at these sharp

corners due to the geometrical discontinuity. In fact, in

order to obtain a better bonding surface as possible, the

bonding surfaces of the adherends are etched; therefore,

this type of sharp corner in the adherends is unusual.

Adams and Harris showed that the stress levels around

the rounded adherend corners are lower than those

around the sharp corners [14]. For this reason, the cor-

ners of both adherends were rounded with a radius of

0.04 mm as shown in Fig. 1. Since the problem requires a

two-dimensional thermal stress analysis, four-noded

quadrilateral and three-noded triangle plane (strain) el-

ements were used to model the upper and lower adher-

ends, and the adhesive layer.

In this study, the effect of the geometrical non-lin-

earity was also considered, thus the small strain–large

displacement analysis of the SLJ was carried out. Since

the stress concentrations occurred around the adhesive

free ends and propagate along the adhesive layer the

mesh refinement of these regions was essential, especially

around the adhesive fillets and along the adhesive layer.

The final mesh and its details are shown in Fig. 2. For

the thermal stress analysis, the (finite element software)

ANSYS 5.3ANSYS 5.3 was used [15].

3. Thermal analysis

In practice, the adhesively bonded joints may expe-

rience complex thermal loads. The adhesive and adher-

ends have different thermal and mechanical properties;

therefore, they exhibit different thermal strain distribu-

tions. Since the thermal strains should be compatible

along the adhesive–adherend interfaces, their thermal–

mechanical mismatches cause thermal stresses in the

joint members, especially in the bonding region. In ad-

dition, the temperature distribution in the adhesive joint

is non-uniform as a result of variable thermal boundary

conditions. This is another factor causing a non-uniform

thermal strain distribution. However, in order to deter-

mine the thermal strain distribution, the final tempera-

ture distribution in the joint should be known.

In the previous studies, the final temperature distri-

butions along the boundary of the adhesive joints and in

Table 1

Mechanical and thermal properties of adherends and adhesive

Medium

carbon

steel (1040)

Rubber

modified

epoxy adhesive

Modulus of elasticity E, GPa 207 3.33

Poisson ratio m 0.30 0.34

Coefficient of thermal

expansion a (�C�1)

11:3� 10�6 45–65� 10�6

Thermal conductivity

k (W/mK)

52 0.19

Yield stress rY, MPa 350 40

Fig. 1. Dimensions of a SLJ.

M.K. Apalak, R. Gunes / Computers and Structures 80 (2002) 85–98 87

the adhesive joints were prescribed and then the thermal

strains were computed using the temperature differences

relative to the initial uniform joint temperature [2–13].

In fact, the materials forming the adhesive joints

have different heat conduction/convection capabilities.

Therefore, the heat transfer through the adhesive joint

members should be analysed in detail for the thermal

conditions allowing different heat transfer mechanisms.

In this study, it was assumed that the SLJ are sub-

jected to the thermal conditions as shown in Fig. 3.

Thus, the upper surfaces of the upper and lower adh-

erends experience an air flow with a temperature of 120

�C and a velocity of 1 m/s. These surfaces are identified

by 1–2–3–4–5 lines in Fig. 3. Whereas the air flow was

normal to the adherend surfaces along 1–2, 3–4 and 4–5

surfaces, it was tangential to the adherend edge along

2–3 surface. Similarly, the lower surfaces were assumed

to have an air flow having a temperature of 20 �C and a

velocity of 1 m/s; the air flow was normal to the ad-

herend surfaces (6–7, 8–9 and 9–10 surfaces), whereas it

was tangential to 7–8 surface. All the joint members

were initially submitted to a uniform temperature of

20 �C.The heat transfer through the adhesive joint occurs

by means of the convection (from fluid to adherends or

adhesive) and the conduction (through adherends and

adhesive). The heat transfer by the convection requires

the computation of the heat transfer coefficients between

the air and the adherends or the adhesive. Their com-

putations are dependent on how the air flows along the

surfaces (vertically or horizontally). Since each surface

has different geometry and especially flow conditions,

the heat transfer coefficients should be computed for

each surface, as follows.

In case of a vertical air flow, the heat transfer coef-

ficient is given as [16]:

hm ¼ 0:205UDeqv

v

� �0:731 k0air

Deqv

ð1Þ

where U is the air velocity (m/s), Deqv is the equivalent

diameter of 50 mm, v is the kinematic viscosity (m2/s)

and k0air is the thermal conductivity of the air (W/m �C).

Fig. 2. Mesh details of the finite element model of a SLJ.

Fig. 3. Thermal boundary conditions of the SLJ.


The air properties are taken based on the air tempera-

ture in case of the vertical air flow.

In case of the horizontal flow, the heat transfer co-

efficient is given as [16]:

hm ¼ NukL

ð2Þ

where k is the thermal conductivity of the air (kcal/

mh �C), L is the plate length (m) and Nu is the Nusselt

number defined as:

Nu ¼ 0:836Re1=2 Pr1=3 ð3Þ

where Re and Pr are the Reynolds and the Prandtl

numbers, respectively and are described as:

Re ¼ ULv

ð4Þ

Pr ¼ cplk

ð5Þ

where cp is the specific heat (kcal/kg �C) and l is the

dynamic viscosity (kg/ms). The previous coefficients

describing the air properties are values based on an av-

erage temperature as follows:

Tf ¼Ta þ Tw

2ð6Þ

where Ta and Tw are the air and the plate surface tem-

peratures (�C), respectively. The thermal coefficients of

the air and the heat transfer coefficients are given in

Table 2.

Assuming a steady state heat transfer, and the final

temperature distribution in the adhesive joint was ob-

tained. This temperature distribution is shown in Fig. 4.

The temperature values in the joint members vary from

59 to 75 �C. The maximum temperature difference is

16 �C, because the adherends are very thin. However,

the maximum temperature difference causing the ther-

mal strains is 55 �C. The higher temperatures arise at the

Fig. 4. The temperature distributions in the SLJ (in �C).

Table 2

Thermal properties of the air and heat transfer coefficients hm (W/m2 �C) along adherend and adhesive outer surfaces

Surface no.

Vertical air flow Horizontal air flow

6–7, 8–9, 9–10 1–2, 3–4, 4–5 7–8 2–3

Ta ¼ 20 �C Ta ¼ 120 �C ðTa þ TwÞ=2 ¼ 20 �C ðTa þ TwÞ=2 ¼ 70 �C

v (m2/s) 15:11� 10�6 25:23� 10�6 15:11� 10�6 19:92� 10�6

k0air (W/m �C) 0.0257 0.0328 – –

k (kcal/mh �C) – – 0.0221 0.0251

cp (kcal/kg �C) – – 0.240 0.241

l (kg/ms) – – 1:82� 10�5 2:05� 10�5

hm (W/m2 �C) 39.4114 35.5782 116.3054 114.9317


right adhesive fillet and decreases through the joint (see

Fig. 4).

Finally, it is obvious that the adhesive SLJ has non-

uniform temperature distribution. Therefore, the devel-

opment and propagation of the thermal strains are

expected to become more complicated.

4. Geometrically non-linear stress analysis

In the elastic analyses of engineering structures, both

displacements and strains are assumed to be small. In

practice such assumptions might be invalid even though

actual strains may be small. In order to compute the

displacements accurately the geometrical non-linearity

may have to be considered [17–21]. Since the adhesive

joints may exhibit geometrical non-linearity under

moderate loads due to large displacements the linear

elastic approach might be misleading in predicting the

stress and deformation states of the adhesive joints.

Some studies on the effect of the geometrical non-lin-

earity on the stresses and deformations in the adhesive

joints have been carried out. First, Sawyer and Cooper

[22] investigated the load transfer of a SLJ in which the

adherends were pre-formed so that the angle between

the line of action of the applied in-plane force and the

bond line was reduced. Since the bending moment is due

to the eccentricity of the loading and to the deformation

of the adherends as the load is applied, they considered

that the dependence of the moment on the applied load

made the problem geometrically non-linear. They found

that pre-forming the adherends reduced the moment

resultant in the adherend at the edge of the overlap re-

gion, which resulted in a reduction in both the peel and

shear stresses and gave a more uniform shear stress

distribution in the adhesive layer.

Adams [23] and Harris and Adams [14,24] analysed

the failure modes and loads of SLJs having adherends

and adhesives with different mechanical properties under

tensile loading. For this purpose they used a non-linear

finite element method including the large displacement

and rotation effects and allowing the effects of non-linear

material behaviour of both adhesive and adherends.

They showed that the mechanical properties of adhesive

and adherends had a considerable effect on the failure

mode and loads. Adams et al. [25] also investigated the

shear and transverse tensile stresses in carbon fibre-re-

inforced plastic/steel double lap joints using the same

finite element approach. They obtained significant in-

creases in the joint strength by modifying the local ge-

ometry of the critical zones at the edges of the overlap

region of the double lap joint.

Reddy and Roy [26] presented a geometrically non-

linear finite element method for the analysis of ad-

hesively bonded joints, and showed that the large

displacement gradients had an evident effect on the

adhesive stresses at the ends of the overlap region of a

SLJ under different loading and boundary conditions.

Czarnocki and Pierkaski [27] also studied the effect of

the joint width on the adhesive stresses treating the ad-

hesive as a non-linear material and considering the effect

of the large displacements. He found that increasing the

bonding width reduced the peak stresses. Edlund and

Klarbring [28] presented a general analysis method for

determining the adhesive and adherend stresses and

deformations in the adhesively bonded joints consider-

ing the geometrical non-linearity and the non-linear

material properties of adherends and adhesive. They

showed that SLJs experienced high geometrical non-

linearity.

Apalak and Engin [29] applied the small strain–large

displacement theory to an adhesively bonded double

containment corner joint using the incremental finite

element method, and showed that the small strain–small

displacement theory may be misleading in predicting the

stress and deformation states of adhesive joints for some

Fig. 5. Boundary conditions of the SLJ: (a) (BC-I), the left edge

of the upper plate fixed and the right free edge of the lower plate

free only in the x-direction, (b) (BC-II), only one corner of the

plates fixed, (c) (BC-III), the free ends of the two plates fixed,

(d) (BC-IV), the left edge of the upper plate fixed and the right

edge of the lower plate was free only in the y-direction.

Fig. 6. Deformed geometries for the four end conditions, BC

(I–IV) (not scaled).


loading and boundary conditions. Apalak [30–33] also

showed that the geometrical non-linearity had a con-

siderable effect in the stress distributions of the adher-

ends and adhesive layer of the different types of corner

joints.

Andruet et al. developed two- and three-dimensional

geometric non-linear finite elements for the analysis of

the adhesive joints [34]. They used these elements in the

analysis of an adhesively bonded SLJ, and also showed

that the SLJ is subjected to high non-linearity. The

studies on the geometrical non-linear analysis of the

adhesive joints showed that the linear elastic analyses

predicted higher stresses and strains in the adhesive layer

and in the adherends. Therefore, the geometrical non-

linear analysis of the adhesively bonded joint was car-

ried out in this section. In the analysis, an updated

Lagrangian method was used, and all computations

were carried out for convergence criteria of 0.05 and

0.001 for forces and displacements, respectively in nearly

350 steps.

In order to determine the behaviour of the adhesive

SLJ for the different end conditions, the adhesive joint

was analysed for four end conditions presented in Fig. 5.

In the first end condition (BC-I), the left edge of the

upper plate was fixed and the uppermost and lowermost

nodes of the right free edge of the lower plate were free

Fig. 7. The von Mises stress distributions of a bonded SLJ for the boundary conditions, I–IV (all stresses in MPa).


only in the x-direction (Fig. 5a). The BC-II assumes that

only one corner of the plates was fixed, therefore, it al-

lows large joint rotations (Fig. 5b). In the BC-III, the

free ends of the two plates were fixed (Fig. 5c), as a re-

sult, larger rotations in the joint region can be expected.

Finally, in case of the BC-IV the left edge of the upper

plate was fixed and the right edge of lower plate was

allowed to move only in the y-direction (Fig. 5d).

As a result of the geometrical non-linear stress

analysis of the SLJ the deformed and undeformed ge-

ometries of the adhesive joint are shown in Fig. 6 for

each of the end conditions. In the cases of the BC-II and

BC-III, considerably large rotations are experienced by

the SLJ (Fig. 6b and c). However, more moderate dis-

placements and rotations occur in the cases of the BC-I

and BC-IV (Fig. 6a and d). Since the high stress and

strain gradients occur in the joint region, the von Mises

stress reqv and strain eeqv contours in the joint region are

shown in Figs. 7 and 8 for the end conditions I–IV, re-

spectively.

In case of BC-I, the free ends of the adhesive layer

experience high stress and strain gradients (Figs. 7a and

8a), whereas the upper adherend is under high stresses.

However, the stress and strain levels are lower than

those in the other boundary conditions. The BC-II re-

sults in higher stress and strain levels; thus, adhesive free

ends are still critical regions, as well as the bottom sur-

face of the upper adherend and the top surface of the

Fig. 8. The von Mises strain distributions of a bonded SLJ for the boundary conditions, I–IV (all strains in m/m).


lower adherend (Figs. 7b and 8b). The similar stress and

strain distributions are observed in the joint region for

the BC-III (Figs. 7c and 8c). The highest stress and

strain levels arise in this case. The BC-IV resulted in a

similar stress and strain distribution to those of BC-I.

However, the both bottom and top surfaces of the upper

and lower adherends are under high stresses (Figs. 7d

and 8d). In all boundary conditions, the common be-

haviour is that the highest strains and stresses occur at

the adhesive free ends. In case of the BC-II and -III, the

bottom surface of the upper adherend, and the top

surface of the lower adherend experience serious stresses

close to yield point. In addition, strain concentrations at

the left and right free ends of the adhesive layer exceed

yield strain for the adhesive material.

In order to determine the critical adhesive locations,

the distributions of the von Mises stress and strain inside

the left and right adhesive free ends (inside adhesive

fillets) are shown in Figs. 9 and 10 for the BC (I–IV),

respectively. These averaged adhesive stress and strain

contours are generated using computed stress and strain

values only in the adhesive elements.

In case of the BC-I, the stress and strain concentra-

tions occur at the free ends of the upper adherend–

adhesive interface and the lower adherend–adhesive

interface in the left and right adhesive fillets, respectively

Fig. 9. The von Mises stress distributions in the left and right adhesive fillets at the left and right columns, respectively for the

boundary conditions, I–IV (all stresses in MPa).


as shown in Figs. 9a and 10a. The von Mises stress and

strains reach higher levels through regions close to ad-

hesive–adherend interfaces in the left adhesive fillet.

The similar stress and strain distributions in the left

and right adhesive fillets arise for the BC-II (Figs. 9b

and 10b). However, a large adhesive region near the

rounded adherend corners experiences high stress and

strains, and the BC-II results in higher stress and strain

levels in the critical adhesive regions.

In case of the BC-III, the stress and strain distribu-

tions in the adhesive fillets are similar to those in the BC-

II, but are relatively higher as shown in Figs. 9c and 10c,

respectively. The stress and strain concentrations occur

at the free ends of the upper adherend–adhesive inter-

face and the lower adherend–adhesive interface in the

left and right adhesive fillets, respectively as shown in

Figs. 9d and 10d. These concentration regions spread

around rounded corners of the lower and upper adher-

ends. The BC-II and -III are critical for the SLJ since the

stress and strain concentrations inside the left and right

adhesive fillets exceed yield point (Figs. 9, 10b and c).

As a result, the critical adhesive regions are the ad-

hesive free ends (adhesive fillets) for all the boundary

conditions. The stresses and strains concentrate on

the free ends of the upper adherend–adhesive interface

and the lower adherend–adhesive interface, and propa-

gate along the adhesive fillets towards the adherend

corners.

Fig. 10. The von Mises strain distributions in the left and right adhesive fillets at the left and right columns, respectively for the

boundary conditions, I–IV (all strains in m/m).


The most serious stresses and strains occur for the

BC-II and -III. The main reason is probably high joint

rotations and that the joint translations are restrained.

Therefore, crack initiations and propagations can be

expected in these critical adhesive locations. Whereas the

previous studies have shown that the stress singularities

occurred at the sharp adherend corners [1], the present

study predicted lower and smoothly distributed stresses

around the rounded adherend corners, as shown in Figs.

9 and 10.

In addition, it is evident that the stresses occurring as

a result of thermal strains arising in an adhesive SLJ

whose translations are partly restrained may become

serious. Therefore, the adhesive joints experiencing the

conductive and convective heat transfers might present

more complicated stress and deformation states.

5. Effect of overlap length

The thermal stress analysis of the adhesively bonded

SLJ for different end conditions showed that the stress

concentrations occurred inside the adhesive fillets and at

the free ends of the adhesive–adherend interfaces. The

geometrical modifications of the joint members may

have an effect, such as overlap length. The previous

studies [1], in which the stress analyses of the adhesive

SLJ subjected to a tensile load were carried out, showed

that the overlap length has a considerable effect in re-

ducing the adhesive stress concentration and that an

optimum overlap length exists.

In order to determine the effect of the overlap length

on the thermal stresses the overlap length/joint length

ratio ‘‘a=L’’ (see Fig. 1) was changed from 0.09 to 0.33

whilst the joint length L is kept constant (55 mm). The

thermal stress analysis of the SLJ was carried out for the

boundary conditions I–IV, respectively.

The variations of the normalized normal and shear

stresses rxx, ryy and rxy and von Mises stress evaluated at

the critical adhesive locations were plotted in Figs. 11

and 12 for the boundary conditions I–IV, respectively.

These critical adhesive locations are shown in figure

insets of Figs. 11a, b, 12a and b. The overlap length had

a negligible effect on the adhesive stresses for the BC-I

(Fig. 11a), and resulted in a decrease of about 18–25% in

the peak adhesive stresses for the BC-II. However, in-

creasing the overlap length had a negligible effect on the

adhesive stresses after a ratio a=L of 0.2 (Fig. 11b). The

overlap length showed a similar effect on the peak ad-

hesive stresses for the BC-III, thus, increasing the

Fig. 11. The effect of the overlap length on the stress compo-

nents ðrxx; ryy ; rxyÞ and on the von Mises stress reqv in the

critical adhesive locations of a bonded SLJ subject to a thermal

load for the (a) BC-I and (b) BC-II.


nents ðrxx;ryy ;rxyÞ and on the von Mises stress reqv in the

critical adhesive locations of a bonded SLJ subject to a thermal

load for the (a) BC-III and (b) BC-IV.


overlap length caused a decrease of 5–15% in the critical

adhesive stresses as shown in Fig. 12a. In case of the BC-

IV, the overlap length had a small effect of 8% on the

adhesive stresses as shown in Fig. 12b. However, the

effect of the overlap length was reasonable in the BC-II

and -III until a ratio a=L of 0.2.

The critical regions in which the peak stresses oc-

curred were also determined in the upper and lower

adherends for the BC I–IV, respectively. The critical

locations of the lower adherend were the lower adherend

zone corresponding to the cap of the left adhesive fillet

for the BC-I (see figure inset of Fig. 13a), and the right

free end of the lower adherend–adhesive interface for the

BC-II, -III, -IV (see figure inset of Fig. 13b). The vari-

ations of the normalized normal rxx, ryy and shear rxy

stresses, and von Mises stress at these critical lower

adherend locations were plotted in Figs. 13 and 14, for

the BC I–II and the BC III–IV, respectively. Increasing

the overlap length/joint length ratio ‘‘a=L’’ between 0.09

and 0.33 resulted in small increases in the stress com-

ponents and in the von Mises stresses in the critical

lower adherend locations for the BC-I, -III and -IV.

Increasing the overlap length resulted in minor increases

of 1–8% for the BC-I, increases of 8–30% for the BC-II

and increases of 15–70% for the BC-IV, respectively in

the stress components and in the von Mises stresses.

In the case of the upper adherend, the critical upper

adherend location corresponds to the left free end of the

upper adherend–adhesive interface for the BC-II, -III, -IV

(see figure inset of Fig. 15b), and the cap of the right

adhesive fillet for the BC-I (see figure inset of Fig. 15a).

The effect of the overlap length on the stress components

and on the von Mises stresses evaluated at the critical

upper adherend locations are shown in Figs. 15 and 16,

respectively. Increasing the overlap length resulted in a

decrease of 20% for the BC-II, an increase of 6% for the

BC-III and an increase of 15% for the BC-IV in the von

Mises stress, respectively whereas its effect on the von

Mises stress for the BC-I is insignificant.

Finally, increasing the support length has a variable

effect on the von Mises stresses at the critical adhesive

and adherend locations, which is dependent on the plate

end conditions. Since the adhesive material is a weaker,

when the von Mises stresses evaluated the critical ad-

hesive locations are considered it is observed that in-

creasing the overlap length has reducing effect of the

peak stresses for the BC-II and -III in which the higher

stresses occurred. In addition, this effect becomes negli-

gible after the overlap length/joint length ratio a=L of



critical lower adherend locations of a bonded SLJ subject to a

thermal load for the (a) BC-I and (b) BC-II.



critical lower adherend locations of a bonded SLJ subject to a

thermal load for the (a) BC-III and (b) BC-IV.


0.2. As seen, increasing the overlap length or bonding

area may not be beneficial in order to reduce thermal

stresses for any structural boundary condition. There-

fore, an optimum overlap length should be verified by

experimental studies and the material non-linear ana-

lyses.

The adhesive joints may experience harder environ-

mental conditions in practice, and their stress and de-

formation states may become more complicated. In this

study, it was intended to show how serious the thermal

stresses and strains might become. The results help us

for predicting the stress and deformation states of the

adhesive joints with complicated geometry.

6. Conclusions

In this study, the thermal stress analysis was carried

out using the ANSYS software. The SLJ was assumed to

be subjected to the air flows with different temperature

and velocity along its outer surfaces. For specific ther-

mal conditions, the thermal analysis of the SLJ was

carried out considering the conductive and convective

heat transfers. The final temperature distribution was

used to compute the thermal strains arising in the SLJ.

The thermal stresses were computed for specific adher-

end end conditions considering the effect of the large

displacements. The analysis showed that the thermal

and mechanical mismatches of the adhesive and adher-

ends caused high strain concentrations through adhesive

regions close to adhesive–adherend interfaces around

the adhesive free ends. The detailed analysis showed that

the peak thermal stresses and strains in the adhesive

layer occurred at the free ends of the upper adherend–

adhesive and the lower adherend–adhesive interfaces,

and that these thermal strain and stress conditions ex-

ceed yield point for some plate end conditions. The

elastic analyses showed that increasing the overlap

length are not beneficial in reducing the peak stresses in

the critical adhesive and adherend regions for all ad-

herend end conditions.

Finally, the thermal and material mismatches of the

adhesive and adherends result in serious thermal strains

along the adhesive–adherend interfaces, especially

around the adhesive free ends. In case the variable

thermal conditions along the outer surfaces of the ad-

hesive joint exist, non-uniform temperature distribution

occurs in the adhesive joint, consequently, non-uniform

thermal strain distribution. This makes thermal stress dis-

tribution in the adhesive joint more complex. Therefore,



critical upper adherend locations of a bonded SLJ subject to a

thermal load for the (a) BC-I and (b) BC-II.



critical upper adherend locations of a bonded SLJ subject to a

thermal load for the (a) BC-III and (b) BC-IV.


variable thermal conditions should also be consid-

ered in the analysis and in the design of the adhesive

joints.

References

[1] Adams RD, Wake WC. Structural adhesive joints in

engineering. London: Elsevier Applied Science; 1984.

[2] Lee SJ, Lee DG. Optimal-design of the adhesively bonded

tubular single lap joint. J Adhes 1995;50(2–3):165–80.

[3] Reedy ED, Guess TR. Butt joint strength and effect of

residual-stress and stress-relaxation. J Adhes Sci Technol

1996;10(1):33–45.

[4] Ioka S, Kubo S, Ohji K, Kishimoto J. Thermal residual

stresses in bonded dissimilar materials and their singular-

ities. JSME Int J Ser A—Mech Mater Engng 1996;39(2):

197–203.

[5] Anifantis NK, Kakavas PA, Papanicolaou GC. Thermal

stress concentration due to imperfect adhesion in fiber

reinforced composites. Compos Sci Technol 1997;57(6):

687–96.

[6] Kim YG, Lee SJ, Lee DG, Jeong KS. Strength analysis of

adhesively bonded tubular single lap steel–steel joints

under axial loads considering residual thermal stresses.

J Adhes 1997;60(1–4):125–40.

[7] Kim YG, Lee DG. Influence of fabrication residual

thermal stresses on rubber-toughened adhesive tubular

single lap steel–steel joints under tensile load. J Adhes

1998;65(1–4):163–85.

[8] Cho JH, Kong DI, Park CE, Jin MY. Effect of curing

temperature on the adhesion strength of polyamideimide/

copper joints. J Adhes Sci Technol 1998;12(5):507–21.

[9] Nakano Y, Katsuo M, Kawawaki M, Sawa T. 2-Dimen-

sional thermal stress analysis in adhesive butt joints

containing hole defects and rigid fillers in adhesive under

non-uniform temperature field. J Adhes 1998;65(1–4):57–

80.

[10] Humfeld GR, Dillard DA. Residual stress development in

adhesive joints subjected to thermal cycling. J Adhes

1998;65(1–4):277–306.

[11] Abedian A, Szyszkowski W. Effects of surface geometry of

composites on thermal stress distribution—a numerical

study. Compos Sci Technol 1999;59(1):41–54.

[12] Nakagawa F, Sawa T, Nakano Y, Katsuo M. A 2-

dimensional finite element thermal stress analysis of

adhesive butt joints containing some hole defects. J Adhes

Sci Technol 1999;13(3):309–23.

[13] Katsuo M, Nakano Y, Sawa T. 2-Dimensional transient

thermal stress analysis of adhesive butt joints. J Adhes

1999;70(1–2):75–93.

[14] Adams RD, Harris JA. The influence of local geometry on

the strength of adhesive joints. Int J Adhes Adhes

1987;7(2):69–80.

[15] ANSYS�. The general purpose finite element software

(version 5.3). Houston, TX: Swanson Analysis Systems,

Inc.

[16] Kakac S, Yener Y. Convective heat transfer. Ankara:

Middle East Technical University Publications; 1980.

[17] Malvern LE. Introduction to the mechanics of a contin-

uous medium. Englewood Cliffs, NJ: Prentice-Hall; 1969.

[18] Saada AA. Elasticity theory and applications. London:

Pergamon Press; 1974.

[19] Zienkiewicz OC, Taylor RC. The finite element method

solid and fluid mechanics dynamics and non-linearity, vol

2. New York: McGraw-Hill Company; 1991.

[20] Crisfield MA. Non-linear finite element analysis of solids

and structures, vol I. New York: Wiley; 1991.

[21] Kleiber M. Incremental finite element modelling in non-

linear solid mechanics. New York: Ellis Horwood; 1989.

[22] Sawyer JW, Cooper PA. Analytical and experimental

results for bonded single lap joints with preformed

adherends. AIAA J 1981;19(11):1443–51.

[23] Adams RD. In: Kinloch AJ, editor. Developments in

adhesives, vol 2. London: Elsevier Applied Science; 1981.

p. 45–81.

[24] Harris JA, Adams RD. Strength prediction of bonded

single lap joints by non-linear finite element methods. Int J

Adhes Adhes 1984;4(2):65–78.

[25] Adams RD, Atkins RW, Harris JA, Kinloch AJ. Stress

analysis and failure properties of carbon fibre reinforced

plastic/steel double lap joints. J Adhes 1986;20:29–53.

[26] Reddy JN, Roy S. Non-linear analysis of adhesively

bonded joints. Int J Nonlinear Mech 1988;23(2):97–112.

[27] Czarnocki P, Pierkaski K. Non-linear numerical stress

analysis of a symmetric adhesive bonded lap joint. Int J

Adhes Adhes 1986;6(3):157–60.

[28] Edlund U, Klarbring A. A geometrically non-linear model

of the adhesive joint problem and its numerical treatment.

Comput Meth Appl Mech Engng 1992;96:329–50.

[29] Apalak MK, Engin A. Geometrically non-linear analysis

of adhesively bonded double containment cantilever joint.

J Adhes Sci Technol 1997;11(9):1153–95.

[30] Apalak MK. Geometrically non-linear analysis of adhe-

sively bonded modified double containment corner joints.

II. J Adhes Sci Technol 1998;12(2):135–60.


sively bonded double containment corner joints. J Adhes

1998;66:117–33.


sively bonded corner joints. J Adhes Sci Technol

1999;13(11):1253–85.


sively bonded modified double containment corner joints.

I. J Adhes Sci Technol 2000;14(9):1159–78.

[34] Andruet RH, Dillard AD, Holzer SM. Two- and three-

dimensional geometrical non-linear finite elements for

analysis of adhesive joints. Int J Adhes Adhes 2001;21(1):

17–34.


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On non-linear thermal stresses in an adhesively bonded single lap joint