Thermal residual stresses in an adhesively bonded functionally graded tubular single lap joint

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doi:10.1016/j.ija

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International Journal of Adhesion & Adhesives 27 (2007) 26–48

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Thermal residual stresses in an adhesively bonded functionally gradedtubular single lap joint

M. Kemal Apalak�, Recep Gunes, Selim Eroglu

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

Accepted 26 September 2005

Available online 30 March 2006

Abstract

Elastic thermal residual stress analysis of an adhesively bonded functionally graded tubular joint was carried out for a uniform

temperature drop using an isoparametric axisymmetric layered finite element. The hoop stress syy, axial stress szz and transverse shear

stress srz were dominant for both free and constrained models of the tubular joint. The stress concentrations occurred around the

adhesive free ends, especially at the left and right free edges of the inner tube–adhesive interface and at the left free edge of the outer

tube–adhesive interface. Increasing layer number played important role on the peak stress values in the ceramic layer of the metal-rich

composition and in the metal layer of ceramic-rich composition since a sudden transition between two phases in these layers, whereas its

effect on the through-the-thickness stress profiles was minor. In addition, the through-the-thickness stress profiles at the critical locations

in both tubes and in the adhesive layer were affected evidently by the directional material composition variation. The compositional

gradient had a minor effect on the through-the-thickness stress profiles whereas only the peak stress values were affected. The hoop, axial

and shear stresses increased with increasing compositional gradient. In the free joint model, the effects of the compositional gradient,

layer number and the directional compositional variation on the through-the-thickness stress variations were apparent whereas the effect

of the boundary condition became noticeable in the constrained joint model.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Functionally graded material; Finite element stress analysis; Joint design

1. Introduction

Adhesively bonding technique has been widely used dueto its ability to distribute a load over a larger area thanmechanical fasteners without holes, high load bearing/weight capability and superior fatigue resistance [1–5].However, some factors affect strongly performance ofadhesive joints, such as surface treatment of adherends,especially the environmental temperature. The residualthermal stresses occur due to the mismatches of the thermaland mechanical properties of adherends and adhesiveduring the curing operation. Consequently, the thermalstress state has attracted the researchers [6]. In practice,adhesive joints may also undergo thermal loads as well asstructural loads. The stress and deformation fields in theadhesive joint due to these thermal loads play an important

e front matter r 2006 Elsevier Ltd. All rights reserved.

dhadh.2005.09.009

ing author. Tel.: +90 352 437 4901; fax: +90 352 437 5784.

ess: [email protected] (M.K. Apalak).

role in the strength of the adhesive joint. Rastogi et al. [7]studied three-dimensional (3D) thermal stress distributionsin aluminium-to-composite, symmetric, double lap jointssubjected to uniform temperature loads. They found thatthe joint corners are critical regions for debondinginitiation. Shin and Lee [8] investigated stress distributionsin co-cured single lap joints subjected to a tensile load usingthe finite element analysis by considering thermal residualstresses resulted from curing process, and showed that theinterfacial tensile stress was a primary factor that causedinterfacial delamination between the steel and compositeadherends in the co-cured single lap joint. Kim and Lee [9]determined thermal characteristics of tubular single lapadhesive joints under axial loads and investigated theeffects of environmental temperature and filler on thetensile modulus and failure strength of the epoxy adhesive.They showed that the average tensile strength of thetubular single lap adhesive joint did not decrease muchcompared to the adhesive properties because the tensile

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ARTICLE IN PRESSM.K. Apalak et al. / International Journal of Adhesion & Adhesives 27 (2007) 26–48 27

residual thermal stress in the joint decreased as theenvironmental temperature is increased, and the mechan-ical properties of the epoxy adhesive decreased sharply inthe vicinity of glass transition temperature.

Adhesive bonding technology has also found a goodplace in joining advanced composite materials. Especially,the adhesive composite lap joints have been used widely invarious engineering structures such as aircraft and auto-motive structures due to its manufacturing simplicity andstructural effectiveness [10]. However, some applicationsrequire high thermal gradients be reduced through thethickness of a structural component. Even a layeredcomposite material may have better mechanical propertiesalong a direction. An abrupt change in their materialproperties across an interface between discrete materialsintroduces large interlaminar stresses that could causedelamination. Therefore, the thermal and mechanicalmismatches of the layered composite materials becomemore important. One way to overcome this adverse effect isto use a functionally graded material (an FGM). FGMs aredesigned as high-performance heat-resistant materialswhich can withstand high-temperature applications. FGMsare microscopically inhomogeneous; thus, their thermo-mechanical properties vary smoothly and continuouslyalong one or more directions [11]. This is achieved with thegradual variation of the volume fraction of the constituentsof FGMs. These constituents are generally ceramics andmetals. The ceramic constituent provides high-temperatureresistance as a thermal barrier due to its low thermalconductivity whereas the ductile metal constituent im-proves the fracture toughness to a probable thermalgradient for short period of time. FGMs were firstintroduced as a concept by a group scientist in Sendai,Japan, in 1984 [12] and then developed by other scientists[13,14]. Stresses in the FGMs under thermal loading havebeen analysed extensively with regard to the elastic–plasticmaterial behaviour in the last years [15–17]. Cheng andBatra [18] analysed thermo-mechanical deformations of alinear elastic functionally graded elliptic plate with rigidlyclamped edges. They used the asymptotic expansionmethod for the analysis and found that the gradients inmaterial properties significantly affect the response of afunctionally graded plate under thermal loads. Reddy andCheng [19] studied 3D thermo-mechanical deformations ofa simply supported functionally graded rectangular plateby using an asymptotic method. They found that theassumption of a constant through-thickness deflectionusually made by 2D plate theories is invalid for the caseof the thermal load. Vel and Batra [20] obtained an exactsolution for 3D deformations of a simply supportedfunctionally graded rectangular plate subjected to mechan-ical and thermal loads. Vel and Batra [21] also presentedanalytical solution for 3D thermo-mechanical deforma-tions of a simply supported functionally graded rectangularplate subjected to time-dependent thermal loads. Cho andOden [22] analysed thermal stress characteristics of FGMs.They investigated the effects of the material variation

through the thickness and the size of the FGM layerinserted between metal and ceramic layers using the finiteelement method. Cho and Ha [23] investigated optimumvolume fraction distribution that minimizes steady-statethermal stresses in FGMs with 1D volume fractionvariation by using both penalty-function and golden-section method. Nemat-Alla [24] introduced two-dimen-sional FGMs (2D-FGMs) to withstand super-high tem-peratures and to give more reduction in thermal stresses,and showed that 2D-FGM has high capabilities to reducethermal and residual stresses than conventional FGM;thus, yielding occurred for a conventional FGM while itdid not for a 2D-FGM. In addition, the zone, in whichmaximum thermal stresses, is smaller for 2D-FGM thanconventional FGM. Apalak and Gunes [25] investigated3D thermal residual stresses occurring in functionallygraded plates subjected to various thermal fields andshowed that a 3D model is necessary in order to under-stand the stress state of the functionally graded platessubjected to different temperature fields, since the lateralstraining effects on the peak stresses are predictedcorrectly.Metal and composite tubes are becoming more popular

in structural applications and transmission shafts. Thesetubes are often connected to each other with an adhesivelayer. The bond strength and its degradation during servicedepend on the mechanical properties of the tubes andadhesive, the geometry of the tubes, voids in the adhesivelayer and loading conditions. There are many studies onthe stress states of tubular joint subjected to structural andthermal loads [26–29]. However, the stress state ofadhesively bonded functionally graded composite lap jointsunder thermal loads has not been studied in detail.In this study, 3D thermal residual stress analysis of an

adhesively bonded functionally graded tubular single lapjoint was carried out for a uniform temperature fieldthroughout the joint and two edge conditions. The outerand inner tubes of the adhesive lap joint are made ofFGMs composed of ceramic (Al2O3) and metal (Ni)phases. The thermal and mechanical properties of theFGM are assumed to vary continuously through the tubethickness between the inner metal and outer ceramic layersaccording to a power-law distribution. An epoxy-basedadhesive was used to bond tubes. The mismatches of thethermo-mechanical properties of the adherends and ad-hesive layer cause high stress discontinuities along theadherend–adhesive interfaces [30,31]. These discontinuousstresses also arise in the functionally graded tubular joints.In order to determine the effect the mechanical propertiesof the layer of each tube having an interface with theadhesive layer on the stress state of the adhesive tubular lapjoint, three types of tubular joints were considered; thus,the tubes are designed such that the through-the-thicknessmechanical properties of the tubes vary continuously fromthe inner metal layer to the outer ceramic layer and inreverse sense. In addition, the effects of layer number andthe compositional gradient exponent on the thermal

ARTICLE IN PRESSM.K. Apalak et al. / International Journal of Adhesion & Adhesives 27 (2007) 26–4828

residual stresses occurring in the adhesive tube joint areinvestigated in detail.

2. Constitutive relations

A graded interlayer (FGM) between the ceramic-richand metal-rich layer of the tubes can reduce andredistribute residual stresses induced by their thermal andmechanical mismatches. In general, the FGM is designedas a layer with continuous composition variation throughthe tube thickness in order to provide a high-temperatureresistance on its face exposed to the high temperature usingthe low thermal conductivity of the ceramic constituent.

An actual FGM consists of ceramic and metal particleswith arbitrary shapes mixed up in random dispersionstructures. Thermo-mechanical properties of a FGM are afunction of this shape and orientation of ceramic and metalparticles, the dispersion structure as well as volumefraction. For this purpose, the simple estimation methodis the linear rule of mixtures in which a generic materialproperty P at any point x in the graded region isdetermined by the linear combination of volume fractionsof ceramic (c) and metal (m) as

PðxÞ ¼ VmðxÞPmðxÞ þ V c xð ÞPcðxÞ, (1)

providing that

Vm þ V c ¼ 1. (2)

This method does not consider the effects of particle anddispersion structure, and the interaction between twoconstituents. The thermo-mechanical behaviour of FGMsis strongly dependent on the accurate estimation of theirmodulus of elasticity and coefficient of thermal expansion[22,32–37].

This study assumes that the functionally graded compo-site tubes are composed of a graded layer between

(a)

(b)

I

M

x3,�

z r

θ

L a

R2i

R2o

�3

x1, r

x2, z

Fig. 1. (a) An adhesively bonded functionally graded tubular sin

homogeneous isotropic ceramic and metal phases as shownin Fig 1a. The volume fraction of the metal phase at anyradial position r through the tube thickness is of the powerlaw type

VmðrÞ ¼t� r

t

� �n

, (3)

where n is the compositional gradient exponent, r thethrough-thickness radial position from the bottom of eachtube and t the tube thickness. Fig 2a shows the through-thickness variation of the volume fraction of the metal forthe compositional gradient exponents n ¼ 0.1, 0.5, 1.0, 2.0,5.0 and 10.0 where the top surface of the functionallygraded tube is ceramic-rich zone and its bottom surface ismetal-rich zone.Tamura et al. [38] proposed a modified rule of mixtures

for the modulus of elasticity as

E ¼qþ Ec

qþ Em

� �VmEm þ ð1� VmÞEc

� �

�qþ Ec

qþ Em

� �Vm þ ð1� VmÞ

� ��1, ð4Þ

where the stress–strain transfer ratio is

q ¼sc � smec � em

where ð0oqoþ1Þ: (5)

The choice of value q affects the averaged modulus ofelasticity based on the modified rule of mixture. For a well-dispersed metal–Al2O3 composite, a value q of 500GPa isrecommended [32].The through-thickness variation of modulus of elasticity

of Ni–Al2O3 functionally graded tube is shown in Fig. 2.The thermal and mechanical properties of constituents ofNi–Al2O3 composite material are given in Table 1. For theisotropic particulate composites, Wakashima–Tsukamoto[36] expressions require that the overall thermal expansion

J

N

1

NlL

100 % ceramic layer

100 % metal layer

FGM

t2

R1i

R1o

CL

�1

�2

gle lap joint and (b) a layered axisymmetrical finite element.

ARTICLE IN PRESS

Table 1

Thermal and mechanical properties of constituents of functionally graded

tubes, and adhesive

Properties Constituents Epoxy

adhesive

Ni Al2O3

Modulus of Elasticity, E (GPa) 199.5 393 4.39

Poisson’s ratio, n 0.3 0.25 0.34

Coefficient of thermal

expansion, a (10�6/1C)

15.4 7.4 40.47

0.25

0.26

0.27

0.28

0.29

0.3

Poi

sson

's r

atio

,v

7.4

9

10

11

12

13

14

15.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Vol

ume

frac

tion

of th

e m

etal

, Vm

200

240

280

320

360

393

(d) (c)

(b) (a)

n=0.1

0.2

0.5

1.0

2.0

10.0n=0.1

0.2

0.5

1.0

2.0

5.0

10.0

n=0.1

0.2

0.5

1.0

2.0

5.0

10.0

n=0.1

0.2

0.5

1.0

2.0

5.010.0

5.0

Distance through tube thickness, (mm)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Distance through tube thickness, (mm)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Coe

ffici

ent o

f the

ther

mal

exp

ansi

on, α

(x10

-6/°

C)

Mod

ulus

of E

last

icity

, E (

GP

a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Fig. 2. Through-thickness variations from bottom to top of (a) volume fraction of metal phase Vm, (b) modulus of elasticity E, (c) Poisson’s ratio n and (d)

the coefficient of thermal expansion a of a Ni–Al2O3 functionally graded tube for different compositional gradient exponents n.

M.K. Apalak et al. / International Journal of Adhesion & Adhesives 27 (2007) 26–48 29

coefficient for dual-phase materials is related to theaveraged bulk modulus using Levin’s relation [39]

a ¼ am þ1=K � 1=Km

� ðac � amÞ

ð1=KcÞ � ð1=KmÞ, (6)

where the overall bulk modulus K is

K ¼ Km þaV cKmðKc � KmÞ

VmKE þ aV cKm. (7)

And the shear modulus m is

m ¼ mm þbV cmmðmc � mmÞVmmc þ bV cmm

, (8)

where a and b are

a ¼Kcð3Km þ 4mmÞKmð3Kc þ 4mcÞ

,

b ¼ð1þ eÞmcmm þ emc

,

together with

e ¼9Km þ 8mm6Km þ 12mm

.

The overall Poisson’s ratio is written as

n ¼3K � 2m2ð3K þ mÞ

. (9)

Figs. 2c and d show the through-thickness variations ofPoisson’s ratio and the coefficient of thermal expansion ofNi–Al2O3 FGMs.

3. Multilayered isoparametric axisymmetrical four-noded

element

The finite element formulation is based on the potentialenergy in a linearly elastic body and more detailedinformation can be found in Ref. [40]. The generalspecifications of the layered finite element are as follows:A four-noded, isoparametric axisymmetrical element hasthe associated local (x1, x2, x3) and global (x1, x2, x3)coordinate systems as shown in Fig. 1b. The element iscomposed of N l layers in its thickness direction x1. Theformulation of the element is based on the axisymmetricproblems of the elasticity theory and is analogous to that ofthe single and multilayered curved isotropic shell elements


given by Ahmad et al. [41] and Yunus and Khonke [42]. Allintegration points along the thickness direction x1 areassumed to have the same material orientation. Since thefinite-element is based on the displacements, the strain–displacement relations can be written in terms of the partialderivatives of the displacement components with respect tothe global coordinates [41–44].

3.1. Stress–strain relations

The elasticity matrix ½D�j for layer j is given as

½D�j ¼

1E1

� n12E1� n13

E10

� n12E1

1E2

� n23E2

0

� n13E1� n32

E3

1E3

0

0 0 0 1G12

2666664

3777775

�1

j

. (10)

In order to provide continuity of stress between thelayers, the elasticity matrix is modified for the layer j as

½D��j ¼

1E�1

�n�12

E�1�

n�13

E�10

�n�12

E�1

1E2

� n23E2

0

�n�13

E�1� n32

E3

1E3

0

0 0 0 1G12

266666664

377777775

�1

j

, (11)

where

n�13 ¼ E�1n13E1

� �j

; n�12 ¼ E�1n12E1

� �j

and

E�1 ¼tPN l

j¼1

tE1

� �j

,

where tj is the average thickness of layer j, N l the number oflayers, t the average total thickness of element, and thesuffices 1, 2 and 3 are coordinate variables z, r and y in thecylindrical coordinates, respectively.

3.2. Stiffness matrix formulation

The element stiffness matrix is written in terms of naturalcoordinates as

½K� ¼ 2pZ 1

�1

Z 1

�1

½B�T½D��j½B�x1 det½J�dx1 dx2, (12)

where ½D�� is the modified stress–strain matrix at the pointof interest within the element. For a layered element, Eq.(12) yields

K½ � ¼ 2pZ 1

�1

XN‘

j¼1

Z ðx1Þtopj

ðx1Þbotj

½B�T ½D��j½B�x1 det½J�dx1 dx2 (13)

[42]. The through-the-thickness numerical integration iscarried out by modifying the variable x1 to (x1)j in the jth

layer such that x1 varies from –1 to 1 in that layer [43–44]:

x1 ¼ �1þ1

t2Xj

k¼1

hk � hjð1� x1jÞ

" #(14)

and

dx1 ¼hj

tdxj . (15)

Substituting Eqs. (14) and (15) into Eq. (13) yields

K½ � ¼

Z 1

�1

XN‘

j¼1

Z 1

�1

½B�T½D��j½B�x1 det½J�hj

t

� �dx1 dx2. (16)

A 2� 2� 2 Gauss integration scheme is needed for eachlayer in order to evaluate the contribution of that layer tothe stiffness of the element.

3.3. Thermal load vector

In order to derive the thermal load vector a biquadraticdistribution of temperatures is assumed on the top (MN)and bottom (IJ) edges and a linear variation through itsthickness (see Fig. 1b). The thermal strain at a point ðx1; x2Þin the layer j is then given as

fethgj ¼ fagj½T1 þ x1T2 � T ref �, (17)

where

fagj ¼ ½ a1 a2 a3 0 0 0 �Tj ;

ða1Þj , ða2Þj and ða3Þj are coefficients of thermal expansionin thex1, x2 and x3-directions, respectively. For the layer j,

T1 ¼T t þ Tb

2and T2 ¼

T t � Tb

2, (18)

where Tt and Tb are, respectively, the temperatures at thetop and bottom edges of the element at the point ðx1; x2Þ.The thermal strain vector in Eq. (17) is written as

fethgj ¼ x1feth1 gj þ x1fe

th2 gj, (19)

where

feth1 gj ¼ fagjðT1 � T ref Þ,

feth2 gj ¼ fagjT2.

The consistent thermal load vector is given as

fF thg ¼

ZV

½B�T½D��fethgdV (20)

and for the layered element as

fF thg ¼XN l

j¼1

ZV

½B�T Tm½ �Tj ½D

��jfethj gdðV jÞ, (21)

where Vj is the volume of layer j and [Tm] is transformationmatrix from the global ðx1;x2;x3Þ to the local ðx1; x2; x3Þcoordinate system [41].


4. Problem description and stress analysis

This study investigates the 3D thermal residual stressesin an adhesively bonded functionally graded tubular singlelap joint for a uniform temperature field under two tube-edge conditions.

The tubular single lap joint consists of epoxy adhesivelayer and two functionally graded tubes composed ofceramic (Al2O3) and metal (Ni). The main dimensions ofthe functionally graded tubular single lap joint are shownin Fig. 1a. An inner radius R1i ¼ 20mm for the inner tube,an inner radius R2i ¼ 22.2mm for the outer tube, a wallthickness of 2mm and a length L ¼ 225mm for inner andouter tubes, the overlap length a of 50mm, and an adhesivethickness t2 ¼ 0.2mm were kept constant throughout theanalysis. The thermal and mechanical properties of thetube material are assumed to vary continuously throughthe tube thickness according to a power-law distribution ofthe volume fraction of the constituents. The thermo-mechanical properties of the graded-tube layers weredetermined by using the Wakashima–Tsukamoto’s estima-tion method as given in Section 2. The thermo-mechanicalproperties of ceramic (Al2O3), metal (Ni) and epoxyadhesive are given in Table 1. The volume fraction of themetal Vm, the modulus of the elasticity E, Poisson’s ratio n,and the coefficient of thermal expansion a variations of thefunctionally graded regions of both tubes for the composi-tional gradient exponents n ¼ 0.1, 0.2, 0.5, 1.0, 2.0, 5.0 and10.0 are shown in Fig. 2. In the analysis, the assumptionsare (i) the joint members, i.e. tubes and adhesive layer, arelinear elastic, (ii) the pure ceramic layer does not includeporosity; therefore, there is no penetration of the adhesivematerial into the ceramic porosity, (iii) no bubble forma-tion occurs in the adhesive layer, (iv) the bonding region isfree of the residual thermal stress occurring during thecuring period, (v) due to the surface treatment of the Nilayer, the changes in the mechanical properties of the Nilayer are ignored and (vi) the interphase formation isignored due to the adhesive and substrate reaction inpractice.

zr

θ

Outer FG Tube

Fig. 3. Finite element model of an adhesively bond

The thermal residual stress analysis of the functionallygraded tubular single lap joint is a 3D problem. Since theadhesive joint has an axisymmetric geometry, one degreesector of the model around the centre axis was considered.The symmetry conditions were applied to side surfaces ofthe tubes and the adhesive layer. Fig. 3 shows the finiteelement mesh of the adhesive joint. An isoparametricaxisymmetrical four-noded layered finite element was usedto model the tubes and an isoparametric axisymmetricalfour-noded structural finite element for the adhesive layer.Both finite elements include two degree of freedoms at eachnode and are formulated based on the 3D elasticity theory.Twenty elements through the thickness of both tubes and10 elements through the adhesive thickness gave stable andenough accurate results. The numerical integration proce-dure through the layered finite elements requires morecomputational time and sources as the total degree offreedoms of the model is increased. Consequently, a seriesof pre-analyses showed that the finite element model of theadhesively bonded tubular single lap joint shown in Fig. 3was successful to obtain accurate results for different layernumbers through the tubes.The thermal residual stress state of the adhesively

bonded functionally graded tubular single lap joint wasanalysed for a uniform temperature field and two differentend conditions as shown in Fig. 4. Both outer and innertubes are graded through the tube thickness between aceramic layer on the top surface and a metal layer on thebottom surface of the tubes. The mechanical properties ofthe functionally graded region between ceramic and metallayers vary based on the compositional gradients shown inFig 2. The outer and inner tubes are joined by means of anepoxy adhesive layer. The tube–adhesive interfaces areregions where the mechanical properties of the materialsare discontinuous. Under the thermal loads, thermalstrains on both sides of the tube–adhesive interfaces willbe different due to different coefficients of the thermalexpansion; therefore, undesirable thermal stresses may takeplace along the tube–adhesive interfaces, especially on theboth free edges of these interfaces. It is evident that the

CL

Inner FG Tube

ed functionally graded tubular single lap joint.

ARTICLE IN PRESS

CCL

(a) (b)θ

rz

Fig. 4. Deformed and undeformed geometries of the adhesively bonded functionally graded tubular single lap joint: (a) BC-I and (b) BC-II.

M.K. Apalak et al. / International Journal of Adhesion & Adhesives 27 (2007) 26–4832

mechanical and thermal properties of the materials on bothsides of the tube–adhesive interfaces play an important roleon the thermal stresses in the vicinities of the tube–adhesiveinterfaces. The grading direction through the tube thick-ness can be changed, for instance, a functionally gradedregion between a metal (M) top surface and a ceramic (C)bottom surface. In addition, a different grading directioncan be considered for each of both outer and inner tubes.In order to determine the effect of the different gradingdirections on the thermal residual stresses, three differentcases of the grading direction in each tube are considered:(a) a functionally graded region between a ceramic topsurface and a metal bottom surface in the outer tube (CM),and an functionally graded region between a ceramic topsurface and a metal bottom surface in the inner tube (CM);hereafter, this combination is denoted by (CM–CM), (b) anouter tube with material variation (CM) and inner tubewith material variation (MC) and (c) an outer tube withmaterial variation (MC) and an inner tube with materialvariation (CM). In addition, the stress analyses werecarried out for the following thermal and structuralboundary conditions: The first case (BC-I) assumes thatall components of the adhesive tubular joint were free andwere cooled from an initial uniform temperature of 100 1Cto a final uniform temperature of 20 1C (Fig 4a), and thesecond case (BC-II) assumes that both the left and rightfree ends of the adhesive tube joint were fixed and allcomponents of the adhesive joint were cooled from aninitial uniform temperature of 40 1C to a room temperatureof 20 1C (Fig 4b). The thermal stress analysis showed thatthe adhesively bonded functionally graded single lap jointhad deformed geometries for both edge conditions asshown in Fig. 4. The overlap region of the adhesive tubularsingle lap joint undergoes considerable deformations forthe BC-II whereas minor deformations in the overlapregion appear for the BC-I.

4.1. Constraint-free model (BC-I)

The adhesively bonded tubular single lap joint wassubjected to a uniform temperature drop from 100 to 20 1Cand is free. In order to determine the general trend of thestress concentrations along the tube–adhesive interfaces,three directional material variations were considered forthe functionally graded outer and inner tubes of theadhesive tubular joint as (i) Ceramic–Metal outer tube and

Ceramic–Metal inner tube (CM–CM), (ii) CM-outer tube,MC-inner tube and (iii) MC-outer tube, CM-inner tube.

4.1.1. Tubular joint with CM-outer tube and CM-inner tube

(CM–CM)

The analysis showed that the hoop stress syy, the axialstress szz and the shear stress srz had higher levels in theouter and inner tubes, whereas the radial stress srr, thehoop stress syy, and the shear stress srz were higher in theadhesive layer. Fig. 5 shows the von Mises stress seqvdistributions in the outer tube, the adhesive layer and theinner tube in the overlap region of the tubular joint(CM–CM) with the metal-rich composition (n ¼ 0.1). Theouter and inner tubes were modelled using 60 layers (threelayers each element) through their wall thicknesses. Thetop ceramic-rich layer of the outer tube experiencesconsiderable von Mises stress seqv distributions (Fig 5a).However, the stress concentrations appeared at the rightcorner of the outer tube due to a sharp change in thematerial composition from the ceramic to the metalthrough the first layer of the outer tube. These stressconcentration regions of the outer tube are far away fromthe tube–adhesive interface and are effective on thedeformation of the outer tube. Since the metal constituentNi is dominant in the through-thickness compositionalvariation of the outer tube, the outer tube experienceslower stress distribution through the remaining tuberegions, except the ceramic-rich layer (Fig 5a). In theadhesive layer, the von Mises stress concentrations areobserved at the left and right free ends of both outertube–adhesive and inner tube–adhesive interfaces (Fig 5b).However, the left free end of the outer tube–adhesiveinterface and the right free end of the inner tube–adhesiveinterface are more critical. Since the inner tube–adhesiveinterface contacts with the ceramic layer of the inner tube,the von Mises stress peaks in the right free end of the innertube–adhesive interface. Therefore, the probable crackinitiation can be expected at the left free end of the outertube–adhesive interface and at the right free end of theinner tube–adhesive interface. The von Mises stressdistributions in the inner tube (Fig 5c) of the overlapregion are similar to those of the outer tube (Fig 5a). Thepeak von Mises stresses appear along the top ceramic layerof the inner tube and suddenly decrease in the vicinity ofthe ceramic layer, and then become minimal in theremaining metal layers of the inner tube. Therefore, the

ARTICLE IN PRESS

12.0

12.012.012.012.0

12.0

12.012.012.012.012.02.0

24.024.024.024.024.0 36.036.036.036.036.0 48.048.048.048.0

80

60.060.060.060.0

600 72.072.072.072.0 84.084.084.084.0 96.096.096.096.096.0

Axial position, (mm)

Rad

ial p

ositi

on, (

mm

)

-25-20-15-10-505101520250.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

8.7

8.7

8.7

8.7

10.3

10.3

10.3

10.310.3

11.9

11.9

11.9

11.9

11.9

11.911.9

11.9

13.5

13.5

13.5

13.5

13.5

13.5

13.5

13.5

15.1

15.1

16.818.420.021.6


Rad

ial p

ositi

on, (

mm

)

2323.223.423.623.82424.224.424.624.8250.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

24.024.024.024.0

24.0

36.036.036.036.0

48.048.048.0

48.0

48.0

60.060.060.060.060.0

72.072.072.072.072.0

84.084.084.0

84.084.0

96.096.096.096.0

108.0108.0108.0108.0108.0


Rad

ial p

ositi

on, (

mm

)

-27-26.7-26.4-26.1-25.8-25.5-25.2-24.9-24.6-24.3-241.80

1.82

1.84

1.86

1.88

1.90

1.92

1.94

1.96

1.98

2.00

13.5

13.5

13.5

13.5

13.5

15.1

15.1

15.1

15.1

15.1

15.1

15.115.1

15.1

16.8

16.8

16.8

16.8

16.8

16.818.4

18.420.0


Rad

ial p

ositi

on, (

mm

)

-25-24.8-24.6-24.4-24.2-24-23.8-23.6-23.4-23.2-230.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

25125.125.125.125.1

48.48.448.448.448.471.771.771.771.771.7

94.994.994.994.994.9

Rad

ial p

ositi

on, (

mm

)

-25-20-15-10-505101520250.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

25.1

48.4

48.448.448.448.4 71.7

71.7

71.771.771.7

94.994.9

94.9

94.994.994.9

118.2118.2

118.2

118.2

141.5141.5

164.8164.8188.0211.3

Axial position, (mm)R

adia

l pos

ition

, (m

m)

-25-24.9-24.8-24.7-24.6-24.5-24.4-24.3-24.2-24.1-241.80

1.82

1.84

1.86

1.88

1.90

1.92

1.94

1.96

1.98

2.00

(b) Adhesive Layer(a)

θ

r

z

(c)

CL

(a)

(b)

(c)


Inner FG tube

Outer FG tube

Fig. 5. von Mises stress seqv distributions in the overlap region of (a) the outer tube, (b) the adhesive layer and (c) the inner tube of an adhesively bonded

functionally graded tubular single lap joint with CM–CM tube configuration for the BC-I (compositional gradient n ¼ 0.1) (all stresses in MPa).



von Mises stresses in the top ceramic layer are 9 timeshigher than those in the bottom metal layer.

The thermal residual stress analysis of the adhesivelybonded functionally graded tubular single lap joint withCM–CM composition for a compositional gradient ex-ponent n ¼ 0.1 showed that the top ceramic-rich layers ofboth tubes experience stress concentrations. In addition,the von Mises stress becomes critical at the left free edge ofthe outer tube–adhesive interface, and at the right free edgeof the inner tube–adhesive interface. The peak residualstresses on the top ceramic-rich layers of both the outer andthe inner tubes decrease as the compositional gradientexponent n is increased. However, the peak residual stressesincrease on the bottom metal-rich layers of the both tubeswith increasing compositional gradient n. The adhesivelayer has considerably high residual stresses for onlyn ¼ 1.0. Accordingly, the stress contours in the overlapregion are not shown here for the compositional gradientexponents n ¼ 1.0 and 10.0 in order to avoid unnecessaryrepetition. For all compositional gradient exponents, thecritical regions are free ends of the tube–adhesive layer.

-15

-12

-9

-6

-3

0

3

6

9

12

-25-20-15-10-50510152025-20

-15

-10

-5

0

5

10

15

20

25

(a)

(c)

Outer FG tube

θ

r

z

Inner tube-adhesive interface

Distance along the overlap, mm


Rad

ial s

tres

s σ rr

, MP

aS

hear

str

ess

σ rz, M

Pa

-25-20-15-10-50510152025

Outer interfaceInner interfaceAdhesive half-thickness


Fig. 6. Distributions of (a) radial stress srr, (b) hoop stress syy, 9c) shear stress sinner tube–adhesive interface and the adhesive half-thickness in an adhesiv

(Ceramic–Metal and Ceramic–Metal) composition for the BC-I (n ¼ 0.1).

Therefore, the radial srr, hoop syy, shear srz and von Misesseqv stress distributions along the both inner and outertubes–adhesive interfaces and the adhesive half-thicknessfor n ¼ 0.1 are shown in Fig. 6. The adhesive layerexperiences radial srr and hoop syy stresses and the shearstress srz concentrations along the left and right free edgesof the adhesive layer. The radial stress srr is negligiblealong a large region in the middle of the overlap regionwhereas it seriously concentrates in tension around the leftfree end of the adhesive layer and in compression aroundthe right free edge (Fig. 6a). The peak radial stress srr

values occur at the left free end of the inner tube-adhesiveinterface. The hoop stress syy also peaks at the left free endof the inner tube–adhesive interface, and remains at a highlevel along the middle overlap region and then decreasessuddenly and becomes negligible at the right free end (Fig.6b). The shear stress srz is very low in comparison with thepeak radial and hoop stress components around theadhesive left and right free edges (Fig. 6c). However, thepeak shear stress appears at the left free end of the innertube–adhesive interface. von Mises stress seqv distribution

0

3

6

9

12

15

18

21

24

27

0

3

6

9

12

15

18

21

24

(b)

(d)

Inner FG tube

Outer tube-adhesive interface

CL

Adhesive half-thickness



Circ

umfe

renc

ial s

tres

s σ θθ

, MP

a

-25-20-15-10-50510152025Distance along the overlap, mm

-25-20-15-10-50510152025


von

Mis

es s

tres

s σ eq

v, M

Pa

rz and (d) von Mises stress seqv along the outer tube–adhesive interface, theely bonded functionally graded tubular single lap joint with CM–CM

ARTICLE IN PRESS

zr

θ CL

Outer FG Tube

Inner FG Tube

Outer FG Tube

Inner FG Tube

A(r,z=25) B(r,z=0) C(r,z=-25)

Adhesive LayerT

hrou

gh-t

hick

ness

dire

ctio

n

Fig. 7. Some selected locations in the outer tube, the adhesive layer and the inner tube, at which through-thickness stress variations were evaluated, along

the overlap region of the functionally graded adhesively bonded tubular single lap joint (all dimensions in mm).


in the adhesive layer is affected by the radial srr and hoopsyy stresses. Both the right and left free ends of the innertube–adhesive interface undergo considerable von Misesstress concentrations (Fig. 6d); therefore, the probablecrack initiation can be expected at these critical regions.

A through-the-thickness material composition variationis achieved using a layered structure in the stress analysis.Each layer is isotropic and homogeneous. Therefore, asmoothed material composition variation was obtainedwith increasing layer number. The effect of the layernumber in each functionally graded tube on its through-the-thickness residual stress profiles and magnitudeswere investigated. For this purpose, the layer numberthrough the adherend thickness was considered as m ¼ 20,40, 60, 100 and 200 corresponding to a layer thickness of0.1, 0.05, 0.033, 0.02 and 0.01mm. As the layer number isincreased, even though the numerical integration throughthe thickness of each element requires more computationalsource and time, the stress profiles and levels becomesimilar, and minor differences were observed for the layernumber X200. The effect of the layer number on thethrough-the-thickness variations of the normal, shear andvon Mises stresses at the selected locations (see Fig. 7) inthe overlap region are shown in Figs. 8–10 for thecompositional gradient exponents n ¼ 0.1, 1.0 and 10.0,respectively.

In case n ¼ 0.1, the normal (syy, szz) and von Mises(seqv) stresses have peak values along the top ceramic layers(Fig. 7) of both the outer (point A) and inner (point C)tubes at which large differences in stress levels appeared(Fig. 8a,b). All stress components decrease considerablyuntil a thickness of 0.2mm from the top ceramic layer andremain almost same through the tube thickness. The shearstress srz becomes maximal in the top ceramic layer of thetubes (C, A), and then becomes negligible in the remainingmetal-rich regions through the thickness of the both outerand inner tubes. As the layer number m is increased, thethrough-thickness stress profiles remain similar after alayer number of 40. However, the peak normal, shear andvon Mises stress levels on the ceramic layer of the outer and

inner tubes become stable considerably after a layernumber of 40. In the adhesive layer, the normal (srr, syy),shear (srz) and von Mises (seqv) stresses reach peak levels atpoint A at the inner tube–adhesive interface (Fig. 8c), andalso become stable as the number of layer is increased.Increasing the layer number does not affect the through-the-thickness stress profiles. However, the peak stresses atthe ceramic-rich region of the tubes are affected by thelayer number since the ceramic layer is very thin forn ¼ 0.1. Therefore, the layer number of 40 should bereasonable in order to predict the stresses in a thin ceramiclayer.In case of a linear material composition n ¼ 1.0, the

compositional gradient exponent affects considerably thestress profiles of the outer and inner tubes (Fig. 9a, b),whereas the through-the-thickness stress profiles of theadhesive layer remain the same (Fig. 9c). Thus, the peakthrough-thickness stress profiles of the outer and innertubes are different from those of n ¼ 0.1 (Fig. 8). The peakvalues of the normal, shear and von Mises stresses increaseslightly and the stress profiles through the tube andadhesive thicknesses become more stable, so that the stresslevels remain the same as the layer number is increased forn ¼ 1.0.The ceramic-rich composition (n ¼ 10.0) causes comple-

tely different stress states in the overlap region of bothouter and inner tubes (Fig. 10) to those in the cases ofn ¼ 0.1 and 1.0 (Figs. 8 and 9) whereas the general trend ofadhesive stress profiles remain the same (Fig. 10). The peaknormal and von Mises stresses concentrate in the bottommetal-rich layers of the tubes and the peak shear stresses ofthe tubes occur near the metal-rich layers whereas the topceramic layers experience lower stresses. However, the peakstress magnitudes in the adhesive layer are negligible in thehalf adhesive thickness whereas peak shear stresses appearon the top and bottom interfaces of the adhesive layer(Fig. 10c). Moreover, increasing layer number through thetube thickness has a similar effect on the stress states of theadhesive and tubes to those in n ¼ 0.1. Since the metal-richlayer is very thin, the layer number should be increased

ARTICLE IN PRESS

-250 -200 -150 -100 -50 0 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-15 -12 -9 -6 -3 0 3 6 -50 0 50 100 150 200 250 300

-6 -3 0 3 6 9 12 15

-20 -15 -10 -5 0 5 10 5 10 15 20 25 30 35-5 0 5 10 15 20 25 30 35 400

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

10 15 20 25 30 35

-250 -200 -150 -100 -50 0 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

mT

hrou

gh-t

hick

ness

dis

tanc

e r,

mm

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

m

Circumferencial stress σθθ Axial stress σzz

Shear stress σrz

Shear stress σrz

von Mises stress σeqv

-50 0 50 100 150 200 250 300



Radial stress σrr

Shear stress σrz

m

-300

-250 -200 -150 -100 -50 0 50

Circumferencial stress σθθ


-300

-300

-250 -200 -150 -100 -50 0 50

Axial stress σzz

-300

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

(a)

(b)

(c)

Fig. 8. Effect of layer number (m ¼ 20, 40, 60, 100 and 200) on normal, shear and von Mises stress distributions through the thickness of (a) the outer

tube, (b) the inner tube and (c) the adhesive layer at the critical locations (see Fig 7) in the overlap region of the tubular single lap joint for the BC-I

(n ¼ 0.1) (all stresses in MPa).


especially in this metal layer in order to predict the peakstress values accurately.

The thermal residual stress analysis of the adhesivelybonded functionally graded tubular single lap joint showedthat increasing layer number in the outer and inner tubeshas a minor effect on the through-thickness stress profiles.Thus, the effect of layer number on the stress levels of boththe tubes and the adhesive layer is minor after 40 layersthrough the thicknesses of the outer and inner tubes. Inaddition, the discrepancies in the peak stresses are very bigfor the metal-rich (n ¼ 0.1) and ceramic-rich (n ¼ 10.0)compositions whereas they disappear for a linear composi-

tion of the ceramic and metal. Consequently, the stressstate of the functionally graded tubes under thermal loads,especially in the vicinity of the ceramic and metal layers forthe metal-rich (n ¼ 0.1) and ceramic-rich (n ¼ 10.0) com-positions, is considerably affected by the layer number.However, the through-thickness stress profiles of theadhesive and tubes remain similar for other compositionalgradients.Increasing the layer number through the tube thickness

has a negligible effect on the through-thickness stressprofiles until a specific layer (m ¼ 40) which can providestable uniform stress profiles for the through-thickness

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 0 40 80 120 160 200

-6 -5 -4 -3 -2 -1 0 1 2 3

-25 -20 -15 -10 -5 0 5 10 15 5 10 15 20 25 30 4035 450 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

10 15 20 25 30 35 40

-150 -100 -50 0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

mT

hrou

gh-t

hick

ness

dis

tanc

e r,

mm

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

m


Shear stress σrz

Shear stress σrz


0 30 60 90 120 150 180



Radial stress σrr

Shear stress σrz

m

-200

-160 -120 -80 -40 400 80 120



-200

Axial stress σzz

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

(a)

(b)

(c)

-150 -100 -50 0 50 100 150-200

-160 -120 -80 -40 400 80 120-200





metal-rich, ceramic-rich and a linear material composition.However, the compositional gradient exponent ‘‘n’’ be-comes more effective on the peak stress profiles andmagnitudes. In order to determine the effect of thecompositional gradient exponent ‘‘n’’ on the through-thickness stress states of the outer and inner tubes and theadhesive layer, the thermal stress analysis of the adhesivelybonded functionally graded tubular single lap jointsubjected to a uniform temperature field was carried outfor the compositional gradient exponents n ¼ 0.1, 0.2, 0.5,1.0, 2.0, 5.0 and 10.0, whilst the layer number was keptconstant at m ¼ 60. The through-thickness thermal and

mechanical properties of corresponding material composi-tions (Ni–Al2O3) were calculated using the formulas (1)–(9)and their variations are shown in Fig. 2.The effect of the compositional gradient exponent on the

through-thickness variations of the normal, shear and vonMises stresses at the critical locations (see Fig. 7) in theoverlap region are shown in Fig. 11 for the CM–CMcomposition. In the outer tube, the axial szz and hoop syystresses are very high in the overlap region. The shear stresssrz among the shear stress components is critical one. Asthe compositional gradient exponent is increased from 0.1to 10.0, the through-thickness profiles of the axial and

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-12 -10 -8 -6 -4 -2 0 2 4 0 30 60 90 120 180150

-4 -2 0 2 4 6 8 10 12

-15 -10 -5 0 5 10 15 8 12 16 20 24 28 32-5 0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

mT

hrou

gh-t

hick

ness

dis

tanc

e r,

mm

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

m


Shear stress σrz

Shear stress σrz


0 30 60 90 120 150 180



Radial stress σrr

Shear stress σrz

m

-50



Axial stress σzz

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

(a)

(b)

(c)

0 50 100 150 200-50

0 50 100 150 200-500 50 100 150 200-50

8 12 16 20 24 28 32





hoop stresses in the outer tube change evidently. Thus, thepeak syyand szz stresses change from compression in theouter ceramic-rich layer to tension in the inner metal-richlayer as the compositional gradient exponent n is increased(Fig. 11a). The shear stress srz is negligible in comparisonwith the axial and hoop stresses, and its peak values appearin the vicinity of the outer ceramic-rich layer and the innermetal-rich layer. The von Mises stress seqv through-the-thickness profiles for each compositional gradient arecompletely different. Thus, the von Mises stress values inthe inner metal-rich layer increase considerably anddecrease in the outer ceramic-rich layer as the composi-tional gradient exponent is increased. It is evident that a

metal-rich composition and a ceramic-rich compositioncause similar through-the-thickness von Mises stressprofiles; thus, they become symmetrical with respect tothe plane at half-tube thickness in the outer tube (Fig. 11a).The critical stress components in the inner tube

(Fig. 11b) have variations similar to those of the outertube (Fig. 11a). As the compositional gradient exponent isincreased, not only the through-thickness profiles but alsothe peak levels of the dominant stresses change (Fig. 11b).Similarly, it is evident that increasing the ceramic phase inthe material composition affects considerably the stressstates of the inner tube and causes an evident change in thepeak stress values.

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-10 -8 -6 -4 -2 0 2 4 0 25 50 75 125 175100 150 200

-6 -4 -2 0 2 4 6 8 10

-25 -20 -15 -10 -5 0 5 10 15-10 0 10 20 30 40 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

5 10 15 20 25 3530 40

-150 -100 -50 0 50 100 2001500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

mT

hrou

gh-t

hick

ness

dis

tanc

e r,

mm

Thr

ough

-thi

ckne

ss d

ista

nce

r, m

m


Shear stress σrz

Shear stress σrz


0 20 40 60 80 100 120 140 160 180



Radial stress σrr

Shear stress σrz

n

-200



Axial stress σzz

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

(a)

(b)

(c)

-150 -100 -50 0 50 100 200150-200

-150 -100 -50 0 50 100 200150-200 -150 -100 -50 0 50 100 200150-200

5 10 15 20 25 3530 40

Fig. 11. Effect of compositional gradient exponent n on the normal, shear and von Mises stress distributions through the thickness of (a) the outer tube,

(b) the inner tube and (c) the adhesive layer at the critical locations (see Fig 7) corresponding to the overlap region of an adhesively bonded functionally

graded tubular single lap joint with CM–CM composition for the BC-I (all stresses in MPa).


In the adhesive layer, the radial srr and hoop syy stressesare very high along the inner tube–adhesive interface(Fig. 11c). However, their values along the outer tube–ad-hesive interface are also high. The shear srz stress reaches apeak level along the inner tube–adhesive interface and allof its profiles are similar for the present compositionalgradient exponents. The peak von Mises stress seqv valuesalso occur along the inner tube–adhesive interface. Thecompositional gradient exponent does not affect the criticalstress profiles through the adhesive thickness except thestress magnitudes. Thus, large differences in the peak stressvalues along the inner tube–adhesive interface are ob-served. The properties of the graded region affect

significantly the response of functionally graded adherendsas well as the adhesive layer to thermal loads.Consequently, the axial szz and hoop syy stresses have an

evident effect on the stress states of the tubes, and changefrom compression in the ceramic-rich layer to tensionthrough the thickness of the outer and inner tubes. Theshear srz stress is negligible in comparison with the axialand hoop stresses in both tubes whereas they becomeimportant in the adhesive layer. Thus, in the functionallygraded tubes, the axial szz and hoop syy stresses are nearly20 times higher than the shear srz stress. However, theshear srz stress is nearly 2 times lower than the radial srr

and hoop syy stresses in the adhesive layer.

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Circ

umfe

renc

ial s

tres

s σ θθ

, MP

avo

n M

ises

str

ess

σ eqv, M

Pa

She

ar s

tres

s σ r

z, M

Pa

Outer FG tube Outer tube-adhesive interface

θ

r

z


CL

Distance along the overlap, mm Distance along the overlap, mm

Rad

ial s

tres

s σ rr

, MP

a


Outer interfaceInner interface

Outer interfaceInner interface

Inner interface


Inner FG tube





(b)(a)

(c) (d)

Fig. 12. Distributions of (a) radial stress srr, (b) hoop stress syy, (c) shear stress srz and (d) von Mises stress seqv along the outer tube–adhesive interface,

the inner tube–adhesive interface and the adhesive half-thickness in an adhesively bonded functionally graded tubular single lap joint with CM–MC

composition for the BC-I (n ¼ 0.1).


4.1.2. CM-outer tube, MC-inner tube

Fig. 12 shows the distributions of the radial srr, hoopsyy, shear srz and von Mises seqv stresses along the outertube–adhesive interface, the inner tube–adhesive interfaceand the adhesive half-thickness in an adhesively bondedfunctionally graded tubular single lap joint with a CM-outer tube and a MC-inner tube subjected to a uniformthermal load for n ¼ 0.1. All stresses remain at a negligiblelevel along the most of the overlap region, but suddenlyincrease near the right and left free edges of the adhesivelayer (Fig. 12). In this type of joint, the top and bottomregions of the adhesive layer interfaces with the metal-richlayers of both outer and inner tubes. The radial srr andhoop syy stress profiles are similar along the outer tube andinner tube interfaces and the half adhesive thicknesses.However, the radial sress srr reaches peak values incompression around the both left and right free ends ofthe adhesive half-thickness (Fig. 12a). On the contrary, thehoop syy stress peaks in tension along the overlap regionexcept the free edges of the adhesive layer (Fig. 12b). Theshear stress srz reaches a peak level at the left and right freeends of both the outer and inner tube–adhesive interfaces,thus becomes very high at both the left free end of the outertube–adhesive interface and the right free end of the innertube–adhesive interface. In addition, it has opposite

directions at the left and right free ends of both theoutermost and innermost interfaces. The shear stress srz isas high as the radial and hoop stresses in the adhesive layer(Fig. 12c). The peak von Mises stresses seqv appear in thevicinity of the left and right free ends of the adhesive layerand remain at a value half of its levels at the left and rightfree ends along the middle region of the adhesive layer (Fig.12d).The thermal residual stress analysis of the adhesively

bonded functionally graded tubular single lap joint boththe CM–CM (Fig. 6) and CM–MC (Fig. 12) jointconfigurations showed that all stress components becomepeak in the vicinity of the left and right free edges of theadhesive layer. In each of two joints, the stress magnitudesand profiles are the same along the adhesive layer exceptthe left free end in which stress magnitudes for theCM–CM (Fig. 6) configuration relatively higher than thoseof the CM–MC (Fig. 12) configuration. Especially, peakstress values appear along the inner tube–adhesive interfaceof the CM–CM configuration since the adhesive layer hasan interface with ceramic-rich layer of the inner tube.

4.1.3. MC-outer tube, CM-inner tube

In order to determine the effect of the thermal andmechanical properties of the tube regions interfacing with

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27R

adia

l str

ess

σ rr, M

Pa

Outer FG tube

Inner FG tube

Outer tube-adhesive interface

θ

r

z



CL



She

ar s

tres

s σ rr

, MP

a

von

Mis

es s

tres

s σ eq

v, MP

aC

ircum

fere

ncia

l str

ess

σ θθ, M

Pa






(a) (b)

(c) (d)

Fig. 13. Distributions of (a) radial stress srr, (b) hoop stress syy, (c) shear stress srz and (d) von Mises stress seqv along the outer tube–adhesive interface,

the inner tube-adhesive interface and the adhesive half-thickness in an adhesively bonded functionally graded tubular single lap joint with MC–CM

composition for the BC-I (n ¼ 0.1).


the adhesive layer on the peak adhesive stresses, anadhesively bonded joint with a MC-outer tube and aCM-inner tube is considered. Fig. 13 shows the distribu-tions of the radial srr, hoop syy, shear srz, and von Misesseqv stresses along the overlap region for n ¼ 0.1. All stresscomponents become peak in the vicinity of the left andright free edges of the adhesive layer. Since the adhesivelayer has an interface with ceramic-rich layers of both theouter and inner tubes, the peak values of von Mises stressesare relatively higher than those of the CM–MC jointconfiguration (Fig. 12). Thus, the peak radial srr stressoccurs in tension at both the adhesive left free end of theinner tube–adhesive interface and the adhesive right freeend of the outer tube–adhesive interface whereas it isnegligible along a large region in the middle of the overlapregion (Fig. 13a). The hoop syy stress becomes maximal atthe similar regions to those of the radial stress, but it is at aconsiderable level in the adhesive middle region (Fig. 13b).The shear stress srz concentrates around the left and rightfree ends of both the outer and inner tube–adhesiveinterfaces (Fig. 13c) and is as effective as the radial andhoop stresses on the stress state of the adhesive layer. Thevon Mises stress seqv peaks at the left and right free ends ofthe adhesive layer (Fig. 13d) and remains at a considerablelevel along the most of the overlap region.

In the case of the MC–CM (Fig. 13) configuration, thepeak values of all stress components are relatively higherthan those of both the CM–CM (Fig. 6) and CM–MC(Fig. 12) configurations. This is because the top andbottom regions of the adhesive layer have interfaces withthe ceramic-rich layers of both outer and inner tubes. Inaddition, the stress magnitudes and profiles are similaralong the most of the overlap region of all joint types.

4.2. Effect of the compositional gradient exponent

Fig. 14 shows the effect of compositional gradientexponent n on the through-the-thickness variations of theeffective stress components in the outer tube, the inner tubeand the adhesive layer of the tubular single lap joints withCM–CM, CM–MC and MC–CM compositions for theBC-I.In the case of the CM–CM configuration (Fig 14(I)), the

axial szz and von Mises seqv stress magnitudes and profilesin the outer tube for different compositional gradientexponent n are similar to those in the inner tube since thecomposition variation directions of both tubes are thesame. In the adhesive layer, the peak radial srr and vonMises seqv stress values occur along the inner tube–adhe-sive interface. The compositional gradient exponent does

ARTIC

LEIN

PRES

S

(a)

(b)

(c)

Fig. 14. Effect of compositional gradient exponent n on the through-the-thickness variations of the axial szz (the radial srr stress for the adhesive layer) and vonMises seqv stresses in (a) the outer tube, (b)

the inner tube and (c) the adhesive layer of the tubular single lap joints with (I) CM–CM, (II) CM–MC and (III) MC–CM compositions for the BC-I (all stresses in MPa).

M.K

.A

pa

lak

eta

l./

Intern

atio

na

lJ

ou

rna

lo

fA

dh

esion

&A

dh

esives2

7(

20

07

)2

6–

48

42


not affect the critical stress profiles through the adhesivethickness whereas the stress magnitudes are considerablyaffected by the compositional gradient exponent.

Fig. 14(II) shows the effect of the compositional gradientexponent n on the critical normal and von Mises stressdistributions through the thickness of both outer and innertubes and the adhesive layer at the specified locations(Fig. 7) corresponding to the overlap region of anadhesively bonded functionally graded tubular single lapjoint with the CM–MC tubes for the BC-I. The stress statesin the outer tube of the CM–MC joint configuration aresimilar to those of the CM–CM joint configuration, andthe compositional gradient exponent n has also a similareffect to the CM–MC joint configuration. Thus, the criticalstress magnitudes and profiles do not change in the outertube of the adhesive tubular CM–MC joint configuration(Fig. 14(II)a). In the inner tube of the CM–MC configura-tion, the critical stresses had a reverse distribution throughthickness due to a changing material composition from theoutermost metal layer to the innermost ceramic layer in theinner tube contrary to the CM–CM joint configuration.However, the peak stress values do not change(Fig. 14b(II)). In addition, the stresses on the innermetal-rich layer of the outer tube become compatible withthose of the outer metal-rich layer of the inner tube.Furthermore, the adhesive layer has an interface with ametal layer with thermo-mechanical properties close tothose of the adhesive epoxy. As a result, the thermal stressdiscontinuities along the tube–adhesive interfaces reduceslightly. In the adhesive layer, the stress profiles havemostly linear variations than those of the CM–CM jointconfiguration. The peak radial srr stresses are compressivealong the inner tube–adhesive interface whereas the hoopsyy stress peaks in tension along the outer tube–adhesiveinterface. The shear srz stress peaks along the outertube–adhesive interface. The discrepancies in peakstresses become more obvious along the lowermost inter-face rather than the uppermost interface of the adhesivelayer (Fig. 14(II)c).

In the case of the MC–CM joint configuration(Fig. 14(III)), the stress characteristics of both the outerand inner tubes for different compositional gradientexponent n are similar to those of both the CM–CM(Fig. 14(I)) and CM–MC (Fig. 14(II)) configurations. Thedirections of the critical stress profiles through thethicknesses of the outer and inner tubes in the MC–CMconfiguration become opposite to those of the CM–MCconfiguration. However, the peak stress values remainthe same. The stress profiles and magnitudes in theadhesive layer for the MC–CM (Fig. 14(III)c) configura-tion are similar to those of the CM–CM (Fig. 14(I)c)configuration.

As a result, the axial szz stresses change from compres-sion in the ceramic-rich layers to tension in the metal-richlayers of the both outer and inner tubes for all jointconfigurations as the compositional gradient exponent n isincreased (Fig. 14). In addition, as the compositional

gradient exponent is increased, von Mises seqv stressdecreases on the ceramic-rich layers whereas increases onthe metal-rich layers. In the case of the CM–MC jointconfiguration, the adhesive layer has an interface withmetal layers of the outer and inner tubes. Therefore, thethrough-thickness stress profiles in the adhesive layerbecome linear for the CM–MC joint configuration(Fig. 14c).

4.3. Constrained model (BC-II)

The response of the adhesively bonded functionallygraded tubular single lap joint was investigated to athermal load for BC-II in which both ends are fixed asshown in Fig. 4b. A uniform temperature distributionwhich cools from 40 to 20 1C was assumed through thejoint. The considerable deformations occurred in theoverlap region of the joint, which cause high stressgradients on the adhesive layer (Fig. 4). The stress andstrain concentrations occurred around the adhesive freeends. In addition, the regions of the outer and innerfunctionally graded tubes close to the adhesive free edgesundergo also significant stress concentrations. First, thethermal stress analysis of the tubular single lap joint withCM–CM configuration was carried out for the composi-tional gradient exponent n ¼ 0.1. In addition, the jointswith CM–MC and MC–CM configurations were investi-gated in order to determine the effect of materialcomposition direction on the stress state of the adhesivetubular lap joint. The outer and inner tubes were modelledusing 60 layers through the tube thickness. The stressanalyses show that the hoop syy and axial szz stresses andthe shear srz stress are dominant in the outer and innertubes whereas the radial srr and hoop syy stresses and theshear srz stress are higher in the adhesive layer.The through-the-thickness von Mises stress seqv dis-

tributions in the outer tube, the adhesive layer and theinner tube in the overlap region for n ¼ 0.1 are shown inFig. 15. In the outer tube, the von Mises seqv stressconcentrations appeared at the left free end of the outertube–adhesive interface (Fig. 15a). In the adhesive layer,the von Mises seqv stress concentrate around the left freeedge of the outer tube–adhesive interface and the right freeedge of the inner tube–adhesive interface. However, thevon Mises stress becomes maximal at the right free edge ofthe inner tube–adhesive interface (Fig. 15b). In the innertube, the peak von Mises stresses appear at the right freeend of the inner tube–adhesive interface which is the mostcritical region of the inner tube (Fig. 15c). The von Misesstress concentrations occur in the top ceramic-rich layer ofthe inner tube and in the bottom metal-rich layer of theouter tube. In addition, the peak von Mises stresses of theinner tube are 1.32 times higher than those of the outertube.It is evident that the free edges of the adhesive layer are

critical since von Mises stresses are peak at the left freeedge of the outer tube–adhesive interface and at the right

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ialp

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0.00

0.25

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1.00

1.25

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2.00

t

(b) Adhesive Layer(a)

θ

r

z

(c)

CL

(a)

(b)

(c)

70.070.0

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ialp

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Outer FG tube

Inner FG tube

Rad

ia lp

ositi

on, (

mm

)R

adia

lpos

ition

, (m

m)

Rad

ia lp

ositi

on, (

mm

)



Axial position, (mm) Axial position, (mm)

Fig. 15. von Mises stress seqv distributions in the overlap region of (a) the outer tube, (b) the adhesive layer and (c) the inner tube of an adhesively bonded

functionally graded tubular single lap joint with CM–CM tube configuration for the BC-II (compositional gradient n ¼ 0.1) (all stresses in MPa).



free edge of the inner tube–adhesive interface. Therefore,the initiation of the local failure in the adhesive layer can beexpected along the left and right free edges of the outer andinner tube–adhesive interfaces.

In the adhesive joint with free edges (Fig. 5), the peakvon Mises stresses appear along the outer ceramic-richlayers of both outer and inner tubes and they suddenlydecrease near the ceramic surface and become minimal inthe bottom metal layers. In addition, the peak von Misesstress concentrations appear at the left and right free edgesof the inner tube–adhesive interface. However, in theadhesive joint with fixed ends, the peak von Mises stressseqv concentrations appear at the left free edge of the outertube–adhesive interface and the right free edge of the innertube–adhesive interface (Fig. 15). The stresses becomemore critical in an adhesive tubular lap joint with fixededges and the effect of the compositional gradient cannotbe seen easily under thermal loads.

Three tube configurations are considered, such as, (i)CM-outer tube and CM-inner tube, (ii) CM and MC tubesand (iii) MC and CM tubes. The critical normal (srr, syy),shear (srz) and von Mises (seqv) stresses distributions alongthe adhesive layer in the joint with CM–CM compositionare shown in Fig. 16. The critical stresses concentrate alongthe left and right free edges of the adhesive layer. The

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Outer FG tube

θ

r

z


Rad

ial s

tres

s σ rr

, MP

a


shea

r st

ress

σrz

, MP

a

25

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25



(a)

(c)

Fig. 16. Distributions of (a) radial stress srr, (b) hoop stress syy, (c) shear stresthe inner tube–adhesive interface and the adhesive half-thickness in an adhe

composition for the BC-II (n ¼ 0.1).

radial stress srr becomes negligible along a large region inthe middle of the overlap region. The left free end of theouter tube–adhesive interface experiences tensile radialstresses whereas the right free end undergoes lowercompressive radial stresses. However, the radial stress hasan opposite variation along the inner tube–adhesiveinterface to those along the outer tube–adhesive interface.The peak radial stresses appear at the right free end of theinner tube–adhesive interface (Fig. 16a). The hoop stresssyy is also peak at the right free end of the innertube–adhesive interface and is minor in the middle of theoverlap region (Fig. 16b). The shear srz stress increasesfrom middle overlap region to the left and right free edgesof the adhesive layer and becomes maximal at the right freeend of the inner tube–adhesive interface (Fig. 16c). The vonMises seqv stress has a similar variation to the shear stressand is maximal in the right free end of the innertube–adhesive interface whereas it is minor in the middleof the overlap region (Fig. 16d). Consequently, the mostcritical zone of the adhesive layer is the right free end of theinner tube–adhesive interface for the BC-II. In addition,the radial srr, hoop syy, shear srz and von Mises seqv stressdistributions along the overlap region of the adhesive layerfor the CM–MC and MC–CM tubes are similar to those ofthe CM–CM (Fig. 16) joint composition. This is because

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Circ

umfe

renc

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tres

s σ θθ

, MP

a

Inner FG tube

Outer tube-adhesive interface Adhesive half-thickness

CL


von

Mis

es s

tres

s σ eq

v, MP

a

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(b)

(d)

s srz and (d) von Mises stress seqv along the outer tube–adhesive interface,

sively bonded functionally graded tubular single lap joint with CM–CM


the edge condition is more effective on the stress states ofthe adhesive tubular joint.

The investigation of the stress magnitudes and profilesthrough the tube and adhesive thicknesses for differentlayer numbers showed that the layer number had a minoreffect on the normal, shear and von Mises stress distribu-tions at specified critical locations for the compositionalgradient exponents n ¼ 0.1, 1.0 and 10.0, respectively. Thestress variations are not shown here in order to avoidunnecessary repetition. Thus, minor differences in the peakstress profiles and magnitudes are observed for theadhesive tubular joint after a layer number of 40. In theBC-I, the critical stress discrepancies near the ceramiclayers in the tubes and the adhesive are more evident thanthose in the BC-II. The stress states of the adhesive joint isaffected considerably by the edge conditions; thus,considerable stresses occurred in the joints with fixed endsdue to thermal loads. Therefore, the effect of layer numberbecomes small on the stress and deformation states of thetubes and the adhesive layer in comparison with the effectof edge conditions.

In addition, the effect of the compositional gradientexponent n on the stress states of the outer and inner tubesand adhesive layer was investigated for the compositionalgradient exponents n ¼ 0.1, 0.2, 0.5, 1.0, 2.0, 5.0 and 10.0,whilst the layer number m was taken as 60. The through-thickness variations of the critical normal and von Misesstresses in the outer and inner tubes and adhesive layer atthe critical locations (Fig. 7) in the overlap region areshown in Fig. 17 for all tube configurations for the BC-II.

In the case of the CM–CM joint configuration(Fig. 17(I)), the axial szz and von Mises seqv stressmagnitudes and profiles were presented in the outer andinner tubes and the adhesive layer for different composi-tional gradient exponent n. In the outer tube, the axial szz

stresses linearly increase from the outer ceramic-rich layertowards the inner metal-rich layer, and their profilesremain the same. In addition, the compositional gradientexponent n has a minor effect on the axial stress. The vonMises seqv stresses are seriously affected by the normalstresses. Therefore, the von Mises stress values on the outerceramic-rich layer decrease considerably as the composi-tional gradient exponent is increased whereas their valuesbecome almost same on the inner metal-rich layer(Fig. 17(I)a). In the inner tube, the axial szz stressesincrease from the inner metal-rich layer towards theceramic-rich outer layer. As the compositional gradientexponent is increased, the axial stress values evidentlyincrease on the inner metal-rich layer whereas the stressvalues are very close on the outer ceramic-rich layer. As thecompositional gradient exponent n is increased, the vonMises seqv stress value decreases on the outer ceramic-richlayer whereas it increases on the inner metal-rich layer andtheir peak values occur on the outer ceramic-rich layer ofthe inner tube (Fig 17(I)b). In the adhesive layer, all criticalstress components increases from the adhesive top surfacetowards the bottom surface in which they reache peak

values. The peak stress values decrease as the composi-tional gradient exponent n is increased (Fig 17(I)c).The effect of the compositional gradient exponent n on

the through-thickness critical stress variations at thespecified critical locations (Fig 7) corresponding to theoverlap region of the adhesive tubular joint with theCM–MC (Fig 17(II)) and MC–CM (Fig 17(III)) tubes aresimilar to those of the CM–CM (Fig 17(I)) joint config-uration. It is evident that the edge conditions of the outerand inner tubes become more effective on the stress statesof the joint.Consequently, the compositional gradient exponent n is

more effective on the stress states of the free joint modelwhereas boundary condition has more effect on the stressstates of the constrained joint model.

5. Conclusions

Based on the thermal residual stress analysis of anadhesively bonded functionally graded tubular single lapjoint subjected to a uniform temperature drop, thefollowing results were concluded:

(i)
The hoop syy and axial szz and the shear srz stresses inthe outer and inner tubes were dominant whereas theradial srr and hoop syy and the shear srz stresses arehigher in the adhesive layer for both the free andconstrained joint models.
(ii)
Stress concentrations occur around the left and rightfree edges of the adhesive layer. The critical stressesbecome peak at the left and right free edges of theinner tube–adhesive interface in the joint with freeedges and at the left free edge of the outer tube–adhe-sive interface. However, the right free edge of the innertube–adhesive interface in the joint with fixed edges ismore critical.
(iii)
The through-the-thickness stress distributions in thetubes and adhesives are affected considerably by thecomposition-variation direction of both tubes. Theceramic layers of the outer and inner tubes are incompression whereas the metal layers are in tension. Inaddition, the stress discontinuities become minimallevels along the tube–adhesive interfaces for theCM–MC tube configuration in the free joint model.However, only hoop syy stress changes from compres-sion in the ceramic layer to tension in the metal layerof both outer and inner tubes for all tube configura-tions (CM–CM, CM–MC and MC-CM). The tubeconfiguration has a minor effect on the through-the-thickness stress profiles and magnitudes.
(iv)
The compositional gradient exponent does not affectthe through-thickness stress profiles of the adhesivelayer for both joint models. However, the peak stressesare also increased with increasing compositionalgradient exponent.
(v)
Increasing the layer number through the tube thicknesshas minor differences in the through-thickness critical

ARTIC

LEIN

PRES

S

(c)

(b)

(a)

Fig. 17. Effect of compositional gradient exponent n on the through-the-thickness variations of the axial szz (the radial srr stress for the adhesive layer) and vonMises seqv stresses in (a) the outer tube, (b)

the inner tube and (c) the adhesive layer of the tubular single lap joints with (I) CM–CM, (II) CM–MC and (III) MC–CM compositions for the BC-II (all stresses in MPa).

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through-the-thickness stress profiles of both models.However, the peak stress levels in the tubes and adhesivelayer exhibit discrepancies near the metal and ceramiclayers until 40 layers for the free joint model.

(vi)
The effect of the compositional gradient exponentbecomes more apparent on the stress profiles andmagnitudes for the free joint model.
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Thermal residual stresses in an adhesively bonded functionally graded tubular single lap joint