1

Sci. Rep. Fukushlma Univ.,No.58(1996) 19

Effects of Anisotropy of an Adhesive Meso-Phaseon the Strength ofAdhesively Bonded Structures Under Thermal Loading.

YoshihitoOZAWA,KatsuoSUGIURA and Koya NoGUcHI*

.Department of Mechanical Engl'neering Faculty of EducationFukttshima University, Matsukawa machi, Fukushlma 960_12, JAPAN.

*Graduate School, Fukushima UniversityMatsukawa-machl, Fukush加na 960-12; JAPAN

(Recetved 10 April, 1996)

ABSTRACTFor the purpose of having the better strength of adhesively bonded structures

under uniform themla1 loading, an edge crack in stnlctufes with adhesive meso-phase is consider,ed theoretically. We introduce a model of anisotropic phase and reduce the mathematical intricacies of the problem. The analysis is based on the two-dimensional theory of thermoelasticity and the singular point method. Numerical calculationsareperfonned in order to clarify the effects of elastic moduli and their configurations on the crack extension in the structures. The strength of adhesively bonded structures is discussed from the results.

1 . INTRODUCTION

The chemical stability of ceramics above the melting point of metal alloys predestines this class of materials for high temperature applications[1], On the other side ceramic components in high temper;ature applications may not survive thermal stresses generated in rapid heating or cooling cycles because of theif inherent brittleness [2,3]. For reliable engineering design and optimum material selection for the space structures and architectures,it is imperative that the thermal stress behaviors of the ceramics are well estimated.

It isa good way to use the structural ceramics combined with metal alloys(Fig.1). In the space environment,large thermal stresses may develop in this kind of adhesively bonded structures owing to the mismatch in the coefficient of thermal expansion(cTE)of adjacent materials. Then, an edge crack should occur in plane approximately Perpendicular to the interface between adfacent material phases. The interfacial

2

20 Ozawa,sugjura and Noguch1:Effects of Anisotropy of an Adhesive MeSo-PhaSe

debondjng between the phases is also observed in fractographyof the St「uCtu「oS. With the Increasingly complex developments in the field of structural ceramics,it became aPPa「ent

that a deeper understanding of thermal shock resistance and damage as Well as the「mat fatigue behavior should result from fracture mechanical analysis combined With Well defined experimental conditions.

For the purpose of having the better strength of adhesively bonded StruCtu「oS undo「 uniform thermal loading,we deal with the growth of an edge crack from the Su「face Of an

outer phase in the structure under uniform thermal loading theoretically (Fig。 2)・ The analysis is based on the two_dimensional theory of thermoelasticity and the Singula「Point method we introduce a model of anisotropic phase. This simple method P「OPOSed could reduce mathematical intricacies of the problem. By 「ePlaCing the C「aCk With continuous distributions of edge dislocations,a system of singular integ「al equations fo「

density functions with cauchy kernels is obtained. The solution is assumed in the fo「met the product of an unknown function and the weight function of Jacobi Polynomials,and iS detemljned by the technique developed by Erdogan, Numerical Calculations a「e performed in order to clarify the effects of elastic moduli and their configu「atiOnSOn the crack extension in the structures. The strength of adhesively bonded St「uCtu「oS iS discussed from the results.

2.MATHEMATICAL FORMULATION OF THE PROBLEM

Using the two_dimensional theory of thermoelasticity,consider the thermal St「eSSeS

around an edge crack in the outer phase l ot the adhesively bonded structure undo「 unite「m thermal loading(Fig.2). This edge crackof lengtha extends from the su「face Of the cute「 phase and jsperpendjcular tothe surface. The outer phases I and Il a「ope「feCtly bonded to the neighboring meso_phase. The outer phases I and II treated he「e a「e assumed to be homogeneous and transverse isotropic and to obey the Duhamel-Neumann 「elation. The thickness of the outer phase f is e,and that of the phase II is d. The meSo-PhaSe *iShomogeneous and orthotropic. The thickness h of the meso-phaSe iS Ve「y Small incomparison with the thickness f . In the analysis, a rectangular CoO「dinate System iSemployed as shown in Fig,2.

For plane defonnation,the stress componentsσ...σy andτ..y,and the displacementgradients u,. and v,.can be expressed, in the absence of body forces, in to「mS of thetemperature potential functionsθ(z)and the elasticpotential funCtiOn0(Z)and、f'(Z)[4]:

σ.1 十of ?1 = 2[0 j (z)十0 J.(Z))] (1)

3

Sci.Rep、Fukushima Univ.,No_58(19961 21

σ ター「t rff = 0 J(Z)十0 j (Z))十Z0? )十1lfi j (Z)) (2)

? (u,.十iv,.)= (3-4v- z)-? )-z0- )-? )十Eja ,e1(z) (3)f

(J=1,II) where z= x;+ iy. E is Young's modulus ; v Poisson s ratio and αthe coefficient of

thermal expansion. The overbar indicates a conjugate complex quantity. In what follows,the subscripts I and II refer to quantities associated with the outer phases I and II, respectively.

The boundaryconditionsof this problem can be written as

(i)From the condition for stresses on the surface of the edge crack C(1x= 0,_a

4

22 Ozawa,sugjura and Noguchi:Effects of Anisotropy of an Adhesive MeSo-PhaSe.

? _(iu =? ? =? (8)a x1 - a x ii x1 ' ?x ? x

In what follows the asterisk will refer to quantities associated with the meSo-PhaSe・The stress c;,,acting in the meso_phase at infinity is given fromfi「St equation Of (8)

by

σ = fc1 _C1221一 σ (9). . l . fc っ:, E ,

Here cリ areelasticconstantsof themeso,phase.Moreover,fromthecontinuitycondlt1onforstressesontheinte「face,We Obtain

_ _ (10)σ yf 一σ y' 一σ 11

The equilibrium of the force due to cfx.and the tangential fo「Ce acting On the jnterfaceof thethinmeso-phaseyieldS

1; - ' 十ha = 0 (11):'yf r yf1 cj x

Let us try to transform Equations (11)in the plain strain condition. The Constitutive equatjonsfortheorthotropicmeso-phasearegiVenby[5]

au* 3v. 3u. (i v.

σ..= ell 一 十C12- , σy.= Cl2- 十C22-:av (12)ax , ・av ax

FromtheconstitutiveequationsfortheouterphaSeI,We Obtain

(jul = 1-Vl σ _Vf(1十Vf_)cf (13)jjx E , ' E , yf

In view of Equations(8),(10)and (l3),eliminating (i v。/(iy from (l2)and Substituting itinto(11),it follows that

τ _ τ 十 aCi lL_ p2C3(:;y1 = 0 (14)Xv i ・y f 1 p i a lt ? x

where

5

pi =

Consequently, the continuity conditions for displacements and stresses on the interface of the thin meso-phase give(8),(10)and (14). By using the complex potential functions,the continuity conditions are written as

? l(3-4Vj)Of(Z)-? )-Z0? )-? )

Re{201(t)十tO;(t)十tf'1(t)}

=Re{20 ,f(t)十te l,(t)十、f'n(t)}

Rei(p1-3p2十It?? )-(pi十p2? tO? )十lf';(t)l}

十Im{?1(t)}=Im? 0;1(t)十?11(t)}

where t= x-ie

Here it should be noted that the problem reduces to the two phases problem of the regions of the phases I and II.

3.SINGULAR INTEGRAL EQUATIONS

We win apply the method of continuous distributions of dislocations to the problem of an edge crack in the adhesively bonded structure.

Let an edge dislocation be situated at a point z= -ls(0 < s< a )in the infinite medium.The slip plane is parallel to the .x:axis. Then the elastic potential functions△0(z) and△平(z)are given by [5]

△0(z)

Δ、f'(z)

Sci.Rep.Fukushima Univ.,No.58(1996)

十E1α1e 1(t)]

=? [(3-4v ,,一 一? )_z0? )_? )

it1 1b

4π(1-vj) z十Is

十o ffαf1θ/J(t)f

23

(15)

(16)

(17)

6

ΔOf(Z)=

/いl;l f 7、=

上 [三 4π(i v1) z十Is

「 z

+ fo-{4(my '- )+B(m? -m(Z-')}dm]

- '、一ノ 411;(1_ v 1)Lz十is (z十is)2 Jo t」L一、ーノ ー ?ーノ」

十[D(m)十imzB(m)ド (Z-S)}dm] (18)

△011(z)=? 「 {E(my '?-S)十F(m? m(Z')? m

Δ、f',,(z)=? Jo?j[G(m)-imzE(m)lorn('+'S)+[ff(m)+imzF(m)? 'M?'S)}dm

where

A(m)= (K2K6- K3K5)/(KIK5- K2K4)

B(m)= (K3K4- KIK6)/(KIK5- K2K4)

C(m)=-A(m)-B(m)+2sm-1D(m)=-A(m)-B(m)+l _ . 、 S ・N

!「(y,+ e-2'm、tfy ,,十e -2dm、一4edm2lA(m、

(y n 十e2dm)(Yf1 十e -2dm)十4 d 2m 2 u、 ' - 八 '' ' 」 、 ノ

_2m[d十(dY1十eY,,十ee-2dm、e-2m? (m)十2dm(2sm- l)

+[j(1_2m(e_s))(yf,+e-2dm)_2dmY,}e2Sm-(Yf,+e-2d'M)ド }S N r,. r , r _ _ . _,? , . m t ,, 、

、 ノー(y lf 十e2dm)(Ylf 十e-2dm)十4lf m2 t- L- L- i - ' - J- 」 、一J

十[(Y1十e2m)(Yf1十e2dm)_ 4edm2? (m)_ (2sm- 1)(yff 十e2dm、e2b'

十{2dm(1- 2m(P- s))十Yf (yf1十e2dm)ド 一2dm}

G(m)=-{l-2m(?+d)}E:(m)-e 2m(- )F(m)

H(m)=_e2- )E(m)-{1十2m(?十d)}F(m) (19)

24 Ozawa,Suglura and Noguchi1Effectsof Anisotropy of an Adhesive Meso・Phase.

where b indicates the Burgers vector of the edge dislocation. By starting with(l7),the required complex potential functions for the edge dislocation at z= -is in the finite-width phase.satisfying(5),(6)and(16),are obtained as follows:

7

Sci.Rep.Fukushima Univ.,No.58(1996) 25

Kt = N・?[(1-2em? -S(2dm十l -e2dm )(Y1(yIf 十e-2dm)_4?dm2)

十S(2dn - l 十e-2dm )(2dmYf 十2em?y11 十e2dm))「

十[-G - S(? m 十l - e2dm )(yu 十e-2d'n)十S .2dm(2dm_ 1十e-2dm)「 }

K2 = N?[? 十S ・2dm(2dm十l -e2d;能)十S(2dm _ l 十e-?m)(y11十e2dm)「 '

十[(2b t十l? 十S(2dm十l - e? X? mYf 十2em(yIf 十e-2dm))

十S(2dm_ l 十e-2dn)(Y1(Ylf 十e2dm)_ 4edm2)? 2em}

K3 = N{-(2sm一中 ・2dm(2abn + 1- e2dm)十S(2dm_ l 十e-2? XY,, 十e2d''')十Gド

+[S?2dm+1-e2?'X2dn,yfe2Sm- f(1-2m(?-sj)e2'解_ 1? ?f +e-2dm))

+S(2dm- l+e-2? XY,(y,J+cum)e2m +2dmj(l_2m(e_sf)e2sm_1})

十(2m?e-s? 2m十1? ド }

(19)

K4 = N?f(2j l(m)十(pi 十p2)m十l)〔ii十S(2dm- 1十e2dWXYf(Y11十e-2dm)_ 4edm2)

十S?2dm十1-e-?m)(2dmY,十2em(Y,,十e2dn))? 4em

十[0(pi 十p2)m- l? 十S(2abn _ l 十e2dつ(】/11十e-2dm)

十S・2dm(2dm十1_e-2dm)「 }

K5 = Nj [ｫpi 十p2)m十l? -S・2dm(2dm_ 1十e2dm)

十S(? m 十l _ e-2dm)(y11 十e2? )? 4em

十[(2f 2(m)十(pl 十p2)m- l? - S(2dm- l 十e2dm X2dmY1十2bn(y11 十e-2dm))

十S(2dm十l -e-2dm)Cy1(Ylf 十e2dn)_ 4?dm2)? 2tm}

K6 = Nj(2sm一中 -2dm(2dm - l 十e2dm)_ S(ylf 十e2dm)(2dm十l _e-2dm)

-(1pl 十p2)m十l? ド

十[S(2dm- 1十e2dm){-2dmYfe2Sm十((l _2m(- ? ?m _ l)(YH十e-2dm)}

十S(2dn 十l-e-2dつ{y1(y11十e? ? 2Sm十2dm((l _2m(e_s)? 2m_ 1)}

-(2f j(m)e'2m十(pl 十p2)m-1? ド }

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26 Ozawa,stlg1ura and Noguchi:Effects of Anisotropy of an Adhesive MeSOphaSe・

N = N, N, = 、j ib N,, = ?i 'i b _Nil 411(1-V1) 41t(1-V11) (19)

j 1十Vf_ s = l 十VffS= St = , 11S i t E 1 E lf

G = (Y11 十e2dn)(Y11十e-2dm)十4d 2m 2

f 1(m)=(pi十p2)?m2十(pl -p2十??

f (m)= -(pi十p2)erm2十(pi -p2十??

f 3(m)=(p,+p2)(e_s)mつ一(2p,+e,-s)m+1Y,= 3-4v, , Yf,= 3-4V,f

To fmd the solution,the edge dislocation wi通be distributed Continuously along the cracks c By paying attention to the behavior of the temperature Potential function at Infinity given by Equation(7),we have the potential functions θ(Z),0(Z)and 'f'(Z)of each phase in the forms.

θ,(z)=△T

Of(z)=? g_? ?:?s?? _Jo°°{一 一 +B.(m?-m(-? m? s

where

(20)

9

Δ=

g = -a ,ΔT' (21)

α'一af

l一

Sci.Rep.Fukushima Univ.,No_58(1996) 27

n e ll

(J = I,ii)

where of(z)and 、'f'(z)include the terms of the density function of edge dislocations b(s) dist「ibuted along C. Once the density functions b(s)has been determined,the thermal stresses can be obtained by substituting Equations(20)into Equation(l).

The potential functionsof Equation(20)satisfy the boundary conditions Equations (5),(6)and(16). SabstitutingEquation(20)into Equation(4).we obtain a set of singular integral equation for the dislocation density

where

K(η,i)=S

T1=- .,a j = -y ,a

1_α・_α f

l十e十d

中 1十一 p)-C(p)? -p(一十[2(1-p?? (p)-D(p)I p(- ? p l 十Vf

4(1-V1? JΔ「 b(s)

(22)

(23)

p=am , A(p)=A(m) l 十

?十d λf 十2 ｵ/

n e l l

The integrals on the left sidesof Equation(22)are considered as the Cauchy principal value We have introduced the non-dimensional notation before setting up singular integral equations.

TosolvedEquation(22)numerically by using the technique developed by Erdogan [6],the singular integral equation(22)should be defined in the normalized interval (_l,l) [7]_The solutions of the singular integral equation (22)are assumed in the form of the product of an unknown function and the weight function of Jacobi polynomials. By noting that0(0)≠0,an appropriate extension of the domain of 0(T1)into the interval (_1,0) may be an even continuation such that

0(T1)=g(T1)(]-T1)o(1十Tl)o , g(「1)=g(-11) (24)

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28 Ozawa,sug1ura and Noguchi:Effects of Anisotropy of an Adhesive MeSo-phaSe

The constant complex value(D must satisfy the relation -l < Rea)

11

Sci.Rep_Fukushima Univ.,No.58(1996) 29

factor of the edge crack and the thermal stresses in the outer phase. We treated the case that the elastic constant c,,is larger than that of the outer phase I. In this case we introduce the following non-dimensional parameter 「 relating to the stiffness of the meso_ phase:

C1-一ｵf

-n一f一一r

(28)

We set that the value of 「 isequal to 10.0; The results are plotted in terms of the various ratios of Young's modulus of the phase I tothatof the phase II E,,/Er

Viewing the reference[10],the Poisson;s ratios of the phases I and II are assumed to be v1=0.4. The ratio ofαJα,is determined to be3.l3x l03because the coefficient of themalexpansionof the direction of x axis in the meso-phase has smaller value than that in the phase I. For the orthotropic meso-phase, c]2/c,1=0.025 and c22/cu=0.085. For the special case that the meso-phase is isotropic,the value of Poissonfs ratio of the meso_ phase is taken as v・= 0.29 and we set that c,1= λ1十2ｵ1, c12= λ1 and c22=cl,. The ratio of the thickness of the phase n tothatof the phase I di e is l.0.

The variation of the stress intensity factor K,at the tip of the edge crack C is

shown versus the ratioof cracklength a/?for 「= 10 and E,JE,= f in Fig.3. As the value of af eincreases, the stress intensity factor K, increases gradually,taking the extreme value, and then tends to zero at a/t = 1. It means that the crack extension terminates at the interface between the outer phase and the meso-phase. We understand from this calculation that the factor K,increases as the Young's modulus of the outer phase E, decreases.

Fig.4 shows the effectof orthotropyof the meso-phaseon stress intensity factors K,. The short dashed line indicates the K,when the adhesive meso_phase is isotropic. It can be recognized from this figure that the K,takes larger values than that in the case of the orthotropic meso-phase. ・

Variations of the value K,are plotted in terms of the crack length al t and E,/E, for E,::=:11.7 Gpa in Fig.5. The shape of each curve in this figure is similar to that shown in

Table t. Mechanical propertiesof thephase[10].

Young's modulus,EYoung's modulus,EPoisson's ratio.vPoisson's ratio,v

Coefficient of Them al Expantion,α, Coefficient of ThemalExpantion,α

138GPa11.7GPa0.20.40.09x10-6/K 28_8x10-6/K

12

30 02awa,sugiura and Noguchi:Effects of Anisotropy of an Adhesive MeSo-PhaSe

Fig 3 The factor K, increases as the Young's modulus ratio E, /E, doc「eases・ lnspectjon of thjs figure reveals that the Young's modulus Enhave a strong effect On the stress intensity factor K,when alt →l_

Fig.6 shows the variations of the value K, against crack length a/f fo「 Va「iOuS E /E, In this calculation,we set that the Young's modulus of the lowe「 Phase II Eu iS equal to 117 Gpa It is easily understood that the variation of K,is very ditto「ent f「Om that in Fig.5. The factor K,takes larger value as the increment of the ratio Of Eu lEI When a/f is small Meanwhile the factor K, takes smaller value as the increment Of E,,/E When ｧｲf → 1 we can recognized from the figure that there appears to have the St「eng effect of the Young's modulus of the outer phase I E,on the factor K,when a/f is Small.

By using the results of the calculations,we can estimate the behaviO「Of the edge crack growth and the temperature△T when the K,attains the fracture toughness Of the cute「 phase Kc The residual stresses in adhessively bonded structures which win be Caused by their manufacturing processes,is also determined by following this analysis.

REFERENCES

(l)Ministry of International Trade and Industry,The Consumer Goods InduSt「ieS Bu「eau,ceramics and construction Materials Division,Fine Ceramics Office,ed.,Fine Ce「amiCS,The Research Institute of International Trade and Industry,1987.

(2)Furukuchj,Hino,Kikuchi and Yata,Transactionof JSME,59-560,A,P.408,l993・(3)Murata and Mukai,Transactionof JSME,58-566,A,P.452,1992.(4)Bogdanoff,J.L.,ASME Journal of Applied Mechanics,Vol。21,P.88,1954.(5)Abe,Hand Sekine,H.,Elastiaty Corona Publishers,Tokyo,P.46,1983・(6)Erdogan,F, Mechanics Today Vol.4, Nemat-Nasser,S.,ed.,Pergamon Press,P.1,

l978. ,

(7) Fu11no, K, sekjne, H.and Abe, H,, ''Analysis of an edge crack in a Semi-infinitecomposite with along reinforced phase 'Inf.J.Fracture,Vol.25.P.81,1984.

(8)Mura T Mjcromechanjcs of Def jects in Solids, 2nd revised ed.., MartinuS NijhOffPublishers,Dordrecht,.p.469,1987.

(9)0zawa, Y, Haraguchj, s. and Sugiura, K., “Ply Cracking in the Longitudinal andTransverse plies of cFRp Laminates Under Thermal Loading,”The72th JSME Fall

Annual Meeting,No.940-30,p.l66,1994.(10)Han Y M Hahn H T and Croman,R_B.,“A Simplified Analysis Of T「anSVe「So Ply

cracking in cross_ply Laminates,” Composites Science and Technology,Vol・23,P・l65.1988_

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Sci.Rep.Fukushlma Univ.,No,58(1996)

Fig.1. Ceramics/Adhesive meso-phase/Metal

y

31

c h

d II

Fig.2. An edge crack in the cater phase and coordinate system

14

32 Ozawa,Sugiura and Noguchi;Effects of Anisotropy of an Adhesive Meso-Phase

Fig_3

0.8

0.0 0.0 0.2 0,4 0.6 0.8 l.0

a/fStress jntensity factors Kl versus crack leagth a/f for variOaS YOung'S modulus E,.

0,8

0.0 0.0 0.2 0.4 0.6 0.8 1.0

a l t

Fig 4 Effectof orthotropyof the meso-phaseon stress intensity facto「S Kt

15

Fig_6

4

2

1

1・

Fig.5

0.0

Sci.Rep.Fukushima Univ.,No.58(1996)

0.0 0.2 0.4 0.6 0.8 1.0a /f

Va「iatiOnSOf thevalueK1againstcracklength? ?for var1o11sE1,/E (E,=11.7 Gpa)

4

2

1

l

1.0

0.8

0.6

0.4

g

与

tv-/n/-/(/、一--)(l一一一-)-f

0.0

E,,/E,:?1/4

11/21 1

1 2 - --1-.4

0.0 0.2 0.4 0,6 0,8 1.0a/f

Va「iationsof thevalueK,againstcracklengtha/a orvarjousE,,/E (E,,=11.7 Gpa)

33