Three-dimensional stress analysis of adhesive butt joints with disbonded areas and spew fillets

Three-dimensional stress analysis of adhesive butt joints with disbonded areas and spew fillet K. T e m m a , * T. Sawa t and Y. Tsunoda t

(*Kisarazu National College of Technology/tYamanashi University, Japan) This paper deals with three-dimensional stress analysis of adhesive butt joints subjected to tensile loads, to help establish fracture criteria. In this analysis, the adherends and adhesive bond are replaced with finite solid cylinders. Stress distributions in adhesive joints are analysed using a three-dimensional theory of elasticity. The effects of a disbonded area and a spew fillet on the principal stress distributions are shown by numerical calculation. Where a joint has a disbonded area, the stress singularity increases with a decrease in the diameter of the spew fillet, and the stress singularity decreases with an increase in the diameter of the adhesive. The analytical result is compared with that obtained by an experiment concerning the strain produced on the adherends in the case of a disbonded area. Both results are satisfactorily consistent.

Key words: elasticity; stress analysis; adhesion; butt joint; tensile load; solid cylinders; disbonded area; spew fillet In the rational design of adhesive joints, it is necessary to know the stress distributions on the joints more precisely, and m a n y investigations 1-s have been carded out on lap, scarf and butt adhesive joints subjected to tensile loads, bending moments and shear loads. Disbonded areas due to lack of adhesive and spew fillets due to extra bonding often occur in adhesive joints. Few investigations 9, however, have been carded out on the effects of disbonded areas and spew fillets on the stress distributions of adhesive butt joints. This paper deals with stress analysis of adhesive butt joints, having the disbonded area at the outer part of the interfaces and the spew fillet subjected to external tensile loads. Replacing adherends and adhesive with finite solid cylinders respectively, stress distributions are analysed strictly as a three-body contact problem using a three-dimensional theory of elasticity. The effects of the disbonded area and spew fillet on the stress distributions are shown by numerical calculation. In addition, the analytical results are compared with experimental ones concerning the strain produced on the adherends, in the case where an adhesive joint has a disbonded area.

cylinders are joined by an adhesive bond, subjected to an axisymmetric external tensile load. Fig. l(a) shows the case where a joint has a disbonded area at the outer part of the interface and Fig. l(b) shows the case where a joint has a spew fillet. Adherends are replaced with finite solid cylinders [I], and the adhesive is replaced with a finite solid cylinder [II]. The diameter of cylinder [I] is designated by 2al, with height 2hi, Young's modulus El and Poisson's ratio vl. Those for cylinder [II1 are designated by 2a 2, 2h 2, E 2 and v2, respectively. It is assumed that external tensile loads F(r) act on both ends of cylinder [I] within the range r < c axisymmetrically in relation to the z I axis. Developing the load distribution F(r) into a series of Be sse 1 fu n ctions Io, the boundary conditions can be expressed as follows:

Theoretical analysis

• On cylinder [Ill (adhesive)

Fig. 1 shows adhesive butt joints, in which two solid 0143-7496/90/040294-07 294

• On cylinder [II (adherend) r = a 1

1 O"r =

I 'g'r'/ ~-" 0 oo

ZI = hi

crz = F(r) = bo + ~

bsJo(ysr)

{1)

s=l

vlr = 0

r = a2

II

II

err -- rrz = 0

© 1990 Butterwonh-Heinemann ktd

INT.J.ADHESION AND ADHESIVES VOL. 10 NO. 4 OCTOBER 1990

(2)

(b) The case where a joint has a spew fillet I

r < aI

II

(az)z, = -h, = ( a t ) z , = h,. II

(r~r)z, = -h, = (rzr).~ = h:

(U%z, = -h, = (uU):~ = h:

(ew'

\ - ~ - r ,1:_,= h=

Or]z,=_ & al < r < a2

Adhesive E2, v 2

(a~

I)z2 = ha =

(4)

(~,113

~" zrlz2 = h2 = 0

where the displacements in the r and z directions are denoted by u a n d w respectively, Ys is the positive root satisfying the equation J l ( Y s a l ) -----0 (s = 1, 2. 3 . . . . ), and bo = a--~l 2

bs =

f0°, F (r)r dr

2 ~ a l J?)(Ysal )

fo'

F(r)rJo(Tsr)dr

(s = I, 2, 3 . . . . )

In the analysis of the finite solid cylinders u n d e r b o u n d a r y conditions (1)-(4), Michell's stress function • is used• Using O, the stresses and the displacements are given as follows II

a

O'r----- ~ - ~ ( V V 2 0 - - 0 2 0 ~

-e-T)

0(

,)

eo--~ vV-'O -r &00 (5) ~r~=

(2 -

v)V20 -

Oz 2 j

r,z =

(1

v)V20

Oz 2 j

Adhesive

q."2

1 -

U =

-

-

v 020

E

OrOz

1+ v ~

w= T

020

~(1 - 2 v ) V 2 0 + ~

( V2V20 = 0,

a2

1 00)

+ r "~-r j"

l&

(6)

a2 )

V2 = ~rr2+ 7 ~r + ~-~z2 w

b Fig. 1 Adhesive butt joints subjected to external tensile loads: (a) joint with disbonded area; (b) joint with spew fillet

• At the interfaces (a) T h e case where a joint has a disbonded area

where J~ (Tsal) = 0, Jl (7sa2) = 0 (s = 1, 2, 3 . . . . ), E is Young's m o d u l u s and v is Poisson's ratio. Michell's stress function O, which is selected from solutions for the method of separation of variables, is set as • I = • II + • 2I + • 3I + O4I to analyse cylinder [I], a n d • 11 as • !1 = OiII + O3II for cylinder [II]. The stress functions • I, O~, O~, O~, O1 ! and O~' are expressed by • -I --! -I --I =I =I ~I EquaUons (7) a n d (8), where A 0, Co, A n, Cs. A n, C s, A o, .~I 7I

~,1 ~ I

~-.I "7II ~ I I "71I ~ I I ";I! ~ , l l ":II

----~1I

~..O,t~l n, l.. s,.q n, l... s,A[O, l..O,.qn, C-.s ,AO, G O.A n a n a t . s =

(r~r)z I

-h I

(u%2, = - , , = ( u " ) , , =

dr ]z, = -h, a2 < r < a I

' (Crz)z, = -h,

(n, s = 1, 2, 3 . . . . ) are u n k n o w n coefficients determined from the b o u n d a r y conditions.

= (~.II) . zr.z 2 = h2

C o , A-'n , -Cs, - I --I o i = o,(a'0,-' ' a l , h l , 3 l , Ys, ~n. ~s, Vl,r, zl)

,~

(3)

=(,w,, I

_,z~

\"~--r /z,. = h,

= (~.)~,

= -hi

=

o

~,r 2

=

y.

~'. A

x ll2(l - v j ) h . + 0~. ~loallo. -- # ~ . d j , . l

I N T . J . A D H E S l O N A N D A D H E S I V E S OCTOBER 1 9 9 0

295

x sin(fiinz~) +

Y~ fls

--II

II

~

~s = f2s = f'/s(ysh2) '

+ Ysht ch(?'shl)} sh(ysz~)

fil

+ 7sZ~ sh(rshl)ch(Ysz~)lJo(Tsr)

~'

'

= f~s = fls(Tshl) = sh(gsht)ch(?'sht)- gshl, = An(fiinal. vl) = [12(1 - v,) + (fi~a,)2lt~a

A~ = ~ L C= s' . a ~ . h ~ . f i , .,.y , .

__

=_

1

: , Ds. = , vt, r, zl) An.

~I1

An = A n ( f n

oo

~'~ 1-12v, c h ( y s h l ) + ?'sht sh(Yshl)l

s=l ~

=

I1'

(I)3(2 ~0, ~~ O , "~~ n , ~~ s ,

a2, v2)

a t , h l , f i n ,['Y s ,

2~v, + ~ ( 2 v ,

TM A"~n , l'~s,

- I) = o,

.~(1 - v , ) + C ~ ) 2 ( 2 - v,) = 0,

v 1, r, Z I )

=Ao-~+

-[I = A n ( f i / l a 2 +v2). An

AIo(1 - v l ) + C ~ 2 ( 2 - v t ) = ao,

× ch(YsZl) + YsZl ch(Yshl)sh(yszl)lJo(¥sr) a,~

I

Substituting Equations (7) and (8) into (5) and (6). the stresses and displacements are obtained. Restricting the stresses and displacements obtained to the boundary conditions (1)-(4), the fo.llowing equations with unknown coefficients are obtained.

I' , , I' -- fin r I l a l l r ] COS(fin Zl )

+ Z

2 2

- (final) loal/flnal. ~ I = An ~! = A n ( f i n i'a l . v 1 ) .

2(l - vl)lla + final loa}lo,

fiP

7s = A/a2,

fls = fls = fls(ysht) = sh(Ysht)ch(ysht)+ ysht,

s=l

0 + = 02(

t

7s = :ts/al,

[-12v~ sh(yshl)

OD

o-~-+

n ~

, (7)

~n

/,

--[{2(l-v])lh~

1)+ Z

~

~

•

j ~ (-°~''~-+ ' /3. lal

n=l

fiin'3

:,

['

.4~v, + a ~ ( 2 v , -

+ Z

I'

+ #in' al Ioa Ilor -- fin rlla 11r] sin(fin Zl)

~ l J ° ( y s a l ) = 0, ~ s Yshl

s=l "1

+ =

~ [{(1 - 2vt)ch(y~hl) Ys lq~

A~t(l - v2) + ~ 2 ( 2

2~l(l - v2) + ~ 1 2 ( 2 - ,'2) -- 0,

+ y, z t ch(ysht)ch(y,z~)]Jo(Ys r)

oo (_l)n + 1 ,4h'v2 + C~'(2v2 - 1) + Z jIn' - n - ~ fin h2

= (I)1(.4 n. Cs. a l . h i . fin. Ys. An. lqn. Vl. r. z l )

-8p = < ~t - . 2 ." n =

1 ,"n

"

-

n=|

+ fiin., I°all°~

v~)I~a

+ Z

--n

I

Qo

(i)~l

=

~1 c,

An+

~ ll(l - 2 v l ) s h ( Y s h l ) s = t Ys f'ls

-

-II'

S=I

nm +

n=l

CsTsm

= O,

s=l oo

~s

,

, "Zll .~II

A~In

+

Z

~I~I CsQ,n = O,

s=l

(8)

, Y s , l:~n, ~l's ,

V2, r, Z2)

Z

~n ~u

(lo)

~ , ,~,, = O,

s=l

where Jp(r) is the first kind of Bess¢l function of order p, I~(r) is the first kind of modified Bessel function of order p and Iv(fina ) is abbreviated as I~a, I~(finr) as I n . I~(fi'nr) as 1'~. and I~(fi'na) as I'va and a is replaced by a~ on cylinder [I] and a2 on cylinder [II]. Sinh is abbreviated to sh and cosh to ch. /'/11'

fiin = fin(h,) = T,'

296

C~Q..=O.

oo

~inl +

fi," = fi.(h:),

An+

s=l

AnS

a2, t12, ]:in

= "*'3~"~ 0 , ~ 0 ,"1 n , C s ,

C.Q.n =O.

oo

+ 7~zl sh(y,h i)sh(y~zl)]Jo(Y~r) ¢ , ( 2 o " t=""O~, "" I=nu, I-'s , a 2 , -n2, o , Pn, Ys,, - .An, -. v2, r, z2) ~ ,.')"II ~ I I '~II ~ I I

ysh2

s=|

-- Yshl ch(yshl)}Ch(YsZl)

¢I l

~'IlJ°(Ys'a2)= 0

~s

I

-- finrll,,Ilrl Cos(finZ l )

+

(9)

.4gv 2 + C~I(2v2 - 1) = 0,

-- Yshl sh(yshl)lSh(YsZl)

= -

ai - v2) = -~a0, a5

fiIn' = fi:(h,)

( 2 n - l)z" = 2h I '

n=l

s=l

Z AnRns + Cs -'-'

ns + Cs = as,

-'

n=l O0

:'

n=| oO

AnRns+Cs+

AnRns+Cs=O

t

fi"' = fin(h2).

INT.J.ADHESION AND ADHESIVES OCTOBER 1990

n=l

n=l

Z-'-'-' Z='=' A . R . s + Cs +

n=l

Cs

n=l cO

-,-, H.~ - Cs Fs + Z

n=l

O0

+ Cm eros=

~II~ll

m=l

,,,,',,

-

v2)

=I = I

=l =I

AnHns + CsFs

~(12)

n=l O0

z{z-.-= -"} n=l

_

OD

y

='

AnRns-

O0

- E

A~l. E~. sI

~i~i ~i~i ~i~i C t D t s -- CsGs + CsGs

- E

n=l

t=l

+ ~[l 2(2 - v2)l z,. O0

E

cO

E

~ I ~ I - C ~ + Z A=I'~I AnRns "~'l n R n s + Cs n=l n=t

-

m=l

m=l

n=l

:{£ m=l

f,.s = O,

"~'ll~'II + ~nrtt

~tI

Hnm-

±

CmF m +

O0

I F=ml + C= m

,-m~...sg... = r / o ( r s a , ) \ G ,

+

n=l co

t=l

=

~s ~s

- c s w~ +

m=l

(ll)

-~]l(Yshl) 2 + (nrr)2] 2 --II = ~s.(h2. Q..

G2 .]"

~I A~I. P~nI m + C--Im Wm

(-1)"

, ~ ] [ ( Y s h l ) 2 + l(2n - 1)n/21212

4 ( - 1 ) n + I(mr)3Jo(Ysal)sh2(Yshl)

=l Qsn

fi]{(Yshl)2 + (///02}2

x

11/@.' + t~) + I/(t~.' - tL)l,

ns(r. + t~)

{

=0

2 ?'sht +

I'-

± --x--,

-t

n=l

AnRnm

F~ [(tz a| )2 +/t212j0(r, al) '

~',,

oO

--II--ll An R n s + C --il s

~0

z{ nz= l- ' - , - '

--

m=l

C m

+

nRnm z --'--' A

n=l

O0

~

"'.".sn=l

-,,

n

=

I'

2 '2

4 ( - 1 ) }t~(tnal) Ila

A~I. l ( t . ra , )2 + ~.~12jo(7,al)"

R.s = R.s(t., A.. Ys. al ).

= l , ' ~ ( l - v , ) + C ~ 2 ( 2 - v,)lVj oO

=

R,,(g,. An.?%ai). --'

~1 , I' ~1 R.. = a..(t., z~.r..a,)

n=l

An Rnm

,

Pm ) ~lI t/m = Snm(tn ,tin), ~II Ys, , a2), Tsm = Tsm(h2"t i nII, l~s, 4 ( _ i ) . + 1~2.~1 -2 --I --I As Dn al lla R., = R . , ( t . , I A., Z,, al ) =

n=l

=l } eros+ t Z - Cm

)~

~sh(ysht)ch(Tsht

rs 11' II

Rn m + C m +

--

+ i y s h l ( 2 n - 1)2rr2Jo(Ysal)ch2(yshl)

Qstt

x

{£ .~,,,-,

(;0

Ys, , G2).

~I I ~I 2(- l )mysJo(ysa l ) Tsm = Tsm(hl ,tim, ~'~s, Ys, a l ) = ~1 2

n=l

m=l

~s, ~I,

I' l = (-- 1 ) m + n + l S~Inm= Snm(tn,tm)

O0

~..

n=l

= 4 ( - 1 )" 7s h I (n rr) 2Jo (Ysa I ) sh 2 (Ts h I )

n=l ~II

~,i ~

-.

~I1 ;'~'.ll/ ' + C m rVmfgms = 0

~"I ~I "~I ~I } ~ItfTll AnPnm-Cm Wm gins + ~s ~ s

-Z , •nI-In s•-IC4-Is ~II Ws

z{ ~

--I Qs. = Qs.(hl,-~].Ys, al)

n=l

Z

G~/

n=l

m=[

=l

Z

CoH'~

where

Y'sdo(Y'sa2)\ GI

{ -C- Im - -U I m - C=m - 'U I - -mI -

(C~

at

n=]

~t ~ts +

1

n=l

QO

+

,

Z ~ iA~n,P . s + C~si ~ ] + E = i =AnPns i

AnHnm

y

tO

- y-,I-,,

+ Cm

,~.=,m -

Cs Us - C=I=, sUs-

~I ~I ~I ~ l } -- C m G m + C m G m gins = 0

+

2",II,~II'~

,= -i -i

A~ln E~nIm -- Z C ~I~I tDt m t=l

n=l

E

n=l

E ~II~ll

ClO

_ E

2.~n

n=l

n=l

n=l

_~"~"

at

--

-- Z

~n~.

Cs = O,

--II = R . d t . .II -i'll , a~). ~II = R ' /R Ii' ~11 , R,,~ /'An, Ys, . R ns "'ns,,t-'n , A n , Ys, a2), --I I --I H.s = H.s(t., A., G I, vl, Ys. a , ) =

2(-1).___~"+ |Y~___/~_£ ( t ~ 2 '~ 2 ~ t ~ . , , , j o ( r . a , ) ( r ~ + t~.)c ' 1 v, + r. + t . I2) • =I I' =! Hns = H.s(~n, A,.Gi.vl.7,.

a l ).

INT.J.ADHESlON AND ADHESIVES OCTOBER 1990

297

--If

H.s

=

II - - I I

'

2 ysJ1 (Ym a I )

H,.(#,,. An, G2, v2. Ys, a2). gins

= Fs(h|. ~s. Gt. vl. Ys)

2a2Jt (Ts a2)

1

27,-~IG~-'-s

-

=1

alJo(Tsal)(y,2m _ y~)

,

{(I

2v,)sh(r,hl)ch(r, hl) - y, ht}.

-

1

- 2ys~-l~G1 I(1 - 2 v t ) s h ( y s h l ) c h ( y s h l ) + Yshl ]. --11

Fs = Fs(h2. Fls, O2. v2. Ys). "I

v;

=

,

:

2

,

7sa2Jo(ysa2)

The suffixes nm and m are replaced with ns a n d s. respectively. By solving 12N simultaneous equations. (10) and (11) in the case where a joint has a disbonded area, and Equations (10) and (12) in the case of a spew • -I ~ll fillet, the u n k n o w n coefficients A- - I n, -Cs . . . . . A~ l nl and C s are determined, respectively, where the n u m b e r of terms is set as N in the numerical calculations. In addition, the u n k n o w n coefficients . ~ . CI0. . . . . .~I01 and ~ I are determined from Equation (9). Using the coefficients determined, the stresses and displacements can be obtained.

=1

F s = F s ( h l , Ds, GI, vt, 7s)

--I1

2a~Ji (7sa=)

v, =

I'

E ns = Ens(h 1,/3n, Vt, GI, Ys, al ) ( - 1 ) n+ I(l - vl)

= (1 + Vl)YsJo(ysal)#l'hiGI " D~s = Dts(hl. vl. Gl. Ys. Yt. al)

(1

-

Experimental

vl)Jo(r, al)

Fig. 2 shows a specimen used in the experiment in case where the joint has a disbonded area. The specimen is made of steel for structural use ($45C, To make the disbonded area, the outer part of the interfaces was cut to a depth of 1 mm. After the specimens were adhesively b o n d e d ( S u m i t o m o 3M Ltd., Scotch-Weld 1838), a tensile load was applied

(1 + vl)hlYsYtJo(Ysal)Gl" ~ll

ll'

,

E ns = Ens(h2. ~ , • v2. G2+ 7s. a2).

~ll

'

'

Dis = Dts(h2. v2. G2, Ys, Yt. a2), ~1 (1 - Vl)Ch2(Yshl) Gs =

(yshl) m I (1 - v l ) s h '"~ Gs = ~t GlYs~s

G,rsfi]

~11 G s

(1 - =

--1

,7"-~II

J|S).

Co. to

038.0

2Ys~l s

(1 -- vl)sh2(yshl)

--ll Us

--I

(I - v2)sh2(Ysh2) ---

--ll

G I l'l s U s

the

' ' v2)ch'(gsh2) G

Us =

G2 l'ls

(1 -- v2)sh 2(ysh2) '

--II

method

=

i16,0f '

--II

G2~s

I

P ns = Pns(fln. An. V!. GI. Ys. al)

.

=

2 ( - 1 ) yslla ~I 2 + 31,2)jo(ysal) GIAnal(Ts

( vl

"f'~s ) , + l ~ n 12

,

100.0

P ns = Pns(fln, An, Vl, GI, 7s, al ), ~11

R1.5

66.0

- 1+

025.0

I1' ~ I I

Pn, = Pns(~.. An. v2. G2. Ys, a2). Ws

Ws

Ys =

X

Fs(h 1,

GI, vl, Ys),

ys X Fs(h I. Ds. G1. vl, '

Ys),

El

Ws

Ys X Fs(h 2,

G2, v2, Ys),

G1 = 2(1 + vl~"

020.0

E2 G2 - 2(1 + v2~)"

1.0 f

R20.0

1

27sa2 Jo(Yma2)J1 (Tsa2) eros = 2 2 2 a iJo(¥+al )(Ys - Y'm2) t eros =

27'salJo(Yrnal)Jl(ysal ) 2 2 , ,2 a2Jo(gsa2)(ys - 72..)

f ms=

2y'ma2Jo(Y'ma2 )Jl (ysa2) 2 2 2 a iJo(Ysal ) ( Y s - r'2m)

O14.0 "

r

2ymalJo(ymal)Jt(Ysa| ) f'ms = 2 2 ' '2 " a2Jo(Ysa2)(y s - 72m) gms =

298

2 y~JI (Y,,,a2 ) ' 2 ' a2Jo(ysa2)(y m _ y~2)

INT.J.ADHESlON AND ADHESIVES OCTOBER 1 9 9 0

Fig.2 Dimensionsof specimen(ram)

the joint using a materials testing machine (Shimadzu Co. Ltd., Servo Pulser EHF-EA10). The strains on the outer circumference of the adherends were measured with strain gauges (Kyowa Electronic Instruments Co. Ltd., KFC-CI-ll), and the load was measured with a load cell. They were recorded with an X - Y recorder through dynamic amplifiers. Analytical experiment

results

and

a comparison

with

In this a n a l y s i s , the stress singularity at the edge of the interfaces is not taken into consideration. Numerical calculations were carried out varying the number N of terms as 100 and 110 in order to examine the effect of N on the stress distributions at the interfaces. It was seen that the difference between both results was less than 3% at r/a = 0.99 and z 2 = h2. Subsequently, numerical calculations were all carried out with N as 100. Fig. 3(a) shows the effect of the diameter of adhesive on the principal stress distribution at the interface (z2 = h2) in the case where a joint has a disbonded area, where the value az/al was varied as 0.7, 0.8 and

1.6

a2/01 = 0 . 7 0 .-

-

1.q.

F

.....

= 0.80

. . . .

=0.90

F

N

02a;

1.2

1.0

0.8 L 0.0

I 0.2

I 0.4

I 0.6

I 0.8

0.9. In the numerical calculations, the assumption was made that tensile loads act uniformly within the region 0 < r < a~ at the upper and lower surfaces of the adherends (zl = hi). In this figure, az,,, represents the mean normal stress. The ordinate represents the ratio tr~/trz,, of the maximum principal stress to the mean normal stress, crt is obtained by taking account of the normal stresses ~, and crz and the shear stress r~. The abscissa represents the ratio r/a2 of the distance r from the centre to the radius a 2 of the adhesive. In these figures stress distributions are shown within the region r/a2 < 0.99. The stress singularity arises near r/a2 = 1 and it increases with a decrease in a2/al. The case of a2/a~ = 1 shows that the joints have neither a disbonded area nor a spew fillet. Fig. 3(b) shows the analytical results in the case where joints have a spew fillet. The value ofa2/a I was varied as 1.04, 1.40 and 2.00. The abscissa represents the ratio r/at of the distance r from the centre to the radius al of the adherends. The stress singularity decreases with an increase in a2/a I near the point

r/a 2 = 1.00. Fig. 4 shows a comparison of the principal stress distributions where a2/al < I (disbonded area), and > I (spew fillet). The abscissa represents the ratio r/a of the distance from the centre to the radius a (a is substituted for a 2 in the case o f a l > a2 and for az in the other cases). The value ofcrl/Crzm is 1.75 for a2/a I < !.0, !.41 for a2/a I = 1.0 and 1.39 for a2/a I > !.0 at the point r/a = 0.99. In the case where a joint has a disbonded area, the stress singularity becomes considerably larger and it is estimated that the strength of the joint decreases. In the case where a joint has a spew fillet, the stress singularity decreases and it is thought that the strength of the joint increases comparatively with the joint having the disbonded area. Fig. 5 shows a comparison of the analytical results with the experimental ones concerning strains produced on the adherends in the case where the joint has a disbonded area. This is the case where the joint

.0

1.8

rio2

a

• z 1 , z2 ....

1.6 ~ r

1.6

~

.....

02/01 = 1.04

m-~

= 1.40 = 2.00

r

1.4

o2/o 1 =0.6

- - = 1 . 0 ~-~=1.4

r 1.4

1.2

c)

~3 ._ _

,/j

1.2

.,

1.0 1.0

O.8t 0.0

b

I 0.2

I 0.4

I 0.6

I 0.8

1.0

0.s 0.0

r/°1

Fig. 3 Effects on principal stress distributions at interfaces w i t h : (a) disbonded area; (b) s p e w fillet

I 0.2

I 0.4

I 0.6

I 0.8

.0

r/a Fig. 4

Comparison of principal stress distributions w i t h varying a2/a 1

INT.J.ADHESlON AND ADHESIVES OCTOBER 1990

299

10

0 Experiment Numerical

,~Z

i x e.

I

I

-10

-5

I

0

5

Distance, z(mm) Fig. 5

distribution on the joints was demonstrated as a three-body contact problem using the threedimensional theory of elasticity. (2) Using method (1), the effects of disbonded areas and spew fillets on the stress distributions at the interfaces were examined by numerical calculation. As a result, it was shown that the joint strength decreased considerably when the joint had a disbonded area with a comparable increase when the joint had a spew fillet. (3) The analytical result obtained by method (1) was satisfactorily consistent with the experimental result concerning the strain produced on the adherends.

Comparison of analytical and experimental results

Acknowledgement is subjected to a tensile load of 0.49 kN. Young's modulus for adherends is taken as 206 GPa and Poisson's ratio as 0.3, and those for adhesive are 3.6 GPa and 0.38, respectively. The height 2hi of adherends is 20 m m and 2h 2 of adhesive was measured as 0.055 ram, and it is assumed that tensile loads act uniformly at the upper surfaces of the adherends (zl = 10 mm). The ordinate represents the strain produced on the adherends in the z direction, and the abscissa represents the distance zj from the centre of the adherends (zl = - 1 0 m m is the interface). The experimental and analytical results are satisfactorily consistent. Next, tensile tests were carried out until specimens ruptured in the cases ofa2/a I = 0.7, a2/a I = 1.04 and az/al = 1.0. In each case, the tests were carried out four times. The mean rupture stresses were obtained as 1.80 k g f m m -2 (17.7 MPa) (a2/al = 0.7), 3.13 k g f m m -2 (30.7 MPa) (a2/al = 1.04) and 1.96 k g f m m -2 (19.2 MPa) (a2/al = 1.0). These results correspond with the analytical ones shown in Fig. 4.

We would like to express our thanks to Prof. K. Ikegami of Tokyo Institute of Technology for his advice on this work.

References 1

Suzuki, Y. Bu/IJSME 27 231 (1984) p 1386

2

Lubkin, J.L JApp/Mech 24 2 (1957) p 2 5 5

3

Renton, W.J. andVinson, J.R. J A p p l M e c h 4 4 ( 1 9 7 7 ) p101

4

Kaplevatsky, Y. and Raevsky, V.J. J Adhesion 6 (1976) p 65

5

Wah, T. I n t J M e c h S c i 1 8 ( 1 9 7 6 ) p233

6

Sawa. T., Nakano, Y, and Ternma, K. J Adhesion 24 (1987) p 1

7

Sawa, T., Iwata, A. and Ishikawa, H. Bull JSME 29 258 (1986) p 4037

8

Alwar, R.S. and Nagaraja, Y.R. J Adhesion 7 (1976) p 279

9

Adams, R.D., Coppendale, J. and Peppiett, N.A. J Strain Anal 13 1 (1978) p 1

10

Gray, A. and Mathew, G.B. ~A treatise on Bessel functions and their applications to physics" (Dover, 1966) p 91

11

Timoshenko, S.P. and Goodier, J.N. "Theory of e/asticiW; 3rcl Edition (McGraw-Hill, 1970) p 380

Conclusions This paper dealt with a three-dimensional analysis of adhesive butt joints, in which two solid cylinders were joined by an adhesive, subjected to external tensile loads when the joints had disbonded areas or spew fillets. The following results were obtained. (1) Replacing adherends and adhesive with finite solid cylinders, a method of analysis of the stress

300

INT.J.ADHESIONAND ADHESIVES OCTOBER 1990

Authors K. T e m m a is in the Department of Mechanical Engineering, Kisarazu National College of Technolo~'. 2-11-1 Kiyomodai-higashi, Kisarazu, Chiba, 292, Japan. T. Sawa and Y. Tsunoda are in the Department of Mechanical Engineering, Yamanashi University, 4-3-11 Takeda, Kofu, Yamanashi, 400, Japan.