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Two-dimensional stress analysis of adhesive butt joints subjected to cleavage loads
Two-dimensional stress analysis of adhesive butt joints subjected to cleavage loads K. Temma,* T. Sawa t and A. Iwata ¢ (*Kisarazu National College of Technology/tYarnanashi University/$Toshiba Machine Co. Ltd., Japan)
This paper deals with the two-dimensional stress analysis of adhesive butt joints subjected to cleavage loads. The purpose of the paper is to contribute to the establishment of fracture criteria of adhesive joints. Similar adherends and an adhesive bond are replaced with finite strips in the analysis. Stress distributions in adhesive joints are analysed using the two-dimensional theory of elasticity. The effects of the ratio of Young's modulus of adherends to that of an adhesive and the thickness of the adhesive bonds on the stress distributions are shown by numerical calculations. For verification, strains produced on adherends are experimentally measured and a photoelastic experiment is carried out. The analytical results are in fairly good agreement with experiment.
Key words: elasticity; stress analysis; adhesion; butt joint; cleavage loads; photoelasticity
Adhesive joints are being used in mechanical structures with improved adhesive properties. At present, however, data for designing these joints are markedly fewer than for conventional joints such as bolted and riveted joints. It is very difficult to estimate stress distributions on the joints and their durability. So it would be useful to have data available for design and to establish an optimal design method. In order to establish a design method for adhesive joints, it is necessary to know the stress distributions on the joints more precisely. Up to now, many investigations have been carded out on butt adhesive joints subjected to tensile stress ~-s, bending moments 6"7 and shear l o a d s8. Furthermore, some investigations 9--22 have been carded out on butt adhesive joints subjected to peel and cleavage loads. However, it seems that few analyses have been made strictly on joints subjected to cleavage and peel loads. This paper deals with stress analysis of adhesive butt joints, in which two similar finite strips are joined by an adhesive, subjected to cleavage loads. In the analysis, replacing adherends and an adhesive with finite strips, the stress distributions on the joints are analysed strictly as an elastic three-body contact problem using the two-dimensional theory of elasticity. Based on the analysis, the effects of Young's modulus,
Poisson's ratio, the thickness of adhesive and the distribution of cleavage loads on the stress distributions at the interface between an adhesive and adherends, are clarified by numerical calculations. For verification, strains produced on adherends are measured experimentally and a photoelastic experiment is also carried out. The numerical result is compared with experiment and the stress distributions are discussed.
Theoretical analysis Fig. 1 shows an adhesive butt joint, in which two similar finite strips are joined, subjected to a cleavage load. To analyse the stress distributions on the adhesive joint, two adherends are replaced with finite strips [I] and the adhesive with a finite strip [II] as shown in Fig. 2, taking into consideration the symmetry of the x-axis in the adhesive. The length of finite strip [I] is designated by 21, the height by 2h I, Young's modulus by El and Poisson's ratio by yr. Those of finite strip [II] are designated by 2/, 2h 2, E2 and v 2, respectively. It is assumed that the cleavage load acts on the upper and lower ends of the joint with respect to they~ axis within the range el < x < e 2.
0143-7496/90/O40285-09 © 1990 Bu~erwo~h-Heinemann Ltd INTJ.ADHESION AND ADHESIVES VOL. 10 NO. 4 OCTOBER 1990
285
Yl ,Yl
!
X = "1-1
F(x)
l
(1.1)(1.2)
(~x = Vxy = 0
Yl = ht h2
[I]
Adherend E1 ,G1, Vl
oo ~
,01 [ttl:
-02
x
~o
as COS
(~y m ao +
x
x
bs sin
+
s=l
x
I
(1.4)
Vxy=O
03
E1 ,G1, vl
Adhesive
[11
E2,G2,v2
where
213
l ['e2 ao = Zt l , F ( x ) d x
F(x]
Fig, 1
/
/
41-
(1.3)
s==l
as = 7
F ( x ) cos
x dx
bs=
F ( x ) sin
x dx,
Adhesive butt joint subjected to a cleavage load
F{x)
(s= 1,2,3 . . . . )
For finite strip [II1 (adhesive) e2
1-
II
X = +_l
II = 0
(2.1)(2.2)
t7x = rxy
For the interface I
el mO
(%)y,
1
X
E1 ,G1 ,Vl
/
/
Finite strip [I] (adherend)
a
Ii
(G),,
h2
(3.1)
x xyJy~=h 2
(3.2)
h, = (%)y~
o -
=
=-h~
=
o., I __ro.,,i Ox /y,=_h, \ OX /y~=h~ (ov, I O X / Y l = - hi
(3.3) (3.4)
k OX ]Y2 = h2
In order to analyse finite strips [I] and [II1 under boundary conditions (1)-(3), Airy's stress function X is used. The stresses and the displacements (plane stress) are given as follows: O2Z
O2Z
trX= ~y2
rry= ~ x 2
O2X rxy =
OxOy
(4)
h2 2Gu =
Y
,t- o 2 -
2Gv =
1
OZ
O0
~ - Ox 1 + v Oy OZ Oy
1
00
l+v
Ox
+ - -
(5)
where the stress function X and the function 0 must be satisfied by the following equations, respectively. ~2 I
b
I
Ox ay
v2z
720 = 0 v2 = ~ x 2 +
Finite strip [11] {adhesive)
Fig. 2 Model for analysis and dimensions: (a) finite strip [I] (adherend); {b) finite strip [11] (adhesive)
Developing the load distribution F ( x ) into a Fourier series, the boundary conditions shown in Fig. 2 can be expressed as follows, where the displacement in the x direction is denoted by u, the displacement in the y direction by v and superscripts I and II correspond to finite strips [I] and [II], respectively. • For finite strip [I] (adherend)
286
yZv2x = 0
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
where G is modulus of elasticity in shear, and v is Poisson's ratio. Airy's stress function XI, which is selected from solutions for the method of separation of variables of biharmonic functions, is represented as Z~t + X~t + X~t + ZI4t + ,t~b + . . . + Xl4b for finite strip [I] and Xn as x~I' + x~3~' + x~~b + x~~b for finite strip [II] 1"7. Stress functions X~k Z~_t. . . . . Z~b, X~b. . . . and Z~". . . . . Zl 'b. . . . are --It --lb expressed by Equations (6)-(17), where A--It ,, B s ..... A, , -Ib ~.b B s .... and_s ( n , s = 1,2,3 . . . . ) are unknown coefficients determined from the boundary conditions.
x~t
,zlb
t . . I t "-;It --It
= X~ ~ao,/in, Alt
I --I ~ I Bs, 1, h l, an,As, An, fls,x, yt)
~
=
b ,'~Ib w l b
= X l {/in , IJs ,
1, h l, a nI, A s , ,
-It
+
,An ,, Ilo'., ch(a'nO + sh(a' O)
=
n = I IXnCtn
/2 Ila~" / sh(a~"/) + ch(a~" O} sh(a~, x) nil
~k~an a n'X
--
sh(a~, x)l cos(a~yO
s
~
I
=
ch(Ashl) + sh(Ash~)} ch(AsYl) - AsYl sh(Ashl) sh(gsyl)] cos(Asx) (6)
o0
(10)
=I =I hl,an,1, As,, An, l'~s,X, Y l )
"" l b
= ~-,=j z.xnanrIAn 1'2 [[a~' l sh(a~' /) + ch(a~' l)} sh(atn'x)
= It
~ (a'.' x) -
-'$''$
b,~Ib =lb = Z 2 ['*in , B s ,I,
v - n , o s~. u l, hl,an,As, I' = I fls,X, =I x~t = Xt2 t~u An, yt)
= -An ~ _ [,.,_ [{a~'lch(a~'l) An a n
IIA'~hlch(X'shD+ sh(A~hl)} ch(AsYl)- XsYl sh(Aj hi) sh(A~Yt )] sin(A~x)
z[b
-----~ l =
~Bs
S = I
[{Ashl
~qsA~
ch(a~/)ch(a.~x)! cos(a~y,)
--Ib
+ E + /~
--I ~ I
A n , ['~s,X, y l )
+ sh(a~'l)}
-- a.I, x ch(a nI' /) ch(al'x)i
,~'/xsh(a'io sh(a~'x)]
sin(a~'yt)
sin(aln'yl)
+ __ ~ ~ + ~ ~ llZshi sh(Ashl) + ch(Xshl)} s = I f~s As
$
==
I
liX.~h, sh(A;hm) + ch(Ash,)}
"-$ "'$
sh(A;yl)- Z;yl ch(A~hi)
sh(Asyl) - Asy~ ch(Ash~) ch(Asyl)] cos(As x) (7)
t
,
#
ch(As Yt )] sm(A,x)
(1 l)
X[ b = Z b t2lb,*-n, -~tlb, s l, hl,a l:As, An, fls, X, Yl) v * n , ~ s~lt , "XI 5 1 ,, X, YlJ" )(,~t..~)(t3i'~It l, ht,an,I' Zs,, t.xn,
¢0
~lb
= _ ->. n=
|
~¢Zn
- - a n Xt,
at,,'x ch(a~'/)sh(atn'x)l
cos (an Yl )
cos(aln'yl)
,:c,
~lb
[Ashl sh(As hi) ch(AsYl)
-Z
~t
~t,'~s
f~' h
s= I lqsAs
1 sh(Ashl)ch(A~Yl)
- Z, yt ch(As hi) sh(AsYt)l sin(Z, x)
s=l
Xsyl oh(A; hi) sh(Ajyl)l cos(X;x) -
z[' =)(,4(An,lJs,l, , . ' T I , ~ I , . hl, a l
(8)
~o
As,. A=n1, %l'~s,X, Y l )
= -
A.
"~I' I2
Ia~ / sh(a I / )
~lb
nZ=, =!A '..i2 n [a~/ch(a'. 0 sh(a', x) A n (A n
ch(aln x) - ~ x sh(a~ o ch(a~' x)l sin(a~ Yl)
I lAtnan
I I anx ch(an/) sh(a~ x)] sin(alyl) -
ao
'~'lt
E S ~
B, |
(12)
n,Bs,/,hl,an,As, An,
=Z4~
~It n=
sh(a~'/) ch(aI.'x)l I,
co
=-
0 sh(a'; x)
~1 ~I,2
/...a
~, A~' [a~ish(a~'l)ch(aln'x) n= , a' a'."-
A°
=lb ~I
2
IA, h, ch(A, hI)Sh(Asy,)
s= I Nsgs
lA'~hlch(A;h,)sh(A',yl)
AsYt sh (As h t ) ch (AsYt )] sin(Asx) -
''S''S
ZsYt sh(Ashi) ch(A;y,)l cos(A;x)
(13)
-
(9)
,~[It
,,~ [All/ 711t 5 l l r =
tao
,an
, Os
,
--II - - I I !, hA, an II, As, An, Ns,X, YA)( 1 4 )
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
287
~,~It
t ,'~IIt a I I t • . II,., "zll ,~II = Z 3 t a n ,/~s , l , n2, a n , A s , / X n , S l s , X ,
, y2)
(15)
~,~Ib =)~Itan b ,-rltb ,~s ~llb ,l, h2, an,Xs, 1I , An, --ll -~tsl X, y2)
(16)
Xs, An, f l $ , x , Y x )
(17)
v-n
,-s
, -, n2, a n ,
where
o0 ~ ~Ib
~lb
An +
00 Ib
Ib
An
Bs Sns = 0 ,
+
s=l
E
~ lb ,~, Ib' B s ~ns
s=l
(18.7), (18.8)
~llt pllt = 0 ~ s --n$
z~llt..[_ E n
=0,
(18.9)
$=1
nrr
i
an =an(hi)=--,
I,
hi
II
_1[,
an = an (h2),
(2n-
,
a n = an(hi)=
fl n
,
= an (h2),
1)zr
o0
2h~
~lit
0O
E
An +
"~ lit A'IIt
=0,
,~tlb _ E ~Hbpub = 0, s=l
s=l
As = s- -n
(18.10), (18.1 I)
1" 00
~.~ _ ( 2 s - l)n 21
.~ub + E
-i = /~n
an (a'. t)
--~"
=I
An
~I
(at.'/),
An
=
=
An
(18.12)
~ub qllb = 0
s=l
sh(at" t) ch(d. t) + at. l, "~In = A n --I, A
o0
--il =An(at"I/), An
o0
E
E
n= [
~II
(18.13)
:"
n= 1
txn = An (at.~' 1), ~
= f~$ (~,s h I ) = sh (As h I ) ch (~,$ h 1) + X$ h I,
e',
fl s = f~s(Xshl), "~'|
t
~'~$I
,
f~s = Os (As h 1) = sh (Xs h I ) ch 0.s h I ) - A,$h I, =
,
--II =
f~s()~shl),~s
I2s()%k2),
~ II =
Os
=
,
l'ls 0.$ h2)
and sinh is abbreviated as 'sh' and cosh as 'ch'. Substituting Equations (6)-(17) into Equations (4) and (5), the stresses and displacements for finite strips [I! and [IIi are obtained. Restricting the stresses and displacements obtained to the boundary conditions, Equations (18) and (19) in terms of the unknown coefficients are obtained. That is, from the boundary condition Equation (1.1), Equations (18.1), (18.2), (18.5) and (18.6) are obtained; from Equation (1.2), Equations (18.3), (18.4), (18.7) and (18.8) are obtained, and Equations (18.13) and (18.14) are obtained from Equation (1.3). Equations (18.15) and (18.16) are obtained from Equation (1.4), from Equation (2.1), Equations (18.9) and (18.11) are obtained, and Equations (18.10) and (18.12) are obtained from Equation (2.2). From Equation (3.1), Equations (18.17) and (18.18) are obtained, from Equation (3.2), Equations (18.19) and (18.20) are obtained, from Equation (3.3), Equations (18.21) and (18.22) are obtained, and Equations (18.23) and (18.24) are obtained from Equation (3.4). oo
-" sE= l -B$ " "Pn$ = 0 ,
An-
E -s-n$
s= l
~
~ It K,/t
(18.1), (18.2)
oo
-Ib
An -
E
o0
o0
+ B$ +
n~]
+ B$
= 0
(18.15)
n=l
E -Abi RIbn--ns + ~Ib + E .~lb_n, .,.
n==l
(18.16)
n=~l
oo
~
E
AItt"jlt
ot~
+ B--1' s + E ~Itf) lt'
- B s=It
n=l
n=l
-
E
"~llt()llt ,-n ~ns
n=l
- -sfiut = 0
(18.17)
O0
o0
"'n ~ n s - B s n=l
+
An ~ , s + B s n=l
0o
-- E
--n~'llbt')llb ~ns + ~Ib = 0
(18.18)
E --nAltRIt--ns+ nlst- E -~ltRlt' ~I' + E ~IItRIl, n - n s - Bs -n --ns n=l
n=l
$=1
(18.19)
aO
oO
n=l
n=l
-~Ib p l b
~s -ns = O,
(18.4), (18.5)
oo
co
= Ib p l b ,
Bs -n~ = 0
(18.6)
+ E n=l
288
(18.14)
+ u.~nt = 0 S
s=l
An +
II,
a,,s
(18.3)
oO
E
b,.
Y "AltRIt - " Y
$=1
= Ib
oo E
n=l
= 0
Zlt ~ , , ~ It ,~It, A n + - s on$ = O,
_ 8$
n=l
O0
~It
t"b . . $ s=l
m=l
oo
An +
An +
-
n=l
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
--AllbRllbn --.s + --s~Itb= 0
(18.20)
oo
oo
~It lilt -- ~ - - I t v l t h l l Bm -m~ms "= R ~/'13 n=l m=l Z
~
+
Experimental method
/~ltIflV n
vns
Strain measured by strain gauges
n=l
co
Z
--
=It Fit, II
Bm-m
~It w i t '
bms-Bs
~'lt [grit,
..s
+Bs
"'s
m=l "~llt ~n$ Hllt + Z - ~ l --rn l t vllt bins I] --n n=l m=l
_ Z
.
II
+ ~ I t wilt
Z
--
m
Z
Uns +
n=l
(18.21)
An vns
v m ares - -
"
m=l
n=l
Photoelastic experiment
oo
Z
+
=lb glb, II
Bm -m
~lb
ares + Bs
lb
Ws
~lb iff:lb,
- Bs
"" s
m=l oo
O0
+ Z
~llb //lib "'t/ v #iS
n=l
~rn Vm a m s - - w s
=0
m=l
(18.22) OO
CO
~ltFlt =ltFItt Z s --s - B s --s +
~[tlglt "n--ns+
n=l oo
-- Z
oo
i~ltl'lt' ares II + Bs --It --ra--m
Results of analysis
oo
~l|t--llt II = 0 tim Jm ares
",~lltldllt n "'ns 1. Z
O0
Fib l -- b~lb Fib, _
=S
~
$
=S
oo
In the present analysis, the stress singulari b' at the edges of the interfaces is not taken into consideration, so that numerical calculations were carded out setting 20
O0
/I
n=l + Z
(18.23)
m=l
m=l
$
Results and discussion
FsIt
mffil
oo
•
"~ltrlt l[ timJmams
In order to examine the stresses on the adhesives, a photoelastic experiment was carded out. Fig. 4 shows the dimensions of the specimen. The adherends are made of aluminium (of Young's modulus 68.7 GPa and Poisson's ratio 0.3) and the adhesive is epoxy resin (Young's modulus 3.5 GPa, and Poisson's ratio 0.376). The adhesive is interposed between two adherends and the interfaces are cemented with adhesive made from the epoxy resin. Isochromatic and isoclinic fringe patterns in the specimen are obtained. The stresses on the adhesion are obtained from the fringe pattern by the shear-difference method 12.
m=l
~'t/./l' "-n "'ns + Z
nml
+ Z
Z
Fig. 3 shows the dimensions of the joint used in the experiment. The adherends are made of steel for structural use ($45C, JIS) and the adhesive is made of epoxy resin (Scotch-Weld 1838, Sumitomo 3M Co. Ltd., Japan). Strain gauges (KFC-CI-I 1, Kyowa Dengyo Co. Ltd., Japan) are attached 2 mm from the interface of the adherends. After joining the two adherends, a cleavage load is applied to the joint using a materials testing machine. The magnitude of the load is measured by a load cell. The outputs from the strain gauges and the load cell are recorded with an X - Y recorder through dynamic amplifiers.
~/IS
+ Z
~
rr7 ~ m
~Plv/$
m=l
¢11
oo
A~lbI.lib' n --ns + Z
n=l
~w ml b /~1?1 I b ' b iIIn + ~llbFllb ~$ $ =
m=l
oo
Go
-
:o n=l
(18.24)
m=l
A~t = ao,
.4[o]' = ,4lot
50
30
( n , s = 1,2,3 . . . . ) (19)
where the constants PUs, Slts . . . . . P ~ , S ~ . . . . are shown in References (1) and (7). Superscripts It and IIt correspond to superscripts I and II in Reference (1) and Ib and IIb correspond to I and II in Reference (7). By solving the infinite set of simultaneous equations Q18), the unknown coefficients ,41nt,~ t . . . . . ,~IIb and B~ rb (n, s = 1, 2, 3. . . . ) are determined. In addition, the unknown coefficients A~ t and .410It are determined from Equation (19). Using these coefficients, the stresses and the displacements are obtained.
,sI
Strain gauge
[]
[]
-
/
[]
-
L.i
/NAA 50 Fig. 3
Dimensions
f
t=8,12
of specimen(in ram)
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 9 9 0
289
W
E,
~
1.0
t "
E2
E 1/E 2 = 3
m
10 100
, ~
i U]
E 0.5 x !
k 30
0.0
I - I .0
15
-0.5
I
I
0.0
0.5
1.0
I 0.0
I 0.5
1.0
3.0
2.0 15
Epoxy-r
I
El
I
1.0 × 15
Aluminit 0.0
- I .0
q~11
I -0.5
I0.0 B
8.0 -
El/E 2 = 3
6.0 -
100
10
E
Fig. 4
-
2.0
-
/,/
Dimensions of joint in photoelastic experiment (in ram)
the number of terms N as 100. Convergence was verified when N was 100. Fig. 5 shows the effects of the ratio EI/E 2 o f Young's moduli of adherends to adhesive on the stress distributions at the interface (Y2 = h2), where El/E2 was set as 3, 10 and 100. It is assumed that a cleavage load acts uniformly within the region 0.8 < x/l < 1.0. In this figure, trym represents the mean normal stress. The ordinate represents the ratio try/trym of the normal stress to the mean normal stress, and the ratio rxy/tyym of the shear stress to the mean normal stress. The abscissa represents the ratio x/l of the distance x from the centre o f the joints to the half length i of adherends and adhesive. It is seen that the distributions o f cry, trx and r ~ tend to increase near the edge, x/l = 1.0, and the stress singularity increases with an increase in EI/E 2. The larger EI/E 2 is, the more uniform the distribution o f stress try becomes. Fig. 6 shows the effect of Poisson's ratio vl/v2 o f
290
4.0
//,
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 9 9 0
_..:')
0.0 -I .0
-0.5
I
I
0.0
0.5
I .0
x/I
Fig, 5 Effect of ratio of Young's modulus on stress distribution at adhesiva-adherend interface. Y2 = h2, h l / I = 0.2, h l / h 2 = 2, V l / V 2 = 1 and F(x) is constant Over the region 0.8 < x / I < 1.0
adherends to that o f adhesive o n the stress distributions at the interface. The distributions o f trx, Cry and rxy increase near the edge x/l = 1.0 with a decrease of the ratio vl/v2. Fig. 7 shows the effect of thickness o f adhesive on the stress distribution at the interface. The ratio hi/h2 o f the height o f adherends, 2hi, to that o f adhesives, 2h2, is varied as 2, 20 and 100. The stress distributions
h2
1.0
1.0 --
E ).. 0.5
_
m
vl/v2
= 1.00
! Ill|
086
i.. :
o.,,
Vl
[
,/
!h I
_
I
0.5
~
E
/
lh I I
W
I I I
h l / h 2 = 100
X t-,
I
20
x i.-,
I
I
I
I I I
0.0
!
0.0
I -1 .0
I
-0.5
I
I
0.0
0.5
I
0.0
0.5
1.0
- I .0
-0.5
1 .0
2.0 1.5
~
-V"2
E
I
i~--"
t
//_.-~. 1.0
1.0 x X 0.5
/// j j,s 0.0 0.0
I ~1.0
-0.5
I
I
0.0
0.5
I •0
1.0
-0.5
I
I
0.0
0.5
1.0
10.0
h l / h 2 = 100
I0.0
d 8.0
Vl/V2
=
•
20
8.0 h
m
2
0.86
E
6.0
6.0
0.75
--
E o
o
4.0
2.0
--
o
4.0
2.0-0.0
--
--
~--
-1.0 0.0
I --1 . 0
-0.5
I
I
0.0
0.5
-2.0 1.0
x/I Fig. 6 Effect of Poisson's ratio of adherends to that of adhesive on stress distribution at adhesive-adherend interface. Y2 = h2, h i / I = 0.2, h l / h 2 = 2, E1/E 2 = 3 and F(x) is constant over the region 0.8 < x / I < 1.0
of ax, Gv and rxy increase near x/l = 1.0 with an increase in the ratio hl/h 2. Fig. 8 shows the effect of the cleavage load distribution F(x) on the stress distribution at the interface. In this case the thickness of adhesives is held constant and the value ofhl/h 2 is set at 5. Numerical calculations were carried out in the cases where the uniform load F(x) acts over the regions 0
-1.0
I
I
I
-0.5
0.0
0.5
1.0
x/I Fig. 7 Effect of thickness of adhesive on stress distribution at adhesiveadherend interface, V2 = h2, h i l l = 0.2, E1/E 2 = 3, v l / v 2 = I and F(x) is constant over the region 0.8 <~x/I < 1.0
can be seen clearly on the stress distribution at the interface. It is found that the stress singularity near the points x/l = - 1.0 and x/l = 1.0 increases in the case where the load distribution F(x) acts over the region 0.8
Fig. 9 shows a comparison of the analytical result with the experimental one with respect to the strains at
INT.J.ADHESION AND ADHESIVES OCTOBER 1 9 9 0
291
1.5 40 O
1.0
-
x
30
.,
Exp Num O
I
E o
x
0.5
X
J
J
0.0
2o -
c10
O 0 W = 0.7 kN
I
-1.0
-0.5
0.0
0.5
-25
I
I
I
I
I
-20
-10
0
10
20
25
Distance, x(mm)
3.0
Fig. 9
Comparison of experiment with analytical results
2.0 E
>.
x o
1.0
0.0
-1.0
-0.5
0.0
5.6"0--0 4.0
~
0.5
]
0.0
--
el/I
=
_
e 1/I
= O. 8
1.0
I~ -i
2.o1 .0
Fig. 10 --
.
~
~
.
Fringe pattern (light field) obtained from photoelastic experiment
~
0.0, -1 .0 ~ ~ ' ~ ' ~ ' ~
-2.0 -1.0
I
I
I
-0.5
0.0
0.5
1.0
x// Fig. 8
Effect of distribution of load on stress distribution at a d h e s i v e interface. )12 = h2, hl/h2 = 5, h 2 / / = 0.1, e 2 / / = 1, E~/E 2 = 65.6 and v l / v 2 = 0.81 adherend
positions 2 m m away from the interface. The results are in the case where the adhesive joint is subjected to a cleavage load of 0.7 kN. In the numerical calculations, Young's m o d u l u s of the s p e c i m e n (adherend) is put at 206 GPa and Poisson's ratio at 0.3, and those of the bond (adhesive) as 3.14 GPa and 0.37, respectively. The thickness 2h2 of the adhesive is measured as 0.055 m m . The distribution F(x) is a s s u m e d to be uniform in the range el = 5 m m and e2 = 25 m m on Yl -- hi o f the specimen. The
292
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
ordinate is the strain of the specimen in the Yl direction, and the abscissa is the distance x from the centre of the specimen. The solid line represents numerical values and O experimental values. They are in fairly good agreement. Fig. I0 shows an isochromatic fringe pattern of the adhesive. As a special case, when hi~h2 is set at unity, a photoelastic experiment was carried out. Fig. 11 shows the comparison between the analytical results and experiment with respect to the normal stress Gv at the middle surface of adhesive. They are also fairly well matched. Conclusions
This paper deals with two-dimensional stress analysis of adhesive butt joints, in which two similar finite strips are joined by an adhesive, subjected to cleavage loads. The following results are obtained. • Replacing adherends and adhesive with finite strips. a method for analysis of the stress distributions on
3.0
/.~
values with the analytical results, it was seen that they were in fairly good agreement.
/_ Num 2.0-
------
Exp
//
Acknowledgement
The authors would like to thank Professor K. Ikegami of Tokyo Institute of Technology and Dr Y. Suzuki of Nippon Sharyo Seizo Co Ltd. for their advice on this work.
D
I .0
/ // /
-
0.0
References 1
W=0.qRN s -
.0
f
I -0.5
I
I
0.0
0.5
1.0
x/I
Fig. 1 1 Comparisonof experimentwith analyticalresults at adhesive middle surface
joints was demonstrated as an elastic three-body contact problem using the two-dimensional theory of elasticity (plane stress). • The effects of the ratio of Young's modulus of adherends to adhesive, the ratio of Poisson's ratio, the thickness of the adhesive and the cleavage load distribution on the stress distributions at the interface were clarified by numerical calculation based on the analytical method mentioned above. • In the experiments, the strains produced on the adherends were measured with strain gauges, and the stresses on the adhesive were measured by a photoelastic experiment. Comparing the measured
Sawa, T. et el. Bull JSME 2g 258 (1986) p 4037
2
Lubkin, J.L Trens ASME E 24 2 (1957) p 255
3
Alwar, R.S. and Nagaraja, Y.R. J Adhesion 7 (1979) p 279
4
Kaplevatsky, Y. endRaevsky, V. JAdhesion6(1976) p65
5
Suzuki, Y. Bu//JSME27 231 (1984) p 1836
6
Wah, T . / n t J M e c h S c i 1 8 ( 1 9 7 6 ) p223
7
Saws, T.,Nakano, Y. andTemma, lLJAdhesion24 1 (1987) p l
8
Renton, W.J. and Vinson, J.R. Trans ASME E 44 1 (1977) p 101
9
Bogey, D.B. TransASMEJApplMech 35 (1968) p 4 6 0
10
Kaelble, D.H. Trens Soc Rheology !V(1981) p 127
11
Crocombe, D.A. etal. JAdhesion 12 (1981) p 127
12
Dally, J.W. and Riley, W.F. "Experimental Stress Analysis', (McGraw-Hill, 1978} p 447
Authors
IC Temma is at Kisarazu National College of Technology, Kiyomidai-higashi 2, Kisarazu, Chiba, 292, Japan. T. Sawa is at Yamanashi University, Japan. A. Iwata is with Toshiba Machine Co. Ltd., Japan.
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 9 9 0
293
This paper deals with the two-dimensional stress analysis of adhesive butt joints subjected to cleavage loads. The purpose of the paper is to contribute to the establishment of fracture criteria of adhesive joints. Similar adherends and an adhesive bond are replaced with finite strips in the analysis. Stress distributions in adhesive joints are analysed using the two-dimensional theory of elasticity. The effects of the ratio of Young's modulus of adherends to that of an adhesive and the thickness of the adhesive bonds on the stress distributions are shown by numerical calculations. For verification, strains produced on adherends are experimentally measured and a photoelastic experiment is carried out. The analytical results are in fairly good agreement with experiment.
Key words: elasticity; stress analysis; adhesion; butt joint; cleavage loads; photoelasticity
Adhesive joints are being used in mechanical structures with improved adhesive properties. At present, however, data for designing these joints are markedly fewer than for conventional joints such as bolted and riveted joints. It is very difficult to estimate stress distributions on the joints and their durability. So it would be useful to have data available for design and to establish an optimal design method. In order to establish a design method for adhesive joints, it is necessary to know the stress distributions on the joints more precisely. Up to now, many investigations have been carded out on butt adhesive joints subjected to tensile stress ~-s, bending moments 6"7 and shear l o a d s8. Furthermore, some investigations 9--22 have been carded out on butt adhesive joints subjected to peel and cleavage loads. However, it seems that few analyses have been made strictly on joints subjected to cleavage and peel loads. This paper deals with stress analysis of adhesive butt joints, in which two similar finite strips are joined by an adhesive, subjected to cleavage loads. In the analysis, replacing adherends and an adhesive with finite strips, the stress distributions on the joints are analysed strictly as an elastic three-body contact problem using the two-dimensional theory of elasticity. Based on the analysis, the effects of Young's modulus,
Poisson's ratio, the thickness of adhesive and the distribution of cleavage loads on the stress distributions at the interface between an adhesive and adherends, are clarified by numerical calculations. For verification, strains produced on adherends are measured experimentally and a photoelastic experiment is also carried out. The numerical result is compared with experiment and the stress distributions are discussed.
Theoretical analysis Fig. 1 shows an adhesive butt joint, in which two similar finite strips are joined, subjected to a cleavage load. To analyse the stress distributions on the adhesive joint, two adherends are replaced with finite strips [I] and the adhesive with a finite strip [II] as shown in Fig. 2, taking into consideration the symmetry of the x-axis in the adhesive. The length of finite strip [I] is designated by 21, the height by 2h I, Young's modulus by El and Poisson's ratio by yr. Those of finite strip [II] are designated by 2/, 2h 2, E2 and v 2, respectively. It is assumed that the cleavage load acts on the upper and lower ends of the joint with respect to they~ axis within the range el < x < e 2.
0143-7496/90/O40285-09 © 1990 Bu~erwo~h-Heinemann Ltd INTJ.ADHESION AND ADHESIVES VOL. 10 NO. 4 OCTOBER 1990
285
Yl ,Yl
!
X = "1-1
F(x)
l
(1.1)(1.2)
(~x = Vxy = 0
Yl = ht h2
[I]
Adherend E1 ,G1, Vl
oo ~
,01 [ttl:
-02
x
~o
as COS
(~y m ao +
x
x
bs sin
+
s=l
x
I
(1.4)
Vxy=O
03
E1 ,G1, vl
Adhesive
[11
E2,G2,v2
where
213
l ['e2 ao = Zt l , F ( x ) d x
F(x]
Fig, 1
/
/
41-
(1.3)
s==l
as = 7
F ( x ) cos
x dx
bs=
F ( x ) sin
x dx,
Adhesive butt joint subjected to a cleavage load
F{x)
(s= 1,2,3 . . . . )
For finite strip [II1 (adhesive) e2
1-
II
X = +_l
II = 0
(2.1)(2.2)
t7x = rxy
For the interface I
el mO
(%)y,
1
X
E1 ,G1 ,Vl
/
/
Finite strip [I] (adherend)
a
Ii
(G),,
h2
(3.1)
x xyJy~=h 2
(3.2)
h, = (%)y~
o -
=
=-h~
=
o., I __ro.,,i Ox /y,=_h, \ OX /y~=h~ (ov, I O X / Y l = - hi
(3.3) (3.4)
k OX ]Y2 = h2
In order to analyse finite strips [I] and [II1 under boundary conditions (1)-(3), Airy's stress function X is used. The stresses and the displacements (plane stress) are given as follows: O2Z
O2Z
trX= ~y2
rry= ~ x 2
O2X rxy =
OxOy
(4)
h2 2Gu =
Y
,t- o 2 -
2Gv =
1
OZ
O0
~ - Ox 1 + v Oy OZ Oy
1
00
l+v
Ox
+ - -
(5)
where the stress function X and the function 0 must be satisfied by the following equations, respectively. ~2 I
b
I
Ox ay
v2z
720 = 0 v2 = ~ x 2 +
Finite strip [11] {adhesive)
Fig. 2 Model for analysis and dimensions: (a) finite strip [I] (adherend); {b) finite strip [11] (adhesive)
Developing the load distribution F ( x ) into a Fourier series, the boundary conditions shown in Fig. 2 can be expressed as follows, where the displacement in the x direction is denoted by u, the displacement in the y direction by v and superscripts I and II correspond to finite strips [I] and [II], respectively. • For finite strip [I] (adherend)
286
yZv2x = 0
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
where G is modulus of elasticity in shear, and v is Poisson's ratio. Airy's stress function XI, which is selected from solutions for the method of separation of variables of biharmonic functions, is represented as Z~t + X~t + X~t + ZI4t + ,t~b + . . . + Xl4b for finite strip [I] and Xn as x~I' + x~3~' + x~~b + x~~b for finite strip [II] 1"7. Stress functions X~k Z~_t. . . . . Z~b, X~b. . . . and Z~". . . . . Zl 'b. . . . are --It --lb expressed by Equations (6)-(17), where A--It ,, B s ..... A, , -Ib ~.b B s .... and_s ( n , s = 1,2,3 . . . . ) are unknown coefficients determined from the boundary conditions.
x~t
,zlb
t . . I t "-;It --It
= X~ ~ao,/in, Alt
I --I ~ I Bs, 1, h l, an,As, An, fls,x, yt)
~
=
b ,'~Ib w l b
= X l {/in , IJs ,
1, h l, a nI, A s , ,
-It
+
,An ,, Ilo'., ch(a'nO + sh(a' O)
=
n = I IXnCtn
/2 Ila~" / sh(a~"/) + ch(a~" O} sh(a~, x) nil
~k~an a n'X
--
sh(a~, x)l cos(a~yO
s
~
I
=
ch(Ashl) + sh(Ash~)} ch(AsYl) - AsYl sh(Ashl) sh(gsyl)] cos(Asx) (6)
o0
(10)
=I =I hl,an,1, As,, An, l'~s,X, Y l )
"" l b
= ~-,=j z.xnanrIAn 1'2 [[a~' l sh(a~' /) + ch(a~' l)} sh(atn'x)
= It
~ (a'.' x) -
-'$''$
b,~Ib =lb = Z 2 ['*in , B s ,I,
v - n , o s~. u l, hl,an,As, I' = I fls,X, =I x~t = Xt2 t~u An, yt)
= -An ~ _ [,.,_ [{a~'lch(a~'l) An a n
IIA'~hlch(X'shD+ sh(A~hl)} ch(AsYl)- XsYl sh(Aj hi) sh(A~Yt )] sin(A~x)
z[b
-----~ l =
~Bs
S = I
[{Ashl
~qsA~
ch(a~/)ch(a.~x)! cos(a~y,)
--Ib
+ E + /~
--I ~ I
A n , ['~s,X, y l )
+ sh(a~'l)}
-- a.I, x ch(a nI' /) ch(al'x)i
,~'/xsh(a'io sh(a~'x)]
sin(a~'yt)
sin(aln'yl)
+ __ ~ ~ + ~ ~ llZshi sh(Ashl) + ch(Xshl)} s = I f~s As
$
==
I
liX.~h, sh(A;hm) + ch(Ash,)}
"-$ "'$
sh(A;yl)- Z;yl ch(A~hi)
sh(Asyl) - Asy~ ch(Ash~) ch(Asyl)] cos(As x) (7)
t
,
#
ch(As Yt )] sm(A,x)
(1 l)
X[ b = Z b t2lb,*-n, -~tlb, s l, hl,a l:As, An, fls, X, Yl) v * n , ~ s~lt , "XI 5 1 ,, X, YlJ" )(,~t..~)(t3i'~It l, ht,an,I' Zs,, t.xn,
¢0
~lb
= _ ->. n=
|
~¢Zn
- - a n Xt,
at,,'x ch(a~'/)sh(atn'x)l
cos (an Yl )
cos(aln'yl)
,:c,
~lb
[Ashl sh(As hi) ch(AsYl)
-Z
~t
~t,'~s
f~' h
s= I lqsAs
1 sh(Ashl)ch(A~Yl)
- Z, yt ch(As hi) sh(AsYt)l sin(Z, x)
s=l
Xsyl oh(A; hi) sh(Ajyl)l cos(X;x) -
z[' =)(,4(An,lJs,l, , . ' T I , ~ I , . hl, a l
(8)
~o
As,. A=n1, %l'~s,X, Y l )
= -
A.
"~I' I2
Ia~ / sh(a I / )
~lb
nZ=, =!A '..i2 n [a~/ch(a'. 0 sh(a', x) A n (A n
ch(aln x) - ~ x sh(a~ o ch(a~' x)l sin(a~ Yl)
I lAtnan
I I anx ch(an/) sh(a~ x)] sin(alyl) -
ao
'~'lt
E S ~
B, |
(12)
n,Bs,/,hl,an,As, An,
=Z4~
~It n=
sh(a~'/) ch(aI.'x)l I,
co
=-
0 sh(a'; x)
~1 ~I,2
/...a
~, A~' [a~ish(a~'l)ch(aln'x) n= , a' a'."-
A°
=lb ~I
2
IA, h, ch(A, hI)Sh(Asy,)
s= I Nsgs
lA'~hlch(A;h,)sh(A',yl)
AsYt sh (As h t ) ch (AsYt )] sin(Asx) -
''S''S
ZsYt sh(Ashi) ch(A;y,)l cos(A;x)
(13)
-
(9)
,~[It
,,~ [All/ 711t 5 l l r =
tao
,an
, Os
,
--II - - I I !, hA, an II, As, An, Ns,X, YA)( 1 4 )
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
287
~,~It
t ,'~IIt a I I t • . II,., "zll ,~II = Z 3 t a n ,/~s , l , n2, a n , A s , / X n , S l s , X ,
, y2)
(15)
~,~Ib =)~Itan b ,-rltb ,~s ~llb ,l, h2, an,Xs, 1I , An, --ll -~tsl X, y2)
(16)
Xs, An, f l $ , x , Y x )
(17)
v-n
,-s
, -, n2, a n ,
where
o0 ~ ~Ib
~lb
An +
00 Ib
Ib
An
Bs Sns = 0 ,
+
s=l
E
~ lb ,~, Ib' B s ~ns
s=l
(18.7), (18.8)
~llt pllt = 0 ~ s --n$
z~llt..[_ E n
=0,
(18.9)
$=1
nrr
i
an =an(hi)=--,
I,
hi
II
_1[,
an = an (h2),
(2n-
,
a n = an(hi)=
fl n
,
= an (h2),
1)zr
o0
2h~
~lit
0O
E
An +
"~ lit A'IIt
=0,
,~tlb _ E ~Hbpub = 0, s=l
s=l
As = s- -n
(18.10), (18.1 I)
1" 00
~.~ _ ( 2 s - l)n 21
.~ub + E
-i = /~n
an (a'. t)
--~"
=I
An
~I
(at.'/),
An
=
=
An
(18.12)
~ub qllb = 0
s=l
sh(at" t) ch(d. t) + at. l, "~In = A n --I, A
o0
--il =An(at"I/), An
o0
E
E
n= [
~II
(18.13)
:"
n= 1
txn = An (at.~' 1), ~
= f~$ (~,s h I ) = sh (As h I ) ch (~,$ h 1) + X$ h I,
e',
fl s = f~s(Xshl), "~'|
t
~'~$I
,
f~s = Os (As h 1) = sh (Xs h I ) ch 0.s h I ) - A,$h I, =
,
--II =
f~s()~shl),~s
I2s()%k2),
~ II =
Os
=
,
l'ls 0.$ h2)
and sinh is abbreviated as 'sh' and cosh as 'ch'. Substituting Equations (6)-(17) into Equations (4) and (5), the stresses and displacements for finite strips [I! and [IIi are obtained. Restricting the stresses and displacements obtained to the boundary conditions, Equations (18) and (19) in terms of the unknown coefficients are obtained. That is, from the boundary condition Equation (1.1), Equations (18.1), (18.2), (18.5) and (18.6) are obtained; from Equation (1.2), Equations (18.3), (18.4), (18.7) and (18.8) are obtained, and Equations (18.13) and (18.14) are obtained from Equation (1.3). Equations (18.15) and (18.16) are obtained from Equation (1.4), from Equation (2.1), Equations (18.9) and (18.11) are obtained, and Equations (18.10) and (18.12) are obtained from Equation (2.2). From Equation (3.1), Equations (18.17) and (18.18) are obtained, from Equation (3.2), Equations (18.19) and (18.20) are obtained, from Equation (3.3), Equations (18.21) and (18.22) are obtained, and Equations (18.23) and (18.24) are obtained from Equation (3.4). oo
-" sE= l -B$ " "Pn$ = 0 ,
An-
E -s-n$
s= l
~
~ It K,/t
(18.1), (18.2)
oo
-Ib
An -
E
o0
o0
+ B$ +
n~]
+ B$
= 0
(18.15)
n=l
E -Abi RIbn--ns + ~Ib + E .~lb_n, .,.
n==l
(18.16)
n=~l
oo
~
E
AItt"jlt
ot~
+ B--1' s + E ~Itf) lt'
- B s=It
n=l
n=l
-
E
"~llt()llt ,-n ~ns
n=l
- -sfiut = 0
(18.17)
O0
o0
"'n ~ n s - B s n=l
+
An ~ , s + B s n=l
0o
-- E
--n~'llbt')llb ~ns + ~Ib = 0
(18.18)
E --nAltRIt--ns+ nlst- E -~ltRlt' ~I' + E ~IItRIl, n - n s - Bs -n --ns n=l
n=l
$=1
(18.19)
aO
oO
n=l
n=l
-~Ib p l b
~s -ns = O,
(18.4), (18.5)
oo
co
= Ib p l b ,
Bs -n~ = 0
(18.6)
+ E n=l
288
(18.14)
+ u.~nt = 0 S
s=l
An +
II,
a,,s
(18.3)
oO
E
b,.
Y "AltRIt - " Y
$=1
= Ib
oo E
n=l
= 0
Zlt ~ , , ~ It ,~It, A n + - s on$ = O,
_ 8$
n=l
O0
~It
t"b . . $ s=l
m=l
oo
An +
An +
-
n=l
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
--AllbRllbn --.s + --s~Itb= 0
(18.20)
oo
oo
~It lilt -- ~ - - I t v l t h l l Bm -m~ms "= R ~/'13 n=l m=l Z
~
+
Experimental method
/~ltIflV n
vns
Strain measured by strain gauges
n=l
co
Z
--
=It Fit, II
Bm-m
~It w i t '
bms-Bs
~'lt [grit,
..s
+Bs
"'s
m=l "~llt ~n$ Hllt + Z - ~ l --rn l t vllt bins I] --n n=l m=l
_ Z
.
II
+ ~ I t wilt
Z
--
m
Z
Uns +
n=l
(18.21)
An vns
v m ares - -
"
m=l
n=l
Photoelastic experiment
oo
Z
+
=lb glb, II
Bm -m
~lb
ares + Bs
lb
Ws
~lb iff:lb,
- Bs
"" s
m=l oo
O0
+ Z
~llb //lib "'t/ v #iS
n=l
~rn Vm a m s - - w s
=0
m=l
(18.22) OO
CO
~ltFlt =ltFItt Z s --s - B s --s +
~[tlglt "n--ns+
n=l oo
-- Z
oo
i~ltl'lt' ares II + Bs --It --ra--m
Results of analysis
oo
~l|t--llt II = 0 tim Jm ares
",~lltldllt n "'ns 1. Z
O0
Fib l -- b~lb Fib, _
=S
~
$
=S
oo
In the present analysis, the stress singulari b' at the edges of the interfaces is not taken into consideration, so that numerical calculations were carded out setting 20
O0
/I
n=l + Z
(18.23)
m=l
m=l
$
Results and discussion
FsIt
mffil
oo
•
"~ltrlt l[ timJmams
In order to examine the stresses on the adhesives, a photoelastic experiment was carded out. Fig. 4 shows the dimensions of the specimen. The adherends are made of aluminium (of Young's modulus 68.7 GPa and Poisson's ratio 0.3) and the adhesive is epoxy resin (Young's modulus 3.5 GPa, and Poisson's ratio 0.376). The adhesive is interposed between two adherends and the interfaces are cemented with adhesive made from the epoxy resin. Isochromatic and isoclinic fringe patterns in the specimen are obtained. The stresses on the adhesion are obtained from the fringe pattern by the shear-difference method 12.
m=l
~'t/./l' "-n "'ns + Z
nml
+ Z
Z
Fig. 3 shows the dimensions of the joint used in the experiment. The adherends are made of steel for structural use ($45C, JIS) and the adhesive is made of epoxy resin (Scotch-Weld 1838, Sumitomo 3M Co. Ltd., Japan). Strain gauges (KFC-CI-I 1, Kyowa Dengyo Co. Ltd., Japan) are attached 2 mm from the interface of the adherends. After joining the two adherends, a cleavage load is applied to the joint using a materials testing machine. The magnitude of the load is measured by a load cell. The outputs from the strain gauges and the load cell are recorded with an X - Y recorder through dynamic amplifiers.
~/IS
+ Z
~
rr7 ~ m
~Plv/$
m=l
¢11
oo
A~lbI.lib' n --ns + Z
n=l
~w ml b /~1?1 I b ' b iIIn + ~llbFllb ~$ $ =
m=l
oo
Go
-
:o n=l
(18.24)
m=l
A~t = ao,
.4[o]' = ,4lot
50
30
( n , s = 1,2,3 . . . . ) (19)
where the constants PUs, Slts . . . . . P ~ , S ~ . . . . are shown in References (1) and (7). Superscripts It and IIt correspond to superscripts I and II in Reference (1) and Ib and IIb correspond to I and II in Reference (7). By solving the infinite set of simultaneous equations Q18), the unknown coefficients ,41nt,~ t . . . . . ,~IIb and B~ rb (n, s = 1, 2, 3. . . . ) are determined. In addition, the unknown coefficients A~ t and .410It are determined from Equation (19). Using these coefficients, the stresses and the displacements are obtained.
,sI
Strain gauge
[]
[]
-
/
[]
-
L.i
/NAA 50 Fig. 3
Dimensions
f
t=8,12
of specimen(in ram)
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 9 9 0
289
W
E,
~
1.0
t "
E2
E 1/E 2 = 3
m
10 100
, ~
i U]
E 0.5 x !
k 30
0.0
I - I .0
15
-0.5
I
I
0.0
0.5
1.0
I 0.0
I 0.5
1.0
3.0
2.0 15
Epoxy-r
I
El
I
1.0 × 15
Aluminit 0.0
- I .0
q~11
I -0.5
I0.0 B
8.0 -
El/E 2 = 3
6.0 -
100
10
E
Fig. 4
-
2.0
-
/,/
Dimensions of joint in photoelastic experiment (in ram)
the number of terms N as 100. Convergence was verified when N was 100. Fig. 5 shows the effects of the ratio EI/E 2 o f Young's moduli of adherends to adhesive on the stress distributions at the interface (Y2 = h2), where El/E2 was set as 3, 10 and 100. It is assumed that a cleavage load acts uniformly within the region 0.8 < x/l < 1.0. In this figure, trym represents the mean normal stress. The ordinate represents the ratio try/trym of the normal stress to the mean normal stress, and the ratio rxy/tyym of the shear stress to the mean normal stress. The abscissa represents the ratio x/l of the distance x from the centre o f the joints to the half length i of adherends and adhesive. It is seen that the distributions o f cry, trx and r ~ tend to increase near the edge, x/l = 1.0, and the stress singularity increases with an increase in EI/E 2. The larger EI/E 2 is, the more uniform the distribution o f stress try becomes. Fig. 6 shows the effect of Poisson's ratio vl/v2 o f
290
4.0
//,
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 9 9 0
_..:')
0.0 -I .0
-0.5
I
I
0.0
0.5
I .0
x/I
Fig, 5 Effect of ratio of Young's modulus on stress distribution at adhesiva-adherend interface. Y2 = h2, h l / I = 0.2, h l / h 2 = 2, V l / V 2 = 1 and F(x) is constant Over the region 0.8 < x / I < 1.0
adherends to that o f adhesive o n the stress distributions at the interface. The distributions o f trx, Cry and rxy increase near the edge x/l = 1.0 with a decrease of the ratio vl/v2. Fig. 7 shows the effect of thickness o f adhesive on the stress distribution at the interface. The ratio hi/h2 o f the height o f adherends, 2hi, to that o f adhesives, 2h2, is varied as 2, 20 and 100. The stress distributions
h2
1.0
1.0 --
E ).. 0.5
_
m
vl/v2
= 1.00
! Ill|
086
i.. :
o.,,
Vl
[
,/
!h I
_
I
0.5
~
E
/
lh I I
W
I I I
h l / h 2 = 100
X t-,
I
20
x i.-,
I
I
I
I I I
0.0
!
0.0
I -1 .0
I
-0.5
I
I
0.0
0.5
I
0.0
0.5
1.0
- I .0
-0.5
1 .0
2.0 1.5
~
-V"2
E
I
i~--"
t
//_.-~. 1.0
1.0 x X 0.5
/// j j,s 0.0 0.0
I ~1.0
-0.5
I
I
0.0
0.5
I •0
1.0
-0.5
I
I
0.0
0.5
1.0
10.0
h l / h 2 = 100
I0.0
d 8.0
Vl/V2
=
•
20
8.0 h
m
2
0.86
E
6.0
6.0
0.75
--
E o
o
4.0
2.0
--
o
4.0
2.0-0.0
--
--
~--
-1.0 0.0
I --1 . 0
-0.5
I
I
0.0
0.5
-2.0 1.0
x/I Fig. 6 Effect of Poisson's ratio of adherends to that of adhesive on stress distribution at adhesive-adherend interface. Y2 = h2, h i / I = 0.2, h l / h 2 = 2, E1/E 2 = 3 and F(x) is constant over the region 0.8 < x / I < 1.0
of ax, Gv and rxy increase near x/l = 1.0 with an increase in the ratio hl/h 2. Fig. 8 shows the effect of the cleavage load distribution F(x) on the stress distribution at the interface. In this case the thickness of adhesives is held constant and the value ofhl/h 2 is set at 5. Numerical calculations were carried out in the cases where the uniform load F(x) acts over the regions 0
-1.0
I
I
I
-0.5
0.0
0.5
1.0
x/I Fig. 7 Effect of thickness of adhesive on stress distribution at adhesiveadherend interface, V2 = h2, h i l l = 0.2, E1/E 2 = 3, v l / v 2 = I and F(x) is constant over the region 0.8 <~x/I < 1.0
can be seen clearly on the stress distribution at the interface. It is found that the stress singularity near the points x/l = - 1.0 and x/l = 1.0 increases in the case where the load distribution F(x) acts over the region 0.8
Fig. 9 shows a comparison of the analytical result with the experimental one with respect to the strains at
INT.J.ADHESION AND ADHESIVES OCTOBER 1 9 9 0
291
1.5 40 O
1.0
-
x
30
.,
Exp Num O
I
E o
x
0.5
X
J
J
0.0
2o -
c10
O 0 W = 0.7 kN
I
-1.0
-0.5
0.0
0.5
-25
I
I
I
I
I
-20
-10
0
10
20
25
Distance, x(mm)
3.0
Fig. 9
Comparison of experiment with analytical results
2.0 E
>.
x o
1.0
0.0
-1.0
-0.5
0.0
5.6"0--0 4.0
~
0.5
]
0.0
--
el/I
=
_
e 1/I
= O. 8
1.0
I~ -i
2.o1 .0
Fig. 10 --
.
~
~
.
Fringe pattern (light field) obtained from photoelastic experiment
~
0.0, -1 .0 ~ ~ ' ~ ' ~ ' ~
-2.0 -1.0
I
I
I
-0.5
0.0
0.5
1.0
x// Fig. 8
Effect of distribution of load on stress distribution at a d h e s i v e interface. )12 = h2, hl/h2 = 5, h 2 / / = 0.1, e 2 / / = 1, E~/E 2 = 65.6 and v l / v 2 = 0.81 adherend
positions 2 m m away from the interface. The results are in the case where the adhesive joint is subjected to a cleavage load of 0.7 kN. In the numerical calculations, Young's m o d u l u s of the s p e c i m e n (adherend) is put at 206 GPa and Poisson's ratio at 0.3, and those of the bond (adhesive) as 3.14 GPa and 0.37, respectively. The thickness 2h2 of the adhesive is measured as 0.055 m m . The distribution F(x) is a s s u m e d to be uniform in the range el = 5 m m and e2 = 25 m m on Yl -- hi o f the specimen. The
292
INT.J.ADHESION AND ADHESIVES OCTOBER 1990
ordinate is the strain of the specimen in the Yl direction, and the abscissa is the distance x from the centre of the specimen. The solid line represents numerical values and O experimental values. They are in fairly good agreement. Fig. I0 shows an isochromatic fringe pattern of the adhesive. As a special case, when hi~h2 is set at unity, a photoelastic experiment was carried out. Fig. 11 shows the comparison between the analytical results and experiment with respect to the normal stress Gv at the middle surface of adhesive. They are also fairly well matched. Conclusions
This paper deals with two-dimensional stress analysis of adhesive butt joints, in which two similar finite strips are joined by an adhesive, subjected to cleavage loads. The following results are obtained. • Replacing adherends and adhesive with finite strips. a method for analysis of the stress distributions on
3.0
/.~
values with the analytical results, it was seen that they were in fairly good agreement.
/_ Num 2.0-
------
Exp
//
Acknowledgement
The authors would like to thank Professor K. Ikegami of Tokyo Institute of Technology and Dr Y. Suzuki of Nippon Sharyo Seizo Co Ltd. for their advice on this work.
D
I .0
/ // /
-
0.0
References 1
W=0.qRN s -
.0
f
I -0.5
I
I
0.0
0.5
1.0
x/I
Fig. 1 1 Comparisonof experimentwith analyticalresults at adhesive middle surface
joints was demonstrated as an elastic three-body contact problem using the two-dimensional theory of elasticity (plane stress). • The effects of the ratio of Young's modulus of adherends to adhesive, the ratio of Poisson's ratio, the thickness of the adhesive and the cleavage load distribution on the stress distributions at the interface were clarified by numerical calculation based on the analytical method mentioned above. • In the experiments, the strains produced on the adherends were measured with strain gauges, and the stresses on the adhesive were measured by a photoelastic experiment. Comparing the measured
Sawa, T. et el. Bull JSME 2g 258 (1986) p 4037
2
Lubkin, J.L Trens ASME E 24 2 (1957) p 255
3
Alwar, R.S. and Nagaraja, Y.R. J Adhesion 7 (1979) p 279
4
Kaplevatsky, Y. endRaevsky, V. JAdhesion6(1976) p65
5
Suzuki, Y. Bu//JSME27 231 (1984) p 1836
6
Wah, T . / n t J M e c h S c i 1 8 ( 1 9 7 6 ) p223
7
Saws, T.,Nakano, Y. andTemma, lLJAdhesion24 1 (1987) p l
8
Renton, W.J. and Vinson, J.R. Trans ASME E 44 1 (1977) p 101
9
Bogey, D.B. TransASMEJApplMech 35 (1968) p 4 6 0
10
Kaelble, D.H. Trens Soc Rheology !V(1981) p 127
11
Crocombe, D.A. etal. JAdhesion 12 (1981) p 127
12
Dally, J.W. and Riley, W.F. "Experimental Stress Analysis', (McGraw-Hill, 1978} p 447
Authors
IC Temma is at Kisarazu National College of Technology, Kiyomidai-higashi 2, Kisarazu, Chiba, 292, Japan. T. Sawa is at Yamanashi University, Japan. A. Iwata is with Toshiba Machine Co. Ltd., Japan.
INT.J.ADHESION A N D ADHESIVES OCTOBER 1 9 9 0
293