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Stress analysis in elastic joint structures
Pergamon
Int. J. Mech. Sci. Vol. 39, No. 7, pp. 873 883, 1997 c~;, 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020 7403/97 $ 1 7 . 0 0 + 0 . 0 0
PII:
STRESS ANALYSIS
S0020-7403(96)00086-0
IN ELASTIC
JOINT
STRUCTURES
A. L. K A L A M K A R O V Department of Mechanical Engineering, Technical University of Nova Scotia, P.O. Box 1000, Halifax, Nova Scotia, Canada B3J 2X4
(Received 26 February 1994; and in revisedform 12 July 1996) Ahstract--A new inverse problem approach is introduced to analyse the stresses in the overlap elastic joint structures consisting of two adherents and an adhesive interlayer. Displacements in the elements of a joint are expressed in terms of some initially unknown contact stresses by solving the boundary-value problem for each element considered separately. The contact stresses are calculated from the matching continuity conditions for the displacements. Contact tearing and shear stresses, three-dimensional stress distribution in the bulk of adhesive interlayer, and the maximum stress values are calculated for different ratios of thicknesses and material properties of the adherents and the adhesive interlayer. The present study represents a new alternative modelling approach to stress analysis in the elastic joint structures. @ 1997 Elsevier Science Ltd. All rights reserved.
Keywords: adhesive lap joints, contact stresses, maximum tearing and shear stresses, inverse problem approach.
NOTATION shear moduli and Poisson's ratios of the materials of upper, lower adherents and of the adhesive interlayer correspondingly f traction force per unit width w of the joint Ti, Mi Qi(i = 1,2) stress and moment resultants, shearing forces tt = 2 h t , t 2 =2h2, to = 2h0 thicknesses of upper, lower adherents and of the adhesive interlayer correspondingly ts = 2h~ thickness of adherents of the symmetric lap joint ax, ay and zxy normal and shear stresses a+(X), a-(X), Z+(X), r_(x) contact stresses u(x, y), v(x, y) displacements q'(x, y) Airy stress function derivative of kth order with respect to x G t , ~,'t, G2, v2 and Go,vo
1. I N T R O D U C T I O N
Adhesive joints would provide attractive solutions to many mechanical engineering problems if their performances and reliability were well known. Owing to the large discrepancy which is usually present between the elastic properties of the adherents and the adhesive stresses are not always precisely evaluated. This represents a drawback for the safe calculation of the limit load which an adhesive joint can sustain. Different methods have been employed for theoretical strength analysis of the joint structures. Most of the basic work in this field, relating to a single lap joint loaded in tension, has been reviewed by Benson [1] and Hart-Smith [-2]. A simple approximate approach, due to Volkersen I-3], considers only shear deformation of the adhesive and elongation of the adherents. The bending effects in the single lap joints were considered first by Goland and Reissner [4]. The asymptotic solution of the problem in the case of a double lap joint was obtained by Gilibert and Rigolot [5]. Rigolot I-6] considered the end-effects and material non-linearity. The fracture of a lap joint was analysed in Ref. [7]. In the present paper, a new inverse problem approach to stress analysis in elastic lap joint structures is introduced. The method is based on the application of some special equations matching contact displacements with normal and shear stresses on the surfaces of a thin strip. These expressions follow from the analysis of the elastic plane strain state of a thin strip. In the suggested method, the elastic equilibrium is considered separately for each element of a joint. Displacements in these elements are expressed in terms of the initially unknown stresses on the contact surfaces by solving the boundary-value problem for each element considered separately. Contact stresses are 873
874
A.L. Kalamkarov
then calculated from the matching continuity conditions for the displacements on the contact surfaces of the elements of a lap joint. Stresses in the bond interlayer are expressed in terms of the contact stresses and are also calculated. The developed inverse problem approach represents a new alternative modelling approach to stress analysis in the elastic joint structures. It provides the remarkable simplification of the initial complex problem and makes it possible to calculate the contact stresses and the stresses in the bulk of the lap joint structure under study. The inverse problem approach was applied earlier by the author in the analyses of the composite reinforced structural elements, see Ref. [9]. 2. E Q U I L I B R I U M E Q U A T I O N S FOR A T H I N LAYER
To proceed to the stress analysis in an elastic joint structure, we first consider a thin elastic layer (strip) of finite length l = 2a and thickness t = 2 h, so that ix b ~< a, IYl ~< h, (Fig. 1). We assume the following notations for the stress and moment resultants and shearing forces:
r(x)=f]hfixdy, M(x)=f yfi~dy, Q(x)=F.
vxy dy.
(1)
Equilibrium conditions for a thin strip as a whole on account of the forces and moments acting on the edges x = _+ a can be formulated as
f~ [fi2(x)- fil(X)] dx + Q2 ~
=0,
Q1
a
[Z'2(X --'rl(X)] dx + T 2 -- T 1
f]
(2)
~ O,
a
--
[T2(x ) -b TI(X)] dx + a(Q2 4- Q1) + M2 - M t
X[fi2(X ) --fix(X)] dx - - h a
= 0.
a
The stress and moment resultants over the arbitrary cross-section x can be calculated from the expressions
T(x) = i X [ % ( { ) - r 1 ( { ) ] d{ + T2, Q(x)= ( x [o"2(3)-fi1({)] d{ + 02, J- a ~ a ~x
M(x)
| o~ a
Ox
(3)
[rz(~) + ~1(¢)] d~ + xQ2 + M2 + aQ2.
(x - ~) [or2({) - fil({)] d{ - h | ,J-
i1
Analysis of the elastic plane strain problem on account of the small thickness of a strip (h ~ a), provides the following equations matching displacements u(x, + h) and v(x, +_h) and stresses on the surfaces y = + h (see Fig. 1) [8]: 4Gh2ula)(x, _ h) = - vh2(fi~2) + fi~22))- (1 - v)h(z] n - z(21)) ± 3(1 - v)(¢2 - rh)
_+ (2 - 3v)h2(rrll2, -- 0"12 2,) ~ 3(1 -- v)h(z]~) + r~~))
4~h~'~(x, +h)= 3(1 - v)(fil - f i j - 4 ( 1 -u)h~(~ ~ -fi~))+3h(1 -~)(~T-%~).
(4)
(5)
Here and in the sequel all the superscripts in the brackets indicate the orders of the derivatives with respect to x. Equations (4) and (5) can be derived from the equilibrium equations for a thin strip by means of the following procedure, see Ref. [8] for the details. First, the longitudinal and transversal components of the displacement vector u(x, y) and v(x, y) are represented in the form of Fourier integrals in variable x. The equilibrium equations yield a system of two ordinary differential equations of the second order for their Fourier transforms. This system can be solved easily and the arbitrary functions entering into the solution can be expressed in terms of Fourier transforms of the normal and shear surface stresses a~ (x), 0"2(X), T I(X ) and %(x) from the conditions on the boundaries of the strip, y = _+ h. After that, the obtained expressions for the Fourier transforms of the displacements are expanded in the power series of a small parameter, c~ = h/a. Finally, we keep only terms of ~X2 order for the longitudinal component and the terms of c~3 order for the transversal component. The application of the inverse Fourier transform to these expansions leads to the above Eqns (4) and (5).
Stress analysis in elastic joint s t r u c t u r e s
875
(~'1
Fig. 1. T h i n layer a n d the c o n t a c t stresses on the u p p e r a n d lower surfaces.
3. A N A L Y S I S
OF THE
LAP
JOINT
PROBLEM
We can proceed now to the stress analysis in an elastic lap joint. F o r the sake of simplicity, we will consider a single lap joint consisting of three elastic layers (Fig. 2): layer 1: al ~< x ~< a, [y[ ~< hi (the upper adherent), layer 0: - a ~< x ~< a, [y[ ~< ho (the adhesive interlayer), and layer 2: - a ~< x ~< ax, [y] ~< h 2 (the lower adherent). The contact stresses a+(x), a_(x), z+(x) and z (x) shown in Fig. 3 can be calculated from the matching continuity conditions for displacements in the interval Ix[ < a. Displacements u(x, +_h) and v(x, +_ h) can be expressed in terms of the initially u n k n o w n contact stresses by solving the boundary-value problem for each of three layers considered separately. The following equilibrium conditions for layers 1 and 2 (Fig. 3) must be satisfied:
f" a r+ ix) dx = f
f ' S +(x) dx = 0 f f a xa+ (x) dx = M
f" z _ ( x ) d x = f
f] . ~ (x)dx=O
--J~Z 1
f] .
(6)
(7)
Equilibrium conditions for interlayer 0 (see Fig. 3),
f~ f
a
[a_ (x) - a+ (x)] dx = 0,
f " [z_ (x) - r+ (x)] dx = 0 a
(8)
" x[o" ( x ) - a + i x ) ] d x + h o f ~ a
a
[z_(x)+z+(x)]dx=0
are satisfied automatically on account of the conditions for layers 1 and 2, i.e. Eqns (6) and (7), and the expression for the m o m e n t M (see Fig. 3),
M = (1/2)f(h2 + hi + 2ho).
(9)
Equations (4) and (5) on the lower surface of the layer 1 (in the interval [ x] < a) yield U(?)(X, - - h i ) - -
1 ~°niA~l" ~ [2(1 - 2v1)hl2 o% (2)(x) - 3(1 - Y1)o'+ i X) - - 4(1 -- Vl)h 1~ ( 1 ) ( X ) ] ,
(lO) [ ( 4 ) i x ' __ h i ) _ 3 ( 1 - - Yl)
4G,h 3 [ - ° ' + i x ) - h l ' C ~ ) ( x ) ] +
( l - Y1) o.(+2)ix )
b~-i
"
On the upper and lower surfaces of interlayer 0, Eqns (4) and (5) yield
1 u~oa)(x, + ho) - 4Goh 2 [2il - 2vo)h 2 o.(2~(x) - 3(1 - Vo)O+(x) - 2(1 - vo)hgo¢z_'(x) + 3(1 -
Vo)a (x) + 4(1 - v o ) h o r + (a)(x) + 2(1 -- vo)hoz_(1)(x)], (11)
V(o4)(x, + ho) - 3(1 -Vo) 4Goh 3 [a+
(x) - e_ (x) - hor~)(x) - hor~)(x)]
(1 - Vo)
876
A.L. Kalamkarov
2hl
2ho
.1]
-.~-
al
Fig. 2. Single lap joint.
1M
1
o÷(x)__
¢_~ ~_~a
2~+
"~(x)
-~ "r.(x)
o-oo/i~ ~ ~~ / -a
Fig. 3. Contact stresses on the contact surfaces of the lap joint elements.
1 U~o3)(x, - h o ) - 4Goh2 E - 2 ( 1 - vo)h2o~ ' (x) - 3(1 - Vo)O+ (x) + 2(1 -2vo)h2o~)(x)
- 3(1 - Vo)a-(x) - 2 ( 1 - vo)hoz{1)(x) - 4 ( 1 - vo)hoz~)(x)], (12)
V(o4)(x, - h o )
-
3(1 - Vo)
4Goh 3 [ o + ( x ) - o _ ( x ) - h o z + , ,
{U,x ~ _ hoz(l_)(x)]
(1 - Vo) [-o~)(x) G~o -°~)(x)]'
Correspondingly, Eqns (4) and (5) yield on the upper surface of the layer 2 (in the interval Ix I < a) 1
{1)t X
ut23'(x, + h2) - 4G2h22 [-2(1 -2v2)h22o{2_)(x) - 3(1 - v 2 ) & ( x ) + 4(1 - 2 v 2 ) h 2 z _ ~
)], (13)
3(1 -- v2) r v{2~)(x, + h2) -
TThT
- hzr~'(x)]
(1 - v2) o~2), , -GTh~ _ tx),
where Gl, vl, G 2 , Y2 and Go, Vo are the elastic characteristics of layers 1, 2 and the adhesive interlayer materials correspondingly. Contact stresses occurring on the contact surfaces of the layers can be calculated now from the following matching continuity conditions for displacements: U(3)~ 1 IX,
-- hi)
=
/,/(3}(X, -I- ho),
v]4)(x, - hD = v(o4}(x, + ho) (14)
U{o3}(x,-ho) = ~?}(~, + h~t, ~{0~'(~,-ho) = ¢~*'(x, + h~). Conditions (14) together with Eqns (10)-(13) yield the following four differential equations: (A 1 - -
Ao) o(_2)(x)
--(B 1 --Bo)o + (x) -[- (ho/2)Coo~)(x) - B o a - (x) - ( C , + Co)r~)(x)
-{1/2)Cor~)(x) = 0, -- (ho/2) Coo+(2}itx) + Boo+ (x) + (Ao - (Co + C2)r~}(x) = 0,
(15) - A2)o~-2)(x) -
- (1)/ (Bo - Be)o- (x) - (Co/2)z+ tx)
(16)
877
Stress analysisin elasticjoint structures (Co + C1)a~+Z'(x) - (Bo/ho + B 1 / h O a + (x)
-
Coa{2_)(x) + ( B o / h o ) a - (x)
- (B, - Bo)v~)(x) - - B o ~ -(1){, x ) = 0,
- - C o a ~ ) ( x ) + B o / h o a + ( x ) + (Co - - B o v ~ ) (x) - ( B o
-
-
+
B2)'f~)(x)
(17)
C2)o(-2}(x) - ( B o / h o
+ B2/h2)a-(x) (18)
= O,
where the following notations are assumed: 3(1 - 2 v l )
1 -- 2vi Ai
-
-
-
,
2Gi
Bi
-
-
-
4Gih ~
1 - vi '
Ci
-
Gihi '
( i = 0 , 1,2).
(19)
Resolving Eqns (15), (16) and(17), (18) about functions zt+l)(x) and z(1)(x), we get following four equations: Vl + 0"+ (X) -~- W1 + O'(_2)(X) Jr- Y ; o'_ (x),
(20)
U1 a+(2)[,x) + V 1- o'+ (x) ~- W 1- O'(_2)(X)-{- Y 1 a - (X),
(21)
v ? ~+(x) + w d ~ ( x ) + Y ; ~_ (x),
(22)
~ ( x ) = u2- ~'+~(x) + V 2 o+ (x) -~ W 2- o(2}(x) -[- Y f a_ (x).
(23)
.C(1)(X) =
Coefficients Us-+, V~-+, W / , Yj-+, ( j = 1, 2) can be easily represented in the form of algebraic functions of the parameters Ai, Bi, C~, (i = 0, 1, 2) defined by the Eqn (19). Combination of Eqns (20), (22) and (21), (23) yields the following system of two linear differential equations for the contact stresses a+ (x) and a (x):
(U?-
" ~ + "~ (2) ~2 +
-
Y;)~_
= o,
(U1-
"~-")a+~2~ + (V1- - V 2 ) a + + ( W F - W2)a~2_ ) + (Y1- u2
Y2)a-
= O.
+ (v~ + -
v;>+
+ (w?
-
w;)~
~+ ( r ?
(24) The general solution of the linear system of Eqns (24) includes four arbitrary constants and the subsequent integration of the expressions (20) and (21) [or (22) and (23) with the same result] will add two more arbitrary constants. To complete the solution of the problem, we must determine these six constants from six equilibrium conditions [Eqns (6) and (7)]. 4. SYMMETRIC LAP JOINT
Let us apply the above general approach to the case of the symmetric balanced single lap joint, shown in Fig. 4 (cf. Fig. 2). We assume that the layers 1 and 2 have the same thicknesses, i.e. hi = h2 = hs = ts/2, and G1 -- G2 = Gs, vl = v2 = vs. Equations (19) yield that A1 = A2 = As, B1 = B2 = Bs and C1 = C2 = C~. It is also clear that a+ (x) = - a_ (x) and r+ (x) = ~_ (x) = z(x). In this case Eqns (20), (21) [or (22), (23) with the same result] can be represented in the following comparatively simple form: [Cs + (3/2)Co]r~l)(x) = - [Ao + (ho/2)Co - A s i a (2)/X + , ) + (2Bo
Bs)a+(x),
(& - 2Bo)rm(x) = - (Cs + 2Co)a(2+)(x) - ( B J h s + 2Bo/ho)a+ (x).
(25) (26)
Equations (25) and (26), after the exclusion of the function ~m(x), yield al+2)(x) -- ~,o'+(x) = 0
(27)
where, on account of Eqn (19), 2(1 - v0) 6
3 3 1 6 2] 1 +5~/+~t/2 +g t/J
(2 C 3--~o)(1 --~s) (28) 6 = (1 - v 0 Go ho ( 1 - vo) Gs h~'
ho tl - hs
(29)
A. L. Kalamkarov
878
[
~ts
J~,o
.
I
~ ~t~ I 1
Fig. 4. Symmetric balanced single lap joint.
After the determination of the normal contact stress a+ (x) from Eqn (27) we can proceed to the calculation of the shear contact stress z(x) = r+ (x) = z_ (x) from the following equation, which can be obtained from Eqns (25) or (26): Z.(II(x) __
1
ho5:20
O'?)(X) ~-
1
~o0
0"+ (X)
(30)
where ¢~2 _
3 (2 +&/z) 4h 2 ( 2 + 6 ) '
0-
(2 - f i r / ) (2+fir/2)
(31)
Equation (27) can be easily solved, but it is necessary to distinguish the cases of positive and negative values of the parameter 7. This parameter is determined by the expressions (28) and (29) and its value depends on the characteristics of the joint, i.e. parameters ho = to~2, hs = ts/2, and G~, v~, Go, Vo. Usually the adhesive material is much softer than the material of the adherents and the thickness of the adhesive interlayer is much smaller than the thickness of adherents, i.e. Go '~ G~ and ho ~ hs, and, consequently, 6 ~ 1 and r / ~ 1. In this case, from Eqn (28) it follows that y > 0, and we denote
3
~/(2-vo)Gsh~ho
2 = ~
l+2r/+4r/e
~
,32,
(1-0.5vo)Eshoh~"
Equations (31) yield, with the same accuracy, ,e/3
0 ~ 1.
(33)
'~ ~ - ~ O '
Equation (27) yields the following general solution: (34)
a+ (x) = C1 cosh(2x) + C2 sinh().x); Equations (30) and (34) yield (0{2 __ /~2)
r(x) -
ho~
(~2 -- /~2)
C1 sinh(2x) + h o ~
C2 cosh(2x) + C3.
(35)
The arbitrary constants C1, C2, Ca must be determined from three equilibrium conditions (6) [or (7) with the same result]. Substituting f = F/w and M = F/w (hs + ho), where F is the applied force, we get from the conditions (6) C1 -~- 0'
C2
Fho~.2 2w[2acosh(2a)-sinh(2a)]'
C3
F 2wa
----
(:~2 -- ~2) sin().a) _ _ C2. ho 0~2 0~. 2 a
Stresses are given then by the following formulae:
Fho ~2 sinh(2x) a+(x) = 2w [2a c o s h ( 2 a ) - sinh(2a)] ' F z+ (x) = r_ (x) = r(x) = - -
2wa
+
a_ (x) = - a+ (x),
F (~{2 __ }2) [2a cosh(ix) - sinh(2a)]
2wa
~20
[2a cosh(2a) - sinh(2a)]
(36)
.
The contact stress distribution on the interface y = ho, 0 < x/a < 1 is shown in Figs 5 and 6.
(37)
Stress analysis in elastic joint structures
879
(5+ / F/(lw)
0.2
~/ to / ts =
0.15
~
o.1 0.0s
~
~= 0.1 t0/ts=0.01
'
~
0.2
0.4
'
'
~'.6
o'.~
'
i
x/a
"
Fig. 5. Contact tearing stress distribution on the interface y = ho, 0 < x/a < 1 in cases of different ratios of layer thicknesses: l/to = 30, Go/G~ =0.025, v0 = 0.4, v~ = 0.3, a+(x) is an odd function in x.
'1;/ F/(lw)
1.9 1.7 1,5 1.3 1.1 0.9 0,7
to / ts = 1
t ~ . ~ j / ~ =
/
0.2
0.4
o'.~
0'.a
i
x/~
Fig. 6. Contact shear stress distributions on the interface y = h o, 0 < x/a < 1 in cases of different ratios of layer thicknesses;/,"to = 30, Go/G~ = 0.025, Vo = 0.4, v~ = 0.3, T(x) is an even function in x.
5. STRESSES IN A D H E S I V E I N T E R L A Y E R T h e a b o v e o b t a i n e d e x p r e s s i o n s [(36), (37)] for the c o n t a c t stresses, i.e. stresses o n the surfaces of the interlayer, c a n be used n o w to d e t e r m i n e the stress d i s t r i b u t i o n in the b u l k of the a d h e s i v e layer. T h e A i r y stress f u n c t i o n in the c o n s i d e r e d case of a t h i n layer ( h o / a ~ 1) c a n be expressed in the f o l l o w i n g a p p r o x i m a t e form [8]: ~Ia(X, y) ~ A o ( x ) y 3 + A l ( X ) y
2 + Az(x)y
+ A3(x).
(38)
Stresses are d e t e r m i n e d as follows: 82W a~, -
ay
~y2 - 6 A o ( x ) v + 2Al(x),
- 82~P - A~o2)(x)Y 3 + A(12)(x)Y 2 + A~22)(x)Y + A
(39)
8 21Y~ z~), -
__ _ _ 3A(ol)(x)y 2 - 2A(ll)(x)y -- A(2U(x). 8 x 8 3,
T h e f o l l o w i n g b o u n d a r y c o n d i t i o n s o n the surface y = +_ ho m u s t be satisfied (cf. Figs 1 a n d 3): o).(x, ho) = a+ (x),
Txy(X, ho) = -
T+(x)
(40) a y ( x , - ho) = e L ( x ) ,
Txy(X, - ho) = - z - ( x ) .
F r e e edge c o n d i t i o n s at the e n d s of the o v e r l a p , x = _+ a, lYl < ho (see Fig. 3), yield
?
-ho
~x( +- a, y) d y = O,
?
-ho
zx, ( _+ a, y) dy = O.
(41)
880
A.L. Kalamkarov
In the case of symmetric lap joint, Eqns (39), (40) and (41) yield: ax = ~o J-,, r(~) dd - h-~o Z.x>: (~
3y2'~
,, (x - ~)o-+ (~) d~x + Cy,
(3Y 2
(42)
3
(43)
2ho)fTo+¢)<'
y3
(44) where C is a constant which can be determined uniquely from the oddness of the function ax(x, y) in x.
The formulae (42), (43) and (44) on account of Eqns (36) and (37) yield the following expressions for stresses in the bulk of the adhesive interlayer, [x I < a, l yl < ho:
x[
sinh o,
ex _6ay ~. sinh(2x) ~ ~2(1- 0 ) - 2 2 +- 1F/(2aw) h~ ~;,tacosh(2a) -sinh(,~a) 2~20 a 2a cosh(2a) - sinh(2a)
~T0 (45)
F/(2aw) - 2a cosh(2a) - sinh(2a) + (1_3y2) Ii_ -~oJ
1 - h~-0J
2cd0
sinh(2a) Za eosh(2a) - sinh(2a)
F/(2aw) - 2a cosh(2a) - sinh(2a)
~oz - 1
- 1 -
0)
20~20
-
2a cosh(£a) J + 2a cosh(2a) - sinh(2a) (46)
2~20
q- 1
.
(47)
Three-dimensional plots of the stresses given by Eqns (45), (46) and (47) at the quarter surface of the interlayer: 0 < x/a < 1, 0 < y/ho < 1, in the case to = 0.5 mm, ts -- 5 mm, l = 15 mm, Go = 712 MPa, v0 = 0.4, Gs = 28.48 GPa, vs = 0.3 are shown in Figs 7, 8 and 9. These plots can be easily extended to the full surface of the interlayer because functions ax(x, y) and ay(x, y) are odd, and the sign of shear stress zxy(x, y) is defined according to Fig. 3. Figures 10 and 11 are plots of the maximum values of the stresses ay/F/(lw) and Zxy/F/lw). The plots are arranged to include the effects of variations in the layer thickness ratio to/ts as well as the effects of variations in the materials of the layers.
0"~/F/Ow)
3.54 10 5
Fig. 7. Distribution of the normal stress crx/F/(Iw).
Stress analysis in elastic joint structures
881
15y / F/(lw)
0 0.0" 0.1 0.0.
~Ty/F/(Iw).
Fig. 8. Distribution of the normal stress
~xy / F/(lw)
Fig. 9. Distribution of the shear stress t~y/F/'(Iw).
Max t5, / F/(zw)
to]ts= 1
0.8
O. 0.2
to/ts = 0.1 ~
,./
0.2
0.4
0.6
Max 'l;xy / F / ( l w )
l'21
0'2
to / ts = 0.01
i Go/Gs values
0.8
to / t~ = 0.01
0'4
0'6
0'8~i
Fig. 10. M a x i m u m values of tearing and shear stresses versus ratio
Go/Gs values
Go/Gs,
Vo = 0.4, v, -= 0.3,
1~to =
30.
882
A.L. Kalamkarov Max Gy / F/(lw) 0.8
0.6
0.4
0.2
/
0.2
0.4
(1.6
0.8
Max %xy/ F/(lw)
/ 1.
G
Go/Gs = 0.001 1
to/tS
values
Go/ Gs = 0.001
o
/ Gs = 0.0~5
1.
1.
1.i
0.2
0.4
0.6
0.8
1 to/ts
values
Fig. 11. Maximum values of tearing and shear stresses versus ratio to,%, Vo = 0.4, v~ = 0.3, I/'to = 30.
6. C O N C L U S I O N S AND D I S C U S S I O N
1. The suggested inverse problem approach represents a new alternative modelling approach to stress analysis in the elastic joint structures. It provides the remarkable simplification of the initial complex problem and makes it possible to calculate the contact stresses on the surfaces of joint elements, as well as the stresses in the bulk of the adhesive interlayer. 2. Volkersen [3] applied a simple approximate approach to analyse the stresses in a single symmetric lap joint. He assumed that the adherents were in a state of pure tension with no bending or shear effects, and the adhesive strip was in a state of pure shear. The following formula for the shear stress was obtained in the framework of these assumptions [8]: v(x)
F/(2aw~)-
)qv]a cosh(Atv]X) sinh(2~vla) ,
~ / Go /qVl= k/2Eshoh~"
(48)
Comparison between the above obtained solution (37) and the approximate solution (48) is illustrated in Fig. 12. A difference between these two solutions is rather remarkable near the ends of the adhesive interlayer (Ix/al = 1). Discrepancy up to a factor of 1.75 occurs in the point x/a = 1. Recognizing this difference is important for determining the adhesive yielding force Fy, and, consequently, for determining the adhesive plastic zone. According to the approximate solution (48) for the joint considered in Fig. 12, the adhesive yielding force is calculated from the condition Fy/(lw) = 0.877ry, where ry is the shear yield stress of the adhesive material. The above obtained solution (37) yields Fy/(lw) = 0.501Vy. Hence, it provides the 1.75 times smaller value of the adhesive yielding force Fy than the approximate solution (48), obtained in [8]. 3. Goland and Reissner [4] obtained a more precise solution than the above mentioned result by Volkersen [3]. They assumed that the stresses were constant across the adhesive thickness and derived the formulae for the contact tearing and shear stresses in the case of a symmetric lap joint. These formulae coincide qualitatively with the above obtained solution [Eqns (36) and (37)]. In particular, in the case of a relatively flexible bond interlayer, Goland and Reissner [4] obtained the
Stress analysis in elastic joint structures
883
"c/F/(lw) i 1.91 1.7 1.5 1.3 1.1 0.9 0.7
J 0.2
0.4
0.6
0.8
1
x/a
Fig. 12. Shear contact stress distribution: curve 1 corresponds to the obtained solution (37), curve 2 to approximate solution (48) obtained by Volkersen [3]; Go/G~ = 0.025, vo = 0.4, v~ = 0.3, to/t~ = 01, I/'to = 30.
following formula for a significant parameter 2 entering into the solution: 2Go 2tOaRl = ~/E~ohs"
(49)
F o r m u l a (49) results from the above obtained formula (28) for ), [or simplified formula (32) for 2] under the following simplifying assumptions: Go/Gs ~ 1, r1 = ho/hs ,~ 1, Vo ~ 0, 1 - vs2 ~ 1, which have quite evident mechanical meaning. It should be noted also that the formula (48), derived in [3], for the parameter 2 gives the result four times smaller than the formulae (28) (our solution) and (49), derived in [4]. 4. The above numerical analysis of the contact tearing and shear stresses (see Figs 5 and 6) and the m a x i m u m values of the stresses in the bulk of adhesive interlayer (see Figs 10 and 11) shows their dependence on the ratios of layer thicknesses and the material properties. In particular, it follows from this analysis that the m a x i m u m value of the stress ay(X, y) increases with ratios Go/Gs and to/t~, whereas the m a x i m u m value of shear stress zxy(x, y) decreases. It is also worth mentioning that the influence of variation of materials properties of the adherents and adhesive interlayer becomes less significant in the case of small ratios of thicknesses of elements of the lap joint. 5. The above introduced inverse problem a p p r o a c h can be successfully applied to stress analysis of the elastic structures with a n u m b e r of contact surfaces and junctions, see Ref. [9], Acknowledgements--This work has been supported by the Natural Sciences and Engineering Research Council of Canada.
REFERENCES 1. Benson, N. K., The mechanics of adhesive bonding. Applied Mechanics Review, 1961, 14, 83 87. 2. Hart-Smith, L. J. Adhesive bonded double lap joints. NASA C.R. 112235, 1973. 3. Volkersen, O., Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschenquerschnitten. Laftfahrtforschung., 1938, 15, 4l 47. 4. Goland, M. and Reissner, E., The stresses in cemented joints. Journal of Applied Mechanics, 1944, 11, A-17-A-27. 5. Gilibert, Y. and Rigolot, A., Analyse asymptotique des assemblages coll+s h double recouvrement sollicit6s au cisaillement en traction. Journal of Applied Mechanics, 1979, 3, 341-372. 6. Rigolot, A., Analytic schemes for interracial stresses in adhesively bonded joints. Mechanical Behaviour of Adhesive Joints. Pluralis, Paris, 1987, pp. 51-70. 7. Hu, G. K., Francois, D. and Schmit, F., Non-linear fracture mechanics for adhesive lap joints. Journal of Adhesion, 1992, 37, 261-269. 8. Aleksandrov, V. M. and Mkhitaryan, S. M., Contact Problemsfor Solids with Thin Coats and Interlayers. Nauka, Moscow, 1983 (in Russian). 9, Kalamkarov, A. L., Composite and ReinJorced Elements of Construction. Wiley, Chichester, 1992.
Int. J. Mech. Sci. Vol. 39, No. 7, pp. 873 883, 1997 c~;, 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020 7403/97 $ 1 7 . 0 0 + 0 . 0 0
PII:
STRESS ANALYSIS
S0020-7403(96)00086-0
IN ELASTIC
JOINT
STRUCTURES
A. L. K A L A M K A R O V Department of Mechanical Engineering, Technical University of Nova Scotia, P.O. Box 1000, Halifax, Nova Scotia, Canada B3J 2X4
(Received 26 February 1994; and in revisedform 12 July 1996) Ahstract--A new inverse problem approach is introduced to analyse the stresses in the overlap elastic joint structures consisting of two adherents and an adhesive interlayer. Displacements in the elements of a joint are expressed in terms of some initially unknown contact stresses by solving the boundary-value problem for each element considered separately. The contact stresses are calculated from the matching continuity conditions for the displacements. Contact tearing and shear stresses, three-dimensional stress distribution in the bulk of adhesive interlayer, and the maximum stress values are calculated for different ratios of thicknesses and material properties of the adherents and the adhesive interlayer. The present study represents a new alternative modelling approach to stress analysis in the elastic joint structures. @ 1997 Elsevier Science Ltd. All rights reserved.
Keywords: adhesive lap joints, contact stresses, maximum tearing and shear stresses, inverse problem approach.
NOTATION shear moduli and Poisson's ratios of the materials of upper, lower adherents and of the adhesive interlayer correspondingly f traction force per unit width w of the joint Ti, Mi Qi(i = 1,2) stress and moment resultants, shearing forces tt = 2 h t , t 2 =2h2, to = 2h0 thicknesses of upper, lower adherents and of the adhesive interlayer correspondingly ts = 2h~ thickness of adherents of the symmetric lap joint ax, ay and zxy normal and shear stresses a+(X), a-(X), Z+(X), r_(x) contact stresses u(x, y), v(x, y) displacements q'(x, y) Airy stress function derivative of kth order with respect to x G t , ~,'t, G2, v2 and Go,vo
1. I N T R O D U C T I O N
Adhesive joints would provide attractive solutions to many mechanical engineering problems if their performances and reliability were well known. Owing to the large discrepancy which is usually present between the elastic properties of the adherents and the adhesive stresses are not always precisely evaluated. This represents a drawback for the safe calculation of the limit load which an adhesive joint can sustain. Different methods have been employed for theoretical strength analysis of the joint structures. Most of the basic work in this field, relating to a single lap joint loaded in tension, has been reviewed by Benson [1] and Hart-Smith [-2]. A simple approximate approach, due to Volkersen I-3], considers only shear deformation of the adhesive and elongation of the adherents. The bending effects in the single lap joints were considered first by Goland and Reissner [4]. The asymptotic solution of the problem in the case of a double lap joint was obtained by Gilibert and Rigolot [5]. Rigolot I-6] considered the end-effects and material non-linearity. The fracture of a lap joint was analysed in Ref. [7]. In the present paper, a new inverse problem approach to stress analysis in elastic lap joint structures is introduced. The method is based on the application of some special equations matching contact displacements with normal and shear stresses on the surfaces of a thin strip. These expressions follow from the analysis of the elastic plane strain state of a thin strip. In the suggested method, the elastic equilibrium is considered separately for each element of a joint. Displacements in these elements are expressed in terms of the initially unknown stresses on the contact surfaces by solving the boundary-value problem for each element considered separately. Contact stresses are 873
874
A.L. Kalamkarov
then calculated from the matching continuity conditions for the displacements on the contact surfaces of the elements of a lap joint. Stresses in the bond interlayer are expressed in terms of the contact stresses and are also calculated. The developed inverse problem approach represents a new alternative modelling approach to stress analysis in the elastic joint structures. It provides the remarkable simplification of the initial complex problem and makes it possible to calculate the contact stresses and the stresses in the bulk of the lap joint structure under study. The inverse problem approach was applied earlier by the author in the analyses of the composite reinforced structural elements, see Ref. [9]. 2. E Q U I L I B R I U M E Q U A T I O N S FOR A T H I N LAYER
To proceed to the stress analysis in an elastic joint structure, we first consider a thin elastic layer (strip) of finite length l = 2a and thickness t = 2 h, so that ix b ~< a, IYl ~< h, (Fig. 1). We assume the following notations for the stress and moment resultants and shearing forces:
r(x)=f]hfixdy, M(x)=f yfi~dy, Q(x)=F.
vxy dy.
(1)
Equilibrium conditions for a thin strip as a whole on account of the forces and moments acting on the edges x = _+ a can be formulated as
f~ [fi2(x)- fil(X)] dx + Q2 ~
=0,
Q1
a
[Z'2(X --'rl(X)] dx + T 2 -- T 1
f]
(2)
~ O,
a
--
[T2(x ) -b TI(X)] dx + a(Q2 4- Q1) + M2 - M t
X[fi2(X ) --fix(X)] dx - - h a
= 0.
a
The stress and moment resultants over the arbitrary cross-section x can be calculated from the expressions
T(x) = i X [ % ( { ) - r 1 ( { ) ] d{ + T2, Q(x)= ( x [o"2(3)-fi1({)] d{ + 02, J- a ~ a ~x
M(x)
| o~ a
Ox
(3)
[rz(~) + ~1(¢)] d~ + xQ2 + M2 + aQ2.
(x - ~) [or2({) - fil({)] d{ - h | ,J-
i1
Analysis of the elastic plane strain problem on account of the small thickness of a strip (h ~ a), provides the following equations matching displacements u(x, + h) and v(x, +_h) and stresses on the surfaces y = + h (see Fig. 1) [8]: 4Gh2ula)(x, _ h) = - vh2(fi~2) + fi~22))- (1 - v)h(z] n - z(21)) ± 3(1 - v)(¢2 - rh)
_+ (2 - 3v)h2(rrll2, -- 0"12 2,) ~ 3(1 -- v)h(z]~) + r~~))
4~h~'~(x, +h)= 3(1 - v)(fil - f i j - 4 ( 1 -u)h~(~ ~ -fi~))+3h(1 -~)(~T-%~).
(4)
(5)
Here and in the sequel all the superscripts in the brackets indicate the orders of the derivatives with respect to x. Equations (4) and (5) can be derived from the equilibrium equations for a thin strip by means of the following procedure, see Ref. [8] for the details. First, the longitudinal and transversal components of the displacement vector u(x, y) and v(x, y) are represented in the form of Fourier integrals in variable x. The equilibrium equations yield a system of two ordinary differential equations of the second order for their Fourier transforms. This system can be solved easily and the arbitrary functions entering into the solution can be expressed in terms of Fourier transforms of the normal and shear surface stresses a~ (x), 0"2(X), T I(X ) and %(x) from the conditions on the boundaries of the strip, y = _+ h. After that, the obtained expressions for the Fourier transforms of the displacements are expanded in the power series of a small parameter, c~ = h/a. Finally, we keep only terms of ~X2 order for the longitudinal component and the terms of c~3 order for the transversal component. The application of the inverse Fourier transform to these expansions leads to the above Eqns (4) and (5).
Stress analysis in elastic joint s t r u c t u r e s
875
(~'1
Fig. 1. T h i n layer a n d the c o n t a c t stresses on the u p p e r a n d lower surfaces.
3. A N A L Y S I S
OF THE
LAP
JOINT
PROBLEM
We can proceed now to the stress analysis in an elastic lap joint. F o r the sake of simplicity, we will consider a single lap joint consisting of three elastic layers (Fig. 2): layer 1: al ~< x ~< a, [y[ ~< hi (the upper adherent), layer 0: - a ~< x ~< a, [y[ ~< ho (the adhesive interlayer), and layer 2: - a ~< x ~< ax, [y] ~< h 2 (the lower adherent). The contact stresses a+(x), a_(x), z+(x) and z (x) shown in Fig. 3 can be calculated from the matching continuity conditions for displacements in the interval Ix[ < a. Displacements u(x, +_h) and v(x, +_ h) can be expressed in terms of the initially u n k n o w n contact stresses by solving the boundary-value problem for each of three layers considered separately. The following equilibrium conditions for layers 1 and 2 (Fig. 3) must be satisfied:
f" a r+ ix) dx = f
f ' S +(x) dx = 0 f f a xa+ (x) dx = M
f" z _ ( x ) d x = f
f] . ~ (x)dx=O
--J~Z 1
f] .
(6)
(7)
Equilibrium conditions for interlayer 0 (see Fig. 3),
f~ f
a
[a_ (x) - a+ (x)] dx = 0,
f " [z_ (x) - r+ (x)] dx = 0 a
(8)
" x[o" ( x ) - a + i x ) ] d x + h o f ~ a
a
[z_(x)+z+(x)]dx=0
are satisfied automatically on account of the conditions for layers 1 and 2, i.e. Eqns (6) and (7), and the expression for the m o m e n t M (see Fig. 3),
M = (1/2)f(h2 + hi + 2ho).
(9)
Equations (4) and (5) on the lower surface of the layer 1 (in the interval [ x] < a) yield U(?)(X, - - h i ) - -
1 ~°niA~l" ~ [2(1 - 2v1)hl2 o% (2)(x) - 3(1 - Y1)o'+ i X) - - 4(1 -- Vl)h 1~ ( 1 ) ( X ) ] ,
(lO) [ ( 4 ) i x ' __ h i ) _ 3 ( 1 - - Yl)
4G,h 3 [ - ° ' + i x ) - h l ' C ~ ) ( x ) ] +
( l - Y1) o.(+2)ix )
b~-i
"
On the upper and lower surfaces of interlayer 0, Eqns (4) and (5) yield
1 u~oa)(x, + ho) - 4Goh 2 [2il - 2vo)h 2 o.(2~(x) - 3(1 - Vo)O+(x) - 2(1 - vo)hgo¢z_'(x) + 3(1 -
Vo)a (x) + 4(1 - v o ) h o r + (a)(x) + 2(1 -- vo)hoz_(1)(x)], (11)
V(o4)(x, + ho) - 3(1 -Vo) 4Goh 3 [a+
(x) - e_ (x) - hor~)(x) - hor~)(x)]
(1 - Vo)
876
A.L. Kalamkarov
2hl
2ho
.1]
-.~-
al
Fig. 2. Single lap joint.
1M
1
o÷(x)__
¢_~ ~_~a
2~+
"~(x)
-~ "r.(x)
o-oo/i~ ~ ~~ / -a
Fig. 3. Contact stresses on the contact surfaces of the lap joint elements.
1 U~o3)(x, - h o ) - 4Goh2 E - 2 ( 1 - vo)h2o~ ' (x) - 3(1 - Vo)O+ (x) + 2(1 -2vo)h2o~)(x)
- 3(1 - Vo)a-(x) - 2 ( 1 - vo)hoz{1)(x) - 4 ( 1 - vo)hoz~)(x)], (12)
V(o4)(x, - h o )
-
3(1 - Vo)
4Goh 3 [ o + ( x ) - o _ ( x ) - h o z + , ,
{U,x ~ _ hoz(l_)(x)]
(1 - Vo) [-o~)(x) G~o -°~)(x)]'
Correspondingly, Eqns (4) and (5) yield on the upper surface of the layer 2 (in the interval Ix I < a) 1
{1)t X
ut23'(x, + h2) - 4G2h22 [-2(1 -2v2)h22o{2_)(x) - 3(1 - v 2 ) & ( x ) + 4(1 - 2 v 2 ) h 2 z _ ~
)], (13)
3(1 -- v2) r v{2~)(x, + h2) -
TThT
- hzr~'(x)]
(1 - v2) o~2), , -GTh~ _ tx),
where Gl, vl, G 2 , Y2 and Go, Vo are the elastic characteristics of layers 1, 2 and the adhesive interlayer materials correspondingly. Contact stresses occurring on the contact surfaces of the layers can be calculated now from the following matching continuity conditions for displacements: U(3)~ 1 IX,
-- hi)
=
/,/(3}(X, -I- ho),
v]4)(x, - hD = v(o4}(x, + ho) (14)
U{o3}(x,-ho) = ~?}(~, + h~t, ~{0~'(~,-ho) = ¢~*'(x, + h~). Conditions (14) together with Eqns (10)-(13) yield the following four differential equations: (A 1 - -
Ao) o(_2)(x)
--(B 1 --Bo)o + (x) -[- (ho/2)Coo~)(x) - B o a - (x) - ( C , + Co)r~)(x)
-{1/2)Cor~)(x) = 0, -- (ho/2) Coo+(2}itx) + Boo+ (x) + (Ao - (Co + C2)r~}(x) = 0,
(15) - A2)o~-2)(x) -
- (1)/ (Bo - Be)o- (x) - (Co/2)z+ tx)
(16)
877
Stress analysisin elasticjoint structures (Co + C1)a~+Z'(x) - (Bo/ho + B 1 / h O a + (x)
-
Coa{2_)(x) + ( B o / h o ) a - (x)
- (B, - Bo)v~)(x) - - B o ~ -(1){, x ) = 0,
- - C o a ~ ) ( x ) + B o / h o a + ( x ) + (Co - - B o v ~ ) (x) - ( B o
-
-
+
B2)'f~)(x)
(17)
C2)o(-2}(x) - ( B o / h o
+ B2/h2)a-(x) (18)
= O,
where the following notations are assumed: 3(1 - 2 v l )
1 -- 2vi Ai
-
-
-
,
2Gi
Bi
-
-
-
4Gih ~
1 - vi '
Ci
-
Gihi '
( i = 0 , 1,2).
(19)
Resolving Eqns (15), (16) and(17), (18) about functions zt+l)(x) and z(1)(x), we get following four equations: Vl + 0"+ (X) -~- W1 + O'(_2)(X) Jr- Y ; o'_ (x),
(20)
U1 a+(2)[,x) + V 1- o'+ (x) ~- W 1- O'(_2)(X)-{- Y 1 a - (X),
(21)
v ? ~+(x) + w d ~ ( x ) + Y ; ~_ (x),
(22)
~ ( x ) = u2- ~'+~(x) + V 2 o+ (x) -~ W 2- o(2}(x) -[- Y f a_ (x).
(23)
.C(1)(X) =
Coefficients Us-+, V~-+, W / , Yj-+, ( j = 1, 2) can be easily represented in the form of algebraic functions of the parameters Ai, Bi, C~, (i = 0, 1, 2) defined by the Eqn (19). Combination of Eqns (20), (22) and (21), (23) yields the following system of two linear differential equations for the contact stresses a+ (x) and a (x):
(U?-
" ~ + "~ (2) ~2 +
-
Y;)~_
= o,
(U1-
"~-")a+~2~ + (V1- - V 2 ) a + + ( W F - W2)a~2_ ) + (Y1- u2
Y2)a-
= O.
+ (v~ + -
v;>+
+ (w?
-
w;)~
~+ ( r ?
(24) The general solution of the linear system of Eqns (24) includes four arbitrary constants and the subsequent integration of the expressions (20) and (21) [or (22) and (23) with the same result] will add two more arbitrary constants. To complete the solution of the problem, we must determine these six constants from six equilibrium conditions [Eqns (6) and (7)]. 4. SYMMETRIC LAP JOINT
Let us apply the above general approach to the case of the symmetric balanced single lap joint, shown in Fig. 4 (cf. Fig. 2). We assume that the layers 1 and 2 have the same thicknesses, i.e. hi = h2 = hs = ts/2, and G1 -- G2 = Gs, vl = v2 = vs. Equations (19) yield that A1 = A2 = As, B1 = B2 = Bs and C1 = C2 = C~. It is also clear that a+ (x) = - a_ (x) and r+ (x) = ~_ (x) = z(x). In this case Eqns (20), (21) [or (22), (23) with the same result] can be represented in the following comparatively simple form: [Cs + (3/2)Co]r~l)(x) = - [Ao + (ho/2)Co - A s i a (2)/X + , ) + (2Bo
Bs)a+(x),
(& - 2Bo)rm(x) = - (Cs + 2Co)a(2+)(x) - ( B J h s + 2Bo/ho)a+ (x).
(25) (26)
Equations (25) and (26), after the exclusion of the function ~m(x), yield al+2)(x) -- ~,o'+(x) = 0
(27)
where, on account of Eqn (19), 2(1 - v0) 6
3 3 1 6 2] 1 +5~/+~t/2 +g t/J
(2 C 3--~o)(1 --~s) (28) 6 = (1 - v 0 Go ho ( 1 - vo) Gs h~'
ho tl - hs
(29)
A. L. Kalamkarov
878
[
~ts
J~,o
.
I
~ ~t~ I 1
Fig. 4. Symmetric balanced single lap joint.
After the determination of the normal contact stress a+ (x) from Eqn (27) we can proceed to the calculation of the shear contact stress z(x) = r+ (x) = z_ (x) from the following equation, which can be obtained from Eqns (25) or (26): Z.(II(x) __
1
ho5:20
O'?)(X) ~-
1
~o0
0"+ (X)
(30)
where ¢~2 _
3 (2 +&/z) 4h 2 ( 2 + 6 ) '
0-
(2 - f i r / ) (2+fir/2)
(31)
Equation (27) can be easily solved, but it is necessary to distinguish the cases of positive and negative values of the parameter 7. This parameter is determined by the expressions (28) and (29) and its value depends on the characteristics of the joint, i.e. parameters ho = to~2, hs = ts/2, and G~, v~, Go, Vo. Usually the adhesive material is much softer than the material of the adherents and the thickness of the adhesive interlayer is much smaller than the thickness of adherents, i.e. Go '~ G~ and ho ~ hs, and, consequently, 6 ~ 1 and r / ~ 1. In this case, from Eqn (28) it follows that y > 0, and we denote
3
~/(2-vo)Gsh~ho
2 = ~
l+2r/+4r/e
~
,32,
(1-0.5vo)Eshoh~"
Equations (31) yield, with the same accuracy, ,e/3
0 ~ 1.
(33)
'~ ~ - ~ O '
Equation (27) yields the following general solution: (34)
a+ (x) = C1 cosh(2x) + C2 sinh().x); Equations (30) and (34) yield (0{2 __ /~2)
r(x) -
ho~
(~2 -- /~2)
C1 sinh(2x) + h o ~
C2 cosh(2x) + C3.
(35)
The arbitrary constants C1, C2, Ca must be determined from three equilibrium conditions (6) [or (7) with the same result]. Substituting f = F/w and M = F/w (hs + ho), where F is the applied force, we get from the conditions (6) C1 -~- 0'
C2
Fho~.2 2w[2acosh(2a)-sinh(2a)]'
C3
F 2wa
----
(:~2 -- ~2) sin().a) _ _ C2. ho 0~2 0~. 2 a
Stresses are given then by the following formulae:
Fho ~2 sinh(2x) a+(x) = 2w [2a c o s h ( 2 a ) - sinh(2a)] ' F z+ (x) = r_ (x) = r(x) = - -
2wa
+
a_ (x) = - a+ (x),
F (~{2 __ }2) [2a cosh(ix) - sinh(2a)]
2wa
~20
[2a cosh(2a) - sinh(2a)]
(36)
.
The contact stress distribution on the interface y = ho, 0 < x/a < 1 is shown in Figs 5 and 6.
(37)
Stress analysis in elastic joint structures
879
(5+ / F/(lw)
0.2
~/ to / ts =
0.15
~
o.1 0.0s
~
~= 0.1 t0/ts=0.01
'
~
0.2
0.4
'
'
~'.6
o'.~
'
i
x/a
"
Fig. 5. Contact tearing stress distribution on the interface y = ho, 0 < x/a < 1 in cases of different ratios of layer thicknesses: l/to = 30, Go/G~ =0.025, v0 = 0.4, v~ = 0.3, a+(x) is an odd function in x.
'1;/ F/(lw)
1.9 1.7 1,5 1.3 1.1 0.9 0,7
to / ts = 1
t ~ . ~ j / ~ =
/
0.2
0.4
o'.~
0'.a
i
x/~
Fig. 6. Contact shear stress distributions on the interface y = h o, 0 < x/a < 1 in cases of different ratios of layer thicknesses;/,"to = 30, Go/G~ = 0.025, Vo = 0.4, v~ = 0.3, T(x) is an even function in x.
5. STRESSES IN A D H E S I V E I N T E R L A Y E R T h e a b o v e o b t a i n e d e x p r e s s i o n s [(36), (37)] for the c o n t a c t stresses, i.e. stresses o n the surfaces of the interlayer, c a n be used n o w to d e t e r m i n e the stress d i s t r i b u t i o n in the b u l k of the a d h e s i v e layer. T h e A i r y stress f u n c t i o n in the c o n s i d e r e d case of a t h i n layer ( h o / a ~ 1) c a n be expressed in the f o l l o w i n g a p p r o x i m a t e form [8]: ~Ia(X, y) ~ A o ( x ) y 3 + A l ( X ) y
2 + Az(x)y
+ A3(x).
(38)
Stresses are d e t e r m i n e d as follows: 82W a~, -
ay
~y2 - 6 A o ( x ) v + 2Al(x),
- 82~P - A~o2)(x)Y 3 + A(12)(x)Y 2 + A~22)(x)Y + A
(39)
8 21Y~ z~), -
__ _ _ 3A(ol)(x)y 2 - 2A(ll)(x)y -- A(2U(x). 8 x 8 3,
T h e f o l l o w i n g b o u n d a r y c o n d i t i o n s o n the surface y = +_ ho m u s t be satisfied (cf. Figs 1 a n d 3): o).(x, ho) = a+ (x),
Txy(X, ho) = -
T+(x)
(40) a y ( x , - ho) = e L ( x ) ,
Txy(X, - ho) = - z - ( x ) .
F r e e edge c o n d i t i o n s at the e n d s of the o v e r l a p , x = _+ a, lYl < ho (see Fig. 3), yield
?
-ho
~x( +- a, y) d y = O,
?
-ho
zx, ( _+ a, y) dy = O.
(41)
880
A.L. Kalamkarov
In the case of symmetric lap joint, Eqns (39), (40) and (41) yield: ax = ~o J-,, r(~) dd - h-~o Z.x>: (~
3y2'~
,, (x - ~)o-+ (~) d~x + Cy,
(3Y 2
(42)
3
(43)
2ho)fTo+¢)<'
y3
(44) where C is a constant which can be determined uniquely from the oddness of the function ax(x, y) in x.
The formulae (42), (43) and (44) on account of Eqns (36) and (37) yield the following expressions for stresses in the bulk of the adhesive interlayer, [x I < a, l yl < ho:
x[
sinh o,
ex _6ay ~. sinh(2x) ~ ~2(1- 0 ) - 2 2 +- 1F/(2aw) h~ ~;,tacosh(2a) -sinh(,~a) 2~20 a 2a cosh(2a) - sinh(2a)
~T0 (45)
F/(2aw) - 2a cosh(2a) - sinh(2a) + (1_3y2) Ii_ -~oJ
1 - h~-0J
2cd0
sinh(2a) Za eosh(2a) - sinh(2a)
F/(2aw) - 2a cosh(2a) - sinh(2a)
~oz - 1
- 1 -
0)
20~20
-
2a cosh(£a) J + 2a cosh(2a) - sinh(2a) (46)
2~20
q- 1
.
(47)
Three-dimensional plots of the stresses given by Eqns (45), (46) and (47) at the quarter surface of the interlayer: 0 < x/a < 1, 0 < y/ho < 1, in the case to = 0.5 mm, ts -- 5 mm, l = 15 mm, Go = 712 MPa, v0 = 0.4, Gs = 28.48 GPa, vs = 0.3 are shown in Figs 7, 8 and 9. These plots can be easily extended to the full surface of the interlayer because functions ax(x, y) and ay(x, y) are odd, and the sign of shear stress zxy(x, y) is defined according to Fig. 3. Figures 10 and 11 are plots of the maximum values of the stresses ay/F/(lw) and Zxy/F/lw). The plots are arranged to include the effects of variations in the layer thickness ratio to/ts as well as the effects of variations in the materials of the layers.
0"~/F/Ow)
3.54 10 5
Fig. 7. Distribution of the normal stress crx/F/(Iw).
Stress analysis in elastic joint structures
881
15y / F/(lw)
0 0.0" 0.1 0.0.
~Ty/F/(Iw).
Fig. 8. Distribution of the normal stress
~xy / F/(lw)
Fig. 9. Distribution of the shear stress t~y/F/'(Iw).
Max t5, / F/(zw)
to]ts= 1
0.8
O. 0.2
to/ts = 0.1 ~
,./
0.2
0.4
0.6
Max 'l;xy / F / ( l w )
l'21
0'2
to / ts = 0.01
i Go/Gs values
0.8
to / t~ = 0.01
0'4
0'6
0'8~i
Fig. 10. M a x i m u m values of tearing and shear stresses versus ratio
Go/Gs values
Go/Gs,
Vo = 0.4, v, -= 0.3,
1~to =
30.
882
A.L. Kalamkarov Max Gy / F/(lw) 0.8
0.6
0.4
0.2
/
0.2
0.4
(1.6
0.8
Max %xy/ F/(lw)
/ 1.
G
Go/Gs = 0.001 1
to/tS
values
Go/ Gs = 0.001
o
/ Gs = 0.0~5
1.
1.
1.i
0.2
0.4
0.6
0.8
1 to/ts
values
Fig. 11. Maximum values of tearing and shear stresses versus ratio to,%, Vo = 0.4, v~ = 0.3, I/'to = 30.
6. C O N C L U S I O N S AND D I S C U S S I O N
1. The suggested inverse problem approach represents a new alternative modelling approach to stress analysis in the elastic joint structures. It provides the remarkable simplification of the initial complex problem and makes it possible to calculate the contact stresses on the surfaces of joint elements, as well as the stresses in the bulk of the adhesive interlayer. 2. Volkersen [3] applied a simple approximate approach to analyse the stresses in a single symmetric lap joint. He assumed that the adherents were in a state of pure tension with no bending or shear effects, and the adhesive strip was in a state of pure shear. The following formula for the shear stress was obtained in the framework of these assumptions [8]: v(x)
F/(2aw~)-
)qv]a cosh(Atv]X) sinh(2~vla) ,
~ / Go /qVl= k/2Eshoh~"
(48)
Comparison between the above obtained solution (37) and the approximate solution (48) is illustrated in Fig. 12. A difference between these two solutions is rather remarkable near the ends of the adhesive interlayer (Ix/al = 1). Discrepancy up to a factor of 1.75 occurs in the point x/a = 1. Recognizing this difference is important for determining the adhesive yielding force Fy, and, consequently, for determining the adhesive plastic zone. According to the approximate solution (48) for the joint considered in Fig. 12, the adhesive yielding force is calculated from the condition Fy/(lw) = 0.877ry, where ry is the shear yield stress of the adhesive material. The above obtained solution (37) yields Fy/(lw) = 0.501Vy. Hence, it provides the 1.75 times smaller value of the adhesive yielding force Fy than the approximate solution (48), obtained in [8]. 3. Goland and Reissner [4] obtained a more precise solution than the above mentioned result by Volkersen [3]. They assumed that the stresses were constant across the adhesive thickness and derived the formulae for the contact tearing and shear stresses in the case of a symmetric lap joint. These formulae coincide qualitatively with the above obtained solution [Eqns (36) and (37)]. In particular, in the case of a relatively flexible bond interlayer, Goland and Reissner [4] obtained the
Stress analysis in elastic joint structures
883
"c/F/(lw) i 1.91 1.7 1.5 1.3 1.1 0.9 0.7
J 0.2
0.4
0.6
0.8
1
x/a
Fig. 12. Shear contact stress distribution: curve 1 corresponds to the obtained solution (37), curve 2 to approximate solution (48) obtained by Volkersen [3]; Go/G~ = 0.025, vo = 0.4, v~ = 0.3, to/t~ = 01, I/'to = 30.
following formula for a significant parameter 2 entering into the solution: 2Go 2tOaRl = ~/E~ohs"
(49)
F o r m u l a (49) results from the above obtained formula (28) for ), [or simplified formula (32) for 2] under the following simplifying assumptions: Go/Gs ~ 1, r1 = ho/hs ,~ 1, Vo ~ 0, 1 - vs2 ~ 1, which have quite evident mechanical meaning. It should be noted also that the formula (48), derived in [3], for the parameter 2 gives the result four times smaller than the formulae (28) (our solution) and (49), derived in [4]. 4. The above numerical analysis of the contact tearing and shear stresses (see Figs 5 and 6) and the m a x i m u m values of the stresses in the bulk of adhesive interlayer (see Figs 10 and 11) shows their dependence on the ratios of layer thicknesses and the material properties. In particular, it follows from this analysis that the m a x i m u m value of the stress ay(X, y) increases with ratios Go/Gs and to/t~, whereas the m a x i m u m value of shear stress zxy(x, y) decreases. It is also worth mentioning that the influence of variation of materials properties of the adherents and adhesive interlayer becomes less significant in the case of small ratios of thicknesses of elements of the lap joint. 5. The above introduced inverse problem a p p r o a c h can be successfully applied to stress analysis of the elastic structures with a n u m b e r of contact surfaces and junctions, see Ref. [9], Acknowledgements--This work has been supported by the Natural Sciences and Engineering Research Council of Canada.
REFERENCES 1. Benson, N. K., The mechanics of adhesive bonding. Applied Mechanics Review, 1961, 14, 83 87. 2. Hart-Smith, L. J. Adhesive bonded double lap joints. NASA C.R. 112235, 1973. 3. Volkersen, O., Die Nietkraftverteilung in Zugbeanspruchten Nietverbindungen mit Konstanten Laschenquerschnitten. Laftfahrtforschung., 1938, 15, 4l 47. 4. Goland, M. and Reissner, E., The stresses in cemented joints. Journal of Applied Mechanics, 1944, 11, A-17-A-27. 5. Gilibert, Y. and Rigolot, A., Analyse asymptotique des assemblages coll+s h double recouvrement sollicit6s au cisaillement en traction. Journal of Applied Mechanics, 1979, 3, 341-372. 6. Rigolot, A., Analytic schemes for interracial stresses in adhesively bonded joints. Mechanical Behaviour of Adhesive Joints. Pluralis, Paris, 1987, pp. 51-70. 7. Hu, G. K., Francois, D. and Schmit, F., Non-linear fracture mechanics for adhesive lap joints. Journal of Adhesion, 1992, 37, 261-269. 8. Aleksandrov, V. M. and Mkhitaryan, S. M., Contact Problemsfor Solids with Thin Coats and Interlayers. Nauka, Moscow, 1983 (in Russian). 9, Kalamkarov, A. L., Composite and ReinJorced Elements of Construction. Wiley, Chichester, 1992.