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Asymptotic modelling of adhesive joints
Mechanics of Materials 28 Ž1998. 137–145
Asymptotic modelling of adhesive joints A. Klarbring a , A.B. Movchan b
b,)
a Linkoping UniÕersity, Linkoping, Sweden ¨ ¨ School of Mathematical Sciences, UniÕersity of Bath, Bath BA2 7AY, UK
Received 19 November 1996; revised 9 December 1996
Abstract An asymptotic approach is presented for modelling of adhesive joints in a layered structure. We consider the case when two elastic layers are bonded together by a thin and soft layer of glue. Differential equations for the displacements Žincluding the displacement jump across the thin interface layer. in a compound thin beam are derived. q 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Asymptotic and numerical modelling of fields in inhomogeneous layered structures is a subject of great interest for many researchers in engineering and mathematics. On the one hand, the work in this direction is motivated by a number of important practical questions. On the other hand, the mathematical properties of the operators of boundary value problems and their solutions attract great attention of applied mathematicians. We would like to refer to the papers by Nayfeh Ž1975, 1980. on heat conduction in laminated composites with bonds, and the papers by Nayfeh and Nassar Ž1978, 1982. on propagation of waves in elastic laminated structures and analysis of interfacial stresses. These papers illustrate very clearly the range of important engineering applications related to the analysis of boundary value problems in laminated composites. Also, we would like to cite the more recent paper by Spencer Ž1990., and the work by Spencer et al. )
Corresponding author.
Ž1993. and by Rogers et al. Ž1995., where the authors analyze three-dimensional elasticity solutions for stretching and bending of inhomogeneous laminated plates. In some particular cases of applied load exact solutions were obtained for normal loading of inhomogeneous and laminated anisotropic elastic plates. The main objective of the present paper is to derive a mathematical model for an adhesiÕe joint in a thin compound beam with a layered structure. To be specific, we assume that the beam includes three layers, with the middle one being infinitesimally thin and soft in comparison with the two others. This situation corresponds, for example, to the presence of an adhesive joint when two elastic layers are bonded together by a thin and soft layer of glue. We note that the formulation includes two small parameters, and different relations between these parameters would lead to different governing equations for the layered structure; in addition, we mention that formally this problem is classified as a singularly perturbed one. Engineers in the aircraft industry Žmodelling of the wing structure. and the car industry Žmodelling of a windscreen. face a question related
0167-6636r98r$19.00 q 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 Ž 9 7 . 0 0 0 4 5 - 8
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
138
to accurate evaluation of the displacement jump across the thin interface layer. It turns out that numerically this is a challenging problem, and the standard finite element packages fail to provide the required accuracy. A problem on the modelling of adhesive joints was considered by Klarbring Ž1991., where an asymptotic approach was based on a variational formulation developed by Ciarlet Ž1990.. In that work the elements ‘glued together’ were treated as threedimensional bodies. In large finite element calculations this is frequently not a satisfactory situation: engineers would rather prefer to model the adherents by a two-dimensional theory, for instance, Kirchhoff’s plate theory. In the present text we resolve the difficulty by deriving a simple and accurate mathematical model for a layered structure that includes a thin and soft adhesive layer. The asymptotic approach, developed in this paper, is based on the results by Nazarov Ž1983a,b., Arutyunyan et al. Ž1987., Movchan and Nazarov Ž1988., Maz’ya et al. Ž1991..
interface boundary includes two parts Sq and Sy specified by < x 1 < - l, x 2 s y´ Ž hr2 y h 2 . q ´ 2 h 0 4 ,
Sqs x : and Sys x :
< x 1 < - l, x 2 s y´ Ž hr2 y h 2 . 4 .
The upper and lower surfaces of the compound region are
Gqs x :
< x 1 < - l, x 2 s ´ 2 h 0 q ´ hr2 4 ,
Gys x :
< x 1 < - l, x 2 s y´ hr2 4 .
The layers V 1 and V 2 are adhesively joined by the thin layer V 0 , which is made of an adhesive material that is much softer than the material of the other two layers. Introduce the variables t 0 s ´y2 Ž x 2 q ´ Ž hr2 y h 2 . y ´ 2 h 0r2 . ,
Ž 2.4 .
t 1 s ´y1 Ž x 2 y ´ 2 h 0 y ´ h 2r2 . ,
Ž 2.5 .
and t 2 s ´y1 Ž x 2 q ´ h1r2 . . In this case
2. The layered structure
t i g w yh ir2, h ir2 x , Consider a thin rectangular domain V´ , which consists of three layers:
V1 s x g R2 :
< x 1 < - l, ´ Ž hr2 y h1 . q ´ 2 h 0 - x 2
- ´ hr2 q ´ 2 h 0 4 ,
V2 s xgR2 :
Ž 2.1 .
i s 1, 2;
t 0 g w yh 0r2, h 0r2 x , within V i, i s 0, 1, 2, and one can see that
E E x2
E s ´y2
E t0
E ,
< x 1 < - l, y´ hr2 - x 2
- y´ hr2 q ´ h 2 4 ,
Ž 2.6 .
E x2
E s ´y1
E ti
,
i s 1, 2.
Ž 2.7 . Ž 2.2 .
and
V0 s xgR2 :
< x 1 < - l, y´ Ž hr2 y h 2 . - x 2
- y´ Ž hr2 y h 2 . q ´ 2 h 0 4 ,
Ž 2.3 .
where the quantities l and h i , i s 0, 1, 2, have the same order of magnitude, and ´ is regarded as a small non-dimensional positive parameter; we also use the notation h s h1 q h 2 . Thus, the upper and lower layers have the thickness ´ h1 and ´ h 2 , respectively, and the middle layer is relatively thin: its thickness is equal to ´ 2 h 0 . The
3. Anti-plane shear First, we study a boundary value problem for the Laplacian that corresponds to the case of an anti-plane shear of the layered structure. We consider different values of the normalized thickness and normalized shear modulus of the middle layer and show that the relation between these two small parameters is important. To make the final formulae simple we also assume here that the quantities h 0 , h1 and h 2 are the same for all three layers H [ h 0 s h1 s h 2 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
139
The displacement vector depends on x 1 , x 2 only, and has the form
value problems on the cross-section of the compound beam
u s Ž 0, 0, w Ž x 1 , x 2 . . ,
E 2 Wi Ž j.
Ž 3.1 .
where the function w satisfies the Poisson equations
E 2 Wi Ž0.
m i Dw Ž x 1 , x 2 . q f i Ž x 1 , x 2 . s 0, in V i ,
i s 0, 1, 2,
E t j2
Ž 3.2 .
and the homogeneous Neumann boundary conditions on the sides G "
Ew E x2
s 0 on G " ,
Ž 3.3 .
where m i are the shear moduli of the materials. Without loss of generality, it can be assumed here that f 0 ' 0. We shall use the superscript index notation w Ž j. to denote the displacement in the region V i . The interface contact conditions on S " have the form
m1
m2
E w Ž1. E x2 E w Ž1. E x2
s m0
s m0
E w Ž0. E x2 E w Ž0. E x2
sy
on Sq ,
on Sy .
w Ž x 1 , "l . s 0.
E x 12 Ž0. E 2 Wiy4
E x 12
y f j d i2
Ž 3.5 .
Ž 3.6 .
w Ž i. Ž x 1 , x 2 . ; W0Ž i. Ž x 1 . q ´ W1Ž i. Ž x 1 , t i .
Ž 3.7 .
and analyze the cases of different relations between the shear moduli of elastic layers:
Ž 3.8 .
where K is a positive integer, and the quantity m has the same order of magnitude as m 1 and m 2 . We could substitute Eq. Ž3.7. into Eqs. Ž3.2., Ž3.3., Ž3.4., Ž3.5. and Ž3.6. and equate the coefficients near like powers of the small parameter ´ . As a result, we obtain the recurrent system of boundary
in V j ,
in V 0 ,
Ž 3.9 . Ž 3.10 .
Ž 3.11 .
On the interface boundary we have the following contact conditions: Wi Ž1. s Wi Ž0. ,
Ž0. m 1 E t1Wi Ž1. s mE t 0WiyK
on Sq ,
Ž 3.12 . and Ž0. m 2 E t 1Wi Ž2. s mE t 0WiyK
on Sy ,
Ž 3.13 .
Ž 3.4 .
We seek the asymptotic approximation for the functions w Ž i. in the form
q ´ W2Ž i. Ž x 1 , t i . ,
Ž j. E 2 Wiy2
and the following boundary conditions E W Ž1. s 0 on Gq ; E t1 i E W Ž2. s 0 on Gy . E t2 i
Wi Ž2. s Wi Ž0. ,
The ends x 1 s "l of the compound beam are assumed to be fixed
m 0 s ´ Km ,
E t j2
sy
where the index K is the same as in Eq. Ž3.8.. Thus, we have the recurrent sequence of the Neumann boundary value problems for ordinary differential equations in the upper and lower layers Žthe longitudinal variable x 1 is included as a parameter. and the sequence of Dirichlet boundary value problems for the middle layer. The Neumann boundary value problems mentioned require certain solvability conditions for the right-hand sides of the equations and boundary conditions. It is verified directly that the solvability conditions for problems associated with W0Ž i. and W1Ž i., i s 1, 2, are satisfied identically, whereas the solvability conditions for problems with respect to W2Ž i. involve the second order derivatives of W0Ž i.. We consider the following cases. Ža. When m 0 s ´m W0Ž0. s W0Ž1. s W0Ž2. Ž 3.14 . Žwe shall, therefore, omit the upper index., the longitudinal displacement jump across the thin layer has the form ´ 2 dŽ x 1 . s ´ 2 ŽW2Ž1. y W2Ž2. ., where m1 m 2 H Hr2 d Ž x1 . s y Ž my1 f Ž x , t . m Ž m 1 q m 2 . yHr2 2 2 1
H
ymy1 1 f 1Ž x 1 , t . . d t
Ž 3.15 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
140
and
and the boundary conditions
E 12 W0 Ž x 1 . s
m
dŽ x1 . m1 H 2 1 Hr2 y f Ž x , t . d t. Ž 3.16 . Hm 1 yHr2 1 1 The discrepancy of order O Ž1. in the boundary conditions at x 1 s "l will be removed if W0 Ž "l . s 0. Ž 3.17 . Thus, the displacement jump across the thin layer is quite small in this case, and the leading order displacement W0 Ž x 1 . is the same in all three layers, and it satisfies the one-dimensional boundary value problem Ž3.16. and Ž3.17.. A similar situation may be observed when m 0 s ´ 2m2 Žhowever, the displacement jump will have the order O Ž ´ . instead of O Ž ´ 2 ... Žb. The most interesting case corresponds to the relation m 0 s ´ 3m. The displacement jump across the thin layer has the order O Ž1.. We shall use the notation d Ž x 1 . s W0Ž1. Ž x 1 . y W0Ž2. Ž x 1 . . Ž 3.18 . The displacements W0Ž1. and W0Ž2. satisfy the following equations, m E 12 W0Ž1. Ž x 1 . s dŽ x1 . m1 H 2 1 Hr2 y f Ž x , t. dt, Ž 3.19 . Hm 1 yHr2 1 1 m E 12 W0Ž2. Ž x 1 . s y dŽ x1 . m2 H 2 1 Hr2 y f Ž x , t . d t. Ž 3.20 . Hm 2 yHr2 2 1
H
d Ž "l . s 0.
Ž 3.23 .
The last case is interesting for engineering applications. In the next section we consider the state of plane strain of the laminated structure.
4. The state of plane strain Here we assume that the elastic materials of the regions V i , i s 0, 1, 2, are characterized by Young’s moduli Ei , i s 0, 1, 2, and by the values n i , i s 0, 1, 2, of the Poisson ratio. Assume that n i , i s 0, 1, 2, have the same order of magnitude for all three materials, and E0 s ´ 3 E,
Ž 4.1 .
where E is comparable with E1 and E2 . Eq. Ž4.1. indicates the softness of the adhesive material in V 0 . By l i , m i , i s 0, 1, 2, we denote the Lame´ constants of the elastic materials
li s
Ei n i
Ž 1 q ni . Ž 1 y 2ni .
,
mi s
Ei 2Ž 1 q n i .
.
H
H
As before, to remove the leading order discrepancy in the boundary conditions at the ends x 1 s "l we set W0Ž i. Ž "l . s 0, i s 1, 2. Ž 3.21 . Consequently, the displacement jump dŽ x 1 . satisfies the following second order differential equation m Ž m1 q m2 . E 12 d Ž x 1 . y d Ž x1 . m1 m2 H 2 1 sy
Hr2
H H yHr2
ž
f 1Ž x 1 , t .
m1
y
f2 Ž x1 , t .
m2
/
dt,
Ž 3.22 .
Also, we use the notations l, m for the normalized Lame´ constants related to the middle layer
ls
En 0
Ž 1 q n 0 . Ž 1 y 2n 0 .
,
ms
E 2Ž 1 q n 0 .
.
Consider the state of plane strain, and assume that the displacement vector has the form t
uŽ i. s Ž u1 Ž x 1 , x 2 . , u 2 Ž x 1 , x 2 . , 0 . , i s 0, 1, 2,
Ž 4.2 .
and the vectors uŽ i. satisfy the homogeneous Lame´ systems
m i= 2 uŽ i. Ž x . q Ž l i q m i . = = P uŽ i. Ž x . s 0,
x g V i , i s 0, 1, 2.
Ž 4.3 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
On the upper and lower surfaces of the compound region V´ we prescribe tractions:
m1
ž
E uŽ1. 2 E x1
m2
ž
E x1
E x2
q
E x2 E x2
E u1Ž1.
/
E x1
s ´ 3 p 2Ž1.
E u1Ž2. E x1
s ´ 3 p Ž2. 2
ž
s m0
ž
E
5. Asymptotic approach for the state of plane strain
Ž 4.5 .
Here we present an asymptotic analysis of the elasticity problem in the thin layered rectangle introduced in the previous section and derive ordinary differential equations including the leading order components of the displacement field.
on Gy .
E x1
E x2
/
E x2
i s 0, 1, 2.
If one substitutes formally this series into Eq. Ž4.3. and boundary conditions Eqs. Ž4.4., Ž4.5., Ž4.6. and Ž4.7., and equates the coefficients near like powers of ´ , then it yields the recurrent system of relations Žfor the sake of convenience we use the notations E t i s ErE t i , E 1 s ErE x 1 .
/
u1Ž0. ,
E x1
u1Ž1.
E
E uŽ0. 2 q l0
E x1
u1Ž0. ,
x g Sq , and
Ž 4.6 .
m i E t2i V 1Ž i , k . q Ž l i q m i . E t i E 1 V 2Ž i , ky1. q Ž l i q 2 m i . E 12 V 1Ž i , ky2. s 0,
Ž2.
Ž 5.2 .
Ž0.
u Ž x. su Ž x. , E Ž2. E Ž2. m2 u2 q u E x1 E x2 1
ž
s m0
x1 , ti . ,
Ž 5.1 .
E
E x2
Ý ´ k V Ži,k. Ž ´ , ks0
uŽ1. 2 q l1
s Ž 2 m 0 q l0 .
Assume that the displacement is approximated by `
E
Ž 2 m 1 q l1 .
5.1. Asymptotic procedure
uŽ i. Ž x . ;
E uŽ0. 2 q
Ž 4.8 .
Ž 4.4 .
We prescribe the special type of external load where the longitudinal external force is greater than the transversal one. The conditions of the ideal interface contact have the form uŽ1. Ž x . s uŽ0. Ž x . , E Ž1. E Ž1. m1 u2 q u E x1 E x2 1
x g V i , i s 0, 1, 2.
on Gq ,
s ´ 2 p 1Ž2. ,
q l2
The ends of a thin compound beam are assumed to be fixed Žclamped.: uŽ i. Ž "l, x 2 . s 0,
s ´ 2 p 1Ž1. ,
q l1
E u1Ž2. E uŽ2. 2
Ž 2 m 2 q l2 .
/
E x2 E uŽ1. 2
Ž 2 m 1 q l1 . E uŽ2. 2
E u1Ž1.
q
141
ž
E
Ž 2 m i q l i . E t2i V 2Ž i , k . q Ž l i q m i . E t i E 1 V 1Ž i , ky1.
/
q m i E 12 V 2Ž i , ky2. s 0 in V i ,
E
E x1
uŽ0. 2 q
E x2
Ž 2 m 2 q l2 .
E x2
mE t20 V 1Ž0, k . q Ž l q m . E t 0 E 1 V 2Ž0, ky2.
E uŽ2. 2 q l2
E x1
u1Ž1.
E s Ž 2 m 0 q l0 . x g Sy .
E x2
Ž 5.3 .
for the upper and lower layers, and
/
u1Ž0. ,
E
i s 1, 2,
q Ž l q 2 m . E 12 V 1Ž0, ky4. s 0,
Ž 5.4 .
E uŽ0. 2 q l0
E x1
Ž 2 m q l . E t20 V2Ž0, k . q Ž l q m . E t 0 E 1 V 1Ž0, ky2.
u1Ž0. ,
Ž 4.7 .
q mE 12 V 2Ž0, ky4. s 0,
in V 0 ,
Ž 5.5 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
142
for the middle layer. On the interface boundary
m 1 Ž E t 1 V 1Ž1, k . q E 1 V 2Ž1, ky1. . s m Ž E t 0 V 1Ž0, ky2. q E 1 V 2Ž0, ky4. . ,
Ž 5.6 .
Ž 2 m 1 q l1 . E t1 V 2Ž1, k . q l1 E 1 V 1Ž1, ky1.
V
Ž 5.7 .
Ž0, k .
Ž 5.8 .
sV
on Sq ,
E t2i V 2Ž i ,0. s 0,
V 2Ž i ,0. s V 2Ž i ,0. Ž x 1 . ,
Ž
V 2Ž0 ,0. s
.
s m Ž E t 0 V 1Ž0, ky2. q E 1 V 2Ž0, ky4. ,
Ž 5.9 .
Ž 2 m 2 q l2 . E t 2 V 2Ž2, k . q l2 E 1 V 1Ž2, ky1. s Ž 2 m q l. V
Ž0, k .
sV
Ž2, k .
E t 0 V 2Ž0 , ky2. q lE 1 V 1Ž0, ky4. , on Sy .
Ž 5.11 .
Ž 5.12 .
5.2. Differential equations of the compound beam We analyze first five terms of the asymptotic series ŽEq. Ž5.1... This will be sufficient to obtain a set of differential equations that constitute a wellposed system including the leading order component. It turns out that i s 0, 1, 2.
V 1Ž0,1. s V 1Ž i ,1.
i s 1, 2 on Gqj Gy .
From these equations we obtain
t 0 q h 0r2 h0
i s 1, 2,
Ž 5.16 .
d1Ž1. q V 1Ž2 ,1. Ž x 1 , h 2r2 . ,
Ž 5.17 .
Ž 5.13 .
In the next subsection we shall analyze the solvability of the problems above, and, in that way derive the ordinary differential equations for the leading order components of the displacement field.
V 2Ž i ,1. s 0,
i s 0, 1, 2,
and the corresponding boundary and interface conditions are
V 1Ž0 ,1. s
Ž 2 m 2 q l2 . E t 2 V 2Ž2, k . q l2 E 1 V 1Ž2, ky1.
V 1Ž i ,0. s 0,
Ž 5.15 .
where d 2Ž0. s V 2Ž1,0. y V 2Ž2,0.. Step 2: For k s 1, Eqs. Ž5.2. and Ž5.4. give
V 1Ž i ,1. s yt i E 1 V 2Ž i ,0. q Õ Ž i. Ž x 1 . ,
m 2 Ž E t 2 V 1Ž2, k . q E 1 V 2Ž2 , ky1. . s d k 3 p 1Ž2. ,
on Gy .
d 2Ž0. Ž x 1 . q V 2Ž2 ,0. Ž x 1 . ,
E t i V 1Ž i ,1. s yE 1 V 2Ž i ,0. ,
Ž 2 m 1 q l1 . E t1 V 2Ž1, k . q l1 E 1 V 1Ž1, ky1.
s d k 4 p 2Ž2.
Ž 5.14 .
i s 1, 2 on Sqj Sy ,
m 1 Ž E t 1 V 1Ž1, k . q E 1 V 2Ž1, ky1. . s d k 3 p 1Ž1. ,
on Gq ,
h0
E t i V 1Ž i ,1. s yE 1 V 2Ž i ,0. ,
On the upper and lower surface we have
s d k 4 p 2Ž1.
t 0 q h 0r2
E t2i V 1Ž i ,1. s 0,
Ž 5.10 .
i s 1, 2,
and
and
m 2 E t 2 V 1Ž2, k . q E 1 V 2Ž2 , ky1.
i s 0, 1, 2.
Taking into account the boundary and interface conditions we obtain
s Ž 2 m q l . E t 0 V 2Ž0 , ky2. q lE 1 V 1Ž0, ky4. , Ž1, k .
Similar assumptions for V 1Ž i,2. and V 2Ž i,3. will be made in steps 3 and 4. Step 1: First, for k s 0, Eqs. Ž5.3. and Ž5.5. yield
with d1Ž1. s V 1Ž1,1. Ž x 1 , yh1r2 . y V 1Ž2,1. Ž x 1 , h 2r2 . , and Õ Ž i. Ž x 1 . being some sufficiently smooth functions. Step 3: From Eqs. Ž5.2., Ž5.3., Ž5.4., Ž5.5., Ž5.6., Ž5.7., Ž5.8., Ž5.9., Ž5.10., Ž5.11., Ž5.12. and Ž5.13. for k s 2 one concludes that V 1Ž i,2., i s 1, 2 are functions of x 1 only. We make the assumption that these functions are zero, i.e. V 1Ž i ,2. s 0,
i s 1, 2.
These assumptions are comparable with the ones made in the first paragraph of this subsection.
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
Next, one concludes, using the results derived in steps 1 and 2, that the following one-dimensional Neumann problem holds for V 2Ž1,2. :
E t21 V 2Ž1 ,2. s
l1
Ž 2 m 1 q l1 .
E 12 V 2Ž1 ,0. ,
Ž 5.18 .
with the boundary conditions
E t1 V 2Ž1 ,2. s y
l1
h1
Ž 2 m 1 q l1 .
2
q
E t1V2Ž1 ,2. s y
2 m q l d 2Ž0. 2 m 1 q l1 h 0
l1
y
Ž 2 m 1 q l1 .
E 12 V 2Ž1 ,0. q E 1Õ Ž1.
on Sq , h1 2
143
the previous steps, that V 2Ž i,3. s 0, i s 1, 2, and that the functions V 1Ž i,3. satisfy the following equations 3 l i q 4m i E t2i V 1Ž i ,3. s t E 3 V Ž0. y E 12 Õ Ž i. , 2 mi q li i 1 2 i s 1, 2. Ž 5.25 . Ž1,3. The boundary conditions for the function V 1 are given by m 1 m 1 E t 1 V 1Ž1,3. s Ž h q h 2 . E 1 V 2Ž0. q Õ Ž1. y Õ Ž2. h0 2 1 y
Ž 5.19 .
m 1 l1
h12
2 m 1 q l1
8
E 13 V 2Ž0. q
m 1 E t 1 V 1Ž1 ,3. s ym 1
Ž 5.20 .
Neumann problems generally do not have solutions unless the data satisfies some solvability condition. This becomes d 2Ž0. s 0.
Ž 5.21 .
Further we shall use the notation V 2Ž0. ' V 2Ž0,0. s V 2Ž1 ,0. s V 2Ž2,0. .
t i2
2 mi q li
2
y
h1 2
l1 2 m 1 q l1
8
E 13 V 2Ž0.
E 12 Õ Ž1. q p 1Ž1. on Gq .
qÕ Ž1. y Õ Ž2. y
h0
p 1Ž1.
m1
,
Ž 5.28 . V 1Ž1,3.
E 12 V 2Ž0. y t i E 1Õ Ž i.
,
i s 1, 2,
where we also have written the corresponding equation for i s 2. Finally in step 3 from Eqs. Ž5.2., Ž5.3., Ž5.4., Ž5.5., Ž5.6., Ž5.7., Ž5.8., Ž5.9., Ž5.10., Ž5.11., Ž5.12. and Ž5.13. for k s 2, we obtain t 0 q h 0r2
Ž 5.27 .
The solvability condition for the problem Eqs. Ž5.25., Ž5.26. and Ž5.27. has the form 4 Ž l1 q m 1 . m 1 h1 E 12 Õ Ž1. s Ž h q h 2 . E 1 V 2Ž0. 2 m 1 q l1 m1 h0 2 1
Ž 5.22 .
Ž 5.23 .
V 2Ž0 ,2. s
E 12 Õ Ž1.
Ž 5.26 . h12
Also, the solution of problem Eqs. Ž5.18., Ž5.19. and Ž5.20. with d 2Ž0. s 0 is
li
2
on Sq ,
E 12 V 2Ž1 ,0. q E 1Õ Ž1.
on Gq .
V 2Ž i ,2. s
h1
Ž V 2Ž1 ,2. Ž x 1 , yh1r2.
V 2Ž2,2. Ž x 1 , h 2r2 . . q V 2Ž2,2. Ž x 1 , h 2r2 . . yV Ž 5.24 . Step 4: Here we shall derive second order differential equations for the functions Õ Ž i. first introduced in Eq. Ž5.16.. When k s 3 one can show, similarly to
and the function has the following representation 3 l1 q 4m 1 t 13 3 Ž0. t 12 2 Ž1. V 1Ž1 ,3. s E V y E1 Õ 2 m 1 q l1 6 1 2 2 y
4 Ž l1 q m 1 . 2 m 1 q l1
q t1
t1
h12 8
E 13 V 2Ž0. y
h1 2
E 12 Õ Ž1.
p 1Ž1.
. Ž 5.29 . m1 In a similar way, we derive for i s 2 that 4 Ž l2 q m 2 . h E 2 Õ Ž2. 2 m 2 q l2 2 1 m 1 sy Ž h q h 2 . E 1 V 2Ž0. q Õ Ž1. y Õ Ž2. m2 h0 2 1 q
p 1Ž2.
m2
,
Ž 5.30 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
144
and V 1Ž2 ,3. s
3 l2 q 4m 2 t 23 2 m 2 q l2 y
6
4 Ž l2 q m 2 . 2 m 2 q l2
y t1
p 1Ž2.
m2
E 13 V 2Ž0. y
t1
h22 8
t 22 2
the ends of the composite beam gives the boundary conditions 1
E 12 Õ Ž2.
E 13 V 2Ž0. y
h2 2
E 12 Õ Ž2.
.
Ž 5.31 .
Eqs. Ž5.28. and Ž5.30. define Õ Ž1. and Õ Ž2., provided V 2Ž0. is known. The next step will give an equation for the leading part of the displacement. Step 5: Here we obtain the fourth order differential equation for the function V 2Ž0.. Consider the case k s 4. The functions V 2Ž i,4., i s 1, 2, satisfy the following equations
Ž
2 m i q l i E t2i V 2Ž i ,4. q
.
l i q m i E t i E 1 V 1Ž i ,3.
Ž
q m i E 2 V 2Ž i ,2. s 0,
.
in V i , i s 1, 2,
Ž 5.32 .
and the boundary conditions
Ž 2 m i q l i . E t i V 2Ž i ,4. q l i E 1 V 1Ž i ,3. s Ž 2 m q l . E t 0 V 2Ž0 ,2.
on S " , i s 1, 2,
Ž 5.33 .
Ž 2 m i q l i . E t i V 2Ž i ,4. q l i E 1 V 1Ž i ,3. s p 2Ž i. on G " ,
i s 1, 2.
Ž 5.34 .
Using the representations of V 1Ž i,3., V 2Ž i,2. in terms of V 2Ž0., one derives the solvability condition for Eqs. Ž5.32., Ž5.33. and Ž5.34.: 1 3
½
m 1 Ž m 1 q l1 . 2 m 1 q l1
q2
½
h13 q
m 2 Ž l2 q m 2 . 2 m 2 q l2
m 2 Ž m 2 q l2 . 2 m 2 q l2
h22 E 13 Õ Ž2. y
5
h32 E 14 V 2Ž0.
m 1 Ž m 1 q l1 . 2 m 1 q l1
V 2Ž0. Ž "l . s E 1 V 2Ž0. Ž "l . s 0,
Ž 5.36 .
Õ Ž i. Ž "l . s 0,
Ž 5.37 .
i s 1, 2.
We note that for the present relation between the orders of magnitude of the longitudinal and transversal components of external load the Eq. Ž5.35. includes both components of the displacement and the coupling between the longitudinal and transversal modes is explicit.
6. Conclusion We have demonstrated an asymptotic procedure which shows to be an efficient tool for derivation of equations of compound structures with thin and soft interface layers. In particular, the asymptotic approach discussed in Section 5 of the paper did not involve any of the intuitive assumptions typically made in a strength-of-materials approach. It is based on the rigorous asymptotic analysis of the Lame´ system in thin regions. This asymptotic algorithm can be easily extended Žwe regard this as a technical exercise. to the case where the thicknesses of the layers depend on the longitudinal variable, to the case of orthotropic materials and to problems involving plates with layered structure.
Acknowledgements We would like to thank Professors V.G. Maz’ya and V.A. Kozlov for fruitful discussions and valuable comments. Financial support of Saab Military Aircraft is gratefully acknowledged.
5
Ž2. Ž1. Ž2. =h12 E 13 Õ Ž1. s p Ž1. 2 y p 2 q h1 E 1 p 1 q h 2 E 1 p 1 ,
Ž 5.35 . where the functions Õ Ž i. satisfy the second order differential Eqs. Ž5.28. and Ž5.30.. The clamping at
1
It should be mentioned that in the general case the boundary layer is to be constructed near the ends of the beam, and conditions of exponential decay of the boundary layer provide the boundary data for V 2Ž0., E 1 V 2Ž0., Õ Ž1. and Õ Ž2. at x 1 s" l.
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
References Arutyunyan, N.Kh., Movchan, A.B., Nazarov, S.A., 1987. Behaviour of solutions of elasticity problems in noncompact domains with parabolic and cylindrical inclusions or cavities. Adv. Mech. 10 Ž4., 3–91. Ciarlet, P.G., 1990. Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis. Masson, Paris. Klarbring, A., 1991. Derivation of a model of adhesively bonded joints by the asymptotic expansion method. Int. J. Eng. Sci. 29 Ž4., 493–512. Maz’ya, V.G., Nazarov, S.A., Plamenevskii, B.A., 1991. Asymptotische Theorie Elliptischer Randwertanfgaben in Singular ¨ Gestorten Gebieten. Akademie Verlag, Berlin, B.1, 1991; B.2, ¨ 1992. Movchan, A.B., Nazarov, S.A., 1988. Stress–strain state of a plane with a thin elastic inclusion. Izv. AN SSSR, Mekh. Tv. Tela ŽMech. Solids. 22 Ž1., 75–83. Nayfeh, A.H., 1975. A continuum mixture theory of heat conduction in laminated composites. J. Appl. Mech. 42 Ž2., 399–404. Nayfeh, A.H., 1980. Continuum modelling of low frequency heat conduction in laminated composites with bonds. J. Heat Transfer 102, 312–319.
145
Nayfeh, A.H., Nassar, E.A., 1978. Simulation of the influence of bonding materials on the dynamic behaviour of laminated wave guides. J. Appl. Mech. 45, 822–828. Nayfeh, A.H., Nassar, E.A., 1982. The influence of bonding agents on the thermomechanically induced interfacial stresses in composite materials. Fibre Sci. Technol. 16, 157–174. Nazarov, S.A., 1983. The structure of solutions of elliptic boundary value problems in slender domains. Vestnik Leningr. Univ., Ser. I, No 15, pp. 99–104. Nazarov, S.A., 1983. Introduction to asymptotic methods of the theory of elasticity. Lecture notes. Leningr. Univ. Press. Rogers, T.G., Watson, P., Spencer, A.J.M., 1995. Exact three-dimensional elasticity solutions for bending of moderately thick inhomogeneous and laminated strips under normal pressure. Int. J. Solids Struct. 32 Ž12., 1659–1673. Spencer, A.J.M., 1990. Three-dimensional elasticity solutions for stretching of inhomogeneous and laminated plates. In: Eason, G., Ogden, R.W. ŽEds.., Elasticity, Math. Methods and Applications, pp. 347–356. Spencer, A.J.M., Watson, P., Rogers, T.G., 1993. Exact theory of heterogeneous anisotropic elastic plates. Mater. Sci. Forum 123–125, 235–244.
Asymptotic modelling of adhesive joints A. Klarbring a , A.B. Movchan b
b,)
a Linkoping UniÕersity, Linkoping, Sweden ¨ ¨ School of Mathematical Sciences, UniÕersity of Bath, Bath BA2 7AY, UK
Received 19 November 1996; revised 9 December 1996
Abstract An asymptotic approach is presented for modelling of adhesive joints in a layered structure. We consider the case when two elastic layers are bonded together by a thin and soft layer of glue. Differential equations for the displacements Žincluding the displacement jump across the thin interface layer. in a compound thin beam are derived. q 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction Asymptotic and numerical modelling of fields in inhomogeneous layered structures is a subject of great interest for many researchers in engineering and mathematics. On the one hand, the work in this direction is motivated by a number of important practical questions. On the other hand, the mathematical properties of the operators of boundary value problems and their solutions attract great attention of applied mathematicians. We would like to refer to the papers by Nayfeh Ž1975, 1980. on heat conduction in laminated composites with bonds, and the papers by Nayfeh and Nassar Ž1978, 1982. on propagation of waves in elastic laminated structures and analysis of interfacial stresses. These papers illustrate very clearly the range of important engineering applications related to the analysis of boundary value problems in laminated composites. Also, we would like to cite the more recent paper by Spencer Ž1990., and the work by Spencer et al. )
Corresponding author.
Ž1993. and by Rogers et al. Ž1995., where the authors analyze three-dimensional elasticity solutions for stretching and bending of inhomogeneous laminated plates. In some particular cases of applied load exact solutions were obtained for normal loading of inhomogeneous and laminated anisotropic elastic plates. The main objective of the present paper is to derive a mathematical model for an adhesiÕe joint in a thin compound beam with a layered structure. To be specific, we assume that the beam includes three layers, with the middle one being infinitesimally thin and soft in comparison with the two others. This situation corresponds, for example, to the presence of an adhesive joint when two elastic layers are bonded together by a thin and soft layer of glue. We note that the formulation includes two small parameters, and different relations between these parameters would lead to different governing equations for the layered structure; in addition, we mention that formally this problem is classified as a singularly perturbed one. Engineers in the aircraft industry Žmodelling of the wing structure. and the car industry Žmodelling of a windscreen. face a question related
0167-6636r98r$19.00 q 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 Ž 9 7 . 0 0 0 4 5 - 8
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
138
to accurate evaluation of the displacement jump across the thin interface layer. It turns out that numerically this is a challenging problem, and the standard finite element packages fail to provide the required accuracy. A problem on the modelling of adhesive joints was considered by Klarbring Ž1991., where an asymptotic approach was based on a variational formulation developed by Ciarlet Ž1990.. In that work the elements ‘glued together’ were treated as threedimensional bodies. In large finite element calculations this is frequently not a satisfactory situation: engineers would rather prefer to model the adherents by a two-dimensional theory, for instance, Kirchhoff’s plate theory. In the present text we resolve the difficulty by deriving a simple and accurate mathematical model for a layered structure that includes a thin and soft adhesive layer. The asymptotic approach, developed in this paper, is based on the results by Nazarov Ž1983a,b., Arutyunyan et al. Ž1987., Movchan and Nazarov Ž1988., Maz’ya et al. Ž1991..
interface boundary includes two parts Sq and Sy specified by < x 1 < - l, x 2 s y´ Ž hr2 y h 2 . q ´ 2 h 0 4 ,
Sqs x : and Sys x :
< x 1 < - l, x 2 s y´ Ž hr2 y h 2 . 4 .
The upper and lower surfaces of the compound region are
Gqs x :
< x 1 < - l, x 2 s ´ 2 h 0 q ´ hr2 4 ,
Gys x :
< x 1 < - l, x 2 s y´ hr2 4 .
The layers V 1 and V 2 are adhesively joined by the thin layer V 0 , which is made of an adhesive material that is much softer than the material of the other two layers. Introduce the variables t 0 s ´y2 Ž x 2 q ´ Ž hr2 y h 2 . y ´ 2 h 0r2 . ,
Ž 2.4 .
t 1 s ´y1 Ž x 2 y ´ 2 h 0 y ´ h 2r2 . ,
Ž 2.5 .
and t 2 s ´y1 Ž x 2 q ´ h1r2 . . In this case
2. The layered structure
t i g w yh ir2, h ir2 x , Consider a thin rectangular domain V´ , which consists of three layers:
V1 s x g R2 :
< x 1 < - l, ´ Ž hr2 y h1 . q ´ 2 h 0 - x 2
- ´ hr2 q ´ 2 h 0 4 ,
V2 s xgR2 :
Ž 2.1 .
i s 1, 2;
t 0 g w yh 0r2, h 0r2 x , within V i, i s 0, 1, 2, and one can see that
E E x2
E s ´y2
E t0
E ,
< x 1 < - l, y´ hr2 - x 2
- y´ hr2 q ´ h 2 4 ,
Ž 2.6 .
E x2
E s ´y1
E ti
,
i s 1, 2.
Ž 2.7 . Ž 2.2 .
and
V0 s xgR2 :
< x 1 < - l, y´ Ž hr2 y h 2 . - x 2
- y´ Ž hr2 y h 2 . q ´ 2 h 0 4 ,
Ž 2.3 .
where the quantities l and h i , i s 0, 1, 2, have the same order of magnitude, and ´ is regarded as a small non-dimensional positive parameter; we also use the notation h s h1 q h 2 . Thus, the upper and lower layers have the thickness ´ h1 and ´ h 2 , respectively, and the middle layer is relatively thin: its thickness is equal to ´ 2 h 0 . The
3. Anti-plane shear First, we study a boundary value problem for the Laplacian that corresponds to the case of an anti-plane shear of the layered structure. We consider different values of the normalized thickness and normalized shear modulus of the middle layer and show that the relation between these two small parameters is important. To make the final formulae simple we also assume here that the quantities h 0 , h1 and h 2 are the same for all three layers H [ h 0 s h1 s h 2 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
139
The displacement vector depends on x 1 , x 2 only, and has the form
value problems on the cross-section of the compound beam
u s Ž 0, 0, w Ž x 1 , x 2 . . ,
E 2 Wi Ž j.
Ž 3.1 .
where the function w satisfies the Poisson equations
E 2 Wi Ž0.
m i Dw Ž x 1 , x 2 . q f i Ž x 1 , x 2 . s 0, in V i ,
i s 0, 1, 2,
E t j2
Ž 3.2 .
and the homogeneous Neumann boundary conditions on the sides G "
Ew E x2
s 0 on G " ,
Ž 3.3 .
where m i are the shear moduli of the materials. Without loss of generality, it can be assumed here that f 0 ' 0. We shall use the superscript index notation w Ž j. to denote the displacement in the region V i . The interface contact conditions on S " have the form
m1
m2
E w Ž1. E x2 E w Ž1. E x2
s m0
s m0
E w Ž0. E x2 E w Ž0. E x2
sy
on Sq ,
on Sy .
w Ž x 1 , "l . s 0.
E x 12 Ž0. E 2 Wiy4
E x 12
y f j d i2
Ž 3.5 .
Ž 3.6 .
w Ž i. Ž x 1 , x 2 . ; W0Ž i. Ž x 1 . q ´ W1Ž i. Ž x 1 , t i .
Ž 3.7 .
and analyze the cases of different relations between the shear moduli of elastic layers:
Ž 3.8 .
where K is a positive integer, and the quantity m has the same order of magnitude as m 1 and m 2 . We could substitute Eq. Ž3.7. into Eqs. Ž3.2., Ž3.3., Ž3.4., Ž3.5. and Ž3.6. and equate the coefficients near like powers of the small parameter ´ . As a result, we obtain the recurrent system of boundary
in V j ,
in V 0 ,
Ž 3.9 . Ž 3.10 .
Ž 3.11 .
On the interface boundary we have the following contact conditions: Wi Ž1. s Wi Ž0. ,
Ž0. m 1 E t1Wi Ž1. s mE t 0WiyK
on Sq ,
Ž 3.12 . and Ž0. m 2 E t 1Wi Ž2. s mE t 0WiyK
on Sy ,
Ž 3.13 .
Ž 3.4 .
We seek the asymptotic approximation for the functions w Ž i. in the form
q ´ W2Ž i. Ž x 1 , t i . ,
Ž j. E 2 Wiy2
and the following boundary conditions E W Ž1. s 0 on Gq ; E t1 i E W Ž2. s 0 on Gy . E t2 i
Wi Ž2. s Wi Ž0. ,
The ends x 1 s "l of the compound beam are assumed to be fixed
m 0 s ´ Km ,
E t j2
sy
where the index K is the same as in Eq. Ž3.8.. Thus, we have the recurrent sequence of the Neumann boundary value problems for ordinary differential equations in the upper and lower layers Žthe longitudinal variable x 1 is included as a parameter. and the sequence of Dirichlet boundary value problems for the middle layer. The Neumann boundary value problems mentioned require certain solvability conditions for the right-hand sides of the equations and boundary conditions. It is verified directly that the solvability conditions for problems associated with W0Ž i. and W1Ž i., i s 1, 2, are satisfied identically, whereas the solvability conditions for problems with respect to W2Ž i. involve the second order derivatives of W0Ž i.. We consider the following cases. Ža. When m 0 s ´m W0Ž0. s W0Ž1. s W0Ž2. Ž 3.14 . Žwe shall, therefore, omit the upper index., the longitudinal displacement jump across the thin layer has the form ´ 2 dŽ x 1 . s ´ 2 ŽW2Ž1. y W2Ž2. ., where m1 m 2 H Hr2 d Ž x1 . s y Ž my1 f Ž x , t . m Ž m 1 q m 2 . yHr2 2 2 1
H
ymy1 1 f 1Ž x 1 , t . . d t
Ž 3.15 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
140
and
and the boundary conditions
E 12 W0 Ž x 1 . s
m
dŽ x1 . m1 H 2 1 Hr2 y f Ž x , t . d t. Ž 3.16 . Hm 1 yHr2 1 1 The discrepancy of order O Ž1. in the boundary conditions at x 1 s "l will be removed if W0 Ž "l . s 0. Ž 3.17 . Thus, the displacement jump across the thin layer is quite small in this case, and the leading order displacement W0 Ž x 1 . is the same in all three layers, and it satisfies the one-dimensional boundary value problem Ž3.16. and Ž3.17.. A similar situation may be observed when m 0 s ´ 2m2 Žhowever, the displacement jump will have the order O Ž ´ . instead of O Ž ´ 2 ... Žb. The most interesting case corresponds to the relation m 0 s ´ 3m. The displacement jump across the thin layer has the order O Ž1.. We shall use the notation d Ž x 1 . s W0Ž1. Ž x 1 . y W0Ž2. Ž x 1 . . Ž 3.18 . The displacements W0Ž1. and W0Ž2. satisfy the following equations, m E 12 W0Ž1. Ž x 1 . s dŽ x1 . m1 H 2 1 Hr2 y f Ž x , t. dt, Ž 3.19 . Hm 1 yHr2 1 1 m E 12 W0Ž2. Ž x 1 . s y dŽ x1 . m2 H 2 1 Hr2 y f Ž x , t . d t. Ž 3.20 . Hm 2 yHr2 2 1
H
d Ž "l . s 0.
Ž 3.23 .
The last case is interesting for engineering applications. In the next section we consider the state of plane strain of the laminated structure.
4. The state of plane strain Here we assume that the elastic materials of the regions V i , i s 0, 1, 2, are characterized by Young’s moduli Ei , i s 0, 1, 2, and by the values n i , i s 0, 1, 2, of the Poisson ratio. Assume that n i , i s 0, 1, 2, have the same order of magnitude for all three materials, and E0 s ´ 3 E,
Ž 4.1 .
where E is comparable with E1 and E2 . Eq. Ž4.1. indicates the softness of the adhesive material in V 0 . By l i , m i , i s 0, 1, 2, we denote the Lame´ constants of the elastic materials
li s
Ei n i
Ž 1 q ni . Ž 1 y 2ni .
,
mi s
Ei 2Ž 1 q n i .
.
H
H
As before, to remove the leading order discrepancy in the boundary conditions at the ends x 1 s "l we set W0Ž i. Ž "l . s 0, i s 1, 2. Ž 3.21 . Consequently, the displacement jump dŽ x 1 . satisfies the following second order differential equation m Ž m1 q m2 . E 12 d Ž x 1 . y d Ž x1 . m1 m2 H 2 1 sy
Hr2
H H yHr2
ž
f 1Ž x 1 , t .
m1
y
f2 Ž x1 , t .
m2
/
dt,
Ž 3.22 .
Also, we use the notations l, m for the normalized Lame´ constants related to the middle layer
ls
En 0
Ž 1 q n 0 . Ž 1 y 2n 0 .
,
ms
E 2Ž 1 q n 0 .
.
Consider the state of plane strain, and assume that the displacement vector has the form t
uŽ i. s Ž u1 Ž x 1 , x 2 . , u 2 Ž x 1 , x 2 . , 0 . , i s 0, 1, 2,
Ž 4.2 .
and the vectors uŽ i. satisfy the homogeneous Lame´ systems
m i= 2 uŽ i. Ž x . q Ž l i q m i . = = P uŽ i. Ž x . s 0,
x g V i , i s 0, 1, 2.
Ž 4.3 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
On the upper and lower surfaces of the compound region V´ we prescribe tractions:
m1
ž
E uŽ1. 2 E x1
m2
ž
E x1
E x2
q
E x2 E x2
E u1Ž1.
/
E x1
s ´ 3 p 2Ž1.
E u1Ž2. E x1
s ´ 3 p Ž2. 2
ž
s m0
ž
E
5. Asymptotic approach for the state of plane strain
Ž 4.5 .
Here we present an asymptotic analysis of the elasticity problem in the thin layered rectangle introduced in the previous section and derive ordinary differential equations including the leading order components of the displacement field.
on Gy .
E x1
E x2
/
E x2
i s 0, 1, 2.
If one substitutes formally this series into Eq. Ž4.3. and boundary conditions Eqs. Ž4.4., Ž4.5., Ž4.6. and Ž4.7., and equates the coefficients near like powers of ´ , then it yields the recurrent system of relations Žfor the sake of convenience we use the notations E t i s ErE t i , E 1 s ErE x 1 .
/
u1Ž0. ,
E x1
u1Ž1.
E
E uŽ0. 2 q l0
E x1
u1Ž0. ,
x g Sq , and
Ž 4.6 .
m i E t2i V 1Ž i , k . q Ž l i q m i . E t i E 1 V 2Ž i , ky1. q Ž l i q 2 m i . E 12 V 1Ž i , ky2. s 0,
Ž2.
Ž 5.2 .
Ž0.
u Ž x. su Ž x. , E Ž2. E Ž2. m2 u2 q u E x1 E x2 1
ž
s m0
x1 , ti . ,
Ž 5.1 .
E
E x2
Ý ´ k V Ži,k. Ž ´ , ks0
uŽ1. 2 q l1
s Ž 2 m 0 q l0 .
Assume that the displacement is approximated by `
E
Ž 2 m 1 q l1 .
5.1. Asymptotic procedure
uŽ i. Ž x . ;
E uŽ0. 2 q
Ž 4.8 .
Ž 4.4 .
We prescribe the special type of external load where the longitudinal external force is greater than the transversal one. The conditions of the ideal interface contact have the form uŽ1. Ž x . s uŽ0. Ž x . , E Ž1. E Ž1. m1 u2 q u E x1 E x2 1
x g V i , i s 0, 1, 2.
on Gq ,
s ´ 2 p 1Ž2. ,
q l2
The ends of a thin compound beam are assumed to be fixed Žclamped.: uŽ i. Ž "l, x 2 . s 0,
s ´ 2 p 1Ž1. ,
q l1
E u1Ž2. E uŽ2. 2
Ž 2 m 2 q l2 .
/
E x2 E uŽ1. 2
Ž 2 m 1 q l1 . E uŽ2. 2
E u1Ž1.
q
141
ž
E
Ž 2 m i q l i . E t2i V 2Ž i , k . q Ž l i q m i . E t i E 1 V 1Ž i , ky1.
/
q m i E 12 V 2Ž i , ky2. s 0 in V i ,
E
E x1
uŽ0. 2 q
E x2
Ž 2 m 2 q l2 .
E x2
mE t20 V 1Ž0, k . q Ž l q m . E t 0 E 1 V 2Ž0, ky2.
E uŽ2. 2 q l2
E x1
u1Ž1.
E s Ž 2 m 0 q l0 . x g Sy .
E x2
Ž 5.3 .
for the upper and lower layers, and
/
u1Ž0. ,
E
i s 1, 2,
q Ž l q 2 m . E 12 V 1Ž0, ky4. s 0,
Ž 5.4 .
E uŽ0. 2 q l0
E x1
Ž 2 m q l . E t20 V2Ž0, k . q Ž l q m . E t 0 E 1 V 1Ž0, ky2.
u1Ž0. ,
Ž 4.7 .
q mE 12 V 2Ž0, ky4. s 0,
in V 0 ,
Ž 5.5 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
142
for the middle layer. On the interface boundary
m 1 Ž E t 1 V 1Ž1, k . q E 1 V 2Ž1, ky1. . s m Ž E t 0 V 1Ž0, ky2. q E 1 V 2Ž0, ky4. . ,
Ž 5.6 .
Ž 2 m 1 q l1 . E t1 V 2Ž1, k . q l1 E 1 V 1Ž1, ky1.
V
Ž 5.7 .
Ž0, k .
Ž 5.8 .
sV
on Sq ,
E t2i V 2Ž i ,0. s 0,
V 2Ž i ,0. s V 2Ž i ,0. Ž x 1 . ,
Ž
V 2Ž0 ,0. s
.
s m Ž E t 0 V 1Ž0, ky2. q E 1 V 2Ž0, ky4. ,
Ž 5.9 .
Ž 2 m 2 q l2 . E t 2 V 2Ž2, k . q l2 E 1 V 1Ž2, ky1. s Ž 2 m q l. V
Ž0, k .
sV
Ž2, k .
E t 0 V 2Ž0 , ky2. q lE 1 V 1Ž0, ky4. , on Sy .
Ž 5.11 .
Ž 5.12 .
5.2. Differential equations of the compound beam We analyze first five terms of the asymptotic series ŽEq. Ž5.1... This will be sufficient to obtain a set of differential equations that constitute a wellposed system including the leading order component. It turns out that i s 0, 1, 2.
V 1Ž0,1. s V 1Ž i ,1.
i s 1, 2 on Gqj Gy .
From these equations we obtain
t 0 q h 0r2 h0
i s 1, 2,
Ž 5.16 .
d1Ž1. q V 1Ž2 ,1. Ž x 1 , h 2r2 . ,
Ž 5.17 .
Ž 5.13 .
In the next subsection we shall analyze the solvability of the problems above, and, in that way derive the ordinary differential equations for the leading order components of the displacement field.
V 2Ž i ,1. s 0,
i s 0, 1, 2,
and the corresponding boundary and interface conditions are
V 1Ž0 ,1. s
Ž 2 m 2 q l2 . E t 2 V 2Ž2, k . q l2 E 1 V 1Ž2, ky1.
V 1Ž i ,0. s 0,
Ž 5.15 .
where d 2Ž0. s V 2Ž1,0. y V 2Ž2,0.. Step 2: For k s 1, Eqs. Ž5.2. and Ž5.4. give
V 1Ž i ,1. s yt i E 1 V 2Ž i ,0. q Õ Ž i. Ž x 1 . ,
m 2 Ž E t 2 V 1Ž2, k . q E 1 V 2Ž2 , ky1. . s d k 3 p 1Ž2. ,
on Gy .
d 2Ž0. Ž x 1 . q V 2Ž2 ,0. Ž x 1 . ,
E t i V 1Ž i ,1. s yE 1 V 2Ž i ,0. ,
Ž 2 m 1 q l1 . E t1 V 2Ž1, k . q l1 E 1 V 1Ž1, ky1.
s d k 4 p 2Ž2.
Ž 5.14 .
i s 1, 2 on Sqj Sy ,
m 1 Ž E t 1 V 1Ž1, k . q E 1 V 2Ž1, ky1. . s d k 3 p 1Ž1. ,
on Gq ,
h0
E t i V 1Ž i ,1. s yE 1 V 2Ž i ,0. ,
On the upper and lower surface we have
s d k 4 p 2Ž1.
t 0 q h 0r2
E t2i V 1Ž i ,1. s 0,
Ž 5.10 .
i s 1, 2,
and
and
m 2 E t 2 V 1Ž2, k . q E 1 V 2Ž2 , ky1.
i s 0, 1, 2.
Taking into account the boundary and interface conditions we obtain
s Ž 2 m q l . E t 0 V 2Ž0 , ky2. q lE 1 V 1Ž0, ky4. , Ž1, k .
Similar assumptions for V 1Ž i,2. and V 2Ž i,3. will be made in steps 3 and 4. Step 1: First, for k s 0, Eqs. Ž5.3. and Ž5.5. yield
with d1Ž1. s V 1Ž1,1. Ž x 1 , yh1r2 . y V 1Ž2,1. Ž x 1 , h 2r2 . , and Õ Ž i. Ž x 1 . being some sufficiently smooth functions. Step 3: From Eqs. Ž5.2., Ž5.3., Ž5.4., Ž5.5., Ž5.6., Ž5.7., Ž5.8., Ž5.9., Ž5.10., Ž5.11., Ž5.12. and Ž5.13. for k s 2 one concludes that V 1Ž i,2., i s 1, 2 are functions of x 1 only. We make the assumption that these functions are zero, i.e. V 1Ž i ,2. s 0,
i s 1, 2.
These assumptions are comparable with the ones made in the first paragraph of this subsection.
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
Next, one concludes, using the results derived in steps 1 and 2, that the following one-dimensional Neumann problem holds for V 2Ž1,2. :
E t21 V 2Ž1 ,2. s
l1
Ž 2 m 1 q l1 .
E 12 V 2Ž1 ,0. ,
Ž 5.18 .
with the boundary conditions
E t1 V 2Ž1 ,2. s y
l1
h1
Ž 2 m 1 q l1 .
2
q
E t1V2Ž1 ,2. s y
2 m q l d 2Ž0. 2 m 1 q l1 h 0
l1
y
Ž 2 m 1 q l1 .
E 12 V 2Ž1 ,0. q E 1Õ Ž1.
on Sq , h1 2
143
the previous steps, that V 2Ž i,3. s 0, i s 1, 2, and that the functions V 1Ž i,3. satisfy the following equations 3 l i q 4m i E t2i V 1Ž i ,3. s t E 3 V Ž0. y E 12 Õ Ž i. , 2 mi q li i 1 2 i s 1, 2. Ž 5.25 . Ž1,3. The boundary conditions for the function V 1 are given by m 1 m 1 E t 1 V 1Ž1,3. s Ž h q h 2 . E 1 V 2Ž0. q Õ Ž1. y Õ Ž2. h0 2 1 y
Ž 5.19 .
m 1 l1
h12
2 m 1 q l1
8
E 13 V 2Ž0. q
m 1 E t 1 V 1Ž1 ,3. s ym 1
Ž 5.20 .
Neumann problems generally do not have solutions unless the data satisfies some solvability condition. This becomes d 2Ž0. s 0.
Ž 5.21 .
Further we shall use the notation V 2Ž0. ' V 2Ž0,0. s V 2Ž1 ,0. s V 2Ž2,0. .
t i2
2 mi q li
2
y
h1 2
l1 2 m 1 q l1
8
E 13 V 2Ž0.
E 12 Õ Ž1. q p 1Ž1. on Gq .
qÕ Ž1. y Õ Ž2. y
h0
p 1Ž1.
m1
,
Ž 5.28 . V 1Ž1,3.
E 12 V 2Ž0. y t i E 1Õ Ž i.
,
i s 1, 2,
where we also have written the corresponding equation for i s 2. Finally in step 3 from Eqs. Ž5.2., Ž5.3., Ž5.4., Ž5.5., Ž5.6., Ž5.7., Ž5.8., Ž5.9., Ž5.10., Ž5.11., Ž5.12. and Ž5.13. for k s 2, we obtain t 0 q h 0r2
Ž 5.27 .
The solvability condition for the problem Eqs. Ž5.25., Ž5.26. and Ž5.27. has the form 4 Ž l1 q m 1 . m 1 h1 E 12 Õ Ž1. s Ž h q h 2 . E 1 V 2Ž0. 2 m 1 q l1 m1 h0 2 1
Ž 5.22 .
Ž 5.23 .
V 2Ž0 ,2. s
E 12 Õ Ž1.
Ž 5.26 . h12
Also, the solution of problem Eqs. Ž5.18., Ž5.19. and Ž5.20. with d 2Ž0. s 0 is
li
2
on Sq ,
E 12 V 2Ž1 ,0. q E 1Õ Ž1.
on Gq .
V 2Ž i ,2. s
h1
Ž V 2Ž1 ,2. Ž x 1 , yh1r2.
V 2Ž2,2. Ž x 1 , h 2r2 . . q V 2Ž2,2. Ž x 1 , h 2r2 . . yV Ž 5.24 . Step 4: Here we shall derive second order differential equations for the functions Õ Ž i. first introduced in Eq. Ž5.16.. When k s 3 one can show, similarly to
and the function has the following representation 3 l1 q 4m 1 t 13 3 Ž0. t 12 2 Ž1. V 1Ž1 ,3. s E V y E1 Õ 2 m 1 q l1 6 1 2 2 y
4 Ž l1 q m 1 . 2 m 1 q l1
q t1
t1
h12 8
E 13 V 2Ž0. y
h1 2
E 12 Õ Ž1.
p 1Ž1.
. Ž 5.29 . m1 In a similar way, we derive for i s 2 that 4 Ž l2 q m 2 . h E 2 Õ Ž2. 2 m 2 q l2 2 1 m 1 sy Ž h q h 2 . E 1 V 2Ž0. q Õ Ž1. y Õ Ž2. m2 h0 2 1 q
p 1Ž2.
m2
,
Ž 5.30 .
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
144
and V 1Ž2 ,3. s
3 l2 q 4m 2 t 23 2 m 2 q l2 y
6
4 Ž l2 q m 2 . 2 m 2 q l2
y t1
p 1Ž2.
m2
E 13 V 2Ž0. y
t1
h22 8
t 22 2
the ends of the composite beam gives the boundary conditions 1
E 12 Õ Ž2.
E 13 V 2Ž0. y
h2 2
E 12 Õ Ž2.
.
Ž 5.31 .
Eqs. Ž5.28. and Ž5.30. define Õ Ž1. and Õ Ž2., provided V 2Ž0. is known. The next step will give an equation for the leading part of the displacement. Step 5: Here we obtain the fourth order differential equation for the function V 2Ž0.. Consider the case k s 4. The functions V 2Ž i,4., i s 1, 2, satisfy the following equations
Ž
2 m i q l i E t2i V 2Ž i ,4. q
.
l i q m i E t i E 1 V 1Ž i ,3.
Ž
q m i E 2 V 2Ž i ,2. s 0,
.
in V i , i s 1, 2,
Ž 5.32 .
and the boundary conditions
Ž 2 m i q l i . E t i V 2Ž i ,4. q l i E 1 V 1Ž i ,3. s Ž 2 m q l . E t 0 V 2Ž0 ,2.
on S " , i s 1, 2,
Ž 5.33 .
Ž 2 m i q l i . E t i V 2Ž i ,4. q l i E 1 V 1Ž i ,3. s p 2Ž i. on G " ,
i s 1, 2.
Ž 5.34 .
Using the representations of V 1Ž i,3., V 2Ž i,2. in terms of V 2Ž0., one derives the solvability condition for Eqs. Ž5.32., Ž5.33. and Ž5.34.: 1 3
½
m 1 Ž m 1 q l1 . 2 m 1 q l1
q2
½
h13 q
m 2 Ž l2 q m 2 . 2 m 2 q l2
m 2 Ž m 2 q l2 . 2 m 2 q l2
h22 E 13 Õ Ž2. y
5
h32 E 14 V 2Ž0.
m 1 Ž m 1 q l1 . 2 m 1 q l1
V 2Ž0. Ž "l . s E 1 V 2Ž0. Ž "l . s 0,
Ž 5.36 .
Õ Ž i. Ž "l . s 0,
Ž 5.37 .
i s 1, 2.
We note that for the present relation between the orders of magnitude of the longitudinal and transversal components of external load the Eq. Ž5.35. includes both components of the displacement and the coupling between the longitudinal and transversal modes is explicit.
6. Conclusion We have demonstrated an asymptotic procedure which shows to be an efficient tool for derivation of equations of compound structures with thin and soft interface layers. In particular, the asymptotic approach discussed in Section 5 of the paper did not involve any of the intuitive assumptions typically made in a strength-of-materials approach. It is based on the rigorous asymptotic analysis of the Lame´ system in thin regions. This asymptotic algorithm can be easily extended Žwe regard this as a technical exercise. to the case where the thicknesses of the layers depend on the longitudinal variable, to the case of orthotropic materials and to problems involving plates with layered structure.
Acknowledgements We would like to thank Professors V.G. Maz’ya and V.A. Kozlov for fruitful discussions and valuable comments. Financial support of Saab Military Aircraft is gratefully acknowledged.
5
Ž2. Ž1. Ž2. =h12 E 13 Õ Ž1. s p Ž1. 2 y p 2 q h1 E 1 p 1 q h 2 E 1 p 1 ,
Ž 5.35 . where the functions Õ Ž i. satisfy the second order differential Eqs. Ž5.28. and Ž5.30.. The clamping at
1
It should be mentioned that in the general case the boundary layer is to be constructed near the ends of the beam, and conditions of exponential decay of the boundary layer provide the boundary data for V 2Ž0., E 1 V 2Ž0., Õ Ž1. and Õ Ž2. at x 1 s" l.
A. Klarbring, A.B. MoÕchanr Mechanics of Materials 28 (1998) 137–145
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