Journal of Elasticity53: 23–35, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

23

Bonded Joints with Nonhomogeneous Adhesives

STEFANO LENCIIstituto di Scienza e Tecnica della Costruzioni, Università di Ancona, Via Brecce Bianche,Monte D’Ago, 60131, Ancona, Italy?

Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie (Paris 6), Tour 66,Case 162, 4 Place Jussieu, 75252 Paris Cedex 05, France

Received: 2 October 1998

Abstract. The problem is considered of two semi-infinite elastic plane regions bonded by a non-homogeneous adhesive along their common linear boundary and subject to constant applied stressesat infinity. The complex variable method has been employed to achieve the closed-form solution forgeneric interface stiffness distributions. Two particular cases of practical interest have been treatedin detail as particular applications of the proposed general solution.

Mathematical Subject Classifications (1991):30E20, 45E05, 73C02.

Key words: nonhomogeneous interfaces, plane elasticity, complex variable method.

1. Introduction

Let us consider two elastic bodies�+ and�− joined along a common surfaceSby a thin layer of a third elastic material and subject to given surface tractionsgon the remaining parts of their boundaries (Figure 1). According to the commontechnical terminology,�+ and�− are calledadherents, the interface body is calledadhesive, or glue, and the two of them together are called abonded joint[1].

In the case illustrated in Figure 1, the stresses and displacements are the solu-tions of a classical elasticity problem. Although it is well-known that a uniquesolution exists, usually its practical determination is quite difficult, even if numer-ical methods are employed. This is due to the fact that the thinness of the adhesiverequires a fine mesh, which in turn implies an increase in the degrees of freedom ofthe system. Furthermore, the adhesive is usually more flexible than the adherents,and this produces numerical instabilities of the stiffness matrix.

To overcome the previous difficulties, some simplified models have been de-veloped in the past, aimed at characterizing the mechanical behaviour of the bondedjoint. One of the most widely used is the so calledspring-layer, which consists insubstituting the adhesive with a weak interface, i.e., with a continuous distributionof elastic spring of adequate stiffness. This model, initially proposed by Golandand Reissner [2] and Volkersen [3], has been deeply studied in the technical and

? Address for correspondence

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24 STEFANO LENCI

Figure 1. An example of the bonded joint.

mathematical literature (see, for example, [4–6]) and it has been applied also toother fields of engineering, such as, for example, to the problem of elastic wavespropagation across ground faults [7], to the determination of the homogenizedbehaviour of composites [8], to the study of spherical inclusions embedded in aninfinite elastic matrix [9] and to the analysis of the interaction between a smallcircular defect and a crack [10].

In the described approximation, the adhesive is treated as a material surface,which is mechanically represented by the elastic energy of the springs (the energyof adhesion). Furthermore, the interface stresses are continuous (by equilibrium)but a jump in the interface displacements is allowed.

The stiffness of the springs is related to the thickness and to the elastic coeffi-cients of the glue by the relation [11]

K = 1

d

E

2(1+ ν)

1 0 0

0 1 0

0 0 21−ν1−2ν

, (1)

whered,E andν are, respectively, the thickness, the Young modulus and the Pois-son coefficient of the adhesive andK is the matrix of proportionality between thestressσe3 = [σ1, σ2, σ3]T and the displacement gapu+ − u− = [u+1 − u−1 , u+2 −u−2 , u

+3 − u−3 ]T at the interface:σe3 = K (u+ − u−). Equation (1) holds in the

case of a flat, homogeneous and isotropic adhesive belonging to the(x1, x2)-plane.More general relations, relaxing all the previous assumptions, are given elsewhere[11].

The mechanical properties of the bonded joint are influenced by several vari-ables. Among the others, however, an important role seems to be played by thesurface of adhesion, which can be shaped to increase the performances of thejoint, and by the stiffness and thickness of the adhesive. In this work we consider

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BONDED JOINTS WITH NONHOMOGENEOUS ADHESIVES 25

Figure 2. The simplified model.

the typical case of a flat surface of adherence and we deal with the problem ofnonconstantK (x1, x2).

Usually, unavoidable imperfections occurring in the creation of the layer pro-duce undesired local strong variations of the adhesive stiffness. Thus, as a rule, theoverall stiffness and strength of the connection are reduced with respect to the caseof constantK , and this phenomenon may be dangerous if not adequately taken intoaccount. Sometimes, however, the disuniformity ofK can be willfully designedto fulfill some technical constraints, as, for example, in the case of the lap joint,where the corners of the adherents are rounded to reduce the stress singularitiesand to increase the strength of the joint [12].

In this paper a simplified model is studied. It is assumed that�+ and�− are twohalf-planes joined along the common boundaryy = 0 (Figure 2). The adherentsare made of the same homogenous and isotropic material, characterized by shearmodulusµ and by Poisson coefficientν, while the glue is orthotropic with normaland tangential stiffnesskN(x) andkT (x), respectively. The nonhomogeneous char-acter ofK is summarized by the dependence of its diagonal componentskN(x) andkT (x) onx.

The considered model permits to determine the main effects due to the vari-ability of the stiffness of the adhesive between two plane bodies, and not simplytwo half-planes, provided that the extent of the surfaceS is sufficiently small withrespect to the extent of the adherents. This can be further justified by noting thatthe disuniformity ofK does not significantly modify the stress of the adherentsfar from the interface, where it is determined only by the averaged interface stiff-ness.

Regarding the load conditions, two cases are considered, both under the hy-pothesis of absence of body force. In the first case, a constant tension normalto the interface is applied at infinity, while, in the second, a constant tangentialshear is studied. It is shown that these problems are governed by the same integro-

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26 STEFANO LENCI

differential equation, which has been analytically solved by the complex variablemethod [13].

The solution has been obtained for an arbitrary interface stiffness distribution.However the two specific casesk(x) = k1x

2 andk(x) = k2x2/(1+ x2) have been

treated in detail as an illustration of the general solution.

2. The General Solution

The solutions of two-dimensional plane elasticity problems can be expressed interms of the two analytic functions8(z) and9(z), which depend on the complexvariable z = x + iy. The stresses and displacements can then be obtained byKolosov’s formulae [13, Sect. 32]

σx + σy = 2[8(z)+8(z)],σy − σx + 2iτxy = 2[z̄8′(z)+ 9(z)], (2)

2µ(u+ iv) = κϕ(z)− zϕ′(z)− ψ(z),whereϕ(z) = ∫ 8(z)dz, ψ(z) = ∫ 9(z)dz,8′(z) = (d8/dz)(z), κ = 3− 4ν inthe case of plane strain andκ = (3− ν)/(1+ ν) in the case of generalized planestress.

When normal tensions are applied at infinity, the problem is symmetric withrespect to the axisy = 0, while it is anti-symmetric in the case of tangentialshear. These properties permit to compute8(z) and9(z) in a single half-space,for example in the lower one�−.

Let us initially consider the case of constant stressesσy = N and τxy = 0applied at infinity and letf (t) represents the vertical displacement of�− at theinterface, which is supposed to be a bounded continuous function which verifiesf (t) = c+O(|t|−λ), λ > 0, for |t| → ∞ and with Lipschitz-continuous derivativevanishing at infinity. Then, according to the theory of plane elasticity in terms ofcomplex potentials, we can representϕ(z) andψ(z) in the regionz ∈ �− in theform

ϕ(z) = − 2µ

π(1+ κ)∫ ∞−∞

f (t)

t − z dt + N2z,

ψ(z) = ϕ(z)− zϕ′(z).(5)

Equations (2), the classical Plemelj formulae for half-planes [13, Sects 68 and71], and the formulae for the derivative of a Cauchy integral [13, Sects 69 and 71]then show that

σy(x, y = 0) = σy(x) = − 4µ

π(κ + 1)

∫ ∞−∞

f ′(t)t − x dt +N,

τxy(x, y = 0) = τxy(x) = 0,

(6)

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BONDED JOINTS WITH NONHOMOGENEOUS ADHESIVES 27

and it is possible to verify that

limy→−∞σy(x, y) = N

limy→−∞ τxy(x, y) = 0.

(7)

Equation (6)1 must be considered in the sense of the Cauchy principal value.The interface boundary conditions are

σy(x) = kN(x)[v+(x)− v−(x)],τxy(x) = kT (x)[u+(x)− u−(x)],

(8)

where the superscripts+ and− denote values of the functions in�+ and�−,respectively. As a consequence of the symmetry,v+(x) = −v−(x) = −f (x) andu+(x) = u−(x), so that (8)2 is identically satisfied (see also (6)2) and (8)1 gives

σy(x) = −k(x)f (x), (9)

where we have definedk(x) = 2kN(x) to simplify the notations. Combining (9)and (6) we obtain the integro-differential equation

k(x)f (x) = γ

π

∫ ∞−∞

f ′(t)t − x dt −N (10)

which governs the problem. In (10) we have denoted byγ = (4µ)(κ + 1) theunique parameter which summarizes all the elastic properties of the adherents.

To compute the solution of (10), we use the complex function

F(z) = 1

2πi

∫ ∞−∞

f (t)

t − z dt − Nz

2γ i, (11)

which is holomorphic on�+ ∪ �−, but not on the whole plane. The properties off (t) assure that the derivative of (11) is given by

F ′(z) = 1

2πi

∫ ∞−∞

f ′(t)t − z dt − N

2γ i, (12)

while a direct application of the Plemelj formulae yields

γ

π

∫ ∞−∞

f ′(t)t − x dt −N = γ i[F ′+(x)+ F ′−(x)],

f (x) = F+(x)− F−(x),(13)

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28 STEFANO LENCI

whereF+(x) = limy→0+

F(x + iy), F−(x) = limy→0−

F(x + iy) and analogous expres-

sions also hold for the derivatives. Equations (13) allows to reduce (10) to[F ′+(x)+ i

γk(x)F+(x)

]−[−F ′−(x)+ i

γk(x)F−(x)

]= 0. (14)

There is a certain freedom about the choice ofk(x). But, in practice, it is suf-ficient to suppose thatk(x) is the trace on thex-axis of a functionk(z) whichis holomorphic in the whole plane except possibly at a finite number of pointszi(i = 1,2, . . . , N), (not belonging to thex-axis) where it has poles. Under thishypothesis, let

G(z) =

F ′(z)+ i

γk(z)F (z), z ∈ �+,

−F ′(z)+ i

γk(z)F (z), z ∈ �−.

(15)

It is then possible to rewrite (14) in the form

G+(x)−G−(x) = 0, (16)

and we can conclude that the functionG(z) is continuous on thex-axis, and there-fore it is holomorphic on the wholez-plane except for the points wherek(z) haspoles.

Whenz tends to infinity, the functionF(z) behaves likeF(z) ' −Nz/(2γ i).It follows thatG(z) tends to infinity as−Nzk(z)/(2γ 2) and therefore it can beexpressed in the form

G(z) = − N

2γ 2zk(z)+ P(z), (17)

whereP(z) satisfies limz→∞P(z)/(zk(z)) = 0 and it is holomorphic on the whole

plane except at most at the pointszi (i = 1,2, . . . , N), where it has the same polesas the function(i/γ )k(z)F (z). Equations (15) and (17) then give

F ′(z)+ i

γk(z)F (z) = − N

2γ 2zk(z)+ P(z), z ∈ �+,

−F ′(z)+ i

γk(z)F (z) = − N

2γ 2zk(z)+ P(z), z ∈ �−,

(18)

or, definingH(z) = F(z)+Nz/(2γ i),

H ′(z)+ i

γk(z)H(z) = −Ni

2γ+ P(z), z ∈ �+,

H ′(z)− i

γk(z)H(z) = −Ni

2γ− P(z), z ∈ �−.

(19)

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BONDED JOINTS WITH NONHOMOGENEOUS ADHESIVES 29

Equations (19) are two ordinary differential equations of the first order, whichcan be integrated directly

H(z) = H+0 e−ib(z) − i N2γ

e−ib(z)∫ z

0eib(t) dt

+e−ib(z)∫ z

0eib(t)P (t)dt, z ∈ �+,

H(z) = H−0 eib(z) − i N2γ

eib(z)∫ z

0e−ib(t) dt

−eib(z)∫ z

0e−ib(t)P (t)dt, z ∈ �−,

(20)

whereb(z) = (1/γ ) ∫ z0 k(t)dt andH+0 andH−0 are two complex constants whichare determined by the boundary conditions. Finally, (13)2 and the definition ofH(z) guarantee that the solution of the problem is given by

f (x) = H+(x)−H−(x). (21)

We have thus obtained the solution of the integro-differential Equation (10)for every distribution of the interface stiffnessk(x). By appropriate choices of thefunction k(x), we are then able to treat various physical situations such as, forexample, the cases of localized and continuously distributed defects, which are ofpractical interest and which will be studied in detail in the following sections.

3. Some Particular Cases

In the particular case of evenk(x), the functionf (x)must also be even, so that thefunctionP(x), the restriction ofP(z) to the real axis, must be real and odd. It isthen possible to simplify (21), which can be expressed in the form

f (x) = A cos[b(x)] + Nγ

∫ x

0sin[b(t)− b(x)]dt

+2∫ x

0cos[b(t)− b(x)]P(t)dt. (22)

The functionP(x) and the unique unknown real constantA are determined bythe boundary condition lim

x→∞ f (x)k(x) = −N , as shown in the following sections.

The constantA, in particular, represents the vertical displacementf (0) on thesymmetry axisx = 0.

The simplest situation corresponds tok(x) = k0 = const. In this exampleP(x) vanishes,A = −(N/k0) and (22) actually gives the trivial solutionf (x) =−(N/k0). More general distributions ofk(x) are analyzed in the next sections.

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30 STEFANO LENCI

3.1. DAMAGE IN A PERFECTLY BONDED JOINT

In a perfectly bonded joint, the jump in the interface displacements vanishes andwe have continuity ofu acrossS. The stress, on the other hand, is also continuousbecause of equilibrium. This case can be described by assumingK →∞.

When damage appears in a perfectly bonded joint, the continuity of the displace-ment is instantaneously relaxed, and a jump takes place. In the first approximation,we can suppose that the interface is suddenly weakened, so that it behaves likea weak interface (1) with finite values of the stiffness. Only successively, if thedamage grows, we will arrive at a localized detachment of the adherents and, ifthe detached part grows again, to a complete separation, in correspondence towhich we must takeK = 0. This shows that the weak interface may be con-sidered as a transitory state between the perfect bond and the complete detachment.Mathematically, this fact is characterized by a decreasing ofK from∞ to 0.

In the intermediate phase after the damage and just before the detachment, wecan thus suppose we have an interface withK →∞ far from the defect andK = 0around the point candidate to be the starting point of the splitting. This situationcan be modeled, for example, by the stiffness distribution

k(x) = k1x2, (23)

which will be considered henceforth.Equation (23) is the trace on thex-axis of the analytical functionk(z) = k1z

2, sothat the only analytical functionP(z) satisfying the required conditions isP(z) =p1z, wherep1 will be determined in due course using the boundary condition.Integrating (23), we obtainb(x) = k1x

3/(3γ ). With these values ofP(x) andb(x), and after some calculations, we see that the solution (22) is given by

f (x) = cosb(x)

{A+ N

γ

γ 1/3

k1/31 32/3

∫ b(x)

0

sins

s2/3ds

+2p1γ 2/3

k2/31 31/3

∫ b(x)

0

coss

s1/3ds

}

+ sinb(x)

{−Nγ

γ 1/3

k1/31 32/3

∫ b(x)

0

coss

s2/3ds

+2p1γ 2/3

k2/31 31/3

∫ b(x)

0

sins

s1/3ds

}. (24)

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BONDED JOINTS WITH NONHOMOGENEOUS ADHESIVES 31

To determine the two unknownsA andp1, we must impose the boundary condi-tion lim

x→∞f (x)k(x) = −N , which in the present case is reduced to limx→∞f (x) = 0

and is equivalent to the two equations (see (24))

A+ Nγ

γ 1/3

k1/31 32/3

∫ ∞0

sins

s2/3ds + 2p1

γ 2/3

k2/31 31/3

∫ ∞0

coss

s1/3ds = 0,

−Nγ

γ 1/3

k1/31 32/3

∫ ∞0

coss

s2/3ds + 2p1

γ 2/3

k2/31 31/3

∫ ∞0

sins

s1/3ds = 0,

(25)

where all the integrals are convergent because the exponent ofs is less than oneand because the integrands are oscillatory for large values ofs. Their values, nu-merically computed, are∫ ∞

0

sins

s1/3ds = 1.16103,

∫ ∞0

sins

s2/3ds = 1.33933,∫ ∞

0

coss

s1/3ds = 0.67706,

∫ ∞0

coss

s2/3ds = 2.32003.

(26)

By solving (25) we obtain the expression

p1 = N

2γ

(k1

3γ

)1/3∫∞

0cosss2/3 ds∫∞

0sinss1/3 ds

,

A = −Nγ

(γ

32k1

)1/3∫∞

0cosss1/3 ds

∫∞0

cosss2/3 ds + ∫∞0 sins

s1/3 ds∫∞

0sinss2/3 ds∫∞

0sinss1/3 ds

,

(27)

and, with the values (26), we arrive at the result

p1 = 0.69276N

γ

(k1

γ

)1/3

, A = −1.29431N

γ

(γ

k1

)1/3

. (28)

We can therefore write the final expression of the solution

f (x)

f (0)= cosb(x)

{1− 0.3714

∫ b(x)

0

sins

s2/3ds − 0.7422

∫ b(x)

0

coss

s1/3ds

}

+ sinb(x)

{0.3714

∫ b(x)

0

coss

s2/3ds − 0.7422

∫ b(x)

0

sins

s1/3ds

}, (29)

wheref (0) = A is given by (28)2. The function (29), sketched in Figure 3, con-firms the expected results that the functionf (x) is regular, it has a maximum forx = 0 and it rapidly tends to zero forx →∞.

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32 STEFANO LENCI

Figure 3. The functionf (x)/f (0), the solution of the problem of the damaged perfectlybonded joint.

3.2. AN ISOLATED LOSS OF THE INTERFACE STIFFNESS

Another case which can be solved on the basis of (22) is that of a homogenousadhesive with an isolated defect. In this case the interface stiffness is constantalmost everywhere, and strongly reduces (until zero, if we are just before the localdetachment) only in the neighborhood of the defect position. This example can becharacterized by the stiffness distribution

k(x) = k2x2

1+ x2, (30)

which is different from a constant only in the neighborhood ofx = 0.The holomorphic function corresponding to (30) isk(z) = k2z

2/(1+z2), whichhas simple poles atz1,2 = ±i. We further obtainb(x) = (k2/γ )(x − arctanx) andnote thatP(z) = p2z/(1+z2). In fact, it vanishes at infinity, it has the same simplepoles ask(z) and its trace onx = 0 is an odd function.

The complete solution of the problem considered here can be achieved by simplyrepeating the same reasoning as above. However, at least when the parameterk2/γ

is equal to an even integer number, the formal expression of the solution can be sig-nificantly simplified. To explain this fact, we will consider the illustrative examplek2/γ = 2.

In this case, the following basic relations hold

sin[b(x)] = − sin(2t)+ 2sin(2t)− t cos(2t)

1+ t2 ,

cos[b(x)] = − cos(2t)+ 2cos(2t)+ t sin(2t)

1+ t2 ,

(31)

and the solution (22) can be put in the form

f (x) = cos[b(x)]{A+ N

γ

[cos(2x) − 1

2+ 2

∫ x

0

sin(2t)− t cos(2t)

1+ t2 dt

]

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BONDED JOINTS WITH NONHOMOGENEOUS ADHESIVES 33

+2p2

∫ x

0

t cos[b(t)]1+ t2 dt

}

+ sin[b(x)]{N

γ

[sin(2x)

2− 2

∫ x

0

cos(2t)+ t sin(2t)

1+ t2 dt

]

+2p2

∫ x

0

t sin[b(t)]1+ t2 dt

}. (32)

The integrals that appear in (32) can be expressed in terms of known transcendentalfunctions by using the classical tables of integrals [14, 15].

The asymptotic behaviour off (x) for x →∞ is given by

f (x) ' − N2γ− cos(2x)

{A− N

2γ+ 2N

γ

∫ ∞0

sin(2t)− t cos(2t)

1+ t2 dt

+2p2

∫ ∞0

t cos[b(t)]1+ t2 dt

}

+ sin(2x)

{−2N

γ

∫ ∞0

cos(2t)+ t sin(2t)

1+ t2 dt

+2p2

∫ ∞0

t sin[b(t)]1+ t2 dt

}. (33)

The boundary condition limx→∞f (x)k(x) = −N , on the other hand, corresponds to

limx→∞f (x) = −(N/k2) = −(N/2γ ), and can be fulfilled if the terms within the

braces in (33) vanish. These are two equations in the two unknownsA andp2,which can easily be solved, since the integrals are explicitely computable, to yield

p2 = N

γ, A = −3

2

N

γ. (34)

Inserting these values in (32) we obtain, after some simplifications, the surprisinglysimple expression of the solution:

f (x) = −Nγ

(1

2+ 1

1+ x2

). (35)

From a mechanical point of view, (35) delivers two conclusions at least:

(i) the loss of interface stiffness which models the defect and the interface dis-placement are governed by the same function 1/(1+ x2). Thus, the effects ofthe local damage are not propagated along the interface but remain circum-scripted in the neighborhood of the defect;

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34 STEFANO LENCI

(ii) the displacement gap atx = 0 is three times as much as the one achievedin the unperturbed part of the adhesive, showing a strong dependance on thedefects in the local joint behaviour.

4. Shear Stress

The problem of constant shear stress applied at infinity,σy = 0 andτxy = T , issimilar to the one analysed in the previous sections. Indeed, the complex potentials

ϕ(z) = i2µ

π(1+ κ)∫ ∞−∞

f (t)

t − z dt − T2iz,

ψ(z) = −ϕ(z)− zϕ′(z),(36)

give

u(x) = f (x),σy(x) = 0,

τxy(x) = − 4µ

π(κ + 1)

∫ ∞−∞

f ′(t)t − x dt + T ,

limy→−∞σy(y, x) = 0,

limy→−∞ τxy(y, x) = T .

(37)

This case is anti-symmetric with respect to thex-axis, and the unique nontrivialinterface boundary condition isτxy(x) = −k(x)u(x), wherek(x) = 2kT (x). Sub-stituting (37) in this relation precisely gives the integro-differential equation (10),which therefore shows an invariant character with respect to the symmetry featuresof the problem. Thus, the complete solution of this problem is again (22), with theobvious re-interpretations of the symbolsN, k(x) andf (x).

It should be noted that the global equilibrium is guaranteed by the fact that thepotential (36) corresponds to vanishing rotations at infinity.

Acknowledgement

The work was done during the author’s permanence at Laboratoire de Modélisationen Mécanique, Université Pierre et Marie Curie (Paris 6), as part of the Trainingand Mobility of Researchers (TMR) programme, contract ERBFMBI CT97 2458.The financial support of this programme is greatly acknowledged.

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BONDED JOINTS WITH NONHOMOGENEOUS ADHESIVES 35

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