Fracture load predictions for adhesive joints

Composites Science and Technology $1 (1994) 587-600 1994 Elsevier Science Limited Printed in Northern Ireland 0266-3538/94/$07.00

ELSEVIER

F R A C T U R E L O A D PREDICTIONS FOR A D H E S I V E JOINTS G. Fernlund, M. Papini, D. McCammond & J. K.

Spelt

Department of Mechanical Engineering, University of Toronto, Toronto, Canada, M5S 1A4 (Received 15 September 1993; accepted 31 January 1994) and strains only. In reality, however, most joints contain significant peel stresses, and this approach is not generally applicable. Other examples of stress- or strain-based failure criteria can be found in the literature. 3-1° The fracture mechanics of adhesive joints has been studied by many investigators. It has been found that, in some cases, the strain energy release rate correlates well with joint fracture, u-2~ Some investigators question the use of linear elastic fracture mechanics because of the fact that commercially available adhesives are often toughened and relatively ductile, and thus do not always exhibit sharp cracks. 2 Anderson et al. 22 have shown that the critical strain energy release rate can be used to predict failure in mode I specimens if a very brittle adhesive is used, and that unflawed mode I specimens could be treated with fracture mechanics by assuming an inherent flaw size due to small bubbles and microcracks. Lee 2a claims that adhesive fracture is controlled by the plastic zone developed at the crack tip, and that a relationship can be found between the fracture energy and certain bulk properties of the adhesive. Groth 24"25 formulated a fracture criterion based on a generalized stress intensity factor derived from material-induced stress singularities at the ends of bonded joints. In summary, there are a large number of approaches and studies in the literature, many with contradictory results. Most suffer from the fact that they are too idealized and cannot be applied easily to many joint geometries and loading conditions. Furthermore, the strength of an adhesive bondline depends not only on the properties of the adhesive, but also on the properties of the adherends, adherend pretreatment, and adhesive cure. Therefore, one must consider a complete 'adhesive system' consisting of adhesive, adherends, pretreatment, bondline thickness, and adhesive cure schedule (i.e. anything that affects the strength of a joint) when attempting to predict failure. This paper presents an engineering approach to fracture load predictions for adhesive joints. There is given an analytical method for calculating the energy release rate, J, of planar lap or strap joints when a crack propagates in the bondline. By using this

Abstract

An engineering approach to fracture load predictions for adhesive joints is presented. The approach is based on the premise that the in-situ strength of the bondline can be characterized by the fracture envelope (critical energy release rate as a function of the mode of loading), for a specific adhesive system. By using the J integral for large deformations together with largedeformation beam theory, a simple closed-form expression is obtained for the energy release rate per unit area extension when a crack propagates in the bondline of a generalized adhesive joint (adhesive sandwich). This technique, together with a published method for mode partitioning, enables fracture load prediction by comparing the calculated fracture parameters with the critical ones from the fracture envelope. The approach is shown to predict fracture loads accurately for a variety of joints including the cracked lap shear (CLS), the single lap shear (SLS) and the double strap (DS) joint. Keywords: adhesive joints, fracture mechanics, failure load predictions, delamination, design 1 INTRODUCTION

The prediction of failure in adhesive joints has been complicated by many factors. Adhesive joints are generally not well behaved from an analytical point of view. They exhibit phenomena such as very complex adhesive-layer stress distributions, bimaterial interfaces, and large deformations. As a result, joints are usually designed by using rules of thumb and general guidelines of the type outlined in Ref. 1. The failure criteria for adhesive joints proposed in the literature usually fall into two classes: those based on the classical strength of materials and those based on fracture mechanics. The former generally utilize a maximum stress or maximum strain failure criterion. For example, Hart-Smith proposed a maximum strain criterion that has been used in the aerospace industry. 2 The approach is based on the assumption that, by proper design of the joint, peel stresses can be reduced to a point where they do not contribute to failure. Thus, this approach considers shear stresses 587

588

G. Fernlund et al.

method, fracture load predictions can be made by comparing the calculated J with an experimentally determined in-situ fracture toughness of the bondline. The applicability of the approach is demonstrated with comparisons between predicted and actual fracture loads for several lap and strap joint geometries.

700

-

T

600500-

E

1

400-

1

300-

1

2 OUTLINE OF A P P R O A C H TO F R A C T U R E LOAD PREDICTIONS

100-

The main criteria for a useful methodology for failure load predictions are that it has to be applicable to a large class of joints, independent of geometry, and that it should be relatively simple. A large class of joints (lap/strap joints) can be reduced to generalized elements by using the concept of the adhesive sandwich 2°'26 where the bonded overlap is isolated from the surrounding structure as a free body. Many joints, including the single-lap-shear (SLS), cracked lap shear (CLS), single strap (SS), and the double strap (DS) joint can be treated in a uniform way by this technique (Fig. 1). The reactions acting at the end of the sandwich can be determined from beam theory. The next step is to determine an appropriate failure criterion. In most cases, a lap/strap joint will either fail in the adherends at the end of the joint overlap, or by crack propagation in the bondline. For adherends with a relatively low fracture toughness there is a possibility of a crack in the bondline propagating into the adherend. This failure mode will not be considered here. With the adherend reactions known, adherend failure can be dealt with readily by standard strength-of-materials techniques. The second failure mode, crack propagation in the adhesive layer, can be addressed by u s i n g f f r a c ~ r e mechanics. In the following it will be assumed tl~t all sandwich elements contain an initial crack. It will be demonstrated experimentally, however, that ~he fracture loads are independent of whether or not/there exists an initial macroscopic crack in the bondline, and hence uncracked joints can be treated as having a fictitious crack tip coinciding with the end of the bondline. The strain energy release rate at the onset of cracking correlates well with fracture for some adhesive systems, but it has often been found to be dependent on the mode ratio (i.e. the ratio of the a)

--.u

b) 4

q

9

@

"

t

c) d)

I

;

(~7 " ) Q

!

t

!

!

, a

j

It,

,

P

Fig. 1. Examples of adhesive sandwich elements: (a) SLS joint; (b) CLS joint; (c) SS joint; (d) DS joint.

0 0

1

3

~

~

xp= atan (Jli/Ji)1/2 [reg.]

Fig. 2. Fracture envelope, Jc versus phase angle % for FPL-etched 7075-T6 aluminium adherends bonded with Cybond 4523GB epoxy adhesive, cured at 150°C for 45 rain, bondline thickness 0-4ram. 2v Error bars correspond to +1 standard deviation.

energy released due to opening versus sliding of the crack). 15-2~ Note that it is in the in-situ fracture resistance of an adhesive system which is of interest, not the bulk fracture resistance of the adhesive. The in-situ fracture resistance of an adhesive system and its dependence on the mode ratio (the 'fracture envelope') can be determined conveniently by using double-cantilever-beam specimens and a specially designed load jig. 27 The advantage of this approach is that one does not have explicitly to consider the mechanisms of failure at the crack tip; all of the relevant characteristics of the adhesive system (including the effects of pretreatment, constraints due to bondline thickness, adhesive cure time, etc.) are built in to the fracture envelope. Figure 2 shows the fracture envelope for the adhesive system: FLP-etched 7075-T6 aluminium bonded with Cybond 4523GB single-part epoxy (American Cyanamid) cured at 150°C for 45min, with a bondline thickness of 0-4 ram. Figure 2 shows that Jc is dependent on the mode ratio with Jiic--3J~c for this adhesive system. 27 The letter J will be used for the energy release rate (rather than G) since many lap/strap joints exhibit a non-linear relationship between load and displacement as discussed in the following section. The present approach to fracture load predictions proceeds by the identification of an adhesive sandwich element in the joint and determination of the adherend reactions at the ends of the adhesive sandwich. If the reactions do not yield the adherends, we then calculate J and the mode ratio. For the calculated mode ratio, we obtain from the fracture envelope the critical energy release rate, J o If the calculated J is less than Jc, the joint will not fail, otherwise failure by crack propagation in the bond!ine will occur.

Fracture load predictions for adhesive joints 3 LARGE-DEFORMATION ADHESIVE JOINTS

A N A L Y S I S OF

Adhesive joints are often subject to remote tensile loading which induces large rotations because of the eccentricity of the centroids. Typical joints which behave in this way are the SLS, CLS and SS joints (Fig. 1). The deformations of such joints can be determined from the yon Karman theory for large deformations of plates. 28 Consider the beam element in Fig. 3 with the resultant forces F and V, and resultant bending moment, M, acting on a cross-section. The von Karman theory2s gives the following relationships between the tensile force, F, the bending moment, M, and the displacements of the beam

F=Eh[auo,+ 1

(Su2~ 2] 2\aal/ J

L aal

M=

(1)

- E1 a2u2 aa 2

(2)

where E is the Young's modulus of the beam, h is the beam height, I is the second moment of area per unit width of the cross-section, u2 the displacement in the 2-direction, and Uol is the displacement of the centroid in the 1-direction. Equilibrium gives

aF aa~

= 0

Fa2u2

(3)

aV

aa~ + ~al = 0

(4)

aM - -

aal

-

v

= o

(5)

Combining eqns (2), (4) and (5) gives the governing equation for transverse displacements of the beam a4u2

aa~

F a2u2 E1 -aa 2=

0

(6)

589

(7) together with the appropriate boundary conditions will give the transverse displacement u2(al). The moment and shear distributions can then be obtained from eqns (2) and (5), respectively, while the axial displacement of the centroid can be determined from eqns (1) and (3). The application of this procedure to the analysis of different adhesive joints is illustrated in later sections.

4 ENERGY RELEASE RATE FOR A CRACKED ADHESIVE SANDWICH UNDERGOING LARGE DEFORMATIONS

A closed-form expression for the energy release rate of a cracked adhesive sandwich undergoing finite (large) deformations will now be developed. Consider the cracked adhesive sandwich subject to the loading condition shown in Fig. 4. Assuming that both the adherends and the adhesive layer are elastic, the energy release rate, J, can be calculated by using the J integral

and W is the strain energy density, n is the outward unit normal vector to the boundary contour in the undeformed state, Fo, and T is the traction vector acting on Fo. It has been shown 29 that eqn (8) is the same for both finite and infinitesimal deformations. In evaluating the integral in eqn (8) we choose the evaluation path, Fo, as a path counter-clockwise along the outer perimeter of the cracked adhesive sandwich shown in Fig. 4 (b-c-d-e-f-g-h-i). A Lagrangian description is used, where stresses (So) and strains (e0) are measured with respect to the undeformed configuration. On the crack faces and on the upper and lower horizontal boundaries of the adherends (b-c, d-e, f-g and h-i), there are no tractions, T, and

with the general solution •

F

u2(al) = Al s,nh( ~ a l )

v3

M

at + A3al + A4

+ A2 cosh

al

VI

F3

(7)

where the Ai are constants to be determined from the boundary conditions of a specific problem. Equation

M2

M3

v~ d vI

v3

a2

T

f3

V

I" al

-

P M

Fig. 3. Beam element.

°ra F

3 v2

Fig. 4. Cracked adhesive sandwich.

G. Fernlund et al.

590

the unit normal vector in the a~ direction, an, is zero, hence the integrand in eqn (8) is identically zero on these parts of the boundary, Fo. The only remaining portions of the contour Fo that will have non-zero contributions to the J integral are the left- and right-hand sides of the adhesive sandwich (g-h, c-d and e-f). If the distances between the left- and right-hand sides to the crack tip are large compared to the total thickness of the sandwich, the members at the cross-sections can be assumed to behave as composite beam elements and their behaviour modeled using the von Karman theory for the large deformation of plates. 28 Due to the assumptions that $22=$33=0 and that plane cross-sections remain plane after deformation (no shear deformations), the strain energy density in a cross-section can be written W

$21 2E

(9)

The stress vector, T, can be expressed in terms of the stress tensor, Sjk

.

c3x,

1", = a;kn; ~a~

(10)

T ,0al .au~ = [S,; + SJkff~ak] auA 0a-' au~-~nJ

(11)

and with x~ = a, + u,

On the vertical cross-sections of interest nz = 0, which yields

(auq ~ ' Oa~ -

au~]. Oal ] 1

neglecting higher order terms. For cross-section i, using eqns (8), (9) and (12)

hi~2 -- ,) Sfl 1 (au~i)~ 2 Ji=J_hi/2 nt [2E+Sll,2\aa] ]

au(i)]

+S'z-~alJda2 (13)

where the superscript i within brackets denotes the respective cross-section. Integration of eqn (13) yields

ji

[F?

M 2.

l(Ou~i)] z

=--n~ i) ~--~, -']"-~-~//-'['-I-V/ ztzn, ' F~ \ Oa~ /

d J t _ _ n ( i ) [ M i V i _ t - ~u~i) a2U~i) V ~2/~/(') (~Vi~u(2i) ] dal1 [ Eli T Fi Oal Oa~ + ' aa~ + Oal 3aj J

(16) However, by eqn (2) the first and third terms cancel, and by eqn (4) the second and the fourth terms cancel, hence, dY/dal = 0 and eqn (15) is seen to be path-independent. The tensile stiffness and flexural rigidity of member i, Eh, and E/~, eqn (14), are strictly the properties of the composite beams (adherend plus adhesive), however, in many adhesive joints, the thickness of the adhesive layer (t) is much less than the adherend thicknesses (hi, h2) hence, the adhesive layer will have a negligible effect on the stiffness of the composite beams at the left- and right-hand side of the adhesive sandwich, and can be neglected in these calculations. Note that eqn (14) requires only that the sandwich behave as a composite beam at locations far from the crack tip, and that no assumptions were made regarding the behaviour close to the crack tip. Equations (14) and (15) show that J is not only dependent on the loads at the ends of the adhesive sandwich, but also on the slopes, OUz/aal, of the beam elements at the respective cross-sections. To calculate the adherend slopes, some simplifying assumptions regarding the behaviour of the adhesive sandwich close to the crack tip will be made. An approximation which, it will be demonstrated, is justified in many engineering applications is to assume that the adherends behave as built-in cantilevers at the crack tip. This assumption implies that the adherend slopes are equal at the crack tip, which, together with equilibrium and path-independence, means that J, eqn (15), can be written 2

f~

m2 ]

~] (14)

where all the loads (F~, M; and V~) and the second moment of area (/~) are defined per unit width. Hence, the total energy release rate, J, can be written

j =j~ +j2 + j 3

It can be shown, as follows, that eqn (15) is path-independent, i.e. that the cross-sections where the ji are evaluated can be chosen in any convenient fashion along the joint. Path-independence requires that dY/dal = 0. Equations (14), (3) and (5) give

(15)

where the Y are given by eqn (14). Note from Fig. 4 that n t l ) = n ~ 2 ) = - I and that nt 3)= +1. For infinitesimal deformations, the third term in eqn (14) will be negligible and eqn (15) reduces to a previously published result. 3°

_

m3

(17)

12Eh3 where the loads, f and rn~, are the loads acting on the respective centroids at the cross-section at the crack tip (Fig. 4). In the case of infinitesimal deformations, it has been shown that eqn (17) is an exact expression for the energy release rate in an infinitely long sandwich when no shear loads are present. 31 An expression very similar to eqn (17) was obtained using other means in the study of the delamination of composite laminates. 3z The assumption that the adherends behave as

Fracture load predictions for adhesive joints built-in cantilevers at the crack tip clearly overestimates the stiffness (and underestimates the deformations) of the adhesive sandwich (Fig. 4). Thus, for a sandwich subject to prescribed loads, eqn (17) will underestimate the energy release rate when shear loads and/or tensile loads and large deformation are present (see eqns (14), (15) and (17)). For a sandwich subject to prescribed displacements, however, the assumption that the adherends behave as built-in cantilevers at the crack tip will overestimate the adherend reactions, and, therefore, eqn (17) will overestimate the energy release rate in this case. For many realistic joints, the boundary conditions at the ends of the adhesive sandwich are dependent on the stiffness of the sandwich, hence, the sandwich is neither subject to prescribed loads nor prescribed displacements. This makes it difficult to analytically assess the net effect of the cantilever assumption on the accuracy of eqn (17) without explicitly considering the effect of the adhesive layer on the sandwich stiffness. The present J-integral formulation of eqn (15), which simplifies to eqn (17) when it is assumed that the adherends behave as built-in cantilevers at the crack tip, has a number of interesting features. In the form of eqn (15), the expression for J is very accurate and convenient to evaluate because it requires only that the adherend reactions and slopes be known far from the singular point at the crack tip. Furthermore, path-independence allows these reaction and slopes to be evaluated at any location along the sandwich, which is a useful feature, especially in a test where the adherend de:formations are measured experimentally. The J-integral formulation also accommodates nonlinear elastic materials 29 which makes it possible to apply eqn (15) to adhesive joints with adhesives exhibiting a non-linear elastic behaviour. For adhesive sandwiches subject to loads other than those shown in Fig. 4, the Jintegral provides a convenient approach to evaluate J. For example, transverse loads applied to the adhesive sandwich can readily be accommodated and incorporated into eqn (15). 30 Finally, as discussed above, the effect of simplifying assumptions regarding the behaviour of the sandwich close to the crack tip can be readily assessed. Note that the following assumptions were made in arriving at the simplified expression for the energy release rate of a cracked adhesive sandwich (eqn (17)): (1) The joint overlap is long enough so that there exists a cross-section to the right of the crack-tip which acts as a slender beam in bending. (2) The adhesive layer is thin and stiff enough to have a negligible effect on the global deformation of the joint. (3) The adherends behave as built-in cantilevers at the crack-tip. These assumptions may seem excessively restrictive, however, it will be demonstrated below that eqn (17) gives results which

591

compare well with finite element analyses, and that the method agrees well with experimental data for joints with aluminium adherends bonded with a structural epoxy resin. It is, of course, possible to relax some of the assumptions above, but the significant increase in computational effort could not be justified in view of these results. 5 M O D E PARTITIONING A N D ENERGY RELEASE RATES---COMPARISON WITH FINITE ELEMENT D A T A There are two methods for the calculation of the mode ratio (and energy release rate) of a cracked homogeneous beam subject to infinitesimal deformations which have been presented in the literature. 31'32 Neither of these methods account for the presence of an adhesive layer. For cracked homogeneous beams with equal adherends, the two methods give virtually identical results, whereas in the unequal adherend case, the methods differ significantly in the prediction of the mode r a t i o . 33 In a previous paper, 27 a third method for mode partitioning denoted Method 3 was discussed, but unfortunately incorrectly referenced. The method uses the nominal adhesive stresses at the end of the bonded overlap, calculated using a beam on an elastic foundation model, 26 to determine the energy release rate and the mode ratio according to the technique described in Ref. 34. Applied to the type of specimen discussed in this paper, the method gives results very similar to the method of Suo and Hutchinson, 31 as shown in Ref. 27. Although the method accounts for the presence of the adhesive layer, it requires more computational effort, and since the results are in good agreement with those of gef. 31, it was not included in the present study. As a test of their applicability to lap/strap joints subject to large deformations, the two methods to calculate the mode ratio were used to analyse the cracked-lap-shear (CLS) joint shown in Fig. 5. The identical problem was studied in an ASTM Round Robin, where nine laboratories world-wide participatedY The results from the Round Robin, which included several non-linear finite element models, provide a useful comparison with the present

I a2 al t t 11

a

~ D

Fig. 5. CLS joint subject to tensile loading as specified in ASTM Round Robin. u

G. Fernlund et

592

methods. 31'32 However, because the joint contains an adhesive layer and undergoes large rotations, the two methods of mode partitioning are strictly not applicable to this problem, so the results are only approximate. It is interesting to note that, because of the relatively large deformations of adhesive joints such as the CLS, there are few reliable studies of energy release rates and mode ratios in the literature. The energy release rate, J, and the JJJn ratio were calculated for the two different ASTM CLS joints, one with equal adherends (hi = h2 denoted CLS-A) and one with unequal adherends (hi =2h2 denoted CLS-B). For both joints the geometry and material properties were: 35 L = 0 . 3 0 5 ( m ) , h 2 = 3 . 1 7 5 x 10E3 (m), t = 0-13 × 10E-3 (m), E1 = E2 = 72.45 (GPa), Vl = v2 = 0.33, Ea = 1"932 (GPa) and va = 0-4. For the CLS-A: hi =3.175 x 10E-3 (m), and for the CLS-B: h l = 6 . 3 5 x 10E-3(m). The applied load per unit width was F = 4.378 x 10E5 (N/m) and plane strain conditions were assumed. To calculate J and Jl/Jn for the CLS joint, it is necessary to know the loads acting on the lower adherend at the crack tip. Since the CLS undergoes large rotations, these reactions were determined using the finite deformation approach discussed above. Treating the joint as a homogeneous beam with a discontinuity of the beam height at the crack-tip, eqn (7) gives •

u~2)(al) = ml slnh( ~ 2

•

F

ai ) al +A3aI+A4

(18)

~ al]'~+ A7al + As

(19)

F

u[3)(a.) = A5 s'nh( ~/~3 a 0 + A6 cosh

where u~2) is the transverse displacement of the joint for ( - a < a i < 0 ) and up ) is the transverse displacement of the joint for (0 < ai < L - a). EI2 and El3 are the flexural rigidities of the lower adherend to the left of the crack tip, and of the composite beam to the right of the crack tip, respectively. In this case, the adhesive layer is so thin that the composite beam to

al.

the right of the crack tip can be approximated as a homogeneous beam of thickness h3=hl+h 2 and consisting entirely of adherend material. The constants A1-A8 can be determined from the boundary conditions• At al = - a the transverse displacements and the bending moment are zero. Using eqn (2), this gives four of the eight required boundary conditions. Continuity and the assumption that the adherends behave as built-in cantilevers at the crack tip implies that the displacements and slopes of the two beam sections be equal at the crack tip. Finally, equilibrium of the cross-section at the crack tip means that the shear forces have to be equal in the two beam sections and that m3 = m2 -f2hl/2 (.1:2= F, see Fig. 4). The last two conditions can be expressed in terms of displacments using eqns (2) and (5). Peforming the displacement analysis, the bending moment, m2, acting on the lower adherend at the crack tip was obtained for different crack lengths, a, for the two CLS joints• Knowing the tensile force, f2 = F, and the bending moment, m2, the energy release rate, J, was calculated using eqn (17), and the JJJn ratio was calculated using the two methods 3I'32 discussed. The results obtained in the ASTM Round Robin 35 showed a large amount of scatter between the different investigators. This was especially true for the unequal adherend CLS-B, where the calculated J and J~/J. ratios differed by as much as a factor of two between the high and low extremes. The conclusion from the Round Robin was that a geometrically non-linear analysis is required for the study of joints of this type. The two geometrically non-linear finite element analyses in the Round Robin (by Dattaguru/Mangalgiri and Everett/Whitcomb) will therefore be used as a comparison with the present results. Tables 1 and 2 show the different J and J~/Jn ratios obtained using the discussed methods together with the results obtained in the Round Robin using geometrically non-linear finite element analyses. Note that using the same reactions at the crack tip, m2 and f2, the two published methods, Method 13I and Method 2, 32 give the same result for J as the present method for both CLS-A and CLS-B. From Tables 1 and 2 it is seen that the energy release rate, J, and the JdJn ratio are largely independent of the crack length

Table 1. J and JdJu ratios for an eqml-adhe~nd CL$ joint (CLS-A) obtained b,v different calculation techniques. Method 1, ~ Method 2, ~ FIEM (D/M): mm~hodof D a t t a g m r u / ~ , ~ ° FEM (E/W): method of Everett/Whitcomb 3s

Crack length

Present

Method 1

Method 2

(mm)

J(J/m2)

JJJ.

Jl/Jn

54 57 76 152

185 186 187 191

0.26 0-26 0.26 0.26

0.26 0.26 0-26 0.26

FEM (D/M)

FEM (E/W)

J(J/m 2)

J,IJ,,

J(J/rn 2)

J,IJ,,

187 187 188 191

0.25 0.25 0.25 0-25

186 187 189 193

0.27 0.26 0.26 0-26

Fracture load predictions for adhesive joints

593

Table 2. $ and J,/Jn ratios for an unequl-adherend CLS joint (CLS-B) obtained by dtiRerent calculation techniques. Method 1, 3n Method 2, ~ FEM (D/M): method of Dattagnra/Mangalgari, ~e FEM (E/W): method of Everett/Whitcomb ~

Crack length (mm) 54 57 76 152

Present

J(J/m 2)

Method 1

Method 2

Ji/Jn

JJJu

0.11 0.11 0.11 0.11

0.78 0.79 0.80 0.77

266 267 270 263

for both specimens. The energy release rate, J, as calculated using the present method is seen to be almost identical to that obtained using non-linear finite element methods for both joints. The Jl/Jn ratio obtained for the CLS-A (Table 1) using either Method 131 or Method 232 also shows a very good agreement with the results from the Round Robin. For the CLS-B (unequal adherends), the two methods give very different results for the JJJn ratio as seen from Table 2. The JJJn ratios obtained using Method 1 were lower, and the results using Method 2 were higher than those obtained using non-linear finite element methods. As mentioned above, however, both Method 131 and Method 232 have been applied to a problem which is strictly outside their range of applicability. These comparisons show that the present method for calculating J for adhesive joints subject to large deformations gives results which are in good agreement with non-linear finite element calculations. They also show that the J~/Jn ratios calculated using either Method 1 or Method 2 are in excellent agreement with the non-linear finite element results for the equal adherend specimen, but for the unequal adherend specimen, Method 1 gives a better agreement with the non-linear finite element calculations. The two methods of mode partitioning are evaluated against experimental data in a later section.

6 EXPERIMENTAL

RESULTS

In order to establish that the present approach enables fracture load predictions for a wide variety of adhesively bonded structures, three different specimen geometries were analysed and tested: the single lap shear joint (SLS), the cracked lap shear joint (CLS), and the double strap joint (DS) (Figs l(a), l(b), l(c), respectively). Each type of joint was evaluated over a wide range of overlap and adherend lengths. 6.1 The cracked lap shear ( C I 3 ) joint

The CLS joint with a pinned-pinned boundary condition was chosen for fracture load testing. In order to perform the large deformation analysis, it was necessary to treat the joint as a homogeneous

FEM (D/M)

FEM (E/W)

J(J/m 2)

JJJn

J(J/m 2)

J~[Jn

273 274 277 262

0.22 0-22 0.22 0.20

271 273 279 263

0.23 0.23 0.23 0-21

[_

Ll

~.

L2

///// Fig. 6, CLS joint pinned at both ends. beam with a discontinuity of the beam height at the crack tip as shown in Fig. 6. The deflection will be governed by eqns (18) and (19), and the constants A1-A8 must be determined using the boundary conditions, which are identical to those given in Section 5 for the pinned-clamped CLS joint with the exception that at a l = L2 the pin implies that the transverse deflection and bending moment are zero. Performing the displacement analysis gives the bending moment, m2, acting on the lower adherend at the crack tip, and knowing the tensile force, f2 = F, the energy release rate, J, can be calculated using eqn (17), and the phase angle, ~p, defined as arctan(Jn/JO ~r2 can be calculated using either Method 131 or Method 2. 32 Both methods give the same results in this case because the adherends are of equal thickness. The predicted fracture load can then be derived by comparison with a curve fit of the fracture envelope, Fig. 2, given by, Jc -- 212 - 2.56~p + 0-143~p2 - 7.75 x 10-4~p3 (20) Equal-adherend CLS joints with a wide variety of lengths L1 and L2 (Fig. 6) were fabricated using the same adhesive system (FPL-etched 7075 T6 aluminium bonded with Cybond 4523GB adhesive with a 0.4mm thick bondline) as was used in defining the fracture envelope, Fig. 2, and their fracture loads were measured quasi-statically. The rate was varied by a factor of ten (from 0.02 mm/min to 0.2 mm/min) with no difference found in the fracture load. A constant adherend width of 20 mm was used since earlier studies showed that the fracture load per unit width was independent of width at this value. 21 The specimens were all given a mode I starter crack made by driving a cold chisel between the adherends. A

G. Fernlund et

594

Table 3. Comparison of predicted and measured fracture loads per unit width for equal adherend CLS joints. The adherend thickness was i inch (~12.54 mm) in all cases. All specimens were given a Mode I starter crack and were 20 mm wide

L1 (m)

Lz (m)

0.160 0.165 0-187 0.182 0-162 0-143 0.213 0-154 0.143 0.220 0.133 0.187 0.242 0-168 0.240

0.187 0.185 0.159 0.154 0.180 0-197 0.127 0.192 0-196 0.118 0.210 0.157 0.101 0.184 0-112

Predicted P(kN/m) 961 952 873 866 944 1002 779 978 1000 760 1043 870 698 943 726

Measured E(kN/m)

displacement of the three beam elements, using eqn (7), can be expressed as,

F

u~l)(aO=Zlsinh(i-E-~la,)

Diff.

(e - E)/E(%)

1100 968 876 915 935 1092 726 1044 1081 769 1226 966 720 960 695 Avg. I(P - E)/EI

-12.6 -1-7 -0-3 -5-3 1-0 -8.2 7.3 -6.3 -7.5 -1.2 - 14.9 -9•9 -3-1 -1.8 4.5 5-7

comparison of predicted and measured fracture loads is given in Table 3. The average difference between the predicted and measured fracture loads for all specimens was found to be 5.7%. Note that the adherend lengths, L1 and L2, were varied such that the fracture loads increased from a low of 695 k N / m to a high of 1100 kN/m. Also, it is noted that the CLS joint is often believed to have an energy release rate which is independent of the lap and strap length (cf. Brussat3S). This may be true when the specimen is very slender, as was the case in the Round Robin s5 example, but is clearly not so in the present ease as seen from Table 3. 6.2 T h e single lap shear (SLS) joint

The pinned-pinned SLS joint, like the CLS joint, experiences large rotations, and thus must be treated using the large deformation analysis. In order to do so, it was necessary to treat the joint as a homogeneous beam with two discontinuities of beam height at either end of the overlap resulting in three beam elements as shown in Fig. 7. The transverse

½

L1 ~i )h3

½

al.

I

Fig. 7. SLS joint pinned at both ends.

+ A2cosh\(

-E-~lal /] + A3a, + A4

(21)

u')(aO=Zssinh(iff~-i~2al) -E-i~2al+A7aI+A8

+A6cosh •

(22)

F

u~3)(a1) ~---A9 slnh( I ~ 3 a 1) +AloCOSh

al +Alia1 +A1:(23)

A system of twelve equations for the twelve unknowns (A1-AI2) results if the boundary conditions are used• At al = 0 and al = L3, the pins cause the transverse deflections and bending moments to be zero. At the two discontinuities of the beam height, continuity, equilibrium, and the cantilever assumption imply that the transverse deflections, shear forces and slopes of the beam elements (1) and (2) at a~ = Ll and (2) and (3) at a~ = L2, are equal. Equilibrium also implies that m(2)(L1 +) = m°)(LF)- FA~ and m(2)(L2 -) = m(3)(L2 ÷) - F A 2 , where the superscripts denote the respective beam sections, Li+ and Li- denote a location just to the right and to the left of the respective discontinuity in the beam thickness, and Ai is the distance between the respective centroids at cross-section i. The crack tip was at ax = L I , the bending moment, m¢l)(L1 -) acting on the lower adherend at the crack tip was found, and the energy release rate, J, was calculated using eqn (17). The phase angle, % was calculated using either Method 13~ or Method 2, 32 with both methods giving the same result for equal adherend specimens. As was the case with the CLS joint, the fracture load of SLS joints fabricated using the FPL-etched 7075T6/4523GB adhesive system was predicted by comparison with the fracture envelope, eqn (20). Fracture load predictions using both methods of mode partitioning were made for a wide variety (differing L~, L 2 and L3) of both equal and unequal adherend SLS joints and compared to the measured fracture loads of actual joints bonded using the adhesive system of Fig. 2. Some of the specimens were given a Mode I starter crack prior to testing, and some of the specimens were left with the adhesive spew fillet intact. For precracked joints, L~ was the adherend length to the crack tip. The results are listed in Tables 4 and 5. For the equal adherend specimens

Fracture load predictions for adhesive joints

m e a s u r e d values (average d i f f e r e n c e = 8 . 7 % ) than M e t h o d 232 (average difference = 30%). As can be seen f r o m the last three specimens in Table 4, the presence of the spew fillet did not significantly affect the fracture load.

Table 4. Comparison of measured and predicted fracture loads per unit width for equnl-adherend SLS joints. Adherend thickness was ~ inch (~12.5 ram). Specimen width was 20mm

L1

Lz

L3

(m)

(m)

(m)

Meas. (E) (kN/m)

Pred. (P) (kN/m)

Diff. (P - E)/E(%)

0.141 0-144 0.178 0.176 0.245 0.265 0.216 0.203 0.185 0-219 0.167 0.134 0.202 0.196 0-216 0.148 0.148 0.148

0.227 0.228 0.260 0-262 0-320 0.320 0.314 0.265 0.294 0-293 0-294 0.190 0-307 0-307 0.308 0-263 0.263 0.263

0.352 0.353 0.401 0.400 0-400 0.400 0.402 0.400 0.391 0.389 0.390 0.298 0-364 0.401 0.390 0-369 0.369 0.369

572" 555" 547b 525b 438a 391~ 497b 551a 544a 469a 565a 530~ 496~ 559~ 525a 574~ 595b 587b

550 543 533 525 417 383 473 468 518 443 563 488 451 511 456 573 575 574 Avg. I(P - E)/EI

-3.8 -2.2 -2.6 -0-0 -4.8 -2.0 -4-8 -15.1 -4.8 -5.5 -0.4 -7.9 -9-1 -8.6 -13"1 -0-2 -3.4 -2.2 5.0

595

6.3 The double strap (DS) joint T h e DS joint is symmetric, greatly reducing the peel stresses at the end of the overlap (Fig. l(d)). H o w e v e r , even in this symmetric joint, there are induced internal bending m o m e n t s in the strap adherends at the ends of the overlap which tend to peel the straps from the central adherend. The DS joint gripped in a pinned-pinned configuration (Fig. 8) was chosen for fracture testing. Because of symmetry, only half of the joint needs to be analysed, and the adhesive sandwich can be isolated as shown in Fig. 9. Far away f r o m the crack tip in either direction, the strain distribution will be uniformly tensile, and thus there will be no induced large rotations as was the case for the SLS and CLS joints. The energy release rate can be calculated using the J integral, eqn (8), along the path shown in Fig. 9. The contributions to the J integral are identical to those derived in Section 4 of this p a p e r with a possible additional contribution along path A-B in Fig. 9. However, along path A-B the slope is zero, and thus, by eqn (8), there is no

a Mode I starter crack. o Fillet left intact.

(Table 4), the average difference between measured and predicted fracture loads was 5 % , over a range of adherend lengths L1, L2, and L3 such that the fracture load changed from a low of 391 k N / m to a high of 5 9 5 k N / m . For the unequal adherend specimens (Table 5), the adherend lengths were varied such that the fracture load changed from 3 4 5 k N / m to 507 k N / m . T h e m o d e partitioning of Method 131 gave predicted fracture loads that were much closer to

Fig. 8. DS joint pinned at both ends.

Table 5. Comparison of measured and predicted fracture loads per unit width for unequal-adherend SLS joints. Method 1, sl Method 2:2 For aH specimens, h~ = ~ in. (=~6.27 ram) and hs = ~ in. (~12.51mm). AH specimens were given a Mode ! starter crack. Specimen width was 20 mm LI (m)

L2 (m)

L3 (m)

Meas.(E) (kN/m)

Pred.(P) (kN/m) Meth. 1

Pred.(P) (kN/m) Meth. 2

Diff. ( P - E)/E Meth. 1 (%)

Diff. (P - E)/E Meth. 2 (%)

0.190 0-192 0.209 0"227 0.205 0.254 0.264 0-260 0.175 0-188 0.201

0.260 0.259 0-307 0.307 0.306 0.324 0.324 0.323 0.256 0-256 0-256

0-405 0.393 0.432 0-430 0.431 0.424 0.434 0.424 0.406 0.369 0-407

429 429 490 445 484 345 355 362 507 448 402

399 383 428 403 433 344 339 332 424 391 378

309 294 331 308 335 258 255 248 331 302 292

-7-0 -10.5 - 12.2 -9.0 -10"3 0-0 -4-3 -8.5 -16-0 -12-4 -5.0

-28-0 -31.5 -32.4 - 30.8 -30.8 -25.2 -28.2 -31"5 -34.7 -32.6 -27-4

8.7

30.3

Avg. I(P - E)/EI

G. Fernlund et al.

596

sandwich giving

l

ffl

lineof symmetry

S(

-~

P ~/h*(hl +h~) F=-~ V h2(hl + h2)

I P/2

[ 12

The m o d e ratio was then calculated using both Method 131 and 2. 32 For large h~, the phase angle, lp, is approximately 52-2 °, using Method 2, whereas for Method 1, pure m o d e II (~p = 90 °) is predicted. Fracture load predictions were made by using eqn (24) in conjunction with the critical energy release rate given on the fracture envelope, eqn (20), using the phase angles predicted by Methods 131 and 232 for a wide variety of DS joints of differing LI = L~, L3, L4 and hi. The adhesive spew fillets were removed with a file on some specimens before they were tested, while others were given a Mode I starter crack, and still others were left with an intact spew fillet. For all specimens, L1 (Fig. 8) denotes the adherend length to the point where load transfer between the adherends start, i.e. the end of the joint overlap or the crack tip depending on the specimen preparation. The experimental results are summarized in Table 6. The fracture loads were found to be relatively uniform for constant strap thickness hi, as predicted. The average difference between the predicted and measured fracture loads was found to be much smaller (3.2%) for the m o d e partitioning of Method 1, 31 than that of M e t h o d 232 (31.6%). As in the case of the SLS joints, it was found that the presence of the adhesive spew fillet did not affect the fracture load significantly.

Fig. 9. DS joint sandwich element.

hi; P/2

h t iP2

~ =

h2

F

h2*

M

Fig. 10. DS joint sandwich element and equivalent large stiffness middle adherend. contribution to the J integral along this portion. The J integral thus reduces to eqn (17), and specifically p2(

J =~

h,

)

(25)

(24)

h2(h-~-4- hE)

for the case depicted in Fig. 9. F u r t h e r m o r e , because the centroid of the central m e m b e r cannot deflect in the a 2 direction, for t h e pui'pose of calculating the m o d e ratio, the joint was modelled as an adherend of thickness hi bonded to a very thick central m e m b e r h~' as shown in Fig. 10, with F adjusted so that the total energy release rate of the right sandwich was equal to energy release rate of the original (left)

6.4 The M o d e II tensile specimen A tensile Mode II specimen (Fig. 11) can be made if the DS joint is prevented from fracturing at the outer

Table 6. Comparison of measured and predicted fracture loads per unit widths for DS joints. Method 1, 3~ Method Specimen width was 20 mm

hi (mm)

6.37 6.37 6.37 6.37 6.35 6-36 6.36 6-37 6.37 6.37 6.36 12.76 12.77 12.77

2h2 (ram)

12.78 12.78 12.78 12-74 12.77 12.76 12.78 12.78 12.78 12-77 12.77 12-74 12.75 12.74

Li = L2 (m)

0.140 0.138 0.141 0.141 0.140 0-140 0.137 0.151 0.152 0.152 0.128 0-138 0.137 O.168

a Mode I starter crack. b Fillet removed with file. c Fillet left intact.

L3 (m)

0.283 0.282 0-282 0.232 0-189 0.189 0-188 0"202 0.203 0.203 0-308 0.280 0.280 0.279

L4 (m)

0.421 0.418 0.420 0.422 0.368 0.369 0.363 0.376 0"379 0.377 0.407 0.416 0-415 0-417

Pred. (P) Meth. 1 (kN/m)

Pred. (P) Meth. 2 (kN/m)

Meas.(E) (kN/m)

1620 1620 1620 1617 1620 1618 1620 1620 1620 1618 1620 1399 1400 1399

2058 2058 2058 2053 2059 2057 2059 2058 2058 2061 2057 1778 1779 1778

1622a 1606h 1626b 1667b 1549~ 1574c 1526c 1542c 1568c 1596c 1731h 1333h 1358c 1354a Avg. I(e - E)/EI

Diff.

Diff.

(P - E)/E

(P - E)/E

Meth. 1 %

Meth. 2 %

-0-1 0.9 -0-4 -3-0 4.6 2.8 6"1 5"0 3.3 1-4 -6.4 5.0 3.1 3-3 3.2

26-9 28-1 26-5 23-2 32.9 30"7 34.9 25-1 31.2 29.1 18"8 33-4 30-9 31-3 31.6

2 . 32

Fracture load predictions for adhesive joints L3 L 1"----'*I~:~ I~

l.al

~hI

Fig. 11. Tensile Mode II specimen pinned at both ends.

597

hiI P/2

=

Fig. 13. Tensile Mode II sandwich element and equivalent large stiffness middle adherend. compressive component will not extend the crack, it was subtracted from the total apparent energy release rate, eqn (26), resulting in

J= Ju=O.624J,pp=O.624 ~E (h,(h~+ h2)) line of symmetry

(27)

S(x 1)

Fig. 12. Tensile Mode II sandwich element.

end of the overlap by the use of two bolts. The specimen has induced bending moments giving rise to compressive transverse stresses at the end of the overlap. This creates a state of pure Mode II with a 'negative Mode I' component which should not contribute to the energy release rate (compressive stresses at the crack tip will not extend the crack). The joint was analysed in a manner similar to that used for the DS joint. Because of symmetry, only half of the joint was considered and the adhesive sandwich was isolated as shown in Fig. 12. The energy release rate was calculated by using the J integral, eqn (8), along the path shown in Fig. 12, as was the case with the DS joint. The contributions to the J integral along path A-B in Fig. 12 were zero because the slope was zero along the line of symmetry. The crack faces were not traction free, but the slope along the faces was zero, and thus there were no contributions from contact between the crack faces using the present beam theory model. The J integral thus reduced to eqn (17), and specifically

am, 8E \hi(hi + h2)]

(26)

for the case depicted in Fig. 12, where the subscript denotes that this is an 'apparent' value, as explained below. Because the centroid of the central member cannot deflect in the a2 direction, for the purposes of calculating the amount of 'negative Mode I', the joint was modelled as an adherend of thickness h l bonded to a very thick central member h~' as shown in Fig. 13. The mode ratio was then calculated using Method 1 (Method 2 cannot account for compressive stresses at the crack tip). For large h~', Method 1 predicted that there was a compressive Mode I component to the energy release rate at the crack tip. Because the

Fracture load predictions were made by comparing eqn (27) with the critical energy release rate given on the fracture envelope for pure Mode II conditions. The predicted fracture loads were compared to the measured fracture loads of actual joints bonded using the adhesive system described by Fig. 2. All specimens were notched with a sharp knife and observed through a microscope (40x magnification) due to the difficulty in observing the propagation of pure Mode II cracks with the naked eye. The results are summarized in Table 7. The average difference between the predicted and measured fracture loads was 11-1%, and thus it can be concluded that the 'negative Mode I' component has a negligible effect on the fracture of tensile Mode II specimens. 7

DISCUSSION

It is interesting to note that the initial conditions at the end of the bondline (spew fillet or Mode I starter crack) did not affect the fracture loads of the SLS and the DS joints. The reason for this is believed to be that the adhesive system exhibits a rising R-curve behaviour, i.e. the critical energy release ratio increases with crack length until steady-state conditions (fully developed failure zone ahead of the crack) are obtained. It was found experimentally that steady-state conditions, which were the conditions used in defining the fracture envelope (Fig. 2), were reached after approximately 5 - 1 0 m m of crack extension. By studying the bondline through a microscope, when testing the SLS and the DS joints, it was seen that a small amount of suberitical crack growth occurred at a load less than the final fracture load where the joints failed by unstable crack propagation. Thus, final fracture always occurred from a macroscopic crack with a fully developed failure zone due to the subcritical crack propagation, and therefore, the presence of an initial spew fillet or Mode I starter crack did not affect the fracture load significantly.

598

G. Fernlund et al. Table 7. Comparison of measured and predicted fracture loads for Mode !1 tensile specimens. All specimens were given a notch with a sharp knife. Specimen width 20 mm

h~ (mm)

2h2 (mm)

L~ = L2 (m)

L3 (m)

L4 (m)

Pred. (P) Meth. 1 (kN/m)

Meas. (E) (kN/m)

Diff. (P - E)/E Meth. 1 %

6.37 6.37 6-37 6-37 6.38 6.38 6.36 6-37 6-37 6.37

12.74 12.65 12.65 12.65 12-66 12.66 12.78 12.67 12-67 12.67

0-140 0.137 0.137 0.137 0.137 0-137 0.137 0.137 0.137 0.137

0-283 0.282 0.272 0-262 0.282 0.272 0.262 0.282 0.272 0.262

0.421 0.473 0-473 0.473 0.473 0.473 0.473 0.473 0.473 0.473

2599 2604 2604 2604 2606 2606 2606 2603 2603 2603

2909 3087 2948 2992 2248 2213 2304 2475 2450 2563 Avg.

-10.7 -15.6 -11-7 -13.0 15.9 17.8 13.1 5.2 6.2 1.6 11.1

l(p -

Batch to batch variability is often of concern in structural adhesive bonding, and was assessed in this investigation by testing multiple batches of typically four specimens. The number of batches for each specimen were: fracture envelope DCB-4, equal adherend CLS-5, equal adherend SLS-8, unequal adherend SLS-4, double strap-6, and tensile Mode II-2. Despite the inherent variability between batches, the overall agreement between the predicted and measured fracture loads was very good.

8 L I M I T A T I O N S OF T H E A P P R O A C H

The present method is intended as an engineering approach to fracture load prediction of structural adhesive joints. Certain assumptions and guidelines must be followed in order for the approach to work. (i) Although the present tests were only done using Cybond 4523GB, a mineral-filled structural epoxy, it is believed that the method will work for many structural adhesives. This has been borne out by subsequent tests with a rubber toughened epoxy. The main criteria for the applicability of the method are that the adhesive system exhibits a fracture envelope, Jc(~P), which is independent of the adherend thickness, and that viscoelastic or time-dependent effects be small. Finally, the adhesive should not contain a filler which settles under gravity during cure. Such an adhesive has marked property variations depending on the crack path. The adhesive used in this study (Cybond 4523GB) can be considered to be brittle and has a relatively low fracture toughness in comparison with rubber toughened epoxies. However, the fracture properties of many rubber toughened epoxies decrease with time, and thus the sustained loading an adhesive joint

E)/E I

can carry is often much smaller than the quasi-static strength. Experiments have shown that Cybond 4523GB exhibits only minor viscoelastic effects, and that the critical energy release rate for joints subject to sustained loading (for a few days at room temperature) is approximately 85% of the quasi-static value (Fig. 2), independent of the phase angle. Thus, the fracture load of a joint subject to sustained loading is less than 10% lower than the quasi-static fracture load for this adhesive system. (ii) The present method can only be used to predict fracture loads for adhesive joints fabricated with the specific adhesive system defined by the fracture envelope. It is thus important that the adhesive system parameters (i.e. adhesive layer thickness, adherend material, adhesive material, pretreatment, adhesive cure schedule) be kept constant for all specimens. Further studies are required to establish the tolerances on each system parameter. For example, earlier experiments showed that the fracture loads of the present adhesive system were virtually unaffected by bondline variations of +0.2 mm. 21 (iii) The present method assumes that the adhesive layer has a negligible effect on the deformation of the joint, so that joint elements may be treated as homogeneous beams and the adhesive layer can be neglected in the calculation of the adherend reactions. A rule of thumb is that the thickness of the adhesive layer divided by the adherend thickness (t/h) should be of the same order of magnitude, or less, than the adhesive modulus divided by the adherend modulus (Ea/E). Thus, the method is valid for many realistic joint geometries. (iv) The overlap (sandwich) length must be sufficiently long (at least two times the adhesive sandwich thickness), so that beam theory strain distributions dominate at some point in the overlap.

Fracture load predictions for adhesive joints

(v) The width of all joints should exceed about 15 mm, to ensure that there are no edge effects. 2~ "Work is presently underway to quantify conditions (iii) and (iv) more accurately. 9 CONCLUSIONS The present method for calculating the energy release rate per unit area of crack extension of a general adhesive joint (adhesive sandwich) was found to correlate well with both non-linear finite element calculations and experimental data. Two methods for mode partitioning were evaluated and comparison with both analytical and experimental data showed that the method of Suo and Hutchinson 31 gave the best agreement. The present analytical approach to fracture load predictions was found to correlate well with experimental data. The average relative difference between the predicted and the measured fracture loads were 5.7% for equal adherend CLS joints, 5.0% for equal adherend SLS joints, 8.7% for unequal adherend SLS joints, 3-2% for DS joints, and 11-1% for Mode II tensile specimens. The closed form nature of the approach together with its consistent and accurate predictions makes it well suited to the design of a large class of structural adhesive joints. ACKNOWLEDGEMENTS Thanks are due to Edward Wang for his assistance during the experiments, and to Nacan (Permabond) Products Ltd, American Cyanamid, Natural Sciences and Engineering Research Council of Canada and Ontario Research Incentive Fund for their financial support. REFERENCES 1. Kinloch, A. J., Adhesion and Adhesives. Chapman and Hall, New York, 1987, pp. 188-99. 2. Hart-Smith, L. J., Designing to minimize peel stresses in adhesive bonded joints, ASTM STP 876, ed. W. S. Johnson, American Society for Testing and Materials, Philadelphia, PA, 1985, pp. 238-66. 3. Harris, J. A. & Adams, R. D., Strength prediction of bonded single lap joints by non-linear finite element methods, Int. J. Adhesion Adhesives, 4(2) (1984) 65-78. 4. Adams, R. D., The mechanics of bonded joints. In Structural Adhesives in Engineering. IMechE Conference publications, Suffolk, 1986, pp. 17-24. 5. Crocombe, A. D., Bigwood, D. A. & Richardson, G., Analyzing structural adhesive joints for failure. Int. J. Adhesion Adhesives, 10(3) (1990) 167-78. 6. Czarnocki, P. & Piekarski, K., Non-linear numerical stress analysis of a symmetric adhesive bonded lap shear joint. Int. J. Adhesion Adhesives, 3 (1986) 157-60. 7. Matsui, K., Size effects on average ultimate shear stresses of adhesively bonded rectangular or tubular lap joints under tension-shear. Int. J. Adhesion Adhesives, 10(2) (1990) 81-9.

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8. Matsui, K., Effects on nominal ultimate tensile stresses of adhesive bonded circular or rectangular joints under bending or peeling load, Int. J. Adhesion Adhesives, 10(2) (1990) 90-8. 9. Kyogoku, H., Uchida, S. & Kataoke, Y., CAD system for strength evaluation of adhesively bonded joints. Advances in Adhesively Bonded Joints, ASME, New York, 1988, 117-28. 10. Crocombe, A. D., Global yielding as a failure criterion for bonded joints. Int. J. Adhesion Adhesives, 9(3) (1989) 145-53. 11. Hamaush, S. A. & Ahmad, S. H., Fracture energy release rate of adhesive joints. Int. J. Adhesion Adhesives, 9(3) (1989) 171-8. 12. Mall, S. & Johnson, W. S., Characterization of mode I and mixed mode failure of adhesive bonds between composite adherends. Composite Materials: Testing and Design (Seventh Conference), ASTM STP 893, ed. J. M. Whitney. ASTM, Philadelphia, PA, 1986, pp. 322-24. 13. Mall, S. & Kochlar, N. K., Criterion for mixed mode fracture in composite bonded joints. IMechE Conference Publications, 1986-8, C159/86, 1986, pp. 71-6. 14. Mall, S. & Kochlar, N. K., Characterization of debond growth mechanism in adhesively bonded composites under mode II static and fatigue loadings. Eng. Frac. Mech., 31 (1988) 747-58. 15. Cao, H. C. & Evans, A. G., An experimental study of the fracture resistance of bimaterial interfaces. Mech. Mater., 7 (1989) 295-304. 16. Suo, Z., Failure of brittle adhesive joints. Appl. Mech. Rev., 5 (1990) 276-9. 17. Thouless, M. D., Fracture of a model interface under mixed-mode loading. Acta Metall Mater., 38 (1990) 1135-40. 18. Suo, Z., Debond mechanics of brittle materials. Scripta Metall Mater., 38 (1991) 1011-16. 19. Wang, J. S. & Suo, Z., Experimental determination of interracial fracture toughness curves using brazil-nutsandwiches. Acta Metall Mater., 38 (1991) 1279-90. 20. Fernlund, G. & Spelt, J. K., Failure load prediction: I. Analytical method. Int. ]. Adhesion Adhesives, 11(4) (1991) 213-20. 21. Femlund, G. & Spelt, J. K., Failure load prediction: II. Experimental results. Int. J. Adhesion Adhesives, 11(4) (1991) 221-7. 22. Anderson, G. P., Brinton, S. H., Ninow, K. J. & DeVries, K. L., A fracture mechanics approach to predicting bond strength. In Advances in Adhesively Bonded Joints, ASME, New York, 1988, pp. 93-101. 23. Lee, S. M., An in-situ failure model for adhesive joints. J. Adhesion, 18 (1985) 1-15. 24. Groth, H. L., A method to predict fracture in an adhesively bonded joint. Int. J. Adhesion Adhesives, 5(1) (1985) 19-22. 25. Groth, H. L., Stress singularities and fracture at interface corners in bonded joints. Int. J. Adhesion Adhesives, 8(2) (1988) 107-13. 26. Bigwood, D. A. & Crocombe, A. D., Elastic analysis and engineering design formulae for bonded joints. Int. J. Adhesion Adhesives, 9(4) (1989) 229-42. 27. Fernlund, G. & Spelt, J. K., Mixed-mode fracture characterization of adhesive joints. Camp. Sci. & Technol. (in press). 28. van Karman, T., Festigkeitsprobleme im maschinenbau. Encyklopadie der matimatischen wissenschaflenn, 4 (1910) 339. 29. Atluri, S. T., Path-independent integrals in finite

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elasticity and inelasticity, with body forces, inertia, and arbitrary crack-face conditions. Engng Fract. Mech., 3 (1982) 341-64. 30. Fernlund, G. & Spelt, J. K., Analytical method for calculating adhesive joint fracture parameters. Engng Fract. Mech., 40 (1991) 119-32. 31. Suo, Z. & Hutchinson, J. W., Steady-state cracking in brittle substrates beneath adherent films. Int. J. Solids Struct., 25 (1989) 1337-53. 32. Williams, J. G., On the calculation of energy release rates for cracked laminates. Int. J. Fract., 36 (1988)

101-19. 33. Charalambides, M., Kinloch, A. J., Wang, Y. & Williams, J. G., On the analysis of mixed-mode failure. Int. J. Fract., 54 (1992) 269-91. 34. Fernlund, G. & Spelt, J. K., Mixed mode energy release rates for adhesively bonded beam specimens. J. Comp. Tech. & Res. (submitted). 35. Johnson, W. S., Stress analysis of the cracked-lap-shear specimen: an ASTM Round Robin, J. Testing Eval., 6 (1987) 303-24.