Precracking behaviour in the single cantilever beam adhesion test

Int J Fract (2011) 169:133–144DOI 10.1007/s10704-011-9586-x

ORIGINAL PAPER

Pre-cracking behaviour in the single cantilever beamadhesion test

S. Chauffaille · J. Jumel · M. E. R. Shanahan

Received: 29 May 2010 / Accepted: 10 January 2011 / Published online: 28 January 2011© Springer Science+Business Media B.V. 2011

Abstract The single cantilever beam adhesion testis a variant of the asymmetric wedge test in which aconstant load is attached to the free adherend end tooriginate the moment required for joint fracture. It hasthe potential disadvantage of leading to an uncontrolla-ble, accelerating crack, due to the constantly increasingapplied couple, but presents the advantage of providingdata on joint behaviour prior to crack initiation. It is thislatter aspect that we consider in this paper. An appar-ently decreasing crack growth rate, as obtained frommeasurements of displacement of the free end of thebeam is attributed to time-dependent adhesive strain.Use of classic, simple beam theory and Winkler, elasticfoundation, equations allows us to assess an effectivelystatic fracture energy, or fracture threshold.

Keywords Cantilever beam adhesion test ·Crack front · Fracture energy · Wedge test

1 Introduction

Several adhesion tests exist in which transfer of theenergy necessary to provoke joint failure occurs viastrain energy stored in either one or both adherends,behaving as bent beams. In the well known double

S. Chauffaille · J. Jumel · M. E. R. Shanahan (B)Institut de Mécanique et d’Ingénierie-Bordeaux (I2M),UMR 5295, Université de Bordeaux, 351 cours de laLibération, 33400 Talence, Francee-mail: [email protected]

cantilever beam (DCB) and tapered double cantileverbeam (TDCB) tests, it is principally external energysupplied from motion of an applied load to the adhe-sive joint by the linkage of the adherend beam(s)which causes failure (Mostovoy and Ripling 1966,1969; Wiederhorn et al. 1968; Kanninen 1974; Mai1976; El-Senussi and Webber 1984; Blackman et al.1991; Troczynski and Camire 1995; Jethwa and Kin-loch 1997; Meiller et al. 1999; Chen et al. 2001; Seneret al. 2002; Xu et al. 2004; Blackman et al. 2008; Bieland Stigh 2008). However, in the case of the wedgetest, in which the adherend(s) is (are) initially forced tocurve, it is the strain energy associated with this cur-vature which is the direct source of energy requiredto propagate fracture (Kanninen 1973; Cognard 1986;Sener et al. 2002; Blackman et al. 2003; Popineauet al. 2004; Sargent 2005; Budzik et al. 2009). Inthe latter case, two rectangular sheets of material arebonded together and a ‘wedge’ inserted at one end ofthe structure, in order to force apart the two adher-ends over a short initial distance. This leads to time-dependent separation in the remaining bonded region.Adhesion, or fracture, energy can be deduced as a func-tion of separation rate, or crack speed. This techniquehas been exploited both experimentally and numeri-cally.

The wedge configuration can be used either in theconstant displacement mode (wedge position fixed withrespect to the bonded joint), as described above, or inthe applied load mode. If an external, separation forceis applied (either by driving the wedge into the joint

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134 S. Chauffaille et al.

at a given rate, or by traction normal to the plane ofthe joint), the test is more akin to a DCB, although thislatter generally uses rather thick adherends. This con-stant speed arrangement is often used (Cognard 2000;Sener et al. 2002), although there is sometimes confu-sion about which analysis is relevant.

Yet another possibility is to apply a constant externalload, rather than constant rate of displacement. This testis the subject of the present paper. As in earlier work,we have opted for an asymmetric configuration, onebeam being relatively flexible, and initially adheringto a more rigid substrate, regarded as inflexible to allintents and purposes. A constant load is attached to thefree end of the flexible beam, and fracture ensues dueto motion of the load. We term this configuration thesingle cantilever beam adhesion test (SCB).

In the present case, we apply it to aluminium sub-strates bonded with an epoxy adhesive. The basicanalysis of this technique is straightforward and rel-atively well known: during crack propagation, theapplied moment increases due to the constant load act-ing through an increasing lever arm length, and, asa result, fracture rate (generally) increases. However,we have observed apparently decreasing crack growthrate. This leads to analysis of pre-cracking, or initiationbehaviour.

2 Experimental

2.1 Materials

The SCB test described in detail below involves theassembly of a thin aluminium plate (aluminium 2024T35) of thickness, h, 1.6 mm, width, b, 5 mm andlength 100 mm, bonded to a thicker support of the samematerial, of thickness, H , 8 mm, width, 29 mm andlength 103 mm, by the intermediary of an epoxy resinadhesive (Bisphenol A Diglycidyl Ether (BADGE)resin with N(3-dimethylaminopropyl)-1,3-propylene-diamine (DMAPPDA) hardener). Flexural rigidity ofthe thin plate is proportional to Eh3, where E isYoung’s modulus, and that of the support E H3. Withsimilar modulus of 73 GPa, the relative rigidity of thetwo aluminium components is given by H3/h3 = 125.This shows that we may reasonably take the thickercomponent to be, to all intents and purposes, rigid com-pared to the thinner member. The flexible adherend wasbonded to the support with a protruding extremity of ca.

18 mm, in order to affix a load and to have a free spacefor contact of the displacement transducer (see below).Bondline was maintained at a thickness of 0.1 mm byuse of PTFE spacers inserted at the extremities of thejoint during assembly. Curing was effected at 24◦Cfor 48 h under an applied pressure of 100 kPa. Prior tobonding, both aluminium surfaces were sand blasted[white corundum BF80 (ca. 200 µm )] and given asulphuric acid-ferric sulphate-based treatment (Digbyand Packham 1995; Prolongo and Ureña 2009). Jointswere left for typically 2 days before testing to equili-brate.

2.2 Method

The basis of the SCB test developed is to block theessentially rigid adherend with the thinner adherendunderneath, with the principal plane of the adhesivehorizontal, as shown schematically in Fig. 1. In thismanner, a constant load can be added at the freeend of the thin adherend causing a moment to act atthe start of the bonded area. A linear displacementtransducer (LVDT Transducer Type AX/5/S, Sensitiv-ity: 104.39 mV/V/ mm, Calibration range: ±5 mm) wasattached near the free extremity in such a way thatthe piston was in direct contact with the upper sideof the load-bearing part of flexible adherend, allow-ing displacement of the load to be measured withtime.

A typical experiment was started by gentle releaseof an attached load [in the range of 0.5–6 kg wt (5–60 N)] placed at 16–25 mm from the bond end. Thelinear transducer signal was recorded with time as thesystem was left to separate at a self-determining rate.The principal results presented here were obtained at atemperature of ca. 20◦C.

Fig. 1 Schematic representation of the single cantilever beam(SCB) adhesion test

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Pre-cracking behaviour in the single cantilever beam adhesion test 135

3 Simple beam theory and results

As shown in Fig. 1, with an applied load, F , acting onthe beam at the origin, x = 0, a moment of M = Fx isexerted at position x . This is true up to the limit of theintact, bonded joint at x = a(t), the time dependence,t , being determined by the position of the crack frontafter bond failure has commenced. The standard, sim-ple Bernouilli beam equation, M = E I/R ≈ E I z′′,where E is Young’s modulus of the beam material, Iis its second moment (bh3/12 : b is width and h thick-ness), R is the local radius of curvature and z is displace-ment perpendicular to the flat, beam plane, can then besolved, with boundary conditions of z(a) = z′(a) = 0.This leads to:

z(x) = F

E I

[x3

6− a2x

2+ a3

3

]

= �

2

[( x

a

)3 − 3( x

a

)+ 2

], (1)

where � = Fa3/(3E I ) represents the normal distanceof the extremity of the flexible member from the ini-tial plane, i.e. z(0). The use of Eq. (1) implies that thebonded part of the joint is rigid, or encastré, whichcan often be a very good approximation. We shall referto this as simple beam theory, or SBT. From a simpleenergy balance approach, it can be readily shown thatwhen such a structure fractures at, or near, the bond-line, the strain energy release rate, G, (and thereforethe fracture energy, Gc) is given by:

G = 6a2 F2

Eb2h3 . (2)

Thus, under conditions of constant load, F , the poten-tial strain energy release rate, G, increases with sep-aration length, a. Since strain energy release rate isequal to fracture energy at failure, and since fracturerate and fracture energy (generally) increase together,the technique above may be expected to produce anaccelerating fracture front, i.e. not only will da/dt bepositive, but also d2a/dt2.

Also of use is the fact that we may estimate cracklength, a, from the above treatment, using the expres-sion:

a =[

Ebh3�

4F

]1/3

, (3)

since both � and F are known.As mentioned above, a potentially serious disadvan-

tage of the SCB test is the rapid increase of energy

release rate, G, with increasing crack length, a. Thismakes the choice of geometrical and loading parame-ters delicate. A careful choice is required in order nei-ther to produce unstable and rapid propagation on onehand, nor very slow propagation in which significantgrowth cannot be detected in a reasonable time on theother. At the same time, we require to maintain theflexible adherend in a purely (linear) elastic state forthe analysis to be valid. Defining σY as the yield stress(or more strictly, the linear elastic limit) of the mate-rial constituting the flexible beam, it is simple from theabove equations to show that the maximal crack length,amax, before exceeding this limit is given by:

amax = bh2σY

6F, (4)

when σY is first encountered at the beam surface (y =± h/2) and at the crack front (x = a), where the bend-ing moment is greatest. Using this relation, it is readilyshown that the maximal permissible energy release rate(or fracture energy) is:

Gmax = hσ 2Y

6E= 6a2

max F2

Eb2h3 . (5)

Clearly, the lower limit of G is set by practical consid-erations of slower crack propagation at low values of a.

The usual way to interpret the experimental resultsobtained from this type of test is to plot a graph of cracklength, a, vs. time, t , obtain crack speed, da/dt, fromestimates of tangents to the a vs. t curve, and calculatestrain energy release rate, G, equal to fracture energy,Gc, by applying Eq. (2). Fracture energy is then givenas a function of crack speed. In the present case, a iscalculated from �, using Eq. (3).

Figures 2 and 3 represent both beam deflection, �,and crack length, a, calculated from SBT as a func-tion of time. In the examples given the loads, F , are28 and 60 N for initial crack lengths of respectively 16and 8 mm. The right hand sections of these graphs areof the form expected, as discussed in terms of the sim-ple theory above. Crack length increases with time andalso d2a/dt2 is positive until final joint failure occurs.However, the left hand parts, corresponding to t <

ca. 20 h (Fig. 2) and 150 h (Fig. 3), are unexpected.Certainly da/dt is positive but d2a/dt2 is negative.This is not consistent with a growing applied fracturemoment and concomitant crack acceleration. Notwith-standing, in Fig. 4 we present apparent fracture energy,GcSBT , (or energy release rate) vs. (log) crack speed,v = da/dt , as calculated from the above for the case

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136 S. Chauffaille et al.

Fig. 2 On left, beam deflection, �, and on right, crack position,a, as a function of time, t , estimated with SBT. The dotted linescorrespond to the initial, physical measurement of a before apply-ing a load (lower) and the initial value calculated by SBT (upper).The applied load, F , is 28 N

Fig. 3 As for Fig. 2, but applied load, F , is 60 N

of F = 28 N. For the higher section, corresponding tolonger times, Gc effectively increases with da/dt , asexpected, but for the lower branch, there is an appar-ent decrease in crack speed with increasing fractureenergy on the lower branch. A similar observation wasmade for F = 60 N. There is no reason to suspect thisbehaviour from the adhesive properties, or from strangephenomenological behaviour [such as ‘stick-slip’behaviour, as in rapidly unspooled pressure sensitiveadhesive tape (Maugis and Barquins 1988)]. We there-fore look deeper at the interpretation of the fundamentaltest.

Fig. 4 Energy release rate, G(= fracture energy, GcSBT ), vs.(log) crack speed, v = daSBT /dt , as evaluated by application ofSBT, for an applied load of 28 N. Note in particular the apparentdecrease in crack speed with increasing fracture energy on thelower branch. The heavy line corresponds to ‘reconstruction’ ofthe data using elastic foundation theory

Fig. 5 Schematic representation of the the adhesive extremity,or pre-crack front, in the elastic foundation model, showing adhe-sive strain perpendicular to plane of bondline

4 Elastic foundation

In SBT, allowance is made for bending of the freebeam, and thus displacement of the applied load, butit is assumed that the intact, bonded part, of the struc-ture is infinitely rigid: an encastré system. This maybe perfectly reasonable in many cases, but certainlynot in all. We therefore allow for the effect of an elas-tic foundation, in other words, strain in the adhesivenormal to the bondline and due to the applied bendingmoment. The general effect is presented schematicallyin Fig. 5. The initial idea for the following analysisstems from Winkler in the nineteenth century (Winkler1867), who studied reactions in railway ballast, but theconcept has since been developed several times in con-junction with adhesive bonds (Gillis and Gilman 1964;Kanninen 1973; Kendall 1973; Williams 1993; Kendall2001; Sargent 2005; Cotterell et al. 2006; Budzik et al.2011).We here present the rudiments necessary for

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Pre-cracking behaviour in the single cantilever beam adhesion test 137

application to the SCB test: more detail is given in(Budzik et al. 2011).

The load applied to the adhesive perpendicularly toits major plane, q(x), is given by q(x) = −kz(x),where k is stiffness (Nm−2) and z(x) is displacement.Application of standard, beam analysis procedure ofequilibrium of force and bending moment leads to thefollowing 4th order differential equations, respectivelyin the bonded and free sections:

d4z

dx4 + kz

E I= 0, a < x < ∞, (6)

d4z

dx4 = 0, 0 < x < a, (7)

recalling that I is the second moment of the beam sec-tion (bh3/12 ). These equations have solutions:

z(x) = eλ(a−x) [A1 cos λ(a − x) + B1 sin λ(a − x)] ,

a < x < ∞, (8)

z(x) = A2x3 + B2x2 + C2x + D2, 0 < x < a, (9)

where only two integration constants are given for thesolution of Eq. (6) since the other terms, in eλ(x−a),have been eliminated due to their having no physicalmeaning (they diverge as x → ∞ ). Equations (8) and(9) were previously obtained by Kanninen (Kanninen1973). The constant, or “wavenumber”, λ, is given by:

λ =√

2

2

(k

E I

)1/4

. (10)

If the adhesive were in a state of plane stress, k would begiven simply k = E Ab/e, where E A and e are Young’smodulus and thickness of the adhesive. However, theadhesive is mainly in plane strain (e/b � 1), so a betterestimate could be:

k = (1 − νA)

(1 − 2νA)(1 + νA)· E Ab

e= m

E Ab

e, (11)

where νA is the Poisson’s ratio of the adhesive. It shouldbe noted that this equation is somewhat approximatenear the crack front since, at best, three out of the foursides [orthogonally, following (x , z) and (joint width,z) planes] are constrained, and thus the zone is nottotally in plane strain. (The adhesive face in contactwith the environment cannot be constrained.) Also, asin classic elasticity theory (Timoshenko and Goodier1951), the denominator tends to zero for νA → 0.5,suggesting an infinite value of k in the case of an incom-pressible adhesive layer. This implies more complexload transfer mechanisms. In a related problem in civil

engineering, the ‘subgrade reaction method’ (SRM)has been used to show that the modulus of the sub-grade reaction is not only a bondline property, butalso depends upon the size and shape of the adher-end contact area. The problem is addressed exten-sively in soil mechanics and civil engineering (Terza-ghi 1955; Melerski 2000). This, however, is beyondthe scope of the present work, which focuses on themacroscopic, phenomenological description of visco-elastic behaviour in adhesive bonded joints. Here, acorrection coefficient, m (Eq. 11), is used to esti-mate the effective bondline tensile stiffness. In prin-ciple, m could be accurately computed using the Vla-sov model or others (Jones and Xenophontos 1976),but this would constitute a considerable and unwar-ranted digression from the main theme here. In adhe-sive bonding testing, the same notation is commonlyused (Kanninen 1973; Yamada 1987; Williams 1989),and m ≈ 2 has been found to be a suitable value whichfits well with experimental data. This increase in bond-line tensile stiffness, due to plane stain effects, will beassumed in the following, but we bear in mind thatthe main aim of this work is to measure the foun-dation stiffness k(t), and its evolution with time. Analternative approach, not employed here, to deal withincompressible, fluid media would be to model thefoundation by a lubrication approximation (more appro-priate for viscous adhesives) as done by (Ghatak et al.2004).

The constants A1, B1, A2, B2, C2, and D2 can befound from boundary conditions of geometry and con-tinuity of functions (Budzik et al. 2011), and deriva-tives up to order 3, leading to the rather complicatedexpressions:

z(x) = Fa3

2E Ieλ(a−x)

[(1 + λa)

(λa)3 cos λ(a − x)

+ 1

(λa)2 sin λ(a − x)

], a < x < ∞ (12)

z(x) = Fa3

2E I

[1

3

( x

a

)3 − x

a+ 2

3

]

+zW + θW (a − x), 0 < x < a (13a)

zW = Fa3

2E I

(1 + λa)

(λa)3 (13b)

θW = Fa2

2E I

(1 + 2λa)

(λa)2 (13c)

In fact, the shape of the free beam shape is unaffected(ignoring any small change in moment due to effective

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138 S. Chauffaille et al.

shortening of the lever arm with respect to the directionperpendicular to applied force, F , caused by rotation).The beam is simply slightly displaced by zW and rotatedby θW with respect to SBT. By letting λ → ∞, it isclear thatEq. (13a)becomesEq. (1).Thiscorresponds toinfinite adhesive modulus. It must also be noted that thevalue of a in the above, denoted henceforth aW , with Wfor Winkler, is appropriate for the elastic foundation cal-culation and is not for the earlier SBT version. The latterestimate will henceforth be referred to as aSBT . Valuesof aSBT and aW are only identical if E A = ∞ (andtherefore λ = ∞). Otherwise, aSBT > aW . The SBTsomewhat overestimates crack length. It can be shown,from the equilibrium in shear (Budzik et al. 2011), that:

F = E Id3z

dx3

= 6E I�

a3W

· λ3a3W

(2λ3a3W + 6λ2a2

W + 6λaW + 3),

0 < x < aW . (14)

A little rearrangement and use of Eq. (3) lead to an eval-uation of the overestimation of crack length by SBT:

ς = aW

aSBT=

[2λ3a3

W

(2λ3a3W + 6λ2a2

W + 6λaW + 3)

]1/3

.

(15)

Again, clearly as λ → ∞, then (aSBT − aW ) → 0.The above-mentioned slight displacement and rota-

tion of the free section of the beam, as found in the elas-tic foundation model compared to SBT, are of coursecloser to reality. More important, the directions of dis-placement and rotation are such that the deflection ofthe load, F (and the displacement transducer), will begreater than predicted by SBT for a given, real, cracklength. (This is the physical reason for SBT overesti-mating crack length.)

We now examine the left hand sections of thegraphs in Figs. 2 and 3, assuming that no crack growthoccurs, but that strain within the adhesive is time-dependent, particularly near the extremity which willlater become the crack front. Even without true crackgrowth (daW /dt = 0), further, time-dependent, strainin the adhesive, due to the local bending moment, isequivalent to reduction in the value of what amountsto the adhesive secant modulus, DA(t) [now replac-ing Young’s modulus, E A(t)]. This will lead to move-ment of the applied load and displacement transducer,the latter giving the impression of crack growth when

applying SBT [daSBT /dt > 0 : cf. Eq. (15) in whicha remains constant but λ diminishes]. In effect, wehave creep, or relaxation, of the polymeric adhesive,assuming all parts of the substrates to be below theirelastic limit. The observed behaviour is qualitativelyconsistent with creep, in which strain rate decreaseswith time (at least, in the primary stage). We shall there-fore attempt to analyse the apparent, or pseudo-crackgrowth behaviour with these basic assumptions.

5 Pseudo-crack and crack propagation

With c = λaW , we may rearrange Eq. (15) into thefollowing cubic equation:

2

[1 −

(aSBT

aW

)3]

c3 + 6c2 + 6c + 3 = 0. (16)

In the present case, aW = ao = 16 mm, which isthe initial, physically measured value of moment armor ‘crack length’, before any propagation. Followingour hypothesis above, although aW remains constant,aSBT will evolve: not because of true crack growthbut because of time-dependent strain in the adhesive.Equation (16) has one real root, but is more conve-niently solved numerically, using the time-dependentvalues of aSBT . We are led to the behaviour shown inFigs. 6 and 7, which corresponds to λ, i.e., c/ao, vs.elapsed time, t , both on logarithmic scales. Applicationof regression analysis shows that a very reasonable phe-nomenological description (better in Fig. 6 than Fig. 7),corresponding to the added line in Fig. 6 (F = 28 N)and Fig. 7 (F = 60 N), is given by:

λ(t) ≈ 0.0028t−0.19, (17a)

and

λ(t) ≈ 0.0020 t−0.16, (17b)

where relations (17a) and (17b) are valid up to, respec-tively, ca. 20 and 150 h, at which times dGc/dv

becomes positive [v is (apparent) crack speed,daSBT /dt]. (Incidentally, since the initial bendingmoments in the two cases are similar, such a differencein induction times is somewhat surprising, and may berelated to (an) individual defect(s) near the adhesiveextremity in the joints studied, being more marked inthe former case. However, this will not be pursued, themajor point here being the reproducibility of the gen-eral phenomenon of decreasing apparent crack rate dueto adhesive strain.)

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Pre-cracking behaviour in the single cantilever beam adhesion test 139

Fig. 6 Elastic foundation parameter, or ‘wavenumber’, λ, vs.elapsed time, t , for the pre-cracking regime (both on logarithmicscales). (F = 28 N, ao = 16 mm )

Fig. 7 Elastic foundation parameter, or ‘wavenumber’, λ, vs.elapsed time, t , for the pre-cracking regime (both on logarithmicscales). (F = 60 N, ao = 8 mm)

It is clear that λ, and therefore the adhesive secantmodulus, DA(t), decrease with time. This is consistentwith creep, or relaxation. Having determined λ(t), wemay return to Fig. 4, insisting that the value of a inquestion is indeed aSBT . From Eq. (15), it is possibleto evaluate an apparent crack speed by differentiation:

daSBT

dt= − ao

ς2 · dς

dt= −ς2a2

o(3 + 4c + 2c2)

2c4 · dλ

dt.

(18)

Since dλ/dt has been obtained empirically (Eqs. 17a,17b), it is possible to ‘reconstruct’ the section of Fig. 4in which dGc/dv is negative, bearing in mind that the

ordinate is the estimate of fracture energy as given byEq. (2), GcSBT (and using aSBT as an estimate of cracklength). The solid line shown in Fig. 4 corresponds tothis reconstruction. As can be seen, the agreement isexcellent. The argument is self-consistent and explainsexperimental measurements, which the earlier, sim-ple theory is incapable of doing. Additional evidencecomes from the fact that optical examination of the‘crack-front’ in the pre-cracking period suggested theabsence of any real cracking.

We should examine the range of values of λ givenin Figs. 6 and 7. (see Sect. 7 for a more detaileddevelopment.) From Eqs. (10) to (14), and the fact thatlocal (elastic) stress at the bond extremity is given byσ(aW , t) ≈ mDA(t)z(aW , t)/e, after some algebra,we find that:

σ(aW , t) = m DA(t)z (aW , t)

e

≈ 2F

baWλaW (1 + λaW ), (19)

where the approximation on the right makes use of thetime-dependent version of Eq. (10) [λ = λ(t), and k =k(t)], consistent with our approach of employing λ val-ues to evaluate the effective, or average, quantities ofthe system. With the F = 28 N case, taking aW =ao = 16mm, and a typical value of the epoxy resinmodulus as E A = 1GPa (tensile tests on dumbbellsamples of the bulk adhesive suggest E A = ca. 0.6, 1and 1.5 GPa respectively, at nominal strain rates of 4 ×10−3, 4 × 10−2 and 4 × 10−1min−1), Eq. (19) gives avalue of local tensile stress of ca. 75 MPa (ca. 66 MPafor νA = 0.3, and ca. 83 MPa for νA = 0.4). In the caseof F = 60 N, we obtain a similar figure of between ca.70 and ca. 79 MPa, depending on νA. This purely elasticcalculation only applies to the instant t = 0, when theload is applied, since it does not take into account anycreep relaxation phenomenon. With this assumption,the strain is found to be ca. 7.5% strain, which is wellabove that expected at the linear elastic limit, or yieldpoint, of an epoxy resin. It is therefore reasonable tosuppose that inelastic, or creep, behaviour occurs vir-tually instantaneously after application of load. Thismay well explain the (relatively) large, apparent, initialcrack length of 20 mm found using SBT (which leadsto an unrealistically low value of E A, of the order of43 MPa).

Unfortunately it is both difficult to apply load‘instantaneously’ (without introducing shock to thesystem), and to estimate consequent loading rate.

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140 S. Chauffaille et al.

Clearly this observation leads to the necessity ofconsidering the equations developed as beingapproximations near the crack front since linear elas-ticity is not strictly obeyed. Note that the estimate ofE A can change radically over a fairly small range, sinceas aSBT → aW , E A → ∞.

Under conditions of ‘pseudo-cracking’, the conceptof fracture energy takes on a different meaning, whichwe shall discuss below. But let us briefly considerthe consequences of application of the modified the-ory, once cracking, or fracture in the bondline, hascommenced. For a linearly elastic system (only bulkbehaviour of the flexible substrate is considered whenevaluating fracture energy), stored strain energy isequal to F�/2. Thus, using Eq. (14), it is possible tocalculate strain energy release rate, GW , leading to:

GW = −1

b· d

daW

[F�

2

]

= 18λ4�2 E I (λaW + 1)2

b(2λ3a3W + 6λ2a2

W + 6λaW + 3)2. (20)

After simplification, this leads to:

GW = 6a2W F2

Eb2h3

[1 + 1

λaW

]2

. (21)

Clearly, Eq. (21) is the same as Eq. (2), but for the cor-rection factor of [1 + 1/(λaW )]2. (The value aW = ao

only applies to the pseudo-crack behaviour). Therefore,provided we have the true value of aW to work with,we obtain the ratio of fracture energies from the twotheories:

η = GW

GSBT=

[1 + 1

λaW

]2

> 1. (22)

Equation (21) has been used to obtain fracture energyfor the elastic foundation model, using the data ofFig. 2. The evolution of GW with crack speed in thetrue cracking regime is presented in Fig. 8.

For ‘credible’ values of crack lengths, the elasticfoundation treatment gives a higher estimate of frac-ture energy. However, if ‘pragmatic’ values are usedwith SBT (i.e. those obtained by applying SBT), therewill be some self-adjustment to the calculated fractureenergy (Budzik et al. 2011). In practice, with relativelyrigid systems (large λ) and/or long cracks, the correc-tion will be small.

Fig. 8 Energy release rate, G(= fractureenergy, GcSBT ), vs.(log) crack speed, v = daW /dt , as evaluated by application ofelastic foundation theory (F = 28 N)

6 Dugdale model for ‘static’ fracture energy

Until fracture commences, although we may estimatea strain energy release rate, G, for the materials andgeometry of the test, we cannot associate this with frac-ture energy, Gc, since fracture has not yet occurred.The above amounts to a global approach to brittle frac-ture. However, local approaches exist, and we shalldevelop one along the lines of (Dugdale 1960), inview of exploiting the pre-crack behaviour observedand explained above. The approach amounts to ‘creep-crack growth’: before the onset of true cracking, time-dependent strain near or at the crack front occurs(Hoff 1954; Barenblatt 1962; Schapery 1965; Knauss1972; Schapery 1975a, b, c; Chudnovsky and Moet1986; Saxena 1986; Bradley et al. 1998; Flavio andDavid 2010).

Now considering that λ(t) is a function of time, andthat the geometrical position of the crack tip, or cracklength, aW , remains constant, relation (13b) indicatesthe evolution of the crack opening displacement.

z(aW , t) = [1 + λ(t)aW ]

[λ(t)aW ]3 · Fa3W

2E I. (23)

The fundamental hypothesis here consists of assum-ing that the kinematic in the process zone remainsunchanged, so that the evolution of the deflection issimilar to that obtained with the Winkler elastic foun-dation model. In the general case, both λ = λ(t) andaW = aW (t). According to the Dugdale model for frac-ture in viscous type media, we may estimate fractureenergy from displacement at the crack tip in the direc-

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Pre-cracking behaviour in the single cantilever beam adhesion test 141

tion of the applied load. Dugdale envisaged the ele-gantly simple expression of σY ·δ where σY is the yieldstress and δ is a (critical) crack opening displacement(COD) (Saxena 1998; Anderson 2005). Our case is alittle more complex, but by considering that the evolu-tion of the secant modulus, DA(t), is controlled by thatof λ(t), according to relation (10), and that reductionof this quantity corresponds to creep (stress relaxation),then we may estimate fracture energy as the local workrequired in deforming the polymer up to the onset of(true) crack growth. This is a slight approximation butgives a tractable form:

Gc =z f∫

zo

z(t, aW )

e· m DA(t) · dz

≈z f∫

zo

z(t, aW ) · 4λ4(t)E I

b· dz, (24)

where z f and zo are displacement values of the adhe-sive, respectively at the onset of fracture and when theload is applied. There is an implicit time dependenceof both z(t) and DA(t) (and also, in principle, m ) inEq. (24). Indeed, to account for the creep/relaxationprocess in the nucleation zone, the empirical Winklerfoundation analysis is used again, replacing E A withsecant modulus DA(t) :

DA(t) ≈ 4eE I

mbλ4(t) (25)

To overcome the difficulty of describing precisely evo-lution of the stress field in the process zone (or zoneof significantly strained adhesive), the local secantmodulus evolution is estimated from that of λ(t).This approximation seems reasonable since the globalresponse is very sensitive to crack tip evolution wheremaximum stress and strain are likely to occur.

By making use of Eqs. (10), (11), and (23), Eq. (24)may be expressed as:

Gc = − F2

E I b

λ f∫λo

[3

λ3(t)+ 5ao

λ2(t)+ 2a2

o

λ(t)

]dλ

= F2

E I b

{3

2

[1

λ2f

− 1

λ2o

]+ 5ao

[1

λ f− 1

λo

]

+ 2a2o ln

[λo

λ f

]}. (26)

Using the definitions co = λoao and c f = λ f ao

respectively for the beginning of loading and the

state just preceding crack initiation, together withI = bh3/12, Eq. (26) may be written as:

Gc = 6a2o F2

Eb2h3

{3

[c2

o − c2f

c2oc2

f

]+ 10

[co − c f

coc f

]

+4 ln

[co

c f

]}, (27)

which bears direct comparison with Eq. (2) (allowingfor the equivalence of G and Gc ). The term in bracesin Eq. (27) is equivalent to a multiplying factor ofEq. (2). Expression (27) may be considered to representthe viscous fracture energy at the onset of crack growth,albeit by a different mechanism. Until the crackingcommences, energy absorption is principally by plasticand/or viscoelastic strain of the adhesive. Once crack-ing is underway, there is also certainly energy absorp-tion by such deformation processes (otherwise fractureenergy would be close to the Dupré, reversible energyof adhesion or cohesion, which rarely exceeds 1 Jm−2,even in the most favourable cases), but since some formof separation (either adhesive or cohesive) takes place,the phenomenon will be different.

It is of interest to estimate a value of Gc fromEq. (27), which we shall now term GcI N I T for fractureenergy up to crack initiation. In the present preliminarystudy, there is some doubt about the values of someparameters, but reasonable estimates will be used.

For the case of λo, Eq. (10) may be expressed as:

λo =[

3m E A

Eeh3

]1/4

. (28)

Despite the rate-dependence of E A, a reasonable esti-mate is ca. 1 GPa. Similarly, the thickness of the bond-line is uncertain, but a value of 1 × 10−4 m is notunreasonable. With values of the other parameters asgiven in the experimental section, we obtain an esti-mate of λo at ca. 500 m−1. (In fact, the value of isnot very sensitive to these values, due to the powerof 1/4). From Figs. 6 and 7, we may estimate λ f atca. 190 and 160m−1. With the initial ‘crack length’ of16 and 8 mm, force F of 28 and 60 N, and other val-ues given before, we estimate, respectively, GcI N I T atca. 1,300 and 2,300 Jm−2. There is some doubt aboutthe initial value of λo to be used in Eqs. (26) or (27),since the initial deformation will be elastic, and we areconcerned with irreversible strain. Notwithstanding,GcINIT is large compared to the range of dynamic valuesof Gc presented in Fig. 4. It would seem that the initi-ation of crack formation requires greater energy input

123

142 S. Chauffaille et al.

than does propagation. This is perhaps not unexpectedsince initiation requires the germination of a stress con-centration, whereas during propagation, the crack, andtherefore the stress concentration, are already present.This is perhaps comparable to the well known distinc-tion between static and dynamic coefficients of fric-tion. In fact, this further need of energy for initiationis quite in agreement with Griffith’s early experimentson drawn glass rods, when fracture mechanics was inits very early stages (Griffith 1921).

As mentioned above, in passing, it should also benoted that this drawing, time-dependent behaviour ofthe adhesive near the crack front should not be restrictedto pre-fracture, but almost certainly continues, albeit toa lesser degree, during fracture. This should be clearfrom the fact that a complete description of the timedependence of z, at given x , is given by:dz(x, t)

dt= ∂z

∂λ· dλ

dt+ ∂z

∂a· da

dt. (29)

However, in the pre-cracking regime, the term in∂z∂a · da

dt is zero. The extent of drawing would be dif-ficult to ascertain in general, since this process occursconcomitantly with fracture.

7 Stress strain behaviour at pseudo-crack front

The time-dependent strain at the pseudo-crack front, oradhesive extremity, before cracking may be consideredin the light of knowledge of the local stress. However,since we are not certain about adhesive thickness, e (atleast in the general case), rather than consider strainrate directly, it is preferable to study rate of changeof thickness, e = z(a0). Substituting c = λaW = λao

into Eq. (23), and differentiating with respect to time, t ,we obtain an expression for z(a), i.e. elongation of theadhesive layer at a = ao and perpendicular to the planeof the bondline:dz(a)

dt= z(ao, t) = − Fa3

o

2E I

[3 + 2c(t)]

c4(t).dc

dt. (30)

At the adhesive extremity, z(ao), prior to its becomingthe place of initiation of separation, the local tensilestress in the adhesive, perpendicular to the bondline isσ(aW ) = m · DA · z(aW )/e [see Sect. 5 and Eq. (19)].Use of Eq. (23), together with relations (10) and (11)and the definition c(t) = λ(t)ao, allows us to write:

σ(ao, t) = [1 + λ(t)ao]

λ3(t).m E A(t)F

2E I e

≈ 2F

aob.c(t) [1 + c(t)]. (31)

Fig. 9 Adhesive elongation rate (left), z(ao, t), and stress (right),σ(ao, t), at the adhesive extremity (a = ao) vs. time, t , duringthe pre-cracking period

Note that the right hand member obviates the need forknowledge of thickness, e.

The time-dependence of both e = z(a0, t) andσ(ao, t) is shown in Fig. 9. Elongation rate is (rela-tively) high at the application of load (ca. 4–5 µms−1),and tails off asymptotically towards 20 h. Local stressduring the pre-cracking period appears to decrease ina similar manner to an asymptote of ca. 6 MPa, dueto relaxation of the adhesive. This is to be expectedboth from geometrical and material considerations ofthe system. Clearly, the onset of cracking will occurhereafter. It would be interesting to consider the time-dependent behaviour of stress at constant strain, as isoften done with polymeric materials, but unfortunately,this type of experiment does not lend itself readily to thepresent type of treatment, since the simultaneous var-iation of stress and strain is an intrinsic feature of theadhesive joint. Notwithstanding, a point made above,but worth reiterating, is that assessment of elongationrate and local stress near the adhesive extremity can bemade without knowledge of adhesive thickness. Thiscould be particularly useful in cases where the pres-ence of adhesive fillets render it difficult to estimateadhesive thickness with any precision.

8 Conclusions

A variation of the wedge test has been investigated,in which a flexible adherend is bonded to an essen-

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Pre-cracking behaviour in the single cantilever beam adhesion test 143

tially rigid member by a structural adhesive, and sep-aration is induced by a constant load attached to anextremity of the former, leading to a bending moment.We refer to this as a single cantilever beam (SCB)test. Crack length was deduced from measured, rela-tive movement of the loaded part of the flexible beam.Classic beam theory was applied to exploit data togive energy release rate, equivalent to adhesive frac-ture energy, as a function of separation rate. For longertimes of load application, the expected behaviour ofincreasing fracture rate due to increasing lever armlength, and thus moment, was observed, but at shortertimes, apparent decrease in fracture rate was found.The simple model used assumes that the intact, bondedzone may be treated as rigid; the beam is thus en-castré. However, a more detailed analysis, using elas-tic foundation assumptions for the intact zone, revealsthat the apparent crack growth, deduced from move-ment of the applied load, is in fact due to gradual dis-placement and rotation of the flexible beam at constantcrack length, associated with time-dependent strain inthe adhesive near the extremity of the bonded zone.Further analysis of the behaviour allowed exploitation,after modification, of the Dugdale model for fractureenergy to be made. Static (as far as crack growth is con-cerned) fracture energy can be determined, and is asso-ciated with viscoelastic or plastic strain of the adhesiveprior to the onset of true separation. In the case stud-ied, the value of this fracture energy is greater thanthe initial cracking energy, and constitutes a thresh-old to bond separation. An analogy with the differ-ence between static and dynamic coefficients of frictionis made.

Acknowledgments The authors thank one of the referees foruseful comments which have allowed us to improve notably thisarticle.

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