Int J Fract (2009) 156:111–128DOI 10.1007/s10704-009-9339-2

ORIGINAL PAPER

Rate effects in mode-II fracture of plastically deforming,adhesively bonded structures

C. Sun · M. D. Thouless · A. M. Waas ·J. A. Schroeder · P. D. Zavattieri

Received: 6 August 2008 / Accepted: 19 March 2009 / Published online: 16 May 2009© Springer Science+Business Media B.V. 2009

Abstract Results from a combined experimental andnumerical investigation into the effects of rate on mode-II fracture of a plastically deforming, adhesively bondedjoint are presented. It is shown that a cohesive-zonemodel has to be modified to include coupling betweennormal and shear modes of deformation when there isextensive shear deformation of the adhesive layer. Asuitable cohesive-zone modeling strategy is described,and the mode-II cohesive parameters determined fromthe model are presented as a function of loading rate.Previous studies of the same system showed that theeffects of rate in mode-I were limited to the proba-bility that a crack growing in a toughened quasi-staticmode would spontaneously make a transition to a brit-tle mode of fracture. No such transitions were foundfor mode-II fracture. Crack growth always occurred in

C. Sun · A. M. Waas (B)Department of Aerospace Engineering, Universityof Michigan, Ann Arbor, MI 48109, USAe-mail: dcw@umich.edu

M. D. Thouless (B) · A. M. WaasDepartment of Mechanical Engineering, Universityof Michigan, Ann Arbor, MI, 48109, USAe-mail: thouless@umich.edu

M. D. ThoulessDepartment of Materials Science and Engineering,University of Michigan, Ann Arbor, MI 48109, USA

J. A. Schroeder · P. D. ZavattieriGeneral Motors Research and Development, 30500 Mound Rd.,Warren, MI 48090, USA

a quasi-static fashion. While there was some evidencethat rate might affect the mode-II fracture parameters,these effects were very limited even up to crack veloci-ties of about 1,000 mm/s. Any possible effects was lim-ited to a very minor increase in toughness and strengthwith increased loading rates. However, the magnitudeof these possible increases were comparable to the mag-nitude of the uncertainties in the measured values.

Keywords Cohesive zone · Bonded joint ·Finite elements · Experiments

1 Introduction

Adhesive bonding exhibits many advantages over tra-ditional joining techniques for automotive applications.These include reductions in weight and, hence, anincrease in fuel efficiency. However, the current useof adhesive bonding in the automotive industry is lim-ited because of the lack of a suitable design meth-odology. The present research is part of an on-goingeffort to develop such a methodology, with an empha-sis on determining mixed-mode cohesive parametersas a function of loading rate. Earlier work of Sun et al.(2008a,b) explored the effects of rate on mode-I frac-ture of plastically deforming, adhesively bonded sheetmetal. The specimens in that study failed either by fullytoughened quasi-static crack growth, or by a “stick-slip” type of behavior with quasi-static crack growthbeing interrupted by intermittent periods of relatively

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brittle dynamic fracture. The experiments indicated thatthe cohesive parameters for mode-I quasi-static crackgrowth were independent of rate, and that quasi-staticcrack growth could occur even at crack velocities ashigh as several meters per second. Effects of rateappeared to be limited to the ease with which a transi-tion to dynamic fracture could be triggered. The presentpaper extends the study to mode-II, whilst a companionpaper (Sun et al. 2009) demonstrates how the mode-Iand mode-II parameters can be combined to exploremixed-mode fracture at different rates.

There is a long history of studying mode-II fracturein adhesive joints (Chai 1988). Much of this work hasfocused on the use of linear-elastic fracture mechanics(LEFM) to develop geometries to measure the mode-II toughness (Carlsson et al. 1986; Edde and Verr-eman 1995; Qiao et al. 2003; Blackman et al. 2006),and on the effects of friction during mode-II fracture(Bulchholz et al. 1997; 2001; Sun and Davidson 2006).In related work, LEFM techniques have also been usedto investigate the effects of rate on the mode-II fractureof composite structures (Jacob et al. 2005). It is notedthat no unambiguous relationship between rate and themode-II toughness resulted from these studies, withdifferent studies showing either increases, decreases orno effect of rate on toughness (Caimmi et al. 2006;Giambanco and Scimemi 2006; Liechti and Wu 2001).

Cohesive models of fracture (Ungsuwarungsri andKnauss 1987; Tvergaard and Hutchinson 1992), pro-vide additional flexibility for fracture analysis by theintroduction of a characteristic strength. Applying acohesive-zone modeling strategy to adhesive joints byidentifying the cohesive parameters with the deforma-tion of the entire adhesive layer, permits fracture to beanalyzed in situations which cannot be described byLEFM. In particular, it is possible to model fracturewhen there is plasticity in the adherends (Yang et al.1999). The major issue is the identification of the cohe-sive parameters. Two general approaches to determinethe cohesive properties and the traction-separation lawsof adhesive joints have evolved in the literature. Inone approach, based on the J -integral, a linear-elasticgeometry is used and the J -integral is calculated fromthe geometry and loads or displacements. This valueof J -integral is plotted as a function of the crack-open-ing displacement, and the slope is taken as a measureof the crack-tip tractions. Examples of this approachfor mode-II studies are given by Leffler et al. (2007)and Zhu et al. (2009). While this approach gives a

rigorous result for the shear tractions as a function ofshear displacement, geometries for which the J -inte-gral is valid are, by definition, geometries that are notsensitive to the magnitude of the cohesive tractions.Furthermore, it is not obvious that cohesive parame-ters must remain unchanged between conditions underwhich they can be determined by use of the J -integraland conditions under which the J -integral may not bevalid, such as in a plastically deforming structure. Analternative approach in which the fracture behavior ismatched empirically to numerical predictions based ondifferent cohesive parameters (Yang et al. 2001) avoidsthis potential problem. However, this latter approachrequires a recognition that there may be multiple setsof cohesive laws that provide fits to single sets of data(Sun et al. 2008b); multiple experimental observationsare then generally required to identify a unique set ofparameters that describes the physical attributes of theadhesive layer.

The present paper uses a cohesive-zone approach toexplore the effects of rate, up to low-velocity impactloading, for mode-II fracture of plastically-deformingadhesive joints. A series of experimental observationsare described from which the mode-II cohesive param-eters are determined. In addition, issues associated withmodeling shear by means of a cohesive-zone model aredescribed. In particular, it is shown that when the adhe-sive layer is relatively thick and the shear strains are rel-atively large, coupling of the normal and shear modes ofdeformation may be required to describe the behaviorof the adhesive layer with sufficient accuracy. A mod-ified cohesive model is developed and used to deter-mine mode-II cohesive-zone parameters as a functionof loading rate. These parameters have subsequentlybeen used in a companion paper to model mixed-modefracture (Sun et al. 2009).

2 Experimental results

The steel/adhesive/steel specimens used in this researchwere made by bonding coupons of a dual-phase steelwith a layer of a commercial, rubber-toughened, epoxy-based adhesive cured at 180◦C for thirty minutes. Thethickness of the adhesive layer was fixed at 0.8 mm bymeans of incorporating glass beads into the adhesive.The properties of the steel coupons (after the curingschedule) and of the bulk adhesive have been summa-rized in a previous paper (Sun et al. 2008a).

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Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 113

2.1 Three-point-bending, end-notched-flexure tests

The deformation at the tip of a crack in an edgenotched-flexure (ENF) specimen is dominated by shear. There-fore, this geometry provides an excellent configurationto study mode-II fracture with elastic (Barrett and Fos-chi 1977; Carlsson et al. 1986) and plastic deformation(Yang et al. 2001). The geometry of the ENF speci-mens used in this study is shown in Fig. 1. A steelwire with a diameter of about 0.8 mm was inserted asa spacer between the steel arms at the crack mouth.Teflon� tape was used to define the initial extent of thecrack, and a scale was attached to the side of the spec-imens. The specimens were tested in a servo-hydrau-lic machine under displacement control. Four tests at aloading rate of 0.1 mm/s, four tests at a rate of 10 mm/s,and five tests at a rate of 200 mm/s were performed.A high-resolution CCD camera was used to monitorcrack propagation and to calibrate the cross-headdisplacement.

Fig. 1 Configuration of the three-point bending, end-notchedflexure (ENF) specimens. The thickness of the steel cou-pon is h2 = 1.42 ± 0.02 mm; the thickness of the adhe-sive is h1 = 0.8 ± 0.1 mm; the width is W = 20.0 ± 0.5 mm;the initial crack length is a = a0 = 10 ± 1 mm; the halfspan is L = 30 ± 1 mm; and the diameter of cylinders isD = 12 ± 0.5 mm

An example of the optical micrographs taken dur-ing the tests is shown in Fig. 2. These optical obser-vations and visual inspections of the specimens aftertesting confirmed that the crack always propagated at(or close to) the interface between the adhesive andthe top adherend (the arm onto which the central loadwas applied). This interfacial mode of failure resultedfrom the mode-II loading on the specimen, and was incontrast to the failure observed in pure mode-I, wherethe crack ran through the middle of the adhesive layer.A crack trajectory along the top interface is consistentwith what would be predicted from a linear-elastic frac-ture mechanics analysis of the ENF geometry. In somespecimens, the initial defect lay along the top interfacealong which it eventually propagated. In other spec-imens (as in Fig. 2), the initial defect lay along thebottom interface and a kink formed to the top inter-face. Since the mode-II toughness of the interface wasmuch higher than the mode-I toughness of the adhe-sive, any kink from the bottom surface was arrestedat the top interface until the load increased to a suf-ficient level for subsequent mode-II delamination tooccur. The mechanics of crack kinking in sandwichedadhesive layers of finite thickness is dictated by thecompetition between the mode-I cohesive toughnessof the adhesive, the interfacial fracture toughness, thegeometry of the specimen and the mode-mixity at thecrack tip, as discussed in Akisanya and Fleck (1992),Xie et al. (2005) and Li et al. (2006a,b).

Plots of the applied load against mid-point deflectionare shown in Fig. 3. As indicated on the plots, the crackswere observed to start growing along the top inter-face just before the peak load. However, as observedin other three-point bending tests (Leffler et al. 2007)there was no obvious effect of this delamination on the

Fig. 2 Micrograph of theexperimental arrangementfor the three-point bendingENF test. This image showsa crack that has kinked tothe top interface

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0.1 mm/s

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Fig. 3 Load plotted against mid-point deflection for the ENFtests at three displacement rates (0.1, 10 and 200 mm/s). Theonset of kinking and crack propagation are marked by arrowsand black dots respectively

load-displacement behavior. The onset of any kinkinghas also been indicated on Fig. 3. As discussed above,this occurred at lower loads than those associated withinterfacial delamination. Figure 3 suggests there maybe a small rate effect on the relationship between loadand displacement. However, this figure shows no evi-dence of the ductile-to-brittle transitions (“stick-slip”)observed in mode-I for this system (Sun et al. 2008a).Nor was there any evidence of this type of transition onthe fracture surfaces.

Plots of crack extension as a function of time afterthe cracks began to grow are given in Fig. 4. These plotsare consistent with quasi-static crack growth and onlyvery limited rate effects, as confirmed by the observa-tion that the crack velocities scale approximately withthe applied displacement rate. The average crack veloc-ities were 0.24 ± 0.06, 20 ± 2 and 350 ± 90 mm/s forapplied displacement rates of 0.1, 10 and 200 mm/s,respectively.

2.2 Clamped, end-notched flexure tests

Clamped, end-notched flexure specimens were testedusing a drop tower to provide high loading rates. Thesespecimens were fabricated in exactly the same fashionas the earlier specimens, and the dimensions are given

0.1 mm/s

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Fig. 4 Crack extension plotted against time for the ENF tests atdisplacement rates of 0.1, 10 and 200 mm/s

Fig. 5 Configuration of the clamped, end-notched flex-ure (ENF) specimens. The thickness of the steel couponis h2 = 1.42 ± 0.02 mm; the thickness of the adhesive ish1 = 0.8 ± 0.1 mm; the width is W = 20.0 ± 0.5 mm; the initialcrack length is a = a0 = 10 ± 1 mm; the distance between theclamp and loading point is L = 30 ± 1 mm; and the diameter ofcylinders is D = 12 ± 0.5 mm

in Fig. 5. The specimens were clamped in the top of afixture, machined and hardened from a 4141-steel, bymeans of a steel block held in place by set screws. Anoptical micrograph of the experimental configurationis shown in Fig. 6. The tests were conducted by drop-ping a steel cylinder with a diameter of 12 ± 0.5 mmattached to a test weight with a mass of 40 kg onto thespecimens from different heights of between 30 and1,000 mm. The distance from the clamp to the loadingpoint on the specimens was 30 ± 1 mm, and the initialcrack length was 10 ± 1 mm from this point (Fig. 5).

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Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 115

Fig. 6 Micrograph of theexperimental setup for aclamped, ENF test. Thespecimen is clampedbetween two blocks of steelin this fixture by means offour screws

A high-speed camera1 was focused on a scale scribedon the side of the specimen. This allowed the positionsof the crack tip and the displacement of the loadingpoint to be monitored as a function of time. The loadtransmitted to the specimen was monitored by meansof a load cell placed between the cylinder and the dropweight. The signal from the load cell was recordedby an oscilloscope. The camera was triggered after afixed delay of 1 µs from when the initial signal from theimpact was received. Figure 7 shows that the velocity ofthe loading point was essentially constant through theentire experiment, except at the lowest loading rate. Theaverage loading velocities were 900 ± 50, 1,800 ± 100and 4,200 ± 50 mm/s for drop heights of 50, 200 and1,000 mm, respectively. The velocity of the loadingpoint varied between about 300 and 600 mm/s for adrop height of 30 mm.

A typical plot of the raw data for the reaction loadand deflection of the loading point is shown in Fig. 8,for a test with a drop height of 200 mm. These data werethen filtered digitally by using a discrete Fourier trans-form to remove the high frequencies that appear to cor-respond to the natural frequency of the clamped beam.This resulted in a relatively smooth load-displacementcurve that has been added to Fig. 8 for comparison.A complete set of all the test data, after smoothingby this procedure, is presented in Fig. 9. Although

1 A Cordin Model 220 gated, intensified CCD camera. This sys-tem could capture a maximum of 12 images in 300 ns, whenneeded.

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Fig. 7 Displacement of the loading point plotted against timefor the clamped, ENF tests with drop height of 30, 50, 200and 1,000 mm. The velocity of the loading point was 900 ± 50,1,800 ± 100 and 4,200 ± 50 mm/s for the last three conditions,and varied from about 300 to 600 mm/s for the lowest drop height

the data from the different tests show a considerablespread, there was no obviously consistent effect of rateon these load-displacement traces over the range ofcrack velocities studied. The onset of kinking and theonset of interfacial crack propagation have been indi-cated on the load-displacement plots of Fig. 9. There

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Fig. 8 A typical plot of load against end deflection for theclamped, ENF tests with a drop height of 200 mm. The curvewas smoothed by digital filtering using a discrete Fourier trans-form

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Fig. 9 Summary of the loads plotted against load point deflec-tions for the clamped, ENF tests with drop height of 10, 30,50, 200 and 1,000 mm. The differences in the load-displacementplots do not appear to be correlated with displacement rates, andare probably associated with minor differences in the specimengeometry and clamping conditions. The onset of crack propaga-tion is marked by solid dots. The onset of kinking is indicated asa range of loads over which this phenomenon was observed. Theinitial crack position is 10.7 ± 2.2 mm for all the tests

was no obvious effect on the load-displacement plotsfrom either event. One significant difference observedbetween the clamped and the three-point bendinggeometries was that crack propagation began after thepeak load for the clamped geometry, but before the peakload for the three-point bend geometry, which will bereferred to as the simply supported geometry

From a mechanics perspective, this clamped ENFgeometry is similar to the simply supported geometry.However, it differs in one respect: the clamping pro-vides a nominal plane of symmetry for the specimenthat is missing from a simply supported specimen witha single crack.2 This difference, in conjunction withthe relatively large deflections associated with plasticdeformation during these tests can influence the behav-ior of the specimens. The other differences betweenthe two geometries are merely cosmetic. The clampedgeometry is inverted, so that the sense of the mode-IIcomponent of the energy-release rate drives the crackalong the interface between the adhesive and the bot-tom adherend. This can be seen from the micrographof Fig. 10, which shows a crack kinking through theadhesive to the bottom interface. Finally, it should benoted that, all else being equal (and for small displace-ments), the loads for the simply supported geometrywould be expected to be approximately double thosefor the clamped geometry (because of where the loadis measured).

Plots of the crack extension against time are shownin Fig. 11. From these data, the crack velocities weredetermined to be approximately 300 ± 60, 500 ± 80,1,100 ± 100 and 2,400 ± 300 mm/s for drop heights of30, 50, 200 and 1,000 mm, respectively. As in the lower-rate tests, there was no evidence of stick-slip behaviorat any velocity in mode-II. This is a significant contrastto the behavior under mode-I conditions for this sys-tem (Sun et al. 2008a) in which at higher crack veloc-ities there was an increased tendency for stick-slip tooccur; this suggests that different toughening mecha-nisms may be operating in the two modes of fracture.

For completeness, companion tests with the clampedgeometry were conducted using a servo-hydraulicmachine under displacement control at rates of 0.1and 200 mm/s. One significant difference was observed

2 The plane of symmetry for a clamped geometry is the verticalwall at the clamping point. A similar plane of symmetry wouldexist in the simply supported geometry if there were cracks atboth ends.

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Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 117

Fig. 10 Micrograph of adeformed clamped, ENFspecimen during thedrop-tower test with a dropheight of 30 mm

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Fig. 11 Crack extensions as a function of time for the clamped,ENF specimens with different drop heights

between these tests and the clamped ENF tests per-formed under impact conditions: the nature of the plas-tic deformation was different. As can be seen from themicrographs of Fig. 12, a plastic hinge was formed nearthe crack tip in the drop-tower tests, and a plastic hingeformed near the clamped root of the specimen in theservo-hydraulic machine tests. Further investigation ofthis phenomenon indicated that it was a robust experi-mental observation, and it was not dependent on uncer-tainties in the boundary conditions or in the details of

Fig. 12 A comparison between the deformed shape of theclamped ENF specimens on a the drop-tower machine with adrop height of 200 mm (corresponding to a displacement rate of1,800 mm/s), and b the servo-hydraulic machine at a displace-ment rate of 0.1 mm/s. The plastic hinge was formed near thecrack tip for the drop-tower tests, and it formed near the clampedroot of the specimen at the lower loading rates associated withthe servo-hydraulic machine

the loading. As will be discussed in the numerical sec-tion, it is believed that the phenomenon is associatedwith a subtle effect of loading rate on the material prop-erties of the adhesive layer.

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Fig. 13 Geometry of the lap-shear test specimens used in thisstudy. The free arm length is L1 = 80 ± 2.0 mm; the overlaplength is L2 = 21.0 ± 1.0 mm; and the out-of-plane width isW = 20.0 ± 0.5 mm

2.3 Lap-shear tests

The single-lap shear geometry is often used to examinemixed-mode crack propagation. However, cohesive-zone calculations indicated that fracture can dependonly on the mode-II cohesive strength, and not on themode-II toughness or the mode-I parameters, if theoverlap length is kept relatively short. Therefore, spec-imens with such a geometry (Fig. 13) were used toexplore whether they could provide a value of the cohe-sive shear strength of the interface. These specimensprovided an analogous test to that used in the earlierwork (Sun et al. 2008b) to determine the mode-I cohe-sive strength for this adhesive system.

The specimens were fabricated following the sameprotocol as used for all the other specimens in this pro-ject; in particular, the thickness of the adhesive layerwas kept at 0.8 mm. Four tests were conducted at eachof three different cross-head displacement rates (0.01,10 and 200 mm/s) in a servo-hydraulic machine. Crackpropagation in all tests was interfacial, with the frac-ture surfaces very similar to those observed in the ENFtests. The crack growth was too rapid to be observedoptically, owing to the difficulty in establishing a suit-able trigger to capture the event with an appropriate

time resolution. The nominal shear strength of the bondwas determined by dividing the maximum load sup-ported by the area of the adhesive. This quantity wasequal to 27 ± 3, 32 ± 3 and 34 ± 3 MPa correspond-ing to displacement rates of 0.01, 10 and 200 mm/srespectively. Based on cohesive-zone models of thisgeometry, these values could be taken to be the mode-II cohesive strengths of the adhesive system. As will bedemonstrated subsequently, these values were consis-tent with the cohesive strengths obtained independentlyfrom the ENF tests.

3 Numerical results

3.1 Numerical simulations for thethree-point-bending, ENF geometry

3.1.1 Continuum calculations

A finite-element calculation using the continuum prop-erties of the adhesive was performed to explore theevolution of the stresses in the adhesive layer priorto the propagation of the crack (Fig. 14). This wasimplemented for the ENF geometry using an ABA-QUS/Standard 2D model, with the dimensions givenin Fig. 1. Both the adherends and the adhesive wererepresented by four-node, plane-strain elements. Whileonly a single layer of elements was used for the 0.8 mmthickness of the adhesive layer, a subsequent mesh-refinement analysis verified that the numerical resultswere essentially unchanged by the use of finer meshes.To prevent interpenetration at the crack mouth, sur-face-based contact elements were used to simulate boththe interaction between the steel spacer and the twoarms. Contact elements were also used to simulate theinteraction between the specimen and the three rollers.

Fig. 14 Geometry of the continuum/cohesive-zone model used to study the ENF tests. In the continuum model, the adhesive layerwas replaced by plain-strain elements with initial thickness of 0.8 mm. In the cohesive-zone model, the adhesive layer was replaced bycohesive-zone elements with an initial thickness of 0.8 mm. The other dimensions are given in Fig. 1

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Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 119

0

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Load

(kN

)

Mid-point deflection (mm)

Continuum model

Experimenal results(deflection rate = 0.1 mm/s)

Fig. 15 Comparison between the load-deflection plot obtainedfrom an ENF test conducted at a deflection rate of 0.1 mm/s andthe numerical predictions of a continuum model with the consti-tutive properties of the adhesive obtained from a tensile test (Sunet al. 2008a)

The coefficient of friction between steel and steel wasassumed to be 0.8; however, numerical calculationsindicated that the precise value for this parameter didnot significantly affect the results. The properties ofthe steel adherends and of the adhesive layer weredescribed by point-to-point representations of the uni-axial stress–strain curves given in Sun et al. (2008a).Isotropic properties, with a von Mises yield criterionand isotropic hardening were assumed for both materi-als. An initial strain rate for the adhesive was assumedso that the appropriate constitutive properties could beused. It was subsequently verified that this assumedrate was consistent with the results of the numericalcalculations.

Figure 15 shows a comparison between the loadversus mid-point deflection plots obtained numericallyand experimentally at an applied displacement rate of0.1 mm/s. Good agreement between the two is seenuntil the peak load in the numerical plot. The shear–strain rate at the crack tip was determined to be about0.0115 s−1 by dividing increments of the calculatedshear displacement across the adhesive layer by theassociated increments of applied displacement, and tak-ing the displacement rate to be 0.1 mm/s. It was this cal-culated strain rate that was verified as being consistent

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Fig. 16 The relationship between shear stress and shear strainfor an adhesive layer at a shear strain rate of about 0.0115 s−1.This curve was derived from the stress–strain relationship of anelement at the crack tip in the continuum analysis of the ENFtest of Fig. 15

with the constitutive properties used for the adhesivein the numerical model. As commented upon earlier,crack propagation was observed before the peak load,3

therefore, the calculations of Fig. 15 indicate that thecontinuum calculations describe the load-displacementbehavior even after the onset of crack propagation. Thisimplies that significant frictional effects allowed thefull load-bearing capacity of the adhesive layer to becontinued even after crack growth had commenced.Indeed, the frictional stress must have been compara-ble to the yield strength of the adhesive layer duringthe initial stages of crack propagation for the load tocontinue tracking with the continuum predictions. Thepeak load in the load-displacement plot must indicatethe point at which the frictional stresses started to dropfrom this high level.

A plot of shear stress against shear strain for theadhesive was obtained by tracking the stress–strainbehavior of the element representing the adhesive atthe crack tip. This curve is shown in Fig. 16. If theco-ordinate axis of this figure were to be converted to

3 Observations of the fracture surface indicated that the crackadvanced further in the middle of the adhesive, than at the edges.Therefore, the observations of crack advance before the peakload were not just a surface phenomenon.

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displacement (rather than strain), the curve could thenbe taken to represent the mode-II cohesive law up tothe point of softening.4 This figure indicates that theshear strength exhibited by the adhesive at an applieddisplacement rate of 0.1 mm/s is about 21 MPa, and thatthe shear modulus of the adhesive at this rate is about350 MPa. These continuum calculations were repeatedfor the tests conducted at higher displacement rates. Fortests with a mid-point deflection rate of 200 mm/s, theshear strain rate at the crack tip in adhesive layer wascalculated to be as high as up to 20/s. This shear strainrate corresponds to a higher normal strain rate than hadbeen obtained in the uniaxial tensile tests describedin Sun et al. (2008a). Therefore, the appropriate shearmodulus could only be estimated by extrapolation ofthe tensile data. It was estimated to be about 520 MPa.The results of the numerical calculations for the ENFtests using this value of shear modulus were not incon-sistent with the experimental results.

3.1.2 Cohesive-zone model

As described in the Appendix, an initial attempt to usethe cohesive-zone model of Li et al. (2006a,b) failed toprovide agreement with the experimental results. Thisoriginal model was one that did not couple the shearand normal modes of deformation. Such an approxima-tion appears to be satisfactory when there are limitedshear strains or when the thickness of the adhesive layeris very much less than the thickness of the adherends.It was apparent that the problem in the present caseresided in the formulation of the numerical model, notin the choice of constitutive properties of the adhesivelayer. The Appendix describes a new formulation forthe cohesive-zone model that couples the shear and nor-mal displacements of the nodes in the cohesive-zone toresolve this discrepancy.

The characteristic traction-separation laws used todescribe the behavior of the adhesive layer are shown inFig. A1. These are approximate representations for theelastic-plastic behavior such as that shown in Fig. 16.Although the geometries presented in this paper areall nominally pure mode-II geometries, a mixed-modeformulation of the cohesive-zone model was used to

4 The agreement between the continuum calculations and theexperimental curve in Fig. 15 implies that the deformation mech-anism of the adhesive layer is not affected by the constraints ofthe adherends.

ensure a general model.5 The mode-I cohesive parame-ters were taken to be the quasi-static mode-I lawobtained in the prior work on this system (Sun et al.2008b). The mode-II parameters were determined by aprocess of fits between numerical results and specificfeatures of the experimental results at different loadingrates. The process is described below using one specificexperimental example at a loading rate of 0.1 mm/s.This process was repeated for every set of experimen-tal data at different rates to determine the ranges ofuncertainty and the effects of rate.

First, the initial slope of the cohesive law wasobtained from the shear modulus of the adhesive layer.The properties of the adhesive, as determined by a ten-sile test (Sun et al. 2008a), were incorporated within acontinuum numerical model of the ENF test. It was ver-ified that the strain rate was consistent with the proper-ties, and that the elastic slope of the load-displacementcurve for the ENF test was correct. Then the shear mod-ulus and the thickness of the adhesive layer were used todetermine the initial slope of the mode-II law to give thecorrect compliance of the adhesive layer. For example,as discussed above, the shear modulus of the adhesivelayer was 350 MPa at displacement rate of 0.1 mm/s(Fig. 16) which, for an adhesive layer of 0.8 mm thick,translates to an initial slope of 440 MPa/mm.

Second, the mode-II cohesive strength, τ̂ , was deter-mined by numerical matches to the portion of the exper-imental load-displacement curves immediately after thedeviation from linearity induced by plastic flow of theadhesive and steel.6 Figure 17 shows the sensitivityto minor changes in this cohesive strength, keeping thetotal area under the curve and the unloading slope of thetraction-separation law fixed. For an applied displace-

5 The mode-I law developed in Sun et al. (2008b) was for acrack within the adhesive layer. In this geometry, mode-II cracksalways grew along the interface. Sensitivity analyses were con-ducted to confirm that the results in this paper were insensitive tothe choice of mode-I cohesive laws. Subsequent work has iden-tified the appropriate mode-I cohesive laws for interfacial failure(Sun et al. 2009), and it was verified that the use of the morecorrect laws did not affect the present results.6 Different forms of non-linear plastic flow that the adhesivemight exhibit did not seem to be a major factor in determin-ing the shape of the load-displacement curve. The characteristicstrength seemed to be the important parameter for capturing thebehavior of this portion of the curve. In other words, the simpleform of Fig. 1 was sufficient for these purposes. There was noneed to mimic the precise shape of the traction-separation curveillustrated in Fig. 17 that was obtained from considerations ofthe continuum properties of the adhesive.

123

Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 121

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8

Load

(kN

)

Mid-point deflection (mm)

τ = 25 MPa

τ = 21 MPa

Experimental results(displacement rate = 0.1 mm/s)

^

^

Fig. 17 A comparison between the results of a cohesive-zoneanalysis and the numerical results for an ENF test at a displace-ment rate of 0.1 mm/s. The cohesive strength is determined bycomparing the point at which the curves deviate from a linearload-deflection relationship

ment rate of 0.1 mm/s, the shear cohesive strength wasdetermined to be 23 ± 2 MPa. Within the uncertaintylevels, this is consistent with the results from the con-tinuum calculation presented above. It is also consis-tent with the results obtained from the lower-rate testsconducted with the lap-shear geometry.

Third, the critical displacement for the onset of crackgrowth, δtc (Fig. A1), was obtained by determining thepoint on the experimental load-displacement curve atwhich crack propagation was observed optically. Iden-tifying this with the equivalent point on the numericalload-displacement curve, allowed the area traced outunder the traction-separation law up to this point tobe determined from the cohesive strength. This area isassociated with the mode-II toughness of the adhesivelayer, as indicated in Fig. A1. For an applied displace-ment rate of 0.1 mm/s, the critical shear displacementwas calculated to be 0.50 ± 0.05 mm. This correspondedto a mode-II toughness of 11 ± 1 kJ/m2.

Fourth, the characteristics of the frictional portionof the traction-separation law were evaluated. The crit-ical displacement at which the traction-separation lawunloads, δt2, was determined from the peak load. Byvarying this displacement in the numerical model untilthe calculated peak load matched the experimental peakload, it was determined to be 0.45 ± 0.05 mm. Finally,the unloading slope of the traction-separation curve was

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

Load

(kN

)

Mid-point deflection (mm)

Numerical results 1 32

Experimental results(displacement rate = 0.1 mm/s)

Fig. 18 The unloading slope of the traction-separation law hasfairly significant effects on the load-displacement curve. Anestimate of the amount of energy dissipated by friction can beobtained by altering the unloading slope in the cohesive law, sothat the unloading portion of the numerical load-deflection curvematches the experimental curve

obtained by varying it until the numerical and exper-imental curves approximately matched after the peakload. Examples of the matching procedure is shownin Fig. 18. In the case being used as an example, theunloading slope was determined to be 40 MPa/mm.Upon completing these calculations it was noted thatcomplete unloading of the traction-separation law atthe original crack-tip had not occurred by the time thecrack tip had run into the compressive region associ-ated with the loading points. Even so, the total energydissipated by the frictional portion of the traction-sep-aration law was 5 ± 1 kJ/m2—a substantial addition tothe cohesive energy associated with mode-II fracture.

This process of analysis was then repeated for all theindividual sets of experimental data, with uncertaintiesin the cohesive parameters being computed for eachrate. A summary of the cohesive strength and the mode-II toughness, and frictional are provided in Figs. 19 and20 as functions of the crack velocity.

3.2 Numerical simulations for the clamped ENFgeometry

The numerical model for the clamped ENF geome-try was similar to that of the three-point-bending ENF

123

122 C. Sun et al.

0

10

20

30

40

50

0.1 1 10 100 1000 104

Strain rate (/s)0.01 0.1 1 10 100

Mod

e-II

cohe

sive

str

engt

h (M

Pa)

Crack velocity (mm/s)

Clamped, ENF tests

Lap shear tests

Three-point-bendingENF tests

Fig. 19 Mode-II cohesive strength plotted as a function of crackvelocity/strain rate for the lap shear tests, three-point-bendingENF tests and the clamped ENF tests

0

5

10

15

20

25

0.1 1 10 100 1000 104

Mod

e-II

toug

hnes

s (k

Jm-2

)

Crack velocity (mm/s)

Three-point-bendingENF tests

Clamped, ENF tests

Fig. 20 Mode-II toughness plotted as a function of crack veloc-ity for both three-point-bending ENF tests and the clamped ENFtests

geometry except for a clamped boundary condition. Itis recognized that this clamping was an approximation,since observations indicated that there was sliding androtation of the specimen in the clamp during impact.The continuum numerical calculations indicated thatthe strain rate of the adherends was about 2/s at the

clamped boundary when the velocity of the loadingpoint was 600 mm/s (corresponding to a drop height of30 mm), and increased to 14/s at the fastest velocities.Therefore, the constitutive properties of the steel weretaken to be the upper bounds of the properties reportedin Sun et al. (2008a). These same continuum calcula-tions were also used to relate the shear strain rate in theadhesive layer to the loading rate. It was determinedthat the strain rate at the crack tip was in the range of30–400/s during the drop-tower tests, compared with0.01–20/s during the tests conducted with the servo-hydraulic machine.

As discussed in Sect. 2.2, the effects of rate in theseclamped-ENF studies were manifested by two differ-ent types of deformation. Figure 12 shows a plastichinge formed at the clamped root in a specimen loadedslowly, and a plastic hinge formed at the crack tip in aspecimens loaded quickly. Continuum numerical cal-culations of the clamped ENF geometry indicated thatthe locations of the plastic hinge depended on the shearstrength of the adhesive. In particular, the hinge formedat the crack tip when the shear strength was greaterthan 32 ± 3 MPa. Experimentally, it was observed thatthe hinge formed at this location at strain rates greaterthan between 20 and 30/s, we can deduce that the max-imum shear strength of the adhesive layer was greaterthan 29 MPa at strain rates above 30/s and <35 MPa forstrain rates below 20/s.

The cohesive parameters at high loading rates werededuced from comparisons between the numerical andexperimental results for the clamped ENF tests in thedrop tower. No further effects of rate were observedfrom the load-displacement traces obtained at any ofthese high rates—specimen-to-specimen variationsdominated any other effect. Therefore, a single set ofhigh-rate mode-II parameters was obtained. Extrap-olation of the uniaxial data from Sun et al. (2008a)to the strain rates appropriate for the drop-tower testsindicated that the shear modulus of the adhesive layerwas about 350–500 MPa. This corresponds to an initialslope for the traction-separation law of 440–625 MPa/mm.7

7 Use of this range in the numerical analysis for the clampedENF geometry resulted in load-displacement predictions thatwere much too stiff, because the numerical boundary condi-tions simulations were too restrictive compared with the experi-mental clamping. Owing to the difficulty in correctly modelingthe boundary conditions, the additional compliance associatedwith slipping at the clamp was incorporated into the initial slope

123

Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 123

The cohesive shear strength was determined fromthe second portion of the experimental load-displace-ment curves, after the deviation from linearity inducedby the plastic flow of the adhesive and steel. (Whilethis region may not be particularly obvious in Fig. 9, adifference in the shapes of the numerical curves couldbe detected as the parameters were changed.) Fits to thenumerical curves suggested that the mode-II cohesivestrength was in the range of 25–50 MPa. However, itshould be noted that, as discussed above, the locationof the plastic hinge at the crack tip imposes a more lim-iting lower-bound of 32 ± 3 MPa for the shear cohesivestrength at the loading rates obtained in the drop-towertests. So, for consistency, the shear cohesive strengthat strain rates above to be between 29 and 50 MPa forstrain rates above 30/s.

The displacement at which unloading of the traction-separation law begins was determined to be 0.25±0.05 mm. This was done by matching the peak loadsof the numerical and experimental load-displacementplots (Fig. 9). The unloading slope of the traction-sep-aration law was determined to be 25–50 MPa/mm. Thiswas done by matching the shapes of the numerical andexperimental load-displacement curves after the peakload. The critical displacement at which the crack beganto grow was determined to be 0.8 ± 0.2 mm, which cor-responds to a mode-II toughness of 17 ± 5 kJ/m2. Thiswas done by matching the points on the load-displace-ment curves at which crack growth was seen during theexperiments to the point at which crack growth began inthe numerical calculations. The remainder of the cohe-sive law, after crack propagation was associated withfrictional dissipation of 5 ± 4 kJ/m2. It will be notedthat the clamped configuration of the ENF geometryresulted in crack growth beginning after the peak load;this is in contrast to the simply supported geometry inwhich crack growth began before the peak load. Thisdifference in behavior is associated with the relativelylarge deformations and the different boundary condi-tions.

Figure 21 shows the results of the numerical simu-lation after this cohesive law had been developed, andcompares the results to the experimental observationsfrom the drop-tower results. The values for the cohesiveparameters used for these high-rate simulations have

Footnote 7 continued(225 MPa/mm) of the mode-II cohesive law while analyzing thisgeometry.

Numerical result

(ΓII = 21 kJm

-2; τ = 50 MPa)

Numerical result

(ΓII = 12 kJm

-2; τ = 25 MPa)

^

^

0

0.5

1

1.5

2

0 2 4 6 8 10 12 14 16

Load

(kN

)

Deflection of load point (mm)

Fig. 21 Comparison between experimental plots of load versusdeflection from the clamped ENF specimens (loaded in the droptower) and upper and lower-bound numerical predictions of thecoupled-cohesive-zone model

been added to Figs. 19 and 20. The deformed shapeof the clamped, ENF geometry using these high-rateparameters is shown in Fig. 22a. It will be noted that theplastic hinge was formed at the initial crack tip, as seenexperimentally. Furthermore, using a mode-II cohesivelaw appropriate to the rates that can be obtained with theservo-hydraulic machine, the plastic hinge was formedat the root of the beam, where the specimen is clamped.The result of this simulation is shown in Fig. 22b. Thisdifference in behavior is consistent with experimentalobservations illustrated in Fig. 12.

3.3 Numerical simulations for the lap-shear geometry

A numerical analysis with continuum elements (planestrain) representing the adhesive layer indicated thatthe displacement rates during the lap-shear tests corre-sponded to shear-strain rates in the adhesive of between0.013 and 250 s−1. A final set of numerical calculationswere performed using the cohesive-zone model for thelap-shear test. These calculations confirmed that forthis geometry and combination of materials, the nom-inal shear strength of the bond was identically equalto the cohesive shear strength. The values for mode-II cohesive strengths obtained from the lap-shear testshave been added to Fig. 19.

123

124 C. Sun et al.

Fig. 22 A comparisonbetween the deformed shapeof the clamped ENF testsimulation witha upper-bound cohesiveparameters(�IIc = 21 kJ m−2 andτ̂ = 45MPa), andb lower-bound cohesiveparameters (�IIc = 8 kJ m−2

and τ̂ = 20 MPa) with theplastic hinge formed nearthe clamped root of thespecimen. The plastic hingeformed at the crack tip whenthe cohesive strength wasgreater than about32 ± 3 MPa

4 Conclusions

Stick-slip behavior was not observed during mode-IIdeformation in this adhesive system. This is a signifi-cant contrast to the mode-I observations for the samesystem, in which random transitions to an untoughenedmode of fracture was observed (Sun et al. 2008a). Crackpropagation was interfacial in mode-II for this system;mode-II crack growth within the adhesive layer wasnot observed. Again, this is in contrast to the behaviorin mode-I where the crack appeared to be stabilizedwithin the adhesive layer in this system.

Cohesive-zone analyses of the data showed that thefracture parameters were slightly rate sensitive. As indi-cated in Figs. 20 and 21, this rate sensitivity was veryslight and not very significant beyond the range ofexperimental uncertainty; it resulted in only oneunambiguous rate effect within the entire range of ratesstudied—a change in the location of the plastic hingeduring the clamped ENF studies. The mode-II

toughness ranged from about 8 kJ/m2 to about 21 kJ/m2

(including uncertainties) at crack velocities between0.4 and 2,400 mm/s for this adhesive. While the tough-ness appears to have increased slightly with rate, anyincrease was within the range of uncertainty. In a simi-lar fashion, the mode-II cohesive strength was between20 and 45 MPa, with any increase with rate being smallcompared to the level of uncertainty.

Acknowledgements C. Sun, M. D. Thouless and A. M. Waasgratefully acknowledge the financial support of General Motors.

Appendix

The mixed-mode cohesive-zone model used in thisstudy is based on that described in Yang and Thou-less (2001) and Li et al. (2006a,b). In those studies,the mode-I and mode-II traction-separation laws areuncoupled in the sense that each law can be describedseparately, with the parameters for each law being

123

Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 125

Fig. A1 A trapezoidal mode-I and mode-II traction-separationlaw. The mode-II law illustrates the form used in this study inwhich there are frictional contributions to the energy dissipatedat the interface

determined by either mode-I or mode-II experiments.During numerical analyses, the energy-release rate foreach mode is calculated independently by integration ofthe opening and shear traction-separation laws (Fig. A1)

GI =δn∫

0

σ(δn)dδn; GI I =δt∫

0

τ(δt )dδt , (A1)

where δn and δt denote the relative normal and sheardisplacements. Coupling between the two modes is pro-vided by the mixed-mode failure criterion. An exam-ple of a simple mixed-mode failure criterion is givenby

GI/�I + GII/�II = 1, (A2)

where �I and �II are the total areas under the curves(integration up to the critical displacements δnc andδtc). In the cohesive-zone analyses, GI and GII areevaluated independently for each element. The ele-ment is assumed to fail when the mixed-mode failurecriterion is met, and the crack advances by one ele-ment. This approach results in a very robust agreementwith mixed-mode LEFM models under appropriate lin-ear-elastic conditions (Parmigiani and Thouless 2007).

Fig. A2 The numerical results from a continuum model for theload-deflection plot of the simply supported ENF geometry ofFig. 1 agree with the experimental results of a test conducted at0.1 mm/s up to the maximum load. However, the predictions ofa cohesive-zone model, with parameters based on an approxi-mate fit to the shear properties of the adhesive given in Fig. 16,diverge from the continuum and experimental results after a lim-ited amount of deflection

It also provides good predictive capabilities for mixed-mode problems when LEFM is not applicable (Yangand Thouless 2001; Li et al. 2006a,b).

The mixed-mode cohesive-zone model describedabove appears to be an excellent numerical tool foradhesively bonded structures in which the adhesivelayer is significantly thinner than the adherends. Underthese conditions, failure to capture accurately the cou-pling between the shear and normal deformations ofthe adhesive is not important, because deformation ofthe adhesive before fracture does not contribute signifi-cantly to the overall behavior of the system.However, when the adhesive layer is relatively thickcompared to the adherends (as in the present study),then an accurate description of the coupling betweenthe shear and normal modes of deformation becomesimportant, as the adhesive has a significant effect onthe overall behavior of the system even before it fails.It is this issue that the modified coupled-cohesive-zonemodel (CCZM) described below addresses.

To illustrate the problem discussed above, the uncou-pled cohesive-zone model was used in a numerical

123

126 C. Sun et al.

Fig. A3 a An undeformedcohesive-zone element.b A deformedcohesive-zone elementsubjected to normal andshear nodal forces, Fy andFx respectively. c Use ofcouple normal forces at thenodes to balancing themoment caused by thenormal nodal forces. d Useof couple normal forces atthe nodes to balancing themoment caused by the shearnodal forces

model of the ENF test at a displacement rate of 0.1 mm/s.The adhesive layer bonding the steel coupons togetherwas replaced by four-node cohesive-zone elements withan initial thickness of 0.8 mm. The mode-II cohesiveparameters were chosen to mimic the shear stress–strain curve of the adhesive given in Fig. 16. As shownin Fig. A2, the uncoupled cohesive-zone model couldnot match the experimental results or continuum cal-culations. In particular, the cohesive-zone model didnot correctly capture the deformation of the specimenbefore fracture. This implies that the cohesive elements,which represented the adhesive layer, were not mim-icking the deformations of the adhesive layer correctly;the problem was independent of the choices made forthe cohesive parameters. The stiffness for the mode-IIlaw was not the cause for this discrepancy—the initialslopes of the load-deflection curves all match. The error

associated with the cohesive-zone analysis becomesmore pronounced as the strains increased.

The poor modeling capacity of the uncoupled mixed-mode cohesive-zone model seen in Fig. A2 arises fromthe moments that develop across a cohesive-zone ele-ment when there is extensive shear deformation. Inuncoupled cohesive-zone elements, the reaction forceson the nodes, Fy and Fx , arise only from the mode-I andmode-II displacements, as described by the separatemode-I and mode-II traction-separation laws (Fig. A3).However, these reaction forces induce a moment Munder the effect of shear deformation. This momentcan be significant if the shear displacements are large.If they are neglected in the formulation of the ele-ment, this moment equilibrium is violated. This canbe neglected if the shear strains are small, or if theadhesive layer is thin compared to the thickness of the

123

Rate effects in mode-II fracture of plastically deforming, adhesively bonded structures 127

Fig. A4 The coupled-cohesive-zone model provides a much bet-ter fit to the results of the continuum calculation for a simplysupported ENF specimen than does the original cohesive-zonemodel. The cohesive parameters for both cohesive zone modelsare identical in this plot

specimen.8 However, as seen in Fig. A2, the geome-try and deformation in the present case are such thatthis deficiency has a significant effect on the overallbehavior of the system.

A modified cohesive-zone element was formulatedthat we refer to as a coupled cohesive-zone element(CCZM) to eliminate the inaccuracies associated withthis unbalanced moment. Figure A3a shows an elementof length (along the interface) Le and height (perpen-dicular to the interface) Te. The element is deformed toa new length Led and height Ted (Fig. A3b). The topand bottom surfaces of the element are sheared rela-tive to each other by Dx . The total moment on the ele-ment that is associated with the normal nodal reactionforces is equal to 2Fy • Dx . This moment can be elimi-nated by applying a couple force, Fyx = Fy • Dx/Led ,to each node, as shown in Fig. A3c. There is also amoment induced by the shear nodal reaction forces of2Fx • Ted . This moment can be eliminated by applyinga couple force, Fxy = Fx • Ted/Led , to each node, asshown in Fig. A3d. This modification to the formula-tion of the cohesive elements eliminates the effects of

8 The range of thickness ratios between the adhesive and adher-end for which these issues are relevant will depend on the prop-erties of the adhesive and the adherends.

Fig. A5 Both the coupled-cohesive-zone model and the originalcohesive-zone model agree with the results of the continuum cal-culations (while the crack does not propagate), provided the shearcontributions are relatively small. The properties of the adhesiveand adherend, and the cohesive parameters are identical to thoseused in Fig. A2, but the thickness of the adherends have beenincreased by a factor of 3–4.2 mm

the moment and couples the shear and normal tractions,without affecting the cohesive parameters. An exampleof the code given in Sun (2007). Simulation of the ENFgeometry using these coupled elements, but using thesame cohesive parameters as for the earlier analysis ofFig. A2 are shown in Fig. A4. Now, it will be observedthat the cohesive-zone model provides good agreementwith the continuum results.

Finally, to verify that the results of the two typesof cohesive models are essentially identical when theadhesive layer is thin enough to avoid the complica-tions discussed above, a similar ENF geometry withthicker arms was analyzed. Both types of cohesive-zonemodel were used, both with identical tractions-separa-tion laws, but with much thicker adherends (the steelarms was increased by a factor of 3–4.2 mm). All otherdimensions and properties were identical to those usedin Fig. A2. Now, as shown in Fig. A5, both models arein agreement with the continuum model, showing thatthe effect discussed in this Appendix is independentof the choice of cohesive parameters and is significantonly when there are relatively large shear deformations.

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128 C. Sun et al.

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