International Journal of Fracture 130: 497–515, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

The effects of bond thickness, rate and temperature on thedeformation and fracture of structural adhesives under shearloading

HERZL CHAIDepartment of Solid Mechanics, Materials and Systems, Faculty of Engineering, Tel Aviv University, Tel Aviv69978, Israel

Received 23 March 2004; accepted in revised form 26 July 2004

Abstract. The deformation and fracture in shear of a structural adhesive undergoing large-scale yielding is studiedas a function of bond thickness, h, temperature, T , and strain rate using the Napkin Ring specimen. The lack ofedges in this test, and the fact that the strain rate can be locally controlled, allow for a meaningful evaluationof the mechanical response throughout the deformation process. In accord with Airing’s molecular activationmodel, the yield stress linearly decreases with T while logarithmically increasing with the strain rate. The ultimateshear strain, γF, is little sensitive to rate while decreasing with h and increasing with T . Some complementaryfracture tests are carried out using the ENF bond specimen in order to explore the relation between the mechanicalproperties of the nominally unflawed adhesive and the mode II fracture energy, GIIC. For sufficiently thin bonds,GIIC/h correlates well with the ultimate energy density (i.e., the area under the stress-strain curve in the NapkinRing test), given, to a first approximation, by τYγF, where τY is the yield stress in shear. Accordingly, the fractureenergy of the bond would be greatly affected by temperature, tending to a small value at the absolute as well as theglass transition temperatures while attaining a maximum in between these two extremes. Because the yield stressdoes not vary much with h, the variation of GIIC with the bond thickness reflects that of γF.

A large-deformation fracture analysis, based on a cohesive zone like model, is developed to account for theobserved variations of γF with h. The analysis assumes that a crack preexist in the bond, either at its center or atthe interface. The results suggest that the observed increase of γF with decreasing h is due mainly to two geometriceffects. The first is due to the interaction of the bonding surfaces with the stress field generated by the crack andthe second has to do with the probability of finding large flaws in the bond to trigger the fracture.

Key words: Adhesive bond, yield, rate, temperature, fracture, flaw distribution.

1. Introduction

The constitutive behavior of polymers in the post-yield regime is an important ingredient inadvanced designs of these materials. Such information is needed, for example, in the analysisof fracture under large-scale yielding, where the material at the immediate crack tip vicinitymay display extensive deformation. While the behavior at yield can be established from simpletensile tests, the response in the post yield regime is more difficult to ascertain due to sucheffects as necking, premature failure from stress concentration sites and variations in the strainrate in the gauge area during the loading. Testing in shear is advantageous in that it reduces oreliminates some of these undesirable effects. G’Sell et al. (1985, 1990) introduced a thicknessgradient parallel to the direction of the applied shear in order to localize the deformation.This led to the realization of considerable plasticity prior to fracture, even in nominally brittleepoxy systems. Bartczac et al. (1994) adopted a similar approach in studying the evolutionof microstructures in the sheared material with load. It was noted by these authors that the

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thickness gradient technique might still lead to inhomogeneous deformation, however. Liangand Liechti (1996) employed the Arcan specimen in studying the post yield behavior in shearof a cross-linked epoxy as a function of strain rate. Using Moiré interferometry, these authorsshow that the shear strain across the specimen may be highly non-uniform, the local strainat the gauge section that may be several times as large as the global shear strain. While theaforementioned testing approaches are useful, their implementation to very thin sections ofmaterials, as in thin-bond adhesives or laminated composites, appears difficult.

Studying the material in adhesive bond form is advantageous in that the plastic deformationis contained. The most popular test of this class is probably the Lap Shear joint. While thefabrication and testing of this specimen is quite straightforward, it may lead to considerablestress concentration at the edges of the bond as well as non uniform stress distribution alongthe bondline. A somewhat similar approach for measuring bond strength is facilitated by theCylindrical Lap joint (e.g., Hylands, 1984), although it is more difficult to maintain bondthickness uniformity in this specimen, particularly for very thin bonds. The aforementionededge effect could be reduced using the Arcan design with a 90◦ opening angle at the bondterminus (Weissberg and Arcan, 1988). A preliminary work by the present author shows,however, that this precaution does not fully guarantee a uniform strain distribution in the deepplastic range. Perhaps the most reliable specimen for evaluating the complete mechanicalresponse in shear is the Napkin Ring adhesive bond test (De Bruyne, 1962), see Figure 1. Thelack of edges in the bond leads to a state of simple shear in the entire joint, irrespective of thedeformation level. Indeed, using this test, the ultimate shear strain of several thermosettingsystems, including a highly cross-linked epoxy resin, was found to be as large as 300% whenthe bond thickness approaches the micrometer range (Chai, 1993, 1994). One of the mostintriguing issues in this or similar tests is the lack of viable explanation for the commonlyobserved increase in the strength of the joint with decreasing the bond thickness (e.g., Bryantand Dukes, 1967, Hughes et al., 1985, Stringer, 1985, Chai, 1993). A prime goal of this workis to attempt to provide a physical rationale for this important phenomenon. Our approachis based on the premise that a crack pre-exists in the bond. The conditions for which thiscrack may propagate are assessed from a large strain, cohesive zone like fracture model (Chai,2003). By matching predictions for various crack lengths from this analysis with the experi-mental data, the critical size of the inherent flaw in the bond could established as a functionof bond thickness. Another prime goal of this work is to explore the possibility of deducingthe shearing fracture energy of the bond from the mechanical properties of the nominallyunflawed material. For this purpose, mode II fracture tests are also performed.

In this work, the stress-strain response of a model structural adhesive deforming in simpleshear under strain-controlled conditions is evaluated as a function of bond thickness (i.e.,2 µm – 600 µm), strain rate (10−4/s. – 1/s.) and temperature (−60 ◦C - Tg) using the NapkinRing test specimen. Mode II fracture tests are also performed on the same adhesive at varioustemperatures and bond thicknesses using the ENF specimen. The experimental apparatus andthe test results are discussed in Section 2 and Section 3, respectively, while the FEM fractureanalysis is given in Section 4.

2. Experimental

The Napkin ring and the ENF adhesive bond specimens are used to study the deformation andfracture characteristics of a model structural adhesive. The Napkin ring test has not been used

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Figure 1. Sectional view of the Napkin Ring test specimen.

as extensively as it probably should have because of technical difficulties such as complexfabrication and testing, sensitivity to misalignments and control of the bond thickness. Acomprehensive effort is undertaken to overcome some of these difficulties. The adhesive isa toughened thermosetting epoxy resin (BP-907, American Cyanamide) having a curing tem-perature of 177 ◦C and a glass transition temperature, Tg, of 98.5◦ (371◦K). The adherends aremade of 5086 (Napkin Ring) or 7075 - T3 (ENF) aluminum alloys; a structural alloy is chosenfor the ENF specimen in order to insure elastic deformation in the adherends throughout thefracture process. The bonding surfaces are cleaned and etched in accordance with the FPLprocedure. The specimens are housed in a test chamber that provides temperature control tobetter than 0.1 ◦C. Temperatures below RT are achieved using liquid nitrogen as a coolingagent. Details of the fabrication and testing of each of the two test specimens follow.

2.1. NAPKIN RING SPECIMEN

Figure 2 illustrates the specimen fabrication process and the test fixture. As shown in Fig-ure 2a, the specimen is made of two right-hand cylindrical rods in which the flat, bondingsurfaces are machined off to form an annular clearance as well as a ring like cylindricalsection, the height of which equals the intended bond thickness, h. After applying the moltenadhesive to the bond area, the two cylindrical halves are aligned by a centering pin beforethey are compressed along the symmetry axis with the aid of a spherical ball. It was noted thatgasses generated inside the clearance cell (Figure 2a) during the curing process may lead to theformation of large voids in the bond. To relieve this undesired gas pressure, a small vent in oneof two specimen halves is created. After curing, the edge of the specimen is turned down usinga lathe until the thickness-controlling step is completely cleared off. This results in a narrowbond (i.e., 2.5 mm) that leads to small variations in the shear stress over the bond annulus. Itwas shown by Liechti and Hayashi (1989) that unless the Napkin Ring specimen is cured at

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Figure 2. The fabrication process of the Napkin Ring specimen and test fixture: (a) and (b) – pre and post curingphases, respectively, (c) – complete test apparatus, including support fixture, extensometer and loading arm. Alldimensions are in mm.

room temperature, residual stresses, both in tension and in shear, may be set up at the innerand outer edges of the bond. Because the outer edge, where the failure is expected to occurfirst, is machined off, and due to the extended cooling period applied (i.e. 2 hours), this effectis believed to be small. The top part of each of the two specimen halves is then connectedto support ends by means of four circumferential threads, see Figure 2b. Figure 2c shows thespecimen and auxiliary devices after it is placed in the support fixture. The supporting endsof the specimen, of which part of their circular section is flattened, are placed inside torqueresistant prismatic grooves that are rigidly connected to the support fixture (see side view). A

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Figure 3. ENF adhesive bond specimen for measuring mode II fracture energy. All dimensions are in mm.

small clearance between the end supports and the supporting grooves helps reduce possiblespecimen bending due to various misalignments in the system.

The specimens are loaded by a compression force that is applied to a loading arm via aspherical surface (see Figure 2c). To reduce friction, a long arm (L = 220 mm) is used. Theshear stress in the bond, τ , is calculated from

τ = PL/(2πbR2) (1)

where P , b and R are the compression load, the bond width and the mean radius of thebond, in that order. To measure shear strain, two arms of length S (= 80 mm) are attachedto the outer diameter of the specimen, one on each side of the bond, approximately 1mmaway from the edge of the bond (Figure 2c). As in the works of Dolev and Ishai (1981) andWycherley et al (1990), among others, the relative rotation of these arms is measured with theaid of an extensometer. The displacement recorded by the latter includes contributions fromthe adherends as well as the adhesive. The former is essentially elastic, and is subtracted offfrom the final results so as to yield the nonlinear part of the adhesive deformation. The totalshear strain in the bond, γ , is obtained by adding this contribution to the calculated linear part.The result is

γ = (�nl/h)(R/S) + τ/G (2)

where G is the shear modulus of the adhesive (i.e., 1.2 GPa) and �nl denotes the nonlinear partof the extensometer displacement. The joints are loaded in an MTS servo hydraulic testingmachine, with the load and extensometer displacement recorded with the aid of a NationalInstrument digital data acquisition package. The output from the extensometer is fed into aclosed-loop PIDP program that controls the shear strain across the bond during the test. Inthis way, constant shear strain rates in the range 10−4/s. to 1/s. are accurately produced duringthe entire deformation history.

2.2. ENF SPECIMEN

As shown in Figure 3, the adherends for this specimen are made of two flat, 20 mm widealuminum bars. A sharp interfacial pre-crack is introduced into the interface of the bond byscribing the crack surface with a soft led pencil. The bond thickness is controlled via spacersthat are placed at several locations in the bond. The fracture energy (per unit area) is calculated

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Figure 4. Stress-strain response in shear for four different bond thicknesses. All curves correspond to a fixed localstrain rate of 0.1/s.

from the following relation, which is derived in accordance with the technical theory of beams(see Chai, 1988):

GII = 4.5(aP )2

Eab2H 3(3)

where Ea, P , a, b and H denote the Young modulus of the adherend, the applied load, in-stantaneous crack length, beam width and specimen thickness, in that order. The specimensare loaded in an MTS servo hydraulic machine at a crosshead speed of 0.017 mm/s. Thecrack tip is observed in-situ with the aid of a video camera that is connected to a long-rangeoptical microscope (Questar, Inc.). To facilitate clear identification of the crack tip duringcrack propagation, a series of fine scratch lines are introduced into the bond prior to theloading.

3. Test results

3.1. NAPKIN RING

Figure 4 shows the stress strain response, up to the point of ultimate failure, for a numberof bond thicknesses. All tests pertain to a fixed strain rate, γ̇ , of 0.1/s. Unloading curves,generated for a number of test specimens but not shown here, generally produce slopes similarto the loading curves, consistent with the common yield behavior of structural materials.The yield stress, τY, defined here as the onset of a noticeable bent in the curve, does notseem to vary much with the bond thickness. No stress softening is observed in any of thetests following the initial yield1. Thereafter, a phase of nearly ideal plastic response occursbefore a steep hardening takes place. Strain hardening is a common phenomenon in polymers,generally attributed to molecular orientation. The results for h > 6 µm show that the ultimateshear stress, τF, and the ultimate shear strain, γF, decrease with increasing h, the effect that

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Figure 5. The effect of strain rate on the yield stress in shear (filled symbols) and the ultimate shear strain (opensymbols). Curves are empirical fits.

seem to result from early termination of the otherwise quite similar mechanical response.The behavior for the 3 µm thick bond is distinguished from that described in that the idealplastic phase extends much further in the strain axis, and that τF is not larger than for the caseh = 6 µm. The behavior in the ultra thin regime is not well understood, however, and it maybe influenced by experimental inaccuracies.

The mechanical response of the adhesive seems to be characterized by three pivotal quant-ities, namely the yield stress, the ultimate shear stress and the ultimate shear strain. Thevariation of these parameters with the test variables is examined in the followings. Unlessotherwise mentioned, room temperature (RT) conditions are considered.

3.1.1. Effect of strain rateFigure 5 shows the dependence of the yield stress (filled symbols) and ultimate shear strain(open symbols) on the strain rate for two sets of bond thicknesses, namely 60 µm and 110 µm.The yield stress increases with the strain rate while the ultimate shear strain appears to berate insensitive over the range studied. Results for other temperatures confirmed the lattertrend. The little rate sensitivity of γF is also supported from tests on the bulk material, e.g.,polycarbonate subject to shear (G’Sell and Gopez, 1985) and PEEK or polyamideinide undercompression (Cady et al., 2003). On the other hand, stress-strain curves for a cross-linkedepoxy resin (Liang and Liechti, 1996) show that the ultimate shear strain is affected by thestrain rate. It is possible, however, that such variations are due, at least in part, to the fact thatglobal rather than local shear strain was considered in those plots.

3.1.2. Effect of bond thicknessAs indicated in Figure 4, the ultimate shear stress generally increases with decreasing the bondthickness; Corresponding data for this material were already reported in Chai, 1993. Similartrends were also found for other adhesive systems by Bryant and Dukes, 1967, Foulkes, et al.,1970, Ikegami and Kamiya, 1982 and Hylands, 1984, among others. We shall thus concentratein the followings on the ultimate shear strain, which received much less attention. Figure 6(open symbols) shows the effect of bond thickness on this quantity. The data includes various

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Figure 6. The effect of bond thickness on the ultimate shear strain (symbols). The strain rate (not specified) is inthe range 0.01–0.3/s, the dashed-line curve is an empirical fit.

Figure 7. The effect of temperature (in Kelvin degrees) on the stress-strain response in shear. All tests pertain toa 40 µm thick bond and a local strain rate of 0.1/s.

strain rates in the range 0.01/s. to 0.3/s; the details are omitted in light of the little rate sensit-ivity seen in Figure 5. Figure 6 shows that within the range studied, the ultimate shear strainincreases with decreasing h. Similar behavior was also reported for a modified epoxy film ina knitted carrier (Stringer, 1985) as well as for a number of unsupported adhesive systems,including a cross-linked and a rubber-modified epoxy resins (Chai, 1993, 1994). An attemptto explain this phenomenon, which is consistent with the common observation that smaller isstronger (e.g., Gao et al., 2003), using fracture mechanics and flaw probability considerations,is given in Section 4.

3.1.3. Effect of temperatureFigure 7 shows the effect of temperature, T , given in Kelvin degrees, on the stress-strainresponse of a 40 µm thick adhesive deforming at a constant strain rate of 0.1/s. The behavioris generally similar to that discussed in Figure 4. Figure 8 shows the variation of the yieldstress (solid symbols) and ultimate shear strain (open symbols) with temperature for a 40 µm

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Figure 8. The effect of temperature on the yield stress in shear (filled symbols) and the ultimate shear strain (opensymbols). All tests pertain to a 40 µm thick bond and a local strain rate of 0.1/s. The curves are empirical fits.

thick bond deforming at a constant strain rate of 0.1/s. The yield stress monotonically in-creases with decreasing the temperature, although one may expect this rise to terminate whenbrittle fracture intervenes. Similar behavior was also found for neat thermosetting resins (e.g.,Carswell and Nason, 1944, Ishai, 1969, G’Sell et al., 1990). In contrast, the ultimate shearstrain decreases with decreasing T , as was found by Huges et al. (1985) in their tests on thickadherend joints made of epoxy or epoxy-nitril adhesives over the temperatures ranging from−55 ◦C to 100 ◦C. Stress-strain plots for polycarbonate as well as some bulk thermosettingepoxies also show that the ultimate shear (G’Sell and Lopez, 1985, G’Sell et al., 1990) ortensile (Carswell and Nason, 1944, Cady et al., 2003) strains increase with temperature. Onthe other hand, results for more brittle systems reported by G’Sell et al. (1990) indicate a morecomplex behavior. This may be due to premature failure from stress risers, however.

3.1.4. Analytical and empirical relationsEmpirical relations are useful for practical applications. Figure 5 and Figure 8 suggest that theyield stress varies linearly with either the logarithm of the strain rate or temperature. Thus,one may write

τY = A1[1 + A2(T /Tg) log(γ̇ /γ̇0)

](4)

As shown by the solid-line curves in Figures 5 and 8, Equation 4 is quite satisfactory providedthat the coefficients (A1, A2, γ̇0) are chosen as (289 MPa, 0.026, 1034.5/s.). A more funda-mental understanding of the dependence of the yield stress on temperature and rate is offeredby Airing’s theory of molecular activation, where the coefficients A1 and A2 are interpreted interms of activation energy and activation volume. It is noted that some authors have proposedthat the effect of temperature and strain rate on the yield stress may be combined into a singleindependent variable2. Consider now the ultimate shear strain. Over the range of data shown,this quantity seems insensitive to rate (Figure 5) while varying linearly with T (Figure 8). Itsvariation with h (Figure 6) is more complex, and is associated with relatively large scatterat both ends of the thickness range. A reasonable approximation for the ultimate shear strainover the range studied is given by

γF = 3.5(T /Tg)(h0/h)c (5)

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Figure 9. (a) the variation of the area under the stress-strain curve in the Napkin Ring test (curves), Ws , approx-imated as τYγF, for various strain rates, and the variation of the experimentally obtained GIIc/h (filled symbols)with the normalized temperature. All data correspond to a 100 µm thick bond. (b) the variation of Ws and theexperimentally obtained GIIc/h at RT with the bond thickness.

with (h0, c) = (1 µm, 0.21). The prediction of Equation 5 is shown as dashed lines in Figures 5,6 and 8.

Another useful quantity that can be extracted from the test results is the ultimate energydensity or the area under the stress-strain curve, Ws, i.e.,

Ws =γF∫

0

τdγ (6)

A straightforward yet effective approximation for this quantity is given by τYγF. (As shownin Figures 4 and 7, this simplification entails a certain underestimation of the true value, theeffect that generally increases with temperature or with decreasing the bond thickness). UsingEquation 4 and Equation 5, one has

WS ≈ τYγF = B1T /Tg[1 + A2T /Tg log(γ̇ /γ̇0)

](h0/h)c (7)

where B1 ≡ 1011 MPa. Figure 9 (curves) shows the variations of WS from Equation 7 withthe temperature (a) or the bond thickness (b) for some given test parameters. Figure 9a showsthat the ultimate energy density is rate sensitive, and that it vanishes both at the absolute

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Figure 10. Video micrographs showing the process of crack propagation in the ENF test for an 80 µm thickbond. The first and last frames correspond to the onset of crack propagation and the onset of steady-state growth,respectively. The letters A and B indicate reference points. The series of scratch lines are introduced prior to thetesting to help view the shear deformation in the bond.

temperature and near Tg, reaching a maximum at T /Tg ≈ 0.6. (It should be noted that our testtemperatures are way off the absolute temperature range, so the corresponding predictions arehighly speculative). This interesting behavior reflects a compromise between the conflictingtemperature dependencies of τY and γF. Figure 9b (solid-line curve) shows that the ultimateenergy density increases with decreasing the bond thickness, similarly to γF. These resultswill be discussed further in connection with the fracture tests.

3.2. ENF FRACTURE TESTS

Figure 10 shows a video sequence of the crack tip region for an 80µm thick bond tested atRT. The interfacial crack increases slowly before rapid or steady-state propagation occurs. Inthe last frame shown, which is just before rapid propagation, the shear strain at the crack tipregion is nearly uniform across the bond. The fracture energy increases monotonically duringthe sub-critical phase before leveling off at steady-state. It is apparent from Figure 10 thatsuch resistance behavior would be a consequence of the evolution of plastic deformation at thecrack tip, with the steady-state growth taking over once a state of homogeneous deformation

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across the bond occurs. This behavior characterizes the results for thin bonds. For relativelythick bonds, it is found that catastrophic fracture occurs prior to the attainment of a stateof homogeneous deformation at the crack tip region. It seem reasonable to assume that thesteady-state mode II fracture energy (per unit area) of thin-bond joints can be approximatedas the area under the stress-strain curve in the Napkin Ring test times the bond thickness, i.e.,

GssIIC = hWs (8)

In order to examine this proposition, ENF fracture tests are conducted at a fixed crosshead dis-placement (i.e., 0.017 mm/s.) for various temperatures and bond thicknesses, and the steady-state values, normalized by h, are shown as symbols in Figure 9a and Figure 9b. The error inthe test data is estimated to be less than 10%. These figures show that the normalized modeII fracture energy agrees reasonably well with Ws corresponding to a strain rate on the orderof 1/s., a value that is reasonable given that crack propagation at steady-state is quite rapid.(The rate of shear strain at the crack tip region at steady-state for these tests could not beeasily determined). As is evident from Figure 9b, the good correlation between Ws and Gss

IIC/his limited to relatively thin bonds (i.e., < 100 µm). For thicker joints, the mode II fractureenergy seems to approach a fixed value independent of the bond thickness (dashed line inFigure 9b). The transition from the ‘thin’ to the ‘thick’ bond regimes appears to be due to atransition from homogeneous to localized plastic deformation at steady-state.

Equation 8 implies that the mode II fracture energy of the joint vanish both at the absolutetemperature and at Tg, reaching a maximum in between these two extremes. As indicatedearlier, the behavior for the lower end of the temperature is quite speculative. It is interestingto note that Mostovoy and Ripling (1972), and Bascom and Cottington (1976), have observedgenerally similar trend in their Mode I fracture study of a rubber-modified epoxy adhesive. Incontrast, Mode II interlaminar fracture tests on laminates constructed from an epoxy matrix(Russell and street, 1982) or PEEK polymer (Hashemi et al., 1990) show that the fractureenergy increases with temperature, the more so as T approaches Tg. This departure appears abit implausible in light of a previous study (Chai, 1990) showing that there exists a good cor-relation between the fracture energy of laminated composites and adhesive bonds constructedfrom the same polymeric binder (provided that the bond thickness is similar to the effectivethickness of the interlaminar resin rich layer). One may explain this apparent discrepancyby the additional energy dissipation in the laminate that occurs well away from the fractureprocess zone. Such contribution depends on the specimen geometry, however.

4. Fracture analysis of the napkin ring specimen

4.1. SCOPE

The main goal of this effort is to determine a rationale for the variation of the ultimate shearstrain (or stress) with bond thickness (Figure 6). For this purpose, it is assumed that the nom-inally unflawed bond contains a distribution of flaws or porosities. Thicker bonds naturallypermit larger flaws, which may reduce the strength of the joint. Flaw statistic arguments arecommonly employed to explain the size effect that is observed in testing brittle materials (e.g.,Sharp et al., 2001) as well polymeric materials exhibiting significant nonlinearity (e.g., Gian-notti, et al., 2003). In the present application, the constraint on deformation that is imposed bythe adherend surfaces is another factor that may be responsible for the variations of the jointstrength with h. Indeed, Erdogan and Gupta (1971), in their fracture analysis of linearly elastic

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Figure 11. FEM mesh at the crack tip region. The crack tip displacements are evaluated from the nodal pointsA and B identified in print (b), where u and v are the corresponding nodal displacements in the horizontal andvertical directions, respectively. The cohesive zone region is indicated by the darkened area in (a).

materials, showed that the mode II stress intensity factor in a sandwich structure containinga center crack and subjected to shear loading decreases rapidly once the bond thickness ap-proaches the crack length. While the flaw distribution and geometric constraint effects notedabove should hold true also under large-scale yielding conditions, relatively little quantitativeinformation is available in this case. The complex nature of the inherent porosities or voidsin the bond, and the large deformation that develop prior to damage growth, renders anystrength analysis difficult and semi-quantitative at best. It is generally recognized, however,that the most dangerous flaws in the population are sharp cracks. Accordingly, it is assumedthat a sharp crack pre-exists in the bond. The shear strain applied to the adherends that causescrack propagation is then determined with the aid of a quasi-static, FEM fracture analysis.Corresponding results for various bond thicknesses and crack lengths can then be assessedrelative to the test data in Figure 6. A prime issue in this undertaken is the determination of asuitable fracture criterion for materials undergoing large deformation. This is discussed next.

4.2. CRACK PROPAGATION CRITERION

A recently proposed cohesive zone like fracture approach is employed (Chai, 2003). As shownin Figure 11a, a fine, square grid is implemented at the crack tip vicinity. A strip of ma-terial there (darken region in Figure 11a) is assumed over which volumetric change duringplastic deformation is allowed. Crack propagation occurs when a certain combination of thenodal displacements in the cohesive zone reaches a critical value. Referring to Figure 11b fornotations, this local criterion is expressed as

ε/εc + γ /γc = 1 (9)

where ε ≡ η/� is the normalized, bond-normal relative displacement of the nodes marked A

and B, γ is the slope of the deformed grid at these points, � is the length of the square grid,and εc and γc are extreme critical values, determined for the present adhesive as 3.3 and 3,respectively (Chai, 2003). This criterion was found to predict well the mixed-mode fracturebehavior of bonded joints, irrespective of the bond thickness or the direction of the appliedshear. While a criterion of this nature is expected to depend on the size of the grid at the cracktip, a study of three different grid sizes, namely 0.5 µm, 1 µm and 1.5 µm, showed little meshsensitivity (Chai, 2003). The following results are performed for the choice � = 0.5 µm.

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Figure 12. FEM prediction of the dependence of the applied shear strain needed to cause crack propagation onthe bond thickness and the crack length (open symbols) for the center crack case. The solid and the dashed linecurves are empirical fits to these and a fit to the test data in Figure 6, respectively.

4.3. FINITE ELEMENT DETAILS

A commercial large-strain Finite Element code (Ansys, Version 5.7) is used to model theNapkin Ring specimen. As indicated by the inserts in Figures 12 and 14, two types of cracksparallel to the interface are considered, one at the bond center and the other at the interface.In accordance with the tests, the adhesive is assumed to be bounded by two thick aluminumblocks that are unconstrained in the bond-normal direction. The blocks are given a relativeshearing displacement V , so that the average shear strain in the bond, γ , equals V/h. Inter-penetration between the two crack faces is eliminated using a built-in contact algorithm. Thefrictional stresses that are developed between the two crack faces are assumed to obey Cou-lomb’s law, with the friction coefficient taken as 0.35. The stress-strain relation assumed forthe adhesive is given in Figure 7 of Chai, 2003. Plane-strain elements are assumed throughoutthe sandwich structure except for the cohesive zone (the darkened strip shown in Figure 11a),where plane stress conditions are considered. (Note that in the case of the interface crack,the cohesive zone is limited to the adhesive material). This choice, while quite simplistic,facilitates volume change during plastic deformation. The post yield behavior of the adhesiveis modeled according to the J2 flow theory, i.e., with isotropic strain hardening and incrementalplasticity theory based on the von Mises invariant. The shear strain is applied incrementally,with the local displacements evaluated in each step. Fracture is assumed when the criteriongiven by Equation 7 is first fulfilled. By repeating this process for various bond thicknessesand crack lengths, the dependence of the applied critical shear strain, γF, on these quantitiescould be established.

4.4. RESULTS

Figure 12 (symbols) shows the results for the center crack case. It is apparent that for agiven crack length, the critical strain needed to cause crack propagation approaches a steady-

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Figure 13. FEM prediction of the evolution of deformation at the crack region for a 20 µm thick bond containinga 6 µm long center crack.

state value once the bond thickness becomes on the order of the crack length, a. Decreasingeither h or a increases the critical shear strain. In the limit h → 0, all data are expectedto converge to γc (i.e., 3), the extreme local shear strain assumed in the fracture criterion.Also shown in Figure 12 are corresponding empirical fits (solid-line curves), given as γF =γc +(a/A)B{exp[−(h/a)C(h/D)E]−1}, with the constants (A, B, C, D, E) chosen as (3.2 µm,0.8, 0.5, 1.37 µm, 0.7). Note that this relation reduces to γF = γc when either a or h approachzero, in consistency with the fracture criterion given in Equation 9. The variation of the crackprofile with the applied shear strain for the center crack case is exemplified in Figure 13 forthe choice a = 6 µm and h = 20 µm. As shown, the crack rotates as the applied shear strainis increased. Contact between the two crack faces seems to occur over the entire crack lengthexcept near the crack tips, where large voids are formed. Figure 14 shows analogous resultsfor the interface crack case, with the fitting constants (A, B, C, D, E) taken as (2.2 µm, 0.55,0.1, 1.37 µm, 0.4). The behavior is generally similar to that for the center crack.

Examination of Figure 12 or Figure 14 against the tests results, shown as dashed lines inthese plots, suggests that a fixed choice of the crack length cannot conclusively explain theexperimental variation of γF with h. A resort to flaw distribution thus seems necessary. Fora given critical crack length aF, the bond thickness that is needed to cause crack propagationis obtained from the intersection of the dashed and the solid lines in Figures 12 or 14. The

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Figure 14. As in Figure 12 for the interface crack case.

Figure 15. FEM prediction of the variation with bond thickness of the critical crack length (open symbols),determined from the intersection of the FEM data in Figure 12 or Figure 14 with the experimental fit in theseplots. The solid-line curve is an empirical fit.

results, shown as symbols in Figure 15, imply that thicker bonds (i.e., larger material volumes)produce larger critical cracks. As shown by the solid-line curve in Fig.15, the variation ofthe critical crack length with the bond thickness can be reasonably well approximated asaF = C1 log(h/h1), where (C1, h1) = (3.2 µm, 0.5 µm).

It is interesting to note a related problem by Ikegami and Kamiya (1982), who studiedthe effect of a pre-embedded interfacial crack in a Napkin Ring joint constructed from abrittle-epoxy resin on the ultimate strength of the joint. Using stress intensity factor solutionsdeveloped by Erdogan and Gupta (1971), these authors were able to predict well the variationof the bond strength with the crack length. Ikegami and Kamiya reported no such success inattempting to correlate the fracture stress of the nominally unflawed adhesive using a crack offixed length, however, in consistency with the present finding.

The effects of bond thickness 513

5. Summary and conclusions

The stress-strain behavior in simple shear of a structural adhesive undergoing large-scaleyielding is studied as a function of bond thickness (2 µm to 600 µm), strain rate (10−4/s.to 1/s.) and temperature (−60 ◦C to Tg) using the Napkin Ring test specimen. The carefuldesign and fabrication of the specimen employed in this work help reduce bond thicknessnon-uniformity, eliminate unwanted bending, and avoid starving joints. This, together with aclosed-loop control of the shear strain across the bond, allow for accurate determination of themechanical response throughout deformation process. In consistency with Airing’s molecularactivation model, the yield stress decreases linearly with temperature and increases logarith-mically with the strain rate. The ultimate shear strain of the joint is found to be insensitiveto the strain rate, decreases linearly with T and increases monotonically with decreasing thebond thickness. Some complimentary fracture tests are carried out using the ENF adhesivejoint specimen in order to examine the feaseability of deducing the fracture energy of the jointfrom more fundamental material properties. The results show that the mode II fracture energyper unit bond thickness at steady-state agrees well with the area under the stress-strain curveof the nominally unflawed adhesive provided that the bond thickness is sufficiently small. Toa first approximation, the ultimate energy density is given by τYγF, where τY and γF are theyield stress and ultimate shear strain, respectively. Accordingly, the so interpreted fractureenergy becomes negligibly small at both the absolute and the glass transition temperatures,reaching a maximum between these two extremes. Simple empirical relations are derived toexpress the variations of the deformation and fracture characteristics of the bond with the testparameters.

A quasi-static, large-deformation analysis is employed to account for the increase in strengthor ultimate shear strain in the Napkin Ring specimen with decreasing the bond thickness. Theanalysis, based on a previously developed cohesive zone like model, assumes that fractureoccurs when a certain interactive relation between the nodal displacements at the tip of apre-existing crack is fulfilled. Two crack systems are considered, namely a center crack andan interface crack, both of which extend in the interface direction. The results show that theapplied shear strain needed to propagate a crack of a given length increases with decreasingthe bond thickness. This geometric effect is insufficient to explain the experimental results,however, and it is necessary to resort to flaw distribution arguments. It is found that the exper-imental variations of γF with h are well predicted by a logarithmic type increase in the criticalcrack length with bond thickness (or material volume).

The results show that thinner bonds lead to superior strength. To realize large toughness,however, thicker bonds are necessary. In this case, a multilayer design approach would bebeneficial since this would reduce the size of the critical flaw in the bond. Another approachfor controlling flaw size is to use a film carrier. Because the yield stress is little sensitive to thebond thickness, the main factor affecting the toughness of the joint appears to be the ultimateshear strain. The Napkin Ring specimen seems well suited to provide accurate measure of thisquantity.

Notes

1. Softening in polymers has been reported in numerous studies, although it is not entirely clear whether this isa conclusive trend or, at least in some cases, a mere artifact of the testing procedure. For example, Liang and

514 H. Chai

Liechti (1996) showed that stress softening could be largely suppressed if local rather than global shear strainis considered.

2. The effect of strain rate and temperature on the yield stress of a rubbery material has been expressed in termsof a reduced variable, ε̇aTY , where aTY is the shift factor for creep (Smith, 1958). This was except for shorttimes or high strain rates, where the material approaches a glass like behavior. Moehlenpah, et al. (1971)proposed that such simplification might also hold true for glassy polymers.

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