Contact Angle and Contact Mechanics of a Glass ... - Springer Link

International Journal of Fracture 86: 361–374, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Contact angle and contact mechanics of a glass/epoxy interface K.M. LIECHTI, S.T. SCHNAPP and J.G. SWADENER Research Center: Mechanics of Solids, Structures & Materials, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas 78712-1085 USA Received: 21 January 1997; accepted in revised form 30 July 1997

Abstract. An attempt was made to extract the thermodynamic work of adhesion from contact angle measurements and contact mechanics in order to compare it with interfacial toughness values of a glass/epoxy interface. The three probe liquid method, in conjunction with laser goniometry, yielded a value of the work of adhesion of 93 mJ/m2 . This was an order of magnitude less than the value extracted from the Maugis solution for contacting spheres with surface interactions. These work of adhesion values were both lower than the 1.5 J/m 2 which was determined in a parallel study of interfacial fracture as a mode-mix independent component of the overall interfacial toughness. Some of the reasons for these differences are explored. Key words: interfacial fracture, thermodynamic work of adhesion, contact angle, contact.

1. Introduction This work was motivated by studies of fracture along glass/epoxy interfaces. The toughness of the interface between a modified bisphenol A epoxy cured with a polyamido amine hardener and BK7 glass has been found to increase with increasing shear component (Liechti and Hanson, 1988; Liechti and Chai, 1992 and Liang and Liechti, 1995) even when the epoxy was sandwiched between glass and aluminum (Liechti and Liang, 1992; Swadener and Liechti, 1996). In the latter case (Swadener and Liechti, 1996), the toughening effect was shown to be due to dissipation in the epoxy, whose mechanical behavior had been characterized over a variety of single and multiple stress states and rates (Liang and Liechti, 1996). Quasi static analyses that modeled the interface as a separate constitutive entity showed that the toughness values at various mode-mixes consisted of an apparently intrinsic component of about 1.5 J/m2 , which was the same for all mode-mixes and a dissipative component which increased with increasing mode-mix. In all these studies, the epoxy was cured directly to the glass without any coupling agents or primers. One might expect that the bonding between the glass and epoxy was due to van der Waals attractions which typically give rise to work of adhesion values of about 100 mJ/m2 (Kinloch, 1987), about a factor of fifteen lower than an already low intrinsic toughness value noted above. In contact experiments between mica and silica, Horn and Smith (1992) found that contact electrification produced interfacial toughness values of up to 6 J/m2 . Discontinuities in the force deflection curves during unloading were attributed to discharges. Similar discharges have been noted by Zimmerman et al. (1991) during the pullout of a steel rod from an epoxy matrix. In light of these developments, it seemed natural to determine the work of adhesion of the glass/epoxy interface on the basis of contact angle measurements and contact experiments. Such comparisons have been made in the past for a variety of systems (Chen et al. 1991;

Engineering Mechanics Research Laboratory Report, EMRL # 96/11.

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Chaudury and Whitesides, 1991; Merrill et al. 1991 and Mangipudi et al. 1996). Work of adhesion values were typically less than 100 mJ/m2 and the difference between values obtained from the two approaches was generally less than 20 mJ/m2 . On the other hand, Maugis and Gauthier-Manuel (1995) have recorded values of almost 500 mJ/m2 for mica to mica contact in humid air due to capillary condensation. This was much closer to the values extracted in this work where contact resulted in a viscous liquid being squeezed from the epoxy. Another reason for performing the contact experiments is that large amounts of shear loading can cause the crack faces to come into contact (Comninou and Schmueser, 1979). Thus an understanding of the characteristics of contact between glass and epoxy will be useful for subsequent modeling of interfacial crack growth under such conditions. The history of the problem of non-sliding contact, accounting for the presence surface interactions, has been well summarized and put in a fracture mechanics context by Maugis (1992). For the purposes of this investigation, we have essentially considered the use of the work of Johnson et al. (1971), JKR theory, and Muller et al. (1980), MYD theory, for extracting the work of adhesion during contact. The latter relaxed the assumption of Derjaguin et al. (1975), DMT theory, that surface interactions outside the contact region affected the stresses near the contact boundary but nonetheless gave rise to a Hertzian displacement profile outside it. As is pointed out by Maugis (1992), the self-consistent MYD approach (Muller et al., 1980) considers a Lennard–Jones potential for the surface interaction and allowed the surfaces to deform accordingly. As a result, the so-called JKR and DMT theories or solutions were reconciled as limiting cases of the MYD theory. Maugis (1992) essentially follows the MYD approach, using a constant traction for the surface interaction, the familiar Dugdale model of fracture, thereby allowing analytical solutions to be obtained. Considering that Lennard–Jones potentials give rise to decidedly non-uniform surface interaction tractions, the question naturally arises as to the validity of models which make use of uniform ones. Ungsuwarungsri and Knauss (1987; 1988) have extracted traction separation laws for cracked joints and crazed cracks. They indicate that certain traction separation laws can affect the stability of crack growth. The question has also been addressed in the work of Tvergaard and Hutchinson who used cohesive zone models to model cohesive (Tvergaard and Hutchinson, 1992) and adhesive fracture (Tvergaard and Hutchinson, 1993; 1994). It has been found that the steady state toughness was relatively insensitive to the shape of the traction separation law. Instead, the two most important parameters were the intrinsic fracture toughness (area under the traction-separation curve) and the maximum traction. Making the extension to contact and surface interactions, we might expect that properly chosen uniform traction and separation displacement values would be sufficient for extracting work of adhesion values. In this study, the thermodynamic work of adhesion was extracted using the three liquid probe method that was developed by Good et al. (1991) and recently applied to fracture problems by Azimi et al. (1995). Contact angles were measured using the laser goniometric method of Israel et al. (1989). Load, gap and contact radius measurements were made in contact experiments with epoxy caps and flat glass surfaces. Crack tip opening displacements and work of adhesion values were extracted using the Maugis (1992) solution. The values extracted from the two approaches differed by roughly an order of magnitude. Possible reasons for this are explored.

Contact angle and contact mechanics of a glass/epoxy interface

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Table 1. Surface energy parameters for the three liquids. Liquid

d , mJ/m2

+ , mJ/m2

, , mJ/m2

Diiodomethane Water Glycerol

50.8 21.8 34.0

0.0 25.5 3.92

0.0 25.5 57.4

Figure 1. Laser goiniometry apparatus for contact angle measurements.

2. Contact angle The work of adhesion of glass and epoxy was determined using the three-liquid probe method (Good et al. 1991). The three-liquid probe method uses three liquids, one apolar and two polar, to determine the contributions of each portion of the total surface energy of each solid and thereby determine the work of adhesion between them. In the case of this study, the two solids were the bisphenol A epoxy and the BK7 glass and the liquids were diiodomethane (apolar), glycerol and distilled water (polar). These liquids were chosen because their acidity and the basicity were known (Good et al. 1996). These values are given in Table 1. Contact angle measurements were made between all of the liquid/solid pairs using laser contact angle goniometry (Israel et al. 1989). In this method, a laser beam is projected at the point of contact between a liquid drop and the substrate. As the laser intersects this corner point, the beam splits into two, forming a measurable angle on a screen located behind the sample apparatus. The arrangement is shown in Figure 1. The angle created by the split laser beam is the contact angle and can be measured using a protractor. The contact angle of at least thirty drops of each liquid (diiodomethane, distilled water, and glycerol) on each solid (epoxy and glass) was measured to obtain an equilibrium contact angle reading and a corresponding standard deviation. Thirty measurements were taken to demonstrate the repeatability of the experiment and to establish the level of accuracy of surface energy interactions and of the contact angle measurement method. The contact angle results are shown in Table 2. The numbers in the parentheses represent the standard deviations. Knowing the contact angle, , for each liquid/solid system the following equations were solved. Equation (1) determines the dispersive component ( d ) and Equations (2a) and (2b) are solved simultaneously to determine the acidic and basic terms ( + and , ) of the surface

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K.M. Liechti et al. Table 2. Contact angle measurements (degrees). Substrate

Diiodomethane

Glass Epoxy

34.7 ( 26.6 (

2.2) 1.8)

Distilled Water 49.5 ( 70.6 (

3.1) 3.9)

Glycerol 48.5 ( 62.5 (

3.5) 2.6)

Table 3. Surface energies of glass and epoxy (mJ/m2 ). Substrate

d

+

,

p

Glass Epoxy Silica (Kinloch et al., 1975) Epoxy (Kinloch et al., 1975)

42.2 45.6 78.0 41.2

0.43 0.02

27.4 10.39

6.9 0.9 209.0 5.0

49.1 46.5 287.0 46.2

energy of each solid. The subscripts s; L1 ; L2 ; L3 refer to the solid phase and diodomethane, distilled water and glycerol, respectively.

sd = 14 Ld 1(1 + cos L1)2 ;

(1)

L2(1 + cos L2) = 2[( sd Ld 2)1=2 + ( s+ L,2)1=2 + ( s, L+2 )1=2 ];

(2a)

L3(1 + cos L3) = 2[( sd Ld 3)1=2 + ( s+ L,3)1=2 + ( s, L+3 )1=2 ]:

(2b)

For each solid, the measured contact angles (Table 2) were used in conjunction with surface energy values of the probe liquids (Table 1) to determine the surface energies of the glass and epoxy which are listed in Table 3. The work of adhesion of glass and epoxy was obtained through

wEG = 2[( Ed Gd )1=2 + ( E+ G,)1=2 + ( E, G+)1=2 ]:

(3)

For the epoxy/glass system used in this study, the work of adhesion was found to be 93 mJ/m2 . Kinloch et al. (1975) analyzed the interactions present between an epoxy adhesive and a silica surface. Although the exact nature of the epoxy and silica are not described, it is assumed that the result should be somewhat comparable with the results obtained in this study. Table 3 shows the specific energy parameters that were obtained from their study. The surface energy values for the epoxy are very close to the results obtained from the contact angle experiments in this study. The silica, however, has a much higher surface energy than the glass that was used here. The corresponding work of adhesion between the two materials using these surface energy parameters was 178 mJ/m2 . It is assumed that the discrepancy between the two studies is a result of the difference in surface energies of the glass (this study) and the silica (Kinloch et al., 1975). Although a closer correlation was expected, these discrepancies may be due to differences in composition of the materials used in contact or differences in their surface preparations.


365

Table 4. Elastic properties. Material

Young’s Modulus (GPa)

Poisson’s Ratio

Epoxy Glass

2.7 69.0

0.34 0.20

3. Contact mechanics As was discussed in the introduction, the work of adhesion also enters analyses of bodies in contact. Because it affects the force displacement response, the pull-off force, the contact radius and the gap profile, it can be extracted from any of these parameters. Due to equipment constraints in this study, only the latter two parameters could be considered as a function of load level. The manner in which the work of adhesion was extracted from contact experiments and analyses is now discussed. 3.1. EXPERIMENTS The experiments consisted of bringing an epoxy hemisphere or cap in contact with a glass optical flat while measuring the load, contact radius and gap. Several methods were considered for producing the epoxy caps (Schnapp, 1996). The most successful method for producing smooth spherical surfaces was by placing liquid drops of uncured epoxy upon a sheet of Teflonr FEP fluorocarbon film manufactured by DuPont. This film had a very low surface energy so that a high contact angle was obtained, and, once the curing took place, a smooth spherical surface remained. This is essentially the scheme that was pioneered by Chaudhury and Whitesides (1991). The elastic material constants for both the epoxy and the glass are listed in Table 4. The epoxy properties were obtained from uniaxial tension tests at a strain rate of 3:3 10,4 s,1; and the values for glass are from manufacturer’s data. In order to be able to adjust the approach distance between the epoxy hemisphere and the glass plane, the plane was fixed in place, while the hemispherical cap was fixed to a rigid member whose elevation could be adjusted as desired (Figure 2). The total load that was transmitted between the epoxy and the glass as the two bodies came into contact was measured using a 50 gram load cell that was placed on the elevating surface and attached to the hemisphere An optical interference technique was used to measure the gap height, or the distance between the two surfaces (Figure 2). To do this, monochromatic light (546 nm) was directed towards the hemisphere/plane contact region at normal incidence using the coaxial illuminator in a Wild Macrosop 420 microscope. As contact was attained, the changing fringe patterns were recorded through a video camera on a 500 line resolution video recorder with a video timer. The video images were subsequently analyzed (fringe locating and counting) using a digital image analysis system. Originally, the resolution in gap corresponding to light to dark fringe transitions was 0.14 m, with fringes being located spatially to within 5 m. Recent developments that make use of measured intensity values between fringes have increased the resolution to about 10 nm. In each experiment, various fixed displacement levels were applied to the epoxy cap and the corresponding load history was recorded using a computer-based data acquisition system

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Figure 2. Apparatus for the contact experiments.

along with the fringe data as described above. It was found that the load first increased (became more compressive) before relaxing (Schnapp, 1996). The loads recorded here were taken 10 minutes after the displacement was applied, once relaxation was essentially complete. Only the loading response was measured so hysteresis effects were not examined. The low resolution version of the interferometry scheme was insufficient to discern any change in conact radius that might have occurred during the period of load transition. Decreasing load levels with increasing contact radius have been recently recorded by Falsafi et al. (1996) for elastomeric contact. The relatively short relaxation times of the polymers involved resulted in immediate relaxation of the load. The occurrence of the load transients, particularly the initial rise in (compressive) load have been confirmed in some recent ongoing experiments with a piezoelectric actuator with a 4 nm displacement control capability. Higher resolution in the contact radius measurements have indicated that it also rises. The rise in compressive load suggests that the surface tractions outside the contact zone are decreasing. If electrostatic charges are being built up during contact, as has been suggested by Derijaguin et al. (1982), Horn and Smith (1992) and Zimmerman et al. (1991), then these may be discharging with a concomitant decrease in surface tractions. The other notable phenomenon in the experiments was the deposition of a viscous liquid residue on the surfaces following contact. Residues have also been alluded to by Ahn and Shull (1996) in their experiments with an acrylic elastomer on a PMMA substrate. In this study, the amount of the residue was reduced considerably when the resin was changed from Ciba Geigy GY 502 to GY 6010. The former used dibutyl phthalate as a plasticizer. In the absence of the plasticizer, the remaining residue from the experiments with the GY 6010 resin is probably uncured resin which formed a meniscus and must, therefore, provide some bridging across the surfaces. All the data provided in this report were obtained from GY 6010 resin, cured with HY 955 hardener with a ratio of 100:35. The process of forming the meniscus may also have contributed to the load transients alluded to above.


367

Figure 3. Contact geometry.

3.2. ANALYSIS Several analytical schemes were considered for extracting the work of adhesion from the contact experiments (Schnapp, 1996). The cohesive zone approach suggested by Maugis (1992) and implemented for the analysis of contact in the presence of liquid menisci Maugis and Gauthier-Manuel (1995) is the most general while maintaining sufficient simplicity that analytical solutions are possible. In this approach, the tractions due to surface interactions are considered to be constant over the cohesive zone. This makes it a special case of the MYD (Muller et al., 1980) analysis, for which only numerical solutions can be obtained and then, as was discovered when attempts were made to implement the scheme for this study, only over very small contact zones due to the extremely large storage requirements for the contact radii being considered here. The relative simplicity of the solution (Maugis, 1992) is also a consequence of all the materials being elastic. Any plasticity would have to be handled using finite element techniques, although it is possible that viscoelastic effects could be handled using correspondence principles (Falsafi et al., 1996). The configuration being analyzed is shown in Figure 3. The constant surface traction, 0, acts over the distance (c , a), where a is the contact radius and c is the outermost extent of the of the cohesive zone. The governing equations are given in the Appendix in the same dimensionless form as the original (Maugis, 1992), which can be referred to for details of derivation and a discussion of its relation to the JKR and DMT limits. The work of adhesion of the glass/epoxy interface, the radius of curvature of the epoxy and the elasticity of the epoxy/glass combination are given by w, R, and K , respectively, where 1=K = (1 , 12)=E1 + (1 , 22)=E2 and Ei and i are the Young’s moduli and Poisson’s ratios of the epoxy (i = 1) and glass (i = 2). In order to extract the work of adhesion and the surface traction or crack opening displacements from the measurements, it is necessary to transform the original equations into dimensional form. The closure condition (A3) and the load vs. contact radius response (A4) become a2 [pm2 , 1 + (m2 , 2) tan,1 pm2 , 1];

t R

16aw p 2 ,1 pm2 , 1 , m + 1] = 1 [ m , 1 tan 3t2 K

(4)

P , PH = ,2a w [ m2 , 1 + m2 tan,1 m2 , 1];

(5)

+

and 2

t

p

p

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K.M. Liechti et al.

Figure 4. Plot for extracting the work of adhesion of the glass/epoxy interface.

where the Hertzian load vs. contact radius response is given by

PH = aRK : 3

(6)

The crack tip opening displacement, t , is a constant and can be determined from gap profiles taken over a range of load levels, particularly if the profiles are plotted in double logarithmic form. Once it has been selected, equations (4–6) can be used to extract the work of adhesion, w, and the extent of the contact zone, m = c=a. The work of adhesion is fixed but the extent of the cohesive zone is a function of the load. The extraction was achieved by first solving Equation (5) for w and then substituting into (4) so that the closure condition becomes

a2 [pm2 , 1 + (m2 , 2) tan,1 pm2 , 1] t R 8(P , PH ) + 3at K

p

p

mp2 , 1 tan,1 m2 , 1 , m + 1 = 1 = 1: p m2 , 1 + m2 tan,1 m2 , 1

(7)

The extent of the contact zone was determined for each load level and then used p to determine w inpa graphical manner by plotting [P , PH ] against the quantity. ,a2 [ m2 , 1 + m2 tan,1 m2 , 1]. The slope, s, of the resulting linear plot is then given by s = 2w=t , from which w may then be determined. In using the JKR formulation, a similar approach is often taken (Schnapp, 1996; Ahn and Shull, 1996) where [P , PH ]2 is plotted against a3 . 3.3. RESULTS The scheme outlined above was applied to two sets of experiments where load conp and 2 2 tact radiusp were measured. The plot of [P , PH ] against the quantity ,a [ m , 1 + m2 tan,1 m2 , 1] is shown in Figure 4. The linearity of the fits was good and the slopes from each experiment were in good agreement. The procedure was applied for several choices of t and the corresponding values of w were extracted.


369

Figure 5. Comparison of measured and predicted load vs. contact radius response for glass on epoxy.

The resulting load vs. contact radius responses are compared with the measured values in Figure 5. The largest predicted values of contact radius for a given load were obtained for the JKR solution ( ! 1) with w = 1:245 J/m2 . The smallest were obtained from the Herz solution. A fit with a crack opening displacement t = 60 nm and a work of adhesion of 823 mJ/m2 gave the best results when compared to the measured values. However, it can be seen that each pair of values of crack opening displacement and work of adhesion has a range of application. The smallest crack opening displacement values resulted in the lowest works of adhesion and yielded the best fit at the lowest load levels. The corresponding fits to the gap profile are shown in Figure 6. The filled data points were obtained from fringe locations, whereas the open symbols represent the data from measurements of intensity values between fringes. The intensity values were extracted from the same already digitized frame that the fringe values were taken from but not necessarily the same radial trace. The far field consistency was good and the intensity measurements provided data within the cohesive zone, closer to the contact radius. Although there is some scatter to contend with, the measured gap profiles suggest that t = 60 nm provided the best fit. The resolution of the gap displacements has recently been improved by increasing the magnification. The independence of t on load level has been established. The corresponding value of 823 mJ/m2 for the work of adhesion is less than double the value determined by Maugis and Gauthier-Manuel (1995), in considering mica/mica contact in humid air. In view of the small meniscus formed by the liquid that was squeezed from the epoxy the association of the two cases seems reasonable. The higher value encountered here might be attributed to the higher viscosity of the residue. The surface traction, 0 , for a work of adhesion of 823 mJ/m2 and a crack tip opening displacement of 60 nm is 13.7 kPa. This is much lower than the 40 MPa tensile yield strength of the epoxy. As a result, the tensile stresses (see A7) in the epoxy near the contact radius would be insufficient to excite any yielding. The largest compressive stress occurs at the center. The contact radius at which yielding would first occur is 140 m for a compressive yield strength of 60 MPa (Liang and Liechti, 1996), which means (Figure 5) that no yielding could have occurred in any of the experiments.

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K.M. Liechti et al.

Figure 6. Comparison of measured and predicted gaps for glass on epoxy.

4. Discussion The first point of discussion is the large difference in the values of the work of adhesion that were obtained from the contact angle and epoxy/glass contact experiments. Each can be compared favorably with values that are available in the literature for relatively stiff materials. The 93 and 823 mJ/m2 from, respectively, the contact angle and epoxy/glass contact measurements compared favorably with the results of Kinloch et al. (1975) for an epoxy/silica interface and Maugis and Gauthier-Manuel (1995), for mica/mica contact in humid air. Closer agreement between work of adhesion values obtained from contact angle and solid/solid contact have recently been reported by Mangipudi et al. (1996) for various elastomers in contact with PMMA. They noted that contact angle and mechanical contact experiments yielded work of adhesion values that were in reasonable agreement, except for interfaces formed by polymers with larger acid-base interactions. However, even in such cases, the work of adhesion values only differed by about 10 mJ/m2 . Similarly small difference levels have been noted by Chen et al. (1991), who also found that contact hysteresis followed the same trends in contact and contact angle experiments. Merrill et al. (1991) report that work of adhesion values that were obtained from pull-off force measurements and the DMT solution were in close agreement with those obtained from contact angle measurements. However, it was also noted that contact radius at pull-off and gap profiles were closer to JKR predictions. Even if JKR pull-off force values had been used, only a 25 percent difference in work of adhesion values would have been found. Chaudhury and Whitesides (1991) found good agreement between work of adhesion values from the JKR solution and contact angle measurements. None of these studies report that any residue was left on the surfaces following contact. Residues were noted in the experiments of Ahn and Scull (1996) when unextracted lenses are used. After the sol fraction has been extracted from the lens, there were no traces of residue left on the substrate. Although no comparison was made with contact angle experiments, the work of adhesion values, particularly on loading, were less than 50 mJ/m 2 for PNBA on PMMA. Neither contact angle or contact mechanics probes are truly non-invasive. Even with glassy organic materials, surfaces can rearrange locally to present different functionalities to the


371

contact interface. Thus, in general, differences in work of adhesion values can be expected to arise. A second notable effect was that work of adhesion values that were obtained from lower load levels were smaller than those obtained at higher load levels. A similar effect has essentially been noted in mica to mica contact in dry air by Maugis and Gauthier-Manuel (1995), who found that the values of the work of adhesion that were extracted from gap profiles were much higher than that which was obtained from the contact radius at zero load. Since mica surfaces are so smooth, this effect cannot be attributed to surface roughness. If the liquid forms a film between the glass and epoxy, as has been reported by Israelashvili et al. (1982), then the dynamics of squeeze films will also be involved. The load vs. contact radius no longer follows the cubic relation of Hertzian contact and could, under certain conditions (Rodin, 1996), even be quadratic. Finally, as was alluded to in the introduction, the mixed-mode toughness of the glass/epoxy interface was obtained through a combination of experiments and analysis where steady state debonding was modeled using a fracture process zone approach. When the nonlinear epoxy constitutive behavior and appropriate traction separation law for the interface (as judged by optical interference measurements of crack opening displacements) was incorporated in the finite element analysis, the interfacial toughness has been shown (Swadener and Liechti, 1996) to consist of a plastic dissipation component which varies with mode-mix and an apparently intrinsic component that was fixed at 1.5 J/m2 . This is almost twice the 823 mJ/m2 from the glass/epoxy contact experiments, which suggests that other dissipative effects, which are independent of mode-mix, are operative during interfacial fracture. As indicated earlier, electrostatic charges have been reported for surfaces in contact by Horn and Smith (1992) and on fracture surfaces by Derjaguin et al. (1982) and Zimmerman et al. (1991). The possible effects of electrostatic charges in the interfacial fracture experiments (Swadener and Liechti, 1996) were therefore investigated. The air in the vicinity of the crack faces was ionized with a Po-210 -radiation source and drawn through the crack with suction. This did not cause any observable changes in the crack face separation. Therefore, if electrostatic charges existed in these specimens with a sufficient magnitude to affect fracture, the charges must have been confined to a region of approximately 1m behind the crack tip. Friction and inelastic material deformation on the nanoscale within the cohesive zone region are possible additional dissipative mechanisms. 5. Conclusions The work of adhesion of a glass/epoxy interface has been determined on the basis of contact angle and contact experiments. The three probe liquid method, in conjunction with laser goniometry, yielded a value of work of adhesion of 93 mJ/m2 . During the solid-solid contact experiments, a viscous liquid (probably hardener) was squeezed from the epoxy, giving rise to a much higher value (823 mJ/m2 ) for the work of adhesion. This was extracted from the Maugis (1992) solution of contact, which models the surface interactions via a cohesive zone with a constant traction. The corresponding crack tip opening displacement was 60 nm. These work of adhesion values are both lower than the 1.5 J/m 2 which was determined in a parallel study of interfacial fracture as a mode-mix independent component of the overall toughness. These results indicate that dissipative effects other than plastic dissipation still need to be accounted for before a link can be made between the thermodynamic work of adhesion and the mixed-mode interfacial toughness of the glass/epoxy interface considered here.

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Appendix This appendix summarizes Maugis’ (1992) solution for contact between spherical surfaces with a constant traction acting between them. The dimensionless form of the contact radius is given by

a = (wR2a=K )1=3 :

(A1)

The parameter is used to represent the surface interactions through 20

= (wK 2 =R)1=3

Rw2 = t K 2 2

!1=3

;

(A2)

since the gap t at the outermost boundary of the cohesive zone r = c is related to the traction and the work of adhesion through t = w=0 . For chosen values of a and , the extent of the cohesive zone m = c=a is determined from the closure condition hp i p 1 2 2 , 1 + (m2 , 2) tan,1 m2 , 1 a m 2

p

p

h

a m2 , 1 tan,1 m2 , 1 , m + 1

4 +3 2

i

=

1:

(A3)

Once m has been determined, the corresponding load is given by

P = a3 , a2 hpm2 , 1 + m2 tan2 pm2 , 1i : P = wR

(A4)

The normalized gap inside the cohesive zone ( < m), where = r=a, is 2 q [uz ] a 2 , 1 2 [uz ] = = , 1 + ( , 2) cos (1=) 2 2 1=3

w R=K ) p 8a p 2 8a q 2 , 1 2 + m , 1 cos (1=) + 3 , 1= m , 1 , mE (; k) : 3 The elliptic integral of the first kind E (; k ) has arguments (

p

2 = sin,1 mp 2 , 1 m ,1

!

and k

=

(A5)

=m:

Outside the cohesive zone ( > m) a2 q2 , 1 + (2 , 2) cos,1(1=) + 8a pm2 , 1 cos,1 (1=) [uz ] = 3 "p ! # 2 2,1 8a m m p + , E (; k) + 1 , 2 F (; k) ; 3 2 , 1

(A6)


373

where the elliptic integrals of the first and second kind have arguments p 2 ! = sin,1 pm2 , 1 and k = m=: m ,1 The normal stress inside the contact zone is given by s 2 3a q z z = (wK 2 =R)1=3 = , 2 1 , 2 + tan,1 m1 ,,21 : Inside the cohesive zone (1 6 6 m); z

=

(A7)

0.

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