## Abstract

Tangential loading in the presence of adhesion is highly relevant to biological locomotion, but mixed-mode contact of biological materials or similar soft elastomers remains to be well understood. To better capture the effects of dissipation in such contact problems owing to viscoelasticity or irreversible interfacial adhesive processes, a model is developed for the combined adhesive and tangential loading of a rigid sphere on a flat half-space which incorporates a phenomenological model of energy dissipation in the form of increased effective work of adhesion with increasing degree of mode mixity. To verify the model, contact experiments are performed on polydimethylsiloxane (PDMS) samples using a custom-built microtribometer. Measurements of contact area during mixed normal/tangential loading indicate that the strong dependence of the effective work of adhesion upon mode mixity can be captured effectively by the phenomenological model in the regime where the contact area stayed circular and the slip was negligible. Rate effects were seen to be described by a power-law dependence upon the crack front velocity, similar to observations of rate-dependent contact seen for pure normal loading.

## 1. Introduction

The coupling between adhesion and friction forces is of increasing significance for contact problems involving compliant materials, such as polymers used in microfabrication or biological materials. These types of contact problems are typically studied analytically to a first approximation using a simple mechanics model of a sphere or cylinder contacting an elastic half-space (or two contacting elastic spheres or cylinders) in the presence of tangential loads or mismatch strains, as is done in the models of Chen & Gao (2006*a*,*b*,*c*, 2007). However, it is unclear whether the simplifying assumptions of linear elasticity and reversible adhesion provide good approximations to the behaviour of inherently viscoelastic elastomers or biological materials subjected to combined normal and tangential contact loading.

Models for such viscoelastic contact problems are not fully developed at present. For pure normal loading, it is well established that the amount of energy dissipation in bulk viscoelastic materials can far exceed the intrinsic work of adhesion between two surfaces (e.g. Kendall 1971, 1975; Gent & Schultz 1972; Andrews & Kinloch 1973; Barquins 1984*a*; Maugis 1987). Thus, viscoelasticity can cause the adhesive pull-off forces to be highly rate-dependent (e.g. Maugis & Barquins 1978; Greenwood & Johnson 1981) and/or introduce significant adhesion hysteresis (e.g. Roberts & Thomas 1975; Meitl *et al.* 2006). As noted by Lin & Hui (2002), the amount of energy available to make or break contact in viscoelastic materials is dependent upon the details of the bonding, debonding and loading history, and is not uniquely determined by the thermodynamic work of adhesion. This history dependence makes rigorous analytical modelling of viscoelastic fracture or contact quite cumbersome, as seen in the work of Schapery (1975*a*,*b*, 1989), Barber *et al.* (1989), Hui *et al.* (1998), Baney & Hui (1999), Lin *et al.* (1999), Lin & Hui (2002), Barthel & Haiat (2002), Haiat *et al.* (2003) and Greenwood (2004). Further complicating the issue, interfacial processes can also cause significant dissipation, even for elastic materials (Silberzan *et al.* 1994; Perutz *et al.* 1998). Kendall (1973) first proposed that interface kinetics can contribute significantly to adhesion hysteresis, and this hypothesis has been verified by several experimental studies (e.g. Chaudhury & Whitesides 1991; Brown 1993; Silberzan *et al.* 1994; Deruelle *et al.* 1995; She *et al.* 1998; Chaudhury 1999). Ghatak *et al.* (2000) noted that a kinetic theory of interface bond failure precludes adhesion reversibility, which has a basis in the so-called Lake–Thomas effect, where the energy required to fracture an elastomeric interface is amplified owing to the stretching and relaxation of polymer chains (Lake & Thomas 1967). Because of the variety and inherent complexity of such dissipation mechanisms, phenomenological models are often used to capture the dependence of the effective work of adhesion upon the rate of change of contact radius, which is analogous to a crack front velocity (e.g. Maugis & Barquins 1978; Maugis 1987; Barthel & Roux 2000; Meitl *et al.* 2006). Such models typically take the form of a power-law dependence upon velocity, and a theoretical basis for such phenomenological modelling has been developed by Greenwood & Johnson (1981), Hui *et al.* (1992), de Gennes (1996), Muller (1996), Saulnier *et al.* (2004) and Persson & Brener (2005), among others.

In addition to the above dissipative phenomena seen in both elastic and viscoelastic materials under pure normal loading, classical studies of interface fracture (e.g. Cao & Evans 1989; Evans & Hutchinson 1989; Evans *et al.* 1990; Jensen *et al.* 1990; Hutchinson & Suo 1992; Liechti & Chai 1992) indicate that, under combined normal and tangential loading (often referred to as ‘mixed-mode’ fracture), dissipative effects are also seen to significantly increase the interface toughness, which is analogous to the work of adhesion in a contact problem. Because of the range of dissipative mechanisms present in such fracture problems, phenomenological models are used to capture the dependence of interface toughness upon the mode mixity of loading. Owing to the similarities between fracture mechanics and contact mechanics, it is likely that such phenomenological mixed-mode models can also capture intrinsic mechanics of adhesive interfaces; however, to date, limited work has been done to investigate mixed-mode fracture in this context. Seminal work on the combined normal and tangential loading of a sphere on an adhesive elastic half-space by Savkoor & Briggs (1977) and Savkoor (1987, 1992) assumed that the work of adhesion was independent of the degree of mode mixity and thus remained constant. Johnson (1996, 1997) introduced an empirical model for mode-mixity-dependent work of adhesion into his analysis of the contact problem, and later Kim *et al.* (1998) showed that such empirical models are valid only in the limit where small-scale yielding applies. Few quantitative experimental results are available for such mixed-mode contact problems. The model of Savkoor & Briggs (1977) underpredicted the area of contact and maximum static tangential loading observed in their experiments, but introducing a mode-mixity-dependent work of adhesion was not considered. Felder & Barquins (1992) performed contact experiments on biaxially stretched rubber sheets and fitted an empirical model for mode mixity to the extracted values of the work of adhesion; however, experiments of this type do not seem to have been pursued further in the literature. Relevant computational work on mixed-mode viscoelastic fracture includes the study by Tang *et al.* (2008), who found that the computed interface toughness is strongly dependent upon mode mixity and is a monotonically increasing function of crack velocity for a fixed mode mixity, reminiscent of the results for pure normal loading of viscoelastic materials cited above.

The objective of the present work is to further develop contact mechanic models for mixed-mode adhesive contact. An analytical framework is developed which includes a generalized phenomenological model of energy dissipation in the form of increased effective work of adhesion with increasing degree of mode mixity. Although the physical mechanisms of energy dissipation are left unspecified, the goal of such modelling is to capture the effects of viscoelasticity or rate-dependent non-equilibrium processes during interfacial separation. To verify the model, mixed-mode experiments are performed on polydimethylsiloxane (PDMS), a compliant elastomer with mechanical properties similar to many biological materials. In the experiments, PDMS demonstrates both adhesion hysteresis and rate-dependent behaviour during normal loading, and measurements of contact area during mixed normal/tangential loading indicate a strong dependence of the effective work of adhesion upon mode mixity, which is captured effectively by the phenomenological model.

## 2. Analytical model for a rigid sphere in adhesive contact with a plane surface

### (a) JKR-type normal and tangential adhesive contact

The starting point for the analytical model is contact between an elastic half-space and a rigid spherical indenter subjected to both normal and tangential loading in the presence of adhesion. Before proceeding, it is important to note that the classical JKR (Johnson *et al.* 1971) adhesion formulation assumes the contact interface is frictionless so that the normal and tangential tractions are uncoupled. However, to capture static friction response under a tangential load, the interface can no longer be modelled as frictionless. In general, when slip is prevented within the contact area, the normal and tangential tractions become coupled and the JKR formulation is no longer valid. However, for the special case where one contacting material is rigid and the other is incompressible, the work of Chen & Gao (2006*a*,*b*,*c*) and Chen & Wang (2006) showed that, even when no slip is allowed in the interface, the normal and tangential tractions decouple and the resulting solution for the normal traction is identical to the frictionless case. Thus, the model presented here will superpose the frictionless JKR solution with a classical no-slip tangential traction solution, providing an exact solution for incompressible half-spaces and an approximate solution if Poisson’s ratio *ν*<0.5. Details and justification for the no-slip condition will be discussed below.

In the contact problem, the profile of the rigid sphere of radius *R* is approximated by the paraboloid (Johnson 1985)
2.1
where *r* is the radial coordinate and *z* is normal to the surface of the half-space, as illustrated in figure 1. The relationship between the applied normal load *P* and contact radius *a* for the contact problem in the presence of JKR adhesion is
2.2
where the work of adhesion is given by *w* and the Hertz contact load *P*_{1}(*a*) for the sphere in contact with the half-space is
2.3
Here, *E**=*E*/(1−*ν*^{2}) is the plane strain modulus of the elastic solid, which is characterized by the elastic modulus *E* and Poisson’s ratio *ν*. For this geometry, the normal contact pressure *p*(*r*) is given by (Maugis & Barquins 1978)
2.4
and the mode I stress intensity factor at the contact boundary is obtained as
2.5

To consider the effect of an applied tangential load *T*, it is assumed that no slip occurs during the static friction stage of tangential loading. For such a case, the tangential traction distribution within the area of contact is given by Johnson (1985) as
2.6
The singular nature of *q*(*r*) at the periphery of contact (*r*=*a*) is problematic for contact in the absence of adhesion. To remove this singularity in non-adhesive contact, traditionally slip is presumed at the periphery and a local Coulomb friction limit is imposed upon the tangential traction distribution (e.g. Cattaneo 1938; Mindlin 1949; Johnson 1985). Historically, the first attempt to study tangential loading in the context of the adhesive contact of soft rubbers was that of Savkoor & Briggs (1977), who used an energy balance approach similar to the JKR method and assumed no slip within the contact area. Later, Johnson (1997) presented a model allowing for slip at the periphery within the context of the Maugis–Dugdale model of adhesion (Maugis 1992), where cohesive zones are introduced to eliminate singularities in the normal and tangential traction distributions. Experiments reported by Savkoor (1987, 1992) showed an initial ‘peeling’ phase of contact, during which reduction of the contact area occurred with increasing load *T* without evidence of slip. Savkoor thus hypothesized that the singularity in tangential traction should be interpreted using fracture mechanics, as is done for the singular normal traction distribution obtained for the JKR adhesion solution. Kim *et al.* (1998) showed that a classical fracture criterion could be used in contact problems when the slip zone near the contact edge is small; in such cases, an effective work of adhesion, which includes all dissipative processes, can be obtained by calculating the energy release rate outside the contact zone. Hence, the present model imposes a no-slip condition within the contact area, with the caveat that there is a small process region at the location of tangential traction singularity in equation (2.6) where the exact details of slip are unknown, just as the exact details of the normal surface displacement at the location of the normal traction singularity are unknown in the JKR adhesion model.

To proceed, Johnson (1997) is followed in using equation (2.6) to write the mode II and mode III stress intensity factors as
2.7
where *θ* is the angle between the radius vector and the direction of *T*. The strain energy release rate can then be expressed as
2.8
To remove the cumbersome *θ* dependence of the above equation, averaging the stress intensity factors around the periphery of the contact area results in the expression
2.9
in which the effective stress intensity factor is given by
2.10
To find a relation between the contact radius *a* and the applied loads *P* and *T*, the strain energy release rate is set equal to the work of adhesion. Hence, setting *G*=*w* and substituting equations (2.5), (2.9) and (2.10) results in
2.11
If the applied normal load *P* is held constant during the tangential loading, equation (2.11) can be rearranged to obtain an expression for the relationship between tangential load and contact radius
2.12
where *P*_{1} is dependent upon *a* through equation (2.3). This expression is equivalent to the result of Savkoor & Briggs (1977), obtained through energy balance.

### (b) Mode-mixity-dependent work of adhesion

In the model presented above, it was assumed that the work of adhesion *w* remained constant throughout the loading, which is the case for a perfectly elastic material with reversible adhesion. However, the experimental evidence discussed in §1 shows that, in general, the effective *w* can increase during contact owing to dissipation from interfacial processes and mode mixity in both elastic and viscoelastic materials. Thus, the goal is to capture such dissipative effects within a mode-mixity-dependent *w* which can then be compared with the strain energy release rate evaluated external to any dissipative process zones, in accordance with the results of Kim *et al.* (1998). The parameterization of Hutchinson & Suo (1992) is chosen so that the mode-mixity-dependent work of adhesion is expressed as
2.13
where *w*_{0} is the work of adhesion for pure mode I loading and *ψ* is the phase angle of mode mixity, defined as
2.14
(Note that this definition of *ψ* is valid for an interface between two materials with identical mechanical properties or an interface between a rigid solid and an incompressible material; the latter is of interest in the context of the present study of glass in contact with PDMS.) The parameter *λ* in equation (2.13) determines the influence of mode mixity and is bounded by 0≤*λ*≤1. Figure 2 shows the variation of *w*/*w*_{0} with *ψ* for a range of *λ*. If *λ*=1, *ξ*(*ψ*)=0 and hence *w*(*ψ*)=*w*_{0} for all *ψ*, which is equivalent to the classical surface energy criterion used in §2*a*. If *λ*=0, the crack is ‘fully shielded’ from any effects of mode II and crack advance depends only upon the mode I component. The *λ*=0 case is thus often referred to as being ‘*K*_{II}-independent’, as rearranging equation (2.9) with *w* defined as in equation (2.13) results in an expression which depends only upon the *K*_{I} term; in other words, equation (2.13) is purposefully constructed in such a way that the *K*_{II} term drops out of equation (2.9) when *λ*=0.

### (c) Dimensionless parameterization of model

Similar to the normalization traditionally done for the JKR problem, the following dimensionless parameters can be used:
2.15
where the bulk modulus *K*=4*E**/3. Thus, equation (2.12) can be expressed in dimensionless form for the incompressible case (*ν*=1/2) as
2.16
where the identity has been used. This formulation allows versus to be plotted for a given and *λ*. Figure 3 shows theoretical curves for ; results are qualitatively similar for all . The initial dimensionless contact radius is ≈1.8 and decreases until a maximum is reached. The portions of the solution curves below this critical point are inaccessible. For higher tangential loading, an equilibrium no-slip solution does not exist; physically, sliding may commence or the interface may separate, depending upon the applied normal load. The exact details are beyond the scope of this model. Increasing *λ* is seen to increase the maximum sustainable static tangential load. The *λ*=0 case is not shown; because of the form of the parameterization in equation (2.13), as the phase angle , so a finite maximum tangential load does not exist for *λ*=0.

## 3. Experimental testing of adhesion and tangential loading using a microtribometer

### (a) Microtribometer set-up

To test the validity of the above model for mixed-mode adhesive contact, a microtribometer was designed and built in-house to apply controlled normal and tangential forces on PDMS samples, which is an approximately incompressible material. In the set-up, a high-strength aluminium (7075) cantilever with four arms acts as a two-dimensional force transducer, with normal and lateral spring constants of 1690 and 3760 N m^{−1}, respectively. Fibre-optic displacement sensors (Philtec D20) were used to measure the deflection of a pair of perpendicular mirrors located on the cantilever, as illustrated in figure 4. The voltage signals from the displacement sensors were digitized, acquired and converted to force measurements using LabVIEW (National Instruments) software routines. The smallest displacement which can be resolved unambiguously by the fibre-optic sensors is 60 nm and thus the smallest measurable forces are 0.1 mN in the normal direction and 0.2 mN in the lateral direction. The cross-talk between the normal and lateral force measurements was determined during calibration to be approximately 1.5 per cent and corrected for within the data acquisition routine. A complete description of the operation of the microtribometer is detailed in Waters (2009).

The spherical probe used for the experiments was a convex borosilicate glass lens of radius 15.5 mm, which was attached to the cantilever end as shown in figure 4 so that its bottom plane lay in the midplane of the cantilever, helping to minimize cross-talk. The glass lens was approximated as rigid, as it is significantly stiffer than PDMS. The contact area was observed through the glass probe using a digital camera (Hitachi K20B) attached to a variable-zoom microscope objective lens (Edmund Optics VZM 1000i). The pixel length in the acquired images was 4 μm with the zoom setting used during the experiments. Samples were mounted on a motorized XYZ stage (Thorlabs T25 XYZ-E/M; displacement resolution 50 nm), allowing for controlled three-dimensional motion to be synchronized with data and image acquisition using LabVIEW.

### (b) PDMS sample preparation

PDMS (Sylgard 184, Dow Corning) samples were prepared on clean 25×76 mm glass slides which were peripherally lined with Scotch (3M) tape in order to create walls for a mould. The resulting PDMS rectangular blocks were approximately 8 mm thick. The elastomer base and curing agent were mixed in a 10:1 ratio by weight and degassed in a vacuum chamber for 1 h. The mixture was then poured into the mould, degassed again for approximately 30 min to remove any air bubbles introduced during pouring and cured in an oven at 150^{°}C for 20 min. The PDMS block was then carefully peeled off the glass slide, flipped over and glued to another glass slide using a thin layer of wet degassed PDMS mixture. This thin layer was cured by placing the sample on a hot plate at 150^{°}C for 15 min. Permanently mounting the PDMS blocks on glass slides in such a manner allowed for the samples to be held more securely during subsequent experimental testing. The exposed top layer of the PDMS blocks inherited the flatness and smoothness of a glass slide, making it ideal for contact experiments.

Sylgard 184 is a PDMS resin which has been shown to demonstrate viscoelastic behaviour (e.g. VanLandingham *et al.* 2005; White *et al.* 2005; Choi *et al.* 2008; Lin *et al.* 2008; Schneider *et al.* 2008). However, Perutz *et al.* (1998) noted that PDMS networks at room temperature are well above their glass transition temperature (*T*_{g}∼−120^{°}C); so for sufficiently low rates of loading/unloading, bulk viscoelastic losses could be minimal.

### (c) Determination of elastic modulus using JKR-type experiments

In order to apply the analytical model derived in §2, first the reduced elastic modulus *E** of the PDMS samples was determined. An indentation and pull-off cycle was performed with a normal displacement rate of 0.5 μm s^{−1} and digital images of the contact area were captured once every 10 s. The resulting data were fitted to the JKR adhesion theory using the method of Chaudhury *et al.* (1996), who rearranged the JKR relation between load *P* and contact radius *a* as
3.1
where *K*=4*E**/3. Thus, a linear relationship is obtained between the variables *P*/*a*^{3/2} and *a*^{3/2}/*R*, with slope 1/*K* and intercept . Data collected during both approach (loading) and withdrawal (unloading) from the PDMS sample are shown in figure 5. The approach data are linear, so equation (3.1) can be used to find *E**=2.3 MPa from the slope of a linear fit. The work of adhesion during loading can then be obtained from the intercept of the linear fit and is computed to be *w*_{l}=25 mJ m^{−2}. The offsets between the approach and withdrawal data are evidence of adhesion hysteresis, and the nonlinear withdrawal data are evidence of rate-dependent behaviour, as the contact area shrinks increasingly rapidly as pull-off is approached. This nonlinearity means that equation (3.1) cannot be applied, as the work of adhesion does not remain constant during unloading. Perutz *et al.* (1998) noted that, if viscoelastic losses were due to bulk viscoelasticity, then *both* the loading and unloading *w* should show evidence of rate dependence. The linearity of the loading data indicates that bulk viscoelastic losses are negligible for this low displacement rate; thus, interfacial dissipation is responsible for the observed adhesion hysteresis and rate-dependent work of adhesion upon unloading. Qualitatively similar results for the loading/unloading behaviour of PDMS were obtained in microtribometer experiments by Galliano *et al.* (2003) and Vaenkatesan *et al.* (2006). Ghatak *et al.* (2000) noted that, even for elastic materials, the work of adhesion measured during unloading is often significantly higher than what is measured during loading, signifying non-equilibrium interfacial processes.

### (d) Tangential loading experiments at constant normal load

Using the microtribometer, tangential loading experiments were performed on PDMS for a range of normal loads. For each test, the PDMS sample was first loaded in compression to 80 mN at a normal displacement rate of 0.5 μm s^{−1} and then unloaded at the same displacement rate, until the desired normal load was reached. Then, a feedback loop was initiated within the LabVIEW software to adjust the *z*-position of the sample as necessary to maintain the normal load. Before application of a tangential load, the sample was held at the normal load setpoint for 5 min to allow the contact area to equilibrate. This step proved crucial, as the contact area tended to shrink significantly during this hold time. At the end of the hold time, with the contact area having stabilized to a constant value, a tangential load was applied by translating the sample in the *x*-direction at a displacement rate of 0.5 μm s^{−1}. The feedback loop continued to maintain the normal load setpoint during this lateral motion. Images of the contact area were recorded throughout the experiment once every 10 s.

Figure 6 shows the results for tangential loading *T* versus time for a range of normal loads. The figure also shows the evolution of the contact area during tangential loading. For tensile normal loading, the contact area shrinks symmetrically in a manner analogous to peeling, and remains circular before separation occurs at a critical tangential load. This behaviour is similar to what is predicted by the theory presented in §2. A series of images of this symmetric peeling are shown in figure 6 corresponding to labels 1–4 for *P*=−5.5 mN. When *T*=0, the contact area is circular (1), and remains symmetric with increasing tangential loading (2) before fluctuations appear around the periphery (3) immediately preceding separation (4). For neutral or compressive normal loading, however, there are three distinct regimes seen during the tangential loading. These are illustrated in figure 6 for the *P*=9.5 mN case in the images A–D. In the images, the direction of sample motion is from left to right, which is equivalent to an indenter moving on a fixed sample from right to left. First, the contact area shrinks symmetrically, as for the tensile case (A). Then, the contact area becomes asymmetric, with significantly more peeling occurring at the trailing edge of contact than at the leading edge, and continues shrinking as *T* continues to increase (B). Finally, at the peak *T*, a slip instability propagates around the periphery of contact (C), and partial reattachment occurs at the trailing edge of contact (D). This cycle of asymmetric shrinking slip instability partial reattachment persists for continued lateral displacement of the PDMS sample. The amplitude of the tangential force oscillations during this cycle is approximately twice as high for the *P*=9.5 mN case as it is for the other compressive loads tested. It is speculated that the inherent instabilities in the system are enhanced for this near-neutral normal loading. Although reattachment cycles for the tensile cases are not seen in these experiments, they may be possible under dead loading; for this set-up, the feedback loop could not correct the normal loading fast enough to maintain contact as the instability was approached.

To better visualize any slip processes occurring within the contact area during tangential loading, a narrow line approximately 10 μm wide was scored on the surface by gently pressing a straight razor blade against the PDMS sample, similar to the method first used by Barquins *et al.* (1975). The PDMS sample was then positioned so that the initial circular contact area would be centred over the line, and the tangential loading experiments were repeated as before. A representative series of images from the experiments are shown in figure 7 for *P*=0.5 mN; results were qualitatively similar for all compressive normal loads. Again, the direction of sample motion in the images is from left to right. Figure 7*a* shows the contact area before the initiation of tangential loading, and figure 7*b* shows the contact area right before the onset of asymmetry. The results showed that there was no slip visible within the contact area while it remained symmetric, as the line appeared to remain straight inside the entire contact area. This verifies that the initial shrinkage in contact area during application of a tangential load is due to peeling and is not due to slip propagation. The absence of measurable slip at the onset of tangential loading is in contrast to results for Hertzian contact from Cattaneo (1938) and Mindlin (1949). As soon as the tangential load was sufficiently high such that slip began to propagate from the periphery, the kink in the line smoothed out and the contact became asymmetric, as seen in figure 7*c*. The presence of the line affected the resulting contact shape after the initiation of slip. However, the line did not affect the data collected during the symmetric peeling phase, as discussed below in §4.

Only data from the initial peeling regime, where there is circular symmetry and negligible slip, will be used for the purposes of comparison with the analytical model. However, behaviour seen in the subsequent regimes merits further comment. Observations of asymmetric contact during tangential loading and the cycle of instability and reattachment have precedence in classical studies of rubber friction. Barquins *et al.* (1975) and Barquins (1985, 1992) observed that slip propagated within the contact area at the onset of tangential loading, leaving a circular no-slip region inside the asymmetric contact area. In their experiments, the size of this no-slip region was found to be greater than that predicted by the Hertzian contact theory of Mindlin (1949). In contrast, experiments reported by Savkoor (1987) showed an initial peeling phase prior to the onset of asymmetric contact and slip. In all of the above cases, instabilities and reattachment folds were seen when indenters with sufficiently large radii of curvature were used, even at low tangential speeds. These instabilities differ from Schallamach waves, which are surface waves which ripple through the contact area like wrinkles in a carpet, transmitting slip (Schallamach 1971). Instead, Barquins (1984*b*) described the reattachment phenomenon seen in his experiments on soft rubbers as being caused by a sudden slip event at the trailing edge of contact, which leads to the formation of a ‘viscoelastic bulge’ located just behind the trailing edge of contact. The height of this bulge is sufficient to cause reattachment. As in the present experiments, the tangential force was seen to increase during the asymmetric peeling phase and suddenly drops as soon as the sudden slip occurs; then, the tangential force began increasing again, peeling of the contact area resumed and the cycle continued. The tangential load corresponding to the initiation of detachment waves was found to depend upon both tangential velocity *v* and radius *R* by Barquins & Courtel (1975) and Barquins & Roberts (1986). More recent studies on the friction of elastomers using microtribometers have reported a wide range of behaviour; contact area reduction owing to tangential loading is universally seen for bulk elastomers, but some authors report steady-state sliding (Galliano *et al.* 2003; Vaenkatesan *et al.* 2006; Varenburg *et al.* 2006) while others report instabilities and reattachment folds in advance of the contact (Rand & Crosby 2006, 2007) or, as in the present case, trailing the contact (Shen *et al.* 2008). Thus, the full range of rubber friction behaviour appears to be specific to the experimental testing apparatus, method and materials used. A thorough investigation of such behaviour is outside the scope of the present study.

## 4. Measuring the mode-mixity-dependence of *w*

Figure 8 shows the data for tangential load *T* versus contact radius *a* at the prescribed constant load *P*, which is used to measure the work of adhesion during the tangential loading experiments. However, only the data points corresponding to symmetric peeling and negligible slip are used. With these data, equation (2.11) can be used to find the effective work of adhesion from the experiment. The initial work of adhesion *w*_{0} is found using the equilibrated *a*_{0} measured before the initiation of tangential loading, when *T*=0. The computed values of *w*_{0} are sensitive to local surface conditions and thus differ for each tangential loading test; *w*_{0} ranged from 80 to 110 mJ m^{−2} for all experiments performed. Note that the computed *w*_{0} was higher than the value of *w*_{l} measured during the loading portion of the JKR experiments, because the tangential loading is applied after unloading has occurred and thus adhesion hysteresis is included in this work of adhesion. Thus, *w*_{0} should not be thought of as the thermodynamic work of adhesion. As *T* increases, the effective value of *w* corresponding to a measured value of *a* can also be found from equation (2.11).

The results for the measured *w*/*w*_{0} for a series of normal loads at a tangential velocity *v*=0.5 μm s^{−1} are shown in figure 9 as a function of the measured phase angle *ψ* during the experiments, defined in equation (2.14). The measured *w*/*w*_{0} is independent of normal load and is seen to increase significantly with *ψ*, clearly indicating that dissipation increases with mode mixity. Using the phenomenological model developed in §2*b*, theoretical results are shown in figure 9 for *λ*=0.15, which provides an acceptable fit to the experimental data. Results from the smooth surface, shown in figure 9*a*, are seen to be virtually identical to results obtained from the cut surface, shown in figure 9*b*. Thus, the presence of the scored line does not affect the observed behaviour during the symmetric peeling phase of contact.

## 5. Rate dependence of measured dissipation

To investigate potential rate effects on the effective work of adhesion, experiments were performed for a series of tangential PDMS sample velocities at a fixed normal load *P*=0.05 mN. The measured *w*/*w*_{0} for these experiments are shown in figure 10*a*. With increasing tangential velocity, *w*/*w*_{0} is seen to increase. It is postulated that, for the quasi-static limiting case of , (or ). As *v* increases, it appears to asymptotically approach the limiting case of . This is illustrated in figure 10*b* by the best fit values of *λ* to the *w*/*w*_{0} data at each tested velocity. The dashed line in the figure is a visual guide for the approximate trend, assuming *λ*=1 at *v*=0. Clearly, the interfacial dissipation is highly dependent upon tangential velocity.

To connect to previous work on rate-dependent dissipation, the dependence of *w*/*w*_{0} on the rate of change of the contact radius (i.e. a crack front velocity) is investigated. As discussed in §1, phenomenological models have been used extensively to model the dependence of the measured work of adhesion on crack velocity in viscoelastic materials. Whether the dissipation mechanism is linked to the bulk viscoelasticity of the sample or is instead an interfacial process, the dissipation itself must be localized to the crack front so that gross displacements are elastic for the definition of strain energy release rate defined in equation (2.8) to be valid.

A general form of the velocity-dependent work of adhesion *w* for viscoelastic contact problems was first introduced by Gent & Schulz (1972) and Andrews & Kinloch (1973)
5.1
where *V* is the crack front velocity and *a*_{T} is the WLF shift factor (Williams *et al.* 1955; Ferry 1980), which allows for viscoelastic data taken at different temperatures to be compared. *φ*(*a*_{T}*V*) is characteristic of the material and independent of the geometry or loading system and knowing *φ*(*a*_{T}*V*) allows one to predict the kinetics of detachment, provided the viscoelastic losses are limited to the crack tip. In their classic study of polyurethane contact, Maugis & Barquins (1978) found that the function *φ*(*a*_{T}*V*) had the form
5.2
Muller (1996) summarized the power-law dependence seen in several experiments as
5.3
where and typical values of *n* fall in the range 0.1<*n*<0.8. Greenwood & Johnson (1981) showed analytically that *n*=0.5 for a simple linear viscoelastic model, while the analysis of Persson & Brener (2005) found *n*=1/3. Such results for the power-law dependence of *w* upon *V* have their origin in the assumption of bulk viscoelasticity. Barthel & Roux (2000) showed that a power law of the form
5.4
can effectively capture experimental trends for interfacial dissipation, where *d**a*/*d**t* is the rate of change of the radius of contact (*d**a*/*d**t*=*V*) and *V*_{0} is a ‘characteristic velocity’ for the onset of dissipation.

Such power-law dependence of *w* upon *d**a*/*d**t* for viscoelastic materials has previously been studied only in the context of traditional JKR-type experiments with pure normal loading. Here, it is investigated whether a power-law dependence also holds for combined normal/tangential loading by using equation (5.4) to fit the experimental data from the tangential displacement rate experiments shown in figure 10. Using data from multiple tangential displacement rates allows a wider range of *d**a*/*d**t* to be measured, as *d**a*/*d**t* generally increases as tangential velocity *v* increases.

The power-law exponent *β* in equation (5.4) can be obtained by performing a linear fit to data presented on a plot of versus , as shown in figure 11*a*. (The absolute value |*d**a*/*d**t*| is used because the contact area is shrinking. Note that some of the scatter in the data is due to the pixel resolution of the digital camera used to record images; often, displacements of less than 1 pixel occurred between recorded frames, leading to round-off error when the corresponding |*d**a*/*d**t*| was computed.) The power-law exponent is found to be *β*=0.39 for the present experiments, which falls well within the range of values reported in the literature. Expressing equation (5.4) as
5.5
gives *v*_{0}=1.4 μm s^{−1} for the present experiments from the intercept of the linear fit. The computed values of *β* and *V*_{0} give a good fit to the experimental data, as shown in figure 11*b*. This indicates that the dissipation for mixed-mode contact problems can be described by a power-law dependence upon crack velocity, just as for the case of adhesion under pure normal loading.

## 6. Concluding remarks

The above results indicate that significant dissipation occurs during mixed normal and tangential contact loading of PDMS, which is not captured in models which assume that the work of adhesion remains constant. Introducing a phenomenological model for the dependence of the work of adhesion upon degree of mode mixity is seen to effectively capture the increase in *w* with increasing *T* which is seen in the microtribometer experiments. Such a model can easily be added to existing analysis of mixed-mode adhesive contact to obtain more quantitatively accurate predictions of behaviour. The parameter *λ* in the phenomenological model is a property of the material interface and loading rate and must be determined via experiment. However, the present experiments show that for very slow loading rates and for high loading rates, giving two limiting conditions.

One limitation of the present work is that the model cannot be applied to the full range of behaviour seen during the tangential loading experiments, as a symmetric contact area with negligible slip is seen only during the initial stages of loading, up to a phase angle of *ψ*≈60^{°}. Thus, one significant limitation of the model is an inability to accurately predict the maximum tangential loads seen in the experiments before the onset of slip instabilities. The model significantly underpredicts the peak experimental tangential loads, and the prediction worsens for increasing compressive normal load. For the *λ*=1 case, equivalent to the classical Savkoor & Briggs (1977) result which assumes that *w* remains constant throughout the loading, putting the measured values of *E** and *w*_{0} into the model yields a predicted maximum which is only 15–35% of what is observed in the experiments. The empirical fit of *λ*=0.15 offers an improvement, but still only predicts the maximum to be 55–85% of the observed values. This indicates that significant energy dissipation is occurring during the asymmetric phase of contact which is not being captured by the symmetric contact model. Different mixed-mode contact problems, such as spherical contact on a membrane which is subsequently biaxially stretched, yield a contact area which remains circularly symmetric throughout the full range of loading and thus present opportunities for validation of the present model at higher degrees of mode mixity.

## Acknowledgements

The authors are grateful to K.-S. Kim and H. Gao for useful discussions and to G. Della Rocca for assistance with the construction of the microtribometer. This work was supported by the Mechanics of Multifunctional Materials and Microsystems programme of the Air Force Office of Scientific Research (grant no. FA9550-05-1-0210; programme manager, Dr Les Lee) and the National Science Foundation (grant no. CMS-0547032).

## Footnotes

- Received September 2, 2009.
- Accepted November 19, 2009.

- © 2009 The Royal Society