## Abstract

For the purpose of determining the relevant scaling, the fundamental and widely applicable problem of adhesive contact between elastic solids is revisited. A comprehensive and accurate finite-element modelling is undertaken. A local contact law, consistent with the current level of modelling, is used. The analysis of the results yields the following conclusions. For a broad range of physically reasonable contact laws, and for low values of the Tabor parameter, a simple modification of adhesive range entering in the Tabor parameter allows for one-parameter scaling of the problem. For high values of the modified Tabor parameter, the problem requires description in terms of two non-dimensional parameters, one of which represents the magnitude of the contact surface stretch. The contact surface stretch correction is significant for a wide range of problems with spheres smaller than the threshold size, which, for a broad range of materials, is 300 nm to 100 μm, depending on the adhesive energy and elastic compressibility.

## 1. Introduction

The problem of adhesive contact has a wide range of applications, including: interactions between colloids (Russell *et al*. 1989), powders (Martin 2004) and biological tissues (Arnoldi *et al*. 1998; Camesano & Abu-Lail 2002); modelling of friction and wear (Johnson 1997); as well as analysis of experiments from surface force apparatus (Israelachvili 1992) and atomic force microscopy (Lantz *et al*. 1996). Therefore, it is not surprising that the problem has received much attention in the last several decades (Barthel 2008). In this communication, we build on the earlier results and address the assumptions that are critical for analytical and numerical modelling on the continuum level. In particular, the following are given.

The analysis of a wide range of physical problems with adhesive contact requires a general numerical method such as the finite-element (FE) method. A proper definition of a contact law linking the surface tractions to the separation between the interacting surfaces is required. We define a local contact model developed based on physical reasoning and mathematical economy, and implement it in the FE model.

The Tabor parameter (Tabor 1976) is often used to establish the applicability of either the Derjaguin–Muller–Toporov (DMT) model (Derjaguin

*et al*. 1975) or the Johnson–Kendall–Roberts (JKR) model (Johnson*et al*. 1971). We show that, for low and intermediate values of the Tabor parameter, a modification of the length scale associated with the range of adhesive tractions results in a better correlation in models employing different contact laws.Moreover, we show that, for large values of the Tabor parameter, a single non-dimensional parameter is insufficient to characterize the problem of adhesive contact. An additional non-dimensional parameter, related to the stretching of the contact patch, appears. Recently, Yao

*et al*. (2007) argued that the validity of the JKR solution is limited by the condition that the average contact traction does not exceed the material strength. In our numerical study, maximum traction cannot exceed the bonding strength, so that the average traction is well below the Yao*et al*. limit.

The paper is organized as follows. In §2, we discuss the local contact law applicable to numerical models, as well as the often-used Derjaguin (1934) approximation. We discuss the FE implementation in §3. In §4, we summarize the current understanding of the benchmark problem—contact between elastic spheres—and compare the results with the FE solutions. In §5, we develop a proper scaling for this class of problems and verify it against the numerical results. The summary and discussion is given in §6.

## 2. Local contact law and the Derjaguin approximation

We focus on the *local* type of contact law, whereby the pointwise surface traction, acting on a surface, is related to a single length that is some measure of the distance to the other surface. The surface interaction is, of course, interatomic in nature, which implies non-local interactions. Recently, progress has been made in that direction (Luan & Robbins 2006; Sauer & Li 2007*a*,*b*, 2008). However, non-local models are computationally demanding, so that, on the side of economy, a local model is preferred, provided that it can describe the relevant physical behaviour.

In defining the contact law, we assume that the traction acting at a point on the surface *S*_{1} is a given function of the distance to the surface *S*_{2}. To fully define the contact traction at a point, one must define: (i) the distance on which the traction depends, (ii) the direction of the traction, and (iii) the area to which the traction is referred.

Physical reasoning requires that the distance

*α*in figure 1*b*, governing the traction*p*that surface*S*_{2}exerts on a point of*S*_{1}, be the shortest distance from the point in question to the smooth surface*S*_{2}, i.e. along the normal to*S*_{2}, and so not, in general, normal to*S*_{1}.To determine the rational direction of the traction

*p*(*α*), consider the atomic forces shown in figure 1*b*. A mildly curved surface can be locally treated as a plane. The forces acting on a point on*S*_{1}, from equidistant neighbours of the central atom on*S*_{2}, are symmetrical and have the resultant directed along*α*. Thus, unless the curvature of*S*_{2}is very high, the direction of the traction on the point on*S*_{1}is reasonably assumed to be along the normal.The reference area for the traction

*p*(*α*) presents a more challenging problem. Two simplest (but not the only) options are the reference (undeformed) area and the current (deformed) area. Various physical arguments can be constructed to support or falsify one or the other assumption. This question must be considered unresolved at present. We opt for the mathematical argument—consistency with the current level of modelling—and that implies the usage of current area as the one to which the traction is referred.

Consider the surface energy density *γ*, given per surface area *A*. When the surface suffers surface strains d*ϵ*_{ij} (*i*, *j*=1, 2), the change in surface energy (Cammarata 1994) can be expressed in terms of surface stressesIn fluids, the normal surface stresses are identical to the surface energy density, , while the shear stresses vanish. For solids, the surface stresses are of the same or lesser order of magnitude as *γ*, but may take different values, including the negative values (Cammarata 1994; Spaepen 2000). To model surface stresses in the present problem, one must define three sets of new parameters for three different interfaces (two solids and the environment). Such complication does not appear to be practical, or particularly beneficial at this stage. The simple and practical option is to assign the surface stresses equal to surface energy *f*_{ij}=*γδ*_{ij}, which then impliesThis, in turn, implies that the surface energy density is given per unit current area.1 Then, following the definition of surface energy as the work of tractions to separate the surfaces in contact, it follows that the traction must also be given per current area.

The successive approximations to a non-local quasi-continuum model (Sauer & Li 2007*a*), developed for the purpose of numerical implementation, indicate that the first two assumptions are exact for flat, parallel surfaces, and are first-order approximations otherwise. By contrast, the Derjaguin (1934) approximation, commonly used in the normal contact between spheres (e.g. Greenwood 1997), implies that

the relevant distance is along the line parallel to the line that connects the centres of the spheres, as shown in figure 1

*a*,consistently, the direction of the traction

*p*(*α*′) is along*α*′ in figure 1*a*, andthe resulting traction is given per unit area of the projection on the mid-plane.

The local contact model used in the FE analysis is a surface-to-point contact law, i.e. the traction at a point on *S*_{1} is the result of action of all (in principle) atoms of solid 2, while the traction at a point on *S*_{2} is the result of action of all atoms of solid 1. Newton's third law must be satisfied by the resultants of the contact forces between the solids. In the present case, this is guaranteed by the equilibrium conditions implicit in the FE formulation.

## 3. Finite-element modelling

With increasing computing power, FE modelling of the contact between solids has become readily accessible. The automatic mesh generation, available in most commercial packages, allows for quick generation of a variable-density mesh and economic runs. This is critical in the current context, since realistic contact laws act on a scale much smaller than the problem size, which demands dense meshing close to the contact edge to ensure an accurate representation of surface tractions.

We use the commercial FE software Abaqus (2004) and its user subroutine capabilities to define the contact law. The interaction between the spheres is specified using a family of Lennard-Jones force laws(3.1)where *c*_{n} is defined so that the total work on the separation is equal to *γ*. In particular, we consider *n*=9 and 10, with *c*_{9}=8*γ*/*α*_{0} and *c*_{10}=9*γ*/2*α*_{0}. In general, the adhesive contact between large compliant spheres is unstable during both initial contact and separation, so that the standard Newton–Raphson technique for implicit integration cannot be used directly. Short of modelling a dynamic problem (with the associated computational cost), the instability can be addressed in at least two ways:

by using the Riks (arc length) method of integration or

by adding a small superficial viscous traction to the surface, in addition to the surface tractions arising from adhesive forces.

The second approach is preferred for economy, particularly if the model contains a large number of degrees of freedom as our dense meshes do. Therefore, we use the superficial viscous damping method, but only after carefully benchmarking it against Riks results and confirming that the associated error is negligible. The key is that the damping coefficient must be small enough so that the total energy damped during the stable portion is negligible compared with the strain energy of the spheres, yet large enough to enable efficient integration in unstable jumps. The accuracy of the method is discussed in §4.

To test the validity of the Derjaguin approximation, we performed FE runs for the following three cases.

*Case 1*. Contact between a deformable sphere and a rigid flat (figure 2).*Case 2*. Contact between a rigid sphere and deformable half-space.*Case 3*. Contact between a deformable sphere and a rigid sphere.

Case 1 satisfies the Derjaguin approximation exactly; the distances and the direction of surface traction are identical to those obtained using the Derjaguin approximation. Cases 2 and 3 do not satisfy the Derjaguin approximation exactly. Note that Greenwood's (1997) and Feng's (2000) analyses of case 2 are based on the Derjaguin approximation.

All the results reported here are for case 1, but *all the runs* were also performed for case 2. Few selected runs were performed for case 3. No measurable differences, distinguishable from numerical errors, were observed between the cases. We conclude that any differences in distances and traction directions, between our contact law and the one based on the Derjaguin approximation, are negligible.

The deformable body is discretized using three-node constant strain elements. Additional runs were performed using six-node triangle elements (modified to support constant face pressure) to ensure that any errors due to discretization are minimized. To ensure proper convergence of the solution in the Newton–Raphson method, at the end of each equilibrium iteration, two checks are performed at all the nodes; the residual force has to be less than 0.5 per cent of the time-averaged force on the structure, and the displacement correction in the last equilibrium iteration in each increment must be less than 1 per cent of the displacement increment.

The mesh for case 1 is shown in figure 2. The meshes for other cases are similar. The discretization errors in the FE contact analysis were discussed by Mesarovic & Fleck (1999). In adhesion problems, the contact radius is defined as the position of the maximum tensile traction; hence, it is of importance to ensure that the surface tractions are accurately represented. While a relatively coarse mesh suffices when the Tabor parameter is small, the problem becomes acute for large values of the Tabor parameter. In the current case, we use a mesh with surface elements comparable in size with the equilibrium distance, *α*_{0} (3.1). The mesh ensures that when contact between the spheres is established, the magnitude of the maximum tensile traction is within 1.5 per cent of the maximum tensile traction *σ*_{0}, given by the Lennard-Jones law (3.1). Results with errors exceeding the limit were not considered.

## 4. Elastic spheres

The problem of adhesive contact between spheres has been analysed in detail (Johnson *et al*. 1971; Derjaguin *et al*. 1975; Maugis 1992) with approximations typical for the Hertzian contact. The significance of this problem is not limited to spheres. For all contacts where the radius of the contact zone is much smaller than the local radius of curvature of the surfaces in contact, the quadratic (paraboloidal) description is the first-order approximation, and, under the same geometrical conditions, a sphere is practically indistinguishable from the paraboloid.2

Let the two spheres in contact have radii *R*_{1} and *R*_{2}, Young's moduli *E*_{1} and *E*_{2} and Poisson ratios *ν*_{1} and *ν*_{2}. The effective radius *R* and effective modulus are defined as (e.g. Johnson 1985)(4.1)and(4.2)In the JKR model (Johnson *et al*. 1971), the Hertz contact pressure is superposed to the singular adhesive tractions, akin to the elastic crack tip stress fields. The Griffith fracture criterion then yields the decohesion criterion. Under the controlled load experiment, the JKR model predicts the magnitude of pull-off , where *γ* is the contact energy density (per unit area), defined as the difference between the interface energy and surface energies of the two free surfaces. In the DMT model (Derjaguin *et al*. 1975), the adhesive tractions are distributed according to the Hertz displacement field. The DMT model predicts the pull-off load of magnitude . Interestingly, both predictions for the pull-off load are independent of the elastic properties of the two spheres.

The relationship between the two models was explained by Tabor (1976). The elastic displacement (with respect to the undeformed spheres) at pull-off,(4.3)roughly represents the height of the neck at the edge of the contact. Tabor compared this length with the equilibrium distance between two surfaces, *α*_{0}, roughly the interatomic distance, and introduced a non-dimensional parameter(4.4)He then argued that the DMT model is a good approximation for small *μ* (small, stiff spheres with weak adhesion), while the JKR model should be a good approximation for large values of *μ* (large, compliant spheres with strong adhesion). The gap between the two models was effectively bridged by Maugis (1992), who used the Dugdale cohesive zone model with uniform traction in the contact zone to describe adhesive tractions outside the contact area. The resulting Maugis–Dugdale approach was later extended to adhesive contact between elastic–plastic spheres (Mesarovic & Johnson 2000).

While the Maugis–Dugdale model uses a simple form of contact law, the contact tractions are realistically represented by the nonlinear Lennard-Jones force law (3.1). Using a numerical approach, with the Lennard-Jones potential, the Derjaguin approximation and the small deformation theory for continua, Greenwood (1997) concluded that the limits of the Maugis–Dugdale model correspond to Tabor's (1976) limits, i.e. the DMT model for small values, and the JKR model for large values of the Tabor parameter. Our numerical approach requires neither the Derjaguin approximation nor the small deformation kinematics.

Figure 3 illustrates the variation of the net load of the spheres with the separation distance between them for small values of the Tabor parameter *μ* (small, stiff spheres and weak adhesion). Consistent with the existing literature, we indicate positive depth *δ*, to correspond to apparent penetration (indentation), while *δ*=0 corresponds to *α*=*α*_{0}. Positive values of the load *P* indicate a net repulsive force between the spheres.

As the value of the Tabor parameter is increased, the load–displacement curves develop a cusp (figure 4) and resulting instabilities. The methods of integrating the motion through the unstable jumps have been discussed in §3. The results shown here illustrate the accuracy of the superficial damping method.

The results in figures 3 and 4 represent the simulation of the displacement-controlled experiment (except for the Riks method). In a load-controlled experiment, the unstable separation occurs at the minimum (maximum tension) force—the *pull-off force*.

A comparison of the pull-off load values, shown in figure 5 for varying values of the Tabor parameter reveals the following two interesting features.

The trend of the normalized pull-off loads for the two different values of

*n*used in (3.1) was identical. However, there was a poor correlation in the pull-off load values for the same value of*μ*.For large values of the Tabor parameter, the FE results predict a trend of pull-off load increasing with

*μ*values, instead of an asymptotic approach to the JKR value.

These observations require that the scaling relationships ((4.1), (4.2) and (4.4)) be revisited.

## 5. Scaling for contact of elastic spheres

The lack of any measurable differences in the load–depth curves, distinguishable from numerical errors for the three cases discussed in §3, implies that the Derjaguin approximation remains a good approximation throughout the range of parameters studied. The sources of discrepancies between solutions using different contact models (I, figure 5), and, of the failure to converge to the JKR model (II, figure 5), must be sought elsewhere.

In his original argument, Tabor (1976) uses the equilibrium distance (*α*_{0} in (3.1)), as the measure of the range of adhesive forces. While such practice has been widely accepted (Greenwood 1997; Johnson & Greenwood 1997), there are at least two strong arguments against it.

While the equilibrium distance and the range may be of the same order of magnitude, they are not proportional. Even for the same family of contact laws, such as the Lennard-Jones family, the same equilibrium distance can correspond to different ranges (figure 6

*a*).The equilibrium distance is irrelevant for continuum mechanics modelling. The change of equilibrium distance (while keeping all other parameters the same) only results in rigid body translation between the solids in contact, as illustrated in figure 6

*c*. Some authors (e.g. Maugis 1992; Barthel 1998) justifiably ignore it, i.e. assume a vanishing equilibrium distance.

How can then the range of adhesive forces be defined in a way that would cover a variety of contact laws? The best agreement3 between different contact laws is achieved by noting a *physically reasonable contact law* that must be similar to the triangular contact law shown in figure 6*b*, i.e. the sharp rise (practically a step) to the maximum traction, then gradual decay in tractions with increasing distance. In such cases, the range of adhesive tractions can be approximated as(5.1)The parameter *α*_{r} is dependent on the value of *n* used in the Lennard-Jones force law. Specifically, *α*_{r} is 1.3645*α*_{0} for LJ_{10} and 1.9486*α*_{0} for LJ_{9}. Using such a definition of the range of adhesive tractions, we now define *the modified Tabor parameter*4 as(5.2)The resulting change in the pull-off force scaling is shown in figure 7. Compared with the scaling shown in figure 5, the results obtained using the two Lennard-Jones contact laws exhibit excellent agreement, both between themselves and with Greenwood's results, at least for low and intermediate values of .

The step model for contact traction, used by Maugis (1992), albeit mathematically convenient, does not fall into the category of physically reasonable contact laws (as we defined them above). To include it in the results shown, the range must be redefined: . With such definition, the Maugis results can also be subjected to the same scaling (figure 7). This rescaling is consistent with Barthel's (1998) analysis of various contact laws. In Barthel's fig. 4, the pull-off force obtained using the step contact law exhibits marked disagreement with these obtained by physically reasonable contact laws (van der Waals and exponential).

We now focus on large values of the modified Tabor parameter, where our numerical results sharply depart from the theory, as well as from other numerical results obtained using small deformation kinematics (Greenwood 1997). By comparing the numerical results for different cases of contact between spheres and half-spaces, we have eliminated the Derjaguin approximation as the source of discrepancy. In fact, we are able to demonstrate unambiguously that it is the large deformation, more precisely the stretching of the contact patch, that results in such dramatic departure from the theory (II in figure 5, figure 7).

In the existing adhesion models (JKR, Maugis, Greenwood), the surface tractions are computed based on the reference configuration of the interacting surfaces. There was no attempt to take into account the stretching of the material on the interacting surfaces. In figure 8, we show numerical results for the distribution of radial stretches on the interacting surface at pull-off, together with the traction distributions. The tractions are heavily concentrated at the area with largest radial stretches. As an approximation, the pull-off force can be thought of as the resultant of the circular line forces, distributed around the contact edge. For a fixed line force density, the stretching of the contact patch has a significant effect on the resultant. In fact, in this simple model, the change in the total force is proportional to the ratio of the radial displacement at the edge of contact *u*_{a} and undeformed radius *a*. To estimate this ratio and to reveal the relevant non-dimensional parameter, we use the JKR model, whereby the radial displacement at the edge of contact (derived in appendix A) is given as(5.3)At the pull-off load, the contact radius is given by(5.4)Therefore,(5.5)Thus, in addition to large , the JKR model requires sufficiently small contact patch stretching(5.6)To verify this conclusion, we note that the radial displacement of the contact edge (5.3) vanishes for incompressible materials, *ν*=0.5. Computationally, this is the easiest way of demonstrating that the condition (5.6) is relevant. In figure 9, we plot the pull-off forces as function of , for two different values of the Poisson ratio *ν*. It is evident that for the nearly incompressible case, *ν*=0.49, the results approach the JKR solution as increases beyond 1. Not surprisingly, the ‘small strain’ theory results by Greenwood (1997) follow the same trend, as such a formulation ignores the radial stretch of the contact patch.

Following (5.6), we introduce the *contact stretch parameter*,(5.7)Then, upon introducing the *uniform traction contact stretch parameter*,(5.8)the modified Tabor parameter can be expressed as(5.9)The parameter *ψ* is twice the contact stretch (*u*_{a}/*a*) under the uniform traction *σ*_{0} on the circular region (Johnson 1985, p. 58), (cf. equation (A 3) in appendix A). Thus, the modified Tabor parameter (5.9) acquires a new physical interpretation as the ratio of the contact stretch under the uniform traction equal to the maximum adhesive traction and the contact stretch under localized (singular, JKR) traction.

While the mathematically strict condition for the validity of the JKR model is , we allow 5–6% error (figure 9) and consider a weaker condition(5.10)In addition to this, condition (5.6) must be satisfied,(5.11)We note that it is the power 1/3 in the definition of *Χ* in (5.7) which makes the condition (5.11) a strong one. The second version of (5.10) can be interpreted as the requirement that the maximum traction be high enough to justify the use of the singular crack tractions in the JKR model. Failing this, the appropriate model is the adhesive zone (Dugdale) model.

Note that the above arguments about large deformation at the contact edge are applicable to the singular model. The effects are much smaller if the adhesive tractions are distributed over a wider area. Consequently, we expect that this effect does not affect the validity of the DMT model, and only marginally affects the Maugis DMT–JKR transition.

We have performed a comprehensive set of FE computations spanning a range of values of *Χ* and *ψ*. Based on these results and the above analysis (5.10) and (5.11), we have developed the map of adhesive contact shown in figure 10.

It is desirable to estimate the range of physical parameters for which contact stretch effects are relevant. For a broad class of materials (e.g. table 1 in Mesarovic & Johnson 2000), the ratio of adhesive energy to elastic modulus can be bounded within an order of magnitude: to 10^{−10} m. Furthermore, for most materials, *κ* is between 1/3 and 1. Then, the condition *Χ*>10^{−2}, implying the need for contact stretch considerations (figure 10), can be recast as(5.12)depending on the relevant values of and *κ*. Thus, the JKR model is strictly applicable to large spheres. For spheres smaller than 300 nm to 100 μm, the correction based on contact stretching is needed.

Given that, for a wide range of parameters the two-parameter problem reduces to a one-parameter problem (modified Tabor parameter), it is desirable to represent the contact map in an alternative parameter space that includes the modified Tabor parameter. After some algebraic manipulation of (5.7)–(5.9),(5.13)The alternative parameter space features the Tabor parameter and the *size-compressibility* parameter *ϕ*, which explicitly measures the size in units of the adhesion range *α*_{r}. The alternative adhesive contact map is shown in figure 11.

## 6. Summary and discussion

First, for the purpose of numerical modelling, we defined a rational local contact law, consistent with the current level of modelling, in particular, with the neglect of distinction between interface stresses and the interface energy. This implies that the tractions are referred to the current (deformed area).

Second, we perform a comprehensive and accurate FE modelling of adhesive contact between elastic spheres. The analysis of the results yields the following conclusions.

For low values of the (modified) Tabor parameter, the problem is governed by a single non-dimensional parameter, the modified Tabor parameter, provided that the contact law belongs to a broad class of physically reasonable laws, with a gradual decay of tractions with the distance between interacting surfaces. For other contact laws (such as the mathematically convenient step law), Tabor's concepts and analysis can be extended, but the range of contact forces must be redefined.

For high values of the modified Tabor parameter, the problem requires description in terms of two non-dimensional parameters, one of which represents a magnitude of surface stretch. The surface stretch correction to the JKR model is significant for a wide range of problems with spheres smaller than 300 nm to 100 μm. The higher threshold applies to compliant, compressible materials and high adhesive energies.

For very small spheres, we expect that the local model will fail (Sauer & Li 2008). Nevertheless, there is a wide range of sizes where the contact stretch effect, as computed here from the local model, is relevant.

We note that the breakdown of Tabor's one-parameter scaling is the result of large deformation combined with the contact law based on the current area. We believe that such a contact law is accurate for compliant, amorphous materials (colloids, polymers biological tissues), and it is just for such materials (and small spheres) that the breakdown occurs. While the definition of the reference area for contact tractions plays an important role, the definition of relevant distances between surfaces and the traction direction (Derjaguin approximation) have no significant effect on the result.

The surface stretch condition ((5.11) and (5.12), figure 11) can be compared with the similar large deformation conditions for non-adhesive, elastic–plastic contact (Mesarovic & Fleck 1999, 2000; Mesarovic 2001). Moreover, a similar condition, but with different physical basis, has been deduced in the analysis of adhesive contact between elastic and plastic solids (Mesarovic & Johnson 2000, fig. 8). There, the relative size of the plastic zone limits the domain of validity of the singular, JKR-like model—a condition known in the fracture mechanics literature as *small-scale yielding*.

It is instructive to revisit the problem from the point of view of simple dimensional analysis (and with hindsight). The simplest dimensional analysis is based on the following two assumptions.

That the shape of the contact law function (equation (3.1), figure 6) has negligible influence on the solutions. Barthel's (1998) and our analyses indicate that this assumption is true for a broad range of physically reasonable contact laws, while for other contact laws, a simple correction suffices, the contact law is then a two-parameter law, say

*α*_{r}and*σ*_{0}(5.1).That deformation is small, and that, consequently, Hertzian scaling ((4.1) and (4.2)) is valid. This reduces the number of additional parameters to two:

*R*and .

The dimensional analysis then requires that two independent non-dimensional parameters govern the problem, or (*Χ*, *ψ*).

From this perspective, Tabor's one-parameter scaling is a remarkable result, and the breakdown of such scaling at some range of parameters should not be surprising. What is interesting is the nature of this breakdown. The one-parameter scaling fails when the small deformation assumption ((ii) above) fails, thus bringing into play, not two but three non-dimensional parameters, say , (*α*_{r}/*R*) and *κ*. That the three parameters combine to give a two-parameter scaling is another remarkable coincidence.

The results shown in figures 10 and 11 are obtained for the contact of a deformable sphere and a rigid flat, so that *κ* (or *ν*) characterize the sphere material. In the general case, when dissimilar spheres are in contact, the question of relevant effective *κ* (or *ν*) remains open. However, our results also indicate no significant differences between the three cases considered in §3 (two deformable spheres, rigid sphere and deformable flat), so that the usage of the lower of the two values of *ν* (the higher value of *κ*) is indicated as the first approximation. Moreover, the experience with elastic–plastic contact, whereby the lower of the two yield stresses proved to be the relevant quantity (Mesarovic 2001), supports the idea that the relevant parameter is the one describing the softer (in this case, the more compressible) material.

## Acknowledgments

The authors are indebted to Dr J. A. Greenwood, University of Cambridge, UK, for providing his data and his numerical code, as well as for the insightful discussions, and to an anonymous reviewer who suggested the alternative scaling shown in figure 11. This work was supported in part by the NSF grant no. CTS-0404370.

## Appendix A

In the JKR model, the total contact traction distribution on the contact of radius *a* is represented as the sum of the Hertzian pressure and the singular traction(A1)The total radial displacement at the edge of the contact is the sum of the displacement arising from the two tractions. The radial contact edge displacement arising from the Hertzian pressure is given in Johnson (1985)(A2)The radial contact edge displacement arising from singular traction (A 1) can be derived using the solution for concentrated force (Johnson 1985, eqn (3.18a)). The elementary radial displacement arising from the force acting on the element of area *s* d*s* d*ϕ* (figure 12) is(A3)so that, using the symmetry about AA',With , we obtain

Then, upon computing the integralwe find(A4)The sum of (A 4) and (A 2) yields the total radial displacement (5.3)(A5)

## Footnotes

↵The accuracy of this assumption depends on the type of materials in contact. We expect that the distinction between surface energy and surface stresses is negligible for amorphous, compliant materials (polymers, colloids, biological tissues). For stiff crystals, more elaborate models may be needed.

↵This ceases to be a good approximation only when the ratio of the contact radius and the radius of the smaller sphere exceeds about 0.4 (Mesarovic & Fleck 2000).

↵We have first tried to define the range for the Lennard-Jones family from an abscissa intercept of the tangent to the maximum slope point in figure 6

*a*. Some improvement in the discrepancy I (figure 5) was achieved, but the gap still remained wide.↵Note that Tabor's (1976) original concept remains unchanged. We modify only the definition of the adhesive range. Nevertheless, the name Tabor parameter has become standard in the literature for the parameter defined by (4.4), so that the non-dimensional parameter in (5.2) must be given a name that draws distinction from (4.4), i.e. the

*modified Tabor parameter*.- Received March 4, 2009.
- Accepted April 1, 2009.

- © 2009 The Royal Society