## Abstract

The adhesion between an elastic punch and an elastic half-space is investigated for plane and axisymmetric geometries. The pull-off force is determined for a range of material combinations. This configuration is characterized by a generalized stress intensity factor which has an order less than one-half. The critical value of this generalized stress intensity factor is related to the work of adhesion, under tensile loading, by using a cohesive zone model in an asymptotic analysis of the separation near the elastic punch corner. These results are used in conjunction with existing results in the literature for the frictionless contact between an elastic semi-infinite strip and half-space in both plane and axisymmetric configurations. It is found that the value of the pull-off force includes a dependence on the maximum stress of the cohesive zone model. As expected, this dependence vanishes as the punch becomes rigid in that case the order of the singularity approaches one-half. At the other limit, when the half-space becomes rigid, the stresses become bounded and uniform and the pull-off force depends linearly on the cohesive stress and is independent of the work of adhesion. Thus, the transition from fracture-dominated adhesion to strength-dominated adhesion is demonstrated.

## 1. Introduction

Although the effect of adhesion on macroscale contacts can usually be neglected, at the micro- and nanoscales, this situation changes dramatically owing to the relatively greater importance of surface forces compared with other forces. Consider, for example, a scaling in which dimensions change from millimetres to either micrometres or nanometres. Body forces, such as those due to the weight of the body, decrease by a factor of 10^{−9} or 10^{−18}, respectively, whereas surface forces decrease by only 10^{−6} or 10^{−12}, respectively, making the relative importance of surface forces between a thousand and a million times more important at these small scales. Furthermore, the pull-off force (i.e. the force required to separate two bodies) varies directly with the linear dimension for a locally spherical contact, making the importance of adhesion even greater at the micro- and nanoscales.

Technologically important examples in which adhesion plays a pivotal role are in microelectromechanical systems (MEMS), computer hard disc drives, particle removal processes in the semiconductor industry and in the emerging field of nanoelectromechanical systems. One of the greatest problems in these areas is the tendency of parts to stick together, sometimes referred to as stiction, which prevents these components from functioning properly.

The effect of adhesion in spherical elastic contacts has been incorporated by Johnson *et al.* [1] and by Derjaguin *et al.* [2]. In these works, adhesion is characterized by the work of adhesion (Δ*γ*) defined as the work per unit area required to reversibly separate two perfectly flat bodies. The work of adhesion (also known as the Dupré energy of adhesion) is related to the surface energies of the contacting bodies by
*γ*_{1} and *γ*_{2} are the surface energies of the two contacting bodies and *γ*_{12} is the interfacial energy. For identical bodies, it is seen that the work of adhesion is equal to twice the surface energy. The surface energies of metals and covalent solids are typically 1–3 J m^{−2}, for ionic solids 0.1–0.5 J m^{−2} and for molecular solids less than 0.1 J m^{−2}. Both the JKR and the DMT theories add the effect of adhesion to Hertz contacts. The JKR model includes the energy of adhesion while neglecting adhesive stresses outside of the contact leading to a pull-off force of 1.5 π*ΔγR*. The DMT theory is equivalent to adding the adhesive force acting outside of the contact region to the applied load, resulting in a pull-off force of 2 π*ΔγR*.

In order to resolve the discrepancy between the JKR and DMT results, Maugis [3] used a cohesive zone model of adhesion. In that model, the stresses in the separation zone are equal to the theoretical stress (*σ*_{0}) in the annular region surrounding the contact in which the separation between the surfaces is less than *h*_{0}. The value of *h*_{0} is taken such that the product of the theoretical stress and *h*_{0} is equal to the work of adhesion, i.e. *σ*_{0}*h*_{0}=Δ*γ*. As a consequence, *h*_{0} is approximately equal to the equilibrium spacing between two half-spaces of the corresponding materials. The results of this model depend upon a parameter, related to the Tabor parameter [4], representing the ratio of the elastic deformation to the range of adhesive forces, and show a smooth transition between JKR and DMT behaviour.

The contact problem of a flat-ended rigid indenter on an elastic half-space has been investigated for plane and axisymmetric geometries [5,6]. These problems give square-root singular stresses at the corners of the indenter. This configuration is mathematically equivalent to an exterior crack in a homogeneous material. Thus, the pull-off force can be found using concepts from fracture mechanics in which unstable separation is determined by relating the stress intensity factor to the work of adhesion. In particular, it is the stress intensity factor (*K*_{I} for mode I, i.e. tensile loading) which governs the local stress field in the immediate vicinity of the indenter/crack. The normal stress acting on the symmetry plane is given by [7]
*K*_{I}≥*K*_{IC} the crack will propagate, then, otherwise, it will not. The quantity *K*_{IC} is known as the critical stress intensity factor, and characterizes the resistance of the material to fracture. When using this concept in adhesion problems, fracture is interpreted as the shrinking of the contact region. The contact region might be expected to smoothly decrease with increasing applied tensile force until a critical condition is reached (i.e. the pull-off force) and the bodies separate suddenly.

Rather than using the stress intensity factor, a related point of view is that given by the energy release rate (*G*) and the related *J*-integral, i.e.
*E**=*E* for plane stress and *E**=*E*/(1−*ν*^{2}) for plane strain, *E* is Young’s modulus and *ν* is Poisson’s ratio. Thus, crack growth will occur when *G* or equivalently *J* reaches a critical value equal to

The problem of finding the pull-off force of an elastic indenter on an elastic half-space is complicated by two factors. One issue is that the solutions for both plane and axisymmetric geometries of the elastic indenter pushed against the half-space are quite complicated. Nonetheless, these solutions do exist. The general plane solution for a semi-infinite elastic strip with arbitrary end conditions was found by Bogy [8] in terms of singular integral equations with generalized Cauchy kernels. Subsequently, Adams & Bogy [9] used that solution along with an elastic half-plane solution in order to determine the stress distribution along the interface between the strip and half-plane. Similarly, Agarwal [10] and Gecit [11] investigated the corresponding axisymmetric problem. It was Gecit who was able to obtain a solution to this problem in terms of a singular integral equation with a generalized Cauchy kernel.

The second complication is that due to the elasticity of the indenter, both the plane and axisymmetric problems have non-square-root singularities at the corners. It is emphasized that unlike the Hertz problem it is not possible to generalize the results of a rigid indenter to that of an elastic one. Furthermore, because the order of the singularity is not one-half, it is not at all obvious how the stress distributions determined by Adams & Bogy [9] and Gecit [11] can inform the determination of the pull-off force. The normal stress acting on the symmetry plane can be written as
*Q* is referred to as the *generalized stress intensity factor* [12]. Thus, for a crack, *Q* as defined by equation (1.4), differs from *K*_{I} by a factor of

The issue of non-square-root stress singularities also occurs in sharp notches. A fracture criterion for sharp notched samples in mode I was given by Gómez & Elices [13]. They related the critical value of the generalized stress intensity factor to that of a crack using a finite-element analysis. Very good agreement with experimental results for a variety of materials was obtained. Cohesive zone modelling was used to determine the initiation of failure over a range of bimaterial corners between epoxy and aluminium [14]. After calibration, the model results agreed well with the measurements. Similarly, Adams & Hills [15] related the critical value of the generalized stress intensity factor to the fracture toughness (critical value of *K*_{IC}) using an analytical procedure based on Mellin transforms. The results agreed very well (within 1%) with the finite-element results of Gómez & Elices [13] which, in turn, agreed very well with experiments.

Technologically relevant areas in which adhesion of elastic cylinders/strips are important include adhesion in biology (e.g. hierarchical structures in Geckos [16]), adhesion at biomimetic fibrillar interfaces [17], stiction in certain MEMS switch contacts [18] and as a limiting case of interaction of a highly worn AFM tip with its scanned surface [19].

## 2. Overview of analysis

The ultimate goal of this analysis is to determine the pull-off force between an elastic punch and an elastic half-space for plane and axisymmetric geometries (figure 1). While solutions exist in the literature for the interface stress distribution for these geometries, these solutions have singularities of order less than one-half. Thus, the path to determining the pull-off force is not clear.

It is noted that solutions exist for a flat-ended rigid punch indenting an elastic half-space in plane and axisymmetric configurations; both have square-root singularities in the stress field. The pull-off force depends only on the work of adhesion, the plane strain elastic modulus of the half-space and the punch width/radius. Again, it is emphasized that it is not possible to take the case of two deformable bodies, considered here, and make it equivalent to a rigid punch on a half-space. This concept of an effective modulus works well for Hertz contact, but does not correctly represent the singular behaviour here owing to the sharp corner of the punch.

In this paper, an asymptotic analysis of the frictionless contact between an elastic quarter-space and an elastic half-space is considered first. The solution for the order of the singularity is given in the literature by Dundurs & Lee [20] and shows that it depends only on the composite material parameter *α*, defined by Dundurs [21]. The fact that the stresses are singular, although not square-root, nonetheless implies the existence of a separation zone near the corner(s) of the contact during tensile loading. An asymptotic analysis of this separation zone using Mellin transforms is then performed and determines the critical value of the generalized stress intensity factor in terms of the material parameter *α*, the work of adhesion and the cohesive stress. This relationship is combined with the results of the contact problems [9,11] in order to determine the relations for the pull-off force of the finite width/radius elastic punch on a half-space.

It is found that the pull-off force depends not only on the material pair, the work of adhesion and the indenter width/radius, but also on the maximum stress of the cohesive zone model. As expected, the dependence on the cohesive stress vanishes as the punch becomes rigid. At the other limit, when the half-space becomes rigid, the pull-off force depends linearly on the cohesive stress and the contact area, and is independent of the work of adhesion. Thus, the transition from fracture-dominated adhesion to strength-dominated adhesion is demonstrated.

## 3. Asymptotic analysis near the contact corner(s)

Consider an elastic quarter-space in frictionless contact with an elastic half-space as shown in figure 2. This configuration represents the asymptotic behaviour near the corner(s) of figure 1. The solution for the order of the singularity was determined by Dundurs & Lee [20] using the method of Bogy [22] which is based on Mellin transforms. For completeness, an analysis of that configuration will also be presented here, but will use the simpler asymptotic approach of Williams [23]. The boundary conditions along the two free surfaces are

These conditions are applied to the stress fields which are summarized by Comninou [24] in a convenient form as
*A*, *B*, *C* and *D*. Setting the determinant to zero yields the result given by Dundurs & Lee [20]
*α* is defined by Dundurs [21] as
*α* for a specified λ. In order to determine λ for a given *α*, the nonlinear equation can be solved or the curve fit given by

The variation of the order of the singularity (λ) with the composite material parameter (*α*) is shown in figure 3. It is noted that *α*=1 corresponds to a rigid quarter-space/punch for which λ=1/2, whereas *α*=−1 corresponds to a rigid half-space for which λ=0. For each value of *α* in the range −1<*α*<1, the numerical solution of equation (3.4) gives one real root in the range 0<λ<1/2 and no complex roots. Roots greater than unity are not allowed based upon energy considerations, whereas negative values of λ lead to bounded stresses as *r*→0 and consequently are not of interest in this study. From figure 3, it is seen that the order of the singularity increases monotonically from zero when *α*=−1 to one-half when *α*=1.

## 4. Analysis of the configuration with a cohesive zone model

It is recognized that the asymptotic solution of the linear elastic problem demands that the stresses be singular as *r*→0, and in particular that the normal stress acting on the plane of contact (*θ*=0) becomes infinite at the corner. Because such infinite stresses cannot exist, including when the applied load is tensile, a simple cohesive zone model is used along the plane of contact in which the normal stress in the separation region is given by

The portion of the separation region for which *h*_{0} is such that the product *h*_{0}*σ*_{0} is equal to the work of adhesion (Δ*γ*). For identical materials, the work of adhesion is equal to twice the surface energy. The cohesive zone extends over the interval (0, *r*_{0}) where the location of the cohesive zone tip (*r*=*r*_{0}) is, at this point in the analysis, unknown. Note that the cohesive zone model eliminates the stress singularity at the corner of the contact without introducing a singularity at *r*=*r*_{0}. According to the Maugis model [3], the quantity *h*_{0} is approximately equal to the atomic equilibrium separation distance between two half-spaces of the contacting materials. Note, however, that in a continuum model the separation is zero when the surfaces touch. Therefore, when the continuum separation is *h*_{0} the atomic planes are separated by the equilibrium spacing plus an additional distance *h*_{0}.

Now, consider the example shown in figure 1 in which an elastic punch of diameter/width 2*a*, is being pulled against adhesion from the surface of an elastic half-space. At the instant of separation, the magnitude of the tension is sufficient to cause the cohesive zone opening at the corner(s) to be equal to *h*_{0}, i.e. *a*. This problem is represented by the sum of two problems—the same problem without the cohesive zone (i.e. continuity along the entire interface; figure 4*b*) and a residual problem with the cohesive zone but without far-field stresses (figure 4*c*). It is this latter configuration that will now be analysed in the immediate vicinity of the cohesive zone.

The corresponding boundary conditions on the free surfaces of this residual problem are given by equation (3.1) and on the frictionless interface are given by
*Q* is the generalized stress intensity factor whose value has been determined from the solution of the problem depicted in figure 1 and is linear in the applied load. The solutions for these problems with plane and axisymmetric geometries are given in §5.

The Mellin transform [25] is a useful tool for analysing the stress and displacement fields in a wedge. Its advantage over the simpler method of Williams is that it allows for the stresses and displacements to be determined away from the tip of the wedge [22,26,27]. In the problem of figure 4*c*, there are two potentially singular points—the corner (at which singular stresses are applied in the residual problem) and the edge of the cohesive zone (at which the stress is bounded).

The Mellin transform of a suitably regular function *f*(*r*) on *s* is a complex transform parameter. Before applying the Mellin transform, we note that the stress and displacement fields can be found from the Airy stress function which is a solution of the biharmonic equation [28], i.e.
*μ* is the shear modulus and *ν* is Poisson’s ratio.

Now *ϕ*(*r*,*θ*), *r*^{2}*τ*_{rr}(*r*,*θ*), *r*^{2}*τ*_{θθ}(*r*,*θ*), *r*^{2}*τ*_{rθ}(*r*,*θ*), *ru*_{r}(*r*,*θ*), *ru*_{θ}(*r*,*θ*), respectively. The application of the Mellin transform to equation (4.7) results in [26]
*a*(*s*), *b*(*s*), *c*(*s*), *d*(*s*) are unknown complex functions to be determined from the boundary conditions. The transforms of the relevant stresses and displacements given by equation (4.7) are
*r*_{0}) according to
*r*=0 and at *r*=*r*_{0}. The values of *M* and *N* will be chosen for convergence. The Mellin transform of the displacement discontinuity

The normal stress acting on the contact plane can now be found using the inverse Mellin transform of *c* is taken such that the integral of equation (4.14) in the complex plane exists, for example, *c*=−1. Consequently, a new integration variable *p* is defined such that *s*=−1+*ip*.

The evaluation of equation (4.14) results in
*f*_{m}(*m*=1,2,…,*M*) and *g*_{n}(*n*=1,2,…,*N*). These unknown constants are determined by applying the first of the mixed conditions of equation (4.3) to the normal stress in equation (4.15). The result is

Equation (4.17) has the additional unknown quantity *Q*’ which gives the length of the cohesive zone (*r*_{0}). A numerical solution is obtained by satisfying this equation at a discrete number of suitably chosen collocation points (*r*_{i}, *i*=1,2,…,*M*+*N*+1) resulting in *M*+*N*+1 linear equations with *M*+*N*+1 dimensionless unknowns (*f*′_{m},*g*′_{n},*Q*′). The length of the cohesive zone is then given by
*Q*′ and *f*′_{1} are determined numerically for a given material combination as described earlier in this section. From the form given by equation (4.19), it is seen that the critical value of the generalized stress intensity factor (*Q*_{C}) can be expressed in terms of the work of adhesion (Δ*γ*) and the cohesive zone stress (*σ*_{0}). However, for the special case of a rigid punch, for which λ=1/2, the dependency on *σ*_{0} vanishes. At the other limit λ→0 when the half-space becomes infinitely stiff compared with the punch, the dependency on the work of adhesion vanishes indicating a transition from fracture-dominated adhesion to strength-dominated adhesion.

Equation (4.19) can also be written in dimensionless form as
*α*) are shown in figure 5 for various values of *α*=1). The corresponding critical length (*r*_{0C}) of the cohesive zone is given by equations (4.18) and (4.19) as
*α*.

## 5. Analysis for a finite width/diameter elastic indenter

The determination of the pull-off force requires the relationship between the applied load and the generalized stress intensity factor to be known for the frictionless contact problem. These results exist in the literature for both the plane strain and axisymmetric cases. In both configurations, the solution can be written in terms of a singular integral equation with a generalized Cauchy kernel. Those equations were solved using the quadrature and collocation method of Erdogan & Gupta [30].

### (a) Plane strain problem

Bogy [8] solved the plane strain problem of a semi-infinite elastic strip with arbitrary end conditions. Adams & Bogy [9] incorporated that solution with a half-plane solution to determine the stress distribution along the contact interface for both frictionless and bonded contact. Because their solution did not specifically give the generalized stress intensity factor, the numerical procedure was repeated in order to determine the generalized stress intensity factor (*Q*_{PS}) versus the composite material parameter (*α*). The results are given by
*g*_{2}(*α*) is shown in figure 7. Note that for plane strain *P* represents the force per unit depth. In the limit as *α*→1(λ→1/2), the generalized stress intensity factor should approach that of a rigid punch, i.e. *α*→−1(λ→0) the stress is uniform over the cross section and *Q*_{PS} simply becomes the stress (*P*/2*a*); the numerical result differs by only 1.6%. In this limit, *Q*_{PS} represents the stress, because λ→0 in equation (2.4).

Setting the value of *Q*_{PS} equal to the critical value leads to the pull-off force for plane strain (*P*_{C}) which is given in dimensionless form by
*α*→1(λ→1/2) and *α*→−1(λ→0) and *P*_{C}→1.992*σ*_{0}*a* which differs from the result for the perfectly rigid half-plane (*P*_{C}=2*σ*_{0}*a*) by 0.4%. A plot of equation (5.2) is shown in figure 8 for fixed *α*=−1, the pull-off force is independent of the work of adhesion, whereas figure 9 demonstrates that at *α*=1 the pull-off force is independent of the cohesive stress.

### (b) Axisymmetric problem

Agarwal [10] investigated the axisymmetric problem of a flat-ended elastic cylinder pressed against an elastic half-space, but did not obtain numerical results. This problem was later solved by Gecit [11] who included a graph of the generalized stress intensity factor as a function of material properties. The results are given by
*g*_{3}(*α*) is shown in figure 7. It is emphasized that these numerical results and the curve-fit given by equation (5.3) were obtained from the graphical results presented by Gecit [11] along with knowledge of the expected behaviour at *α*=±1 which helped to reduce the maximum error at the endpoints to 4.9% at *α*=−1 to 1.6% at *α*=1.

Setting the value of *Q*_{axi} equal to the critical value leads to the pull-off force (*P*_{C}) for the axisymmetric configuration as
*α*→1, *P*_{C} should approach the rigid punch solution of *P*_{C} differs by 0.6%. At the other limit when *α*→−1, *P*_{C} should approach the value for an elastic cylinder on a rigid surface (*P*_{C}=*σ*_{0}*πa*^{2}); in this case, the result differs by 5.8%. A plot of equation (5.4) is shown in figure 10 for fixed *α*=−1, the pull-off force is independent of the work of adhesion, whereas figure 11 demonstrates that at *α*=1 the pull-off force is independent of the cohesive stress.

### (c) Possible inclusion of friction

The effect of friction has not been included in this analysis nor has it been included in studies of adhesion in spherical contacts [1–3]. In order to account for friction, several significant changes would need to be made. In the asymptotic problem, both normal and shear stresses would need to be included in the separation region. The adjacent contact region would have to be modified to be either a perfect bond (zero relative normal and tangential displacements) or a region of frictional slip followed by a perfect bond. In the former case, the analysis of bonded wedges with dislocations of Hein & Erdogan [31] would no doubt be useful. Then, a mixed-mode cohesive zone failure criterion of a simple form would be required. Finally, the problems of Adams & Bogy [9] and Gecit [11] would have to be reformulated and solved so as to allow for frictional slip. Such an undertaking, while potentially useful, is well beyond the scope of this paper.

## 6. Conclusion

The adhesion between an elastic punch and an elastic half-space has been investigated for plane strain and axisymmetric configurations. The pull-off force was determined for a range of material combinations. The critical value of the generalized stress intensity factor was related to the work of adhesion by using a cohesive zone model in an asymptotic analysis of the separation near the elastic punch corner. These results were used in conjunction with existing results in the literature for the contact between an elastic semi-infinite strip and half-space in plane and axisymmetric configurations. It was found that although the pull-off force depends on the maximum stress of the cohesive zone model, this dependence vanishes as the punch becomes rigid. At the other limit in which the half-space becomes rigid, the stresses become bounded and uniform, and the pull-off force depends on the cohesive stress and is independent of the work of adhesion. Thus, the transition from fracture-dominated adhesion to strength-dominated adhesion was demonstrated.

- Received April 16, 2014.
- Accepted June 5, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.