## Abstract

The unilateral axisymmetric frictionless adhesive contact problem for a toroidal indenter and an elastic half-space is considered in the framework of the Johnson–Kendall–Roberts theory. In the case of a semi-fixed annular contact area, when one of the contact radii is fixed, while the other varies during indentation, we obtain the asymptotic solution of the adhesive contact problem based on the solution of the corresponding unilateral non-adhesive contact problem. In particular, the adhesive contact problem for Barber’s concave indenter is considered in detail. In the case when both contact radii are variable, we construct the leading-order asymptotic solution for a narrow annular contact area. It is found that for a v-shaped generalized toroidal indenter, the pull-off force is independent of the elastic properties of the indented solid.

## 1. Introduction

In recent years, bioinspired mechanical aspects of adhesion have received much attention [1,2]. It was found [1] that a toroidal contact geometry with an annular contact area should lead to better attachment than that of a convex contact geometry with a single-connected area of contact. In particular, based on the two-dimensional model for line adhesive contact [3] the pull-off force, *F*_{c}, for a narrow rigid toroidal indenter in contact with a semi-infinite elastic medium is given by the formula
*E** is the effective elastic modulus of the elastic medium, *Δγ* is the work of adhesion, *R* and *ρ* are the major and minor torus radii, respectively (figure 1*a*).

At the same time, it was observed that besides contact size, contact shape can exert a strong influence on adhesion and, therefore, other contact geometries, which produce a narrow annular contact area, represent an undoubted interest especially for developing bioinspired structured interfaces [4].

On the other hand, since a toroidal indenter exhibits a comparatively large pull-off force, it can be employed for effective measuring adhesive properties of flat surfaces. Therefore, in order to implement depth-sensing adhesive indentation techniques [5,6], it is necessary to obtain the force–displacement relationship. At low loads, the force–displacement curves reflect not only elastic properties, but also adhesive properties of the contact, and therefore characteristics of adhesion, such as the work of adhesion and the pull-off force, can be extracted. Note also that though the rate effect may be essential in some biological applications, our analysis of adhesive contact is quasi-static, and the Johnson–Kendall–Roberts (JKR) is used.

The frictionless contact problem for a flat-ended annular punch was solved in a number of papers [7–9]. An exact analytical solution of this contact problem was obtained in [10]. Based on an analytical solution of this problem and using the variational method developed by Barber & Billings [11], which is based on Barber’s theorem [12] about the contact area that maximizes the value of the contact force, one can solve the axisymmetric unilateral contact problem with an annular contact area, whose contact radii are unknown *a priori*. In this way, in particular, the frictionless non-adhesive contact problem for a concave punch was solved [13].

The axisymmetric JKR [14] adhesive contact problem with an annular contact area was studied by Kesari & Lew [15], who reduced the contact problem to an integral equation, which defines the inverse Abel transform of the surface normal displacement. They solved this equation numerically for a concave punch (figure 2), but the force–displacement relationship was not investigated.

The rest of the paper is organized as follows. In §2, we solve the JKR adhesive contact problem with an annular contact area under the assumption that one of its radii is fixed and the solution of the corresponding non-adhesive contact problem is known, being parametrized by the variable contact radius. This solution is applied in §3 to the adhesive contact problem for Barber’s conically concave indenter [13]. In §4, we consider the axisymmetric JKR adhesive contact problem with an arbitrary annular contact area and provide an approximate solution using the method of dimensionality reduction (MDR) [16] and assuming that the annular contact region is relatively narrow. It is worth noting here that the case of the semi-fixed annular contact cannot be straightforwardly treated as a special case of the general case considered in §2, because there we introduce the so-called effective contact radius, whose evaluation is as yet a more demanding problem. In §5, by means of the variational method of Barber & Billings [11], we consider the unilateral non-adhesive contact problem for a generalized toroidal indenter and construct the leading-order asymptotic solution, which afterwards (in §6) is used for deriving an analytical solution of the corresponding JKR adhesive contact problem. In §§7 and 8, we investigate the behaviour of the pull-off force and the force–displacement relation due to the variation of the punch’s shape parameter. Finally, in §9, we formulate our conclusions.

## 2. Asymptotic solution of the Johnson–Kendall–Roberts adhesive contact problem with a semi-fixed annular contact area

Consider indentation of a profile *z*=*φ*(*r*) into an elastic medium with plane surface, which produces an annular contact area of radii *a*_{1} and *a* (such that *a*_{1}<*a*). Without loss of generality, it is assumed that *a*_{1} is fixed, while *a* varies during the indentation (figure 2).

We assume that the *non-adhesive normal contact problem* for this shape and this medium has been solved, so that the dependence of the normal force *F* and of the contact area *A* on the indentation depth *δ* is known. Each of these three quantities determines uniquely two others, so that we can consider the normal force *F*_{na}(*a*) and the indentation depth *δ*_{na}(*a*) as known functions of the variable contact radius *a*, too. We further can define the potential energy of the non-adhesive contact, *U*_{na}(*a*) and the contact stiffness
*a*.

Now let us consider an adhesive contact under assumptions of the JKR theory (range of interaction of adhesive forces much smaller than any characteristic size of the problem, that is to say zero) and characterize adhesion with the work of detachment of surfaces per unit area, Δ*γ*.

If we indent the profile up to the contact radius *a*, then the potential energy in this state will be *U*_{na}(*a*) and the indentation depth *δ*_{na}(*a*). The force in this moment will be *F*_{na}(*a*). Now let us lift the indenter by Δ*l* *without changing the contact area*. During this process, the stiffness of the contact remains constant and equal to *k*_{na}(*a*). Therefore, the force will change according to
*l*. Inserting the result it into (2.3), we get
*A*(*a*)=*πa*^{2} is the contact area.

The equilibrium value of *a* corresponds to the minimum of this energy with respect to *a* for a constant indentation depth *δ*. To determine the minimum, we let the derivative
*l* provides the following equation:
*A*(*a*)/d*a*=2*πa*.

In view of (2.9), equations (2.2) and (2.4) take the form
*l*_{c}(*a*) given by equation (2.9) and provide the solution to the JKR adhesive contact problem.

Note that in the case of an annular contact area of radii *a*<*a*_{1}, we will have *A*(*a*)/d*a*=−2*πa*. However, formula (2.9) still holds, because d*k*_{na}(*a*)/d*a*<0 in this case.

## 3. Johnson–Kendall–Roberts adhesive contact of Barber’s conically concave indenter

Let the indenter shape function (figure 2) is given by the following formula:
*E**=*E*/(1−*ν*^{2}) is the effective elastic modulus, and
*n*=1,2,3,4, while the functions *Ξ*_{n}(*ε*) are given by (3.11), (3.12), (3.14) and (3.17), and the following notation has been introduced (see (3.5)):
*μ* (see formula (3.16)), a good coincidence is observed. We note that for the validity of the analytical solution, the quantity *μ*, we extend the range of applicability of the solution when

Direct three-dimensional simulations have been carried out using the complete boundary element formulation for normal and tangential contact problems proposed in [17]. This method uses the known fundamental solutions by Boussinnesq and Cerruti connecting surface forces and surface displacements, while the corresponding integral transformation between distributions of stress and displacements in the contact plane are carried out using the fast Fourier transform. The latter is implemented on a graphic card allowing massive parallelization of this time-consuming step. Furthermore, for simulation of adhesive contacts, a stress-based criterion for detaching of surface elements has been used as suggested in [18]. The criterion is based on equating the increasing surface energy and decrease of elastic energy due to detaching of a surface element. The numerical procedure is the following. Outgoing from some reference configuration, the indentation depth is firstly reduced by a small value. The contact area is kept unchanged at this step and the stresses are calculated by the BEM as presented in [17]. Then the stress of each contact element is compared with the detachment criterion as described in [18]. If the criterion is fulfilled, the grid element detaches and this point is removed from the contact zone. The equilibrium contact configuration is determined by repeating this procedure. The detailed derivation on the criterion, numerical algorithm and verification of the method can be found in [18].

Figure 4 shows the effect of the parameter *μ* on the force–displacement curve. Observe that, in view of (3.18), the pull-off force is approximately *μ* decreases).

## 4. General case of Johnson–Kendall–Roberts adhesive contact: approximate treatment

If the set of contact configurations of adhesive contact would repeat the set of the contact configurations of the normal contact for the same shape (which regrettably will generally not be the case!), then the adhesive contact could be solved in the following way. For simplicity, we consider here homogeneous media. We assume that the normal contact problem was solved so that the dependence of the normal force *F*_{na} and contact area *A* on the indentation depth *δ* is known:
*α* has the dimension of length and in the special case of homogeneous isotropic elastic half-space coincides with the harmonic capacity radius defined as *α*=2**c**/*π*, where **c** is the harmonic capacity of the contact area [20,21].

The non-adhesive normal contact problem now can be described by the MDR [16,22] with the equivalent profile *z*=*g*(*x*), where the function *g*(*x*) is defined according to
*δ*.

The condition for the equilibrium of an adhesive contact can be obtained from the standard balance of energy at small variation of the contact radius. We assume that the boundary springs (in the MDR setting) detach when they achieve the critical length Δ*l*_{c}, which is determined by equating the relaxed elastic energy
*α*=*a* and, of course, trivially *A*=*πa*^{2} and d*A*/d*a*=2*πa*. Thus, formula (4.9) implies that

## 5. Approximate solution of the non-adhesive contact problem with a narrow annular contact area

Recall that the leading-order asymptotic solution for a frictionless flat-ended annular narrow indenter, which is pressed into an elastic half-space to the unit depth, is represented by the following asymptotic formula [23]:
*n*=*r*−*R* is the distance from the annular indenter centreline, *h* is the indenter’s half-width, *P*_{1} is the line contact pressure density, which is related to the contact force, *F*_{1}, by the following formulae [24]:
*a*) is taken in the form
*R* is the radius of the toroidal indenter’s centreline, *Λ* and λ are constant parameters.

By Mossakovskii’s theorem [25], in view of (5.4) and (5.1), the contact force under the toroidal indenter can be evaluated as follows:
*B*(*x*,*y*) being the Beta function)
*h* (figure 5*b*) can be constructed by the variational method of Barber & Billings [11] by maximizing the functional of contact force *F*(*h*), based on the theorem of Barber [12]. So, taking into account (5.3), we arrive at the problem of maximizing the function
*f*^{′}(*x*)=0, we obtain

## 6. Approximate solution of the adhesive contact problem with an annular narrow contact area

According to (5.3), the harmonic radius of the narrow annular domain *ω* is given by
*ω* is
*L* was introduced in (3.5), and correspondingly we obtain
*h*=*εR* according to (6.6) into equation (5.10), we get

Thus, according to (2.10), (2.11) and (4.9) the approximate solution of the adhesive contact problem is given by the formulae

## 7. Pull-off force for the generalized toroidal indenter with a narrow annular contact area

According to equations (5.11) and (6.9), we have
*ε* and equating the result to zero gives the critical value
*θ* is the angle of the wedge. It is interesting to observe that the pull-off force (7.6) is independent of the elastic properties of the indented elastic medium.

Now, let us compare the values of the pull-off force (7.5) for different values of the shape parameter λ under the assumption that the contact area is the same at the instant of indenter detachment from the surface. In other words, we fix the value (7.4), so that *A*_{c}≃2*πR*^{2}*ε*_{c} is the same for different values of *m*_{λ}. In this way, expressing the shape parameter *m*_{λ} from equation (7.4) in terms of *ε*_{c} and substituting the result into equation (7.5), we arrive at the formula

### Remark

Let us put λ=2 and *Λ*=1/(2*ρ*) (figure 1*b*). Then, *m*_{2}=1/(4*ρ*) and equations (5.12) and (5.13) reduce to the following ones:

## 8. Force–displacement relation for the generalized toroidal indenter

In view of (7.4) and (7.5), let us introduce dimensionless contact force and indenter displacement
*ε*_{c}.

Figure 8 illustrates the difference of the solution with and without considering the adhesive force for a toroidal indenter (λ=2, *Λ*=1/(2*ρ*), and *m*_{2}=1/(4*ρ*)). The dashed line corresponds to the non-adhesive solution. It is necessary to note that the same normalization (8.1) is used.

## 9. Conclusion

In this paper, we have developed an analytical approach to solve the JKR adhesive contact problem with an annular contact area. The method allows to construct an asymptotic solution in the case of a semi-fixed contact area, while the accuracy of the solution depends on the accuracy of the analytical solution of the corresponding non-adhesive contact problem. In the case of a fully unknown annular contact area, the MDR-based approach allows to obtain an approximate solution, which coincides with the leading-order asymptotic solution for a singularly perturbed narrow contact region. One of the major findings is that the pull-off force for a v-shaped toroidal indenter is independent of the elastic properties of the indented solid.

## Authors' contributions

I.A. carried out the theoretical analysis, designed the study and drafted the manuscript; Q.L. carried out the numerical simulation; R.P. designed the initial numerical programme; V.L.P. conceived of the study, coordinated the study and helped draft the manuscript. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

I.A. is grateful to the DAAD (German Academic Exchange Service – Deutscher Akademischer Austausch Dienst) for financial support during his stay at the TU Berlin.

- Received March 28, 2016.
- Accepted June 23, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.