## Abstract

If a rigid punch is perfectly bonded to an elastic half-plane, the stress state possesses a well-known oscillating singularity. Because the shear and normal stresses are out of phase with each other, the application of a frictional slip model is expected to result in a slip zone at each of the corners. A solution exists in the literature if the punch is subjected to a normal load. It was shown that the extent of the slip zone is an eigenvalue which depends upon Poisson’s ratio and the coefficient of friction, but is independent of the magnitude of the applied load. In this investigation, the extent of the slip zone as well as the slip displacement is determined from the perfect bond solution. The analysis is valid if the length of the slip zone is small compared with the punch width. However, the results are shown to be in excellent agreement with the solution in the literature even when the total length of the slip zones is equal to half of the punch width. A solution is then obtained for combined normal and tangential loading. This work, and its extensions, is expected to be applicable in the study of the mechanics of fretting.

## 1. Introduction

The plane strain elasticity problem of a flat-ended rigid punch bonded to and pressed against an elastic half-space is well known to have an oscillatory singularity at the corners of the punch [1]. This singularity is characterized by stresses which become infinite with order one-half, and with sign which changes with ever increasing frequency, as either corner is approached. This behaviour is, of course, physically unrealistic. In order to alleviate this anomalous behaviour, Spence [2] modelled this problem with Amontons–Colomb friction which resulted in a finite-width frictional slip zone near each corner. For any value of the friction coefficient, the size of these slip zones was shown to be the solution of an eigenvalue problem. Thus, the extent of the slip zones is independent of the magnitude of the applied force. Results were presented in tabular form for the stick zone length versus the friction coefficient for two different values of Poisson’s ratio.

This problem is one example of an elasticity problem in which the solution with a perfect bond is known, but the solution for the corresponding problem with frictional slip is sought. The question becomes *can the extent of the frictional slip zone be inferred from the solution of the corresponding problem with a perfect bond?* In this paper, it is shown that such a solution can be obtained from an asymptotic analysis of a residual problem in the vicinity of the slip zone. A comparison of these results with the known solution of Spence shows that this solution provides an excellent approximation even if the lengths of the sticks and slip regions are comparable. The procedure is then generalized to include a tangential as well as a normal force. The slip displacements are also determined for each of these cases. It is anticipated that this work, and its extensions to other problems, will be relevant in the understanding of the mechanics of fretting.

The general method of solution involves the application of the Mellin transform to a residual problem defined as the difference between that of the perfect bond and the frictional slip problems. It is noted that this general method was used in a different context for problems with a non-square-root (and non-oscillatory) singularity. Three such examples follow in which the solution for a perfect bond was used to infer the existence and characterization of a cohesive zone. Adams [3] determined the pull-off force for an elastic punch in frictionless contact with an elastic half-space. In that case the solution of the problem with a perfect bond (no separation, but with slip allowed [4]) was used to infer the size of cohesive zones near the punch corners. The results showed a transition between the rigid punch result (square-root singularity) and the strength-based non-singular limit (rigid half-space). Similarly, Adams & Hills [5] used this technique to determine the onset of fracture at a finite angle notch in a tension specimen, i.e. the critical value of the generalized stress intensity factor was determined. The results agreed very well (within 1%) with the finite-element results of Gómez & Elices [6] which, in turn, agreed very well with experiments. The problem of a crack perpendicular to and touching a bimaterial interface also possesses a non-square-root singularity. The critical value of applied pressure needed for crack extension was determined by Adams [7] by using the results without a cohesive zone of Chen [8].

In the current investigation, a rigid punch is in frictional contact with an elastic half-space. The punch is subjected simultaneously to a normal and tangential force. By subtracting the problem of a rigid punch which is perfectly bonded to a half-space, a residual problem is obtained which is solved using Mellin transforms. In §2, the solution for the perfect bond is reviewed. The formulation and solution of the residual problem is presented in §3. Results for the slip zone lengths and slip distances are presented and discussed in §4 for both normal loading and combined normal and tangential loadings. Finally, conclusions are given in §5.

## 2. A rigid punch perfectly bonded to an elastic half-space

Consider the plane strain problem of a flat-ended rigid punch of width 2*a* which is perfectly bonded to the surface of an elastic half-space. A normal force per unit depth *P* and a tangential force per unit depth *T* are applied to the punch as shown in figure 1. The solutions for the surface displacement derivatives of an elastic half-space subjected to arbitrary distributions of compressive normal tractions *a*<*x*<*a* are given by Barber [9]
*x*- and *y*-directions, respectively, the constants *A*=4(1−*ν*)/*μ*, *B*=2(1−2*ν*)/*μ*, *ν* is Poisson’s ratio and *μ* is the shear modulus. The boundary conditions are given by

From Muskhelishvili [1], the solution of equations (2.1) subject to (2.2) and (2.3) can be found by defining the complex function

Note from equation (2.1) that for *v*≠1/2, a tangential traction produces a normal displacement. Thus, a reactive moment must be exerted on the half-space by the rigid punch in order to prevent rotation. The magnitude of this reactive moment can readily be calculated by *v*=1/2, tangential/normal tractions no longer produce relative normal/tangential displacements, the singularity is no longer oscillatory, and the reactive moment vanishes.

## 3. Analysis of the configuration with a frictional slip zone

It is recognized that the solution of the linear elastic problem with a perfect bond demands that the stresses be singular as *x*→±*a*, and in particular that both the normal and shearing tractions acting on the interface plane become infinite as indicated by equation (2.8). Because these tractions also oscillate as the corner is approached, some frictional slip is expected to occur, as shown in figure 3, near each corner for any finite value of the coefficient of friction.

The solution which is sought *u*,*v*) according to
*f* is the coefficient of friction and sgn(*x*) is the signum function equal to +1 when *x* is positive and −1 when *x* is negative. It is noted that *f*>0 implies inward slip. That inward slip is anticipated, comes from the solution of the *frictionless* case obtained by setting *T*=0), the slip zones are located symmetrically about the middle of the punch, i.e. *c*_{1}=*c*_{2}=*c*.

Rather than attempting a solution of the mixed boundary-value problem, an asymptotic solution of the residual problem is determined near each of the corners. Consider the configuration in the vicinity of the corner at *x*=*a* (i.e. *r*=0) as shown in figure 4. The boundary conditions in the slip zone and beyond become
*x*=*a*−*r* has been used to relate the two coordinate systems of figures 1, 3 and 4. Thus, the slip zone extends over the interval (0,*r*_{0}) where the location of the slip-to-stick transition point (*r*_{0}=*a*−*c*_{2}) is, at this point in the analysis, unknown.

The Mellin transform [11] is a useful tool for analysing the stress and displacement fields in a wedge. Its advantage over the simpler method of Williams [12] is that it allows the stresses and displacements to be determined away from the tip of the wedge (e.g. Bogy [13], Tranter [14]). Here there are two potentially singular points—the punch corner and the end of the slip zone (*r*=*r*_{0}) at which the stress is bounded.

The Mellin transform of a suitably regular function *χ*(*r*) on

where *s* is a complex transform parameter. Before applying the Mellin transform, it is noted that the stress and displacement fields can be found from the Airy stress function which is a solution of the biharmonic equation [15], i.e.
*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 (3.5) results in [14]
*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 equations (3.6) are
_{3−5} and the combination of that result with equation (3.8) gives
*u*_{r}(*r*,0)) is now represented by a finite series in the interval (0,*r*_{0}) according to
*r*=0 and at *r*=*r*_{0}. The values of *M* and *N* will be chosen for convergence. The values of λ_{1} and λ_{2} are determined from an asymptotic analysis using the simpler method of Williams [12]. It can be shown (see appendix A) that λ_{1} is the negative root of smallest magnitude of

whereas λ_{2} is the smallest positive root of
_{1}=−λ_{2}. Thus the stresses are seen to be bounded at *r*=*r*_{0} but the oscillatory singularity at *r*=0 has been replaced by a non-square-root singularity or order λ_{2}.

The Mellin transform of the radial displacement (3.11) is
_{1}=−λ_{2} was used for simplification. The integrals given in equation (3.15) as well as other calculations in this work were evaluated using the symbolic interpreter language Mathematica^{®} (v. 9.0, 2012; http://www.wolfram.com).

The normal and shear stresses acting on the interface can now be found using the inverse Mellin transforms of *c** is taken such that the integrals in equations (3.16) in the complex plane exist, e.g. *c**=−1. Consequently, a new integration variable *p* is defined such that *s*=−1+*ip*.

The evaluation of equation 3.16 results in
*x*=*a*−*r*,

It is further noted that the linear unknowns on the left side of equation (3.198) are *g*′_{m}(*m*=1,2,…,*M*) and *h*′_{n}(*n*=1,2,…,*N*). An additional unknown is *r*_{0}/*a*, which gives the ratio of the slip zone length to the half-width of the punch. A numerical solution of equation (3.19) is obtained by satisfying this set of equations 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 (*g*′_{m},*h*′_{n},*r*_{0}/*a*). For an initial guess of *r*_{0}/*a*, *M*+*N* of the equations (3.19) can be solved for *g*′_{m},*h*′_{n} and the extra equation of (3.19) can be used as a nonlinear equation to determine *r*_{0}/*a*.

## 4. Results and discussion

### (a) Normal loading

Consider first the case of normal loading, i.e. *T*=0 in which case the problem is symmetric with *c*_{1}=*c*_{2}=*c*. The slip zone length (*r*_{0}/*a*) is independent of the value of the applied force per unit depth *P*. A plot of *r*_{0}/*a*=(*a*−*c*)/*a* versus the friction coefficient ( *f*) is given in figure 5 for different values of Poisson’s ratio. Also included are the results from Spence [2], which are shown with the symbol ‘x’. Note the excellent agreement between the results of Spence and the asymptotic analysis presented here, even when the slip zone length is a significant fraction of the punch dimensions. It is noted that Spence [2] also presents the results of an asymptotic analysis which are in excellent agreement with the complete solution for *r*_{0}/*a*<0.293. As expected as the friction coefficient increases the length of the slip zone decreases, but it is always finite for Poisson’s ratio *ν*≠1/2. However, in the limit as Poisson’s ratio approaches one-half, tangential displacements decouple from normal tractions and hence there is no tendency to slip even if *f*=0. This trend is evident in that for the same value of the friction coefficient, larger values of Poisson’s ratio tend to give smaller slip zones than do smaller values.

In figure 6 is shown the dimensionless slip displacement, i.e. *u*_{r}(0,0)*μ*/*P* versus the friction coefficient ( *f*) for different values of Poisson’s ratio. Note that the slip displacement increases with decreasing friction coefficient. This behaviour is, in large part, due to the increase in the slip zone length with decreasing friction coefficient. The effect of decreasing Poisson’s ratio is also to increase the slip displacement in a similar manner to how it increased the slip zone length. It may seem peculiar that the slip displacements do not appear to scale with the punch width. However, the average pressure on the punch is *P*/2*a*, and so the displacements do scale as the product of this average pressure and the punch width. Alternatively for fixed punch width the displacements scale with the magnitude of the applied force.

### (b) Normal and tangential loading

Consider now the case in which there is both a normal and a tangential force per unit depth. Owing to the nature of friction, the extent of the slip zones can depend upon the loading history [16,17]. Here the case in which both forces are applied simultaneously is considered, i.e. a single force of magnitude *φ* taken positive counterclockwise with respect to the vertical. In this case, the magnitudes of the slip zones are independent of the magnitude of the force.

A plot of (*a*−*c*_{2})/*a* and (*a*−*c*_{1})/*a* versus *T*/*P* is given in figure 7 for *ν*=0.3 and various values of the friction coefficient. The solid lines refer to the right corner, whereas the dotted lines pertain to the left corner (obtained by changing the sign of *f*). Recall that for *T*=0, the slip is inward. Thus, the tangential force acting to the right tends to increase the size of the right slip zone but it decreases the length of the left slip zone. Hence the tangential force does not cause reversal of the direction of slip, as anticipated based upon the asymptotic analysis presented in appendix A. Note that the slip zones encompass half the punch width well before *T*/*P*=*f*. In figure 8 is shown the dimensionless slip displacements, i.e. *u*_{r}(0,0)*μ*/*P* versus *T*/*P* for *ν*=0.3 and various values of the friction coefficient. The trend is similar to that observed in figure 7. With increasing *T*/*P* the slip displacement in the right slip zone increases, the slip displacement in the left slip zone decreases.

## 5. Conclusion

The frictional behaviour of a flat-ended punch pushed against an elastic half-space has been investigated using plane strain elasticity. The method involves an asymptotic analysis of a residual problem using the Mellin transform. For normal loading the size of the frictional zone agrees well with an existing solution. The slip displacements are also determined. When additionally a tangential force is exerted, the sizes of the two slip zones (one at each corner) are determined as are the slip displacements.

The method developed demonstrates that it is possible to infer the size of a slip zone and the corresponding slip displacement from knowledge of the solution for a perfect bond. For example, if a finite-element analysis is performed for a more complicated geometry, the mode I and mode II stress intensity factors could in principle be determined. Such a procedure may be complicated by the existence of the oscillating singularity. An equivalent value of the loading angle *φ* can then be readily determined from equation (2.8) and (2.9). Finally, the use of the figures 7 and 8 will give the length of the slip zone and the corresponding slip displacement. Future work will focus on the interface between arbitrary material pairs at which the bonded solution typically has a non-oscillatory and non-square-root singularity. Thus, it is anticipated that this method can be used for a variety of other problems involving fretting in which the perfect bond solution is known but the frictional slip solution is sought.

## Data accessibility

The data are generated only by solving the equations described in the paper and have been presented in the graphs of figures 2 and 5–8.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Appendix A: asymptotic analysis in the frictional slip zone

In figure 3 is shown the rigid punch in frictional contact with the elastic half-space. The stick zone extends from −*c*_{1}<** x**<

*c*

_{2}, whereas the slip zones extend over −

*a*<

*x*<−

*c*

_{1}and

*c*

_{2}<

**<**

*x**a*. An asymptotic analysis using the simpler method of Williams [12] will now be performed at each of the two ends of the right slip zone. The appropriate boundary conditions will be applied to the stress fields which are summarized in a convenient form by Comninou [18] as

_{2}is the smallest positive root of equation (3.13), i.e. the stress field is singular of order λ

_{2}at the corner of the punch.

At the transition from stick to slip, the origin of the coordinate system of figure 4 is translated to *r*=*r*_{0} and the boundary conditions
_{1} as a root of equation (3.12) and
_{1}<0 and inward slip ( *f*>0,*u*_{r}(*r*,*π*)<0) leads to *τ*_{rθ}(*r*,*π*)>0 which is compatible with inward slip. From equation (21), it then follows that −1<λ_{1}<−1/2. However, outward slip ( *f*<0,*u*_{r}(*r*,*π*)>0) also leads to *τ*_{rθ}(*r*,*π*)>0 which is incompatible with outward slip. Thus, only inward slip is possible at a transition from stick to slip.

- Received May 16, 2016.
- Accepted June 13, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.