## Abstract

The problem of normal contact with friction of a rigid sphere with an elastic half-space is considered. An analytical treatment of the problem is presented, with the corresponding boundary-value problem formulated in the toroidal coordinates. A general solution in the form of Papkovich–Neuber functions and the Mehler–Fock integral transform is used to reduce the problem to a single integral equation with respect to the unknown contact pressure in the slip zone. An analysis of contact stresses is carried out, and exact analytical solutions are obtained in limiting cases, including a full stick contact problem and a contact problem for an incompressible half-space.

## 1. Introduction

The first solution to the problem of contact interaction of two bodies (the problem of normal elastic contact of two spheres) was constructed by Hertz (1881). The key assumptions that allowed for a simple closed-form solution to the normal contact problem were the small size of the contact area compared with the dimensions of contacting bodies and their relative radii of curvature, absence of friction and adhesion in the contact, and the presumed linear elastic material response. Further progress in contact mechanics was associated with relaxing these assumptions and developing more realistic physical models of contact interaction. Cattaneo (1938) and Mindlin (1949) gave a solution to the problem of contact of elastic spheres under both normal and tangential forces. The contact area was divided in two zones, the inner circular (stick zone) and outer annular (slip zone), where different types of contact interaction occurred. It was assumed that in the inner circular zone, there were no relative displacements of the points on the contact surfaces (‘stick’ or ‘adhesive’ zone), and over the outer annular zone friction on the interface was sufficient to cause slip (‘slip’ zone). The boundary between stick and slip zones was unknown and had to be determined from the solution.

The presence of slip in the contact area is a natural consequence of the action of the tangential force. However, slip can be present even under the normal contact, if elastic properties of the contacting bodies are different. If the friction coefficient between the surfaces is large enough, there is no mutual sliding in contact and such contact is also called full stick (adhesive) contact. Galin’s classical work (Galin (1953), see also a recent revised translation by Galin & Gladwell (2008), and a review in Zhupanska (2008)) on the contact of a rigid flat-ended indenter with an elastic half-plane with partial slip was the first successful attempt to take into account friction in the problem of normal contact. Galin considered a two-dimensional (2D) problem of normal contact with friction of a rigid flat punch and an elastic half-plane. He divided the contact zone into the stick and slip zones with unknown boundary between them, and assumed that in the stick zone tangential force was insufficient to displace the points of the elastic body relative to the punch, and in the slip zone the slip of the punch relative to the elastic body occurred. The formulated mixed boundary-value problem was solved using the combined method of the Hilbert problem and conformal mapping. A completely independent approach to Galin’s stick–slip problem was proposed by Antipov & Arutyunyan (1991), whose method is based on the reduction of the original boundary-value problem to a special Riemann problem for two functions. The Riemann problem is subsequently reduced to an infinite system of algebraic equations, whose solution can be obtained with any prescribed accuracy. This method was further developed by Antipov (2000) for the solution of the contact problem for a periodic system of rigid punches pressed into an elastic half-plane by normal forces applied to each punch. An effective analytical solution procedure for 2D stick–slip contact problems was suggested by Zhupanska & Ulitko (2005), who gave an exact solution to the normal contact with friction of a rigid cylinder with an elastic half-space. The corresponding boundary-value problem was formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation was constructed using the Wiener–Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement and relative slip zone sizes.

Mossakovski (1954, 1963) and Goodman (1962) studied axisymmetric problems of normal adhesive (full stick) contact of elastic bodies using similar incremental approaches where a step-by-step computation of the contact stresses is performed simultaneously with the increase of the contact area. One of the noticeable contributions of Goodman was the introduction of the model of contact interaction in which the normal displacement caused by shear stress is neglected and only the normal displacement caused by the normal stress is taken into consideration. In such a way, the normal stress is always distributed according to the Hertzian theory of smooth contact.

Spence (1968*a*, 1975) originated a self-similarity approach to the contact problems, which yields geometrically similar stress and displacement fields at each step of application of progressive load for any indenter having profile *y*=−*A*|*x*|^{n} (2D case) or *z*=−*B**r*^{n} (axisymmetric case). A self-similarity approach changed the way of treatment for contact problems that involve moving contact boundaries and allowed one to avoid an incremental step-by-step computation of the contact stresses performed simultaneously with the increase of the contact area (e.g. Mossakovskii 1954, 1963; Goodman 1962). The full stick axisymmetric contact problem was reduced by Spence (1968*a*) to an integral equation, which was subsequently solved using the Wiener–Hopf technique (Spence 1968*b*). In the case of the normal axisymmetric contact with friction, Spence found that the slip radius is the same for all power-law indenters, which follows from the possibility of transforming the equations and boundary conditions for power-law indenters into those for a flat punch. The problem was reduced to a dual system of integral equations with respect to the unknown contact stresses, which was then solved analytically for the case when the influence of friction on the normal stress is neglected, due to uncoupling of equations in the system. In the case when the influence of friction on the distribution of the normal stress was preserved, a numerical solution was developed. Nowell *et al.* (1988) used a self-similarity approach to construct a solution of the contact problem for dissimilar cylinders. Furthermore, the self-similarity argument was exploited in the solutions of contact problems with finite friction not only in the case of elastic but also visco-elastic and elasto-plastic material response. Particularly, we can mention the contribution to the self-similarity approach by Borodich and Galanov. Galanov (1981) and Borodich (1983) extended the self-similarity approach to a three-dimensional (3D) case of frictional elastic contact and later to 3D case of frictional and adhesive contact (Borodich 1993). They also applied their self-similar approaches to contact between plastic and viscoelastic solids with power-law constitutive relations and to non-convex punches (see references in Borodich & Galanov 2002). Using the Mossakovski (1963) approach, Borodich & Keer (2004) gave the exact solution to the axisymmetric adhesive elastic contact problem and obtained the exact relation between the depth of indentation and the external load, similar to the formulae obtained earlier by Galin (1953) in the case of frictionless contact.

In the present paper, we consider the problem of indentation with friction of a rigid sphere into an elastic half-space. The self-similarity condition of Spence is used to model the effects of friction in the contact zone, and the influence of shear stress on the normal stress in contact is preserved in the problem formulation. The employed solution procedure, which includes the use of toroidal coordinates, Papkovich–Neuber functions and the Mehler–Fock integral transform, allowed us to reduce the boundary-value problem to a singular integral equation (as compared with dual integral equations obtained by other authors (Spence 1975; Nowell *et al.* 1988)) with respect to the unknown normal stress in the slip zone. This, in turn, allowed for constructing analytical expressions for the components of stress and displacement fields in the contact area. The proposed solution approach was used to construct exact solutions for the limiting cases of the contact of a rigid sphere with an incompressible elastic half-space and the contact problem of full stick between the sphere and half-space.

The rest of the paper is organized as follows. §2 introduces the boundary-value problem under consideration. §3 describes the solution procedure that includes reduction of the original elasticity problem to a boundary-value problem of the potential theory, application of the Mehler–Fock integral transform to the boundary-value problem and reduction to the integral equation. An exact solution in the case of an incompressible half-space is constructed in §3*d*. An exact solution to the limiting case of full stick between the sphere and half-space is furnished in §3*e*.

## 2. Formulation of the boundary-value problem

Consider an elastic half-space (*z*≥0) indented by a rigid sphere of radius *R* (figure 1). The force *P*_{0} applied to the centre of the sphere is normal to the surface of the half-space.

The difference in material properties of the rigid sphere and elastic half-space leads inevitably to non-equal lateral displacements of the sphere and half-space in the area of contact, *r*≤*b* (see figure 1). This is known as the *slip* phenomenon, which is confined within an annulus *a*<*r*≤*b*. The central circular part of the contact area, namely the region *r*≤*a*, is called a *stick zone*, for the absence of relative lateral displacements on the interface. Obviously, the contact and stick–slip boundaries do not remain stationary under progressively increasing load. In accordance to Spence’s self-similarity condition (Spence 1968*a*), the contact stress and strain fields are geometrically similar, and the proportion in which the contact area is divided in stick and slip zones is constant. This requires an additional assumption on the displacement in the stick zone, which in our case takes the form
2.1
Here *u*_{r} is the radial displacement, *C*_{0} is a non-positive constant that we call the frozen-in strain constant, since it ensures that the lateral strain at any given point of the stick zone does not change when the boundary of the contact area increases due to a progressive load, which explains the term ‘frozen-in’. At the same time, the ratio of the slip and stick zones sizes *b*/*a* stays constant, and the stress field remains self-similar as well. The value of the frozen-in strain constant *C*_{0} is unknown and has to be determined as a part of the solution.

The assumption on the relatively small size of the contact zone (*b*≪*R*) allows us to write the boundary condition for normal displacement *u*_{z} in the entire contact area as
2.2
where *δ* is the axial displacement of the half-space. We also assume that in the slip zone shear stress *τ*_{rz} is related to normal stress *σ*_{z} by the Coulomb law of dry friction
2.3
where *μ*_{0} is the coefficient of interfacial friction between the half-space and sphere, and the direction of the shear stress opposes the direction of slip. In the stick zone the magnitude of the shear stress is insufficient for causing slip, thus
2.4
Outside the contact area the surface of the elastic half-space is free of stresses:
2.5
The equilibrium condition leads to the equality to be satisfied by the normal stress *σ*_{z} and the applied force *P*_{0}:
2.6
Equations (2.1)–(2.5) constitute a mixed boundary-value problem of axisymmetric elasticity. Our objective is to construct a comprehensive analytical solution of the problems (2.1)–(2.5) without imposing any additional restrictions on the behaviour of contact stresses and displacements.

## 3. Solution procedure

### (a) Reduction of the original elasticity problem to the boundary-value problem of the potential theory

It is known that the choice of representation of a general solution of elastostatics equations plays a major role in the analytical treatment of the boundary-value problems of the theory of elasticity. Typically, a general solution of elasticity equations represents the components of displacement and stress fields by a combination of functions that are solutions of some other equations (usually, Laplace equations). Such a combination must satisfy identically the equations of elastostatics, thereby reducing the original problem to fulfilling only the boundary conditions. For instance, the Papkovich–Neuber general solution (Sokolnikoff 1956; Uflyand 1965; Gladwell 1980) is known to provide advantages in solving boundary-value problems of elasticity for bodies bounded by infinite planar surfaces (half-space, layer, wedge, etc.). Generally, the Papkovich–Neuber solution has the form (e.g. Gladwell 1980)
3.1
Here **u** is the displacement vector, **B** and *B*_{0} are vector and scalar harmonic functions, **x** is the position coordinates vector and ν is the Poisson ratio. For an elastic half-space *z*≥0 in the axisymmetric state, the components of elastic displacement and stress are represented by two harmonic Papkovich–Neuber functions *f* and *Φ* (Uflyand 1965) as follows
3.2
where *G* is the shear modulus of the elastic medium. Moreover, the harmonic function *F* is introduced as
3.3
The boundary surface *z*=0 in boundary-value problems (2.1)–(2.5) is divided into three regions with different boundary conditions, namely *r*≤*a*,*a*<*r*≤*b* and . Given the specific structure of the boundary conditions, it is advantageous to reformulate the boundary-value problem in toroidal coordinates (*α*,*β*,*φ*) that relate to the cylindrical coordinates (*r*,*φ*,*z*) as (e.g. Morse & Feshbach 1953)
3.4
Here the metric parameter *a* equals the radius of the stick zone (note that due to axial symmetry of the problem, the solution is independent of the polar angle *φ*). The toroidal coordinate system is shown in figure 2.

Observe that the surface *β*=0 corresponds to the region *r*≤*a* on the surface of the half-space (*z*=0); the surface *β*=π corresponds to the exterior of the circle of radius *a*, namely on the surface of the half-space (*z*=0), *α*=0 corresponds to the line *r*=0 and the ‘infinite’ point corresponds to *r*=*a*. Now, it is apparent that boundary conditions (2.1)–(2.5) in the toroidal coordinates can be reformulated in two infinite regions, namely and , which facilitates substantially the solution procedure.

Denoting
3.5
where *σ*(*r*) is the unknown contact pressure in the slip zone, we rewrite boundary conditions (2.1)–(2.5) in toroidal coordinates as
3.6
The value of the constant *α*_{0} is
3.7
and depends on the size ratio of the stick and the slip zones and, therefore, has to be determined from the solution of the problem.

Next, we reformulate the boundary-value problems (2.1)–(2.5) in terms of harmonic functions *F* and *Φ* in the toroidal coordinates. First, we apply the operator to the first equation of (3.6) simultaneously expressing *u*_{r} in terms of harmonic functions *f* and *Φ* according to the first equation of (3.2). Since the function *f* is harmonic, these operations lead to
3.8
Furthermore, integrating the last equation of (3.6) with respect to *r* in the interval from *r* to , the boundary-value problem (3.6) is reduced to the problem of finding two harmonic functions, *F* and *Φ*, in the half-space *z*>0 with the following boundary conditions at its surface *z*=0:
3.9
where the functions *F*_{2}(*α*),*F*_{3}(*α*),*F*_{4}(*α*),*F*_{5}(*α*) are defined as follows:
3.10
In such a way, we have reduced the boundary-value problems (2.1)–(2.5) of elasticity theory to the boundary-value problem (3.9) of potential theory. The next step will be to reduce the boundary-value problem (3.9) to an integral equation. This will be accomplished using the Mehler–Fock integral transform, which is known to be particularly useful in the analytical treatment of mixed boundary-value problems with a circular boundary (Uflyand 1965).

### (b) Application of the Mehler–Fock integral transform and reduction to integral equation

The Mehler–Fock integral transform of a function *f*(*α*) defined on an interval has the form
3.11
where is the Legendre function of the first kind. The inversion formula for *f*(*α*) is
3.12
To satisfy the definition (3.11) and the inversion formula (3.12), the function *f*(*α*) must be continuous and have bounded variation in an interval [*α*,*A*], where , and also satisfy the following convergence condition
3.13
We apply the Mehler–Fock integral transform to construct the solution of the boundary-value problem (3.9). Following Uflyand (1965), we represent the harmonic Papkovich–Neuber functions *F* and *Φ* in the toroidal coordinates in the form
3.14
Such representations reduce boundary conditions (3.9) to equations of the form
3.15
which is basically the inversion formula (3.12) of the Mehler–Fock integral transform (3.11). If the function *g*(*α*) satisfies the condition of convergence (3.13), then the function *G*(*τ*) can be written in the form
3.16
If the function *g*(*α*) is such that
3.17
then *G*(*τ*) is determined as (Uflyand 1965)
3.18
where the integrand function satisfies (3.12). Note that representation (3.18) is obtained using the following equality (Bateman 1953)
3.19
Now with the help of the Mehler–Fock integral transform, we derive the integral equation of the problem with respect to the unknown contact pressure *σ* in the slip zone. First, functions *F* and *Φ* in the first four equalities of equation (3.9) are replaced with their integral expansions (3.14). After this, using the inversion formulae (3.16) and (3.18), we obtain the unknown densities *A*(*τ*),*B*(*τ*),*C*(*τ*) and *D*(*τ*) in the form
3.20
where
3.21
Note that the inversion formula (3.16) was applied to the first and the third equations of (3.9); identity (3.19) along with its derivative with respect to *β* were employed to invert the second equation of (3.9), whereas inversion formula (3.18) was applied to the fourth equation of (3.9) since the function *F*_{4}(*α*) is such that
3.22
Substitution of the densities (3.20) in the last equation (3.9) results in an integral equation with respect to the unknown contact pressure *σ* in the slip zone:
3.23
where the right-hand side has the form
3.24
Further, the integral equation (3.23) is reduced to the Fredholm integral equation of the first kind with respect to the unknown function *σ*(*y*):
3.25
where
3.26
and
3.27
The details of derivation of the Fredholm integral equation (3.25) are presented in appendix A. Once the contact pressure in the slip zone is found as a solution of the Fredholm equation (3.25), harmonic functions *F* and *Φ* are determined by equation (3.14), where the integrand densities are calculated using equation (3.20), and the stress and displacement fields may be found by employing the general solution (3.2). Note that the integral equation (3.25) contains unknown constants *α*_{0},*C*_{0},*δ* and *a* that have to be determined as a part of the solution of the boundary-value problem. Equations for determination of these constants will be obtained in the next section from the analysis of the contact stresses.

Further analytical treatment of the Fredholm equation (3.25) does not look feasible in the most general case of the axisymmetric contact problem with friction, but simple analytical solutions can be obtained in some limiting cases, which will be discussed later in the paper. Nevertheless, it is important to emphasize that the use of the Papkovich–Neuber solution (3.2) and toroidal coordinates (3.4) has allowed us to reduce the boundary-value problems (2.1)–(2.5) to a single integral equation (3.25) with respect to the unknown contact pressure in the slip zone. In contrast, previous approaches to contact problems with partial slip resulted in dual systems of integral equation with respect to the unknown normal and shear stresses in the contact zone (Spence 1975; Nowell *et al.* 1988).

### (c) Analysis of contact stresses

In this section, we derive expressions for normal and shear stresses in the stick zone in terms of the contact pressure in the slip zone and analyse the asymptotic behaviour of contact stresses across the stick–slip boundary. The stress distributions in the stick zone are obtained by substituting densities (3.20) of the functions *F* and *Φ* into the general solution for stresses (3.2). The normal stresses in the stick zone have the form
3.28
and the shear stresses in the stick zone are
3.29
where , with *α*=0 corresponding to the centre of the contact area (*r*=0) and corresponding to the boundary *r*=*a* between the stick and slip zones.

To determine how contact stresses change across the stick–slip boundary, we evaluate the asymptotic behaviour of equations (3.28) and (3.29) as , i.e. at the boundary *r*=*a* between the stick and slip zones. To this end, the following asymptotic representation of the Legendre function as is used (Bateman 1953)
3.30
which allows us to write the following asymptotic expression for the normal stress
3.31
as well as an asymptotic expression for the shear stress in the form
3.32
The integrand functions in integrals with respect to *τ* in the expressions (3.31) and (3.32) have singular points in the upper half-plane of the complex plane *z*=*τ*+*i**η*. Simple poles are located at , and the poles of the second order are located at *z*_{m}=*i**m*,*m*=1,2,3,…. Calculating residues of these poles allows for writing asymptotic representations of (3.31) and (3.32) in the form of infinite series
3.33
Moreover, the contact stresses must be finite and integrable over the entire contact area. Therefore, in equation (3.33) the sum of coefficients *a*_{01}+*a*_{02} and the sum of coefficients *r*_{01}+*r*_{02}, which both are proportional to residues at , must be equal to zero. This leads us to the complex-valued condition of finiteness of the contact stresses across the stick–slip boundary *r*=*a*
3.34
Moreover, at the boundary of the contact zone *r*=*b*, the contact stresses must vanish
3.35
Four equations of (3.34), (3.35) and (2.6) must be used to determine uniquely the unknown constants of the contact problem: the radius of the contact zone *b*, the radius of the stick zone *a*, the frozen-in strain constant *C*_{0} and the axial displacement constant *δ*.

Having derived the integral equation of the problem together with the expressions for the contact stresses in the stick zone and equations for determination of unknown constants, we turn our attention to the analysis of the limiting cases of the contact problems (2.1)–(2.5) that admit exact analytical solutions. In particular, the next two sections discuss a contact problem for an incompressible half-space and a rigid sphere and a full stick contact problem for a compressible half-space indented by a rigid sphere.

### (d) Exact solution in the case of an incompressible half-space

Contact of a rigid sphere with an elastic incompressible half-space is one of the limiting cases of the presented contact problem that admits a simple solution. When the half-space is incompressible, the value of Poisson’s ratio is ν=0.5, and the integral equation (3.25) turns into the Abel integral equation
3.36
where
3.37
The solution to the Abel equation (3.36) has the form (Gakhov 1966)
3.38
which delivers immediately the distribution of the unknown contact pressure *σ*(*y*). Furthermore, the integral in the right-hand side of (3.38) can be calculated explicitly as
3.39
The axial displacement *δ* in equation (3.37) is found from the condition of vanishing of the contact stress at the boundary of the contact zone (equation (3.35))
3.40
Moreover, for an incompressible half-space, the complex-valued condition of finiteness of the contact stresses across the stick–slip boundary (3.34) becomes
3.41
Finally, satisfying the last equation while taking into account equations (3.39) and (3.40) enables us to determine the values of the constants and *C*_{0}=0. At the same time, according to equation (3.7) one has , which immediately yields *b*=*a*. Therefore, in the case of an incompressible half-space, the slip zone is absent and there are no radial displacements *u*_{r} in the contact zone. Then stress distributions in the contact zone are defined by equations (3.31) and (3.32) and are
3.42
Hence, in spite of the differences in the elastic properties of the half-space and the sphere, normal contact of a rigid sphere with an incompressible elastic half-space is frictionless, shear stresses are absent in the contact zone and the normal stress distribution follows the Hertzian stress distribution (Johnson 1985).

### (e) The limiting case of full stick

Full stick contact between a rigid sphere and an elastic half-space is another limiting case of the presented contact problem with friction that admits an exact solution. The boundary conditions (2.1)–(2.5) in the case of full stick take the form
3.43
where the slip zone is now absent, *a*=*b*, and the entire contact zone *r*≤*b* represents the stick zone. Formulating boundary conditions (3.43) in the toroidal coordinates and accounting for *a*=*b*, we determine that as follows from equation (3.7). Therefore, the stress distributions in the contact zone are obtained from the general expressions (3.31) and (3.32) for stresses in the stick zone by discarding terms with integrals over the interval
3.44
3.45
The complex-valued condition of finiteness (vanishing) of the contact stresses at the edge of the contact zone (equation (3.34)) now takes the form
3.46
and is used to determine the unknown constants *C*_{0} and *δ*. As one can see, selection of the general solution in the form of Papkovich–Neuber functions and use of a special curvilinear coordinate system, namely the toroidal coordinates, allows for obtaining a simple analytical solution to the axisymmetric full stick contact problem by means of the Mehler–Fock integral transform.

Lastly, we transform the expressions for normal (3.44) and shear (3.45) stresses to a form more suitable for numerical calculations. Namely, the Legendre function is replaced with the integral representation (A 1), then integrals with respect to *τ* are calculated using equation (A 4). After somewhat tedious computations, the normal stress distribution is reduced to the form
3.47
and the shear stress distribution is as follows:
3.48
where
3.49
and *K*=0.915965594 is Catalan’s constant (Abramovitz & Stegun 1964). As one can see, in the centre of the contact zone (*α*=0), the shear stress is zero. Note that infinite series in expression (3.48) is a result of representation of elliptic integrals in the form of hypergeometric Gauss functions (Abramovitz & Stegun 1964)
3.50
Moreover, in numerical calculations of equations (3.47) and (3.48), it is convenient to use the series representation for the following integral
3.51
that is obtained from Bateman (1953)
3.52
It is worth mentioning that both normal (equation (3.47)) and shear (equation (3.48)) contact stresses have finite-amplitude oscillations when approaching the edge of the contact zone. These oscillations in contact stresses demonstrate that the full stick boundary condition is physically impossible for the normal contact of solids. Hence, the presence of slip at the edge of the contact zone is inevitable. The presence of oscillations in stresses in full stick problems was mentioned for the first time by Abramov (1937).

Finally, we present the graph of distributions of contact stresses for the value of the Poisson ratio ν=0.3. Curves 1 and 2 in figure 3 correspond to distributions of the normal contact stress and shear stress given by formulae (3.47) and (3.48), respectively. Curve 3 shown in grey corresponds to the Hertzian distribution of normal stress (Johnson 1985)
3.53
Lastly, we note that a full stick axisymmetric problem for an elastic half-space and a rigid polynomial indenter has been considered previously by Spence (1968*a*), who reduced the problem to an integral equation whose complex solution procedure by means of the Wiener–Hopf technique was presented in a separate paper (Spence 1968*b*). Our solution procedure for the full stick problem is much simpler, does not require solution of any integral equation, reduces only to inversion of the Mehler–Fock integrals and yields an explicit solution. It is also worth mentioning that an exact solution to the full stick problem for a rigid cylinder and an elastic half-space (plane strain formulation) was obtained in a similar fashion by Zhupanska & Ulitko (2005) by employing bipolar coordinates and the Fourier integral transform.

## 4. Conclusions

In this work, an analytical approach to the treatment of axisymmetric problems of contact with friction is presented. In particular, the problem of contact of a rigid sphere with an elastic half-space has been considered, where the influence of the shear stress on the normal displacement in the contact zone was preserved in the formulation. Using a general solution in the form of Papkovich–Neuber functions, toroidal coordinate system and the Mehler–Fock integral transform, the problem has been reduced to a single singular integral equation with respect to the unknown contact pressure in the slip zone. In two limiting cases, namely full stick contact between a rigid sphere and an elastic half-space and contact between a rigid sphere and an incompressible elastic half-space, exact analytical solutions have been obtained.

## Appendix A

The integral equation (3.23) was transformed into the Fredholm integral equation of the first kind (3.25) in the following way. First, the Legendre functions in equation (3.23) are replaced by the following integral representation (Lebedev 1965)
A 1
The resulting equation is considered as the Abel integral equation (Gakhov 1966), inversion of which leads to the following equation
A 2
where
A 3
which is exactly equation (3.26). After this, the integral representation (A 1) for the Legendre functions is used in equation (A 2). Moreover, the integrals with respect to *τ* in equation (A 2) are calculated using the following integrals
A 4
Integrals (A 4) were evaluated by the calculus of residues. In addition, the following integrals were employed (Bateman 1953)
A 5
In derivation of equations (A 2) and (A 3), the following integral representations were useful (Lebedev 1965):
A 6

In such a way, the integral equation (3.23) was transformed to the Fredholm integral equation of the first kind A 7 where A 8 The integrals in the right-hand side of equation (A 8) were calculated using equation (A 4) and the formula A 9 resulting in equation (3.26).

## Footnotes

- Received February 26, 2009.
- Accepted May 7, 2009.

- © 2009 The Royal Society