## Abstract

In this paper, a contact problem is considered with a circular indenter pressed normally against a semi-infinite elastic composite that consists of a contact layer with uniform thickness welded together with another dissimilar medium. The indentation is modelled by means of a real continuous function *g*(*t*), and an integral equation representation for the displacement is derived in terms of *g*(*t*) on the contact boundary. The integral equation is evaluated numerically for *g*(*t*) when the contact layer is finite. In the case when the thickness of the contact layer becomes small compared with the radius of the contact region, a singular perturbation technique is used to derive an asymptotic expansion for *g*(*t*). This leads to a Wiener–Hopf equation formulation for the semi-infinite geometry in the Fourier-transformed domain of the inner coordinates. Subsequently, Van Dyke's matching principle is used to match the inner with the outer solution of *g*(*t*). The results are illustrated and verified with a simple limit solution derived using an integral argument.

## 1. Introduction

Contact mechanics is a sub-branch of continuum mechanics in which the stress field beneath an indentor and the associated deformation are used to interpret and explain a variety of phenomena. Such contact may result in fracture that is intentional as in drilling, for example, or to be avoided in order to resist wear. Precise modelling, for each of these, requires adequate constitutive equations describing the medium (and maybe the indentor) as well as the methods for solving the consequent boundary-value problems. One area for such application is in coating technology, a growing fraction of materials technology.

For homogeneous materials, the measured load or penetration response during elastic loading of an indentation test can be used to estimate the effective modulus. For the indentation of a coated system (i.e. a coating of thickness in the range of 0.2–5 μm), the effective modulus measured depends in an unknown way on the elastic properties of the coating and the semi-infinite substrate, although substrate properties are more likely sensed when the coating is very thin than those sensed when the coating is much thicker. Various finite-element solutions already exist; however, an analytic approach is still lacking. In the previous works by Chen & Atkinson (Atkinson & Chen 1997; Chen & Atkinson 2005), a variety of semi-analytical methods have been developed to solve fracture problems in layers, which can be further extended here to treat such contact problems. In particular, it should be possible to use the singular perturbation theory to obtain relatively explicit formulae for the situation when the layer is very thin and nano-indentation is used.

Recently, the nano-indentation technique has been established as the primary tool for investigating the hardness of small volumes of materials (Malzbender *et al.* 2002; Cheng & Cheng 2004). The atomic force microscope (AFM) has been used as a microindenter of thin biological samples to determine the local elastic moduli of the tissue (Mahaffy *et al.* 2000; Dimitriadis *et al.* 2002). The microscope provides the signals for force and displacement, and hence the force–indentation relationship for the material, which can then be used to estimate the elastic moduli, provided that a theoretical model describing the contact is available. Thus, a simple-to-use model will be useful in the routine AFM work and indentation tests in general. A model that provides an approximate solution for the indentation of a thin compressible elastic layer by a flat-ended cylinder has been developed by Haider & Holmes (1997). By considering the axisymmetric indentation of a thin incompressible elastic layer, Chadwick (2002) provided estimates for the force and contact radius by asymptotically matching a lubrication-type expansion in the contact region to the far field of an edge layer expansion.

For our proposed model here, a perfectly rigid solid of revolution of prescribed shape is considered, which is pressed normally against the boundary *z*=0 of a semi-infinite (*z*>0) elastic composite with its axis of revolution coinciding with the *z*-axis. The composite is that of a thin layer of uniform thickness welded together with another dissimilar medium. A flat-ended circular cylinder of radius *a* is considered, which penetrates a small depth *δ*_{0} below the level of the boundary *z*=0. The stresses produced are those in the half-plane *z*>0 that is bounded by the plane *z*=0. The origin in this case is situated in the centre of the disc-shaped indentation and axisymmetry is assumed. Figure 1 demonstrates this situation. The indentation is modelled by means of a real continuous function *g*(*t*) (see Sneddon 1966; Green & Zerna 1968) such that the normal stress on the surface is represented by(1.1)(‘Re’ is used here and elsewhere to mean the real part). This ensures the condition that the normal stress on the surface is zero exterior to the contact region. Subsequently, an integral representation is derived for the displacement *u*_{3} on *z*=0 for *r*<*a* in an approach similar to that described for the penny-shaped crack (Chen & Atkinson 2005). The function *g*(*t*) is evaluated numerically with *u*_{3} given on *z*=0 for *r*<*a*, which, in turn, gives the loading on the boundary.

Note that the general equations for the three-dimensional deformations of sandwiched structures have been derived previously by Chen & Atkinson (2005), which, for this paper to be self-contained, will be included in the appendix. More specific assumptions are made later, such as the axisymmetry geometry or the uniform indentation within the contact region. These equations can therefore be employed to solve other similar cases, i.e. elliptic cracks or possibly some other contact problems with different shapes of indenters. For simplicity of the notation, the coordinates (*x*, *y*, *z*) are used instead of (*x*_{1}, *x*_{2}, *x*_{3}), and the suffices (1, 2, 3) of stress and strain thus correspond to (*x*, *y*, *z*).

## 2. Formulation of the problem

The stress–strain relationship for the homogeneous isotropic elastic media is given by(2.1)for small strain, *λ* and *μ* being the Lamé constants. The equations of motion ignoring the inertia term are(2.2)where the repeated suffices sum over 1, 2 and 3. Substituting equation (2.1) into the above equation gives(2.3)This is then differentiated with respect to *x*_{i} and rearranged to obtain the Laplace equation(2.4)Taking the double Fourier transform over *x* and *y* of the above equation gives(2.5)with the double Fourier transform defined asSolving the equation for both regions subject to the continuity condition of stresses and displacements together with other appropriate conditions on the contact boundary *z*=0 and at the far field, a relationship between stress and displacement on *z*=0 is given by(2.6)with and denoting the normal stress and displacement of region (2) (indicated by the superscript) on the boundary *z*=0. The double overbar of *u* and represent the double Fourier transform of stress and displacement over *x* and *y* coordinates with *ξ* and *ζ* being the corresponding Fourier transform variables. The function is given by(2.7)where *λ*_{2} and *μ*_{2} are the Lamé constants and is Poisson's ratio of region (2) (indicated by the subscript). *T*_{2}, *T*_{3} and *ρ* are the functions of *ξ* and *ζ*, particularly, ; the details of these functions together with the derivation of equation (2.6) are given in appendix A.

To proceed, we first note that the mixed boundary condition is such that the normal stress outside the disc-shaped contact region and the indentation is known and assumed uniform within the region. The problem here is the one with dual characteristics to that of a penny-shaped crack in a sandwiched composite medium. Thus, a similar approach is taken here to model the problem with a continuous function *g*(*t*), which results in an integral representation for in equation (2.9) (cf. Green & Zerna 1968). To achieve this, it is assumed that on the boundary *z*=0 with , which has double Fourier transform . From equation (2.6), the displacement (denoted by on *z*=0) can be written in terms of the stress (denoted by on *z*=0)(2.8)An integral expression for is derived by first inverting in the above equation, which is then multiplied by a continuous function *g*(*t*) and integrated over *t* from 0 to *a* and then taking the real part to give(2.9)where(2.10)and(2.11)with given by (2.7). The real part is taken to ensure that the stress vanishes for *r*>*a*. Axisymmetry is assumed for the circular cylindrical indentor and with the substitution that(2.12)the above function becomes(2.13)with(2.14)By writing the first integral on the right-hand side of equation (2.9) as , i.e. for , we have(2.15)which is then multiplied by and integrated over *r* from 0 to *u*. Changing the order of integration gives(2.16)and the function *g*(*u*) can be written as(2.17)If we further assume that the displacement , a constant, within the contact region, then with(2.18)the function *g*(*u*) satisfies the equation(2.19)for , where(2.20)Note that we have used the result(2.21)

It can be easily shown that is exponentially small in the limit and tends to a constant as ; thus the integral exists and can be evaluated numerically. The function *g*(*u*) can now be evaluated numerically by dividing the range of integration from 0 to *a* in equation (2.19) into *n* intervals. This then gives us a system of equations and unknowns. These unknown 's, are found by inverting a matrix.

## 3. Singular perturbation

When the thickness of the inside layer becomes very small compared with the radius of the contact region, i.e. , the numerical solution for *g*(*u*) evaluated using equation (2.19) diverges at the point *u*=*a*. Thus, to derive the solution for *g*(*u*) at *u*=*a* in the limit , we define an inner coordinate system(3.1)and with . Note that *η* used below is simply a dummy variable in the integrals. Thus, equation (2.19) written in the inner coordinates is now(3.2)for , where and(3.3)derived from of equation (2.20) with . We have written in the inner coordinates since is even in *ρ* and thus(3.4)(3.5)Extending the definition of , we define, for *X*>0,(3.6)Thus, in the limit , we assume that equation (3.2) is defined from . Then taking the Fourier transform of over wherewe obtain(3.7)where(3.8)using the result(3.9)(3.10)We have ignored the Fourier transform of the term , assuming that it is of higher order. The subscripts ‘+’ and ‘−’ of the half Fourier transform and denote the regions of upper and lower half-planes of regularity, respectively. Rearranging equation (3.7) gives(3.11)ignoring the error term Particularly, is given by(3.12)where 's are the functions of the elastic constants (see appendix A for details). The function is factorized into the product of plus and minus functions by applying Cauchy's integral theorem to the logarithm of the function , i.e.(3.13)with the principal valued branch of log. The integration paths are taken such that from above and below for the plus and minus functions, respectively. The asymptotic expansions for and are derived by closing the contour from below and above, respectively. Subsequently, equation (3.11) is rewritten to give the Wiener–Hopf equation(3.14)Note that is given by equation (3.19). Using analytic continuation, the function is thus analytic in the whole *ξ*-plane and both sides of the equation bounded for large *ξ*. Using Liouville's theorem with and since as , we obtain, in the limit , the leading order solution(3.15)Inverting this gives, in the limit ,(3.16)which will be verified in §4 using a simple limit solution via an integral (see equation (4.6)). To obtain higher order solutions, we add an eigen solution by setting the function in the Wiener–Hopf equation as . The constant *A*, which we expect to be of the order *ϵ*, is unknown. Note that by rescaling with the inner coordinates the far-field boundary condition can no longer be applied, a consequence of the singular perturbation technique, and hence the presence of an unknown constant in the expression for is expected. To find the unknown constant in , matching with the outer solution will be required in order to supply the missing boundary condition. In what follows, we will employ Van Dyke's matching principle to match the *n*-term inner expansion of the *m*-term outer with the *m*-term outer expansion of the *n*-term inner (see Van Dyke 1975). We thus require an expansion of *g*(*u*) in terms of the outer coordinate *u* for small *h*, which will be termed as the outer solution. For the inner solution, we seek an expansion of in the inner coordinate *X* for small , and hence an expansion for in the limit is required. To derive such an expansion for , we first need to find an expansion for for . From Cauchy's integral theorem above (equation (3.13), the expansion for is given by(3.17)where *m*_{2} is unknown (determined later through matching) and(3.18)with and(3.19)the detailed derivation of can be found in Atkinson & Chen (1997). An expansion for from the Wiener–Hopf equation (3.14) can now be derived, thus in the limit ,(3.20)Fourier inversion gives the two-term outer solution with ,(3.21)where is the Dirac delta function. Note that Fourier inversion gives(3.22)and, asymptotically, Fourier inversion of and in the limit gives, to the leading order,(3.23)Writing in the outer coordinates gives up to terms indicating that a three-term outer expansion is sought in order to match with the inner solution above. Note that equation (3.21) indicates that we require and terms in the outer solution expansion.

### (a) Outer solution

To determine the unknown constant *A*, matching to the outer region is required. The outer solution is derived here by iterating the equation for *g*(*u*), namely, equation (2.19), in the outer coordinate to obtain an expansion for small *h* (). We recall that the function *g*(*u*) satisfies the equation(3.24)We first expand the function in small *h* giving(3.25)where(3.26)with *m*_{1} given by equation (3.18). From equation (3.4) with replaced by the expansion above, we have(3.27)since(3.28)Substituting the expansion for into equation (2.19), the expansion for *g*(*u*) for small *h* is given by(3.29)This gives the anticipated term for matching. To obtain the term, we solve a correction term for *g*(*u*) with satisfying the equation(3.30)The unknown constant , which we anticipate , will be determined through matching with the inner solution. Thus, we write . Iterating with given by equation (3.27), we obtain the expansion for for small *h* as(3.31)Thus, the three-term outer solution for *g*(*u*), denoted by , is given by(3.32)with *α*, *β* and *γ* being unknown.

### (b) Matching

We can now apply Van Dyke's matching principle by matching the three-term outer solution written in the two-term inner coordinate with the two-term inner solution written in the three-term outer coordinate. Thus, in the two-term inner coordinate is(3.33)Matching equation (3.21) with equation (3.33), we require(3.34)(3.35)(3.36)(3.37)for the four unknowns *A*, *m*_{2}, *α* and *β*. Hence,(3.38)Thus, the improved solution of from the Wiener–Hopf equation (3.14) is now(3.39)Taking the limit inverting above, we obtain(3.40)

## 4. Results

In this section, the results derived from the analysis of §§ 2 and 3 are illustrated and verified with Poisson's ratio held equal for both regions. This is assumed for the convenience and clarity in verifying our results, and the analysis from §§ 2 and 3, however, can be applied to materials with different values of Poisson's ratio. The solution for *g*(*a*) evaluated using the integral equation (2.19) is illustrated in figure 2 for *h*>1 and in figure 3 for *h*<1. The solution diverges for *h*≪1 as shown in figure 3; the asymptotic solution derived using the singular perturbation method is then applied for the region when *h*≪1.

Before we verify these results, we first derive *g*(*a*) by considering simple limit solutions via the energy release rate arguments. The same argument has been used for the two-dimensional plane-strain case (Atkinson & Chen 1997) and in the three-dimensional case of a penny-shaped crack (Chen & Atkinson 2005; see also Eshelby 1970; Budiansky & Rice 1973). The integral used in the three-dimensional case is given by(4.1)where the repeated suffices sum over 1, 2 and 3. This integral is invariant provided the medium is homogeneous. Thus, in the limit , we are able to relate the contribution to from infinity to that on *z*=0. This is reduced to the local integrals taken around since all the shear stresses vanish on the boundary. In addition, for *r*>*a* and for . This results in an energy release rate type of expression ^{*}(4.2)where *C* is a constant. When the layer thickness, *h*, tends to 0, we imagine that the problem is that of an indentation in medium (1) alone. However, the local integrals taken around are in fact in medium (2). Using the above two arguments gives(4.3)where is that derived from the approximation by setting *h*=0 while *g*(*a*) is the solution required as . Thus,(4.4)By setting is found to be(4.5)subsequently, *g*(*a*) is derived, namely,(4.6)this agrees with the leading order solution of (equation (3.16)), which is derived using the singular perturbation method in §3.

We now compare the above *g*(*a*) obtained via the energy release rate argument with the result obtained from the integral equation. In figure 2, the function *g*(*a*), evaluated from the integral equation, is normalized by *g*_{∞} and plotted against *h* with(4.7)which is given by *g*(*a*) in the limit . We consider two cases and with Poisson's ratio taken to be identical in both regions (constant). In both cases, we thus expect in the limit , as illustrated in figure 2, while in the limit , using *g*(*a*) from the energy release rate argument (equation (4.4)) gives(4.8)which agrees with the asymptotic result shown in figure 3. Note that if but if , and hence we expect the curve to rise as *h* increases for and to drop when .

The distribution of the function *g*(*u*) along the radius of the contact region is shown in figure 4, again, for and . In each case, *g*(*u*) is evaluated for different values of *h*, namely, while Poisson's ratio is taken to be identical for both regions; approximately as since the values of *h* chosen here are small (*h*<1). Note that *a*=1 is the radius of the contact region and the indentation is assumed in all figures.

## 5. Force versus displacement

To evaluate the total force *P* exerted by the punch on the elastic composite, we require(5.1)With the formulation of §2,(5.2)thus(5.3)We can express this as(5.4)where *δ* is such that . This means that and as but *δ* is otherwise arbitrary; one could choose for example (compare Atkinson & Leppington (1983) for other examples where this idea is used). The outer approximation is used in the first integral and the inner approximation in the second.

Using the outer approximation from equation (3.33), we have(5.5)To evaluate , we rewrite this in the inner coordinates, so(5.6)and we have the expansion as . With this in mind, it is convenient to write(5.7)for *X*<0, where(5.8)and is understood to be non-zero at to be consistent with equation (3.21). It follows that(5.9)Writing(5.10)then(5.11)Taking the limit of as and using the result of equation (3.20) together with the asymptotic results (see appendix B)(5.12)and(5.13)gives(5.14)Thus, substituting in equation (5.9), we get(5.15)When we combine this with equation (5.5), we get(5.16)and . When one views equation (3.33) correct to order *h* (neglecting the term) and equation (3.36), equation (5.16) gives(5.17)Significantly, the result of equation (5.17) is independent of *δ* assuming only that .

## 6. Conclusion

In this paper, a problem of contact mechanics is considered where a perfectly rigid indenter with a prescribed shape is pressed normally against a semi-infinite elastic composite that consists of a contact layer with uniform thickness welded together with another dissimilar medium. In particular, a circular indentation of uniform depth is considered. An approach similar to that of a penny-shaped crack is taken and the indentation is modelled by means of a real continuous function *g*(*t*) leading to an integral representation for the displacement in terms of *g*(*t*) on the contact boundary. Applying the mixed boundary condition of uniform displacement in the contact region and zero stress outside, the integral equation is evaluated numerically for *g*(*t*) when the contact layer is finite. In the case when the width of the contact layer becomes small compared with the radius of the contact region, the singular perturbation technique is used to derive an asymptotic expansion for *g*(*t*). This leads to a Wiener–Hopf formulation for the semi-infinite geometry in the Fourier-transformed domain of the inner coordinates. An asymptotic expansion for *g*(*t*) in the inner coordinates is derived using Liouville's theorem from the Wiener–Hopf equation with some unknown constants that arise as a consequence of the singular perturbation technique. This missing boundary condition is supplied by Van Dyke's matching principle that matches the inner with the outer solution. The results are illustrated and the figures show good matching between the outer and inner solutions. The results in the limiting cases are also verified with a simple limit solution derived using the integral argument. In deriving the asymptotic formula for *g*(*u*) near *u*=*a* when *h* is small, we have employed the techniques that we previously developed when evaluating the stress intensity factor for the crack problems (Atkinson & Chen 1997; Chen & Atkinson 2005). To demonstrate the applicability of the techniques for dealing with the contact problems here, axisymmetry is assumed for simplicity. The general equations derived, however, are valid for the three-dimensional bimaterial structures and can thus be applied to solve other similar cases such as those with different shapes of indenters.

## Acknowledgments

The first author would like to thank the National Science Council of Taiwan for their financial support.

## Footnotes

- Received November 2, 2007.
- Accepted January 29, 2008.

- © 2008 The Royal Society