Special Issue Paper Received 18 August 2015 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.3782 MOS subject classification: 74M15; 74G10 An effective asymptotic method in the axisymmetric frictionless contact problem for an elastic layer of finite thickness I. I. Argatov * † Communicated by E. Sanchez-Palencia A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth-order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd. Keywords: contact problem; asymptotic method; elastic layer; integral characteristics 1. Introduction Contact problems of the linear theory of elasticity for an elastic layer loaded by a rigid indenter have been examined for more than a half century, and asymptotic methods played an important role in this perspective. Such problems contain a dimensionless parameter " D a=h, where h is the layer thickness and a is a characteristic size of the contact area, so that their solutions can be sought for in the form of an asymptotic series in ascending powers of the small parameter ". Note that the case of large values of " (that is the case of a relatively thin elastic layer) requires another type of asymptotic Ansatz (see, in particular, [1–7]). Asymptotic solutions of the linear contact problem with a fixed circular contact area have been constracted in a number of papers [8–11] under the assumption that the relative thickness of the elastic layer is sufficiently large, that is, for large values of the ratio  D h=a. In the non-axisymmetric case of an elliptical indenter, the ‘method of large ’ was applied by Aleksandrov and Vorovich [12] and later was further developed by Aleksandrov [13–15]. The so-called large  method was used in a number of papers to solve two-dimensional [14, 16–18] and three-dimensional [19–22] contact problems. It should be noted that usually, an asymptotic method allows to construct only a small number of terms in the asymptotic series. For certain types of integral equations, the method of large  was generalized by Chebakov with co-workers [23–25] for constructing complete asymptotic expansions. The case of unilateral contact problem (with a priori unknown contact area) was first considered by England [26] and later was studied in detail by Vorovich et al. [10] using the large  method. An extended analysis of the contact problem for an arbitrary axisymmetric indenter pressed against a transversely isotropic elastic layer of finite thickness was recently performed by Argatov and Sabina [27]. The constructed asymptotic solutions allowed to investigate the thickness effect [28,29] and the substrate effect [30] in the indentation testing for elastic layers of finite thickness. Yet, another asymptotic approach to the contact problems for an elastic layer of finite thickness was developed by Argatov both in the linear case (with a fixed contact area) [31] and in the nonlinear case (with a variable contact area) [32–35] in the framework of the method of matched asymptotic expansions [36–38]. In the present paper, the method of Argatov and Sabina [27] has been further enhanced by highlighting the role of integral characteristics (so-called polymoments [31]) of the contact pressure in constructing the asymptotic approximations in the case of relatively small contact radius. As it was first emphasized by Sabina [39], the leading asymptotic terms in Rayleigh’s solution [40] to the problem of the scattering of scalar waves by a soft obstacle (which was given in terms of the harmonic capacity) is, in fact, expressed Institut für Mechanik, Technische Universität Berlin, 10623 Berlin, Germany * Correspondence to: I. I. Argatov, Institut für Mechanik, Technische Universität Berlin, 10623 Berlin, Germany. † E-mail: ivan.argatov@campus.tu-berlin.de Copyright © 2015 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2015