Z. Angew. Math. Phys. (2018) 69:80
2018 Springer International Publishing AG,
part of Springer Nature
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Justiﬁcation and reﬁnement of Winkler–Fuss hypothesis
J. Kaplunov , D. Prikazchikov and L. Sultanova
Abstract. Two-parametric asymptotic analysis of the equilibrium of an elastic half-space coated by a thin soft layer is
developed. The initial scaling is motivated by the exact solution of the plane problem for a vertical harmonic load. It is
established that the Winkler–Fuss hypothesis is valid only for a suﬃciently high contrast in the stiﬀnesses of the layer and
the half-space. As an alternative, a uniformly valid non-local approximation is proposed. Higher-order corrections to the
Winkler–Fuss formulation, such as the Pasternak model, are also studied.
Mathematics Subject Classiﬁcation. 74G10, 74K35.
Keywords. Soft thin coating, Asymptotic, Winkler–Fuss model, Contrast, Substrate.
The simplest and apparently the most popular structural model of an elastic foundation is based on
the Winkler–Fuss assumption treating a continuum as a system of vertical springs. As mentioned in
, Fuss came to this hypothesis studying a beam ﬂoating along the surface of an incompressible ﬂuid,
whereas Winkler  considered a foundation as a mattress consisting of non-connected elastic springs,
see also  for more detail. It is crucial that, according to the Winkler–Fuss hypothesis, the resulting
displacements occur only under the applied force, with the neighbouring area not aﬀected. For further
historical remarks on the Winkler–Fuss model, the reader is referred to . A detailed critical review
of the Winkler–Fuss, as well as other elastic foundation models, can be found in . Another useful
review of foundation models is presented also in , taking into account viscoelastic behaviour. One
of the focuses of the recent monograph  is on the asymptotic analysis of layered and inhomogeneous
foundations. In addition, we mention  as an example of modelling thin interlayers using Winkler–Fuss
The Winkler–Fuss approximation is widely used in civil engineering [9,14,21,25,28,30] and also in
contact mechanics, including, in particular, analysis and interpretation of the experimental results for a
depth-sensing indentation, see  and references therein. We specially mention its potential in nanoin-
dentation modelling . As a rule, the Winkler–Fuss model may be implemented provided that there
is a high contrast in the stiﬀnesses of a substrate and a coating, when the substrate is much harder. At
the same time, for a thin coating supported by an inﬁnite substrate, a limited amount of eﬀort has been
made to provide a mathematical justiﬁcation of this model, see, for example, , as well as [4,19,23].
These contributions appreciate the general challenge; however, they do not take into consideration a
two-parametric nature of the problem, involving geometric and material parameters, expressing relative
thickness and stiﬀness, respectively. The Winkler–Fuss formulation was only fully validated for a layered
foundation clamped at the bottom, although without a special emphasis on the contrast properties of the
layers, see .