Acta Mech 228, 3003–3017 (2017)
Gökhan Adıyaman · Erdal Öner · Ahmet Birinci
Continuous and discontinuous contact problem
of a functionally graded layer resting on a rigid foundation
Received: 17 October 2016 / Revised: 19 April 2017 / Published online: 19 May 2017
© Springer-Verlag Wien 2017
Abstract In this study, the continuous and discontinuous contact problem of a functionally graded (FG) layer
resting on a rigid foundation is considered. The top of the FG layer is subjected to normal tractions over a
ﬁnite segment. The graded layer is modeled as a non-homogenous medium with a constant Poissons’ ratio and
exponentially varying shear modules and density. For continuous contact, the problem was solved analytically
using plane elasticity and integral transform techniques. The critical load that causes ﬁrst separation and contact
pressures is investigated for various material properties and loadings. The problem reduced to a singular
integral equation using plane elasticity and integral transform techniques in case of discontinuous contact.
The obtained singular integral equation is solved numerically using Gauss–Jacobi integral formulation, and
an iterative scheme is employed to obtain the correct separation distance. The separation distance and contact
pressures between the FG layer and the foundation are analyzed for various material properties and loading.
The results are shown in Tables and Figures. It is seen that decreasing stiffness and density at the top of the
layer result in an increment in both critical load in case of continuous contact and separation distance in case
of discontinuous contact.
In problems in which body forces are neglected, after the application of the load, the contact area between the
layer and the substrate diminish to a ﬁnite size independent of the magnitude of the applied load. However, in
reality, it is expected that the separation depends on the applied load and the layer will remain in contact with
the substrate at some point because of gravity.
Civelek and Erdogan  studied the frictionless contact problem of an elastic layer under gravity. Civelek et
al. studied the interface separation for an elastic layer loaded by a rigid stamp. Continuous and discontinuous
contact problems for strips on an elastic semi-inﬁnite plane were investigated by Cakiroglu and Cakiroglu .
Birinci and Erdol  studied a frictionless contact problem for two elastic layers with vertical body forces
supported by a Winkler foundation. Oner and Birinci  studied the continuous contact problem for two elastic
layers resting on an elastic half-inﬁnite plane. An analysis of continuous and discontinuous cases of a contact
problem using analytical method and FEM was performed by Birinci et al .
Parallel to developments in sciences, new materials such as functionally graded materials (FGMs) in which
material properties vary smoothly along a spatial direction are developed as an alternative to homogeneous
G. Adıyaman (
) · A. Birinci
Department of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey
Department of Civil Engineering, Bayburt University, 69000 Bayburt, Turkey