Axisymmetric circular indentation of a half-space reinforced by a buried elastic thin film

The analytical treatment of an axisymmetric rigid punch indentation of an isotropic half-space reinforced by a buried extensible thin film is addressed. With the aid of appropriate displacement potential functions, Hankel transforms, and some mathematical techniques, the mixed boundary value problem under consideration is reduced to a Fredholm integral equation of the second kind. The most interesting results of the problem, including the equivalent normal stiffness of the system and the contact stress distribution beneath the rigid punch, are expressed in terms of the solution of the obtained Fredholm integral equation. Some limiting cases corresponding to inextensible and extremely extensible thin films, a surface-stiffened half-space, axisymmetric surface loading, and infinite embedment are studied and the available results in the literature are used for verification purposes. Some dimensionless plots are provided to show the effects of extensibility of the membrane on the system stiffness. It is observed that neglecting the elastic behavior of thin films and utilizing the simplified inextensible assumption for their modeling can lead to unrealistic predictions. It is shown that the axisymmetric problems concerning the elastic half-spaces including surface effects can be equivalently modeled by a surface-reinforced elastic medium. Taking the surface effects into account, the problem of an isotropic half-space under the action of a surface patch load is treated by employing the present approach.