Contact problem for wavy surfaces in the presence of an incompressible liquid and a gas in interface gaps

This paper presents a study on smooth elastic contact between two semi-infinite elastic bodies, one of which has a wavy surface, for the case when there are an incompressible liquid, not wetting the surfaces of the bodies, at the central region of each interface gap and a gas under constant pressure at the edges of each gap. Due to the surface tension of the liquid, a pressure drop occurs in the liquid and the gas, which is described by the Laplace formula. The formulated contact problem is reduced to a singular integral equation (SIE) with the Hilbert kernel, which is transformed into a SIE with the Cauchy kernel for a derivative of a height of the gaps. A system of transcendental equations for a width of each gap and a width of the gap region filled with the liquid is obtained from the condition of boundedness of the contact stresses at the gap ends and the condition of liquid amount conservation. It is solved numerically, and the dependences of the width and shape of the gaps, the width of the gap regions filled with the liquid and the contact approach of the bodies on the applied load and the surface tension of the liquid are analyzed.

Contact interaction between bodies with wavy relief taking into account interstitial compressible liquid

The non-frictional contact between two semi-infinite elastic bodies, one of which has a wavy surface, is considered for the case of interface gaps filled with a compressible barotropic liquid. The contact problem formulated is reduced to a singular integral equation (SIE) with the Hilbert kernel, which is transformed into a SIE with the Cauchy kernel for a derivative of a height of the gaps. A system of transcendental equations for a width of the gaps and a pressure of the liquid is obtained from the condition of boundedness of the SIE solution at the integration interval ends and the equation of state of a compressible barotropic liquid, and then it is solved numerically. The dependences of the width and shape of the gaps, the pressure of the liquid, the average normal displacement and contact compliance of the bodies on the applied load and bulk modulus of the liquid are analyzed.

Singular Integro-Differential Equations with Hilbert Kernel and Monotone Nonlinearity

Методом максимальных монотонных операторов в вещественных пространствах Лебега доказываются теоремы о существовании и единственности решения для различных классов нелинейных сингулярных интегро-дифференциальных уравнений с ядром Гильберта. Приведены следствия, иллюстрирующие полученные результаты.

Effect of compressible liquid on the contact between an elastic body and a rigid base with a periodic array of quasielliptic grooves

The frictionless contact between an elastic body and a rigid base in the presence of a periodically arranged quasielliptic grooves with in interface gaps in the presence of a compressible liquid is modeled. The contact problem formulated for the elastic half-space is reduced to a singular integral equation (SIE) with Hilbert kernel for a derivative of a height of the interface gaps, which is transformed to a SIE with Cauchy kernel that is solved analytically, and a transcendental equation for liquid’s pressure, which has been obtained from the equation of compressible barotropic liquid state. The dependences of the pressure of the liquid, shape of the gaps, average normal displacement and contact compliance of the bodies on the applied load and bulk modulus of the liquid are analysed.