Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary

An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

Cooperative Transmission in Uplink NOMA Networks with Wireless Power Transfer

5G networks and wireless power transfer are the topics that have attracted both academic and industry communities in recent years. In this paper, we study the cooperative transmission of uplink non-orthogonal multiple access (NOMA) network with wireless power transfer in terms of performance analysis. Specifically, energy-constrained amplifyand-forward relay cooperates with two users that applying NOMA scheme to transmit the message to base station by using the energy harvested from base station. For performance analysis, we derive the closed-form expressions of outage probability and throughput for two users based on the statistical characteristics of signal-to-noise ratio (SNR) and signalto-interference-plus-noise ratio (SINR) by using the Gaussian-Chebyshev quadrature method. To understand more detail of the behaviour of this considered system, the numerical results are provided according to the system key parameters, e.g., transmit power, distances. Furthermore, the theoretical results are also verified by the Monte-Carlo simulation.

Existence, Uniqueness, and Approximation for Solutions of a Functional-integral Equation in Lp Spaces

In this work we consider the general functional-integral equation: \begin{equation*}y(t) = f\left(t, \int_{a}^{b} k(t,s)g(s,y(s))ds\right), \qquad t\in [a,b],\end{equation*}and give conditions that guarantee existence and uniqueness of solution in $L^p([a,b])$, with {$1<p<\infty$}.We use Banach Fixed Point Theorem and employ the successive approximation method and Chebyshev quadrature for approximating the values of integrals. Finally, to illustrate the results of this work, we provide some numerical examples.

Micro/Nanocontact Between a Rigid Ellipsoid and an Elastic Substrate With Surface Tension

For micro/nanosized contact problems, the influence of surface tension becomes prominent. Based on the solution of a point force acting on an elastic half space with surface tension, we formulate the contact between a rigid ellipsoid and an elastic substrate. The corresponding singular integral equation is solved numerically by using the Gauss–Chebyshev quadrature formula. When the size of contact region is comparable with the elastocapillary length, surface tension significantly alters the distribution of contact pressure and decreases the contact area and indent depth, compared to the classical Hertzian prediction. We generalize the explicit expression of the equivalent contact radius, the indent depth, and the eccentricity of contact ellipse with respect to the external load, which provides the fundament for analyzing nanoindentation tests and contact of rough surfaces.