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In this paper, we study the propagation of Rayleigh waves in a nonlocal orthotropic elastic half-space subject to impedance boundary conditions using the weakly nonlocal elasticity model which is introduced recently by the first two authors of the paper. The half-space may be compressible or incompressible. Our main aim is to derive explicit secular equations of Rayleigh waves and based on them examine the effect of nonlocality, impedance boundary conditions and incompressibility on the Rayleigh wave characteristics, such as the velocity and the displacement components at the half-space surface. The secular equation for the compressible half-space is derived using the surface impedance matrix method. The secular equation for the incompressible half-space is then obtained by employing the incompressible limit method. It is shown numerically that: (i) While the Rayleigh wave velocity decreases when increasing the nonlocal and impedance parameters, the incompressibility makes it increasing. (ii) The nonlocality affects very strongly the displacement amplitudes at the half-space’s surface, the incompressibility and the impedance boundary conditions affect them considerably.

A coupled contact problem involving the interaction of a punch with a half-space under a follower load has been considered. The punch is pressed against the elastic half-space by a force that acts parallel to the punch axis and is offset from the center of the punch base. The direction of the force remains unchanged relative to the punch as the force increases, thus defining it as a follower force.Amathematical model has been formulated, and an analytical solution has been obtained. The specific motion behavior of the punch under increasing force has been analyzed, including the determination of the maximum angle of rotation. The variation in the horizontal motion of the punch depending on the applied force has been studied. Additionally, having considered a modification of the problem where the force direction is reversed, significant differences in the punch displacement have been found.

We report on the resolution of the vibration problem for a homogeneous and isotropic elastic half-space (the Lamb problem), with application to the seismic tensorial force. We assume a homogeneous and isotropic half-space with a localized force which produces vibrations. The solution is achieved by introducing vector plane-wave functions. Explicit results are given for an isotropic tensorial force and a half-space with free surface. The contribution of the Rayleigh surface waves to vibrations is analyzed in the special case of a temporal-impulse force, where the solution exhibits unphysical features, as expected: it extends over the entire free surface and time domain, with a (scissor-like) double-wall propagating both in the future and the past.

The indentation of a periodic system of axisymmetric indenters into an elastic half-space is analytically investigated taking into account the mutual influence of the contact spots. The system consists of two levels of indenters with a given height difference and is loaded by the given nominal pressure. For definiteness, it is assumed that at each level the indenters are located at the nodes of the quadratic lattice. Using the localization principle, a system of equations is derived to determine the loads applied to the indenters of each level and the real contact area. The influence of the height difference of the indenters and the distance between them on the contact characteristics is analyzed. The dependence of the nominal pressure providing a two-level contact on the geometric parameters of the system is also analyzed. The effect of the mutual influence of the indenters on contact characteristics is investigated by comparison with the results following from the Hertz theory. It is concluded that neglecting the mutual influence of the contact spots can lead to significant errors in determining the characteristics of the contact interaction particular for the high values of the nominal pressure.

Machines operate under increasingly harsher contact conditions, causing significant wear and contact fatigue. Sub-surface stresses are responsible for the premature contact fatigue of rolling element bearings, meshing gears, and cam–follower pairs. Surface protection measures include hard, wear-resistant coatings. Traditionally, contact integrity has been predicted using classical Hertzian contact mechanics. However, the theory is only applicable when the contact between a pair of ellipsoidal solids of revolution may be considered as a rigid indenter penetrating a semi-infinite elastic half-space. Many coatings act as thin bonded elastic layers that undergo considerably higher pressures than those predicted by the classical theory. Furthermore, inelastic deformation of bonded solids can cause plastic flow, work-hardening, and elastoplastic behaviour. This paper presents a comprehensive, integrated contact mechanics analysis that includes induced sub-surface stresses in concentrated counterformal finite line contacts for all the aforementioned cases. Generated pressures and deformation are predicted for hard coated surfaces, for which there is a dearth of relevant analysis. The contact characteristics, which are of particular practical significance, of many hard, wear-resistant advanced coatings are also studied. The paper clearly demonstrates the importance of using efficient semi-analytical, detailed holistic contact mechanics rather than the classical idealised methods or empirical numerical ones such as FEA. The novel approach presented for the finite line contact of thin-layered bonded solids has not hitherto been reported in the open literature.

This paper examines the generic case of nonlinear mechanical oscillation under the influence of Hertzian-type restoring forces, a model relevant to phenomena involving elastic contact. The study addresses the complexity of strongly nonlinear systems by focusing on the differential equation governing the oscillation of a rigid sphere interacting with an elastic half-space, which includes a full series expansion to account for large deformations. Since no closed-form solution exists for the amplitude-dependent oscillation period, a new approximate analytical approach is introduced. This method preserves the system’s dominant Hertzian scaling while incorporating higher-order corrections through an averaged factor. For amplitudes where the deformation is less than or equal to the sphere’s radius, this approximation is nearly identical to the numerical solution. For larger amplitudes, the accuracy is further enhanced by introducing a semi-empirical linear adjustment to the relative error. This framework provides a reliable analytical description of the system’s behavior, offering a useful tool for theoretical studies and comparison with numerical results.

According to the basic solution of viscoelastic and saturated soil subjected to moving load on the surface and circular uniform harmonic load on the inside, the integral equation of soil-pile interaction under moving load in frequency domain is established by using Muki virtual pile method, and then the time-space dynamic response characteristics of a single pile in viscoelastic saturated soil to moving load vibration source are analyzed by using inverse Fourier transform. The factors that affect the dynamic response of single pile, such as load speed, length of single pile, saturated soil parameters, vibration source properties, etc., are analyzed and compared with the known results in related literature.

— Using generalized functions, Green’s functions for homogeneous elastic isotropic half-planes and half-spaces are constructed. Airy and Maxwell stress functions are used to find Green’s functions. One-dimensional and two-dimensional integral Fourier transforms are used to solve the boundary value problems. Taking into account the properties of generalized functions with a point support, singular components of displacement images are distinguished. It is shown that they correspond to the rigid-body displacement. If there are no singular components, then the stresses and di-splacements coincide with the known classical solutions of the Flamant, Boussinesq, and Cerutti problems.

The subject of this paper is an analysis of the influence of circumferential prestressing on the interaction of cylindrical silos and tanks with the subsoil. The behaviour of the shell structures of RC and PC cylindrical silos or tanks (with circumferential pre-tensioning), and particularly of the ground slab interacting with subsoil, depends largely on the function graphs of the subsoil reactions on the foundation surface. Distributions of the subbase reactions on the ground slab in such structures as silos and tanks have a significant impact on the behaviour of not only the slab itself, but also the interacting shell structure. An analysis of these structures with walls fixed in a circular ground slab and foundation ring was carried out taking into consideration the elastic half-space model using the Gorbunov-Posadov approach and the two-parameter Winkler model. In the computational examples of RC and PC silos and tanks with walls fixed in the circular ground slab or foundation ring, the eventual effects of prestressing obtained as a result of the superposition of internal forces were examined. Although the results for both subsoil models proved to be divergent, the conclusions that follow are fairly important for the engineering practice.

In elastic wave systems, combining the powerful concepts of resonance and spatial grading within structured surface arrays enable resonant metasurfaces to exhibit broadband wave trapping, mode conversion from surface (Rayleigh) waves to bulk (shear) waves, and spatial frequency selection. Devices built around these concepts allow for precise control of surface waves, often with structures that are subwavelength, and utilise Rainbow trapping that separates the signal spatially by frequency. Rainbow trapping yields large amplifications of displacement at the resonator positions where each frequency component accumulates. We investigate whether this amplification, and the associated control, can be used to create energy harvesting devices; the potential advantages and disadvantages of using graded resonant devices as energy harvesters is considered. We concentrate upon elastic plate models for which the A0 mode dominates, and take advantage of the large displacement amplitudes in graded resonant arrays of rods, to design innovative metasurfaces that trap waves for enhanced piezoelectric energy harvesting. Numerical simulation allows us to identify the advantages of such graded metasurface devices and quantify its efficiency, we also develop accurate models of the phenomena and extend our analysis to that of an elastic half-space and Rayleigh surface waves.