Semi-infinite Elastic Solids Research Articles

On the adhesion between thin sheets and randomly rough surfaces

Thin, elastic sheets are well known to adapt to rough counterfaces, whereby adhesive interactions and pull-off stresses σp can be significant, yet no generally applicable, quantitative guideline has been suggested hitherto as to when a sheet should be considered thin enough to be sticky. Using computer simulations, we find that the dependence of σp on surface energy γ has a high and a low-pull-off-stress regime. For randomly rough surfaces, we locate the dividing line at the point, where γ is approximately half the elastic energy per unit area needed to make conformal contact, which is the same ratio as for semi-infinite elastic solids. This rule of thumb also applies to a certain degree for single-wavelength roughness, in which case the transition from low to high stickiness occurs when at the moment of maximum tension contact is not only broken at the height maxima but also at the saddle points.

SH surface wave in multicracked solids

We focus on the dispersion spectra of surface coherent waves in damaged semi-infinite elastic solids. The damage region is located within the surface layer containing depth-depending concentration of parallel closed cracks. The dispersion equation is similar to that for the Love waves in a coated substrate, except that the layer's effective elastic properties, being deduced from multiple scattering theory, are frequency-dependent and complex. For healthy elastic solids, the grazing SH wave is nondispersive. The dispersion under study is induced by the presence of cracks. The effect of both the magnitude of crack-concentration and its profile into the depth is investigated. The reliability of the semianalytical predictions is also discussed through a comparison with the numerical results obtained by using a finite-difference code.

Theory of images in plane-layered axisymmetric elastostatics

A fundamental theorem which automatically transforms the solution due to the presence of an arbitrary axisymmetric singularity in an unbounded homogeneous isotropic elastic solid into the corresponding solution for two perfectly bonded isotropic semi-infinite elastic solids is systematically applied in a stepwise fashion to obtain the complete image system when an arbitrary axisymmetric singularity is operative in or near a thick elastic layer which separates two other dissimilar isotropic semi-infinite elastic solids. The solution is closed and well structured. As an illustration, the distant effect in the interface layer produced by an influencing normal point force is luminously revealed to be two-dimensional, consisting of a combination at the origin of a bending hot spot and an infinite line of centres of dilatation. We conclude the paper with a complete theory of images for a free elastic layer under the influence of both axisymmetric and asymmetric singularities. We find that if the influencing displacement field is the gradient of a harmonic function, then the calculation of the induced elastic field reduces to the operation of differentiation only.

Practical Green’s function approach to the simulation of elastic semi-infinite solids

This paper is concerned with the principles of Green's function-based molecular dynamics (GFMD) simulations of semi-infinite elastic solids and their application to various contact mechanical problems. A methodology to compute the (renormalized) elastic interactions of surface atoms is presented first. It is based on the fluctuation-dissipation theorem, with the help of which thermal fluctuations of atomic displacements can be related to the elastic Green's functions and thus to the effective coupling between surface atoms. We suggest a sparse representation of these renormalized spring constants and present numerical results for some simple two- and three-dimensional lattices. The renormalized elastic interactions can be obtained for relatively small systems and then be extrapolated to large systems. They incorporate the full elastic response of semi-infinite solids in a way that only surface atoms have to be considered in molecular dynamics simulations. The usefulness of GFMD is demonstrated by applying it to various idealized contact models, such as nonadhesive Hertzian contacts as well as nonadhesive contacts between flat, semi-infinite elastic solids and a self-affine, rigid substrate. In all cases, a zero probability density $P(p)$ for infinitesimally small contact pressures $p$ is found, as predicted theoretically. If the self-affine, nonadhesive surfaces are under such high loads that the contact is complete, the pressure histogram can be represented by a Gaussian also in accordance with theoretical predictions. However, if the topography of the substrate resembles that of industrial steel surfaces and the loads are moderate, $P(p)$ decays exponentially for medium and large $p$ in contradiction to theoretical predictions for randomly rough surfaces.

A two-dimensional theory for the analysis of surface acoustic waves in finite elastic solids

The analysis of surface acoustic waves in elastic solids started from a semi-infinite isotropic elastic body with solutions and techniques dated back to Lord Rayleigh. These solutions have explained the surface acoustic wave phenomena and guided its engineering applications in many fields. Research work followed have been using the same semi-infinite model with solution techniques for approximate and exact results in both analytical and numerical manners for many application problems involving finite elastic solids. On the other hand, we have noticed that various two-dimensional theories, notably plate theory by Mindlin, have been derived to study the bulk wave propagation in finite elastic solids like plates and bars with satisfactory results. The lack of such a two-dimensional theory has made the analysis of surface acoustic wave propagation in various waveguides primitive and difficult because the precise solutions cannot be obtained through any analytical effort and numerical solutions are also difficult to obtain because of the complicated nature of problems. To meet the need of a simplified analytical method for surface acoustic waves in finite elastic solids, we start the derivation from the well-known three-dimensional solutions of semi-infinite elastic solids. Using exact solutions as the basis for the two-dimensional expansion, we found that the usual procedure of dimension reduction works perfectly in this case because the unknown wavenumber is eventually removed in the expansion process, and the depth of the solid is considered through integration constants that are exponentially decaying functions of the wavenumber, which is to be specified. In addition, we also found the two-dimensional equations give the exact phase velocity without corrections of any kind in the limiting case in which the solid is semi-infinite. With these equations, two-dimensional solutions of mechanical displacements and phase velocity can be obtained analytically from four coupled differential equations that are similar to Mindlin plate equations. We demonstrated the applications of the two-dimensional theory with the analysis of surface waves in isotropic elastic rectangular strips. The velocity and displacement results offer a chance to understand surface acoustic waves in finite solids and the presence of overtone modes and transition zones.

Development of 3‐D finite element model for flexible pavements

This paper presents the development of a three‐dimensional finite element model for flexible pavements. The procedures of building a model and performing static analysis with the ABAQUS package are introduced. The element shapes and the size of the finite element model were studied to improve analysis efficiency. The model should at least consist of finite elements up to three times the loading diameter if centered at the loading position. Infinite elements should be used beyond the boundary of finite elements. The accuracy was justified with Boussnisq solutions for semiinfinite elastic solids. The surface layer was modeled with creep test data of asphalt concrete. Viscoelastic behavior was observed for the pavement model under wheel loading. The model was also validated by comparing computed results and field test data. Further calibrations of the model are underway by comparison with laboratory test results. The application of the model is introduced with an example of placing a geogrid‐reinforced asphalt concrete overlay on rigid pavement. Significant reduction of tensile strains at the bottom of the asphalt concrete was found with geogrid reinforcement.

Physics of lubricated impact of a sphere on a plate in a narrow continuum to gaps of molecular dimensions

This paper investigates the phenomenon of lubricated impact dynamics of ellipsoidal bodies upon semi-infinite elastic solids, giving rise to Hertzian contact conditions. The analysis conforms to the numerical predictions and experimental findings of others, when the physics of motion of the lubricant can be described through Newtonian continuum mechanics, with the dominant viscous action embodied in the transient solution of Reynolds' equation. The equivalence of squeeze film action under impacting conditions with that of a converging gap in pure entraining motion is shown. This concept is extended to study the accelerative nature of the lubricant film surface, and its concordance with Reynolds' assumption through use of a relativistic frame of reference and hyperbolic geometry.When the investigation is extended to the case of ultra-thin film conjunctions of the order of a few to several molecular diameters of the intervening fluid layer, the physics of fluid film motion through impact involves more complex kinetic interactions. These include the effect of structural force of solvation, as well as that of a meniscus force, formed in such narrow conjunctions. The former, through active dispersion, tends to promote a structureless environment, whilst the latter through wetting action encourages the formation of a coherent film. This paper shows the interplay between these competing kinetics.

Elastic fields due to centers of dilatation and thermal inhomogeneities in plane-layered solids

A n image method for obtaining the solution for a center of dilatation in a three-layer elastic solid with planar interfaces is presented. The three-layered elastic solid consists of an elastic slab sandwiched between two semi-infinite elastic solids. The three elastic solids are perfectly bonded together at the two planar interfaces. The solution is given in terms of Galerkin vectors which are in terms of an infinite series of the Newtonian potential function of a mass point at the center of dilatation, its mirror images and their derivatives. As an application, the solution for the center of dilatation is used to obtain the elastic solution due to thermal inhomogeneities. The thermoelastic solution is obtained by a method which is based on the integration of properly weighted centers of dilatation over the volume occupied by the inhomogeneity. The potential functions for the problem solved are the harmonic potential functions of attracting matter filling the volume of the thermal inhomogeneity and its mirror images. The solution for the thermal elastic stresses due to an expanding (or contracting) thermal inhomogeneity (inclusion) of any shape embedded in one of the solids is given as an example. Numerical results for a spherical inclusion with pure dilatation eigenstrain are also presented and discussed.

Effect of bearing deformation on the characteristics of squeeze films between circular and rectangular plates

The effect of bearing deformation on squeeze films between two circular plates and between two infinitely long rectangular plates is analysed. The lower plates which bear the load are assumed to be semi-infinite elastic solids. Under high loads the bearing surfaces deform producing a wedge effect in the lubricant film and this deformation increases the squeeze.

Reflection, Refraction, and Absorption of Elastic Waves at a Frictional Interface: SH Motion

The reflection, refraction, and absorption of obliquely incident plane harmonic antiplane strain (SH) waves at a frictional interface between dissimilar semi-infinite elastic solids is investigated by an approximate analytical approach. The frictional stress at the interface is assumed to depend on the normal stress and the relative slip across the interface, but remains otherwise arbitrary throughout the analysis. General expressions are developed for the transmission and reflection coefficients, and the partitition of incident wave energy into reflection, transmission, and absorption. The special case of bonding by Coulomb friction is examined in detail as an example of application of the general procedure. An exact solution is also presented for the case of bonding by Coulomb friction, and a comparison between approximate and exact solutions provides an indication of the accuracy of the approximate method of analysis.

Displacement Discontinuity on the Interface of Two Joined Dissimilar Semi-Infinite Elastic Solids

Abstract The Papkovitch functions are given for a displacement discontinuity over a portion of the interface of two dissimilar isotropic semi-infinite elastic solids joined across a plane.

Thermal Stresses in Transversely Isotropic Semi-Infinite Elastic Solids

Abstract A general method of solution of the steady-state thermal-stress problem of a transversely isotropic semi-infinite elastic solid is given in this paper.

Force in the Plane of Two Joined Semi-Infinite Plates

Abstract The stress due to a force parallel to the plane of two joined semi-infinite plates is obtained from the three-dimensional state of stress produced by a force operative at a point interior to one of two joined semi-infinite elastic solids. The variation in the traction on the interface of a steel and aluminum plate is shown graphically.