Semi-infinite Elastic Half-space Research Articles

Elastic and Elastoplastic Contact Mechanics of Concentrated Coated Contacts

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.

Analytical Elastostatic Contact Mechanics of Highly-Loaded Contacts of Varying Conformity

In applications requiring high load carrying capacity, conforming contacting pairs with a relatively large contact footprint are used. These include circular arc, Novikov, and Wildhaber gears found, for example, in helicopter rotors. Closely conforming contacts also occur in many natural endo-articular joints, such as hips, or their replacement arthroplasty. The main determining factors in contact fatigue are the sub-surface shear stresses. For highly loaded contacts, classical Hertzian contact mechanics is used for many gears, bearings, and joints. However, the theory is essentially for concentrated counterforming contacts, where the problem is reduced to a rigid ellipsoidal solid penetrating an equivalent semi-infinite elastic half-space. Applicability is limited though, and the theory is often used inappropriately for contacts of varying degrees of conformity. This paper presents a generic contact mechanics approach for the determination of sub-surface stresses, which is applicable to both highly conforming as well as concentrated counterforming contacts. It is shown that sub-surface shear stresses alter in magnitude and disposition according to contact conformity, and lead to the different modes of fatigue failure noted in practice.

Random vibration of train-track-ground system with a poroelastic interlayer in the subsoil

A semi-analytical train-track-ground model considering rail irregularity is developed to study random vibration of the system. The ground is modeled as a layered half-space comprising two elastic layers, a poroelastic interlayer and a semi-infinite elastic half-space from top to bottom. Solution to the ground is derived adopting ‘adapted stiffness matrix method’. The ground is coupled with the track according to continuity conditions at the ballast-ground interface. The track-ground system is coupled with the train adopting Hertzian contact spring model. Pseudo-excitation method (PEM) is introduced to solve the random vibration induced by rail irregularity. The method is compared with existing research. Dynamic responses of the system such as amplitude, power spectral density (PSD) and standard deviation (SD) of displacement, velocity and acceleration are investigated. The existence of groundwater significantly influences vertical displacement for locations far from track center and long wavelength of rail irregularity, while it mainly affects the vertical velocity and acceleration for locations far from track center but short wavelength of rail irregularity. The maximum SD of vertical displacement increases with increasing speed, and no ‘critical speed’ exists for it when 0 m/s <c<200 m/s, which is different from the variation of vertical displacement with speed due to axle loads. High frequency components of the PSD of maximum vertical displacement, velocity and acceleration attenuate faster from the rail to the ground and with the distance from the track center on ground surface.

Propagation of ultrasonic Love waves in nonhomogeneous elastic functionally graded materials

This paper presents a theoretical study of the propagation behavior of ultrasonic Love waves in nonhomogeneous functionally graded elastic materials, which is a vital problem in the mechanics of solids. The elastic properties (shear modulus) of a semi-infinite elastic half-space vary monotonically with the depth (distance from the surface of the material). The Direct Sturm–Liouville Problem that describes the propagation of Love waves in nonhomogeneous elastic functionally graded materials is formulated and solved by using two methods: i.e., (1) Finite Difference Method, and (2) Haskell-Thompson Transfer Matrix Method.The dispersion curves of phase and group velocity of surface Love waves in inhomogeneous elastic graded materials are evaluated. The integral formula for the group velocity of Love waves in nonhomogeneous elastic graded materials has been established. The effect of elastic non-homogeneities on the dispersion curves of Love waves is discussed. Two Love wave waveguide structures are analyzed: (1) a nonhomogeneous elastic surface layer deposited on a homogeneous elastic substrate, and (2) a semi-infinite nonhomogeneous elastic half-space. Obtained in this work, the phase and group velocity dispersion curves of Love waves propagating in the considered nonhomogeneous elastic waveguides have not previously been reported in the scientific literature. The results of this paper may give a deeper insight into the nature of Love waves propagation in elastic nonhomogeneous functionally graded materials, and can provide theoretical guidance for the design and optimization of Love wave based devices.

Group and Phase Velocity of Love Waves Propagating in Elastic Functionally Graded Materials

AbstractThis paper presents a theoretical study of the propagation behaviour of surface Love waves in nonhomogeneous functionally graded elastic materials, which is a vital problem in acoustics. The elastic properties (shear modulus) of a semi-infinite elastic half-space vary monotonically with the depth (distance from the surface of the material). Two Love wave waveguide structures are analyzed: 1) a nonhomogeneous elastic surface layer deposited on a homogeneous elastic substrate, and 2) a semi-infinite nonhomogeneous elastic half-space. The Direct Sturm-Liouville Problem that describes the propagation of Love waves in nonhomogeneous elastic functionally graded materials is formulated and solved 1) analytically in the case of the step profile, exponential profile and 1cosh2type profile, and 2) numerically in the case of the power type profiles (i.e. linear and quadratic), by using two numerical methods: i.e. a) Finite Difference Method, and b) Haskell-Thompson Transfer Matrix Method.The dispersion curves of phase and group velocity of surface Love waves in inhomogeneous elastic graded materials are evaluated. The integral formula for the group velocity of Love waves in nonhomogeneous elastic graded materials has been established. The results obtained in this paper can give a deeper insight into the nature of Love waves propagation in elastic nonhomogeneous functionally graded materials.

Buckling of Piles under Liquefied Soil Conditions

The paper pertains to the analysis of piles embedded in liquefiable soils to predict its’ critical buckling load under partial to full loss of lateral support over a portion of the pile length. The analysis is based on extension of Mindlin solution for a point load acting inside a semi infinite elastic half space. Degenerated solutions obtained by using the developed method compares very well with reported results. Parametric studies showed that the depth of liquefiable soil, degradation of soil strength on liquefaction, slenderness ratio, pile stiffness factor and end conditions have significant influence on the buckling behavior of the piles.

Elastic PMLs for resonator anchor loss simulation

AbstractElectromechanical resonators and filters, such as quartz, ceramic, and surface‐acoustic wave devices, are important signal‐processing elements in communication systems. Over the past decade, there has been substantial progress in developing new types of miniaturized electromechanical resonators using microfabrication processes. For these micro‐resonators to be viable they must have high and predictable quality factors (Q). Depending on scale and geometry, the energy losses that lower Q may come from material damping, thermoelastic damping, air damping, or radiation of elastic waves from an anchor. Of these factors, anchor losses are the least understood because such losses are due to a complex radiation phenomena in a semi‐infinite elastic half‐space. Here, we describe how anchor losses can be accurately computed using an absorbing boundary based on a perfectly matched layer (PML) which absorbs incoming waves over a wide frequency range for any non‐zero angle of incidence. We exploit the interpretation of the PML as a complex‐valued change of co‐ordinates to illustrate how one can come to a simpler finite element implementation than was given in its original presentations. We also examine the convergence and accuracy of the method, and give guidelines for how to choose the parameters effectively. As an example application, we compute the anchor loss in a micro disk resonator and compare it to experimental data. Our analysis illustrates a surprising mode‐mixing phenomenon which can substantially affect the quality of resonance. Copyright © 2005 John Wiley &amp; Sons, Ltd.

Reconstruction of Layered Elastic Bottom Characteristics by the Frequency Dependence of Sound Reflectivity

The relation between the sound reflection losses measured at fixed grazing angles and the characteristics of the sediment layer and underlying half-space is considered. Based on this relation, a method of the reconstruction of the sea bottom characteristics is developed for a ocean bottom consisting of a single sediment layer overlaying a semi-infinite elastic half-space. Using this bottom model, the reconstruction of the characteristics of a layered elastic bottom is performed from the numerically simulated data with induced synthetic error.

RECONSTRUCTION OF LAYERED ELASTIC BOTTOM CHARACTERISTICS BY THE FREQUENCY DEPENDENCE OF SOUND REFLECTIVITY

The relation between the sound reflection losses measured at fixed grazing angles and the characteristics of the sediment layer and underlying half-space is considered. Based on this relation, a method of the reconstruction of the sea bottom characteristics is developed for a ocean bottom consisting of a single sediment layer overlaying a semi-infinite elastic half-space. Using this bottom model, the reconstruction of the characteristics of a layered elastic bottom is performed from the numerically simulated data with induced synthetic error.

Effects of soil–structure interaction on longitudinal seismic response of MSSS bridges

This paper presents the results of a comprehensive study on the effects of soil–structure interaction on longitudinal seismic response of existing bridges. FHWA's guidelines for footing foundation on semi-infinite elastic half-space are used to determine translational and rotational stiffnesses at the base of bridge abutments and piers. Similarly, stiffness and strength of abutment backfill soil are determined based on FHWA's procedures. Various stiffnesses at the abutments are then lumped (condensed) into one translational spring at the point of impact between the abutment and the deck. Translational springs at the abutments are bilinear with their yield strength in compression determined based on Mononobe–Okabe method. In tension it is equal to the friction force at the footing. Among the parameters considered is the case of damaged back wall, where it is assumed that due to shear failure at the juncture of the back wall and breast wall the abutment strength and stiffness, as well as mobilized abutment mass, have changed. Results indicate that soil–structure interaction (SSI) has a significant effect on the seismic response in the longitudinal direction. Abutment strength is the most critical parameter. Impact force, deck sliding, and SSI affects all plastic rotations at the base of columns. Thus, it is important that analytical models used in seismic evaluation of bridge systems explicitly consider SSI.

The effect of substrate on the elastic properties of films determined by the indentation test — axisymmetric boussinesq problem

T he elastic solution of an axisymmetric mixed boundary value problem is considered. An elastic layer is assumed to be either in frictionless contact or perfectly bonded to a semi-infinite elastic half-space. The elastic field caused by the indentation of the elastic layer, by a rigid indenter is solved for spherical, conical and flat-ended cylindrical indenters. The results are obtained by solving a Fredholm integral equation of the second kind with a continuous symmetrical kernel which depends on the bonding conditions. Numerical results are given for several combinations of film and substrate elastic moduli and film thicknesses. These results provide a guideline for selecting the appropriate film thickness and substrate in order to determine the elastic constants of thin films.

A Fast Method for Computing Load Induced Stress in the Earth

Summary Deformation of the Earth by a large load is usually modelled by the loading of a semi-infinite elastic half space. Solution of this Boussinesq problem can be achieved by numerical integration of the equations for deformation due to a point load, or by integral transform methods. The solution for the deflections is found by Fourier transforms using the fast Fourier transform algorithm for determining the reservoir topography spectrum. After using a well-known solution due to Love on the average load, the stresses and displacements can be calculated at the surface. The method is illustrated by applying it to the Kariba Reservoir, and it is found the method is up to twenty times more efficient than numerical integration. Hydrostatic loading of the Earth is often, but not always associated with an increase in regional seismic activity, (Gough & Gough 1970a, b; Lane 1971; Ellis er al. 1974; Howells 1972; Rotlie 1970). This paper is concerned with a method of calculating the stress in the Earth due to such loading. This loading is only one of several causes of induced seismic activity but other causes will not be examined here. Gough (1969) and Gough & Gough (1970a) presented two papers in which they examined the stress pattern below lakes. They represented the load due to the bathymetry of Lake Kariba as a system of point forces and computed the stress and displacement within the lithosphere by an application of the solution to Boussinesq’s problem. (See for instance Landau & Lifshitz 1957.) They acknowledge that their method is very expensive in computer time, and we present here a more efficient method for this computation. Our numerical results reproduce theirs but we can extend our computations to zero depth whereas Gough & Gough were limited by their method of representing the load. Our theoretical formulation also shows promise of generalization to layered, porous, water filled media. Landau & Lifshitz (1957, pp. 26-29), as well as numerous other texts, show that for a semi-infinite elastic half-space of Poisson’s ratio 0, and Young’s modulus E, the equilibrium equation for small deformation is

Transmission of Plane Waves through Layered Linear Viscoelastic Media

The two-dimensional propagation of time-harmonic plane waves through a plane horizontally layered viscoelastic medium is discussed. The problem is formulated as the equivalent elastic plane-strain case with modified Lamé constants, which are complex and frequency dependent, replacing the usual elastic Lamé constants. Rather than use potentials, incident angles, etc., we formulate the problem directly in terms of stresses and displacements and solve it by using matrix methods. This approach is felt to be more direct and leads to some interesting conclusions. If the incident wave is not attenuated in the direction parallel to the layering, interface waves can be generated only if one of the layers is “pseudoelastic,” i.e., has at least one real wave speed. In this case, the interface waves are generated in the same manner as in the purely elastic case. Such a physical problem would exist, for example, if the incident waves were to travel through a semi-infinite elastic half-space before striking the plane viscoelastic layers. If the incident wave is attenuated in the direction parallel to the layering, interface waves can be generated at specific angles of incidence and specific combinations of material parameters.

On the Unbonded Contact Between Plates and an Elastic Half Space

In this paper an approximate solution is presented for the radius of contact between an elastic plate and a semi-infinite elastic half space. The plate is assumed to rest on the supporting half space without bond, and to be pressed against the elastic region by a concentrated load. In the absence of bonding no tensile stress can be transmitted across the interface between the plate and its elastic support so that contact takes place only within a circle centered about the concentrated load. Outside of this circle the plate lifts up and is no longer in contact with the elastic region.

Elastic Rayleigh Wave Effects Due to Nuclear Blasts

Analysis of stresses and displacements due to Rayleigh waves that are produced by nuclear burst on surface of semi-infinite elastic half space causing time decaying pressure pulse that acts over circular surface area of increasing radius; numerical example is presented in which Rayleigh stresses at various depths and locations are given; use for practical design purposes, such as in design of underground facilities.

Seepage Requirements of Filters and Pervious Bases

A nuclear burst on the surface of a semi-infinite elastic half-space produces a time decaying pressure pulse that acts over a circular surface area of increasing radius. This paper presents results for the stresses and displacements due to Rayleigh waves that are produced by such a transient normal surface pressure distribution.

Elastic Rayleigh Wave Effects Due to Nuclear Blasts

Analysis of stresses and displacements due to Rayleigh waves that are produced by nuclear burst on surface of semi-infinite elastic half space causing time decaying pressure pulse that acts over circular surface area of increasing radius; numerical example is presented in which Rayleigh stresses at various depths and locations are given; use for practical design purposes, such as in design of underground facilities.