Rigid Half-space Research Articles

The Interactions between Spheres and between a Sphere and a Half-Space, Based on the Lennard–Jones Potential

This paper investigates the interaction between two rigid spheres and between a rigid sphere and a rigid half-space, based on the Lennard–Jones potential. By using the divergence theorem and integrating the Lennard–Jones potential over the surfaces, the analytical forms of the surface tractions were obtained, and by integrating the Lennard–Jones potential over the volumes, the analytical form of the total force between two rigid spheres was obtained. The results are compared to those with the Derjaguin approximations and with the parabolic approximation for the sphere profile. The accuracy of the Derjaguin approximation and the parabolic approximation were estimated. The analytical surface traction can be used for the adhesive contact between an elastic body and a rigid one.

Friction-induced vibration and stick-slip waves

This paper presents a short review and new results about the self-excited responses under the form of stick-slip regimes. First, the Van-der Pol oscillator with one degree of freedom is considered. Then it is shown that it is possible to build semi-analytical and numerical (by the FEM.) solutions of stick-slip-separation waves for a brake-like system. Then, we present new results concerning the mechanical model composed of a rigid half space in frictional sliding with an elastic half-space. The method of solution, based on periodic complex Radoks potentials, is novel and differs from those in literature. Besides, in contrast with many works, we shall consider the longitudinal elongation which plays a crucial rule in the solution procedure. A unique and weakly singular solution is found and satisfies all stick-slip conditions except over a narrow zone at transition points which implies a cracklike behaviour at the stick-slip borders.

A thermal creep model for the contact of nominally flat surfaces: Jeffreys' linear visco-elastic model

This work considers the mechanics of contact of thermo-visco-elastic materials. In particular the creep behavior of a nominally flat rough surface in contact with a rigid half space is studied. The rough surface is modeled using fractal geometry. A synthesized profile, a Cantor structure, is utilized to model the surface. Such a profile has two scaling parameters and different heights for each generation of asperities. The effect of temperature will be included through the concept of activation energy using the Arrhenius equation. The objective of this model is to study the normal creep approach of the surface (punch) as a function of the applied creep load, time, and temperature. The material of the punch is assumed to behave according to Jeffreys' model. Such a model is an arrangement of springs and dashpots in parallel and/or in series. The creep approach of linearly visco-elastic materials is explored using elastic–visco-elastic correspondence analysis. An asymptotic power law is obtained, which relates the force and the bulk temperature acting on the punch to its approach. This model is valid only when the approach between the punch and the half space is in the range of the roughness size. The proposed model admits an analytical solution for the case when the deformation is linear thermo-visco-elastic. The obtained model shows a good agreement when compared with experimental results from literature.

Vibrations of a deformable ball rolling on a lubricated rigid half-space

The problem of vertical vibrations of a deformable ball rolling on a lubricated rigid half-space has been stated and numerically solved. It is assumed that the force, which causes the ball’s elastic deformation in the zone of lubricated concentrated contact, is applied to the ball. The problem’s mathematical model is reduced to the system of nonstationary integrodifferential equations and inequalities with initial and boundary conditions. The computational algorithm of the problem’s solution was based on Newton’s method and application of an implicit scheme to approximate nonstationary equations. Vibrational solutions, which describe the behavior of the system consisting of the ball and the lubrication layer both under excitation of the initial conditions and under the excitation force, have been obtained. It is shown that as opposed to the traditional elastohydrodynamic lubrication theory approach, in which contacting bodies are considered as massless ones, taking the ball’s mass and corresponding inertia into account makes it possible to extend the category of obtained solutions and investigate vibrational processes in the contact zone. The results may be used to analyze processes occurring in lubricated concentrated contacts of gears and rolling bearings.

Friction-induced vibration and stick-slip waves

This paper presents a short review and new results about the self-excited responses under the form of stick-slip regimes. First, the Van-der Pol oscillator with one degree of freedom is considered. Then it is shown that it is possible to build semi-analytical and numerical (by the FEM.) solutions of stick-slip-separation waves for a brake-like system. Then, we present new results concerning the mechanical model composed of a rigid half space in frictional sliding with an elastic half-space. The method of solution, based on periodic complex Radoks potentials, is novel and differs from those in literature. Besides, in contrast with many works, we shall consider the longitudinal elongation which plays a crucial rule in the solution procedure. A unique and weakly singular solution is found and satisfies all stick-slip conditions except over a narrow zone at transition points which implies a cracklike behaviour at the stick-slip borders.

A New Numerical Approach for Adhesive Contacts of Real Engineering Surfaces

Microelectromechanical system (MEMS) and nanotechnology are important directions on the development of the science in twenty-first century. Some of the effects, such as viscous force, surface force, electrostatic force, friction etc., which can be usually ignored on the traditional scale, have become noticeable when the scale has turn to micro or nano scale. Nanotribology is one of the main areas of the indispensable researches on the basic theory and methodology of the effects. The micro/nano adhesive contact which is the foundation of nanotribology is studied in this paper. The earliest study on adhesive contact was done by Bradley. He presented an expression of adhesive force of two contacting rigid spheres. Derjaguin, Muller and Toprov (DMT) gave the relation of the contact area and the applied load of the adhesive contact of two spheres, but they did not consider the elastic deformation due to the adhesive force of the bodies. Johnson, Keudall and Roberts (JKR) provided a theory of the adhesive contact of two elastic spheres. Tabor gave a parameter (Tabor parameter) to interpret the ratio of the elastic deformation with the adhesive force of two contacting bodies. That is to say the DMT model corresponding to small Tabor parameter(<0.1) and the JKR model to large Tabor parameter(>5). Maguis gave a DMT-JKR transition using the Dugdale model in fracture mechanics (M-D model) in the intermediate region between the DMT model and the JKR model. A numerical algorithm of elastic adhesive contact based on the meshless method is presented in this paper. This make it possible to solve the adhesive contact with more complex surface topography and to consider more intricate factors, such as thermal stress, friction, elasto-plastic deformation etc. in the further studies on micro/nano scale adhesive contact problems. The meshless method seems to be a promising approach for contact analyses because of its flexibility in domain descritization and versatility in node arrangements. It can be used to solve a variety of complicated engineering problems. A numerical example of adhesive contact between a micro elastic cylinder and a rigid half-space is carried out to show the feasibility of the algorithm. In the simulation, an effective method of the M-D model is used to save the cost of computation. Compared with the existed solutions, the results solved by the presented algorithm are reasonable.

Nonexistence of Solution for Horizontally Rigid Half-Space

The Westergaard expressions for stresses and displacements in a half-space of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the half-space. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic half-space deduced from Michell’s expressions. A surprising conclusion is drawn, namely, that there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic half-space. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of the theory of elasticity.

Nonlinear site response evaluation procedure under the strong motion: A case study of Miyaneh, Azarbayjan Sharghi Province, Iran

The objective of a site response analysis is to estimate the ground shaking intensity during an earthquake. The Miyaneh city and its suburban areas are located in the northwest of Iran in Azarbayjan-Sharghi province. This area is prone to high seismic risk due to the presence of several active faults. In this paper, a case study on ground response analysis of a geotechnical site in Miyaneh region during the Ardabil earthquake (28 February, 1997, Mw6.1) is presented. For sitecharacterization, deep site investigations have been undertaken and a seismic geotechnical procedure for the proposed bridge over the rivers at the mentioned site, which is performed for Iran railway network, subjected to earthquake provokes, has been notified. The effect of nonlinearity on site response analysis for the selected site with the assumption of elastic and rigid half space bedrock (by use of nonlinear standard hyperbolic model) was evaluated and the results obtained over a number of geotechnical areas were compared to each other. Test of the capability of designed computer codes by authors, namely; “Abbas converter”, which describes and evaluates the nonlinearity of the subsurface soil conditions encountered at the sites to analyze, evaluate the obtained test, site response and quantify the site effect on the surface over a number of geotechnical areas, were the targets of this study. The results clearly showed that the effect of bedrock and local soil conditions on soil behavior under the earthquake excitation is one of the main effective factors on computed response spectra in ground response prediction. The key factor in this work was to develop and use “Abbas converter”. It has several advantages, which can make and render the study easier, than previous studies done, and take over the encountered problem. Key words: Miyaneh city, “Abbas Converter”, Ardabil earthquake, site response, site amplification.

Adhesion-Induced Instability in Asperities

Adhesive forces between two approaching asperities will deform the asperities, and under certain conditions this will result in a sudden runaway deformations leading to a jump-to-contact instability. We present finite element-based numerical studies on adhesion-induced deformation and instability in asperities. We consider the adhesive force acting on an asperity, when it is brought near a rigid half-space, due to van der Waals interaction between the asperity and the half-space. The adhesive force is considered to be distributed over the volume of the asperity (body force), thus resulting in more realistic simulations for the length scales considered. Iteration scheme based on a “residual stress update” algorithm is used to capture the effect of deformation on the adhesion force, and thereby the equilibrium configuration and the corresponding force. The numerical results are compared with the previous approximate analytical solutions for adhesion force, deformation of the asperity and adhesion-induced mechanical instability (jump-to-contact). It is observed that the instability can occur at separations much higher, and could possibly explain the higher value of instability separation observed in experiments. The stresses in asperities, particularly in case of small ones, are found to be high enough to cause yielding before jump-to-contact. The effect of roughness is considered by modeling a spherical protrusion on the hemispherical asperity. This small-scale roughness at the tip of the asperities is found to control the deformation behavior at small separations, and hence are important in determining the friction and wear due to the jump-to-contact instability.

Love wave dispersion in anisotropic visco-elastic medium

The paper presents a study on Love wave propagation in a anisotropic visco-elastic layer overlying a rigid half space. The characteristic frequency equation is obtained and the variation of the wave number with frequency under the combined effect of visco-elasticity and anisotropy is analysed in detail. The results show that the effect of visco-elasticity on the wave is similar to that of anisotropy as long as the coefficient of anisotropy is less than unity.

Evaluation of a nonlinear seismic geotechnical site response analysis method subjected to earthquake vibrations (case study: Kerman Province, Iran)

Equivalent static load and dynamic analyses methods are usually used for designing structures under and subjected to earthquake excitations. Estimation of site response from an earthquake is fundamental step to anticipate the possible damages and then to try to mitigate them. In this paper, the effect of nonlinearity on site response analyses summarized and evaluating ground surface response taking into account the local soil and subsurface soils properties for the proposed bridge over the river at Sirdjan Boulevard road subjected to earthquake vibration and provokes with assumption of rigid (viscoelastic) and elastic half space bedrock and quantify the site effect on the surface over a number of geotechnical areas has been notified. First, by field investigation, the required data were collected and by primary processing the acceptable data were selected. Then, in nonlinear analysis, for elastic and rigid half space bedrock, standard hyperbolic model was selected and executed, and then the results were compared to each other. The critical point of this work was to develop and use a computer code by the authors, named the “Abbas Converter”, with several advantages, such as work and quick installation, operating as a connecter function between the used softwares and generating the input data corresponding to defined format for them. Its output results can easily be exported to the other used softwares in this study. This code can make and render this study more easily than the previous softwares have done, and take over the encountered problem. This study clearly showed the applicability of the “Abbas Converter” for evaluation of site response with bedrock-type assumption on soil behavior under the earthquake excitations. The proposed scheme is used to analyze the ground motion data from the Bam earthquake in Kerman Province, Iran (2003, Mw 6.5).

Fractal Geometry-Based Hypergeometric Time Series Solution to the Hereditary Thermal Creep Model for the Contact of Rough Surfaces Using the Kelvin-Voigt Medium

This paper aims at constructing a continuous hereditary creep model for the thermoviscoelastic contact of a rough punch and a smooth surface of a rigid half-space. The used model considers the rough surface as a function of the applied load and temperatures. The material of the rough punch surface is assumed to behave as Kelvin-Voigt viscoelastic material. Such a model uses elastic springs and viscous dashpots in parallel. The fractal-based punch surface is modelled using a deterministic Cantor structure. An asymptotic power law, deduced using approximate iterative relations, is used to express the punch surface creep which is a time-dependent inelastic deformation. The suggested law utilized the hypergeometric time series to relate the variables of creep as a function of remote forces, body temperatures, and time. The model is valid when the approach of punch surface and half space is in the order of the size of the surface roughness. The closed-form results are obtained for selected values of the system parameters; the fractal surface roughness and various material properties. The obtained results show good agreement with published experimental results, and the methodology can be further extended to other structures such as the Kelvin-Voigt medium within electronic circuits and systems.

Approximation and numerical realization of 3D contact problems with given friction and a coefficient of friction depending on the solution

The paper presents the analysis, approximation and numerical realization of 3D contact problems for an elastic body unilaterally supported by a rigid half space taking into account friction on the common surface. Friction obeys the simplest Tresca model (a slip bound is given a priori) but with a coefficient of friction \( \mathcal{F} \) which depends on a solution. It is shown that a solution exists for a large class of \( \mathcal{F} \) and is unique provided that \( \mathcal{F} \) is Lipschitz continuous with a sufficiently small modulus of the Lipschitz continuity. The problem is discretized by finite elements, and convergence of discrete solutions is established. Finally, methods for numerical realization are described and several model examples illustrate the efficiency of the proposed approach.

A comparative case study on time domain nonlinear site response analysis subjected to earthquake excitation with elastic and rigid half space (site of Damghan earth dam-Semnan province-Iran)

The aim of this study was evaluate the site response with consideration rigid and elastic half space as the bedrock and quantify the site effect on the surface over a number of geotechnical areas. The authors have an intention to establish a comparative study on time domain analysis with consideration of bedrock type as the location of input motion and its effect on top layers over it. For this aim the site of Damghan earth dam in Seman province of Iran was selected. At the first by field investigation, the required data were collected and by primary processing the acceptable data were selected. Then in nonlinear analysis, for elastic and rigid half space bedrock, Standard Hyperbolic Model was selected and performed and the results of them were compared to each other which clearly showed the effect of bedrock on soil behavior under the earthquake excitation is one of the main factors on computation of response spectra in ground response prediction. A critical element in this work was to develop and use a computer code that could be installing quickly and act as a logic relation link between used softwares. This produced computer code which namely as “Abbas Converter” can change and support various recorded format to applicable form for used softwares can make easy and render this analysis with generation and producing the required data for used softwares and take over the encountered problem. Key word: Nonlinear time domain analysis, “Abbas converter”, elastic and rigid half space bedrock.

Mathematical modeling of the thermal relaxation of nominally flat surfaces in contact using fractal geometry: Maxwell type medium

This work aims at studying the stress relaxation behavior of a nominally flat (rough) surface of a viscoelastic material in contact with a rigid half space. The effect of temperature will be included through the concept of activation energy using Arrhenius's equation. A synthesized Cantor–Borodich ( C B) profile is used to construct the rough surface. C B profile has two scaling parameters, a and b, and different heights h i for each generation of asperities. This simple model is applicable for fractal surfaces in which a single exponent (the fractal dimension) is enough to describe their quality. The surfaces in contact are viscoelastic, and they are assumed to behave according to the linear Maxwell model. An asymptotic power law is obtained, which relates the force and the bulk temperature acted on the punch to its approach. This model is valid only when the approach between the punch and the half space is in the range of the roughness size. The proposed model admits an analytical solution for the case when the deformation is linear thermo-viscoelastic. The obtained model shows a good agreement when compared with the experimental results obtained by Handzel-Powierza et al. [Handzel-Powierza Z, Klimczak T, Polijaniuk A. On the experimental verification of the Greenwood–Williamson model for the contact of rough surfaces. Wear 1992;154:115–24].

Transversely isotropic elastodynamic solution of a finite layer on an infinite subgrade under surface loads

The dynamic response of a transversely isotropic, linearly elastic layer bonded to the surface of a half-space of a different transversely isotropic material under surface loads is considered. An analytical formulation is presented for the determination of the displacements and stresses in both regions in frequency domain. By means of a complete representation using two displacement potentials, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order and a second-order partial differential equation. Using Fourier expansions and Hankel transforms in cylindrical coordinates, the formulation includes a complete set of transformed displacement-potential and stress-potential relations. As illustrations, the cases of a finite layer bonded to (i) an identitcal underlying half-space, (ii) on a rigid half-space, and (iii) an elastic half-space of different material moduli are considered. Typical numerical results on the response characteristics as a function of material anisotropy, in both the frequency and spatial domains are included.

On a mixed nonstationary problem of heat conduction for a half-space

The paper deals with some mixed nonstationary problem of heat conduction for a homogeneous rigid half-space. The body is assumed to be initially at zero temperature, the boundary plane is kept at a known temperature f(t) for time 0≤t≤t0 and is insulated for t>t0. The exact solution is obtained by using the integral Fourier method. Some special case is solved in details.

Energy-rate consideration in wafer bonding

In the present study, we consider the bonding process of a pair of wafers with rough surfaces via an energy-rate concept in which “rate” refers to “area rate”. A finite element numerical simulation for a rough surface with periodical distributed asperities bonded onto a rigid half-space is carried out for demonstrating the energy-rate concept. The simulation reveals various phenomena in the wafer bonding, such as self-activated bonding, asperity shape effect and plastic deformation effect in the bonding process.

An EFG-FE Coupling Method for Microscale Adhesive Contacts

An elastic adhesive contact model based on the element-free Galerkin-finite element (EFG-FE) coupling method is presented in this paper. The model is first validated though comparison to theoretical solutions. A numerical simulation of the adhesive contact between a microelastic cylinder and a rigid half-space is then conducted. The adhesive contact characteristics of three metals (Al, Cu, and Fe) are studied at different Tabor parameters. The relationships of the applied load and contact half-width of the adhesive contacts are analyzed. Contact pressures, stress contours and deformed profiles of different cylinder sizes and applied loads are illustrated and discussed. The results are compared to published solutions, and good agreements are observed.

Adhesion and friction of an elastic half-space in contact with a slightly wavy rigid surface

The paper deals with the estimation of the pressure distribution, the shape of contact and the friction force at the interface of a flat soft elastic solid moving on a rigid half-space with a slightly wavy surface. In this case an unsymmetrical contact is considered and justified with the adhesion hysteresis. For soft solids as rubber and polymers the friction originates mainly from two different contributions: the internal friction due to the viscoelastic properties of the bulk and the adhesive processes at the interface of the two solids. In the paper the authors focus on the latter contribution to friction. It is known, indeed, that for soft solids, as rubber, the adhesion hysteresis is, at least qualitatively, related to friction: the larger the adhesion hysteresis the larger the friction. Several mechanisms may govern the adhesion hysteresis, such as the interdigitation process between the polymer chains, the local small-scale viscoelasticity or the local elastic instabilities. In the paper the authors propose a model to link, from the continuum mechanics point of view, the friction to the adhesion hysteresis. A simple one-length scale roughness model is considered having a sinusoidal profile. For partial contact conditions the detached zone is taken to be a mode I propagating crack. Due to the adhesion hysteresis, the crack is affected by two different values of the strain energy release rate at the advancing and receding edges respectively. As a result, an unsymmetrical contact and a friction force arise. Additionally, the stability of the equilibrium configurations is discussed and the adherence force for jumping out of contact and the critical load for snapping into full contact are estimated.