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.

Dynamic response of a homogeneous hygrothermoelastic slab sandwiched between elastic half-spaces

In the current study, the authors discuss a homogeneous hygrothermoelastic slab of thickness 2h sandwiched between two similar elastic half-spaces. The hygrothermoelastic slab is subjected to a mechanical source of constant magnitude. The source is applied along the interface of the slab and upper elastic half-space. The analytical expressions of the components of displacement, stresses, moisture concentration, and temperature field are obtained by the normal mode analysis technique. The analytical results are used to solve the problem numerically by taking a wood slab as a porous material. The graphical results elaborate the effect of the thickness of a slab on the physical quantities. The problem of hygrothermoelastic layers sandwiched between elastic half-spaces is a novel idea, which finds its applications in various engineering models. The results found are interesting in the context of the problem. It is observed that the thickness of a hygrothermoelastic slab affects the deformation in the medium and the values of physical quantities decrease with an increase in the thickness of the slab.

Analytical prediction and numerical verification of stress concentration profile around an in-situ tow break in resin-impregnated filament-wound composites

A new empirical analytical approach is developed for predicting the stress concentration profile around an in-situ tow break in filament-wound composites. A shear-lag analysis is firstly performed to solve for the perturbational axial displacement of the broken tow and resultant debonding lengths. Solution of stress field caused by tangential load on the surface of an elastic half space is then utilized in combination with superposition concepts to obtain the overload magnitudes in the neighboring tows. Subsequently, high-fidelity finite element analysis on a representative uni-directional laminate model under different stress states is performed, and excellent overall agreement is observed between analytical and numerical predictions. The proposed method takes into account essential aspects such as transversely isotropic material properties, in-situ stress states and their effect on the interfacial frictional forces in the debonded interfaces, and thus provides a convenient way to evaluate the stress concentration factors in damaged filament-wound composites. In addition, this approach can be applied to yield auxiliary failure evaluation criteria for statistical strength prediction or finite element modeling of filament-wound composites or similar structures.

Surface acoustic waves in laterally periodic superlattices

The existence of surface acoustic waves is studied in laterally periodic superlattices, modelled as an elastic half-space of general anisotropy with an arbitrary periodic modulation of material properties in the direction (call it X) parallel to the surface. Unlike a homogeneous half-space, a laterally periodic one may support more than one (subsonic) surface wave. Specifically, it is shown that the maximum number of surface waves travelling along X in any superlattice with a generic (asymmetric) profile of periodic stratification is three. What is rather remarkable is that the same upper bound applies to the total number of waves in two superlattices with mutually inverse profiles or, which is the same, to the number of counter waves travelling in the opposite directions of a given superlattice, i.e. their overall number cannot exceed three either. At least one surface wave exists in one of the two superlattices unless the bulk-wave threshold is of the so-called exceptional type. In the particular case of the symmetric stratification profile, i.e. the one invariant to the inversion of the axis X, the surface wave remains unique, like in the case of a homogeneous half-space. The above general results are illustrated by the perturbation theory derivations under the assumption of weak periodic modulation. Explicit leading-order formulas are obtained for the frequencies of the two quasi-Rayleigh waves evolving from the Rayleigh wave by one for the mutually inverse profiles and for the frequency of a quasibulk wave evolving from the exceptional transonic state for one of the profiles (but not for both).

Integrating Tensometer Measurements, Elastic Half-Space Modeling, and Long-Term Pavement Performance Data into a Mechanistic–Empirical Pavement Performance Model

Pavement performance models (PPMs) are utilized to predict pavement network conditions which is an essential part of any sustainable pavement management system (PMS). The reliability of a PMS and its outputs is proportional to the reliability of the PPM used. This article describes a mechanistic–empirical pavement performance model based on pavement response parameters—strains calculated in the pavement layers measured by tensometers embedded in the pavement surface and verified by calculations in the elastic half-space model and supplemented by empirical data from long-term pavement performance monitoring and accelerated pavement testing. Hence, the herein described PPM combines pavement serviceability evaluation, pavement bearing capacity, and the physico-mechanistic properties of paving materials. The analytical methods which were used to ascertain the physico-mechanistic characteristics, the material fatigue degradation model, and the surface degradation, unevenness in particular, are described. A comparison of the empirical PPM created in the last century used by the national road administrator to this day and the newly created PPM is presented. The comparison shows the difference in the calculated socio-economic benefits and subsequent cost–benefit analysis results. The comparison shows that the use of the old PPM may have produced false economic evaluation results that have led to poor decision making, partially explaining the unsustainable trend of road network management in our country.

A complex approach to solving the problem of interaction between a rigid double-connected punch and an elastic half-space

A complex approach to solving the problem of interaction between a rigid double-connected punch and an elastic half-space

Modelling deformable boundary by spherical particle for normal contact

The normal contact of the elastic spherical particle with deformable boundary is investigated in terms of the Discrete Element Method (DEM). The particle of the prescribed radius is moving under gravity and the initial velocity. The deformable boundary is treated as rigidly fixed spherical particle with variable elasticity modulus and variable radius. The limit case, approaching the infinite radius presents an elastic half-space, while increasing of the elasticity modulus presents the rigid boundary, respectively.
 The linear model and the nonlinear Hertz contact model used in the discrete element method are investigated numerically by applying the 5th-order Gear’s  predictorcorrector integration scheme. The numerical model is tested by comparing it with analytical solution. The time variations of the particle positions, velocities and  accelerations are presented. On the basis of simulation results the limit values of the boundary particle parameters are evaluated and recommendations for the boundary  article parameters required in DEM simulation are drown.

Numerical Analysis of Unsteady Vibrations of a Plate Resting on an Elastic Isotropic Half-Space

The paper is devoted to the numerical solution of the problem of vibrations of an infinite elastic plate resting on an elastic isotropic half-space using the analytical method based on the ray method with its numerical realization via the Maplesoft package. Unsteady oscillations are caused by the action of instantaneous loads on the plate, resulting in the appearance of two plane wave surfaces of strong discontinuity in the elastic half-space, behind the fronts of which, up to the contact boundary, the solution is constructed using ray series. The unknown functions entering the coefficients of the ray series and the equation of plate motion are determined from the boundary conditions of the contact interaction between the plate and the half-space. Previously, the approximate solution of this problem was obtained analytically without using mathematical packages, and the dynamic deflection of the plate involving only the first three terms of the ray series was written down. In this work, a two-layer medium with different properties was investigated using an algorithm developed to solve contact dynamic problems related to the occurrence and propagation of strong and weak discontinuity surfaces.

Three-dimensional frictional contact within the framework of couple stress elasticity

This work investigates a three-dimensional size-dependent frictional contact problem of a rigid spherical punch sliding on an elastic half-space. The size-dependent material behaviors are governed by the couple stress theory, and the corresponding frequency response functions, which include two additional length-related elastic parameters, are obtained. Based on the conjugate gradient method and the discrete convolution-fast Fourier transform algorithm, a semi-analytical model is established to predict the frictional size-dependent contact behaviors and then the detailed parameter analysis is performed. The transverse effects of the couple stress, which are induced due to the considerations of the two additional elastic parameters in couple stress theory, are investigated in the contact problems for the first time. And the dependences of the size-dependent contact behavior on the coupling effects of the friction and the two additional elastic parameters are presented.

Rayleigh Wave Scattering by a Near-Surface Inclusion in Elastic Half-Space

Rayleigh Wave Scattering by a Near-Surface Inclusion in Elastic Half-Space

Improved modeling of anisotropic effects on seismic waves in layered transversely isotropic half-spaces: Implications for velocity profile inversion challenges

In this paper, we first derive the dynamic solution in a transversely isotropic (TI) elastic and layered half-space, induced by a time-harmonic vertical load on its surface. The solution is obtained via the efficient and unconditionally stable dual variable and position (DVP) matrix method along with the fast Fourier-Bessel series (FBS) expansion approach. In order to study the effect of material anisotropy, the degree of anisotropy, namely the ratio of the horizontal Young's modulus over vertical Young's modulus, is introduced as the key parameter, with a very reasonable assumption on other involved elastic constants in the layered TI half-space. This enables us to investigate the effect of material anisotropy on the wave Green's functions, dispersion curves, ellipticity, and polarity. By comparing with the effect of Poisson's ratio in the corresponding layered isotropic (ISO) half-space, we observe that nearly all the wave features with varying degrees of anisotropy in the TI layered half-space are similar to those with varying Poisson's ratio in the corresponding ISO half-space. This indicates the challenge or non-uniqueness in the inversion of TI velocity profiles using vertical surface loading only.

Adhesive contact between a rigid toroidal indenter and an elastic half-space

The adhesive contact between a rigid toroidal indenter and an elastic half space is investigated numerically. The traction is modeled using the Lennard-Jones potential with Derjaguin’s approximation. It is found that toroidal Johnson–Kendall–Roberts (JKR) model can predict the adhesive contact for large toroidal Tabor parameters and larger radius of the cross section of the torus, while toroidal Bradley model can predict the adhesive contact for small toroidal Tabor parameters. Toroidal indenter can generate larger adhesion than spherical indenter. The contour for the ratio of the pull-off forces between toroidal indenter and spherical indenter is plotted. The pull-off force is larger for larger radius of the cross section of the torus. Thus, the pull-off force of volcano-shaped indenter is larger than that of donut-shaped indenter.

Scattering of plane waves from stress-free boundary surface of a microstretch elastic solid half-space containing voids

Abstract Scattering phenomena of plane coupled waves from stress-free boundary surface of a microstretch elastic solid half-space containing uniform distribution of voids have been investigated. Using appropriate boundary conditions and Snell's law, the reflection coefficients corresponding to various reflected \textcolor{red}{sets of coupled} waves and their corresponding energy ratios have been presented. One of the boundary conditions leads to the vanishing of a cross-coupling parameter between curvature tensor and voids. Numerical computations have been carried out for a specific model and the variation of all the reflection coefficients and \textcolor{red}{their} corresponding energy ratios is depicted graphically against the angle of incidence. Some special cases have been reduced from the present model and discussed.

Estimating the slip rate in the North Tabriz Fault using focal mechanism data and GPS velocity field

Abstract This study delves into slip distribution on the North Tabriz Fault (NTF), a critical aspect of seismic hazard analysis due to its proximity to the Tabriz metropolis. The study operates within a uniform elastic half-space, maintaining constant values for fault geometry and regional rheological parameters throughout the research. To calculate strain boundary conditions, permanent and periodic global positioning system (GPS) data from the northwest region were utilized. The fault was constrained perpendicularly while allowing tangential movement, facilitating the determination of its annual slip rate using the boundary element method, with the Okada analytical model serving as the fundamental solution. The findings underscore the intricate relationship between the fault’s slip rate and boundary conditions, revealing a predominant right-lateral strike-slip motion. The study offers two slip rate assessments, obtained through earthquake focal mechanisms and GPS velocity data, yielding values of 5 and 5.5 mm / year {\rm{mm}}/{\rm{year}} , respectively. Importantly, the alignment of these calculated slip rates with paleo-seismological data underscores the credibility of the results generated via the boundary element method, distinguishing it as a reliable approach when compared to other numerical and analytical techniques. This research provides valuable insights into the behavior and slip dynamics of the NTF, which is pivotal for assessing seismic risks.

Modeling the line contact on an elastic half-space with the statistical approach: Self-affine fractal roughness and numerical framework

This paper presents a line contact model for describing the normal contact between rough surfaces with self-affine fractal roughness based on the statistical approach. An iterative algorithm combining the Levenberg-Marquardt (LM) method with penalty function is proposed to overcome the limitation of restricted indentation depth. Results show a competitive relationship between the plasticity index and the fractal dimension on the real contact area. Besides, nonlinear characteristics between force and area appear when the elastic modulus of the substrate is larger than that of the asperity. Furthermore, the wavy machining texture leads to a great increase in the maximum pressure. The current work can be used for modeling the transmission interfaces involving line contact, such as gears and cylindrical rotor bearings.

Inverse problem for Love waves in a layered, elastic half-space

In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves.

A framework for computing directivities for ultrasonic sources in generally anisotropic, multi-layered media

The knowledge of the directivity of ultrasonic sources is an important element in the design of non-destructive evaluation inspection. The commonly used directivity model of piezoelectric devices is the superposition of out-of-plane force monopoles on the stress-free surface of an isotropic elastic half-space. However, this does not cover a growing range of ultrasonic generation scenarios, as new ultrasonic technologies such as laser ultrasound (LU) emerges, and the need for inspection of materials such as carbon fibre reinforced polymers (CFRPs) rises. A directivity calculation framework based on the reciprocity theorem is developed in the current paper, which accommodates various combinations of source types, material properties, and boundary conditions, for example, a localised heating under a surface layer over an anisotropic half-space. The calculation is applicable to materials with weak anisotropy or wave modes that do not have cuspoidal wavefronts, and was validated with finite element models on key representative cases. The analytical solution for LU excitation in a CFRP sample also shows a good fit to the experimentally measured directivity of the quasi-longitudinal wave mode.

Modeling of the Combined Effect of the Surface Roughness and Coatings in Contact Interaction

The model of indentation of a spherical punch with a rough surface into a thin elastic layer lying on an elastic half-space has been developed. A numerical-analytical solution is suggested based on the two-scales approach. At macroscale, the integral equation of the second kind is reduced to calculate the nominal contact characteristics, taking into account the surface layer thickness and its mechanical characteristics, as well as additional compliance due to surface roughness calculated from the microscale analysis. The influence of the punch roughness and the surface layer mechanical and geometrical characteristics on the nominal contact pressure distribution, on the nominal contact area and the indentation depth, as well as on the real contact area and pressures at the individual contact spots, is analyzed. The developed contact model can be used to analyze the indentation of the punch into the layered elastic base, taking into account the roughness of the punch surface, and can also be used to give a complete analysis of the roughness effect on the contact process at both scale levels. The results can be used to control the indentation of the rough-coated bodies.

Modeling of shear horizontal waves propagation in magneto-electro-elastic substrate loaded with complex conductive fluid

In this study, the dissipation and dispersion of shear horizontal (SH) wave in magneto-electro-elastic (MEE) materials overlying an elastic half-space loaded with a complex conductive fluid have been investigated. The complex dispersive relation of SH waves for magnetic short and magnetic open cases has been derived using an exact analytical approach. The derived dispersive relations have been theoretically and graphically validated against previous work. The study explores the impact of geometrical parameters, frequency, conductivity, and permittivity of the complex conductive fluid on the phase velocity and attenuation of SH waves.

Analysis of the discrete contact characteristics based on the Greenwood-Williamson model and the localization principle

The contact of a rigid body with nominally flat rough surface and an elastic half-space is considered. To solve the contact problem, the Greenwood-Williamson statistical model and the localization principle are used. The developed contact model allows us to investigate the surface approach and the real contact area with taking into account the asperities interaction. It is shown that the mutual influence of asperities changes not only contact characteristics at the macroscale, but also the contact pressure distribution at the microscale. As follows from the results, the inclusion in the contact model of the effect of the mutual influence of asperities is especially significant for studying the real contact area, as well as the contact characteristics at high applied loads. The results calculated according to the proposed approach are in a good agreement with the experimentally observed effects, i.e., the real contact area saturation and the additional compliance exhaustion.