Rolling contact of a rigid sphere/sliding of a spherical indenter upon a viscoelastic half-space containing an ellipsoidal inhomogeneity

In this paper, the frictionless rolling contact problem between a rigid sphere and a viscoelastic half-space containing one elastic inhomogeneity is solved. The problem is equivalent to the frictionless sliding of a spherical tip over a viscoelastic body. The inhomogeneity may be of spherical or ellipsoidal shape, the later being of any orientation relatively to the contact surface. The model presented here is three dimensional and based on semi-analytical methods. In order to take into account the viscoelastic aspect of the problem, contact equations are discretized in the spatial and temporal dimensions. The frictionless rolling of the sphere, assumed rigid here for the sake of simplicity, is taken into account by translating the subsurface viscoelastic fields related to the contact problem. Eshelby's formalism is applied at each step of the temporal discretization to account for the effect of the inhomogeneity on the contact pressure distribution, subsurface stresses, rolling friction and the resulting torque. A Conjugate Gradient Method and the Fast Fourier Transforms are used to reduce the computation cost. The model is validated by a finite element model of a rigid sphere rolling upon a homogeneous vciscoelastic half-space, as well as through comparison with reference solutions from the literature. A parametric analysis of the effect of elastic properties and geometrical features of the inhomogeneity is performed. Transient and steady-state solutions are obtained. Numerical results about the contact pressure distribution, the deformed surface geometry, the apparent friction coefficient as well as subsurface stresses are presented, with or without heterogeneous inclusion.     

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

This paper studies the contact vibration problem of an elastic half-space coated with functionally graded materials (FGMs) subject to a rigid spherical punch. A static force superimposing a dynamic time-harmonic force acts on the rigid spherical punch. Firstly, we give the static contact problem of FGMs by a least-square fitting approach. Next, the dynamic contact pressure is solved by employing the perturbation method. Lastly, the dynamic contact stiffness with different dynamic contact displacement conditions is derived for the FGM coated half-space. The effects of the gradient index, coating thickness, internal friction, and punch radius on the dynamic contact stiffness factor are discussed in detail.     

Resonant phenomena in a cylindrical shell containing a spherical inclusion and immersed in an elastic medium

A method is proposed to investigate the behavior of an axisymmetric system consisting of an infinite thin elastic cylindrical shell immersed in an infinite elastic medium, filled with a perfect compressible fluid, and containing an oscillating spherical inclusion. The system is subjected to periodic excitation. The task is to detect so-called resonant phenomena, to establish conditions that cause them, and to examine the possibilities of using the characteristic parameters of such a hydroelastic system to influence these conditions. The method allows transforming the general solutions of mathematical physics equations from one coordinate system to another to obtain exact analytic solutions (in the form of Fourier series) to interaction problems for systems of rigid and elastic bodies __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 82–97, July 2006.     

On the contact of a spherical membrane enclosing a fluid with rigid parallel planes

The mechanical response of an inflated spherical membrane-fluid structure in contact with rigid parallel planes is studied. The membrane is assumed to be a two-dimensional non-linear elastic and isotropic structure, while no assumption is imposed on the fluid. A numerical procedure is employed to compute the equilibrium configurations of the membrane-fluid structure. This study provides information regarding the contact force, stress distribution and pressure in the membrane and in the enclosed fluid, respectively. It was observed that a transition between unwrinkled to partially wrinkled configurations of the membrane occurs subjected to the loading conditions. Further investigation of the wrinkled configurations is presented.     

Adhesion in the contact of a spherical indenter with a layered elastic half-space

With the emergence of micro- and nano-technology, the contact mechanics of MEMS and NEMS devices and components is becoming more important. Thus it is important to gain a better understanding of the role of coatings and thin films on micro- and nano-scale contact phenomena, and to understand the interactions of measurement devices, such as an atomic force microscope (AFM), with layered media.More specifically, in this work the frictionless contact, with adhesion, between a spherical indenter and an elastic-layered medium is investigated. This configuration can be viewed as either a single contact model or as a building block of a multi-asperity rough surface contact model. As the scale decreases to the nano level, adhesion becomes an important issue. The presence of adhesion affects the relationships among the applied force, the penetration of the indenter, and the size of the contact area. This axisymmetric problem includes the effect of adhesion using a Maugis type of adhesion model. This model spans the range of the Tabor parameter between the JKR and DMT regions. The key parameters in this analysis are the elastic moduli ratio of the layer and the substrate, the dimensionless layer thickness, and the Maugis adhesion parameter. The results can be applied to a rigid or to an elastic indenter.     

Resonance phenomena in cylindrical shell with a spherical inclusion in the presence of an internal compressible liquid and an external elastic medium

In the present paper a method is proposed to investigate the behaviour of the axisymmetric system consisting of an infinite thin elastic cylindrical shell submerged in an unbounded elastic medium, filled with an ideal compressible liquid and containing a vibrating spherical inclusion, under periodic dynamic action. The goal is the analysis of the so-called “resonance” phenomena; namely: finding conditions for their appearance, and possible control by means of characteristic parameters of the hydroelastic system under consideration. The technique presented in this work was developed during the realization of a project on elaboration of methods of renewal of oil production in foul wells at the Theory of Vibration Department of the S.P. Timoshenko Institute of Mechanics of the Ukrainian Academy of Science. This mathematical technique allows rewriting the general solution of the corresponding mathematical physics equations from one coordinate system to another, so as to get an exact analytical solution (as a Fourier series) of the interaction problem for a collection of rigid and elastic bodies.     

The stiffness of an elastic solid with an embedded,nominally spherical inclusion subjected to a small arbitrary motion

The article examines the problem of translation and rotation of a nominally (slightly deformed) spherical rigid inclusion embedded into an unbounded elastic medium. To the first order in the small parameter characterizing the boundary perturbation, explicit expressions are deduced for the induced displacement field as well as for the net force and net torque required to produce the applied translation and rotation.     

Effect of the physical nonlinearity of a material on the stress state of a medium with an elastic spherical inclusion under uniform loading

We have used the perturbation method as the basis for obtaining an approximate solution of the three-dimensional problem for a physically nonlinear elastic medium with an elastic inclusion under uniform tension— compression. From this solution, we can obtain as a special case a solution for an elastic medium with a stress-free cavity and for an elastic medium with a rigid inclusion. We have plotted the normal and tangential stresses as a function of the radius and the ratio of shear moduli for the inclusion and the medium. We have investigated their behavior under different loading conditions. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 46–51, November, 1998.     

Axial displacement of a rigid elliptical disc inclusion embedded in a transversely isotropic elastic solid

The present paper examines the axial translation of a rigid elliptical disc inclusion which is embedded in bonded contact with a transversely isotropic elastic medium of infinite extent. The load-displacement relationship for the embedded elliptical inclusion is evaluated in explicit closed form.     

On the load transfer from a rigid cylindrical inclusion into an elastic half space

The present paper examines the problems related to the axial, lateral, and rotational loading of a rigid cylindrical inclusion which is embedded in bonded contact at the boundary of an isotropic elastic half space. The rigid inclusion is modeled as a field of distributed forces which represent the normal and shear tractions that act on the inclusion-elastic-medium interface. The intensities of these distributed tractions are determined by enforcing displacement compatibility conditions at discrete locations of the interface. These compatibility conditions are derived from rigid-body displacement modes appropriate for each loading. The results derived from this numerical scheme are compared with equivalent results derived via analytical techniques which focus on the solution of the governing integral-equation schemes and other approximate-solution schemes. The numerical results presented in the paper illustrate the manner in which the generalized stiffnesses of the embedded inclusion are influenced by its geometry and Poisson's ratio of the half-space region.     

Contact analysis in presence of spherical inhomogeneities within a half-space

This paper presents a fast method of solving contact problems when one of the mating bodies contains multiple heterogeneous inclusions, and numerical results are presented for soft or stiff inhomogeneities. The emphasis is put on the effects of spherical inclusions on the contact pressure distribution and subsurface stress field in an elastic half-space. The computing time and allocated memory are kept small, compared to the finite element method, by the use of analytical solution to account for the presence of inhomogeneities. Eshelby’s equivalent inclusion method is considered in the contact solver. An iterative process is implemented to determine the displacements and stress fields caused by the eigenstrains of all spherical inclusions. The proposed method can be seen as an enrichment technique for which the effect of heterogeneous inclusions is superimposed on the homogeneous solution in the contact algorithm. 3D and 2D Fast Fourier Transforms are utilized to improve the computational efficiency. Configurations such as stringer and cluster of spherical inclusions are analyzed. The effects of Young’s modulus, Poisson’s ratio, size and location of the inhomogeneities are also investigated. Numerical results show that the presence of inclusions in the vicinity of the contact surface could significantly changes the contact pressure distribution. From a numerical point of view the role of Poisson’s ratio is found very important. One of the findings is that a relatively ‘soft’ and nearly incompressible inclusion – for example a cavity filled with a liquid – can be more detrimental for the stress state within the matrix than a very hard inclusion with a classical Poisson’s ratio of 0.3.     

Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

The paper establishes a relationship between the solutions for cracks located in the isotropy plane of a transversely isotropic piezoceramic medium and opened (without friction) by rigid inclusions and the solutions for cracks in a purely elastic medium. This makes it possible to calculate the stress intensity factor (SIF) for cracks in an electroelastic medium from the SIF for an elastic isotropic material, without the need to solve the electroelastic problem. The use of the approach is exemplified by a penny-shaped crack opened by either a disk-shaped rigid inclusion of constant thickness or a rigid oblate spheroidal inclusion in an electroelastic medium __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 47–60, July 2008.     

Diffraction of a plane acoustic wave by a rigid sphere in a cylindrical cavity: An axisymmetric problem

The paper studies the interaction of a rigid spherical body and a cylindrical cavity filled with an ideal compressible fluid in which a plane acoustic wave of unit amplitude propagates. The solution is based on the possibility of transforming partial solutions of the Helmholtz equation between cylindrical and spherical coordinates. Satisfying the interface conditions between the cavity and the acoustic medium and the boundary conditions on the spherical surface yields an infinite system of algebraic equations with indefinite integrals of cylindrical functions as coefficients. This system of equations is solved by reduction. The behavior of the system is studied depending on the frequency of the plane wave     

On certain bounds for the in-plane translational stiffness of a disc inclusion at a bi-material elastic interface

This paper presents a set of bounds that can be used to estimate the in-plane translational stiffness of a rigid circular disc inclusion that is embedded at the interface between two dissimilar elastic half-space regions.     

Onset of residual stress fields near solitary spherical inclusions in a perfectly elastoplastic medium

When computing residual stresses in deformable solids, one has to use the theory of elastoplastic solids, because the final level and distribution of residual stresses is determined exactly by the accumulated reversible strains. In turn, to compute the elastic strains, one needs to determine the displacement field. The problem of determining displacements in statically determinate problems of the theory of perfect elastoplastic solids was considered for the first time in [1, 2]. The techniques proposed there permitted solving the problem of finding the residual stresses near a cylindrical cavity in a perfectly elastoplastic medium [3]. It was shown that secondary plastic flow [4] may arise in the unloading processes, which significantly redistributes the final residual stresses. In the present paper, we consider the loading and unloading problems for a ball with a rigid or elastic spherical inclusion. We study the onset of secondary plastic flow under unloading and compute the residual stresses. Thus, we model the onset of the residual stress field near a more rigid inhomogeneity. The case of a softer inhomogeneity was essentially considered in [3], where the onset of the residual stress field near a continuity flaw was studied.     

The effect of contact conditions and material properties on plastic yield inception in a spherical shell compressed by a rigid flat

The onset of plastic yielding in a spherical shell loaded by a rigid flat is studied for stick and slip contact conditions using finite element analysis. The effect of various material properties on the critical normal load, critical interference and critical contact area at the onset of plastic yielding is investigated and the location where plastic yielding first occurs is determined. A comparison is made with results obtained previously for slip contact condition. Substantial differences are found at low to medium Poisson’s ratio values, while some similarities are found to occur for high Poisson’s ratio values. In particular, a spherical shell is more prone to yield under stick than under slip contact condition.     

Problem of a crack on the boundary of a rigid inclusion in an elastic plate

The problem of equilibrium of a thin elastic plate containing a rigid inclusion is considered. On part of the interface between the elastic plate and the rigid inclusion, there is a vertical crack. It is assumed that, on both crack edges, the boundary conditions are given as inequalities describing the mutual impenetrability of the edges. The solvability of the problem is proven and the character of satisfaction of the boundary conditions is described. It is also shown that the problem is the limit problem for a family of other problems posed for a wider region and describing equilibrium of elastic plates with a vertical crack as the rigidity parameter tends to infinity.     

Influence of penetration depth and mechanical properties on contact radius determination for spherical indentation

Knowledge of the relationship between the penetration depth and the contact radius is required in order to determine the mechanical properties of a material starting from an instrumented indentation test. The aim of this work is to propose a new penetration depth–contact radius relationship valid for most metals which are deformed plastically by parabolic and spherical indenters. Numerical simulation results of the indentation of an elastic–plastic half-space by a frictionless rigid paraboloïd of revolution show that the contact radius–indentation depth relationship can be represented by a power law, which depends on the reduced Young’s modulus of the contact, on the strain hardening exponent and on the yield stress of the indented material. In order to use the proposed formulation for experimental spherical indentations, adaptation of the model is performed in the case of a rigid spherical indenter. Compared to the previous formulations, the model proposed in the present study for spherical indentation has the advantage of being accurate in the plastic regime for a large range of contact radii and for materials of well-developed yield stress. Lastly, a simple criterion, depending on the material mechanical properties, is proposed in order to know when piling-up appears for the spherical indentation.     

Elastic interaction of spherical nanoinhomogeneities with Gurtin-Murdoch type interfaces

A complete solution has been obtained for the problem of multiple interacting spherical inhomogeneities with a Gurtin-Murdoch interface model that includes both surface tension and surface stiffness effects. For this purpose, a vectorial spherical harmonics-based analytical technique is developed. This technique enables solution of a wide class of elasticity problems in domains with spherical boundaries/interfaces and makes fulfilling the vectorial boundary or interface conditions a routine procedure. A general displacement solution of the single-inhomogeneity problem is sought in a form of a series of the vectorial solutions of the Lame equation. This solution is valid for any non-uniform far-field load and it has a closed form for polynomial loads. The superposition principle and re-expansion formulas for the vectorial solutions of the Lame equation extend this theory to problems involving multiple inhomogeneities. The developed semi-analytical technique precisely accounts for the interactions between the nanoinhomogeneities and constitutes an efficient computational tool for modeling nanocomposites. Numerical results demonstrate the accuracy and numerical efficiency of the approach and show the nature and extent to which the elastic interactions between the nanoinhomogeneities with interface stress affect the elastic fields around them.     

The rotation of a rigid elliptical disc inclusion embedded in a transversely isotropic elastic solid

This paper examines the problem of asymmetric rotation of a rigid elliptical disc inclusion embedded in bonded contact with a transversely isotropic elastic solid of infinite extent. The moment-rotation relationship for the embedded inclusion is evaluated in explicit closed form.