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.     

The second Eshelby problem and its solvability

It is still a challenge to clarify the dependence of overall elastic properties of heterogeneous materials on the microstructures of non-elliposodal inhomogeneities (cracks, pores, foreign particles). From the theory of elasticity, the formulation of the perturbance elastic fields, coming from a non-ellipsoidal inhomogeneity embedded in an infinitely extended material with remote constant loading, inevitably involve one or more integral equations. Up to now, due to the mathematical difficulty, there is almost no explicit analytical solution obtained except for the ellipsoidal inhomogeneity. In this paper, we point out the impossibility to transform this inhomogeneity problem into a conventional Eshelby problem by the equivalent inclusion method even if the eigenstrain is chosen to be non-uniform. We also build up an equivalent model, called the second Eshelby problem, to investigate the perturbance stress. It is probably a better template to make use of the profound methods and results of conventional Eshelby problems of non-ellipsoidal inclusions.     

Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

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 stress analysis of an elastic half-plane containing multiple inclusions

This paper presents a two-dimensional contact stress analysis to investigate the effects of multiple inclusions on the contact pressure and subsurface stresses in an elastic half-plane. The boundary element method is used to analyze the contact problem where a set of integral equations is derived on the contact region and the matrix–inclusion interfaces. As the contact region is unknown a priori, an iterative procedure is implemented to determine the actual contact region and the contact pressure, and the tractions and displacements on the matrix–inclusion interfaces are obtained by solving the integral equations numerically. Numerical results show that the inclusions near contact surface could cause significant alterations in the contact pressure distribution. The stiff inclusions could toughen the surrounding material and reduce the internal stresses while the soft inclusions could increase the subsurface stresses.     

Self-consistent field method in the problem about the effective properties of an elastic composite

This paper is devoted to the calculation of effective elastic properties of a medium containing a random field of ellipsoidal inhomogeneities. It is assumed that the centers of the inclusions (the inhomogeneities) form a random spatial lattice, i.e., the field of inhomogeneities considered is strongly correlated. The interaction between the inhomogeneities is taken into account within the frame-work of the self-consistent field approximation. It hence turns out that the symmetry of the tensor of the elastic properties of the medium is determined by the symmetry of the elastic properties of the inclusion matrix, as well as by the symmetry of the spatial lattice formed by the mathematical expectations of the centers of the inclusions.     

Stress disturbances caused by the inhomogeneity in an elastic half-space subjected to contact loading

Inhomogeneities can increase localized stress and cause microstructural alterations to initiate fatigue failures in rolling elements under cyclic contact loading. To study the stress disturbances created by the inhomogeneity, a two-dimensional contact stress analysis is presented for a cylindrical indenter sliding on an elastic half-space containing an inhomogeneity of arbitrary shape. The boundary element method is used to analyze the contact problem, where actual contact boundary, contact pressure as well as tractions and displacements at inhomogeneity–substrate interface are determined by solving a set of integral equations numerically. Numerical results are presented to investigate effects and the stress disturbances caused by the inhomogeneity with various locations, sizes and material properties of inhomogeneity. The results also show that hard inclusions are more detrimental than soft deformable particles in rolling contact elements.     

Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems

At small length scales and/or in presence of large field gradients, the implicit long wavelength assumption of classical elasticity breaks down. Postulating a form of second gradient elasticity with couple stresses as a suitable phenomenological model for small-scale elastic phenomena, we herein extend Eshelby’s classical formulation for inclusions and inhomogeneities. While the modified size-dependent Eshelby’s tensor and hence the complete elastic state of inclusions containing transformation strains or eigenstrains is explicitly derived, the corresponding inhomogeneity problem leads to integrals equations which do not appear to have closed-form solutions. To that end, Eshelby’s equivalent inclusion method is extended to the present framework in form of a perturbation series that then can be used to approximate the elastic state of inhomogeneities. The approximate scheme for inhomogeneities also serves as the basis for establishing expressions for the effective properties of composites in second gradient elasticity with couple stresses. The present work is expected to find application towards nano-inclusions and certain types of composites in addition to being the basis for subsequent non-linear homogenization schemes.     

Effect of pore shape on effective porothermoelastic properties of isotropic rocks

The present work is devoted to the determination of the macroscopic poroelastic and porothermoelastic properties of geomaterials or rock-like composites constituted by an isotropic matrix with embedded ellipsoidal inhomogeneities and/or pores randomly oriented. By considering the solution of a single ellipsoidal inhomogeneity in the homogenization problem it is possible to observe the significant influence of the shape of inhomogeneities on the effective porothermoelastic properties. In the particular case of microscopic and macroscopic isotropic behaviors, a closed form solution based on analytical integrate of the Eshelby solution for the single ellipsoidal inhomogeneity can be obtained for the randomly oriented distribution. This result completes the well known solutions in the limiting cases of spherical and penny shape inhomogeneities. Based on recent works on porous rock-like composites such as shales or argillites, an application of the developed solution to a two-level microporomechanics model is presented. The microporosity in homogenized at the first level, and multiple solid mineral phase inclusions are added at the second level. The overall porothermoelastic coefficients are estimated in the particular context of heterogeneous solid matrix. Numerical results are presented for data representative of isotropic rock-like composites.     

The effective thermoelectroelastic properties of microinhomogeneous materials

The paper is concerned with composite materials which consist of a homogeneous matrix phase with a set of inclusions uniformly distributed in the matrix. The components of these materials are considered to be ideally elastic and exhibit piezoelectric properties. One of the variants of the self-consistent scheme, the Effective Field Method (EFM) is applied to calculate effective dielectric, piezoelectric and thermoelastic properties of such materials, taking into account the coupled electroelastic effects. At first the coupled thermoelectroelastic problem for a homogeneous medium with an isolated inclusion is solved. For an ellipsoidal inclusion and constant external field the solution of this problem is found in a closed analytic form. This solution is then used in the EFM to derive the effective thermoelectroelastic operator for the composite containing a random array of ellipsoidal inclusions. Explicit formulae for the electrothermoelastic constants are given for composites, reinforced by spheroidal inclusions.     

An efficient approximate numerical method for modeling contact of materials with distributed inhomogeneities

This research explores the influence of distributed non-interpenetrating inhomogeneities on the contact of inhomogeneous materials via a new efficient numerical model based on Eshelby’s Equivalent Inclusion Method. The half-space contact of a sphere with an inhomogeneous material is considered, and the solutions take into account interactions between all inhomogeneities. The efficiency and solution accuracy of the proposed method are demonstrated through comparative studies with those of an existing numerical method and the finite element method. The influence of spatial inhomogeneity orientations on the contact elastic field is investigated and parametric studies are conducted for the effect of arbitrarily distributed inhomogeneities on the stress field of the materials. The significance of the influences of inhomogeneity distribution parameters on the inverse volumetric stress integral is quantified and the corresponding data are fitted into selected several formulas as a step towards understanding the rolling contact fatigue life of the materials.     

The averaged mechanical properties of prismatic lattice formed by ellipsoidal inclusions

Summary The objective of this paper is to evaluate the averaged elastic properties of 3-D grained composites in which identical inclusions form a prismatic network interacting with the matrix material. The inclusions are of ellipsoidal shape with transverse circular sections located at the nodes of a doubly-periodic lattice with an orthogonal elementary cell. When the arrays of inclusions are set at equal spacings in normal directions through the thickness of the matrix, the material formed is an anisotropic composite with tetragonal symmetry at planes transverse to the fiber axis. The longitudinal and transverse elastic and shear moduli as well as the longitudinal Poisson's ratios of such composites are evaluated in this paper. The averaged properties are studied in terms of the aspect ratio and volume fraction of the inclusions as well as the relative rigidity of the constituent phases. Employing the Eshelby's theory for the stress field around a single ellipsoidal inhomogeneity, which is surrounded by the effective anisotropic material, and considering the Mori-Tanaka's concept for the mutual interaction of the neighboring inclusions, we may evaluate the averaged elastic properties of grained composites with aligned ellipsoidal inclusions at finite concentrations. The results provided in a closed-form solution concern the stiffness of 3-D grained composites with parallely dispersed ellipsoidal inclusions forming a prismatic network inside the principal material. It is shown that the stiffness is affected by both the geometry of the inclusions and their concentration. The use of different composite models in the analysis shows that intense variations of stiffness occur mainly in hard composites weakened by soft ellipsoidal inclusions. These findings come in full verification with experimental or theoretical results from the literature. Received 10 February 1998; accepted for publication 27 November 1998     

Elastic fields generated by inhomogeneities: Far-field asymptotics,its shape dependence and relation to the effective elastic properties

We discuss connections between the effective elastic properties of a solid with inhomogeneities and the far-field asymptotics of the elastic fields generated by them. We focus on the dependence of the far-field asymptotics on the inhomogeneity shape. This shape dependence in the inhomogeneity problem is in contrast with shape independence of the far field in the eigenstrain problem. For the latter, the far field applies to inclusions of any shape. We show that the external fields in the eigenstrain – and the inhomogeneity problems are interrelated by the same tensor that characterizes the compliance contribution of an inhomogeneity.     

Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

A transversely isotropic composite with a statistical distribution of spheroidal inclusions: a geometrical approach to overall properties

In the literature, the determination of global elastic properties of composites with ellipsoidal inclusions is based on the averaged stress, strain and elastic-energy fields (e.g. Compos. Sci. Technol. 27 (1986) 111). These are related to the local fields of the inclusion, the matrix, and the inclusion-matrix interface. In this study, we propose a method to obtain the global elastic properties of any transversely isotropic composite directly from the elastic properties of the matrix and the inclusions. Thus, it is not necessary to refer to the stress and strain applied globally or generated locally. The inclusions can have any transversely isotropic probability distribution of orientation. The problem is entirely geometrized and is treated in terms of averages of Walpole's (Adv. Appl. Mech. 21 (1981) 169) components of the fourth-order tensors describing the problem. We give a general numerical solution for any transversely isotropic statistical distribution of orientation, and also provide a validation of our method by applying it to some known cases and by retrieving the known exact solutions from the literature.     

热对流条件下考虑球形夹杂分布的材料热弹接触摩擦热影响分析

机械传动关键活动零部件接触副往往受到力载荷和摩擦热载荷的耦合作用,使得接触界面间的接触力学行为的分析变得极其复杂. 利用基于等效夹杂方法建立的考虑热对流非均质材料热弹接触力学分析模型研究不同摩擦系数、夹杂位置和材料属性等参数对材料表面及内部温升及热应力分布影响规律. 此外,进一步分析了接触副材料中含分布球形夹杂时摩擦热造成的影响. 结果表明:接触副表面温升梯度受热对流系数的影响较大;下表面温升和热应力随摩擦系数增大而增大;分布夹杂则将接触副材料下表面温升及热应力分布变得更为复杂.      

Stick-slip analysis of a circular point contact between a rigid sphere and a flat unidirectional composite with cylindrical fibers

The stick-slip contact problem is investigated here when at least one of the contacting bodies behaves as an ideal composite material with long fibers perpendicular to the direction of movement. Cylindrical inhomogeneous inclusions within a homogeneous media and with axes parallel to the contact surface are considered. The Eshelby’s equivalent inclusion method is used to solve the problem numerically. Interactions between close inclusions are taken into account in the numerical procedure, as well as the coupling between the normal and tangential contact problems. It is found that the presence of heterogeneities in the vicinity of the surface contact affects significantly the contact pressure distribution and subsequently the distribution of shear and slip at the interface.     

Effective thermoelastic properties of graded doublyperiodic particulate matrix composites in varying externalstress fields

We consider a linear elastic composite medium, which consists of a homogeneousmatrix containing aligned ellipsoidal uncoated or coated inclusions arranged in a doubly periodicarray and subjected to inhomogeneous boundary conditions. The hypothesis of effective fieldhomogeneity near the inclusions is used. The general integral equation obtained reduces theanalysis of infinite number of inclusion problems to the analysis of a finite number of inclusions insome representative volume element (RVE) . The integral equation is solved by a modifiedversion of the Neumann series; the fast convergence of this method is demonstrated for concreteexamples. The nonlocal macroscopic constitutive equation relating the cell averages of stress andstrain is derived in explicit iterative form of an integral equation. A doubly periodic inclusion fieldin a finite ply subjected to a stress gradient along the functionally graded direction is considered.The stresses averaged over the cell are explicitly represented as functions of the boundaryconditions. Finally, the employed of proposed explicit relations for numerical simulations oftensors describing the local and nonlocal effective elastic properties of finite inclusion pliescontaining a simple cubic lattice of rigid inclusions and voids are considered. The local andnonlocal parts of average strains are estimated for inclusion plies of different thickness. Theboundary layers and scale effects for effective local and nonlocal effective properties as well as foraverage stresses will be revealed.     

On the solution of the dynamic Eshelby problem for inclusions of various shapes

In many dynamic applications of theoretical physics, for instance in electrodynamics, elastodynamics, and materials sciences (dynamic variant of Eshelby’s inclusion and inhomogeneity problems) the solution of the inhomogeneous Helmholtz equation (‘dynamic’ or Helmholtz potential) plays a crucial role. In materials sciences from such a solution the dynamical fields due to harmonically transforming eigenfields can be constructed. In contrast to the static Eshelby’s inclusion problem (Eshelby, 1957), due to its mathematical complexity, the dynamic variant of the problem is comparably little touched. Only for a restricted set of cases, namely for ellipsoidal, spheroidal and continuous fiber-inclusions, analytical approaches exist. For ellipsoidal shells we derive a 1D integral representation of the Helmholtz potential which is useful to be extended to inhomogeneous ellipsoidal source regions. We determine the dynamic potential and dynamic variant of the Eshelby tensor for arbitrary source densities and distributions by employing a numerical technique based on Gauss quadrature. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method is especially useful to be applied in self-consistent methods (e.g. the effective field method) if one looks for the effective dynamic characteristics of the material containing a random set of inclusions.