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Contact Problems for a Porous Composite in the Presence of Friction Forces
Abstract
A contact problem is considered for a heterogeneous fluid-saturated half-space considering friction forces in the contact domain originated from the movement of a die with a flat or and parabolic base. For to consider the internal microstructure of the base, Biot's model is used. The boundary problem is reduced with the help of the Fourier transformation to a first-kind integral equation with a kernel having a logarithmic singularity. The solution of the integral equation has been constructed using a collocation method. The effect of porosity and friction coefficient exerted on the contact stresses in the oil-filled phenylone-based composite has been studied. The mechanical moduli of the composite have been determined using the methods of micromechanics and finite-elemental simulation, and compared with experimental results.
INTRODUCTION Contact problems and the applications thereof to tribology are attracting the attention of many researchers for a long time [1-5]. The complications in the formulations and approaches that arise when solving contact problems are described in detail in a fundamental monograph [1]. The contact problems in a quasistatic formulation for homogeneous viscoelastic media are considered by the authors of [1-5]. It should be noted that the properties of the contacting surfaces significantly affect the friction force. The microgeometry influence of the contacting surfaces on the friction force has been investigated by the authors of [4, 5]. In this paper, we consider the contact problem in a quasistatic formulation for the motion of a rigid die under friction on the base considering the microstructure of the base. The internal microstructure of the base consisting of a viscoelastic skeleton and a filler fluid is considered by using the equations of a heterogeneous two-phase Biot's medium as determining ones [6-9]. The determination of the mechanical moduli for Biot's medium is a separate and very important problem. The bulk compression moduli of a saturated and drained medium have been determined experimentally, including by the nanoindentation method [10-12], as well as based on micromechanics and finite-element simulation methods. The cases of dies with flat and parabolic bases are considered. The problems of friction force determination are relevant in designing antifriction composites based on a viscoelastic matrix and a fluid filler [11-13].
... In [10], a contact problem with a variable friction coefficient is presented. Contact problems for heterogeneous fluid-saturated media and stamps with different base shapes are considered in [11]. These investigations emphasize the relevance and necessity of a comprehensive theoretical and experimental study using modern experimental means and new mathematical models of the physical processes proceeding in composite materials. ...
... The solution to problem 1 is based on the Fourier integral transform with transition to a moving coordinate system x x Vt y x 1 2 , whose origin is at the center of the stamp. As a result of the transformations described in detail in [11,25], we arrived at the integral equation for normal contact pressures q( ) ξ ...
... where V i is the propagation speed of longitudinal and transverse waves in the heterogeneous medium; m k k , , = 1 2 , are roots of the quadratic equation [11] whose coefficients depend on the mechanical parameters of the Biot medium. ...
The physical and mechanical properties of antifriction oil-filled composites with a viscoelastic nanomodified matrix are investigated on the basis of mathematical models. The identification and prediction of their mechanical properties is performed using a micromechanics model with account of experimental data obtained in nanoindentation. Solutions of new contact problems in a quasi-static formulation of the motion of a punch with a flat base into a heterogeneous oil-saturated half-space with allowance for friction in the contact area are constructed. The multiphase heterogeneous medium is described, first, within the framework of the Biot–Frenkel model and second, using the concept of effective homogeneity. The contact problem for the Biot medium is reduced to an integral equation of the first kind with a differential kernel and a logarithmic singularity. After regularization, the numerical solution of the integral equation is constructed by the boundary element method. The solution to the boundary value problem was found by the finite-element method in the ANSYS software package for an equivalent homogeneous medium. A comparative analysis of two approaches to modeling the microstructure of a heterogeneous medium is presented. The influence of mechanical properties of the composite on its stress-strain state is investigated. The magnitude of the friction force arising in the contact area of the medium is studied. Such studies are of great practical importance in investigating new nanomodified antifriction composite materials. For this purpose, numerical calculations for an oil-filled composite with a phenylone matrix and nanosize additives are presented. The influence of porosity, fluid saturation, and friction coefficient on the tangential contact stresses is also examined.
... Much work on the influence of microgeometry of contacting surfaces on the friction force was carried out by [9]. Consideration of the base microstructure as a whole for contact stresses was presented in [10][11][12]. In these studies, contact problems in quasistatic and dynamic formulations were considered, and the microstructure of a heterogeneous medium was described in terms of the Biot-Frenkel model. ...
... Note that at low porosity ( 0.2 m < ), the relative error of the two calculation methods did not exceed 4%. Thus, the coefficients of equations (1) for the known bulk modulus of a viscoelastic matrix s K , drained porous medium b K , fluid f K , and porosity m were calculated by the formulas [10]. ...
... Moreover, the stress distribution over the contact area is asymmetric, which is also characteristic of contact problems of the theory of elasticity with allowance for the friction forces. It is important to note that for the case of dynamic loading, the influence of the coefficient of friction on contact pressures is more pronounced than in problems in a quasistatic formulation [7,10]. ...
Based on the solution of the dynamic contact problem of vibration of a rigid punch on a heterogeneous half-space, taking into account friction in the contact area, the tribological properties of an oil-filled composite material with a microstructure are modeled. The microstructure of the base is taken into account in the framework of the Biot-Frenkel model. The boundary-value problem is reduced to the integral equation, its approximate solution is constructed, which describes contact stresses, tangential displacements. The dependencies of the friction forces on the microstructure of the composite, the viscosity of the fluid filling the pores, and the degree of phase interaction are investigated. Keywords: dynamic contact problem with friction, oil-filled composite 1. Introduction Currently, there is a considerable amount of concepts, hypotheses, and research on diverse friction and wear-related issues. Many attempts have been made to study the properties, structure, and state of the surface layers of tribological conjugations at the atomic and molecular levels. The core problem of surface engineering (for example, metal friction units) is the synthesis of coating technologies and materials with specified wear-resistant properties. The most remarkable result to emerge from our study [1] is that the methods of vacuum ion-plasma treatment (PVD-method) and atomic modifications of diamond-like coatings (DLC) are the most appropriate among the wide array of methods aimed at hardening surface and improving tribological properties. However, along with increasing the strength properties of the materials, such technologies contribute to the formation of a damping layer with residual compressive stresses inside and a large number of stress micro-concentrators at the boundary with the main material. The above-listed aspects predetermine the formulation and solution of the dynamic contact problem, taking into account the friction forces on tribocontact. This problem is attracting an increasing interest due to the widespread application of polymer composite materials [2]. Recently, oil-filled nanocomposites, whose components are characterized with viscoelastic properties and the properties of viscous-fluid filler, have become extensively used in units and parts of tribotechnical purposes [3-5]. Constructing new composite materials with given physical and mechanical properties poses an actual, yet practically unexplored task to study the influence of dynamic effects that are caused by vibration on the antifriction properties of these materials. We undertook this study to investigate the patterns of changes in the stress-strain state of a composite material depending on its composition and dynamic loading conditions by solving a dynamic contact problem. The latest takes into account friction in the contact area for a base with a microstructure. The
... content of Ti/Al in TiAlN coatings. Furthermore, Eq. 1 allows simulating various real forms of inclusions, incorporating limiting acicular and plate-like forms, which are peculiar to nanotubes and graphene fragments [13,14], spherical nanoparticles [15][16][17], and superellipsoidal pores [18]. To verify the findings obtained by Eq. 1-based calculations, the program was tested on special cases for the components of the Eshelby's tensor for an isotropic medium with spherical and thin pennyshaped inclusions. ...
The paper outlines two approaches (the ANSYS finite element modeling and the differential scheme of the self-consistency method) to model the mechanical properties of TiAlN multilayer coatings. To validate the experimental results, these findings were compared with the nanoindentation test results for single and multilayer coatings. This work has revealed the effect of Ti/Al ratio on the mechanical properties of multilayer coatings. The given approaches could be applied to calculate, with an acceptable margin of error, the mechanical properties of TiAlN multilayer coatings.
In this paper, the study of the tribological properties of self-lubricating composites is based on a mathematical model taking into account experimental data. We considered self-lubricating composite materials based on phenylone, which is modified with ultrafine PTFE powder. Theoretical studies were carried out on the basis of a numerical solution of the quasi-static contact problem on the sliding of a punch with a spherical base along the boundary of a microinhomogeneous medium. It was assumed that the coefficient of friction depends on the path traveled by the punch. The microstructure of the base was taken into account within the framework of the concept of equivalent homogeneity. The numerical solution of the contact problem was carried out in the finite element complex Ansys. The stress state of the composite depending on the variable coefficient of friction and the mechanical properties of the medium was investigated. The dependence of the coefficient of friction on the distance traveled was determined on the basis of laboratory experiments. Functional dependences of stresses in the contact area, internal stresses in the projection of the contact area, maximum shear stresses depending on the mechanical properties of the heterogeneous medium and the variable coefficient of friction are established.KeywordsVariable coefficient of frictionSelf-lubricating compositeQuasi-static contact problem
The problem of determining the stress-strain state of highly flexible rods is relevant when using them in damping devices. Rods of high flexibility, non-linearly resisting external load, experience large movements during longitudinal bending. The method for determining the stress-strain state of highly flexible rods developed on the basis of the elliptic parameter method allows us to analyze the results obtained, taking into account various boundary conditions.KeywordsHigh flexibility rodsStress-strain stateElliptic parameter method
The tasks of statistical analysis of data obtained during applied research in various industries are described. The main tasks that allow estimating distributions, as well as selecting parameters of approximating dependencies, and checking the fulfillment of assumptions about the method of obtaining data are considered. Materials of theoretical and experimental research on the development of mathematical models of torsion of cylindrical metal working elements of systems of protection against dynamic loads for the purpose of their application in calculations of vibrations of mechanical systems are proposed. Mathematical models of torsion of the considered elements are proposed, which will allow us to fairly accurately describe the relationship between stresses and deformations, both at the boundaries of the limit hysteresis loops and inside them.
This study aims to examine multilayer TiAlN coatings. The mechanical properties of the coatings were obtained based on micromechanics methods and finite element modeling of a representative volume of a layered medium in ANSYS. Our findings are in line with the data on coating nanoindentation. Besides, we have succeeded in finding the dependence of the mechanical properties of multilayer coatings on various Al/Ti ratios.
In this paper, the problems of directional modeling of the tribological properties of composites by introducing oils into them are considered. The contact problem of moving a flat punch, taking friction into account, on a heterogeneous semi-infinite media is considered. The Biot-Frenkel equations are taken into account. Mechanical constants included in the defining relations are found on the basis of the effective medium method and confirmed by numerical finite element method simulation. The solution of the contact problem is reduced to an integral equation of the first kind with a logarithmic kernel, the numerical solution of which is based on an iterative algorithm. The dependences of the contact stresses for the oil-filled composite on the porosity, the volume of applied filler, and the speed of the movement of the punch are investigated.
The article suggests the technology of modifying a polymer matrix by microencapsulation, i. e. the introduction of microparticles (lubricants with nano-additives in polymer shells) into nanocomposites matrix, to form multilevel structures on the tribounit surface. Besides, it suggests the method of predicting the operational elastic properties of multicomponent matrix composites with microcapsules, filled with a liquid substance. The method is based on the generalized singular approximation of the theory of random fields and allows, taking into account the geometric dimensions of the inclusions in the shell. It contains the results of numerical modelling of the effective elastic characteristics (Young’s modulus and Poisson’s ratio) of composites, based on phenylone with dispersed inclusions (microcapsules), which are glycerin-filled spherical shells of the kapton. The paper investigates the effect of the geometric dimensions of microcapsules and the volumetric content of components on the operational elastic properties of tribocomposites. The developed antifriction nanomaterials with microcapsules are able to create an oriented lubricating coating on the friction surfaces, apply lubrication to a certain friction area and carry out the lubrication portion wise precisely in the necessary contact zone of the bodies.
Contact of viscoelastic materials with complicated properties and surface topography require numerical solution approaches. This paper presents a 3-D semianalytical contact model for viscoelastic materials. With the hereditary integral operator and elastic-viscoelastic correspondence principle, surface displacement is expressed in terms of viscoelastic creep compliance and contact pressure distribution history in the course of a contact process. Through discretizing the contact equations in both spatial and temporal dimensions, a numerical algorithm based on the robust Conjugate Gradient method and Fast Fourier transform has been developed to solve the normal approach, contact pressure, and real contact area simultaneously. The transient contact analysis in the time domain is computationally expensive. The fast Fourier transform algorithm can help reduce the computation cost significantly. The comparisons of the new numerical results with an analytical viscoelastic contact solution for Maxwell materials and with an indentation test measurement reported in the literature has validated and demonstrated the accuracy of the proposed model. Moreover, the present model has been used to simulate the contact between a polymethyl methacrylate (PMMA) substrate and a rigid sphere driven by step, ramped, and harmonic normal loads. The validated model and numerical method can successfully compute the viscoelastic contact responses of polymer-based materials with time-dependent properties and surface roughness subjected to complicated loading profiles. [DOI:10.1115/1.4004928]
The behavior of hysteretic, coupled elastic and fluid systems is modeled. The emphasis is on quasistatic equilibrium in response to prescribed chemical potential (μ) protocols and prescribed stress (σ) protocols. Hysteresis arises in these models either from the presence of hysterons or from the presence of self-trapping internal fields. This latter mechanism is modeled in finite element calculations which serve to illustrate the creation of hysteresis in a range of circumstances that go from conventionally hysteretic systems, a sandstone, to systems like a wood fiber. An essential ingredient in the behavior of these systems, the interaction between the mechanical variables and the fluid variables, is accorded special attention. The proper venue for the exploration of these systems is (μ,σ) space and appropriate μ protocols, σ protocols, and combined μ-σ protocols.
In this paper, we consider a mathematical model that describes the contact interaction of a semi-bounded medium with a microstructure and a rigid planar punch with allowance for friction in the contact area. The equations of a heterogeneous bi-phase Biot medium are used as determining ones to take into account the internal microstructure of a semi-limited medium. The boundary-value problem is reduced to an integral equation of the first kind with a kernel having a logarithmic singularity. The influence of a two-phase medium and the speed of the punch on the stress-strain state in the contact area is investigated.
Contact problems on the surface interaction of rigid stamps with a deformed layered medium are considered provided that the variable friction coefficients arise in the contact zone as a function of the coordinate under the horizontal motion of stamps. The cause of the variable friction coefficients arising may be surface phenomena induced by a complex rheology of the deformed-medium surface, the chemical reactions proceeding, or a change in the properties of the contact surface of the stamps, for example, as a result of the presence of separate particles of the wear contact surface of the stamp and the base.
The problems of interaction between railway track and a train at high speeds have become particularly relevant with the advent of high-speed trains and rail lines. On the basis of the mathematical model of the elastic half-space it has been theoretically determined and experimentally confirmed that the propagation velocity of surface waves is critical in increasing the speed of the train, causing a sharp increase of oscillations in a system of railway-the soil medium. In this work, heterogeneous layer consisting of an elastic skeleton and pores filled with liquid is considered as a model of soil medium to determine the degree of influence of heterogeneity structure and saturation of the underlying environment in the upper structure of the railway on its deformation during high-speed movement. This allows you to more accurately describe the deformation process in saturated by air and moisture soils. The passage of a train at the rail-sleeper lattice is modeled by an equivalent moving load, distributed in a rectangular region. Special attention is paid to the study of critical speeds of movement resulting from properties of the ground soil. The decision of model boundary value problem is constructed by applying the integral transformation of Fourier to equations Bio, determining heterogeneous medium. After the exact satisfaction of the boundary conditions, the integral formulas, describing the stress-strain condition in the layer in a moving coordinate system connected with the moving load, is constructed. The movements in a heterogeneous layer near the load zone are calculated using a numerical algorithm, in the far zone an effective asymptotic formula is constructed. The appearance and mechanical properties of the base significantly determine the critical speed of wave propagation in the medium, which depends on the porosity and the water content of the base. Experimental data, confirming the theoretical conclusions are given.
The solution of the problem of interaction with regard to the forces of adhesive (molecular) attraction of two nominally plane half-spaces one of which is elastic and the surface of the other has a regular relief is presented. The surface mutual approach dependence on the applied nominal pressure and the effective specific work of adhesion are analyzed for various parameters of adhesive interaction and micro-geometry of the surfaces.
The behaviors of dilatational wave propagation and attenuation through eleven different types of saturated soils (sand, loamy sand, sandy loam, loam, silt loam, sandy clay loam, clay loam, silty clay loam, sandy clay, silty clay, and clay) whose pore spaces are completely occupied either by water or by air were investigated numerically in this study based on the celebrated Biot model equations of poroelasticity. The corresponding frequency equation was first derived and then solved numerically to determine the phase velocity and attenuation coefficient of the Biot fast (P1) and slow (P2) dilatational waves in the lower-frequency range (50 to 250 Hz). Our results show that the phase velocity of P1 wave is equal to the square root of the ratio of an effective bulk modulus to an effective density of the fluid-containing porous medium, regardless of excitation frequency and pore fluid. In the water-saturated case, the P1 wave propagates slowest in silt loam whereas in the air-saturated case clay has the smallest velocity. Among eleven soil texture classes, clay also has the greatest difference in phase velocity of the P1 wave between the air-saturated and water-saturated cases. The attenuation coefficient of the P1 wave is found to be approximately proportional to the square of the excitation frequency. The magnitude of the phase velocity of attenuation coefficient of the P2 wave increases with the square root of the excitation frequency.
An exact stiffness formalism is presented to study harmonic and transient wave propagation in multilayered dry, saturated and unsaturated isotropic poroelastic media. Smeulders’ extension of Biot’s poroelastic theory is used to incorporate unsaturated porous media with a small gas fraction. A wide range of problems in geophysical and civil engineering can be treated, ranging from amplification of plane harmonic waves, dispersion and attenuation of surface waves to transient wave propagation due to a forced excitation. The effect of full or partial saturation on wave propagation in a poroelastic layered halfspace is demonstrated in a numerical example, in which the layering is caused by a moving ground water table.
Experiments on explosive (impulsive) loading of composite rings fabricated by winding are performed. Special loading conditions were used to excite simultaneous axisymmetric and flexural vibrations. Averaged damping characteristics of axisymmetric motion and the eigenfrequencies of axisymmetric and flexural vibrations of glass–reinforced, acrylic, and carbon–filled plastic rings were obtained from experimental results.
A unified treatment of the mechanics of deformation and acoustic propagation in porous media is presented, and some new results and generalizations are derived. The writer's earlier theory of deformation of porous media derived from general principles of nonequilibrium thermodynamics is applied. The fluid‐solid medium is treated as a complex physical‐chemical system with resultant relaxation and viscoelastic properties of a very general nature. Specific relaxation models are discussed, and the general applicability of a correspondence principle is further emphasized. The theory of acoustic propagation is extended to include anisotropic media, solid dissipation, and other relaxation effects. Some typical examples of sources of dissipation other than fluid viscosity are considered.
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