Some Criteria for Coating Effectiveness in Heavily Loaded Line Elastohydrodynamically Lubricated Contacts—Part I: Dry Contacts

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Ilya I. Kudish ◽  
Sergey S. Volkov ◽  
Andrey S. Vasiliev ◽  
Sergey M. Aizikovich
Keyword(s):  
New York ◽  
Asymptotic Methods ◽  
Elastohydrodynamic Lubrication ◽  
Contact Problems ◽  
Functionally Graded ◽  
Frictional Stress ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Lubrication Film ◽  
Lubricated Contacts

Contacts of indentors with functionally graded elastic solids may produce pressures significantly different from the results obtained for homogeneous elastic materials (Hertzian results). It is even more so for heavily loaded line elastohydrodynamically lubricated (EHL) contacts. The goal of the paper is to indicate two distinct ways the functionally graded elastic materials may alter the classic results for the heavily loaded line EHL contacts. Namely, besides pressure the other two main characteristics which are influenced by the nonuniformity of the elastic properties of the contact materials are lubrication film thickness and frictional stress/friction force produced by lubricant flow. The approach used for analyzing the influence of functionally graded elastic materials on parameters of heavily loaded line EHL contacts is based on the asymptotic methods earlier developed by authors (Kudish, 2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman & Hall/CRC Press, New York; Kudish and Covitch, 2010, Modeling and Analytical Methods in Tribology, Chapman & Hall/CRC Press, New York; and Aizikovich et al., 2006, Contact Problems of Elasticity for Functionally Graded Materials, Fizmatlit, Moscow, Russia). More specifically, it is based on the analysis of contact problems for dry contacts of functionally graded elastic solids and the lubrication mechanisms in the inlet and exit zones as well as in the central region of heavily lubricated contacts.

Some Criteria for Coating Effectiveness in Heavily Loaded Line Elastohydrodynamically Lubricated Contacts—Part II: Lubricated Contacts

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Ilya I. Kudish ◽  
Sergey S. Volkov ◽  
Andrey S. Vasiliev ◽  
Sergey M. Aizikovich
Keyword(s):  
Analytical Solution ◽  
Contact Problem ◽  
Half Space ◽  
Contact Problems ◽  
Functionally Graded ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Lubricated Contacts ◽  
Dry Contact ◽  
Appl Math

Contacts of indentors with functionally graded elastic solids may produce pressures significantly different from the results obtained for homogeneous elastic materials (Hertzian results). It is even more so for heavily loaded line elastohydrodynamically lubricated (EHL) contacts. The goal of the paper is to indicate two distinct ways the functionally graded elastic materials may alter the classic results for the heavily loaded line EHL contacts. Namely, besides pressure, the other two main characteristics which are influenced by the nonuniformity of the elastic properties of the contact materials are lubrication film thickness and frictional stress/friction force produced by lubricant flow. The approach used for analyzing the influence of functionally graded elastic materials on parameters of heavily loaded line EHL contacts is based on the asymptotic methods developed earlier by the authors such as Kudish (2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman & Hall/CRC Press, Boca Raton, FL), Kudish and Covitch (2010, Modeling and Analytical Methods in Tribology, Chapman & Hall/CRC Press, Boca Raton, FL), Aizikovich et al. (2002, “Analytical Solution of the Spherical Indentation Problem for a Half-Space With Gradients With the Depth Elastic Properties,” Int. J. Solids Struct., 39(10), pp. 2745–2772), Aizikovich et al. (2009, “Bilateral Asymptotic Solution of One Class of Dual Integral Equations of the Static Contact Problems for the Foundations Inhomogeneous in Depth,” Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, p. 317), Aizikovich and Vasiliev (2013, “A Bilateral Asymptotic Method of Solving the Integral Equation of the Contact Problem for the Torsion of an Elastic Halfspace Inhomogeneous in Depth,” J. Appl. Math. Mech., 77(1), pp. 91–97), Volkov et al. (2013, “Analytical Solution of Axisymmetric Contact Problem About Indentation of a Circular Indenter Into a Soft Functionally Graded Elastic Layer,” Acta Mech. Sin., 29(2), pp. 196–201), Vasiliev et al. (2014, “Axisymmetric Contact Problems of the Theory of Elasticity for Inhomogeneous Layers,” Z. Angew. Math. Mech., 94(9), pp. 705–712), Aizikovich et al. (2008, “The Deformation of a Half-Space With a Gradient Elastic Coating Under Arbitrary Axisymmetric Loading,” J. Appl. Math. Mech., 72(4), pp. 461–467), and Aizikovich et al. (2010, “Inverse Analysis for Evaluation of the Shear Modulus of Inhomogeneous Media in Torsion Experiments,” Int. J. Eng. Sci., 48(10), pp. 936–942). More specifically, it is based on the analysis of contact problems for dry contacts of functionally graded elastic solids and the lubrication mechanisms in the inlet and exit zones as well as in the central region of heavily loaded lubricated contacts. The way the solution of the EHL problem for coated/functionally graded materials is obtained provides a very clear structure of the solution. The solution of the EHL problem in the Hertzian region is very close to the solution of the dry contact problem while in the inlet and exit zones the solutions of the EHL problem with the right asymptotes coming from the solution of the dry contact problem can be related to the solutions of the classic EHL problem for homogeneous materials.

Effectiveness of Coatings With Constant, Linearly, and Exponentially Varying Elastic Parameters in Heavily Loaded Line Elastohydrodynamically Lubricated Contacts

2016 ◽  
Vol 139 (2) ◽  
Author(s):  
Ilya I. Kudish ◽  
Sergey S. Volkov ◽  
Andrey S. Vasiliev ◽  
Sergey M. Aizikovich
Keyword(s):  
Asymptotic Methods ◽  
Functionally Graded ◽  
Problem Solution ◽  
Elastic Materials ◽  
Elastic Parameters ◽  
Graded Materials ◽  
Lubrication Film ◽  
Lubricated Contacts ◽  
Homogeneous Materials ◽  
The One

In contacts of functionally graded elastic solids, the conditions produced are significantly different from the ones in similar contacts of homogeneous elastic materials. Especially it is true for heavily loaded lubricated contacts. The situation is even more diverse due to different dependences of the material elastic parameters on material depth. In the previous papers, the cases of lubricated contacts with coatings made of homogeneous materials are considered using asymptotic and semi-analytical methods. The goal of the paper is to determine the behavior of the coating effectiveness criteria in heavily loaded elastohydrodynamically lubricated (EHL) contacts for coatings with elastic modulus varying linearly and exponentially across the coating thickness as well as to compare the results with the case of coatings made of homogeneous materials. The above criteria include the criteria on the lubrication film thickness and friction force. The approach used for analyzing the influence of functionally graded elastic materials on parameters of heavily loaded line EHL contacts is based on the asymptotic methods earlier developed by the authors. The analysis is based on splitting the problem into two distinct parts: the problem for dry (nonlubricated) contacts and a problem for lubricated contacts. The bridge between the two problems is the asymptotic behavior of pressure in the vicinity of the end points of the contacts. More specifically, in the central part of the contact the solution of the EHL problem for functionally graded materials is close to the one for the dry contact of these materials while in the narrow zones near the inlet and exit points of the contact the lubrication effects become comparable to the effects due to the elasticity of the solids. This approach to the EHL problem solution reveals its structure.

Lubricated point heavily loaded contacts of functionally graded materials. Part 2. Lubricated contacts

2017 ◽  
Vol 23 (7) ◽  
pp. 1081-1103 ◽  
Author(s):  
Ilya I Kudish ◽  
Sergey S Volkov ◽  
Andrey S Vasiliev ◽  
Sergey M Aizikovich
Keyword(s):  
Functionally Graded Materials ◽  
Central Region ◽  
Contact Problems ◽  
Functionally Graded ◽  
Problem Solution ◽  
Elastic Materials ◽  
Graded Materials ◽  
Lubrication Film ◽  
Lubricated Contacts ◽  
Lubricated Contact

The paper presents a study of the effectiveness of functionally graded materials in heavily loaded point elastohydrodynamically lubricated contacts with straight lubricant entrainment. As part of the study, some criteria of coating effectiveness are introduced and discussed. The approach used for achieving this goal is based on the recently developed asymptotic solution method for steady isothermal elastohydrodynamically lubricated problems of heavily loaded point contacts, as well as on a semi-analytical method for the solutions of contact problems for elastic functionally graded materials without the presence of lubrication in the contacts (dry contacts). For line elastohydrodynamically lubricated contacts such a problem has already been studied. The solution process can be subdivided into two stages: the solution of the contact problems for dry contacts and a solution of the problem for lubricated contacts. The solution of dry contact problems for functionally graded materials is described in detail in Part 1 of the paper. As in the case of homogeneous contact materials in the analysis of the elastohydrodynamically lubricated problem for functionally graded elastic materials, it is shown that the whole contact region can be subdivided into three subregions: the central one which is adjacent to the other two regions occupied by the ends of the zones. The central region can be subdivided into the Hertzian region and adjacent to it the inlet and exit zones which, in turn, are adjacent to the inlet and exit boundaries of the contact, respectively. In the Hertzian region the elastohydrodynamically lubricated problem solution is very close to the solution of the corresponding dry (i.e. non-lubricated) contact problem for functionally graded elastic materials which have been analyzed in Part 1 of the paper. In the central region in the inlet and exit zones of the heavily loaded point elastohydrodynamically lubricated contact, the elastohydrodynamically lubricated problem for functionally graded elastic materials using certain similarity transforms can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of the heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials. Therefore, many of the well known properties of heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials are also valid for heavily loaded point elastohydrodynamically lubricated contacts for functionally graded elastic materials. These asymptotically valid equations can be analyzed and numerically solved based on stable methods using a specific regularization approach which was developed for lubricated line contacts. Also, this asymptotic analysis leads to the easy analytical derivation of equations for the lubrication film thickness which take into account the inhomogeneity of the elastic materials. As a result of this analysis, criteria for an increase in lubrication film and friction force reduction are proposed. These criteria depend on lubricant properties, as well as the properties of functionally graded elastic materials involved in lubricated contacts.

Lubricated point heavily loaded contacts of functionally graded materials. Part 1. Dry contacts

2017 ◽  
Vol 23 (7) ◽  
pp. 1061-1080 ◽  
Author(s):  
Ilya I Kudish ◽  
Sergey S Volkov ◽  
Andrey S Vasiliev ◽  
Sergey M Aizikovich
Keyword(s):  
Contact Problem ◽  
Film Thickness ◽  
Functionally Graded Materials ◽  
Functionally Graded ◽  
Elastic Materials ◽  
Graded Materials ◽  
Lubrication Film ◽  
Lubricated Contacts ◽  
Dry Contact ◽  
Lubricated Contact

Over the last couple decades coatings attract more and more attention in practical applications. The present study addresses a question which is not well studied: how coated surfaces behave in lubricated contacts? In other words, this is a study of the effectiveness of functionally graded materials in heavily loaded point elastohydrodynamically lubricated contacts with straight lubricant entrainment. As a part of the study, some criteria of coating effectiveness are introduced and discussed. More specifically, the behavior of main parameters such as the lubrication film thickness and the frictional force in point heavily loaded lubricated contacts of functionally graded elastic materials are considered. The problem is studied based on the method of matched asymptotic expansions which allows us to split the problem into two separate problems: a dry contact problems for functionally graded elastic materials and an elastohydrodynamically lubricated problem for functionally graded materials. The elastohydrodynamically lubricated problem uses as input data not only the operational and physical parameters of the materials and lubricant but also the asymptotic behavior of the dry contact problem solution near the contact boundaries. Therefore, a sequence of two problems must be solved: the dry contact problems for functionally graded elastic materials and the elastohydrodynamically lubricated problem for functionally graded materials. Similar methods have been used for the analysis of an elastohydrodynamically lubricated problem for heavily loaded line contacts of functionally graded materials. The dry contact problem will be analyzed in Part 1 of the paper based on a semi-analytical bilateral method which produces correct asymptotic solutions for thin and thick coatings. The analytical expressions for contact pressure are obtained and analyzed for various combinations of coating thicknesses and elastic properties. The elastohydrodynamically lubricated problem will be considered in Part 2 of the paper based on the method of matched asymptotic expansions. In the analysis of the elastohydrodynamically lubricated problem, as in the case of homogeneous contact materials, it is shown that the whole contact region can be subdivided into three subregions: the central one which is adjacent to the other two regions occupied by the ends of the zones. The central region can be subdivided into the Hertzian region and then adjacent to it inlet and exit zones which, in turn, are adjacent to the inlet and exit boundaries of the contact, respectively. In the Hertzian region the elastohydrodynamically lubricated problem solution is very close to the solution of the corresponding dry (i.e. non-lubricated) contact problem for functionally graded elastic materials which have been analyzed. In the central region in the inlet and exit zones of a heavily loaded point elastohydrodynamically lubricated contact, the elastohydrodynamically lubricated problem for functionally graded elastic materials using certain scaling transforms can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials. Therefore, many of the well known properties of heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials are also valid for heavily loaded point elastohydrodynamically lubricated contacts for functionally graded elastic materials. These asymptotically valid equations can be analyzed and numerically solved based on stable methods using a specific regularization approach, which were developed for lubricated line contacts. Also, this asymptotic analysis leads to an easy analytical derivation of formulas for the lubrication film thickness which take into account the inhomogeneity of the elastic materials. As a result of this analysis, some criteria for lubrication film thickness increase and friction force reduction are proposed. These criteria depend on lubricant properties as well as the properties of functionally graded elastic materials involved in lubricated contacts. Such a sequential solution of the elastohydrodynamically lubricated problem for functionally graded materials makes it perfectly clear what the dependence is of elastohydrodynamically lubricated contact parameters on the solid material (including the coating) and lubricant properties.

Effect of Roughness Orientation on the Elastohydrodynamic Lubrication Film Thickness

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Dong Zhu ◽  
Q. Jane Wang
Keyword(s):  
Film Thickness ◽  
Stochastic Models ◽  
Deterministic Model ◽  
Elastohydrodynamic Lubrication ◽  
Contact Problems ◽  
Operating Conditions ◽  
Orientation Effect ◽  
Line Contact ◽  
Lubrication Film ◽  
Dry Contact

Effect of roughness orientation on lubricant film thickness has been an important issue of surface design, attracting much attention since the 1970 s. A systematical study, however, is still needed for various contact types in an extended range of operating conditions, especially in mixed lubrication cases with film thickness to roughness ratio (λ ratio) smaller than 0.5. The present study employs a deterministic mixed elastohydrodynamic lubrication (EHL) model to investigate the performance of lubricating films in different types of contact geometry, including the line contact, circular contact, and elliptical contacts of various ellipticity ratios. The speed range for analyzed cases covers 11 orders of magnitude so that the entire transition from full-film and mixed EHL down to dry contact (corresponding λ ratio from about 3.5 down to 0.001 or so) is simulated. Three types of machined surfaces are used, representing transverse, longitudinal, and isotropic roughness, respectively. The line contact results are compared with those from the stochastic models by Patir and Cheng (“Effect of Surface Roughness Orientation on the Central Film Thickness in EHD Contacts,” Proc. 5th Leeds-Lyon Symp. on Tribol., 1978, pp. 15–21) and the influence of roughness orientation predicted by the deterministic model is found to be less significant than that by the stochastic models, although the basic trends are about the same when λ > 0.5. The orientation effect for circular or elliptical contact problems appears to be more complicated than that for line contacts due to the existence of significant lateral flows. In circular contacts, or elliptical contacts with the ellipticity ratio smaller than one, the longitudinal roughness may become more favorable than the isotropic and transverse. Overall, the orientation effect is significant in the mixed EHL regime where theλratio is roughly in the range from 0.05 to 1.0. It is relatively insignificant for both the full-film EHL (λ > 1.2 or so) and the boundary lubrication/dry contact (λ < 0.025 ∼ 0.05).

Uniform Equations in the Inlet and Exit Zones of Heavily Loaded Point and Line Elastohydrodynamically Lubricated Contacts Involved in Various Steady Motions

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Ilya I. Kudish
Keyword(s):  
Film Thickness ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Point Contacts ◽  
Lubrication Film ◽  
Lubricated Contacts ◽  
Asymptotic Equations ◽  
Line Contacts ◽  
Steady Motions ◽  
Numerical Approaches

Heavily loaded point elastohydrodynamically lubricated (EHL) contacts involved in steady purely transitional, skewed transitional, and transitional with spinning motions are considered. It is shown that in the central parts of the inlet and exit zones of such heavily loaded point EHL contacts the asymptotic equations governing the EHL problem along the lubricant flow streamlines for the above types of contact motions can be reduced to two sets of asymptotic equations: one in the inlet and one in the exit zones. The latter sets of equations are identical to the asymptotic equations describing lubrication process in the inlet and exit zones of the corresponding heavily loaded line EHL contact (Kudish, I. I., 2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman and Hall/CRC). For each specific motion of a point contact, a separate set of formulas for the lubrication film thickness is obtained. For different types of contact motions, these film thickness formulas differ significantly (Kudish, I. I., 2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman and Hall/CRC). For heavily loaded contacts, the discovered relationship between point and line EHL problems allows to apply to point contacts most of the results obtained for line contacts (Kudish, I. I., 2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman and Hall/CRC; Kudish, I. I., and Covitch, M. J., 2010, Modeling and Analytical Methods in Tribology, Chapman and Hall/CRC).

Revision of a Fundamental Assumption in the Elastohydrodynamic Lubrication Theory and Friction in Heavily Loaded Line Contacts With Notable Sliding

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Ilya I. Kudish
Keyword(s):  
Stress Reduction ◽  
Elastohydrodynamic Lubrication ◽  
Soft Materials ◽  
Solid Surfaces ◽  
Frictional Stress ◽  
Friction Modeling ◽  
Practical Applications ◽  
Lubricated Contacts ◽  
Line Contacts ◽  
Improved Model

An analysis of the classic friction modeling in lubricated contacts is conducted. Its major deficiency for soft materials (low-elastic moduli) leading to significant overstating of friction in heavily loaded isothermal and thermal lubricated contacts is revealed. An improved model of friction in a heavily loaded lubricated contact is proposed. The model is based on incorporating the tangential displacements of the solid surfaces in contact, leading to a significant reduction of the frictional stress due to the decrease of the actual sliding of lubricated surfaces. Generally, this frictional stress reduction increases with the slide-to-roll ratio, and it is extremely important for high slide-to-roll ratios for which classic approaches lead to unrealistically overestimated values of frictional stresses. The high slide-to-roll ratio values can be found in many practical applications, such as clutches. Several examples of the frictional stress calculated based on this model as well as the comparison with the classical results are given for the case of smooth solid surfaces and lubricants with Newtonian rheology. Also, the results allow to take a look at the role and the necessity of considering thermal and lubricant non-Newtonian effects on solution of various EHL problems for such heavily loaded contacts.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

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