FROM WINKLER’S FOUNDATION TO POPOV’S FOUNDATION

In recent years, the method of dimensionality reduction (MDR) has started to figure as a very convenient tool for dealing with a wide class of elastic contact problems. The MDR modeling framework introduces an equivalent punch profile and a one-dimensional Winkler-type elastic foundation, called henceforth Popov’s foundation. While the former mainly accounts for the geometry of contact configuration, the Popov foundation inherits the main characteristics of both the contact interface (like friction and adhesion) and the contacting elastic bodies (e.g., anisotropy, viscoelasticity or inhomogeneity). The discussion is illustrated with an example of the Kendall-type adhesive contact for an isotropic elastic half-space.

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Inclined Flat Punch of Arbitrary Shape on an Elastic Half-Space

Journal of Applied Mechanics ◽

10.1115/1.3171861 ◽

1986 ◽

Vol 53 (4) ◽

pp. 798-806 ◽

Cited By ~ 10

Author(s):

V. I. Fabrikant

Keyword(s):

Integral Representation ◽

Half Space ◽

Arbitrary Shape ◽

Contact Problems ◽

Elastic Half Space ◽

Elastic Contact ◽

Circular Sector ◽

Flat Punch ◽

Circular Segment ◽

Arbitrary Planform

A new method is proposed for the analysis of elastic contact problems for a flat inclined punch of arbitrary planform under the action of a normal noncentrally applied force. The method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier. Some simple yet accurate relationships are established between the tilting moments and the angles of inclination of an arbitrary flat punch. Specific formulae are derived for a punch whose planform has a shape of a polygon, a triangle, a rectangle, a rhombus, a circular sector and a circular segment. All the formulae are checked against the solutions known in the literature, and their accuracy is confirmed.

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Self similar solutions to adhesive contact problems with incremental loading

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1968.0105 ◽

1968 ◽

Vol 305 (1480) ◽

pp. 55-80 ◽

Cited By ~ 150

Keyword(s):

Integral Equation ◽

Boundary Value Problem ◽

Half Space ◽

Boundary Value ◽

Contact Problems ◽

Adhesive Contact ◽

Elastic Half Space ◽

The Body ◽

The Use of Various Types of Pressure Element in Some Elastic Contact Problems

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/pime_proc_1984_198_107_02 ◽

1984 ◽

Vol 198 (3) ◽

pp. 189-196 ◽

Cited By ~ 4

Author(s):

A Mostofi ◽

R Gohar

Keyword(s):

Contact Problem ◽

Three Dimensional ◽

Contact Problems ◽

Elastic Half Space ◽

Considerable Improvement ◽

Percentage Error ◽

Elastic Contact ◽

Rectangular Section ◽

Gear Teeth ◽

Triangular Elements

The numerical solution to the problem of the contact between an elastic half-space and a punch of rectangular section, is compared with the classical solution. The numerical method uses both triangular and rectangular pressure elements in order to find which gives the more accurate results, especially in the vicinity of the singularity at the punch edge. It is found that rectangular elements generally give a smaller percentage error than do triangular elements. When, however, the pressure elements are used in conjunction with a power law singularity element at each edge of the punch, a considerable improvement in overall accuracy results. Three-dimensional singularity elements are then used in the contact problem of overriding conformal gear teeth, where the normal footprint ellipse has been truncated.

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Explicit transformation between non-adhesive and adhesive contact problems by means of the classical Johnson–Kendall–Roberts formalism

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2020.0374 ◽

2021 ◽

Vol 379 (2203) ◽

pp. 20200374

Author(s):

Nikolay V. Perepelkin ◽

Feodor M. Borodich

Keyword(s):

Rotational Symmetry ◽

A Priori ◽

Contact Problems ◽

Adhesive Contact ◽

Elastic Half Space ◽

Principle Of Superposition ◽

Linear Elastic ◽

Thin Elastic Layer ◽

Force Displacement ◽

Superposition Of Solutions

The classic Johnson–Kendall–Roberts (JKR) contact theory was developed for frictionless adhesive contact between two isotropic elastic spheres. The advantage of the classical JKR formalism is the use of the principle of superposition of solutions to non-adhesive axisymmetric contact problems. In the recent years, the JKR formalism has been extended to other cases, including problems of contact between an arbitrary-shaped blunt axisymmetric indenter and a linear elastic half-space obeying rotational symmetry of its elastic properties. Here the most general form of the JKR formalism using the minimal number of a priori conditions is studied. The corresponding condition of energy balance is developed. For the axisymmetric case and a convex indenter, the condition is reduced to a set of expressions allowing explicit transformation of force–displacement curves from non-adhesive to corresponding adhesive cases. The implementation of the developed theory is demonstrated by presentation of a two-term asymptotic adhesive solution of the contact between a thin elastic layer and a rigid punch of arbitrary axisymmetric shape. Some aspects of numerical implementation of the theory by means of Finite-Element Method are also discussed. This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction’.

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Effect of Material Characteristics on Viscoelastic Frictionless Contact Configuration

STLE/ASME 2008 International Joint Tribology Conference ◽

10.1115/ijtc2008-71125 ◽

2008 ◽

Cited By ~ 1

Author(s):

Fatin F. Mahmoud ◽

Ahmed G. El-Shafei ◽

Mohamed A. Attia

Keyword(s):

Computational Model ◽

Viscoelastic Material ◽

Contact Problems ◽

Contact Interface ◽

Viscoelastic Materials ◽

Frictionless Contact ◽

Material Parameters ◽

Contact Configuration ◽

Linear Behavior ◽

Vital Effect

The tribological status of contact systems is affected by the contact configuration; contact stress distribution throughout the contact interface. Viscoelastic materials have the capability of storing and dissipating energy. When the contacting bodies are made of viscoelastic material, viscous and elastic properties of the material have a vital effect upon the contact pressure distribution and the extent of the contact interface. This paper illustrates the effect of viscoelastic material parameters on the contact configuration. Two material parameters are considered; the ratio of the delayed elasticity to the instantaneous elasticity and the material relaxation time. The results are obtained by using a time-dependent nonlinear computational model capable of analyzing quasistatic viscoelastic frictionless contact problems. This computational model adopts the Wiechert model to simulate the linear behavior of viscoelastic materials and the modified incremental convex programming method to accommodate the contact problem of viscoelastic bodies.

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METHOD OF DIMENSIONALITY REDUCTION IN CONTACT MECHANICS AND FRICTION: A USER’S HANDBOOK. III. VISCOELASTIC CONTACTS

Facta Universitatis Series Mechanical Engineering ◽

10.22190/fume180327013p ◽

2018 ◽

Vol 16 (2) ◽

pp. 99 ◽

Cited By ~ 3

Author(s):

Valentin L. Popov ◽

Emanuel Willert ◽

Markus Heß

Keyword(s):

Dimensionality Reduction ◽

Three Dimensional ◽

Contact Problems ◽

Method Of Dimensionality Reduction ◽

Mapping Rules ◽

Elastic Bodies ◽

Elementary Calculus ◽

Algebraic Operations ◽

Three Dimensional Elasticity ◽

Large Groups

Until recently the analysis of contacts in tribological systems usually required the solution of complicated boundary value problems of three-dimensional elasticity and was thus mathematically and numerically costly. With the development of the so-called Method of Dimensionality Reduction (MDR) large groups of contact problems have been, by sets of specific rules, exactly led back to the elementary systems whose study requires only simple algebraic operations and elementary calculus. The mapping rules for axisymmetric contact problems of elastic bodies have been presented and illustrated in the previously published parts of The User's Manual, I and II, in Facta Universitatis series Mechanical Engineering [5, 9]. The present paper is dedicated to axisymmetric contacts of viscoelastic materials. All the mapping rules of the method are given and illustrated by examples.

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A Review of the Method of Dimensionality Reduction in Contact Mechanics: Applications for Structural Damping, Wear and Adhesion

Proceedings of the 8th International Scientific Conference "BALTTRIB 2015" ◽

10.15544/balttrib.2015.17 ◽

2015 ◽

Author(s):

V.L. Popov ◽

M. Heß ◽

M. Popov

Keyword(s):

Dimensionality Reduction ◽

Three Dimensional ◽

Contact Problems ◽

Adhesive Contact ◽

Tangential Direction ◽

Fretting Wear ◽

Normal Contact ◽

Method Of Dimensionality Reduction ◽

Frictional Damping ◽

Arbitrary Axis

In the method of dimensionality reduction (MDR), contacts of three-dimensional bodies are mapped to the contact problem with a one-dimensional elastic or viscoelastic foundation. This is valid for the normal contact, the tangential contact and the normal contact of viscoelastic bodies. For the above classes of contact problems, several examples are considered and discussed in detail. This includes: (a) Fretting wear for arbitrary histories of loading (for simultaneous oscillations both in normal and horizontal directions); (b) Frictional damping under the influence of oscillations in normal and tangential direction as well as normal and torsional loading; (c) Adhesion of bodies of arbitrary axis-symmetric shape with extension to the adhesive contact of elastomers.

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Adhesion at the Wavy Contact Interface Between Two Elastic Bodies

Journal of Applied Mechanics ◽

10.1115/1.1794702 ◽

2004 ◽

Vol 71 (6) ◽

pp. 851-856 ◽

Cited By ~ 21

Author(s):

George G. Adams

Keyword(s):

Integral Transform ◽

Mixed Boundary ◽

Contact Region ◽

Applied Pressure ◽

Contact Length ◽

Contact Interface ◽

Elastic Bodies ◽

Contact Configuration ◽

Complete Contact ◽

Two Bodies

The plane strain elastic contact of two bodies with a wavy contact interface is investigated. The effect of adhesion is accounted for by using the Maugis model. This periodic mixed boundary value problem is solved using integral transform techniques. Results are obtained for the extent of the contact region as a function of the dimensionless applied pressure and for various values of the dimensionless adhesive stress and peak-to-valley height. These contact length versus applied pressure curves are characterized by discontinuities and hysteresis. A finite contact region exists at zero load, with further loading causing one or more jumps into a complete contact configuration. Unloading is also characterized by one or more jumps before pull-off occurs suddenly with a finite contact length and tensile pressure loading.

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