A new approach for closed-form analytical solution of two-dimensional symmetric double contacts and the comparison with finite element method

Abstract

In this paper, a new procedure is developed for the solution of a general two-dimensional uncoupled symmetric double contact problem with smooth contact zones in which the indenter geometry is described by a piecewise biquadratic function. This procedure gives an approximate closed-form solution for any smooth indenter profile. In order to evaluate the accuracy of this approach, it is applied to the symmetric indentation of a flat surface by two rigidly interconnected parabolic indenters and results are compared with the exact unclosed-form solution. Moreover, this procedure is applied to the symmetric indentation of a flat surface by two rigidly interconnected cylinders to compare the results with the finite element solution obtained by the finite element method software, ABAQUS. The results showed that in comparison with the finite element method, this procedure is a fast and highly accurate method with low complexity that makes feasible the possibility of determining approximate closed-form solution for a wide range of indenter geometries with a concavity between two symmetric contact zones; hence it can be useful in practical issues.

Keywords

Uncoupled plane contact problems analytical approach closed-form solution piecewise biquadratic function finite element solution symmetric double contacts

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References

Scthayerman IYa. Contact problem of the theory of elasticity. Gostekhteoretized, Leningrad, 1949. Available from the British Library in an English Translation by Foreign Technology Div. FTD-MT-24-61-70, 1970.

Galin

LAL

. Contact problems in the theory of elasticity, USA: North Carolina State College, 1961.

Muskhelishvili

. Some basic problems of the mathematical theory of elasticity, The Netherlands: Noordhoff Ltd, Groningen, 1963.

Gladwell

GML

. Contact problems in the classical theory of elasticity, The Netherlands: Monograph and textbooks on mechanics of solids and fluids, Sijthoff, Noordhoff, 1980.

Johnson

. Contact mechanics, Cambridge, UK: Cambridge University Press, 1985.

Hills

Nowell

Sackfield

. Mechanics of elastic contacts, Oxford, UK: Butterworth-Heinemann, Oxford Press, 1993.

Hills

Fleury

RMN

Dini

. Partial slip incomplete contacts under constant normal load and subject periodic loading. Int J Mech Sci 2016; 108–109: 115–121.

Fleury

RMN

Hills

Barber

. A corrective solution for finding the effects of edge-rounding on complete contact between elastically similar bodies. Part I: contact law and normal contact considerations. Int J Solids Struct 2016; 85–86: 89–96.

Fleury

RMN

Hills

Barber

. A corrective solution for finding the effects of edge-rounding on complete contact between elastically similar bodies. Part II: near-edge asymptotes and the effect of shear. Int J Solids Struct 2016; 85–86: 97–104.

10.

Hills

Ramesh

Fleury

RMN

et al.

A unified approach for representing fretting and damage at the edges of incomplete and receding contacts. Tribol Int 2017; 108: 16–22.

11.

Fleury

RMN

Hills

Ramesh

et al.

Incomplete contacts in partial slip subject to varying normal and shear loading, and their representation by asymptotes. J Mech Phys Solids 2017; 99: 178–191.

12.

Shahani

Adibnazari

Naderi

. Non-symmetrical plane contact of a wedge indenter. Proc IMechE, Part C: J Mechnaical Engineering Science 2007; 221: 1233–1239.

13.

Yaylacı

Öner

Birinci

. Comparison between analytical and ANSYS calculations for a receding contact problem. J Eng Mech-ASCE 2014; 140: 04014070–04014070.

14.

Birinci

Adıyaman

Yaylac

et al.

Analysis of continuous and discontinuous cases of a contact problem using analytical method and FEM. Lat Am J Solids Struct 2015; 12: 1771–1789.

15.

Barber

. Indentation of the semi-infinite elastic solids by a concave rigid punch. J Elasticity 1976; 6: 149–159.

16.

Barber

. The solution of elasticity problems for the half-space by the Green and Collins. Appl Sci Res 1983; 45: 135–157.

17.

Barber

. A four-part boundary value problem in elasticity-indentation by a discontinuosly concave punch. Appl Sci Res 1983; 40: 159–167.

18.

Westergaard

. Bearing pressures and cracks. J Appl Mech 1939; 6: A49–A53.

19.

Sundaram

Farris

. Numerical analysis of double contacts of similar elastic materials. J Appl Mech-T ASME 2008; 75: 061017–061017. (9 pages).

20.

Ghanati

Adibnazari

. Two-dimensional symmetric double contacts of elastically similar materials. Proc IMechE, Part C: J Mechnaical Engineering Science 2016; 230: 1626–1633.

21.

Jäger

. A new principle in contact mechanics. J Tribol-T ASME 1998; 120: 677–684.

22.

Ciavarella

. The generalized cattaneo partial slip plane contact problem, I: Theory. Int J Solids Struct 1998; 35: 2349–2362.

23.

Ciavarella

. The generalized cattaneo partial slip plane contact problem, II- Examples. Int J Solids Struct 1998; 35: 2363–2378.

24.

Davies

Hills

. Closed-form solutions for a piecewise quadratic half-Plane contact problem. J Strain Anal Eng 2010; 45: 505–511.

25.

Milne-Thomson

Elliptic integrals. In: Abramowitz

Stegun

(eds). Handbook of mathematical functions, New York: Dover publications Inc., 1965, pp. 587–626.

26.

ABAQUS Documentation 6.14, DS SIMULIA, http://abaqus.software.polimi.it/v6.14/index.html.