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Friction 2024, 12(2): 340-355 https://doi.org/10.1007/s40544-023-0785-z
Research Article | Open Access | Issue | Published: 24 August 2023

Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies: A practical tool for design engineers and tribologists

Show Author's Information Valentin L. POPOV( ),  Qiang LI( ),  Emanuel WILLERT( )
Institute of Mechanics, Technische Universität Berlin, Berlin 10623, Germany
Keywords:
normal contact, non-axisymmetric indenter, extremal principle, generalized method of dimensionality reduction (MDR), functional elastic grading
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POPOV VL, LI Q, WILLERT E. Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies: A practical tool for design engineers and tribologists. Friction, 2024, 12(2): 340-355. https://doi.org/10.1007/s40544-023-0785-z
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Abstract Full text About this article

In two recent papers, approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested, which provide explicit analytical relations for the force–approach relation, the size and the shape of the contact area, as well as for the pressure distribution therein. These solutions were derived for profiles, which only slightly deviate from the axisymmetric shape. In the present paper, they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM). Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.


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About this article

Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies: A practical tool for design engineers and tribologists

Show Author's information Valentin L. POPOV( ), Qiang LI( ), Emanuel WILLERT( )
Institute of Mechanics, Technische Universität Berlin, Berlin 10623, Germany

Abstract

In two recent papers, approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested, which provide explicit analytical relations for the force–approach relation, the size and the shape of the contact area, as well as for the pressure distribution therein. These solutions were derived for profiles, which only slightly deviate from the axisymmetric shape. In the present paper, they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM). Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.

Keywords: normal contact, non-axisymmetric indenter, extremal principle, generalized method of dimensionality reduction (MDR), functional elastic grading

References(27)

[1]
Hertz H. Über die Berührung fester elastischer Körper. J Reine Angew Math 92: 156–171 (1882) (in German)
DOI Google Scholar
[2]
Popova E, Popov V L. Ludwig Föppl and Gerhard Schubert: Unknown classics of contact mechanics. Z Angew Math Me 100(9): e202000203 (2020)
DOI Google Scholar
[3]
Föppl L. Elastische beanspruchung des erdbodens unter fundamenten. Forschung auf dem Gebiet des Ingenieurwesens A 12(1): 31–39 (1941) (in German)
DOI Google Scholar
[4]
Schubert G. Zur frage der druckverteilung unter elastisch gelagerten tragwerken. Ing Arch 13(3): 132–147 (1942) (in German)
DOI Google Scholar
[5]
Sneddon I N. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Eng Sci 3(1): 47–57 (1965)
DOI Google Scholar
[6]
Barber J R, Billings D A. An approximate solution for the contact area and elastic compliance of a smooth punch of arbitrary shape. Int J Mech Sci 32(12): 991–997 (1990)
DOI Google Scholar
[7]
Shield R T. Load–displacement relations for elastic bodies. Z Angew Math Phys 18(5): 682–693 (1967)
DOI Google Scholar
[8]
Barber J R. Determining the contact area in elastic-indentation problems. J Strain Anal Eng 9(4): 230–232 (1974)
DOI Google Scholar
[9]
Popov V L. An approximate solution for the contact problem of profiles slightly deviating from axial symmetry. Symmetry 14(2): 390 (2022)
DOI Google Scholar
[10]
Popov V L. Correction: Popov, V.L. An approximate solution for the contact problem of profiles slightly deviating from axial symmetry. Symmetry 2022, 14, 390. Symmetry 14(10): 2108 (2022)
DOI Google Scholar
[11]
Cattaneo C. Sul contatto de due corpi elastici: Distribuzione locale deglisforzi. Rendiconti dell Accademia Nazionale dei Lincei 27: 342–348 (434–436, 474–478) (1938) (in Italian)
Google Scholar
[12]
Mindlin R D. Compliance of elastic bodies in contact. J Appl Mech 16(3): 259–268 (1949)
DOI Google Scholar
[13]
Fabrikant V I. Flat punch of arbitrary shape on an elastic half-space. Int J Eng Sci 24(11): 1731–1740 (1986)
DOI Google Scholar
[14]
Willert E. On Boussinesq’s problem for a power-law graded elastic half-space on elliptical and general contact domains. Materials 16(12): 4364 (2023)
DOI Google Scholar
[15]
Mossakovskii V I. Compression of elastic bodies under conditions of adhesion (axisymmetric case). J Appl Math Mech 27(3): 630–643 (1963)
DOI Google Scholar
[16]
Barber J R. Contact Mechanics. Cham (Switzerland): Springer Cham, 2018.
[17]
Booker J R, Balaam N P, Davis E H. The behaviour of an elastic non-homogeneous half-space. Part II—Circular and strip footings. Int J Numer Anal Met 9(4): 369–381 (1985)
DOI Google Scholar
[18]
Popov V L, Heß M, Willert E. Handbook of Contact Mechanics: Exact Solutions of Axisymmetric Contact Problems. Berlin (Germany): Springer Berlin Heidelberg, 2019.
DOI
[19]
Popov V L, Heß M. Method of Dimensionality Reduction in Contact Mechanics and Friction. Berlin (Germany): Springer Berlin Heidelberg, 2015.
DOI
[20]
Pohrt R, Li Q. Complete boundary element formulation for normal and tangential contact problems. Phys Mesomech 17(4): 334–340 (2014)
DOI Google Scholar
[21]
Li Q, Popov V L. Boundary element method for normal non-adhesive and adhesive contacts of power-law graded elastic materials. Comput Mech 61(3): 319–329 (2018)
DOI Google Scholar
[22]
Lee E H. Stress analysis in visco-elastic bodies. Q Appl Math 13(2): 183–190 (1955)
DOI Google Scholar
[23]
Khazanovich L. The elastic-viscoelastic correspondence principle for non-homogeneous materials with time translation non-invariant properties. Int J Solids Struct 45(17): 4739–4747 (2008)
DOI Google Scholar
[24]
Jäger J. A new principle in contact mechanics. J Tribol 120(4): 677–684 (1998)
DOI Google Scholar
[25]
Ciavarella M. The generalized Cattaneo partial slip plane contact problem. I—Theory. Int J Solids Struct 35(18): 2349–2362 (1998)
DOI Google Scholar
[26]
Ciavarella M. Tangential loading of general three-dimensional contacts. J Appl Mech 65(4): 998–1003 (1998)
DOI Google Scholar
[27]
Paggi M, Pohrt R, Popov V L. Partial-slip frictional response of rough surfaces. Sci Rep 4: 5178 (2014)
DOI Google Scholar
Publication history
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Publication history

Received: 17 February 2023
Revised: 13 April 2023
Accepted: 31 May 2023
Published: 24 August 2023
Issue date: February 2024

Copyright

© The author(s) 2023.

Acknowledgements

This work has been conducted under partial financial support from Deutsche Forschungsgemeinschaft (DFG) (Grant Nos. PO 810/66-1 and LI 3064/2-1).

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