AI Chat Paper
Note: Please note that the following content is generated by AMiner AI. SciOpen does not take any responsibility related to this content.
{{lang === 'zh_CN' ? '文章概述' : 'Summary'}}
{{lang === 'en_US' ? '中' : 'Eng'}}
Chat more with AI
Powered by AMiner. Clicking this button will take you away from SciOpen.
Home Friction Article
PDF (1.7 MB)
Cite
EndNote(RIS) BibTeX
Collect
Submit Manuscript AI Chat Paper
Show Outline
Outline
Show full outline
Hide outline
Outline
Show full outline
Hide outline
Research Article | Open Access

Interior-stress fields produced by a general axisymmetric punch

Longxiang YANG1, Zhanjiang WANG2, Weiji LIU3, Guocheng ZHANG4, Bei PENG4,5( )
School of Automotive Engineering, Geely University of China, Chengdu 641423, China
Department of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China
School of Mechatronic Engineering, Southwest Petroleum University, Chengdu 610500, China
School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
Center for Robotics, University of Electronic Science and Technology of China, Chengdu 611731, China
Show Author Information

Abstract

This work is a supplement to the work of Sneddon on axisymmetric Boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile. A novel set of mathematical procedures is introduced to process the basic elastic solutions (obtained by the method of Hankel transform, which was pioneered by Sneddon) and the solution of the dual integral equations. These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space. The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study, and the deduced results are verified by comparing them with the classical results. Finally, these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.

Keywords

axisymmetricBoussinesq probleminterior stressHankel transformdual integral equations

References

[1]
Sneddon I N. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary prole. Int J Eng Sci 3(1):47-57 (1965)
Crossref Google Scholar
[2]
Boussinesq J. Application des Potentiels a L'etude de L'equilibre et du Mouvement des Solides Elastique. Pari (France): Gauthier Villars, 1885.
[3]
Lamb H. On Boussinesq's problem. Proc Lond Math Soc 34(1):276-284 (1901)
Crossref Google Scholar
[4]
Schubert G. Zur Frage der Druckverteilung unter elastisch gelagerten Tragwerken. Ingenieur-Archiv 13(3):132-147 (1942)
Crossref Google Scholar
[5]
Selvadurai A P S. On Boussinesq's problem. Int J Eng Sci 39(3):317-322 (2001)
Crossref Google Scholar
[6]
Hertz H. On the contact of elastic solids. J Reine Angew Math 92:156-171 (1882)
Crossref Google Scholar
[7]
Johnson K L. Contact Mechanics. Cambridge (UK): Cambridge university press, 1987.
[8]
Barber J R. Contact Mechanics. Cham (Switzerland): Springer International Publishing AG, 2018.
Crossref
[9]
Popov V L, Heß M, Willert E. Handbook of Contact Mechanics: Exact Solutions of Axisymmetric Contact Problems. Berlin (Germany): Springer-Verlag GmbH Deutschland, 2019.
Crossref
[10]
Huber M T. On the theory of elastic solid contact. Annln der Physik 14:153-163 (1904)
Crossref Google Scholar
[11]
Hills D A, Nowell D, Sackeld A. Mechanics of Elastic Contacts. New York (USA): Butterworth-Heinemann, 1993: 25, 203-204.
Crossref
[12]
Love A E H. IX. The stress produced in a semi-infinite solid by pressure on part of the boundary. Phil Trans R Soc London A 228(659-669):377-420 (1929)
Crossref Google Scholar
[13]
Love A E H. Boussinesq's problem for a rigid cone. Q J Math os-10(1):161-175 (1939)
Crossref Google Scholar
[14]
Harding J W, Sneddon I N. The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch. Math Proc Camb Phil Soc 41(1):16-26 (1945)
Crossref Google Scholar
[15]
Sneddon I N. Boussinesq's problem for a flat-ended cylinder. Math Proc Camb Phil Soc 42(1):29-39 (1946)
Crossref Google Scholar
[16]
Sneddon I N. Boussinesq's problem for a rigid cone. Math Proc Camb Phil Soc 44 (4):492-507 (1948)
Crossref Google Scholar
[17]
Papkovich P. Solution generale des equations diffentielles fondamentales de-lasticite exprimee par trois fonctions harmoniques. CR Acad Sci Paris 195(3):513-515 (1932)
Google Scholar
[18]
Papkovich P. Expression for the general integral of the principal equations of the theory of elasticity by harmonic functions, Akad Nauk SSSR ser fiz-met 10(14):1425-1435 (1932)
Google Scholar
[19]
Neuber H. Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel. Z Angew Math Mech 14(4):203-212 (1934)
Crossref Google Scholar
[20]
Green A, Zerna W. Theoretical Elasticity. Oxford (UK): Oxford University Press, 1968.
[21]
Sneddon I N. Fourier Transforms. New York (USA): McGraw-Hill, 1951: 453-455, 459.
[22]
Muskhelishvili N I. Some Basic Problems of the Mathematical Theory of Elasticity. Dordrecht (the Netherlands): Springer Science, 1977.
Crossref
[23]
Gladwell G M L. Contact Problems (The Legacy of Galin L A). Dordrecht (the Netherlands): Springer Science, 2008: 105-111, 266, 267.
Crossref
[24]
Lur'e A I. Three-Dimensional Problems of the Theory of Elasticity. New York (USA): Interscience Publishers, 1964.
[25]
Sneddon I N. Mixed Boundary Value Problems in Potential Theory. Amsterdam (the Netherlands): North-Holland Publishing Company, 1966.
[26]
Gladwell G M L. Contact problems in the classical theory of elasticity. Alphen aan den Rijn (the Netherlands): Springer Science & Business Media B.V., 1980.
Crossref
[27]
Barber J R. The solution of elasticity problems for the half- space by the method of Green and Collins. Appl Sci Res 40(2):135-157 (1983)
Crossref Google Scholar
[28]
Payne L E. On axially symmetric punch, crack and torsion problems. J Soc Ind Appl Math 1(1):53-71 (1953)
Crossref Google Scholar
[29]
Segedin C M. The relation between load and penetration for a spherical punch. Mathematika 4(2):156-161 (1957)
Crossref Google Scholar
[30]
Deich E G. On an axially symmetrical contact problem for a non-plane die circular in plan. J Appl Math Mech 26(5):1404-1409 (1962)
Crossref Google Scholar
[31]
Deresiewicz H. Bodies in contact with applications to granular media. In R.d.mindlin & Applied Mechanics. New York (USA): Pergamon Press, Inc., 1974: 105-147.
Crossref
[32]
Woirgard J, Audurier V, Tromas C. Elastic Stress Field beneath anarbitrary axisymmetric punch. Philos Mag 88(10):1511-1523 (2008)
Crossref Google Scholar
[33]
Hill R, Storakers B. Elasticity: Mathematical Methods and Applications (The Ian N. Sneddon 70th Birthday Volume). Eason G, Ogden R W, Ed. Chichester (UK): Ellis Horwood Publishers, 1990: 199-210.
[34]
Geike T, Popov V L. Mapping of three-dimensional contact problems into one dimension. Phys Rev E 76(3):036710 (2007)
Crossref Google Scholar
[35]
Popov V L, Heß M. Method of Dimensionality Reduction in Contact Mechanics and Friction. Berlin (Germany): Springer- Verlag Berlin Heidelberg, 2015.
Crossref
[36]
Willert E, Forsbach F, Popov V L. Stress tensor and gradient of hydrostatic pressure in the contact plane of axisymmetric bodies under normal and tangential loading. Z Angew Math Mech 100(7):1-13 (2020)
Crossref Google Scholar
[37]
Sneddon I N. The elementary solution of dual integral equations. Glasgow Math J 4(3):108-110 (1960)
Crossref Google Scholar
[38]
Gradshteyn I S, Ryzhik I M. Table of Integral, Series, and Products. Burlington (USA): Elsevier Inc., 2007.
[39]
Bower A F. Applied Mechanics of Folids. Boca Raton (USA): CRC Press, 2010: 309-310.
Crossref
Friction
Volume 10 Issue 4,
April 2022
Pages 530-544
DOI: 10.1007/s40544-020-0478-9
Cite this article:
YANG L, WANG Z, LIU W, et al. Interior-stress fields produced by a general axisymmetric punch. Friction, 2022, 10(4): 530-544. https://doi.org/10.1007/s40544-020-0478-9

841

Views

39

Downloads

4

Crossref

3

Web of Science

3

Scopus

0

CSCD

Altmetrics
Received: 27 November 2019
Revised: 28 December 2019
Accepted: 23 November 2020
Published: 05 April 2022
© The author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Return