Interior-stress fields produced by a general axisymmetric punch

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.

[1]  H. Deresiewicz,et al.  Bodies in Contact with Applications to Granular Media , 1974 .

[2]  A. Love.  The Stress Produced in a Semi-Infinite Solid by Pressure on Part of the Boundary , 1929 .

[3]  The relation between load and penetration for a spherical punch , 1957 .

[4]  I. N. Sneddon.  The elementary solution of dual integral equations , 1960, Proceedings of the Glasgow Mathematical Association.

[5]  J. Woirgard,et al.  Elastic stress field beneath an arbitrary axisymmetric punch , 2008 .

[6]  J. Barber,et al.  The solution of elasticity problems for the half-space by the method of Green and Collins , 1983 .

[7]  Gerhard Schubert,et al.  Zur Frage der Druckverteilung unter elastisch gelagerten Tragwerken , 1942 .

[8]  R. Knops,et al.  Three-Dimensional Problems of the Theory of Elasticity , 1967, The Mathematical Gazette.

[9]  On an axially symmetrical contact problem for a non-plane die circular in plan , 1962 .

[10]  J. Gillis,et al.  Mixed boundary value problems in potential theory , 1966 .

[11]  V. Popov.  METHOD OF DIMENSIONALITY REDUCTION IN CONTACT MECHANICS AND FRICTION: A USERS HANDBOOK. I. AXIALLY-SYMMETRIC CONTACTS , 2014 .

[12]  Stress tensor and gradient of hydrostatic pressure in the contact plane of axisymmetric bodies under normal and tangential loading , 2020, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik.

[13]  A. E. H. Love,et al.  BOUSSINESQ'S PROBLEM FOR A RIGID CONE , 1939 .

[14]  I. N. Sneddon,et al.  Boussinesq's problem for a rigid cone , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  N. Muskhelishvili.  Some basic problems of the mathematical theory of elasticity , 1953 .

[16]  Thomas Geike,et al.  Mapping of three-dimensional contact problems into one dimension. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  M. Huber,et al.  Zur Theorie der Berührung fester elastischer Körper , 1904 .

[18]  Elasticity: Mathematical Methods and Applications (The Ian N. Sneddon 70th Birthday Volume) , 1993 .

[19]  I. N. Sneddon,et al.  The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch , 1945, Mathematical Proceedings of the Cambridge Philosophical Society.

[20]  G. Gladwell.  Contact Problems in the Classical Theory of Elasticity , 1980 .

[21]  A.P.S. Selvadurai,et al.  On Boussinesq's problem , 2001 .

[22]  Hertz.  On the Contact of Elastic Solids , 1882 .

[23]  I. N. Sneddon.  The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile , 1965 .

[24]  David A. Hills,et al.  Mechanics of elastic contacts , 1993 .

[25]  H. Neuber.  Ein neuer Ansatz zur Lösung räumlicher Probleme der Elastizitätstheorie. Der Hohlkegel unter Einzellast als Beispiel , 1934 .

[26]  I. N. Sneddon.  Boussinesq's problem for a flat-ended cylinder , 1946, Mathematical Proceedings of the Cambridge Philosophical Society.