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.
Highlights
Sneddon [1] obtained a seminal and compact solution of the displacement–load relationship for the axisymmetric Boussinesq problem [2,3,4,5] involving a punch with an arbitrary profile
The present work explicitly provides the analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space involving axisymmetric punch profile z Arm/n, (m = 0, 1, 2, 3, ...; n = 1, 2, 3, ...)
The methods adopted in this study for the axisymmetric Boussinesq problem are the Hankel transform and the theory of dual integral equations, which have been pioneered by Sneddon and developed to solve the related problems
Summary
Sneddon [1] obtained a seminal and compact solution of the displacement–load relationship for the axisymmetric Boussinesq problem [2,3,4,5] involving a punch with an arbitrary profile. Except for the known expressions of load P, indentation depth D, and normal surface stress obtained from the aforementioned studies, the explicit and concise formulas of interior-stress fields have not been integrally derived These subjects may either have not been pursued, only a speculative sketch was presented, the implicit expressions were given, the mathematical procedures were quite complicated, or a few special cases of punch profiles were considered. H0 g( ); x , to deal with the arbitrary profile of a punch Instead of utilizing such complicated formula with its cumbersome derivations, a novel set of simpler and more ingenious mathematical methods are employed in the present study for this special issue, which enable us to directly obtain results similar to those by Sneddon [1] and to further fully obtain the interior-stress fields. 1, 4/3, 2, 4 are computed according to the explicit and analytical expressions of the interior-stress components in the present work
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