An Elastic Half Space with a Moving Punch

The dynamic problem of a punch with rounded tips moving in an elastic half-space in a fixed direction has been considered. The static problem of determining stress component under the contact region of a punch has also been solved. Fourier integral transform has been employed to reduce the problems in solving dual integral equations. These integral equations have been solved using Cooke’s [1] result (1970) to obtain the stress component. Finally, exact expressions for stress components under the punch and the normal displacement component in the region outside the punch have been derived. Numerical results for stress intensity factor at the punch end and torque applied over the contact region have been presented in the form of graph.

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.

Contact interaction of a predeformed plate which lies without friction on rigid base with a parabolic indenter

Within the framework of linearized formulation of a problem of the elasticity theory, the stress-strain state of a predeformed plate, which is modeled by a prestressed layer, is analyzed in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.

An Axisymmetric Contact Problem of an Elastic Layer on a Rigid Circular Base

An analytical solution is presented to a doubly mixed boundary value problem of an elastic layer partially resting on a rigid smooth base. A circular rigid punch is applied to the upper surface of the medium where the contact is supposed to be smooth. The case of the layer with a cylindrical hole was considered by Toshiaki and all [5]. The studied problem is reduced to a system of dual integral equations using the Boussinesq stress functions and the Hankel integral transforms. With the help of the Gegenbauer formula we get an infinite algebraic system of simultaneous equations for calculating the unknown function of the problem. The truncation method is used for getting the system coefficients. A closed form solution is given for the displacements, stresses and the stress singularity factors. The stresses and displacements are then obtained as Bessel function series. For the numerical application we give some conclusions on the effects of the radius of the punch with the rigid base and the layer thickness on the displacements, stresses, the load and the stress singularity factors are discussed.