Elastic solutions Boussinesq

Boussinesq and Cerruti made use of potential theory for the solution of contact problems at the surface of an elastic half space. One of the most important results is the solution to the displacement associated with a concentrated normal point load P applied to the surface of an elastic half space. As presented in Johnson [49]... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

Contact problems have their origins in the works of Hertz (1881) and Boussinesq (1885) on elastic materials. Indentation problems are an important subset of contact problems (17,18). The assessment of mechanical properties of materials by means of indentation experiments is an important issue in polymer physics. One of the simplest pieces of equipment used in the experiments is the scleroscope, in which a rigid metallic ball indents the surface of the material. To gain some insight into this problem, we consider the simple case of a flat circular cylindrical indentor, which presents a relatively simple solution. This problem is also interesting from the point of view of soil mechanics, particularly in the theory of the safety of foundations. In fact, the impacting cylinder can be considered to represent a circular pillar and the viscoelastic medium the solid upon which it rests. 

An active abrasive grit is subject not only to a normal load as in static indentation, but also to a tangential load in the direction of motion [29-32]. Analogous to the Boussinesq solution, the elastic stress field due to both a normal force component P and tangential component P acting at a point on the surface has been modeled using the Michell solution [29] ... 

We will write down the displacement-traction relationship on the boundary that will form the basis of the considerations of this chapter. This is essentially the solution of the stress boundary value problem, discussed in Sect. 3.2 in the plane case. We shall neglect surface shear, however, so that the required relationship is a generalization to Viscoelasticity of the classical Boussinesq relationship. Its form follows directly from the elastic result by invoking the Classical Correspondence Principle. A more explicit derivation may be found in Hunter (1961) and also Golden (1978), who includes a shear traction term. Letting... 

The most basic configuration for an elastic contact is the indentation of an elastic halfspace by a rigid axisymmetric frictionless punch. Frictionless means that we assume that no shear stress can develop between the punch and the half-space. While historically the first solution of such a problem was given by Hertz for the case of a spherical indenter [846], we will start with a flat rigid cylindrical punch (Figure 8.3) that was first worked out by Boussinesq in 1885 [847] and solved in all details by Sneddon in 1946 [848]. 

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