Indentation and Impact Problems

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. 

The problem is specified as the determination of the state of stress and the deformation produced in viscoelastic half-space (z 0) by a circular punch of radius a whose force is P (Fig. 16.4). As is well known (Ref. 15, p. 25), the displacement of a half-space caused by forces applied to its free surface with the condition of null deformation at infinite distance is given by 

Since the deflection of all the points of the punch are the same, it follows from Eq. (16.177) that 

Changing Eqs. (16.178) and (16.179) to polar coordinates and solving the resulting expression, we obtain 

If the mean pressure P %c is denoted by p, then the distribution p r) can be expressed as 

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