The Plane Normal Contact Problem

Consider a rigid smooth punch pressed into a viscoelastic half-space under a varying load. There is no tangential movement of the punch and the contact is lubricated. For simplicity, it will be assumed that the punch is symmetrical, though this is not in fact necessary. We choose the origin so that the contact interval is [ a t), a(t)]. The symmetry assumption then implies that the point of deepest indentation is at the origin. We have the fundamental relationship (3.2.10) between displacement and pressure. In the contact interval, u x, t) is the derivative of the punch profile. It is time-independent and will be denoted by u x). 

As the load varies, the contact interval will pass through a series of states characterized uniquely by (/), so that one can plot its history by plotting a t). This problem falls into the category discussed at the end of Sect. 2.6, referred to as involving repetitive expansion and contraction. We now apply the general method developed in that section to this particular case. 

Let a(t) pass through a series of maxima and minima before the current time t. First consider the case where it is contracting at time t (Fig. 3.2a). The sets Bfjit) in this case are C(r), C t). The times 0i(t), given by (2.6.6, 7), are here defined as 

The fact that p(x, t ) is zero outside of C(t ) has been used in (3.10.5), in order to validate the interchange of order of integration see (2.6.12). On using (3.2.10) again, we obtain 

This quantity is known, since u (x) is simply the slope of the punch profile. In fact, P(Xy t) is proportional to the elastic pressure. 

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