The Normal Contact Problem under Varying Load

2 The Normal Contact Problem under Varying Load 

Consider the problem of a rigid indentor, pressed into a viscoelastic half-space, under a varying load. Friction between the indentor and the half-space is neglected here Graham (1980) considers this problem, in the case of complete and partial adhesion, and Graham and Sneddon (1981) examine the problem with rotational friction. As noted above, our approach will be based on (5.1.2). The strategy will be to reduce it to a form analogous to the elastic equation, and then use the familiar elastic solutions to determine the viscoelastic quantities. 

As the load varies, it will be assumed that the contact patch will pass through a one-parameter family of states, as shown schematically in Fig. 3.2. This assumption will be justified later on the basis that it enables the problem to be solved. Furthermore, it will be shown that the one-parameter family of states is in fact the family of possible elastic states. The fact that C t) is a one-parameter family means that the explicit formalism developed for repetitive expansion and contraction in Sects. 2.6 and 3.10 may be used, as opposed to the more general method summarized in Sect. 2.6 in the context of the Extended Correspondence Principle, which is applicable to any situation where the boundary regions are expanding and contracting in time. 

In the contact region, the normal displacement, which we denote by i/(r, t), dropping the subscript z, will have the form 

Before considering a general history of loading, two simple cases will be discussed. First consider the situation where C(t ) is non-decreasing at all times up to the current time t. This is covered by the Extended Correspondence Principle discussed in Sect. 2.6 see also Sect. 3.9. Rather than invoke the Principle directly, it is instructive to give an explicit solution. Let us replace C(t) 

---