The General Mixed Boundary Value Problem

We will consider the problem of a series of rigid indentors pressed into a viscoelastic half-space (y 0) and moving across it. If plane strain conditions are to hold, the indentors must be infinitely long in one direction, taken to be the z direction, and of uniform cross-section. Also the loading distribution must be uniform along each punch. We consider a typical cross-section of this configuration. All subsequent discussion refers to this cross-section of which the material occupies the half-plane y 0. 

Let us denote the contact region by C(0, and its complement on the boundary by C (0- Initially the character of C(t) will not be restricted. If may consist of a series of separate intervals. The general method developed applies in principle to such general contact regions. However, the only detailed solution presented applies to the case where C(t) is a single interval, corresponding to a single load. 

The boundary stresses are zero on Inside C(0, the vertical displace- 

Let the contact be frictional and let the indentor be moving in the negative X direction. We denote the x derivative of normal displacement on the boundary by u (x,t) and the complex stress by Z(xJ), From (3.1.6), we have that 

Relation (3.3.2) constitutes a Hilbert problem if v(x,t) is known. This however is not generally the case for problems with varying boundary regions, which is the source of the added difficulty of non-inertial viscoelastic problems over elastic problems. Nevertheless, in this section, we will proceed as if v(x,t) were known. 

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