Displacement-Traction Relationships on the Boundary

In order to deal with moving contact problems, we consider as before a halfplane occupying 0, consisting of a homogeneous isotropic linear viscoelastic material. In this section, it will be assumed that the stresses are known on the x-axis, and are zero at infinity. Consider the two-dimensional version of (1.9.18)  

The quantities Cj (o), Cl(cu) are generalizations of the transverse and longitudinal speeds of sound in the medium. Note that they are complex since Cj, Cl are proportional to the complex moduli. The presence of an imaginary part leads to exponential decay of sound waves in the medium, as a consequence of energy dissipation. 

If the second equation of (7.1.3) is to have a solution, the determinant of the matrix acting on Uj k,a ) must be zero. The gives an equation which has solutions 

Only the positive signs are acceptable, since the others give exponentially increasing solutions, as will be seen in a moment. We therefore have 

The steady-state assumption means that all quantities, expressed in terms of space and time variables are functions of x+Vt rather than x, t separately. 

---

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... 

---