Displacement-traction relationship

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... 

I. Displacement-Traction Relationships. Displacement-traction relationships, in terms of Fourier transformed quantities, on the surface of a half-space, under plane strain conditions, are derived in Sect. 7.1 and given by (7.1.15) for steady-state uniform motion. A similar relationship between the Fourier transform of the displacement and tearing stress is given by (7.1.23) for tearing mode fracture, along the line of the crack. These have the same form as the equivalent elastic relations, with moduli replaced by complex moduli, as required by the Classical Correspondence Principle. 

Take a variation of the equilibrium Equations (25)-(30) and then apply the virtual displacements principle using the Ritz variational technique, incorporating the constitutive relationships, using the section properties parameters, adopting a second order approximation for displacement components and internals actions, and evaluating the conservative surface tractions at the boundaries, for monosymmetric beams, consider the case of no initial force and ignoring the axial displacement terms, the second variation of Total Potential equation can be reduced to ... 

---