The authors study the global existence of classical solutions of a certain problem which occurs in the theory of nonlinear vibrations of finite rods with nonlinear viscoelasticity. This phenomenon is described by an initial-boundary value problem (with the boundary conditions \(u(t,a) = 0\), \(u(t,b) = 0\), for all \(t)\) for the integro-partial differential equation \[ u_{tt} - u_{xx} + \int^ t_ 0 \lambda (t - s) \sigma (u,u_ x)_ x ds = f(t,x,u,u_ t). \] Under certain conditions on \(\sigma\) and \(f\), the unique existence of the global classical solution is proved.