Abstract
A non-stationary dynamic problem of linear viscoelasticity is considered under the condition of a limited perturbation propagation domain. The connection of such a problem with the problem of free oscillations of the body under consideration is established. The structure of the set of eigenvalues of the spectral problem of free oscillations is investigated. The cases when the Poisson's ratio of the material is constant, as well as the case when it depends on time, are considered. A method for finding eigenvalues in the vicinity of the limit points of the spectral set is proposed. It is assumed that relaxation kernels are represented as a finite sum of exponents. For the case when the Poisson's ratio depends on time, as an example, the solution of the problem of the propagation of unsteady waves in the cross section of a linear viscoelastic cylinder is constructed. The resulting solution is valid over the entire time range and is convenient for numerical implementation. An example of calculations performed on the basis of the constructed solution with specific source data is given.
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