НЕСТАЦИОНАРНЫЕ ВОЛНЫ В ЛИНЕЙНО-ВЯЗКОУПРУГОМ ЦИЛИНДРЕ С ЖЕСТКИМ ВКЛЮЧЕНИЕМ

Abstract

We have constructed a solution to a non-stationary dynamic problem for a linear-viscoelastic homogeneous infinitely long cylinder with a rigid axial inclusion exposed to an axisymmetric radial load uniformly distributed along the element of cylinder. On the contact surface with a rigid inclusion, the displacements are equal to zero. The hereditary properties of the material of the cylinder are taken into account using the Boltzmann–Volterra linear integral relation, and the Poisson's ratio of the material is considered time-independent. The integral Laplace transform in time is applied to the initial problem and the analysis of the solution in images is carried out. In the case when the hereditary kernel is exponential two-parameter, originals of the displacement and stresses are constructed in a form of series. Asymptotic formulas for the stresses behind the front that first came from the loaded boundary are obtained. The constructed solution to the non-stationary problem is valid over the entire time range and does not require that the viscosity be small. With the help of constructed solution for the case of an exponential relaxation kernel, investigations of wave process in cylinder cross-section with various initial parameters are carried out. It is established that in the case of a compressive external load, significant tensile stresses occur at certain moments at the contact boundary with a rigid inclusion. They decrease with the growth of the parameter characterizing the viscosity of the material. With a stepwise change in time of the external load, the dependence of the maximum tensile stresses at the boundary with a rigid inclusion on the inclusion relative radius and the viscosity parameter of the material is investigated.