A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions technique to justify the change of integration contour in the complex Laplace inversion formula. The second integral representation for the relaxation modulus is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail: a model with fractional derivatives of uniformly distributed order and a model with general fractional derivatives, the kernel of which is a multinomial Mittag-Leffler-type function. To illustrate the analytical results, some numerical examples are presented.

Dynamic problems describing transient wave processes in linearly viscoelastic solids are considered for bounded domains of perturbation propagation and bounded creep of the material. The integral Laplace transform with respect to time is applied to the original problem, and several statements about the properties of Laplace transforms simplifying the construction of the original functions are stated. Relations establishing a correspondence between relaxation kernels that belong to various function classes but nevertheless affect the transient processes in a similar way are proposed. The results justifying these relations in a certain range of the input data are presented.

Transient wave processes in viscoelastic structures built from functionally graded material (FGM) still remain almost unexplored. In this article, the problem of the propagation of nonstationary longitudinal waves in an infinite viscoelastic layer of a FGM with plane–parallel boundaries is considered. The physical and mechanical parameters of the FGM depend continuously on the transverse coordinate, while the wave process propagates along the same coordinate. The viscoelastic properties of the material are taken into account employing the linear integral Boltzmann–Volterra relations. The viscoelastic layer of the FGM is replaced by a piecewise-homogeneous layer consisting of a large number of sub-layers (a package of homogeneous layers), thus approximating the continuous inhomogeneity of the FGM. A solution of a non-stationary dynamic problem for a piecewise-homogeneous layer is constructed and, using a specific example, the convergence of the results with an increase in the number of sub-layers in the approximating piecewise-homogeneous layer is shown. Furthermore, the problem of the propagation of nonstationary longitudinal waves in the cross section of an infinitely long viscoelastic hollow FGM cylinder, whose material properties continuously change along the radius, is also considered. The cylinder composed of the FGM is replaced by a piecewise-homogeneous one, consisting of a large number of coaxial layers, for which the solution of the non-stationary dynamic problem is constructed. For both the layer and the cylinder composed of a viscoelastic FGM, the results of calculating the characteristic parameters of the wave processes for the various initial data are presented. The influence of the viscosity and inhomogeneity of the material on the dynamic processes is demonstrated.

The propagation of nonstationary disturbances in a plane-parallel layer of infinite extent, which arise as a result of normal dynamic loading of one of its surfaces when the other surface is stationary, is investigated. The material of the layer is assumed to be functionally graded and, at the same time, viscoelastic. The hereditary properties of such a material are taken into account using linear integral Boltzmann–Volterra relations with regular kernels in the form of partial sums of the Prony's series. The peculiarity of this work is that here the parameters of the relaxation kernels are considered continuous functions of the transverse coordinate in the same way as other physical and mechanical characteristics. A method was used for the study, which consists in replacing a functionally graded material with an approximating layered homogeneous structure with continuity conditions at the contact of homogeneous layers. The solution of the nonstationary dynamic linear viscoelasticity problem for a package of plane homogeneous viscoelastic layers is presented in a special form, which greatly simplifies its numerical implementation, especially with a large number of layers with different hereditary properties. This made it possible to successfully apply this method and carry out a series of calculations using an efficient algorithm. Transient wave processes are investigated in the case when the parameters of a functionally graded material are non-monotonic functions of the transverse coordinate, symmetrical with respect to the middle surface of the layer. A comparison of transient wave processes for different types of these functions is carried out. The convergence of the calculation results with an increase in the number of approximating homogeneous layers with a continuous dependence of the external load on time is confirmed. The significant influence of both heterogeneity and viscosity of the material on nonstationary wave processes has been established.

The paper considers the problem of unsteady wave propagation in the cross section of an infinite hollow cylinder made of a viscoelastic functionally graded