Григоренко А.Я., Григоренко Я.М., Лоза И.А. «Численный анализ динамических процессов в неоднородных пьезокерамических цилиндрах (обзор)» Прикладная механика, 56, № 5, с. 3-55 (2020)

The review of works is given, which are devoted to the numerical investigations of the new problems of the theory electroelasticity. Namely, they are devoted to the determination of dynamical characteristics of the inhomogeneous piezoceramic circular waveguides and the inhomogeneous piezoceramic finite-length cylinders. An effective numerical-analytical approach is proposed in these works. The proposed method is based on uniting the different analytical transforms (apparatus of special functions, expansion in the Fourier series, and spline-approximations method with collocation method) allows reducing the initial three-dimensional equations of electroelasticity theory in the partial derivatives to the boundary value problem for the system of ordinary differential equations. The obtained one-dimensional problem is solved by the method of discrete orthogonalization. Basing on the obtained solutions, the new regularities of spectral characteristics with the inhomogeneous structure are studied, with allowance for the coupled electric field of the piezoceramic layers. Also, the study is carried out for an effect of the inhomogeneity and coupled electric field on the dynamic characteristics of the bodies under study. Significant attention is paid to the validation of the reliability of the results obtained by the numerical calculations.

Рущицкий Я.Я., Юрчук В.Н. «О влиянии третьего приближения при анализе эволюции нелинейно упругой Р-волны. Часть 1» Прикладная механика, 56, № 5, с. 65-77 (2020)

The nonlinear plane longitudinal elastic wave of displacement is studied theoretically and numerically within the framework of the Murnaghan model for two forms of initial profile – harmonic and bell-shaped ones. The basic novelty consists in that the evolution of waves is analyzed by the approximate methods by taking into account the first three approximations. The analysis of the harmonic wave is considered for the only comparison with the new results for the bell-shaped wave. Some essential distinctions between the evolution of waves are shown. First, the symmetric initial profiles are transformed owing to evolution into the distorted ones in a different way: symmetrically for harmonic profile and asymmetrically for the bell-shaped profile. Second, the third approximation introduced the fourth harmonic for a harmonic wave when this wave being analyzed by the method of successive approximations, whereas the bell-shaped wave is characterized in the third approximation in a very different way when being analyzed by the method of restrictions on displacement gradient. On the sufficiently long distances of wave propagation, the one-hump bell-shaped wave is transformed into the two-hump one. These humps adjoint one to another ad decrease in twice their bottoms. The third approximation permits to observe the new wave effects: non-symmetry of the left and right humps relative to their picks and non-symmetry of the humps relative to each other – sinking of the left hump and elevation of the right one. The findings are commented on.

Луговой П.З., Мейш В.Ф., Орленко С.П. «Численное моделирование динамики трехслойных сферических оболочек с дискретным ребристым заполнителем при действии ударной волны» Прикладная механика, 56, № 5, с. 78-88 (2020)

The equations of oscillations of a three-layer spherical shell with a discrete ribbed filler under the unsteady loading are obtained. In analyzing the elements of the elastic structure, the models of the nonlinear theory of shells and Timoshenko rods are used in the quadratic approximation for each layer. The numerical method for solving the obtained equations is based on the application of the integro-interpolation method for constructing the finite-difference schemes for equations with the discontinuous coefficients. The problem of the dynamic behavior of a three-layer spherical shell under unsteady loading is solved taking into account the discreteness of the ribbed filler.