No 6 (2024)

. A polymer gel is considered as a mixture consisting of a highly elastic elastic material and a liquid (solvent) dissolved in it. Based on the generalized Mooney-Rivlin model, an expression of free energy is proposed that describes the deformation behavior and thermodynamic properties of polymer gels. In this model, it is assumed that the Mooney-Rivlin “constants” depend on the concentration of the liquid dissolved in the polymer. From this expression, the defining relations for the stress tensor, the chemical potential of the solvent and the osmotic stress tensor are obtained. On their basis, an experimental study of the deformation properties of mesh elastomers swollen in a solvent of various chemical nature has been performed. In particular, the dependence of the elastic properties of elastomers on the solvent concentration has been studied and the parameters describing this dependence have been determined.

The paper analyzes the influence of nonlinear (cubic) internal damping (in the Kelvin–Feucht model) and cubic nonlinearity of elastic forces on the dynamics of a rotating flexible shaft with a distributed mass. The shaft is modeled by a Bernoulli–Euler rod using the Green function, the discretization and reduction of the problem of rotating shaft dynamics to an integral equation are performed. It is revealed that in such a system there is always a branch of limited periodic movements (self-oscillations) at a supercritical rotation speed. In addition, with low internal damping, the periodic branch continues into the subcritical region: when the critical velocity is reached, the subcritical Poincare–Andronov–Hopf bifurcation is realized and there is an unstable branch of periodic movements, below the branch of stable periodic self-oscillations (the occurrence of hysteresis with a change in rotation speed). With an increase in the internal friction coefficient, the hysteresis phenomenon disappears and at a critical rotation speed, a soft excitation of self-oscillations of the rotating shaft occurs through the supercritical Poincare–Andronov–Hopf bifurcation.

The Jacobi stability analysis of the nonlinear dynamical system on base of Kosambi–Cartan–Chern theory is considered. Geometric description of time evolution of the system is introduced, that makes it possible to determine five geometric invariants. Eigenvalues of the second invariant (the deviation curvature tensor) give an estimate of Jacobi stability of the system. This approach is relevant in applications where it is required to identify the areas of Lyapunov and Jacobi stability simultaneously. For the nonlinear system – the double pendulum – the dependence of the Jacobi stability on initial conditions is investigated. The components of the deviation curvature tensor corresponding to the initial conditions and the eigenvalues of the tensor are defined explicitly. The boundary of the deterministic system transition from regular motion to chaotic one determined by the initial conditions has been found. The formulation of the inverse eigenvalue problem for the deviation curvature tensor associated with the restoration of significant parameters of the system is proposed. The solution of the formulated inverse problem has been obtained with the use of optimization approach. Numerical examples of restoring the system parameters for cases of its regular and chaotic behavior are given.

The problem of free spatial vibrations of an overhead power line wire with an asymmetric mass distribution over a cross-section caused by ice deposits on its surface, which give the cross-section an asymmetric shape, is considered. As a result, an eccentricity is formed between the centers of torsional stiffness and mass in the cross section and a dynamic connection of vertical, torsional and “pendulum” vibrations occurs with the output of the wire from the sagging plane. The wire is modeled by a flexible heavy elastic rod that resists only stretching and torsion. The case of a weakly sagging wire is investigated, when the tension and curvature of its centerline can be considered constant within the span. It is also believed that the elasticity of the ice casing is small compared to the elasticity of the wire. The mathematical model is constructed taking into account the interaction of longitudinal, torsional and transverse waves polarized in the vertical and horizontal planes. The relations of the phase velocities of all types of waves are analyzed and a group of particular subsystems determining partial oscillations is identified. The partial and natural frequencies and waveforms of the wire are studied. Analytical solutions to the problem of determining the spectrum of natural frequencies and forms of spatial vibrations are obtained. The effect of the ice casing on the vibration spectrum of the wire is studied. The dependence of the wave number of torsional vibrations on the frequency has been found, which is determined not only by the elastic-inertial, but also by the gravitational factor, which is strongly manifested for wires in long spans, especially those prone to Aeolian vibration (galloping). This circumstance is essential for the analysis of the Aeolian vibration phenomenon from the positions linking the occurrence of dancing by the convergence of the frequencies of torsional and transverse modes during the icing of the wire. It has been shown that the ratio of these frequencies, which cause an auto-oscillatory process, turns out to be significantly more complex.

The spectrum of frequencies and shapes of bending vibrations of a rectangular plate in contact with a liquid or gas are determined. A derivation of the expression for the distributed transverse load on a plate movably embedded along the contour is given. The surfaces of the plate are in contact with media of different densities and pressures. The medium can be compressible during surface deformation and incompressible. The influence on the bending of the interaction of average pressure and changes in the curvature of the middle surface, as well as the added mass of the gaseous medium, is determined.

In this paper, the mechanics of micropolar elastic solids is extendded to more general thermoelastic media in order to take account of the effect of temperature on their states and mechanical behavior. Since a thermoelastic micropolar medium conducts heat, it is required to include one or another mechanism of thermal conductivity in the basic equations of micropolar thermoelasticity. A model of thermoelastic micropolar medium CGNII is developed on ground of the wave principle of heat transfer (i.e., thermal conductivity of the second type known from previous discussions by Green and Naghdi), characterized by zero internal entropy production. All the basic equations of the theory presented in this study are derived from the conventional equations of balance of continuum mechanics and the fundamental thermodynamic inequality. Constitutive equations for a linear anisotropic thermoelastic micropolar medium (CGNII) are obtained by using a quadratic energy form for the Helmholtz free energy. Special attention is paid to hemitropic micropolar medium, when the components of one of the fourth rank constitutive pseudotensors demonstrate sensitivity to mirror reflections of three-dimensional space. A closed system of coupled differential equations is given in terms of translational displacements, microrotations and temperature displacement. It is important since can be used in formulations of applied problems of thermomechanics regarding to the wave heat transfer mechanism in micropolar elastic media.