Izvestiâ Akademii nauk. Rossijskaâ akademiâ nauk. Mehanika tverdogo tela.

A mechanical system consisting of a rigid body and a material point is considered. They interact with each other by means of internal forces whose physical nature is not defined. Both objects are affected by external forces specified as functions of time. The task is to construct a trajectory for a point mass such that the solid body, under the action of the force of interaction with this mass, changes its orientation in space according to a predetermined program. Based on the theorem on the change in the relative moment of momentum, a second-order vector differential equation, unresolved with respect to the highest derivative, is obtained that describes the system. A change of variables is found that allows replacing the original equation with a first-order vector differential equation resolved with respect to the derivative. All special cases arising from the use of the new equation are considered. The obtained relationships can be used to control spacecraft and robotic systems.

Investigation of the conditions of conventional stability of steady-state motions of a mechanical system, which assumes existence of an additional partial V.A. Steklov’s integral, has been conducted. Analysis of conventional stability is conducted by Chetayev’s technique with the Lyapunov’s function represented by a full bundle of first integrals of the disturbed motion. It has been ascertained by analytical computations that, in case of a full bundle of integrals, it is possible to formulate sufficient stability conditions, which are the most close to the necessary conditions. It has been demonstrated for the scrutinized system that the boundary conditions of stability may be reached under the limit values of some Langrange multipliers. This allows the researcher to consider the bundles of integrals in the Routh-Lyapunov method and in the Chetayev’s technique differently.

The influence of shield around crystal on gas-and hydrodynamics, heat transfer and defects formation in Czochralski silicon crystal growth process is studied by means of mathematical modeling. The domestic silicon crystal growth furnace Redmet-90M is considered, which allows the silicon single crystal growing with 200 mm in diameter and 1.5 m in length. The growth process occurs under argon flow pumping in the rarefied atmosphere of growth camera. High-temperature heating ensures silicon melting in crucible and silicon ingot crystallization by Czochralski pulling from a melt. The mathematical model considers the conjugation of heat exchange and silicon monoxide transfer processes. The effect of shield around crystal on axial temperature distribution in grown single crystal is considered. For dislocation-free single crystals this characterizes the type of forming intrinsic point defects. The international verification results of the shield around crystal influence on axial temperature distribution in the growing crystal for Czochralski crystal growth furnace EKZ-1300 are discussed. Also, the verification results in application to Czochralski crystal growing on EKZ-1600ELMA are presented, which illustrate the application of defect formation theory of V.V. Voronkov in dislocation-free single silicon crystals.

The purpose of the work is to develop a model of viscoelastic thermomechanical behavior of cross-linked polyethylene (CLPE) products with a shape memory effect. The main area of application of this class of materials is heat-shrinkable tubes and their modifications. They are characterized by initial expansion values of 100 percent or more. Simo and Holzapfel proposed modified defining relations representing a synthesis of the viscoelastic Proni model and the hyperelastic model based on the elastic energy potential. The published work proposes a methodology for identification of both damping and nonlinear elastic material constants in case of large deformations. An experimental program is developed and implemented to determine the material constants of cross-linked polyethylene. It includes the procedure of reducing the dimension of the nonlinear optimization problem when finding exponential approximation coefficients of the relaxation function. A description is given of the developed algorithm for searching for the material constants of a hyperelastic model based on the results of an experiment on cyclic loading of a sample with large deformations in the range of highly elastic behavior. An adaptation of the hyperelastic model in ANSYS is made taking into account the presence of relaxation properties. As an illustration of the application of the updated physical model, an example of a finite element calculation of the pressure of heat-shrinkable tubes when deposited on a rigid cylindrical surface is given. An engineering technique is also proposed that allows similar calculations to be made in the Excel spreadsheet.

Analytical solutions have been obtained for tensors of effective electromagneticmotive properties of a composite with a quasi-periodic structure, taking into account the anisotropy of the disordered inclusions relative to the corresponding ideal periodic structure in a generalized singular approximation of the method of periodic components of statistical mechanics. The magnitude and anisotropy of the disordered inclusions (fibers) are taken into account in the solution through the coefficients of periodicity and anisotropy of the disordered inclusions relative to the corresponding ideal periodic structure, the effective properties of which are considered known. For the case of isotropic disordered inclusions, analytical solutions for tensors of effective electromagneticmotive elastic and pyroelectric properties of a composite with a quasi-periodic structure are obtained as simple linear decompositions of the desired solutions through known tensors of corresponding effective properties for a periodic structure and for a statistical mixture using the periodicity coefficient. The results of numerical modeling of the frequency dependences of the electromagnetic coupling coefficients are presented, taking into account a given initial stress state and the Maxwell-Wagner relaxation of the electric fields of the structural elements of a quasi-periodic composite.

In the approximation of the Föppl—von Kármán model, which takes into account the presence of plastic deformations, the solution of the problem of elastoplastic bending of a thin plate was obtained, under the boundary conditions of the type of rigid, or generalized elastic embedment type. The model of ideal plasticity with the Tresca—Saint-Venant yield surface was used. The use of the standard Kirchhoff—Love hypotheses allowed to reduce the problem to a system of ordinary differential equations. The numerical solution of this system for the boundary conditions of the generalized elastic embedment was obtained. The solution for the boundary conditions of rigid clamping is obtained as a particular case.

This paper addresses the problem of altering the natural vibration frequencies of an elastic body with embedded piezoelectric elements by applying an electric potential to them. Presented mathematical formulation of the problem based on the principle of virtual displacements for a piecewise-homogeneous electroelastic body. Finite deformations are represented as the sum of linear and nonlinear parts, which are linearized with respect to a state featuring a small deviation from the initial equilibrium position caused by the reverse piezoelectric effect. Provided experimental and numerical results validate the reliability of the numerical algorithm based on the finite element method. Using a plate with an embedded piezoelectric element as an example, presented numerical results demonstrate the influence of various parameters on the change in natural vibration frequencies: the stiffness characteristics of the elastic body; the dimensions, location, and number of piezoelectric actuators; the area ratio of the piezoelectric element to the elastic body; and the magnitude and sign of the electric potential.