Том 215, № 2 (2016)

We study a three-dimensional problem of the theory of elasticity for a rectangular parallelepiped in the case of steady-state forced vibrations. By the method of superposition, we reduce the problem to an infinite system of linear algebraic equations for the coefficients of double Fourier series. For this infinite system, we prove that the conditions of quasiregularity are satisfied and that the bounded solution exists. We also construct the asymptotics that describes the behavior of unknowns in the infinite system. The method is illustrated by several numerical examples.

We solve the problem of coupled vibrations for an adhesive lap joint of two rods. The connecting layer is modeled by a multiparameter elastic foundation and the outer layers are regarded as Timoshenko beams. This approach describes, with high accuracy, the stressed state of the connecting layer and enables one to satisfy the boundary conditions on its free boundary. We solve a model problem and compare our results with the data of numerical analysis carried out by the classical method.

We consider the method of cell functions allowing one to compute the temperature and thermal fields in 2D periodic composites. The coefficients of macroscopic heat conduction are determined as the integrals of cell functions obtained as a result of the solution of a family of boundary-value problems in a periodic cell.

We propose a procedure of getting analytic expressions for the description of axisymmetric stationary thermal fields, axisymmetric static or quasistatic stress and strain fields in long hollow multilayer cylinders made of thermally sensitive materials with constant normal loads and arbitrary classical conditions of heat exchange specified on the bounding surfaces. The problem of construction of the solution of the problem of heat conduction is reduced to the determination of one constant of integration and all other constants from a nonlinear algebraic equation are determined via the indicated constant. The problem of thermoelasticity is thus reduced to the solution of a system of Volterra integral equations of the second kind with the corresponding integral conditions. As a result of application of the proposed methods for the solution of the system of integral equations, we deduce formulas for the evaluation of the characteristics of the stress-strain state in the form of functional dependences on temperature, mass forces, thicknesses of the layers, surface loads and the temperature dependences of the mechanical characteristics of the materials of layers.

We study the thermal stressed state of a finite cylindrical shell (with conditions of sliding restraint imposed on its end faces) caused by the difference of temperatures of the ambient medium on the front faces of the shell and the coordinate-dependent heat-transfer coefficients on these faces. We propose a method for the reduction of the boundary-value problem of heat conduction to a coupled system of Fredholm integral equations of the second kind and present the results of numerical analysis of the distributions of mean temperature and temperature moment and the values of parameters induced by these distributions, namely, the deflection, elongation, forces, and bending moments.