Article outline

An overview of the studies carried out by the staff of the Laboratory “Solid Mechanics” for 2020-2022 is given. The objects of research are: a cantilever rod, a cylindrical shell, an underwater gas pipeline, a pipeline with vibrating supports, a resonator with rectangular and circular cross sections on hinged supports, a rod of micro- and nanometer cross-section clamped at the ends. In this case, all objects are in contact with liquid or gas.

The purpose of the research is to study the statics and dynamics of a rod, pipeline, carbon nanotubes (CNTs), nanoresonators and nanostrings interacting with the medium. To achieve this goal, new tasks were formulated and solved.

Methods. In the presented works, it is assumed that the objects under consideration are deformed within the limits of Hooke's law. In this case, the deformations of thin-walled structural elements obey Kirchhoff's hypotheses. Continuous media (liquid or gas) interacting with elastic objects are considered ideal. The motion of research objects is described by both linear and non-linear partial differential equations. To solve them, for example, in the study of nonlinear bending vibrations of a pipeline, the Bubnov-Galerkin variational method and the Runge-Kutta numerical integration method are successively applied. The analysis of numerical calculations is performed using the discrete Fourier transform and the Poincaré mapping.

The results of the studies carried out are of both theoretical and practical interest. First, the development of the theory of direct and inverse problems of stability and oscillations in aerohydroelastic systems containing thin-walled elements, including those with micro and nano sizes, has been obtained. Secondly, the results of studying the statics and dynamics of thin-walled structural elements interacting with continuous media can be used in power engineering, the oil and gas industry, technological machines, vacuum, aviation, and rocket and space technology. In particular, some of the results related to nanoelements are needed for the design of terahertz resonators based on CNTs and high-precision mass and force nanosensors based on the electromechanical coupling effect that CNTs exhibit.

Conclusions. The review presents brief contents and main conclusions from the results of solving a number of new problems of the interaction of thin-walled elastic structural elements with a liquid or gas.

An experimental and theoretical study of the linear stability of an elastic cylindrical shell containing an incompressible ideal fluid under impact on the end in the axial direction is carried out under the assumption of an instantaneous establishment of compression along the entire length. A qualitative agreement between experimental and theoretical data is shown.

The eigenfrequencies of bending vibrations of a wire with a cross section of micro- and nanosizes are determined. A significant effect of the difference in the elastic characteristics of the main volume of the resonator and its near-surface layer on the frequency spectrum has been established. In addition, the dependence of the frequency spectrum on the average pressure of the medium (such as a light gas) on the resonator surface is shown.

A review of studies on three approaches to solving problems of identifying local defects in rods and pipelines is given. In the first approach, local defects are modeled by conjugation conditions, in the second one, by delta functions included in the differential equation, in the third approach, longitudinal cavities and cracks are modeled by continuous sections of the rod with changed stiffness, cross-sectional area, and density.

The influence of the conditions of fastening along the contour of a round plate under pressure on its axisymmetric bending is studied. Deflections are determined in linear and nonlinear formulations.

The dynamics of a liquid in a cylindrical container and the impulse action on the bottom when hitting its upper end are considered. The case was studied when cavitation occurs in the column of liquid near the bottom upon impact. The study was carried out under the conditions of well-known experiments on the destruction of a bottle as a result of a blow to its upper end. It is shown that the destruction of the container in the experiments occurs due to water hammer caused by the collapse of the cavitation cavity.

The areas of attraction of the deflection to the upper and lower equilibrium positions of a two-support pipe during its spatial bending-rotational vibrations are determined. It has been established that the prevailing influence on the bending of the internal pressure of the medium in the pipeline over its weight and the excess of the moment of vibration forces of supports over the moment of gravitational forces determine the presence of upper and lower equilibrium positions and the corresponding areas of attraction.

The interaction of forced and parametric oscillations of the pipeline has been studied. Approximate analytical expressions are obtained that determine the amplitudes of nonlinear oscillations in the vicinity of a linear resonance of forced and main resonance of nonlinear parametric oscillations of a pipe. It is shown that the results of the analytical and numerical solutions of the problem are in satisfactory agreement. It has been established that in the oscillatory movements of the pipe there are oscillations with both the fundamental frequency and fractional frequencies.

In static and dynamic formulations, the linear bending of a cantilever bar loaded with all-round pressure and longitudinal force is considered. The action of the transverse distributed force on the rod, which occurs during bending as a result of the formation of the difference in the areas of the convex and concave parts of the surface, is taken into account.

The effect of the internal added mass of a gas-liquid medium with an annular flow shape on the nonlinear bending vibrations of a pipeline has been studied. It has been established that for certain values of the input data, this influence can be significant. The total attached mass of the internal and external continuous media of a pipe moving with acceleration in the transverse direction is determined. It is shown that the flow of particles of the liquid phase in the cross sections of the pipeline causes a change the attached mass of the transported medium. It has been established that the ratio of frequencies of natural vibrations with and without taking into account the effective added mass depends on the ratio of the densities of the liquid and material, the wall thickness and the inner radius of the annular region of the liquid phase to the radius of the pipe. When the pipe is completely filled with liquid, these masses are equal to each other.

The radial dynamics of a single-layer carbon nanotube is considered. The dependence of the modes of oscillatory motion and the exponential increase in the deflection on the ratio of the values of the external pressure on the surface and the critical static pressure is established.

The eigenfrequencies of bending vibrations of a micro- and nanometer-sized rod clamped at the ends are calculated. From the solution of the inverse problem for the changed values of natural frequencies, the coordinate and value of the added mass are found. Linear vibrations of micro- and nanostrings with varying pressure in a gaseous medium are also considered, taking into account surface effects. The eigenfrequencies of bending vibrations of carbon nanotubes (CNTs) of different diameters are calculated using a molecular dynamics model with a reduced number of degrees of freedom under the conditions of a flat deformed state.

The simplest model of the ascent of a span of an underwater gas pipeline has been constructed. The model describes the rise of the pipeline to a position that coincides with its upper part with the water surface. The above model of ascent can be generalized taking into account the elastic nonlinearity of the pipeline and the compliance of supports, its thermal expansion, fluctuations in gas pressure with the frequency of operation of pumping stations. The conditions for the rise of the pipeline are determined.

The static interaction of instabilities is considered depending on the compression force, internal and external pressures, fluid flow velocity, axisymmetric expansion of the pipeline, and changes in the temperature of its wall. It is shown that bending stiffness, tensile forces and external hydrostatic pressure stabilize, while compression forces, internal hydrostatic pressure, movement of the transported liquid at any speed and an increase in wall temperature destabilize the pipeline.