No 2 (2024)

The article is an analytical review and is devoted to the theory of thin, isotropic elastic plates. There are basic relations of the theory based on the kinematic hypothesis confirming that the tangential displacements are distributed linearly along the thickness of the plate and its deflection does not depend on the normal coordinate. As a result, a system of equations of the sixth order with respect to two potential functions – the penetrating potential, which determines the plate deflection, and the boundary potential which makes it possible to set three boundary conditions on the plate edge and eliminate the known contradiction of Kirchhoff's plate theory was obtained. Problems that have no correct solution in the framework of Kirchhoff's theory – cylindrical bending of a plate with a free edge, bending of a rectangular plate with non-classical hinge fixing, torsion of a square plate by moments distributed along the contour, and bending of a plate by a rigid die – are considered. In conclusion, a brief historical review of the papers devoted to the theory of plate bending was presented.

The recent earthquake in Turkey and Syria occurred on February 6, 2023. In the Kahramanmaras region the earthquake had an estimated magnitude of Mw7.8, and reached intensity XI by the modified Mercalli scale. The earthquake had catastrophic consequences, leading to the deaths of more than 52,800 people, as well as the multiple damages of the structures and buildings. The paper relates to the analyses of the seismograms and the detection of an unusually strong delta impulse, which can be associated with the arrival of a horizontally polarized S-wave.

A schematic diagram and mathematical model of functioning of a new piezoelectric membrane (MDS) actuator with double spiral (DS) electrodes on the upper and/or lower surfaces of a thin piezoelectric layer with axisymmetric and periodic (with a small period) in radial coordinate mutual reversed electric polarization are presented. The polarization of the layer was realized as a result of connecting the polarizing electric voltage of the appropriate value to the outputs of the double spirals of the electrodes. The electrodes of each (upper and lower) double spiral of the MDS-actuator are made in the form of electrodeposited ribbon coatings on the surfaces of the piezoelectric layer in close proximity to each other (due to the small spiral pitch) to create high values of electric field strength along the lines of force in localized areas of the piezoelectric layer between them when an alternating or constant control electric voltage is connected to the electrodes, in particular, with positive and negative values of the electrical potentials. Importantly, the electric field force lines and, as a consequence, the polarization of the piezoelectric layer of the MDS actuator are oriented mainly along (i.e. towards or against) the radial coordinate of the membrane, in contrast to many conventional actuator schemes. The results of numerical modeling for a circular elastic membrane with piezoelectric actuators installed on its upper and lower surfaces confirmed the effectiveness of the proposed piezoelectric MDS-actuator when it functions according to the “bimorph” scheme, including the use of the proposed new structural element (section) – a piezoelectric “compression ring” MDS at various geometric and control parameters. The effect of a significant increase in the membrane deflection with installed piezoelectric MDS-actuators compared to the use of traditional homogeneous plate piezoelectric actuators of bimorph type for different conditions of the membrane fixation, in particular, stationary (rigid) fixation of its center is revealed. For a hybrid piezoelectric MDS-actuator including independent concentric round and circular (i.e. “compression ring”) sections, the non-monotonic nature and numerical analysis of the nonlinear dependence of the largest deflection at the center of a hinge-immobile membrane fixed at the edge on the ratio of the radii of its round and circular MDS sections were revealed. The cases in which the effect of the “compression ring” is manifested, i.e. when the maximum deflection of a membrane with the “compression ring” exceeds the best possible value of the deflection of this membrane without its use in the traditional “bimorph” scheme, are identified. The new piezoelectric MDS-actuator can be used in micromechanics, controlled optics, sensor technology, acoustics, in particular, in the manufacture of piezoelectric acoustic or sensor elements of membrane type, electromechanical transducers for vibration energy collection.

The research is relevant for the trajectory movement design modeling of the of vehicles on deformable support wheels. The purpose of the research: to obtain a theoretical dependence for calculating the effective normal stiffness of a deformable wheel with an inclined rotation axis. Mathematical relationships between this stiffness and the rotation axis tilt angle of the wheel are defined. It is determined that the effective normal stiffness changes by a factor of Kaz at specified tilt angle. A theoretical dependence for calculating the correction factor Kaz was obtained. The dependence is a function of the wheel design parameters and the rotation axis tilt angle. The dependence is correct at angles up to 10 degrees. At tilt angles that are permissible under the conditions of wheel wear (up to 5 degrees), the effective normal stiffness is significantly reduced, for example, for the study object, up to 25%. The validity of the theoretical dependence is confirmed by laboratory experiments on a specially designed stand for measuring the parameters of the elastic properties of the deformable wheel at different positions of its rotation axis. Based on the processing of the experimental data, the calculation error of Kaz according to the obtained theoretical dependence was determined, which does not exceed 6%.

An exact solution to the contact problem of indentation of an absolutely rigid punch with a straight base, taking into account friction, into one of the edges of a finite crack located in a homogeneous elastic plane was derived. It is assumed that shear contact stresses are directly proportional to normal contact pressure. In this case, it is assumed that the friction coefficient is directly proportional to the coordinates of the contacting points of the contacting surfaces. The governing system of equations for the problem was derived in the form of the heterogeneous Riemann problem for two functions with variable coefficients and its closed solution is constructed in quadratures. Simple formulas for contact stresses and the normal dislocation component of displacements of crack edge points were obtained. The patterns of changes in contact stresses and crack opening depending on the maximum value of the friction coefficient have been studied.

The problem of the necessary class of smoothness of solutions to quasi-static problems of deformable solid mechanics in terms of displacements was discussed. It is shown that in order for the equations of compatibility of deformations to become identities when displacements are substituted in them, the existence of some third mixed derivatives of displacements is required. A counterexample for a linearly elastic compressible isotropic elastic medium was given. In this counterexample, the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lame equations in the entire domain, is not a solution to the displacement problem at all points in this domain.

The principle of modification of mechanical properties of thin-film membrane structures of arbitrary shape by non-contact method was proposed, realized and explained for the first time. The idea was tested on an aluminum thin-film membrane formed by magnetron method on a silicon substrate. The external influence was realized by means of cyclic loading in the form of discharge and supply of excess air pressure to the membrane. As a result of repeated impacts, physical properties of materials (grain size and roughness) and mechanical properties (internal mechanical stresses and critical overpressure) are changed. Changing the magnitude of residual mechanical stresses in the membrane material allows the formation of a surface with a desired curvature value. In this work, after cyclic loading with pressure equal to half of the critical pressure, the following effects were revealed: the deflection of the membrane in the absence of external influence increased by more than an order of magnitude, the structure shifted to the plastic type of deformation, the critical rupture pressure decreased by several tens of percent. Application of this methodology allows to create new materials with unique mechanical properties.

The rationale for the prospects of using the physical and mathematical theory of metal creep in creep computations is carried out by a comparative analysis of the classical phenomenological and physical and mathematical metal creep theories. On the example of the description by both theories specific results of non-stationary creep experiments and analysis of the theories equations it is shown that implementing the physical kinetic equation for the actual structural parameter of the material, namely the scalar density of immobile dislocations, makes the physical and mathematical theory universal for solving non-stationary metal creep problems with multiaxial loading, when change, including abruptly, temperature, forces and loading rates.