A refined mathematical model is constructed to describe the process of geometrically and physically nonlinear deformation of a test specimen made of fibrous composite of rectangular cross-section, having thin elastic side linings in the attachment areas in the fixture for axial compression testing. The latter are intended to transfer external load to the specimen under kinematic loading (due to friction forces arising between the linings and rigid elements of the test fixture), ensuring the implementation of one of the possible loading schemes in accordance with existing testing standards. The specimen with linings in the attachment areas in the fixture is represented as a three-layer rod, for which the S.P. Timoshenko shear model is adopted for the linings, taking into account transverse compression, and for the middle layer along the thickness — a linear approximation for deflection and a cubic approximation for axial displacement. The kinematic relations and equilibrium equations of the theory are obtained based on the geometrically nonlinear relations of elasticity theory written in a simplified quadratic approximation. They retain such geometrically nonlinear terms that, having the necessary degree of accuracy and content, allow identifying both classical flexural and non-classical transverse-shear buckling modes of specimens during compression testing. For fibrous composites with unidirectional structure, physical nonlinearity is taken into account only in the relationship between transverse shear stress and the corresponding shear strain, and for composites with a ±45° structure, also in the relationship between normal stress in the cross-section of the specimen and the corresponding axial strain.

The article considers an optical stand designed to study the conditions for the occurrence of cavitation phenomena in melts of optically transparent polymer materials and other optically transparent media. The operation of the stand is based on the modulation of optical radiation in the infrared range passing through a transparent polymer due to oscillations of cavitation bubbles arising at the moment of transition of the initially solid polymer to a liquid state. The study of cavitation in polymer melts is of great practical importance, since on the one hand, cavitation and the secondary hydrodynamic phenomena it generates can contribute to the acceleration of diffusion processes occurring at the interface of layered materials during their ultrasonic welding. On the other hand, emerging cavitation bubbles subsequently transform into non-collapsing gas cavities, the presence of which weakens the strength of the welded joint upon polymer cooling. Thus, during ultrasonic welding, in addition to the amplitude of the ultrasonic action, the compression force of the welded materials, and the time of ultrasonic action, the presence or absence of cavitation phenomena in the polymer melt can be attributed to the factors determining the quality of the formed welded joint. The result of the stand operation is the signal spectra from the optical sensor output, which reflect the presence or absence of cavitation bubble pulsations, as well as the frequencies at which these pulsations occur. The operability of the stand is tested both on liquid media and on the example of an optically transparent polymer material (acrylic glass). The presented stand will allow studying and fully revealing the potential of ultrasonic welding of polymer materials, and, in particular, will allow developing methods for implementing ultrasonic welding of dissimilar polymer materials.

The relations of a mathematical model for the analysis of static strength and stability of structurally anisotropic composite panels of aircraft with skins of medium and large thickness, taking into account transverse shear deformations, are presented. A method for solving stability problems based on a new tenth-order partial differential governing equation is developed. The equation is constructed by the method of symbolic integration of a system of five differential equilibrium equations derived within the framework of an improved model that takes into account transverse shears according to first-order shear theory, where the plane problem and the bending problem are coupled. In the case of a thickness-asymmetric orthotropic structure of the composite panel, the linear differential operator of the governing equation contains even-order derivatives with respect to each coordinate. Analytical constraints for the mass objective function in the optimal design of skins of medium and large thickness, based on the refined stability theory, are obtained. The result of the analytical solution of the stability problem is an analytically formulated constraint in the form of an inequality when constructing the final relation for the critical load parameter in explicit form. The influence of design parameters on critical loads is investigated. The applicability limits of first-order shear theory and classical thin plate theory for longitudinally compressed flat carbon fiber skins of rectangular shape are determined. The possibility of using analytical solutions of stability problems based on linear differential operators of the eighth and tenth orders as design constraints in the creation of advanced aviation composite panels is proven. When designing root zones of panels in the area of attachment to the side spar, constraints on the mass objective function should be introduced according to the compressive strength criterion; in this case, the average critical stresses across the laminate reach values that provide a significant stability margin.

The article is devoted to mathematical modeling of vibrations of rod systems made of domestic high-damping steel 01Yu5T. The damping properties of steel 01Yu5T are of magnetostrictive nature, are characterized by a pronounced dependence on the amplitude of stresses acting in the element, and at the same time do not depend on the vibration frequency. The peak value of the damping capacity is achieved at a stress amplitude of 12 MPa and amounts to 37%. To describe the dynamic behavior of rod systems made of high-damping steel, a non-local in time model of dynamic deformation of the material is used in the work. The model is built on the assumption that the state of the system at the current moment is influenced by the entire history of its deformation. To ensure the applicability of the model for solving applied problems, it is integrated into the finite element method algorithm. The level of dissipative properties in the material is determined by the scale parameter of the non-local model, which characterizes the rate of decay of the kernel function. Since the level of acting stresses may differ in different elements of the rod system, a kernel function matrix is introduced into the equation of motion formulated in a non-local finite element formulation. When solving the equation of motion using an implicit scheme at each step of the iterative process, the scale parameters of the elements of the kernel function matrix are determined depending on the stresses acting in the elements. As a numerical example, a truss structure of five rods loaded by a concentrated force varying randomly in time is considered. The results of numerical simulation for the considered structure made of steel 01Yu5T are compared with the results obtained for steel 3.

A variational formulation of the Nth-order theory of generalized-shallow elastic orthotropic shells of variable thickness is proposed. A generalized-shallow shell is defined as a three-dimensional body bounded by two non-intersecting smooth face surfaces that allow unambiguous parametrization in a coordinate system normally connected to the plane, and a piecewise-smooth ruled lateral surface. In accordance with the variational formalism of analytical mechanics of continuous systems, a three-dimensional shell model specified by the displacement vector, the spatial density of the Lagrangian functional and its boundary density is taken as the basis for further calculations. By differentiating the spatial density of the Lagrangian with respect to the covariant derivatives along one of the families of coordinate lines of the curvilinear coordinate system on the reference plane, the components of the stress vector on areas orthogonal to this direction are determined. By the Legendre transformation of the Lagrangian functional of the three-dimensional shell model with respect to the conjugate pairs of covariant derivatives of the displacement vector components and the contravariant components of the conjugate stress vector, a mixed functional is obtained, interpreted by analogy with the mechanics of discrete systems as a generalized Rouse functional. The spatial reduction of the three-dimensional model and its reduction to the Nth-order shell theory consists in expanding the components of the displacement and stress vectors in terms of functions of some biorthogonal system that forms a basis in the Hilbert space over the domain of definition of the dimensionless normal coordinate, and calculating the corresponding integrals. The obtained shell model as a continuum-discrete mechanical system is defined by a 6(N+1)-dimensional state space with 3(N+1) first-kind field variables (expansion coefficients of the displacement vector) and 3(N+1) generalized forces conjugate to them (moments of the stress vector), the surface density of the generalized Rouse functional defined on the reference plane and independent of covariant derivatives along the selected direction, and its contour density. The extremal conditions corresponding to the Lagrange variational principle reduce to a system of 6N+6 equations resolved with respect to first-order covariant derivatives along the selected coordinate direction, similar to the Rouse equations of a discrete mechanical system. The generalized Rouse equations are given for the case of a homogeneous isotropic shell of variable thickness on a circular plan, which is a model of an intraocular lens.

Currently, the manufacture of parts by additive technologies based on metamaterials is gaining popularity. A subclass of such materials are periodic lattice structures. Direct calculation of such structures by numerical methods is quite difficult or practically impossible, which forces engineers and researchers to use various averaging methods for further calculation based on classical approaches. The work is devoted to determining the elastic properties of a three-dimensional rectangular lattice by two independent methods; cross-sections in the form of a circle and a square are considered for the ribs. The first method for determining elastic properties consists of modeling the lattice ribs using the Timoshenko beam technical theory, the second method models the ribs of the same lattice using the apparatus of continuum mechanics (three-dimensional elasticity theory). Averaging is implemented using periodic boundary conditions, which allows obtaining the effective properties of a periodic medium from a problem on only one cell. A comparison of such approaches and the possibility of their applicability is carried out. The use of beam models allows obtaining effective moduli in the form of finite formulas, which provides great convenience. It is known that the use of beam models gives underestimated stiffness values for the effective material due to the simplified modeling of the lattice rib joint zone. However, in such works, no study was conducted regarding the change in the ratio of rib lengths between different sides of the periodicity cell. Of particular interest are the shear properties of such a material. This paper provides an analysis for rectangular periodicity cells. Analytical formulas obtained on the basis of beam models and their verification using numerical methods using three-dimensional elements modeling a continuous medium are proposed.

A previously unknown exact solution of the boundary value problem on the tension of an infinite elastic strip with a stiffening rib perpendicular to the strip axis (the rib length equals the strip width) is constructed. The stiffening rib works only in tension-compression. The solution includes two stages. At the first stage, using the Papkovich orthogonality relation, a solution to the inhomogeneous problem for the strip is constructed, in which the external load represents the unknown contact stresses between the rib and the plate acting along the rib-plate contact line. At the second stage, the contact stresses are determined from the equilibrium condition of an elementary section of the rib, which are then substituted into the solution of the inhomogeneous problem. The problem of determining contact stresses is solved using a Lagrange series expansion in terms of the system of Papkovich-Fadle eigenfunctions of the linear transverse displacement on the vertical axis of the strip. This is one of the key points in the method for solving the problem. At each stage of the solution, the expansions are represented by series in terms of Papkovich-Fadle eigenfunctions, the coefficients of which are determined explicitly. Thanks to this, it is possible to construct an exact solution of the problem. The expansions using the Papkovich orthogonality relation and the Lagrange series expansions are related to each other. This relationship is based on the existence of nontrivial expansions of zero in series of Papkovich-Fadle eigenfunctions. It consists in the fact that the series constructed using the Papkovich orthogonality relation equals the sum of the Lagrange series and some series representing a nontrivial expansion of zero. The obtained solution is compared with the numerical solution obtained on the basis of the finite element method.

In the first part of the article, for power actuators with a working body in the form of a rod made of shape memory alloy and linearly elastic displacement bodies, analytical dependencies were obtained between the initial deformation of the working body and the ratio of the stiffnesses of the working body and the displacement body, ensuring the implementation of a closed two-way shape memory effect during heating and cooling of the working body. Analytical dependencies were obtained for the minimum possible values of the ratios of the displacement body length to the working body length, cross-sectional areas, and volume ratios of the same elements under this requirement. It was established that when using titanium nickelide as the material for the working body and structural metals and alloys for manufacturing displacement bodies in the form of rods or coil springs, the dimensions, volumes and mass of the displacement bodies must many times exceed the same quantities for the working body, which indicates the irrationality of the corresponding design. In this work, a similar conclusion is obtained for a displacement body in the form of a metal bendable beam. To solve the arising problem, a criterion for selecting material for creating displacement bodies for power actuators of the considered type is proposed. According to this criterion, polymers have some advantages over metals and alloys when creating displacement bodies. However, the most rational designs are obtained when creating displacement bodies from unidirectional polymer composites with glass or organic fibers. Specific design parameters of power actuators of this type and their output data are presented.

In this work, elastomer composites developed for use in vibration damper structures were investigated. The mechanical properties of these composites were evaluated depending on the composition and addition of shungite rock particles of various dispersity. The hysteresis (dissipative) properties of the composites, as well as the creep of these materials during nanoindentation, were studied. The micromechanical characteristics of elastomer composite samples, as well as creep curves before and after ultraviolet irradiation, were evaluated. For the indentation depth versus time dependencies obtained during the experiment, an approximating mathematical function was selected whose parameters are the actual viscoelastic and viscoplastic characteristics of the material. This approximating function corresponds to the function of the four-parameter Maxwell-Voigt mechanical model. Elastomer composites with improved composition and enhanced mechanical characteristics for use as structural elements of vibration dampers were developed. It was established that the required set of mechanical characteristics of these materials is achieved by using a combination of butyl rubber and EPDM, as well as the addition of submicron shungite rock particles (5 vol.%). It is shown that the addition of submicron shungite rock particles in elastomer composites based on a mixture (combination) of butyl rubber and EPDM allows increasing the relative hysteresis (up to 39%) of these materials while maintaining elastic-strength properties. A method for quantitative assessment of the effect of UV radiation on the properties of elastomer composites based on analysis of the kinetics of crack and cluster formation on the sample surface and experimental evaluation of micromechanical properties by nanoindentation was developed. It was found that UV exposure affects the surface structure and micromechanical characteristics of the samples differently depending on the composition. When studying samples of elastomer composites based on EPDM and a combination of butyl rubber and EPDM, it was found that the addition of micro- and especially submicron shungite rock particles increases the resistance of these composites to UV exposure and allows increasing the performance of these materials as structural elements of vibration dampers. For the studied samples of elastomer composites, when analyzing creep curves during nanoindentation in the case of a four-parameter model, the following characteristics were obtained: elastic components (E₁, E₂), viscosity components (η₁, η₂). It was found that after UV exposure on the composite, the elastic component E₁ decreases noticeably (by 15%), the viscosity component η₁ (by 10%), and the viscosity component η₂ changes most significantly (up to three times).