An analysis of the three-dimensional equations of elasticity theory formulated for the case of finite displacements and deformations of solids is carried out. It is established that for small deformations, their generally accepted simplifications are insufficiently justified, which is confirmed by the formulation and solution of model problems based on them both for determining the stress-strain state and for critical loads and buckling modes. All known equations in the mechanics of thin-walled structures for finite displacements and small deformations, based on determining deformations as increments of the metric tensor components, are also insufficiently justified. Such geometrically nonlinear equations of elasticity theory for finite displacements and small deformations are constructed that are correct and consistent. They also turned out to be absolutely accurate in the case of large shear-free deformations of the body. For straight elastic rods subjected to conservative external forces and made of orthotropic material, based on the use of the Timoshenko model, a reduction of three-dimensional linearized equations of elasticity theory to one-dimensional equations is carried out in two variants. The first of them corresponds to the use of a consistent variant of three-dimensional geometrically nonlinear relations compiled in an incomplete quadratic approximation and the Timoshenko model without taking into account transverse elongation deformations, and the second — to three-dimensional relations compiled in a complete quadratic approximation and the Timoshenko model taking into account transverse elongation deformations. In order to conduct a qualitative analysis of their content, a number of new non-classical rod stability problems are formulated on their basis and their analytical solutions are found. These include, in particular, problems of torsional, flexural and shear buckling modes of a rod under longitudinal axial, bilateral transverse and triaxial compression of the rod, on flexural-torsional buckling under pure bending and axial compression together with pure bending, as well as on flexural buckling of a rod under torsion and flexural-torsional buckling under pure shear.

The paper investigates the stability of modified laminated composite plates that use modified fibers containing an additional whiskerized layer. It is assumed that the fibers are modified with carbon nanotubes grown perpendicular to the fiber surface. Composites modified in this way have slightly increased transverse strength characteristics, which makes them attractive from the point of view of damage accumulation. The paper compares the stability of plates made of modified composites with the corresponding stability estimates of laminated plates made of traditional fibers. The effective mechanical properties of the modified composite were preliminarily determined using the Christensen-Eshelby three-phase method. In the general case, laminated plates simply supported along the contour under the action of forces acting in the direction of one of the axes were considered. The dependence of the stability of a laminated composite formed by traditional and modified fibers on the fiber lay-up angle and on the volume fraction of the inclusion is investigated. A direct numerical calculation of the stability of plates by the finite element method is presented for two types of laminated plates: for a plate reinforced with carbon fiber and for a plate reinforced with modified carbon fiber. For the same plates, a calculation is performed using formulas that allow estimating the influence of transverse shear deformation on the critical buckling load. The discrepancies in the results of analytical analysis and numerical analysis by the finite element method amounted to no more than 10%. The differences in numerical and analytical values increase with increasing fiber volume fraction. The possibility of increasing the mechanical properties of a composite plate due to fiber whiskerization is assessed. Calculations show that the studied plates created using modified fibers are capable of withstanding higher loads compared to plates made of traditional fibers.

The problem of optimal design of a multilayer composite shell subjected to pressure and non-uniform heating is considered in deterministic and stochastic formulations. The vector of varied variables includes layer thicknesses, and the objective function is the shell mass. The search domain for the optimal design within the deterministic formulation is defined by geometric constraints on the varied variables, temperature and strength constraints. The temperature constraint ensures the absence of overheating of the inner layer. When formulating the stochastic optimal design problem, a vector of random variables is introduced containing elastic moduli, shear moduli, Poisson’s ratios, technical constants of linear expansion, strength limits of the layer materials, and the strength constraint is replaced by its probabilistic analogue. A computational algorithm is developed for the numerical solution of the optimal design problem of a multilayer composite shell. Using the Monte Carlo method, the estimated value of the probability of failure-free operation SP is assessed and a deterministic equivalent of the original stochastic optimization problem is constructed. Subsequently, the general deterministic nonlinear programming problem is transformed using the external penalty method into a sequence of unconstrained minimization problems, each of which is solved using the Nelder-Mead algorithm. The finite difference method is used to determine the temperature field in the designed structure, and the finite element method is used to analyze the stress-strain state of the multilayer composite shell. The influence of statistical scatter of elastic and strength properties of layer materials on the weight efficiency criterion of a multilayer shell is investigated. Dependences of the minimum shell mass on the coefficients of variation of physical and mechanical properties of materials for various levels of the required probability of failure-free operation SP are constructed. Based on the analysis and comparison of the obtained calculated data, the correspondence of the parameters of the deterministic and stochastic optimal design problems is established.

An approach is proposed for studying the influence of mechanical properties of metamaterials on the transition of potential energy into kinetic energy in them based on a numerical method of continuum mechanics — the dynamic finite element method. In this approach, instead of traditional modeling of a representative volume of a metamaterial with real geometric structure, modeling of the stress-strain state of the metamaterial is carried out under the assumption of a continuous medium. In this case, the integral mechanical characteristics of the entire structural element made of metamaterial are taken into account — average density, elastic constants calculated using technical elastic characteristics, complexes of elastic constants. Total stresses and strains in the metamaterial are determined using the generalized Hooke’s law in a three-dimensional formulation, which allows modeling elastic deformations in metamaterials taking into account three different values of Poisson’s ratios, including in complete auxetics. The problem of joint deformation of a layer of metamaterial and an upper thin aluminum layer constituting a sandwich panel is considered. The loading source is considered to be a constant velocity of the upper layer of the sandwich panel. The source of constant velocity of the upper layer of the sandwich panel is the action of a blast wave. Dependences of changes in the velocities of the centers of mass of the upper aluminum plates, characterizing their kinetic energy as a result of interaction with the metamaterial layer, are obtained. The level of residual kinetic energy in the metamaterial and the aluminum plate is investigated depending on the technical elastic characteristics of the metamaterial. It is shown that during compression deformation of metamaterials as part of sandwich panels, the most important characteristic determining the deformation process is the linear compression modulus of the metamaterial determined in the direction of the compression axis, which is especially important for the considered class of metamaterials — complete auxetics.

The limit state of mixed-mode loading of a crack-like defect of an extremely thin adhesive layer of a composite is considered. The problem of loading a crack-like defect was solved numerically using the finite element method implemented in the Ansys software package, within the framework of a crack model in the form of a mathematical cut corresponding to zero thickness of the adhesive layer. The result of the simulation is the calculation of the components of the J-integral by modes I, II at the tip of the mathematical cut. A fracture criterion is proposed that takes into account the decomposition of the J-integral into terms responsible for the specific (per unit surface) energies of the volume change and shape change energy types. It is assumed that for mode I, the decomposition of the J-integral into additive terms is determined by the solution for an extremely thin adhesive layer and is related to its mechanical characteristics, which sets a certain range of changes in specific energies. For mode II, the absence of diagonal components of the stress tensor associates the J-integral with the specific shape change energy. Based on experimental data on the fracture of a sample within the framework of mixed-mode loading I+II and known critical values of J-integrals of modes I and II, the parameters of the criterion dependence are found for a certain combination of decomposition of the J-integral for mode I. A comparison of the results with the known fracture criterion with a change in crack length is carried out. The influence of the representation of the J-integral through its components of mode I and mode II is investigated. It is assumed that the term determining mode I is subject to decomposition with a weight coefficient, and the term responsible for mode II forms the specific shape change energy. For a number of values of the decomposition weight coefficient, the model parameters are determined from the analysis of limit states of normal fracture and mixed loading within the framework of mode I+II at a fixed crack length.

In remote regions of the Far North and Far East of Russia, the use of wind power plants, the main structural elements of which are made of polymer composite materials, is promising. The use of PCMs in the structures of blades and generator fairings solves a number of problems associated with reducing the weight of units, increasing resistance to various weather and climatic factors and, as a result, increasing the service life of the installation. The main limitation for the use of these materials for the manufacture of blade structures is their high sensitivity to impact, for example, hail or pieces of ice that break off when the blades are heated [1]. Climatic factors have a significant effect on PCMs during long-term operation of products. Temperature, relative humidity, precipitation, solar radiation, cyclical changes in ambient temperature can cause a decrease in the strength characteristics of materials [2]. The purpose of the work is to use various diagnostic tools to study the degradation of the strength properties of wind turbine blades under impact damage and exposure to humid climate. This article provides a comparative assessment of the impact strength of carbon fiber reinforced plastic before and after exposure to humid climate. Before the start of the tests, the dimensions of the damage were determined, recording the depth and area of the indentation depending on the impact energy. Samples with damage caused by energy of 10 J, 20 J, 30 J were transferred to the Russian-Vietnamese Tropical Center for exposure in three climatic zones of Vietnam with humid climate at the sites Hoa Lac (Hanoi), Can Gio (Ho Chi Minh City), Dombay (Nha Trang). The article presents the results of a study of the physical and mechanical properties of B180 grade carbon fiber reinforced plastic after exposure in natural conditions for 3 years. The geometric dimensions of impact damage caused to samples with different impact energies are investigated, and the effect of relaxation on them as a result of climatic aging is established.

The paper develops an analytical approach to solving the problem on a cell in the asymptotic averaging method of Bakhvalov, which makes it possible to determine the effective characteristics of composite materials with a high degree of accuracy. The asymptotic averaging method provides a rigorous mathematical algorithm for calculating both stiffness and strength characteristics based on solving a series of elasticity theory problems on a cell with periodic conditions. If the structure of the composite material does not satisfy the periodicity conditions, then a representative volume element should be taken as the periodicity cell. For fibrous and dispersed materials, the cell is a parallelepiped with a cylindrical or spherical inclusion with an intermediate interphase layer, which may have a complex structure with varying characteristics (gradient interphase layer). In this work, some symmetry in the arrangement of inclusions is assumed, which corresponds to an orthotropic material. This means that the inclusion is not necessarily located in the center of the periodicity cell, but the entire material structure is obtained by repeated reflection of this inclusion relative to all faces of the parallelepiped. As a result, the problem on a cell with periodic conditions reduces to a boundary value problem with some conditions of symmetric/antisymmetric reflection of individual displacement components, and the effective stiffness tensor is guaranteed to have an orthotropic structure (9 independent stiffness moduli). By shifting the center of inclusions in the cell, it is possible to effectively simulate the process of particle aggregation and the effects that arise, and by changing the characteristics of the interphase layer, various scale effects of nanocomposites, in particular, the anomalous hardening of nanomaterials with ultra-small concentrations of inclusions. In general, the exact solution of the problem on a cell allows not only to accurately determine all moduli of the orthotropic material, but also to accurately estimate the stress concentration at the local level near the inclusions, which is very important for the strength characteristics of fibrous and dispersed materials. The finite element method is not suitable for solving this problem, therefore this article develops a special analytical method for solving the problem on a cell with periodic or boundary conditions based on representing the solution by T-complete systems of functions that analytically exactly satisfy the original equations with precise consideration of contact conditions at the interphase boundaries based on the Papkovich-Neuber representation. The solution is represented as a series in these systems of functions, and the periodic or boundary conditions are satisfied using the direct Trefftz method, which consists in minimizing the energy norm of the deviation of the approximation in the form of a series from the exact solution. The resulting system of equations has a number of properties that allow reducing the number of problems to be solved necessary for the complete determination of the orthotropic stiffness matrix to a minimum. The analytical scheme developed in this article is used to model the anomalous properties of polymethyl methacrylate filled with multilayer carbon nanotubes.