### №4-2020

Full versions of the articles

https://elibrary.ru/contents.asp?id=44525959

**Gaganova N.V.**

## Superelasticity description based on the combined model of shape memory alloys deformation considering development of the martensitic elements

The article considers a constitutive model for shape memory alloys, which allows to take into account the differences between phase and structural transformation. The model reflects the fact that hardening effect is typical for structural transformation, but not for phase transformation. Deformation due to structural transformation is described with the use of loading surface by analogue of the plasticity theory with isotropic hardening. The deformed state is determined by one parameter, which can be changed by phase or structural deformation. Inelastic deformation due to structural transformation in the active process is subject to the associated flow rule. The article examines the possibilities of the model for describing the phenomenon of superelasticity. The temperature of the phase transition in shape memory alloys depends on the operating stress, so the phase transition can occur at the constant temperature. In a certain range of stresses, dependence of deformations on stresses becomes nonlinear. This phenomenon can be explained by a phase-structural transition. In this paper, a proportional monotonic loading at a constant temperature and phase transitions caused by increasing and decreasing stresses. The model is extended to the case of the development of martensitic elements during the phase and structural transition. Phase-structural and phase deformations plots are provided. It is shown that the model allows to describe the phenomenon of superelasticity correctly. The obtained results are compared for different material functions that determine the relationship between the processes of origin and development of martensitic elements. It is shown that under the considered loading conditions, phase deformations increase with temperature. The values of phase deformations are higher for material functions that take into account the development of martensitic elements.

Pages: 441-454 Elibrary

**Khokhlov Andrew V.**

## Exact solution for strain and stress fields in a multilayer thick-walled tube of non-linear viscoelastic materials under given internal and external pressures

We construct the exact solution of the quasi-static boundary value problem for a multilayer thick-walled tubes made of physically non-linear viscoelastic materials obeying the Rabotnov constitutive equation with two arbitrary material functions for each layer (a material creep compliance and a function which governs physical non-linearity). We suppose that every layer material is homogeneous, isotropic and incompressible and that a tube is loaded by time-dependent internal and external pressures (varying slowly enough to neglect inertia terms in the equilibrium equations) and that a plain strain state is realized, i.e. zero axial displacements are given on the edge cross sections of the tube. We obtained the closed form expressions for displacement, strain and stress fields via the single unknown function of time and integral operators involving this function, pairs of (arbitrary) material functions of each tube layer, preset pressure values and ratios of tube layers radii and derived integral equation to determine this unknown function. To derive it we split and solve the set of non-linear integral equations for unknown functions of time governing strain fields of every tube layer and for unknown interlayer normal stresses (depending on time). Assuming material functions are arbitrary, we proved that the total axial force at a tube cross section doesn’t depend on a number of layers, their thicknesses and pairs of material functions governing their mechanical behavior and on a history of loading although stresses and strains do. The axial force depends only on a tube radii and current values of given pressures. It proved to be equal to the axial force calculated for homogeneous linear elastic tube although axial stress depends on radial coordinate in the case of non-linear viscoelastic materials. Assuming the material functions that govern non-linearity of each layer material coincide with a power function with a positive exponent and assuming their relaxation moduli are proportional to a single (arbitrary) function of time, we constructed exact solution of the resolving functional equation, calculated all the integrals involved in the general representation for the tube stress field and reduced it to simple algebraic formulas convenient for analysis.

Pages: 455-476 Elibrary

**Blohin V.V., Kulakov V.V., Lisin A.N., Mozalev V.V., Pankov M.I., Sivurova V.A.**

## Evaluation of capacity for work of carbon/carbon composite materials for aircraft brake disks

An analysis of the effectiveness of using carbon-carbon composite materials for the manufacture of aircraft brake discs is carried out. Materials made using fundamentally different technologies were considered: materials with a pitch matrix formed by the liquid-phase method, with graphitized, carbonized and with a different ratio of carbonized and graphitized fibers, as well as materials made by the needle-stitching method, with a pyrocarbon matrix reinforced with tapes and felt. The performance of the brake discs was evaluated by testing the disc on a device that allows you to determine the strength of the most vulnerable zone of the disc – the spike. The disk loading scheme is close to real conditions, i.e. the groove is engaged with a power finger, which acts as a guide for the drum, and the load force is directed in the circumferential direction. During the testing process, the current loading parameters are continuously monitored up to failure. In addition, tests of samples of materials for tensile strength when the interlaminar shear and bending. Samples for testing the bending strength were cut from the disk along the radius with the application of force in the circumferential direction. The regularity of changes in strength during interlayer shear, bending, and spiking of the disk from the apparent density of the material used is determined. It was found that the strength of materials during interlayer shear is practically independent of the apparent density of the material, while the Flexural type of material is most preferable for improving the performance of the brake discs. strength and strength of the force elements of the disk structure correlates with the density of the material. Set the type of material is most preferable for improving the performance of the brake discs.

Pages: 477-489 Elibrary

**Azarov A.V., Babichev A.A., Razin A.F.**

## Optimal design of an airplane wing composite lattice panel under axial compression

Composite materials are now widely used in aircraft structures. However, the main structural concept for composite wing panels is a traditional stringer structure. Such structure consists of a load-bearing skin and reinforcing longitudinal ribs-stringers. Due to the special properties of polymer composites, this design usually does not reduce the composite panel’s weight compared to the metal prototype. An alternative to structures with a load-bearing skin is a composite lattice structure, where the main load-bearing elements are intersecting diagonal and transverse ribs. The ribs are formed from unidirectional CFRP, which has high specific strength and stiffness. Such a design scheme allows us to realize the high longitudinal properties of the composite material and to ensure a high weight efficiency of the structure. The paper is concerned with design of a flat rectangular panel consisting of a system of intersecting ribs made of unidirectional composite material by filament winding or lay-up processes. The panel is a structural element of an airplane wing. The panel is simply supported and compressed in the longitudinal direction. The relations which specify the optimal structural parameters of the lattice panel, i.e., the panel thickness, the ribs orientation angle, the ribs thickness and spacing are obtained. The optimal parameters provide the panel minimum mass under strength and buckling constraints. Optimization is undertaken by the method of minimization of the safety factors corresponding to possible modes of the panel failure. The method allows us to reduce the constrained optimization problem to the problem of conditional minimization. Two types of panels for which the length-to-width ratio is more or less then unity are considered. Optimization of a carbon-epoxy lattice panel is presented as an example.

Pages: 490-500 Elibrary

**Firsanov Vic.V.**

## A variant of refinement of the classical theory of thin plate bending

The classical theory of bending of thin plates based on Kirchhoff’s hypotheses about the absence of normal stresses in the transverse to the bases direction, invariability of the length of the normal element to the middle plane of the plate, which means the invariability of the thickness and lack of linear deformation in the transverse direction, the absence of shear strains in planes perpendicular to the bases of the plate. At the same time, in the equilibrium equations, both normal stresses in the transverse direction and tangential stresses associated with shear deformations by physical relations remain, but the physical connections are obviously broken. Refinement of the classical theory is usually associated with the rejection of all Kirchhoff hypotheses, which significantly complicates such a model, or the rejection of one or two kinematic hypotheses. For example, you can set a displacement in the transverse direction as a power series along the transverse coordinate. In this case, if the degrees are even, the linear deformation in the transverse direction is different from zero, but the normal element connecting the bases of the plate does not change its length, which is not in contradiction with the Kirchhoff hypothesis. However, this approach may not lead to significant refinements of the classical model, so for a more or less significant refinement, it is assumed that the most acceptable rejection of the hypothesis of the absence of shear deformations in the planes transverse to the plate bases. In this case, the physical relationship between shear and stress is restored. Accounting for these shear deformations is especially important for materials with low shear stiffness in the transverse directions. Another reason for refining the classical model of plate bending is that some boundary conditions are not satisfied accurately enough, which is due to the introduction of a generalized Kirchhoff shear force into the calculation model, which consists of a purely shear force and an increment along one of the plane coordinates of the torque. With certain refinements, it is possible to solve the problem of three boundary conditions on the free edges of the plate.

Pages: 501-512 Elibrary

**Belov P.A., Lurie S.A.**

## Variational formulation of coupled of heat/mass transfer and thermoelasticity problem

A variational formulation of a coupled system of equations of thermoelasticity, heat and mass transfer is given. A special case of the gradient model of the Mindlin-Tupin medium is proposed, when the gradient component of the potential energy depends only on the gradients of the constrained dilation. In general, a generalized variational model is considered, in which the gradient variational model is expanded by taking into account the potential energy of defective media with dilatational damage, combining two types of free (incompatible) dilations: free dilations associated with a change in volume due to temperature effects, and free dilations associated with the concentration of impurities due to diffusion processes. As a result, the equations of motion included in the coupled system of equations are a special case of the gradient theory (dilation model) in the part of the differential operator over displacements. The heat and mass transfer equations have the same structure and reflect the diffusion-wave mechanism of evolution in a continuous medium of temperature and impurity. It was found that the coupled system of equations decomposes into three independent boundary value problems with respect to displacements, a free (incompatible) change in volume associated with temperature loading, and a free (incompatible) change in volume associated with the diffusion (concentration) process, when the tensors of physical properties are spherical and the corresponding connectivity coefficients are equal to zero. The consistency equations obtained from the generalized equations of Hooke’s law for force factors and their fluxes by eliminating kinematic variables give a whole spectrum of laws of heat conduction, diffusion and thermoelasticity, including the laws of Fourier, Maxwell-Cattaneo, Soret and Dufour.

Pages: 513-527 Elibrary

**Afanasyeva E.A., Dolgova E.V., Goryashnik Yu.S., Krivushina A.A.**

## Change of epoxy polymer properties after mold fungi influence depending on initial epoxy resin structure

The reaction activity of mixtures based on epoxy resins of bisphenol A as well as 4,4′-methylenedianiline tetraglycidyl ether with an amine hardener were studied. The gelation of the mixtures was determined. The curing reaction parameters of epoxy compositions are determined by differential scanning calorimetry. The temperature-time regime of specimen curing was selected by means of mathematical modeling implementing and the experimental data studying of the curing kinetics obtained the thermoanalytical method. Given regime was recommended for polymer matrices with a predictable high degree of reactive groups conversion obtaining. A mixture of bisphenol A diglycidyl ether epoxy oligomer (ED-22 brand resin) with an amine hardener was chosen as a model sample for mode selecting. When using the proposed mode, the degree of curing of epoxy matrices of all samples under study exceeded 90%. A high completeness of the reaction was suggested. The fungal resistance of epoxy polymers has been studied. Microstructural studies of the samples were performed by means of optical microscopy. The predicted damage of the samples surfaces of all epoxy matrices exposed to micromycetes was found. It could be associated with the frequency of polymer chains cross-linking, which directly depends on the value of the initial resin epoxy number. To assess the change in the characteristics of polymers after microbiological tests, the mechanical and thermal properties of the cured epoxy matrices were carried out. The research of heat resistance and static bending strength of samples after exposure to micromycetes was performed. The dependence of the change in the properties of epoxy polymers after tests for resistance to mold fungi on the frequency of cross-linking of polymer matrices was revealed.

Pages: 528-543 Elibrary

**Fedotenkov G.V., Lokteva N.A., Serdyuk D.O., Skopintsev P.D.**

## Transient stress-strain state of a composite cylindrical shell

This work is devoted to the description of an approach to studying the propagation of non-stationary disturbances, stresses and strains in a thin elastic composite cylindrical shell. The shell is accepted unlimited, with a constant thickness. An aggregate of non-stationary moving pressures affects along the normal to the outside surface of the shell. The shell is assumed to be unbounded, with a constant thickness. The outer face of the shell is subjected to a set of non-steady moving loads. It is assumed also that the composite material of the shell is linearly elastic, with a lamination scheme symmetric with respect to the midsurface. The shell model is based on the Kirchhoff-Lowe hypotheses, while the load instantly applied to the shell are modeled by Dirac functions. The study of non-steady deformation of the shell is carried out using the transient function, which is a normal displacement that occurs as a response to a single load concentrated in time and coordinates. The transient function is constructed using exponential Fourier series expansion, Laplace integral transformations in time domain and Fourier transforms with respect to the longitudinal coordinate. The inverse Laplace transform is performed analytically, whereas the original of the Fourier transform is found by using the numerical method of integrating rapidly oscillating functions. The non-steady normal deflection of the cylindrical shell is represented as a triple convolution of the transient function with the functions defining the moving concentrated loads with time-varying amplitudes and coordinates of the impact. The convolution integrals are evaluated using rectangle quadrature formulae. The study of the space-time stress-strain state of an unbounded thin elastic composite cylindrical shell becomes possible after constructing a non-steady deflection function with further use of constitutive and kinematic relations to obtain the stress state of the shell. In the study of the non-steady stress-strain state of the composite shell, the given technical constants determined through the generalized stiffness of the material are used. As an example, the space-time dependences of the non-steady deflection, the distribution of stresses and deformations on the outer surface of the polymer composite shell are constructed. Non-steady impact was considered as a set of moving concentrated loads.

Pages: 544-559 Elibrary

**Yankovskii A.P.**

## Modeling of viscoelastic-plastic bending behavior of cylindrical shells with spatial reinforcement structures

Based on the method of time steps, a numerical-analytical model of viscoelastic-plastic deformation of circular cylindrical shells with spatial reinforcement has been developed. Instant plastic deformation of the materials of the components of the composition is described by the theory of flow with isotropic hardening. The viscoelastic deformation of these materials is described by the governing equations of the Maxwell – Boltzmann model of the body. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The possible weak resistance of composite shells to transverse shears is taken into account on the basis of the Ambardzumyan theory. The developed model of the mechanical behavior of materials of the components of the composition is adapted to the use of an explicit numerical “cross” type scheme. The viscoelastic-plastic and elastoplastic dynamic and quasistatic deformation of thin flexible fiberglass cylindrical shells under the influence of internal pressure, as well as rectangular elongated plates under the action of a uniformly distributed transverse load, are studied. The constructions have traditional reinforcement structures with orthogonal laying of fibers on equidistant surfaces or have spatial reinforcement structures. It is demonstrated that calculations performed on the theory of elastoplastic and rigid-plastic deformation of reinforced shells and plates do not give even an approximate idea of the residual states of composite constructions under their dynamic loading. It is shown that, after viscoelastic-plastic dynamic deformation, relatively thin reinforced constructions acquire corrugated residual forms. With quasistatic transverse loading of composite plates, the residual deflection has a traditional form, i.e., folds are not formed.

Pages: 560-578 Elibrary