### №2-2019

Full versions of the articles

https://elibrary.ru/contents.asp?id=39106220

**Firsanov V.V., Gavva L.M.**

## The parametric analysys system of pre-critical stress-strain state of structurally-anisotropic panels made of composite materials

The mathematical model relations for investigation of pre-critical stress-strain state of structurally-anisotropic panels made of composite materials are presented. The pre-critical stressed state of plane angle laminated panels made of polymer fiber composite materials with eccentric longitudinally-lateral stiffening set is considered for further development of refined buckling problems. The new mathematical model for stress-strain state investigation of structurally-anisotropic composite panels is designed. The mathematical model of stiffening rib being torsioned under one-side contact with the skin is refined. Further development of the theory of thin-walled elastic robs related to the contact problem for the skin and the rib with improved rib model reflects the scientific novelty of the research. One takes into account the influence of panel production technology: residual thermal stresses and reinforcing fibers preliminary tension. The resolved equation of eight order differential operator and natural boundary conditions are obtained with variation Lagrange procedure. The schematization of the panel as structurally anisotropic one has been proposed as a design model. Exact analytical solutions for edge problems are considered accordingly general treatment of physical boundary conditions for structure components. The solution in closed form is designed by unitary trigonometric series for the particular case of conformable boundary conditions on two opposite sides. All possible combinations for lateral edges boundary restrictions are examined when bending is integral with plane stress state problem. Computer program package is developed using operating MATLAB environment. The influence of the structure parameters on the inner force factors distribution by the length of carbon-plastic panels has analyzed while longitudinal compression.

Pages: 145-153 Elibrary

**Yankovskii A.P.**

## Modeling of elasto-plastic dynamic behavior of flexible cylindrical spatial-reinforced shells in the framework of refined bending theory

A mathematical model is proposed for elastic-plastic deformation of flexible cylindrical shells with spatial reinforcement structures, adapted to the use of a numerical scheme of the «cross» type. Inelastic behavior of the materials of the phases of the composition is described by equations of flow theory with isotropic hardening. The geometric nonlinearity of the problem is considered in the Karman approximation. The possible weakened resistance of reinforced shells to transverse shear is taken into account. The initial-boundary value problems, allowing determining with different accuracy the stress-strain state in the phases of the composition of fibrous shells, are formulated. The equations, boundary and initial conditions of the traditional non-classical Reddy theory follow from the obtained relations in the first approximation. The dynamic elastic-plastic flexural behavior of unidirectional-, flat- and spatial-reinforced closed cylindrical shells made of fiberglass plastic under the action of explosive type loads is investigated. It is shown that calculations on the Reddy theory can lead not only to quantitatively unacceptable, but even qualitatively incorrect results. The difference in the calculations performed by Reddy’s theory and refined theory increases with the increase of the calculated time interval. It is demonstrated that, according to the calculations by the refined theory for closed shells with a relative thickness less than 1/10, the structure with «flat» 2D-reinforcement is rational. It is shown that due to the geometric nonlinearity of the problem under study, the maximum modulo deflections in thin reinforced shells may appear much later than the stopping of the short-term dynamic load.

Pages: 154-172 Elibrary

**Gamlitsky Y.A., Litvinova I.A., Veselov I.V.**

## Interpretation of test results rubber with non-traditional fillers using the strength theory

The report presents the results of testing rubber for massive tires with different types of fillers. Based on the molecular-phenomenological model of the phenomenon of amplification, the behavior of the stretching curves and hysteresis losses are explained. One of the main components of rubber compounds, affecting the output characteristics of the product, is a filler. As the investigated fillers were selected: 1. iron oxide fillers (modified by fullerenes and unmodified) with different particle sizes; 2. Taunit filler (carbon nanotubes). In the course of the work, it was revealed that modified taunit nanotubes and fullerene-containing iron oxide filler (JON) in certain concentrations lead to an increase in strength and a decrease in hysteresis losses. When added to rubber for massive tires JON, modified with fullerene, in an amount of 20 mph, strength does not decrease and heat generation decreases by 24%. Adding other fillers in such quantities is always accompanied by a significant deterioration in performance. When adding this JON in the amount of 5 m. there is an increase in strength and a decrease in heat generation, which indicates the possibility of including this filler as an active. The article also explains the features of the stretching curves for rubbers with different types of filler based on the enhancement model. The model uses the idea of the presence on the surface of active filler particles of a thin layer of a rubber matrix that is in a pseudo-glassy state, which, as it moves away from the surface of the filler, becomes a highly elastic state. When the composite is deformed, the most stretched macromolecules are not torn, but are pulled out of the pseudoglass layer by a mechanism close to the phenomenon of forced elasticity (cold flow). The main thermal losses in the composite occur in the transition zone from the pseudo-glass to the highly elastic states.

Pages: 173-191 Elibrary

**Dudarkov Yu.I., Levchenko E.A., Limonin M.V.**

## Some features of cfrp stringer panels load bearing capacity estimation

The aim of this paper is to show some features of aircraft stringer panels load bearing capacity estimation, associated with carbon fiber reinforced plastic (CFRP) application. Skin local buckling is commonly allowed for fuselage and tail panels, provided their strength up to the ultimate load level. This makes it possible to increase the weight efficiency of panels and a whole structure. For metal alloy panels, methods for accounting for skin local buckling are well established and are widely used in engineering practice. In general, these methods consist of reduction factors calculation that characterizing the degree of skin stiffness decrease in the postbuckling state, and assess the strength and global buckling of the panel under these conditions. However, direct application of these methods to CFRP panels in some cases lead to incorrect results. This applies both to the calculation methods, the strength criteria, and the methods of obtained results analysis. This is demonstrated with virtual simulation of stringer CFRP panels’ uniaxial compression and shear tests. Analytical methods as well as numerical methods were used for analysis. Paper contains models, calculation and analysis procedures that allow estimating panel load bearing capacity under strength and global bulking conditions for both, isolated compression, shear loading and there combination. A comparison of the calculated and available experimental data on the failure loads is given. It is shown for the panel under consideration that the main factor determinates its’ load capacity is the skin postbuckling strength.

Pages: 192-206 Elibrary

**Grishanina T.V., Russkikh N.M.**

## Analysis of influence of the incompessible flow unsteadyness on the bending-torshional aeroelastic vibrations of oc large elongation wing

Forced harmonic bending-torsional vibrations of a straight large elongation wing in incompressible flow of ideal liquid (gas) is considered. Hypothesis of plane non-tear flow over thin wing the cross sections is used. The aerodynamic load acting on a thin vibrating airfoil in incompressible flow of ideal gas for small harmonic vibrations of the wing in determined exactly using unsteady linear theory and as well quasi-steady theory. A wing is considered as supported by longitudinal elements (spars, stringers) a thin-walled beam with a single-closed or multi-closed cross-section contour, which are considered to be non-deformable in their planes. Elastic displacements of the wing for bending-torsional vibrations are determined by the Ritz method in series of basic functions with unknown coefficients which are considered as generalized coordinates. The equations of the wing aeroelastic vibrations under action of transverse harmonic force with prescribed frequency are obtained as the Lagrange equations and solved in complex variables. The purpose of the work is the comparison of the calculation results for the amplitude-frequency characteristics of the forced bending-torshional vibrations of the wing obtained by use of the unsteady and quasi-steady aerodynamic theories. Calculations were made for a wing model of constant cross section, where one generalized coordinate represents the wing bending and the other – torsion. On the basis of the obtained results it was shown that for small reduced oscillation frequencies a simple (from the point of view of laboriousness of computations) quasi-steady theory makes it possible to obtain solutions with quite acceptable accuracy. The influence of the added air masses, which was taken into account in the unsteady theory, is very small.

Pages: 207-218 Elibrary

**Kolmakov O.V., Kostyukov E.N., Nikiforov I.I., Nikiforova M.S., Vakhmistrov S.A.**

## Prediction of strength characteristics of parts made of energy materials with use of acoustic emission method

Capabilities assessment for predicting strength characteristics of energy materials specimens by the results of monitoring parameters of acoustic emission in the preliminary test of specimens by loads that don’t exceed the limit value is the main purpose of the research, the results of which are discussed in this article. The experimental work has been fulfilled regarding the study of a fracture process and mechanical properties of specimens made of plasticized HMX during the trials under the conditions of compression and tension by using an acoustic emission method. The load-strain diagrams « » and the diagrams of acoustic emission parameters have been analyzed for a search of «exceptional» points in the acoustic emission diagrams, which can be used as a basis by developing a method of prediction. It has been revealed that the most informative parameters according to use for predicting of the mechanical characteristics limit of specimens are the number of acoustic emission events and the activity of acoustic emission, significant variation of which are observed in the elastic range of deformation of specimen. An «Exceptional» point in the form of a maximum of the activity of acoustic emission diagram allows to monitor the moment when the critical load is reached during the experiment visually. A possibility to reduce a load of the preloading has been studied. Based on a detailed analysis of the diagrams of acoustic emission parameters and load-strain diagrams and structuring by frequency of array data of acoustic emission it was been found that using of the high-frequency components of the acoustic emission for the mechanical properties of specimens analysis makes it possible to reduce the preload load to the level of compliance; it will allow to use tested specimens for other types of research. A set of check experiments was carried out, whose results provide support for a possibility to determine strength of parts made of plasticized HMX according to the results of preloading with an error comparable to a strength determination error under mechanical tests.

Pages: 219-227 Elibrary

**Firsanov Vic.V.**

## A computational model of an mechanicaly incompressible round plate under bending load temperature

The condition of incompressibility for an isotropic linearly elastic material seriously limits the application of the classical hypotheses of the theory of bending of plates, formulated for small strains and displacements. It is assumed that such a strong kinematic condition as a condition of invariability of the volume must certainly be met. When bending an incompressible round plate by a force load, it is shown that the application of certain Kirchhoff hypotheses is associated with the boundary conditions of the problem: for some conditions it is possible to use separate hypotheses, for others it is necessary to completely abandon, and to obtain relatively simple solutions to build models of calculation using other hypotheses that are not inconsistent with the condition of invariability of the volume. So, for example, at rigidly fixed of a plate on two contours (or on one for a continuous plate) absence of deformation in the transverse direction in relation to bases should be considered not as a hypothesis, and a consequence of an incompressibility condition. Hypotheses of no shift in the plane and normal stress are not used because of their incompatibility with the incompressibility condition. The temperature load has its own characteristics, which is why the temperature problem requires a non-standard approach in its solution. Note, first of all, that the invariability of the volume is kept for the elastic component of the total deformation, but to the temperature component has nothing in relation. Therefore, the total deformation of the volume change is not zero, since it is related to the temperature field. Therefore, this connection is not a condition of incompressibility, but it is an important kinematic condition for the incompressible plate. The principle of using different hypotheses under different boundary conditions is kept here. At the same time, the absence of transverse deformation is a hypothesis even under the condition of rigidly fixed of the contours due to the variability of the volume of the plate under bending temperature loading.

Pages: 228-239 Elibrary

**Zhavoronok Sergey I.**

## An application of the nth order extended plate theory in the wave dispersion problem for a functionally graded layer

A solution of the wave dispersion problem for a functionally graded plane layer is based on the extended plate theory of I.N. Vekua – A.A. Amosov type satisfying the boundary conditions on the faces exactly within an arbitrary approximation order. A variational problem’s statement corresponding to the heterogeneous plate theory of Nth order is is given by a set of field variables of the first kind being the displacement expansion factors with respect to biorthogonal function system, the surface Lagrangian density, and the non-holonomic constraints following from the boundary conditions on the faces. The generalized Lagrange equations of the second kind for a 2D continuum are obtained. The spectral problem for normal waves in the functionally graded layer is formulated as a constrained stationary problem for two quadratic forms and solved by the Golub approach. The phase locking frequencies and wave forms for the asymmetric layer with power gradation law are computed, as well as the corresponding stress distributions across the thickness. The convergence of approximate locking frequencies is analyzed for different gradation laws. The minimum order of theory required to secure the convergence correspond to the homogeneous layer if the stiffer phase prevails; the waveforms are close to the ones in the homogeneous layer. The softer phase prevailing leads to minimum orders exceeding the ones of the homogeneous layer for some modes; the wave forms for higher frequencies differ significantly from the ones of homogeneous layer. The stress distributions across the thickness are significantly asymmetric, especially for higher frequencies.

Pages: 240-258 Elibrary

**Khokhlov A.V.**

## Applicability indicators of the linear viscoelasticity theory using creep curves under tensile load combined with constant hydrostatic pressure

The Boltzmann-Volterra linear constitutive equation for isotropic non-aging visco-elastic materials (with an arbitrary shear and bulk creep compliances) is studied analytically in order to find out its capabilities to provide an adequate qualitative description of rheological phenomena related to creep under uni-axial loading combined with constant hydrostatic pressure and to outline the material functions governing abilities, to indicate application field boundaries of the relation and to develop identification and verification techniques. The constitutive equation doesn’t involve the third invariants of stress and strain tensors and implies that their hydrostatic and deviatoric parts don’t depend on each other. It is governed by two material functions of a positive real argument (that is shear and bulk creep compliances); they are implied to be positive, differentiable, increasing and convex up functions. General properties and characteristic features of the creep curves for volumetric, longitudinal and lateral strain generated by the linear equation under constant tensile load and constant hydrostatic pressure are investigated. Their dependences on pressure and tensile stress levels and the material functions qualitative characteristics, the conditions for creep curves monotonicity and for existence of extrema and sign changes of strains are in the focus of attention. In particular, it is proved that the linear relation is able to simulate non-monotone behavior and sign changes of lateral and axial strains. The analysis reveals a number of specific features and quantitative characteristics of the theoretic creep curves which can be employed as the applicability or non-applicability indicators of the linear viscoelasticity theory. They are convenient to check using data of a material creep tests with various levels of pressure and tensile stress. More detailed and precise indication of phenomenological restrictions for the linear viscoelasticity theory and enclosure of linear behavior range of a rheonomic material is also significant for identification of non-linear models and simulation of a material behavior in non-linear range. A simple and effective identification technique is developed. It is based on two creep tests with different pressures and implies measurement of axial compliance in each test (i.e. axial creep curve divided by the stress level). The explicit formulas are derived to determine the shear and bulk creep compliances via experimental axial compliances.

Pages: 259-280 Elibrary

**Saganov E.B.**

## Numerical modeling of the phenomena of martensite inelasticity in shape memory alloys with account of their tension-compression asymetry

The work is devoted to the numerical simulation of the phenomenon of martensitic inelasticity in shape memory alloys (SMA), taking into account their tension-compression asymmetry. Under tension-compression asymmetry is meant the dependence of the stress-strain state of these alloys on the type of stress state. The parameter associated with the third invariant of the stress deviator is used as a parameter of the type of stress state. Numerical simulation of the process of martensitic inelasticity is carried out in the framework of the model of nonlinear deformation of SMA in phase and structural transformations. In the course of the work, the integration of relations describing the behavior of the SMA under monotonous active loading in the low-temperature martensitic phase into the finite-element complex of Simulia Abaqus was performed using the procedure for creating user material. For this purpose, relations are reversed that establish the relationship between the components of the tensors of total deformations and stresses in order to obtain a tangent stiffness matrix used as a physical law for each element of the model. The process of validation of the module, developed for the finite-element analysis of the stress-strain state of structures containing SMA, was performed using the available experimental data for uniaxial tension and compression of samples from these alloys. A comparison is made of the deformation diagrams obtained by finite-element modeling of the phenomenon of martensitic inelasticity in a three-dimensional in space formulation with experimental curves. As part of the work, the stress-strain state of a spherical thick-walled shell from SMA under the action of internal and external pressure was calculated. It was established that in the process of deformation of the shell by internal pressure for each element through the thickness of the shell, the parameter of the type of stress state takes on the value corresponding to pure compression, and for the case of external pressure, pure tension.

Pages: 281-298 Elibrary