### №4-2022

https://elibrary.ru/contents.asp?id=50000890

**Saganov E.B., Sharunov A.V.**

## NUMERICAL ANALYSIS OF THE DISTRIBUTION PROCESS OF A CYLINDRICAL SHELL MADE OF A SHAPE MEMORY ALLOY USING THE EFFECT OF ACCUMULATION OF DEFORMATIONS OF DIRECT TRANSFORMATION, TAKING INTO ACCOUNT THE INFLUENCE OF THE TYPE OF STRESS STATE

The work is devoted to the numerical simulation of the distribution process of a thick-walled cylindrical shell-coupling with constant internal pressure. The deformation of the shell occurs due to the phenomenon of accumulation of deformations of direct martensitic transformation. Numerical simulation of the distribution is performed taking into account the resistance of shape memory alloys (SPF). Resistance to corrosion is understood as the dependence of the stress-strain state (VAT) 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 shell distribution is performed within the framework of the model of nonlinear deformation of SPF during phase and structural transformations. This model of SPF behavior is integrated into the finite element complex Simulia Abaqus through the procedure of creating custom material and explicitly defining the tangent stiffness matrix. The process of deformation of the shell is considered in a once-coupled thermomechanical formulation, taking into account the effect of the acting voltage on the values of the phase transition temperatures. The distribution problem is solved in plane-stressed and plane-deformed formulations. The problem is considered in a three-dimensional spatial formulation. In the course of the work, the stress-strain state of a thick-walled cylindrical shell made of SPF, which is under the influence of internal pressure and undergoes cooling through the temperature range of direct martensitic transformation, was analyzed. Plots of radial and annular stresses along the shell section are obtained, and the actual radial distribution of the shell is determined. The dependences of the stress intensity on the value of the phase composition parameter are given. It is established that during the cooling process, the parameter of the type of stress state changes non-monotonously along the cross section and becomes alternating during the cooling process of the shell.

Pages: 437-448 Elibrary

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**Yamilinets S.Yu., Konyukhov V.Yu., Kondratov A.P.**

## PHYSICO-CHEMICAL RESISTANCE OF MULTILAYER POLYMER COMPOSITES

The article is devoted to the urgent problem of the stability of the mechanical properties of multilayer polymer composites to the periodic effects of aqueous solutions and organic solvents. The constancy of the mechanical characteristics of the canvases during prolonged contact with water and periodic surface treatment with solvents is an important condition for high-quality replication of multicolor printing products and the durability of printing equipment. The composition of solvents and aqueous solutions used in printing production depends on the chemical composition of polymer layers and printing inks, the range and qualitative composition of which is periodically changed by manufacturers within wide limits. The choice of solvents and the justification of the stability of multilayer polymer composites to liquids of known polarity and thermodynamic affinity for the polymer is an important and urgent task. To solve the problem, methods were used that do not require complex laboratory equipment that is not available for mass printing production. Sorption measurements were carried out according to ISO 11358-1:2014 using analytical scales, accelerated video recording and a modified thickness gauge available in many printing houses and TSL manufacturers of printing consumables. The analysis of measurement results and prediction of composite properties in a liquid is based on the theory of polymer solutions by Florey-Huggins and a computational and analytical apparatus developed using the Kelvin-Feucht mathematical model of an elastic body. A parameter is proposed for estimating the hysteresis of compression of multilayer polymer composites, reflecting the features of the internal friction of the layers during compression relaxation and recovery. It is proposed to use the relaxation time of compression and recovery to assess the operational properties of the web. Composite rubber fabrics coated with butadiene-nitrile rubber (NBR) are the least resistant to the action of dichloroethane, and fabrics with layers of polyolefins (EPDM) are unstable in ethyl acetate. The most vulnerable are the tissue layers, which significantly change their physical properties in water. The novelty of the study lies in the application of the Florey-Huggins parameter, previously used to assess the thermodynamic affinity of liquids and monolithic polymers, to predict the physico-chemical resistance of multilayer polymer composites.

Pages: 449-464 Elibrary

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**Movchan A.A.**

## THREE-LAYER STRIP WITH THIN OUTER LAYERS OF SHAPE MEMORY ALLOY AND VISCOELASTIC INNER LAYER

The mechanical behavior of a three-layer strip with outer thin layers of shape memory alloy (SPF) and an inner viscoelastic layer is investigated. The simplest kinematic hypotheses are accepted about the uniform distribution of longitudinal deformation and stresses over the thickness of the outer active layers and the linear distribution of longitudinal deformation and stress over the thickness of the inner layer. Shear, transverse normal deformations, stresses and transverse shifts are not taken into account. A model of a standard linear body is used to describe the behavior of a viscoelastic layer. Before combining with the inner layer, the material of the outer layers is transferred to the martensitic state with a tensile phase-structural deformation of 0ε. It is assumed that this deformation does not change when combined, and the stresses in the layers after combination are zero. The heating process of one (active) outer layer is considered, accompanied by a reverse thermoelastic phase transformation into an austenitic state. In an analytical form, a solution to the corresponding problem is obtained in the first approximation without taking into account the variability of the elastic modules of the SPF during the phase transition, in an unrelated formulation and under the assumption of the absence of structural transformation in the outer layers. For the case of a time-uniform increase in the temperature of the active layer, analytical dependences of the curvature, average deformation of the package, stresses in the active layers and parameters of the linear stress distribution in the inner layer as functions of the temperature of the active layer are obtained. It is shown that the graphs of all these dependencies are located between two linear asymptotics. The first corresponds to the solution under the assumption of the elastic behavior of the middle layer with a Young’s modulus equal to the instantaneous modulus of a viscoelastic material. The solution of the initial problem tends to this asymptotic behavior while striving for infinity of the heating rate. The second corresponds to the solution under the assumption of the elastic behavior of the middle layer with a Young’s modulus equal to the long-term modulus of a viscoelastic material. The solution of the initial problem converges to this asymptotic when the heating rate tends to zero.

Pages: 465-487 Elibrary

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**Khvostunkov K.A.**

## STRESS INTENSITY COEFFICIENT FOR A TWO-LAYER BEAM WITH A DIFFUSE LAYER UNDER THREE-POINT LOADING

The problem of obtaining new layered composite materials based on ceramics/intermetallic with a given regular structure and improved physical and mechanical characteristics is solved by optimizing the proportions of layer thicknesses within the framework of the technology of self-propagating high-temperature synthesis – SHS technology. The problem of brittle fracture of a two-layer beam with a section under three-point loading is considered. A calibration function for the stress intensity coefficient is obtained, taking into account the proportion of thicknesses and elastic properties of both layers of the beam and the intermediate diffuse layer.

Pages: 488-494 Elibrary

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**Gaganova N.V.**

## DESCRIPTION OF MARTENSITIC INELASTICITY AFTER ORIENTED TRANSFORMATION, TAKING INTO ACCOUNT THE DEVELOPMENT OF MARTENSITIC ELEMENTS WITHIN THE FRAMEWORK OF THE COMBINED DEFORMATION MODEL OF SHAPE MEMORY ALLOYS

To describe martensitic inelasticity after oriented transformation, a combined model of phase-structural deformation of shape memory alloys with a non-integral hardening parameter is used. The model describes both the phase and structural mechanisms of inelastic deformation changes, as well as the effect of the first mechanism on the second, taking into account isotropic and translational hardening. In most existing models, only the process of formation of new martensitic meso-elements is taken into account, but the process of development of meso-elements formed earlier is not taken into account. Meanwhile, experiments show that the development of martensitic elements can significantly affect the values of deformations. Consideration of this factor is necessary for the model to correctly describe the phenomenon of oriented transformation quantitatively and qualitatively, as well as processes in which direct thermoelastic transformation occurs at stepwise or smoothly decreasing stresses. The propagation of a combined model of phase-structural deformation of shape-memory alloys with a non-integral hardening parameter in the case of taking into account the development of martensitic elements is considered. A simulation of martensitic inelasticity after oriented transformation for proportional loading is given, taking into account isotropic and translational hardening. It is shown that the critical value of stresses at which the development of deformations according to the structural mechanism begins after direct transformation in this combined model does not depend on taking into account the development of martensitic elements, which is consistent with experimental data. At the same time, taking into account the development of martensitic elements significantly affects the magnitude of phase-structural deformations, as well as the radius and position of the center of the loading surface.

Pages: 495-510 Elibrary

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**Kriven G.I., Shavelkin D.S.**

## OSCILLATION OF A BEAM CONSISTING OF A FIBROUS COMPOSITE WITH A VISCOELASTIC COATING

The paper considers transverse vibrations of a generally hinged layered composite strip. It is assumed that the layers of the strip are made of fibrous material with different fiber orientations in different layers of the system, and the fibers have a thin viscoelastic coating, the properties of which differ from those of the matrix providing solidity. The beam layers are considered to be transversely isotropic. The viscoelastic coating can provide high dissipative properties of the composite laminated strip material. The natural frequencies and loss coefficients are investigated depending on the thickness of the viscoelastic coating. The technique of complex elastic modulus is used to evaluate dissipative properties. The effective properties of layers containing fibers with a viscoelastic layer are determined using the three-phase method using the self-consistent Eshelby method, which allows to obtain analytical estimates of the modules and effectively use the method of complex elastic modules. It is shown that for a refined estimation of natural frequencies and loss coefficients for quasi-stationary oscillations, the most accurate rod model should be used, otherwise the errors in estimating the desired parameters are unacceptably large. One of the objectives of the work is to assess the effect of the thickness of the viscoelastic coating on the loss coefficients and determine the optimal coating values, at which the loss coefficients become significant, but the effective stiffness characteristics remain high. As an example, we consider rods made of three different layered composite: 1) the rod is made of layers with longitudinal laying of modified fibers, 2) the rod is made of flat layers with fibers laid in an orthogonal direction with respect to the axis of the rod, 3) the laminated material of the rod is made of a system of two layers with fiber laying ± 45o.

Pages: 511-523 Elibrary

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**Volkov-Bogorodsky D.B.**

## APPROXIMATIONS BASED ON THE GENERALIZED TREFFZ METHOD IN PROBLEMS OF GRADIENT THEORY OF ELASTICITY

The paper proposes a scheme of finite element approximation in problems of gradient elasticity theory, which does not assume continuity of approximating functions (shape functions) at the boundaries between elements, which in this case are considered as independent bases of the solution. This scheme is based on the Papkovich-Neiber analytical representation for displacements, which allows us to construct complete systems of approximating functions that analytically accurately satisfy the initial fourth-order equations, and on a generalization of the Trefftz method for a system of subdomains-blocks, which is a finite element grid. It is shown that the generalized Trefftz scheme allows, simultaneously with minimizing the energy functional, to stitch all the necessary quantities at the block boundaries: displacements, surface forces, and for fourth-order gradient equations also displacement derivatives and cohesive moments, which is achieved solely due to the analytical design of the functions used. All the results are transferred to the classical equations of elasticity theory, since they are a special case of gradient equations. The analytical representation of the solution opens up the possibility of constructing approximations on unstructured grids and inconsistent shape functions, and can be considered as a new technology of finite element approximations that does not assume the consistency of a finite element grid and the continuity of shape functions between elements.

Pages: 524-542 Elibrary

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**Sidorov V.N., Badina E.S., Detina E.P.**

## NUMERICAL SIMULATION OF VIBRATIONS OF COMPOSITE FRAME STRUCTURES TAKING INTO ACCOUNT TIME-NON-LOCAL DAMPING

The article is devoted to the problem of finite element analysis of the oscillatory process of a composite frame structure with modeling of the dissipative properties of the material as non-local in time. The internal friction in the material is assumed to depend not only on the value of the deformation rate of the frame material at the current time, but also on the values of the deformation rates throughout the history of the oscillatory process. The decrease in the influence of time points on each other with increasing distance between them – namely, the degree of non-locality of the material – is determined by the scale parameter. The damping matrix of a finite element model of a frame structure is obtained from the condition of stationarity of the total deformation energy of a moving mechanical system. The solution of the equation of motion of the computational model of the structure by the Newmark method is implemented in the MATLAB software package. A two-dimensional computational model constructed without taking into account the orthotropic properties of the material is considered. Two U-shaped frames with different upper crossbar spans made of vinyl-etheric fiberglass were considered as examples of numerical calculation. The scale parameter of temporal nonlocality is determined by the least squares method based on the results of a numerical experiment implemented in the verified SIMULIA Abaqus calculation complex in a three-dimensional formulation with modeling of orthotropic properties of a composite material. The results obtained using a calibrated time-non-local damping model are presented in the article in comparison with the data of a three-dimensional numerical experiment. It is shown that the nonlocal model is calibrated with adequate accuracy for a frame with a relatively low first natural oscillation frequency. The results obtained using a calibrated time-non-local damping model can be used in cases where the use of detailed solid-state models is ineffective.

Pages: 543-552 Elibrary

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**Antufyev B.A., Orekhov A.A., Rabinsky L.N.**

## THE EFFECT OF A LOCAL PRESSURE PULSE ON A SHELL WITH A COMPOSITE THERMAL PROTECTIVE COATING

An approximate solution is given to the problem of the effect of a local pressure pulse on a thin carrier shell with a composite thermal protective coating. This coating is considered as an inertial layer, which mainly changes the dynamic characteristics of the structure. In case of local damage, the aerodynamic pressure is transferred to the bearing layer, causing a local perturbed dynamic state in it. Thus, the problem is reduced to the equations of bending of the shell under the action of an arbitrary local dynamic load. These partial differential equations according to the Bubnov method are transformed in spatial coordinates into equations of motion of the shell, but in ordinary derivatives. An exact solution has been obtained for them. As a side result, an exact formula for the natural oscillation frequencies of a two-layer composite shell is obtained. Examples are considered in which the influence of the size of the pulse site on the dynamic deformed state of the structure is analyzed. It is noted that a violation of the continuity of the thermal protective coating or the appearance of operational defects in it leads to the evolution of the characteristics of thermal conductivity, heat capacity, emissivity and catalytic activity in the “carrier material – coating” system. This has a significant effect on the change in the thermal and stress-strain states of the structure.

Pages: 553-560 Elibrary

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**Gritskov A.V., Kireenkov A.A., Fedotenkov G.V.**

## GREEN’S FUNCTIONS FOR A TIMOSHENKO BEAM ASSOCIATED WITH A DEFORMABLE BASE

Nonstationary fundamental solutions (Green’s functions) of the problem for an unbounded Timoshenko beam associated with deformable bases of various types are constructed and investigated. The one-parameter Winkler base, the two-parameter Pasternak base model and the three-parameter Pasternak inertial base model are considered as base models. To construct the solution, the apparatus of integral Laplace transformations in time and Fourier transforms in spatial coordinate is used. Two methods of analytical treatment are proposed for constructing the original Fourier and Laplace images. The first of them is based on the connection of the integral Fourier transform with a series expansion over a variable interval. The second method is based on the decomposition of images into series by degrees of rational functions and is applicable only if there is no basis. Integral representations of the solution are constructed. The integral operators included in them have the form of convolutions with kernels in the form of found influence functions. Examples of calculations are given.