### №3-2022

https://elibrary.ru/contents.asp?id=49513606

**Yankovskii A.P.**

## Refined model of thermo-elastic-plastic deformation of flexible reinforced plates

A coupled initial-boundary value problem of non-isothermal elastoplastic deformation of flexible reinforced plates is formulated using the refined theory of bending. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The temperature of structures over the thickness is approximated by high-order polynomials. An explicit * Работа выполнена в рамках государственного задания (№ гос.регистрации 121030900260-6). 294 numerical scheme is used to solve the formulated nonlinear two-dimensional problem. Thermo-elastoplastic deformation of plane-cross and spatially reinforced metal-composite and fiberglass plates dynamically bent under the action of an air blast wave has been studied. It is shown that in order to adequately calculate the temperature in thin-walled structures, it must be approximated by polynomials of the 7th order in thickness; to adequately determine the strain of the composition components, it is necessary to use the refined theory of plate bending, the simplest version of which is the Ambartsumian theory. For fiberglass plates, the temperature increment during their dynamic bending is 3-18oC, and for metal-composite plates 30-35oC. Therefore, the dynamic elastic-plastic calculation of fiberglass structures under explosive loads can be carried out without taking into account the heat release in them. In similar calculations of metal-composite plates, the thermal sensitivity of the composition materials can be ignored, but the thermal effect must be taken into account.

Pages: 293-312

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**Chernov M.I., Movchan A.A.**

## The effect of the rate of temperature change on the stress – strain state and functional properties of a composite with fibers from a shape memory alloy and a viscoelastic binder

The deformation properties of composites (CM) with shape memory alloy (SMA) fibers and a linearly elastic binder are significantly limited by small deformations of the binder. One of the ways to overcome this disadvantage is the use of viscoelastic binders with limited creep and deformations reversible after unloading over time, the magnitude of which is comparable to the deformations of SMA returned during the reverse phase transformation. CM with shape memory alloy elements and viscoelastic binder exhibit rheonomic (i.e. time-scale-dependent) behavior. Therefore, the analysis of the influence of the rate of temperature change of fibers on the functional properties of such CM is an urgent task. The article describes the results of solving such a problem for unidirectional CM in the framework of the model of nonlinear deformation of SMA during phase and structural transformations and the model of a standard linear body for a viscoelastic binder. Special attention is paid to the study of the influence of the rate of temperature change and the viscoelastic properties of the binder on the possibility of implementing a closed two way shape memory effect in CM. It is established that when the rate of temperature change of SMA fibers tends to infinity, the behavior of CM with a viscoelastic binder tends to the behavior of the same composite, but with an elastic binder if the instantaneous modulus of the viscoelastic binder is equal to the Young’s module of the elastic binder. At the same time, when the rate of temperature change tends to zero, the behavior of KM with a viscoelastic binder tends to the behavior of KM with an elastic binder, the Young’s modulus of which is equal to the long-term modulus of the viscoelastic binder. A method is proposed for determining the filling coefficient that ensures the implementation of a closed two way shape memory effect in KM with a viscoelastic binder by solving the problem for the case of an elastic binder.

Pages: 313-338

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**Abdusattarov А., Leonenko D.V., Starovoitov E.I.**

## Bending of a three-layer plate in a temperature field alternating ring load

The bending of an elastic-plastic three-layer circular plate under alternating loading by an axisymmetric ring load is investigated. The effect of the temperature field is taken into account. The plate package is asymmetrical in thickness. It is assumed that its deformation obeys the polyline hypothesis. The outer bearing layers are assumed to be thin, and Kirchhoff’s hypotheses are valid for them. The materials of the bearing layers are elastic-plastic. In a thicker rigid filler, Timoshenko’s hypothesis about the straightness and incompressibility of the deformed normal is fulfilled. The change in radial displacements is assumed to be linear in the thickness of the layer. The filler material is non-linearly elastic. The plate is assumed to be thermally insulated at the end and the outer surface of the lower bearing layer. The heat flow incident perpendicular to the upper layer creates a temperature field in the plate. The formula for its calculation is obtained by averaging the thermophysical characteristics of the materials of the layers over the thickness of the package. The influence of temperature on the elastic and plastic characteristics of the materials of the plate layers is taken into account. The Lagrange variational method is used to derive differential equations of equilibrium under primary loading of the plate. The work of tangential stresses in the filler in the tangential direction is taken into account. Boundary conditions are formulated on the contour of the plate. The case of an annular uniformly distributed load is considered. To solve the corresponding boundary value problem, an approximate method based on the Ilyushin method of elastic solutions is applied. The resulting iterative analytical solution is written out in Bessel functions. The iterative analytical solution is written out in Bessel functions. In case of repeated alternating loading, Moskvitin’s theory of alternating loading was used. The hardening of the material of the bearing layers is taken into account. For the obtained analytical solutions, a numerical analysis of the dependence on the physical equations of state, temperature, and boundary conditions is carried out.

Pages: 339-358

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**Sharunov A.V.**

## Numerical analysis of the dorning process of a connecting cylindrical coupling made of an alloy with shape memory, taking into account the influence of the type of stress state on the behavior of the material

The work is devoted to the numerical analysis of the dorning process of a cylindrical coupling made of shape memory alloy (SMA) in the mode of martensitic inelasticity (MI). The problem is considered within the framework of the model of nonlinear deformation of the SMA during phase and structural transformations. The resulting solution takes into account both elastic deformations and the property of the material’s resistivity. The resistance of the materials is understood as the dependence of the material constants of these alloys on the parameter of 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.

Pages: 359-373

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**Mishustin Ilya V.**

## Estimation of the limit strain tensor of a polycrystalline shape memory alloy

For a more complete study of the resistance asymmetry of polycrystalline shape memory alloys (SMAs) without texture, an algorithm for calculating the tensor of limit inelastic strain accumulated in the process of a complete direct phase transition under the action of a constant stress, was used. Known data on the geometry of the crystal cells of the phases and the ways of their transformation are used in the calculation. The algorithm is based on the assumption of a uniform distribution of orientations of the austenite phase cells in a representative volume of the material and consists in choosing for each of these orientations the most energy efficient variant of martensite orientation with subsequent averaging of the corresponding shape changes. Using equiatomic titanium nickelide as an example, the dependence of the limit inelastic strain deviator on the directing deviator of external stress and the relationship between strain and stress states were studied. It is shown that the crystallographic features of the SMA lead to its resistance asymmetry. When describing the phase transition, small strain tensor, Cauchy-Green finite strain tensor, and Hencky logarithmic strain tensor were used. In addition to averaging the specified strain tensors of the selected variants of martensite orientation, the deformation gradient tensors were averaged, followed by the calculation of the limit strain tensors of the specified types. Alternative methods of averaging led to close results. The deviator of the limit strain of the phase transition calculated by the described method is uniquely associated with the directing deviator of the external stress, regardless of its orientation. Their principal axes coincide, but the principal values are proportional only in uniaxial tension or compression. The limit phase transition strain deviator is represented as the sum of two terms, one of which is proportional to the directing deviator of the external stress, and the other term is proportional to the directing deviator, orthogonal to the first one. The dependence curves of the intensities of the limit strain deviator terms on the loading type parameter are plotted.

Pages: 374-386

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**Babaitsev A.V., Rabinsky L.N., Shesterkin P.S.**

## Determination of the damping properties of a 3m metal tape based on the solution of the inverse problem of the dynamics of a three-layer rod and experimental data

One of the methods for increasing the service life of stress-cycled structural elements for various purposes is to improve the damping capacity. There are various ways to increase damping, one of which is the use of damping layers. The article investigates the effect of 3M damping tape on the three-layer clamped-free beam dynamic characteristics. The results of natural frequencies experimental studies and logarithmic decrements for free vibrations of an aluminum beam-plate without damping layers (single-layer beam) and with damping layers glued to the front surfaces (three-layer beam) are presented. The dynamic characteristics are calculated based on the amplitude-frequency characteristics analysis obtained by the fast Fourier transform method. The samples’ physical constants are predetermined in static tests. For a given three-layer Euler-Bernoulli beam, its lowest natural frequency is theoretically determined and compared with the frequency calculated on the basis of the Timoshenko model. To determine the three-layer beam decrement with damping layers, a numerical solution of the inverse problem is obtained.

Pages: 387-398

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**Kurbatov A.S., Zhavoronok S.I.**

## The generalized routh equations in the orthotropic shell theory of n-th order and their application to normal wave dispersion problems

A new formulation of the initial-boundary value problem of the N-th order orthotropic shells theory based on the methods of analytical mechanics of continuum systems is proposed below. Some smooth base surface is introduced, and curvilinear coordinates are determined on the two-dimensional manifold corresponding to the base surface. The model of a non-thin shell as a three-dimensional elastic body is defined by a set of field variables of the first kind, and the spatial and boundary densities of the Lagrange functional. As a field variable of the first kind, a pseudovector of displacement is considered, given by covariant components of the vector of true displacement in the basis of the tangent bundle of a two-dimensional manifold. The components of the generalized stress pseudovector on sections with normal unit co-directed to the base vector of one of the curvilinear coordinates are defined by differentiating the spatial density of the Lagrange functional by covariant derivatives of the components of the displacement pseudovector in the selected direction. By Legendre transformation of the spatial density of the Lagrange functional with respect to to the covariant derivatives of the field variables, the spatial density of the mixed functional is obtained, depending on the state variables – the introduced stress pseudovector, displacement and velocity pseudovectors, as well as covariant derivatives of the components of the displacement pseudovector in the second coordinate direction. A biorthogonal base system of functions of the dimensionless normal coordinate is introduced, a system of new state variables defined on the tangent bundle of a two-dimensional manifold corresponding to the basis surface is determined by the series coefficients of state variables with respect to the biorthogonal base system. The surface and contour densities of the mixed functional defining the two-dimensional shell model are obtained by the corresponding reduction of the spatial and boundary densities of the mixed functional while retaining first N+1 series coefficient. The Euler-Lagrange equations, which follow from the Hamilton-Ostrogradsky principle, are resolved hence with respect to covariant derivatives of new state variables and, in a certain sense, are equivalent to the known Routh equations of the analytical mechanics of discrete systems. The application of the generalized Routh equations of the N-th order shell theory to problems of normal wave dispersion leads to a spectral problem linear with respect to the wavenumber, which allows one to construct as well real as imaginary and complex branches of dispersion curves corresponding to propagating and evanescent wave modes. A solution of the spectral problem for an isotropic plane layer is obtained and the solution convergence based on the N-th order theory to the exact solution in the second quadrant of the complex plane at some nodes of the Mindlin grid is studied for imaginary branches.