### №2-2020

Full versions of the articles

https://elibrary.ru/contents.asp?id=43105472

**Rybakov L.S.**

## Linear elastic analysis of a plane orthogonal rod system with different models of deformation of elements

A linear theory of a plane regular elastic system formed of horizontal and vertical rods and ascending and descending inclined rods, rigidly connected to each other at the intersections of elastic lines of the rods of the first two families, is presented. The peculiarity of the system under study is the combination of different models of rod deformation. According to the assumption, all rods work for tension-compression, and horizontal rods are also endowed with the ability to perceive transverse bending loads. When constructing the theory, the gluing method is applied. By analyzing the behavior of isolated elements and the geometric conditions of their coupling, it is shown that the deformation of the system is described by nodal displacements and rotations, complete deformations and the initial internal force factors of the rods. All these dependent variables turned out to be functions of integer parameters used for numbering the elements of the system, and are related to each other by geometric and physical relations of the theory. The rest of its defining dependencies are derived from the Lagrange and Castigliano variational principles, which are based on the discrete analog of the calculus of variations, in which, unlike its classical version, functionals are formed by sums and depend on functions of discrete arguments. Static equations are identified from the Lagrange principle and the formulation of the discrete boundary value problem in nodal displacements and rotations is given. The general solution of static equations is presented up to three functions of integer parameters called force functions. In the constructed theory, they play the same role as the stress functions in the mechanics of elastic bodies. Using force functions, the compatibility equations for the complete deformations of the rods are derived from the Castigliano principle and the formulation of the discrete boundary value problem in the initial force factors and in the force functions is given.

Pages: 157-173 Elibrary

**Sharunov A.V.**

## Solution of loading problems on thin-walled spheres and cylinders of shape memory alloy, taking into account stress state influence in the martensitic inelasticity mode

In the work, an analytical solution of the problem on the stress-strain state (SSS) of a thin-walled sphere and a cylinder of a shape memory alloy (SMA), which is under the influence of internal or external pressure, is loaded in the mode of martensitic inelasticity (MN) or in the process direct transformation without taking into account elastic deformations and taking into account the property of material tension-compression asymmetry. Under the property, tension-compression asymmetry refers to the dependence of the material constants of these alloys on the type parameter of the state of stress. The parameter associated with the third invariant of the stress deviator is used as a parameter of the type of stress state. In the framework of the work, a linear dependence of material constants on the type parameter of the stress state is taken. The solution was obtained on the basis of the model of nonlinear deformation of SMA during phase and structural transformations. When solving the problem without taking into account elastic deformations, the provision on active processes of proportional loading is used. In the framework of the deformation process under consideration, the influence of the SMA diversity resistance on the distribution (compression) of thin-walled structures is demonstrated. The distribution and compression of thin-walled structures are simulated taking into account axial symmetry. Thin-walled cylinders are considered under the assumption of plane deformation (PD) and plane stress state (PSS). It has been established that the parameter of the state of stress of a thin-walled cylinder under the assumption of PD is the same as under pure shear, both under internal and external pressure, and under the assumption of PSS, it is the same as under uniaxial tension at internal pressure and uniaxial compression at external . It was established that under the same loading (internal or external pressure), the parameter of the state of stress for a thin-walled sphere and a thin-walled cylinder has different values.

Pages: 174-189 Elibrary

**Baranov A.V.**

## Non-isothermal flow of reacting viscoelastic polymer compounds in a flat channel

Mathematical model of non-isothermal flow of non-Newtonian fluid in a flat channel is presented. Many assumptions were made on the basis of the fact that the flow occurs at low values of the Reynolds number and at a high Peclet number. This allows us to neglect inertia terms in the equation of motion and ignore axial thermal conductivity in the energy equation. Phan-Thien-Tanner model is used as a rheological model. Thermal boundary conditions of the first kind and the energy dissipation are taken into account. The flow is accompanied by a chemical reaction that leads to a sharp increase in viscosity. The viscosity is considered to depend on the temperature and the degree of conversion. This, in turn, led to the inclusion of the kinetic equation of a chemical reaction in the mathematical model. It is believed that a chemical reaction takes place in one stage and can be described using a single parameter – the degree of conversion. When a certain critical degree of conversion is reached, the viscosity rushes to infinity and the compound loses its fluidity. The fluid temperature at the inlet of the channel and the temperature of the walls of the channel are different. This means that the composition in the channel will be heated both because of hot channel walls and due to energy dissipation. The heat output at a chemical reaction is not taken into account. The solution was analyzed numerically by the finite difference method according to the iterative scheme. Results of calculations have been presented. The significant influence of various factors on the velocity profiles, as well as on the distribution of pressure and mass-average temperature along the channel is shown. From the made calculations, it can be seen that the dependence of viscosity on temperature and the degree of conversion can significantly change the entire hydrodynamic and thermal situation in the channel. Thus, when calculating the mass-average temperature, ignoring the energy dissipation and the dependence of the viscosity on the degree of conversion leads to a significant error, which is growing as the dimensionless length increases.

Pages: 190-199 Elibrary

**Firsanov Vic.V.**

## Bending beams made of a material with an unchangeable volume

Materials that do not change their initial volume under the action of a force load are called incompressible. These are usually low-modulus rubber-like materials, the feature of which is an infinitely large volume module that characterizes the resistance of the medium to changes in the volume of the material. Therefore, of the two undependable physical characteristics (modules) for incompressible materials, only one module remains, which characterizes the resistance of the medium to shape change. There is no Poisson’s ratio equal to 0.5 in the defining relations of the problem. The product of an infinitely large modulus on the deformation of the volume change, equal to zero, is an uncertainty that is replaced by some force function, which is an additional unknown. The term “low-modulus material” does not contradict the property of infinitely large resistance to volume change, since there is no volume module in the defining relations of the mechanics of incompressible media. In all these relationships, the shear modulus appears, which is much smaller than similar modules of widely used construction materials. The additional ratio, which is the absence of volume change, calls into question some classical hypotheses, such as the Kirchhoff hypotheses in plate theory and the Bernoulli hypotheses in beam bending theory. The hypothesis of non-compressibility of fibers in the transverse direction is not of great importance for the construction of determining relations, and the other two hypotheses about the absence of linear deformation in the transverse direction and shear in the plane may lead to an unacceptable solution. Here and further is the longitudinal coordinate that coincides with the neutral line of the beam, and is the transverse coordinate to the neutral line. The origin of coordinates for symmetrical loads and boundary conditions at the ends is located in the middle of the neutral line, and if there is no symmetry at one of the ends of the beam. The problem of bending an incompressible beam is constructed in displacements, although for such a beam the term “in displacements” is conditional, since the physical relations of the incompressible material, known in the scientific literature as “neo-hook” relations, contain a force function that cannot be expressed in terms of deformation or displacement. To define an additional unknown, an incompressibility condition is added to the defining relations of the problem.

Pages: 200-211 Elibrary

**Babaitsev A.V., Dobryanskiy V.N., Lurie S.A., Ripetskiy A.V., Shumskaya S.A., Solyaev Yu.O.**

## Impact strength of the three-layered lattice beams with non-local mechanical metamaterials in the core

The paper presents the test results for the three-layered structures with lightweight lattice cores made by three-dimensional printing using selective laser sintering technology with polyamide. The used core structures correspond to the so-called pantographic mechanical metamaterials, in which two systems of parallel beams are spaced apart by a small distance and connected to each other by transverse pivots at the intersections. For such materials, it is known that to describe their equivalent mechanical characteristics, it is necessary to use non-classical models of the theory of elasticity taking into account the nonlocal nature of the deformations of the structure of the material under loading. In this paper, we consider three options for transverse connections in the structure of the metamaterial, in which the transverse pivots provide the transition of both forces and moments (rigid joints), only forces (pinned joints) or are simply absent. Such cores are compared with a conventional lattice core, in which the intersecting beams form a rigidly connected system such as a flat frame. The fabricated samples were tested for the impact resistance according to the double-support shock bending scheme using pendulum impact testing machine. It has been established that with the same cross-sectional dimensions of the rods in the core, the samples with pantographic core with rigid transverse joints have the greatest bload earing capacity under impact. However, specimens with pinned joints exhibit unusual fracture mechanisms in which the damage zone is greatest and damage develops with the formation of many small fragments that impede the passage of the projectile through the structure and increase its specific energy absorption, which makes such core options potentially promising for creating energy absorbing structures.

Pages: 212-223 Elibrary

**Khokhlov Andrew V.**

## Characneristic features of stress-strain state evolution in thick-walled tubes of non-linear viscoelastic material subject to constant pressures

We construct and study analytically the exact solution of the boundary value problem for a hollow cylinder (a tube) made of physically non-linear viscoelastic material obeying the Rabotnov constitutive equation with arbitrary material functions. We suppose that a material is homogeneous, isotropic and incompressible and that a tube is loaded with constant internal and external pressures and a plain strain state is realized, i.e. zero axial displacements are given on the edge cross sections of the tube. We obtain explicit closed form expressions for displacement, strain and stress fields via the single unknown function of time and integral operators involving this function, two material functions of the constitutive relation, preset pressure values and radii of the tube and derive functional equation to determine this unknown resolving function. Assuming material functions are arbitrary, we prove that strains (creep curves) increase with time but the axial force at cross section doesn’t depend on time and material functions although stresses and strains do. The axial force proved to be equal to the one calculated for linear elastic tube although axial stress depends on radial coordinate in the case of non-linear viscoelastic material. We show that for special choices of material functions the strain and stress fields coincide with classical solutions in the frames of linear viscoelasticity, elasticity or elastoplasticity with hardening or without it. Fixing the material function governing non-linearity to be power function with any positive exponent and assuming creep function is arbitrary, we construct exact solution of the resolving functional equation, calculate all the integrals involved in the general representation for strain and stress fields and reduce it to simple algebraic formulas convenient for analysis. The stresses in this case don’t depend on time and creep function and coincide with classical solution for elastoplastic material with power hardening. We obtain criteria for increase, decrease or constancy of stresses with respect to radial coordinate in form of inequalities for the exponent value.

Pages: 224-246 Elibrary

**Abdurakhmanov A.A., Bogacheva V.E., Glagolev V.V.**

## To evaluating the limit state of the adhesion layer in a layered composite

The model of deformation of a layered composite in the form of a double-console plate is studied on the basis of the concept of an interaction layer in a linearly elastic formulation. It is assumed that the adhesive layer has a finite thickness and does not bind the cantilevers along their entire length. Layer thickness is considered as a linear parameter. The stress-strain state of a layer is considered on the basis of average thicknesses and boundary stresses related by equilibrium conditions. The use of stresses average in thickness allows us not to consider the shape of the end of the layer and to remain within the framework of the regular distribution of the stress field in the end region of the layer. Using the variational formulation of the problem containing a linear parameter, a finite element solution is constructed based on the quadratic distribution of the displacement field on the element. The numerical solution is compared with its analytical approximation with normal separation. The analytical solution was based on hypotheses of the Tymoshenko type taking into account shear deformations in consoles and in the absence of compression deformations. The energy product is introduced into consideration in the form of the product of the increment of the specific free energy by the layer thickness. The energy product of the adhesive layer was investigated as a function of the linear parameter for loading, such as normal detachment and mixed loading mode. It is shown that with a decrease in the linear parameter, the convergence of the energy product takes place. The effect of simplifying hypotheses on the distribution of the displacement field in consoles on the limiting value of the energy product is shown. The limiting value of the energy product at a critical external load is proposed to be considered as a criterial value. In this case, it is possible to isolate the range of values of the linear parameter at which the value of the external critical load will be practically constant.

Pages: 247-258 Elibrary

**Khomchenko A.V., Matrirosov M.I., Medvedsky A.L.**

## Behavior of layered structural elements made of polymer composite with internal defects under nonstationary influences

Increased requirements for new models of equipment in various fields of mechanical engineering lead to the need to use modern structural materials, which by their characteristics should exceed traditional materials. The use of polymer composite materials (PCM) in the creation of structural elements has become widespread. With all the advantages, PCM products are sensitive to internal defects that can appear at various stages of production and operation. The paper describes General principles of modeling layered structures with consideration for internal defects between layers using software systems (LS-DYNA, Siemens Femap) based on the finite element method (FEM) using an explicit scheme for integrating a complete system of FEM equations. The results of calculations for layered thin-walled structures made of a high-modulus polymer composite and subjected to non-stationary loads are presented, namely: a rectangular plate and a smooth flat cylindrical panel under the influence of a pressure field, a reinforced flat cylindrical panel and a smooth cylindrical shell under the action of an explosive spherical wave. The behavior of the above-mentioned layered structures with and without elliptical defects is analyzed. The fields of stress, strain, and displacement in the mono-layers at various times are determined. Safety factors coefficients are calculated using the following criteria: Hashin, Chang-Chang, Puck, LaRC03, Fischer. The degree of influence of interlayer defects is estimated. The developed method allows us to consider the influence on the strength of interlayer defects of any shape, size and location between the layers of the composite package (CP).

Pages: 259-268 Elibrary

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

## Continuum model of the lattice composite structure

The continuum model of the lattice composite structure of a cylindrical shell consisting of a system of helical and hoop ribs made by automated filament winding is proposed and discussed. Such lattice shells are used as load-bearing elements in rocket and spacecraft structures. The model is based on the analysis of the stress-strain state of the elementary lattice cell formed by the ribs and the averaging of the obtained results for a regular system of ribs. The symmetric cell consisting of a pair of helical ribs and a hoop rib passing through the middle of the segments of helical ribs between the nodes of their intersection is considered. The cell is loaded with normal and tangential stresses acting in the plane of the structure. The forces and moments acting in the ribs and the displacements of the ribs are determined and stiffness coefficients of the lattice structure allowing for axial strains and bending of the ribs in the plane of the structure are obtained. The obtained results allow us to write down the constitutive equations of the applied theory of lattice composite shells, including membrane and bending stiffness coefficients. The equilibrium and kinematic equations retain the traditional shape and taking into account the bending stiffness of the ribs does not lead to an increase in the order of equations. Bending stresses in the ribs are also determined. Bending stresses can make a significant contribution to the total stresses acting in the ribs of the lattice structure. The obtained analytical results are compared with the results of the finite element analysis of the lattice structure stiffness and the compression test results of the lattice element. The obtained expressions for the stiffness coefficients of the lattice structure and the stresses in the ribs can be used for the design and analysis of composite lattice structures.

Pages: 269-281 Elibrary