### №3-2019

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https://elibrary.ru/contents.asp?id=41164324

**Beliakova T.A., Goncharov I.A., Khokhlov A.V.**

## The impossibility of modelling of sigmoid superplasticity curves using only parallel or series connections of power-law viscous elements

Models for superplastic deformation of materials are applicable only in case they are able to describe properly the characteristic qualitative features of test data (the main effects observed), in particular material stress-strain curves features and sigmoid shape of their dependency on strain rate (in logarithmic coordinates). One of the traditional approaches to superplasticity modelling commonly used for more than half a century is based on structural rheological models, consisting of connections of various viscous and plastic elements, particularly connections of non-linear power-law viscous elements which are governed by two material parameters. In the present paper it is proved that it is impossible to describe the sigmoid shape of superplasticity curve by only parallel or only series connections of any number of non-linear viscous elements with arbitrary parameters. It results in the necessity to combine both parallel and series connections of power-law viscous elements in the model or to add elements of other types. The analysis showed that the shape of the strain rate sensitivity curves generated by the mixed connection of three power-law viscous elements as well as the model ability to provide a sigmoid strain rate sensitivity curves depend significantly on the relation between the strain rate exponents of the elements involved. The possibility of qualitative simulation of sigmoid strain rate sensitivity curve as well as providing high values of the strain rate sensitivity index within the framework of the linear viscoelasticity theory is shown without any complex restrictions on the relaxation modulus. It means that linear integral operators of the viscoelasticity theory may be used as an “element” in construction of constitutive equations for a material with a sigmoid strain rate sensitivity curve, in particular in modelling superplastic deformation of materials.

Pages: 299-315 Elibrary

**Baranov A.V.**

## Dissipative flow of viscoelastic polymer compound in a porous channel

The heat exchange at the flow of non-Newtonian fluid in a flat channel filled with porous material is investigated. Brinkman model is taken as the equation of motion. 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. The flow is described by the Brinkman equation. Phan-Thien-Tanner model is used as a rheological model. There are no transverse normal stresses and no transverse velocity component. When writing the energy equation, a single-temperature model is used. This approach assumes a local thermal equilibrium between the liquid and solid phases. Thermal boundary conditions of the first kind and the energy dissipation are taken into account. The temperature dependence of the viscosity is not considered. 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 solution was analyzed numerically by the finite difference method. Results of calculations have been presented. Accounting for viscoelasticity makes the velocity profile even more flat. A significant effect of Weissenberg and Brinkman numbers on the temperature profile and Nusselt number distribution along the channel has been shown. It was also noted that the inclusion of viscoelasticity with significant values of Weissenberg number tends to reduce dissipative heating of the liquid. This is reflected both in the temperature profiles and the local heat transfer on the channel wall. The calculations show that the impact of elastic properties is so great that neglect of viscoelastic effects can result in significant error.

Pages: 316-326 Elibrary

**Basalov Yu.G., Fomin L.V., Lokoshchenko A.M.**

## On accouning the influence of corrosion processes on delayed fracture of a rod under creep condition

The question of the influence of corrosion processes on ensuring the safe operation of materials and structural elements is considered on the basis of the formulation of the problem of the long-term destruction of a tensile rod that is in a creep condition. To solve the problem, a mechanical-mathematical model has been developed, including a modified diffusion equation, a kinetic equation for the accumulation of damage, and a relationship for the chemical interaction parameter. The parameters of this model are determined on the basis of the experimental dependence of the thickness of the corrosion film on time. A multi-stage process of destruction of corrosion layers under the action of increasing effective stress is considered.

Pages: 327-335 Elibrary

**Rybakov L.S.**

## About one approximate discrete-continual theory of a plane stress state

An approximate linear discrete-continuum analytical theory described by differential-difference equations is constructed for a rectangular elastic isotropic plate of constant thickness using the finite element method. The plate was represented by a regular discrete-one-dimensional elastic system composed of the same rectangular elements in the plan. Their length coincided with one of the dimensions of the plate, and the transverse size was determined by the ratio of the other size of the plate to the specified number of elements. The plane stress state was taken as the initial model of deformation of the plate and finite elements. The displacements of all the finite elements were approximated in the transverse direction linearly so that the geometric conditions of conjugation of the adjacent elements were satisfied. This made it possible, at the initial stage, to reduce the two-dimensional continuum theory of a plane stress state to a discrete-continuum theory described by functions of two arguments. One of them is the continual variable (the longitudinal Cartesian coordinate), and the other is an integer parameter, by means of which the elements of the elastic system are numbered. A rigorous discrete-continual analysis of the finite-element elastic system based on the gluing method and the variational principles of Lagrange and Castigliano allowed us to reveal generalized displacements, deformations, internal and external forces of the theory under study and establish its defining geometric, physical and static relationships, including the equations of compatibility of deformations. Within the framework of the theory, alternative formulations of differential-difference boundary-value problems in generalized displacements and internal forces are given. When the problem is formulated in the internal forces, force functions (discrete-continual analogs of stress functions) are introduced, which have made it possible to reduce the number of differential-difference resolving equations. The application of the theory is illustrated on a flat rod for which the simplest statically indeterminate model of deformation is constructed, with Bernoulli and Tymoshenko models as special cases.

Pages: 336-353 Elibrary

**Atlukhanova L.B., Karnet Yu.N., Kozlov G.V.**

## The structure of nanofiller and interfacial contacts in nanocomposites polymer/carbon nanotube

The structural aspects of interfacial adhesion in nanocomposites polymer/carbon nanotube on the base of polyamide-6 were studies. As it is known, carbon nanotubes in polymer matrix of nanocomposites create annular formations, which are a structural analogue of macromolecular coils of branched polymers. It has been shown experimentally that the indicated annular formations are fractal objects. Since macromolecular coils of polymer matrix have fractal structure also, then formation of interfacial contacts in the considered nanocomposites can be simulated as result of interaction of two mentioned above fractal objects. With this purpose the fractal model, based on the main postulates of Flory theory, was used. It is proposed that the level of interfacial adhesion in polymer nanocomposites is defined by number of contacts polymer matrix-nanofiller. This assumption is confirmed by linear correlation of the indicated contacts number and dimensionless parameter , characterizing the level of interfacial adhesion. The obtained relationship allows to establish approximately linear correlation between number of contacts polymer matrix-nanofiller and radius of annular formations of carbon nanotubes. The indicated circumstances give possibility to obtain percolation relationship, in which at fixed content of nanofiller reinforcement degree is defined by number of contacts polymer matrix-nanofiller only. Besides, these equations demonstrate that reinforcement degree of polymer nanocomposites is controlled by structure of nanofiller in polymer matrix. The proposed model assumes that for enhancement of reinforcement degree of nanocomposites simultaneous increasing of radius of annular formations of carbon nanotubes, their dimension and dimension of macromolecular coil of polymer matrix is required. Since simultaneous increase of all indicated parameters is impossible, then search of their optimal combination is necessary. And at last, let us note the strong dependence of number of contacts polymer matrix-nanofiller on level and type of interactions between carbon nanotubes them self. At transition from repulsive interactions to attractive ones the indicated contacts number reduces sharply.

Pages: 354-364 Elibrary

**Suknev S.V.**

## Nonlocal criteria for quasi-brittle fracture of structurally inhomogeneous materials with stress concentrations

The analysis of nonlocal fracture criteria which have recently developed in the framework of the theory of critical distances is carried out. A common property of these criteria is the introduction of the intrinsic length of the material characterizing its microstructure, which allows one to describe the size effect under stress concentrations, thereby expanding the scope of application in comparison with conventional criteria. At the same time, it is marked that this region is limited to cases of brittle or quasi-brittle fracture with a small fracture process zone. To expand the scope of the criteria for cases of fracture with a developed fracture process zone, it is proposed to abandon the hypothesis of the size of this zone as a material constant, associated only with the microstructure of the material. New nonlocal criteria for quasi-brittle fracture are proposed, which are the development of the average stress criterion, and point stress criterion, and which contain a complex parameter that characterizes the size of the fracture process zone and accounts not only for the material microstructure, but also ductile properties of the material, geometry of the specimen, and its loading conditions. The size of the fracture process zone is represented as the sum of two terms, one of which characterizes the material microstructure itself, and the other one refers to the zone of inelastic deformations, whose size is determined by the features of the structural member (notch shape, notch size, loading conditions) and ductile properties of the material. Formulae are obtained for the critical stress in the problem of tension or compression of the plate with a circular hole. The results of calculations are in good agreement with the experimental data.

Pages: 365-380 Elibrary

**Dumanskii S.A., Kazarina S.A., Movchan A.A., Sil’chenko A.L.**

## Experimental data and results of simulation the phenomena of buckling caused by phase and structural transformations in shape memory alloys

The article contains a review of publications devoted to the analysis of stability of model objects, rods, plates and shells made of shape memory alloys (SMA) loaded in the modes of martensitic inelasticity or superelasticity. Particular attention is paid to the phenomenon of buckling caused by martensitic phase and (or) structural transformations in these materials. Experimental data are presented to show that the critical buckling loads of this type can be many times lower than the Euler critical loads of elastic buckling corresponding to the minimum (martensitic) values of elastic modules. Various concepts are discussed (fixed or variable phase composition, fixed or variable load, fixed or variable temperature, adiabatic or isothermal buckling) within which these effects can be described. Formulated uncoupled, once coupled and double-coupled formulation of appropriate stability problems. Analytical solutions of boundary value problems of buckling caused by phase and (or) structural transformations for Schenley column on SMA rods, operating on tension – compression and bending of SMA rod, plates and cylindrical shells are obtained. It is shown that the lowest values of the critical parameters are obtained in solving problems in a once-coupled formulation within the framework of the concept of variable load and the assumption of the isothermal nature of buckling (the concept of a fixed temperature). It is established that for the loss of stability of plates from SMA caused by direct phase transformation under the action of bilateral two-parameter compression, the proposition of the convexity of the stability region on the plane of loads is not valid.

Pages: 381-393 Elibrary

**Karpov E.V., Malikov A.G., Orishich A.M.**

## The effect of thermomechanical treatment on the mechanical and microstructural properties of the laser welded seam of aluminum-lithium alloys containing mg and cu

Complex experimental studies of the deformation and destruction of laser welds of modern aviation aluminum-lithium alloys doped with Mg and Cu have been carried out. The influence of mechanical and heat treatment of welded joints on the structure, strength and deformability of the weld, as well as the effect of temperature within the operating range (from -60 to +85C) on the destruction of the original and treated weld and its mechanical characteristics under static and low-cycle stretching, is investigated. The boundaries of the possible hardening of the seam by static pressing were determined for alloy 1420. It has been shown that, for alloys containing Cu, the pressing of a weld does not give results, and that with an increase in temperature, the non-uniformity of deformation of the untreated weld increases. This leads to a decrease in the ultimate strength and ultimate deformation of welded joints, and also increases their sensitivity to damage accumulation under low-cycle loading. The effect of quenching on the microstructure and mechanical properties of alloy B-1461 has been investigated. For it, the heat treatment mode of welded joints is determined, at which the ultimate load reaches 96%, and the ultimate deformation – 150% of the corresponding characteristics of the original alloy.

Pages: 394-400 Elibrary

**Motorin A.S., Ostropiko E.S., Razov A.I.**

## Functional properties of tini alloy after high-strain rate deformation at different temperatures

The paper presents the results of a study of the effect of the strain rate at various temperatures on the one-way and a two-way shape memory effects in a TiNi alloy. Preliminary high-strain rate deformation was carried out according to the Kola method for the split Hopkinson rod with a strain rate of about 10s. Preliminary quasistatic deformation was conducted using the Instron universal testing machine, with a strain rate of 10s. It is shown that the preliminary high-strain rate tension does not lead to an improvement in the functional properties of TiNi alloy. However, the preliminary high-strain rate compression can lead to improvement of functional properties of TiNi alloy. Presented results show that the values of the one-way shape memory effect and a two-way shape memory effect of the martensitic type after preliminary high-strain rate compression in the temperature range 20-60°C are higher than after quasistatic compression. The two-way shape memory effect of the austenitic type after preliminary high-strain rate compression is always greater than after quasistatic compression.

Pages: 401-407 Elibrary

**Kazarina S.A., Mishustin I.V., Sil’chenko A.L.**

## Modeling of shape memory alloys resistance asymmetry on a basis of experimental data

Experiments were carried out with uniaxial tension and compression of shape memory alloys (SMA), corresponding to the processes of martensitic inelasticity, the accumulation of direct transformation strains, cross hardening, stress relaxation and limited creep. Both quantitative and qualitative differences of the deformation diagrams under tension and compression have been established. To take them into account in the nonlinear model of phase-structural deformation of SMA, it is assumed that material functions depend on dimensionless parameters of the stress and strain state type.

Pages: 408-415 Elibrary

**Volkov-Bogorodsky D.B.**

## Structure of the generalized echelby problem solution and gauss representation for homogeneous polynomials

Solutions of the generalized Eshelby problem with polynomial displacements at infinity are investigated in elasticity theory for spherical and cylindrical multilayer inclusions. Such problems arise in the asymptotic averaging method for the viscoelasticity equations with rapidly oscillating coefficients. They are used for accurately calculating effective characteristics of the composite materials. To solve this problem we use the Gauss representation for homogeneous polynomials and it’s related potentials of Papkovich-Neuber representation that resolve the Eschelby problem in finite algebraic form.

Pages: 416-422 Elibrary

**Babaytsev A.V., Inyukhin A.V., Lisitsyn A.V., Mossakovskii P.A., Rabinskiy L.N., Solyaev Y.O.**

## Influence of fiber curvation on the carbon durability during high-speed loading

The indirect laying of reinforcing fibers in polymer composite materials can be both a consequence of technological defects and a targeted constructive solution aimed at optimizing the local characteristics of the material. Fiber curvatures can occur as technological defects in the case of lay-out errors, uneven impregnation and improper polymerization conditions, uneven shrinkage, etc. This paper presents the results of an experimental determination of the tensile strength and ultimate strains during compression of unidirectional carbon fiber specimens with straight-line (ordinary) and wave-like laying of fibers under high-speed loading. Dynamic tests were carried out according to the Kola method using a split Hopkinson rod on a high-speed system “StrainMaster High-Speed 3D DIC”. Static tests were performed on an Instron universal testing machine. The wave-like laying of fibers was created purposefully to study the effect of fiber curvature on the mechanical properties of the composite. For each type of fiber laying, a test of cylindrical samples cut in the directions of the axes of orthotropy of the material was performed. Three samples were tested for each type and each direction. For comparison, the strength of the samples under quasi-static loading was determined. As a result of the tests, a decrease in the compressive strength of the composite, in the presence of fiber curvature by 5-32%, and an increase in the high-speed strength characteristics, in comparison with quasi-static, by 30-60% were found. An increase in the ultimate strains of a composite with curved fibers under quasi-static loading has been established. The change in the nature of the destruction of the samples in the presence of fiber curvature under static and high-speed loading is investigated.

Pages: 423-433 Elibrary

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

## Variation statement of related dissipative tasks of the mechanics of a continuous media

The effectiveness of using the variational approach, formulated in the general case for the spacetime transversely-isotropic four-dimensional continuum together with the generalized Sedov’s variational equation is shown for the formulation of reversible and irreversible processes of deformation and heat transfer in both linear and nonlinear formulations. We use the procedure proposed by the authors for constructing a non-integrable variational form, which allows us to extend the formal variational models of the mechanics of deformable media to irreversible processes of deformation. Non-integrable variational forms determine the possible channels of dissipation depending on the list of generalized variables for a particular model of the medium. It is proved that if the reversible part of the processes under consideration is physically linear, then it can be distinguished in the Sedov’s equation as a variation of a separate functional. In this case, the Sedov’s variational equation can always be represented as the sum of the variation of the Lagrangian of the reversible part and the linear combination of dissipation channels of irreversible physically nonlinear processes. The importance of introducing of transverse isotropy properties in describing the connected processes of thermodynamics of deformation is noted. Such a generalization not only expands the physical models of the media under consideration, but is also necessary to obtain a consistent statement of the problem in generalized voltages for the generalized vector of momenta and the heat flux vector. The possibility of generalizing the variational approach to for formulating non-linear models of dissipative processes, including non-linear Navier-Stokes equations, is shown. In particular, examples of using the variational approach to describe hydrodynamic models in the absence of heat transfer in the medium are considered. The corresponding variational models are constructed: Darcy hydrodynamics, Navier-Stokes linear hydrodynamics, Brinkman hydrodynamics, gradient hydrodynamics and some generalization of the classical non-linear Navier-Stokes hydrodynamics.

Pages: 434-447 Elibrary