### №4-2018

https://elibrary.ru/contents.asp?id=36685755

**Dimitrienko Yu.I.**

## Modeling of incompressible elastic composites with finite deformations by the method of asymptotic homogenization

The method of calculating the effective deformation diagrams for incompressible laminated composites with finite deformations are suggested. The method is based on the application of the theory of asymptotic expansions over small geometrical parameters to a general non-linear elasticity equations with finite deformations of inhomogeneous media with a periodic structure. It was used suggested earlier the universal representation of constitutive relations for nonlinear elastic compressible media with finite deformations. A solution of the local problems over the periodicity cell in a formal explicit form of interconnected systems of nonlinear algebraic equations was received. It was mathematically rigorously proved that if the phase of the composite are incompressible, the composite as a whole is also an incompressible material. The example of numerical calculation of the effective deformation diagrams multilayer composite fibers are described by the model of nonlinear elastic incompressible material such as Money.

Pages: 503-520 Elibrary DOI: 10.33113/mkmk.ras.2018.24.04.503_520.01

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**Yankovskii A.P.**

## Modeling of unsteady creep of sandwich panels with thin reinforced bearing layers of nonlinear hereditary materials

On the basis of the method steps in time the problem of unsteady creep are formulated for the bending sandwich panels with thin reinforced bearing layers. The weakened resistance of the filler to transverse shears is taken into account in the framework of the Reissner theory. The mechanical behavior of the component materials of the composition of the layers is modeled by the ratio of the nonlinear hereditary Rabotnov theory of creep. It is shown that in discrete moments of time the mechanical condition of such structures are formally described by the defining equations for composite laminated plates of nonlinear-elastic anisotropic materials with known initial states of stress. The linearization of the formulated problem at each time step is carried out by the method of successive approximations. In the case of cylindrical bending the unsteady creep of elongated rectangular composite sandwich panel with weak honeycomb core are investigated. At short-term and long-term loadings of such plates, the analysis is held for the dependence of flexibility on the parameters of the reinforcement bearing layers. It is shown that using the classical theory of bending of layered plates leads to the prediction of an unjustifiably low flexibility of composite sandwich panels, especially under conditions of long-term loading. It was obtained that when the account of weakened resistance of the filler to the transverse shifts, the two different mechanisms of flexural deformation of three-layer plates can be realized: “classic”, when the flexural strain state prevails, and “non-classical”, when the transverse shear of the filler has a main influence on the deflection. In the latter case, in the vicinity of the supported edges of plate the edge effects, describing a shift of the sandwich panels in the cross direction may arise.

Pages: 521-538 Elibrary

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**Garishin O.K., Izyumov R.I.**

## Modeling of the spherulitic structure of thermoplastic as a composite system with elastic-plastic properties

Computer model of a polyolefin (thermoplastic) describing its mechanical properties at micro and macro levels is proposed. It takes into account the presence of supramolecular crystallite formations (spherulites) in the structure. These materials (polyethylene, polypropylene, etc.) are semi-crystalline polymers, so their structure is strongly heterogeneous. Spherulites consist of lamellas (flexible thin plates composed of macromolecules laid in transverse loops) and an amorphous phase between them. Lamellas emanate radially from one common nucleus, filling the spherical space around it. Direct analysis of spherulite deformation, taking into account the detailed morphology of the structure, is practically impossible because of its extremely complex geometry. Therefore, a phenomenological approach was used while modeling. Spherulite was represented as a radially anisotropic inclusion, in which the mechanical properties in radial direction were determined by lamellas, and in tangential – by amorphous phase. The spherulitic structure was modeled as a regular lattice of radially anisotropic elastic or elastoplastic inclusions. The hypothesis of the affine nature of structure deformation was also used. That is, it was assumed that the inner regions of the spherulite change their shape as well as the whole one. The degree of spherulite crystallinity (the ratio of crystalline and amorphous phases) was varied by changing its radial and tangential stiffness. Computer simulation was carried out on a hexagonal cell of periodicity. Boundary problems of cell elongation (both in elastic and elastic-plastic formulation) were solved numerically – using the finite element methods. As a result, the dependencies allowing an assessment of the influence of the spherulite structural characteristics on the effective mechanical properties of thermoplastic were constructed.

Pages: 539-554 Elibrary

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**Shchetinin V.N., Sokolov A.P.**

## Identification of elastic properties of the adhesion layer of dispersed-reinforced composite materials from experimental data

A numerical technique for identifying the elastic properties of an adhesive layer of composite materials from experimental data is presented. The problem is posed as the “inverse homogenization problem”. Homogenization problem was solved by the asymptotic averaging method. A wide literature review of papers devoted to solving and investigating the problem of inverse homogenization is presented. Finite element method was used for solving “local problem” on periodic cell of composite. The inverse problem was reduced to the optimization problem, for which the method of sequential quadratic programming was applied. Tikhonov’s regularization with a classical apriori approach to the choice of the regularization coefficient was used for regularize the solution. The method of homogenization taking into account the adhesive layer of the composite material is considered. The adhesive layer is introduced as an additional isotropic phase of the composite. The Monte Carlo method was used to analyze the stability of the identification problem solution to the errors in the experimental data. Dispersed-reinforced composite filled with glass microspheres is considered in computational experiments. A comparison with the experimental data is given. The necessity of including of an adhesive layer in model for calculating the effective elastic characteristics of dispersed-reinforced composite materials is substantiated. Approach of the elastic properties of the adhesive layer identification from experimental data taking into account the noise of mathematical models and experimental errors is approved.

Pages: 555-581 Elibrary DOI: 10.33113/mkmk.ras.2018.24.04.555_581.04

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**Bochkov A.V., Kozirev J.M., Ponomarev A.V., Raisson G.**

## Effect of viscosity on hip process of long cylindrical blanks

At present, powder metallurgy is becoming more and more widespread due to the construction of large gasostatic extruders, which make it possible to produce a wide range of products of large geometric dimensions with high performance characteristics. In this paper, we investigate the effect of viscosity on the process of hot isostatic pressing of a long cylindrical billet, which is one of the typical problems solved in production. The insistence of the problem is due to the peculiarities of the hot isostatic compaction process, associated with significant distortions in the shape of the product during the process, as well as the high complexity and cost of further processing. Also, because of the high cost of raw materials, carrying out field experiments is seriously hampered. Under these conditions, it is important to consider all the factors affecting the final form of the product, one of which is the dependence of the yield stress on the strain rate intensity, which is assumed to be given in power form in the work. Powder material in the work is considered as a single plastically compressible medium under conditions of an inhomogeneous non-stationary temperature field. A general formulation is considered that includes the equilibrium equation, the fluidity equation, the associated flow law, the ideal plasticity condition and the incompressibility condition, the continuity equation. The problem of hot isostatic pressing of a long cylindrical billet, which is a capsule with a powder material placed in it, is also considered. The influence of the capsule on the ends of the cylinder is neglected. In this formulation, the Green’s fluidity condition is used to describe the mechanical properties of the powder material. Relations are obtained that make it possible to evaluate the qualitative picture of the processes occurring at different stages of the hot isostatic pressing process (HIP). Calculations are made for the ratio of the initial cylinder sizes to the final ones, depending on the parameters.

Pages: 582-596 Elibrary DOI: 10.33113/mkmk.ras.2018.24.04.582_596.05

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**Safronov P.A.**

## Solution the coupled direct phase transition problem in bending beam from shape memory alloy with taking into account tension-compression asimmetry

In this paper, a solution of the problem of direct martensite transitions (DT) in a beam of a solid rectangular cross section from shape memory alloy (SMA) under the constant bending moment was obtained. An account has been taken of the property of the tension-compression asymmetry (TCA) of the SMA, which consists in a significant discrepancy between the stress-strain curves under tension and compression of samples from the SMA. The solution is obtained database on the model of nonlinear deformation of the SMA in phase and structural transitions in a single coupled thermomechanical formulation. An account has been taken of the heterogeneous hardening of elementary volume of SMA in DT. Slow processes are considered – the distribution of the temperature field over the height of the beam section is considered uniform. The Bernoulli-Euler hypotheses were accepted on the physical side of the beam bending process. Within the framework of the considered process, DT shows the effect of TCA of the SMA in compression and tension on the distribution of normal stresses and the phase composition parameter in the beam section, on the position of the boundaries of the beginning and ending of the phase transition in the stretched and compressed regions of the beam, and on the compliance of the beam. The position of the neutral plane at each point of the DT process is established. The influence of taking into account both homogeneous and heterogeneous hardening of the elementary volume of the SMA on the solution of the problem is shown. The range of values of the bending moment is determined, for which the influence on the solution of the problem of the phenomena of TCA and inhomogeneity of the hardening of the representative SMA volume during the DT turns out to be maximum. The convergence of the solution of the DT problem in the beam from SMA under a constant bending moment in a coupled formulation to the solution of a similar problem in a non-coupled formulation is shown.

Pages: 597-615 Elibrary

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**Romanova T.P.**

## Modeling of behavior of rigid-plastic doubly-connected curvilinear composite plates in viscous medium under action of explosion loads

Based on the model of a perfectly rigid-plastic body and a structural model of the composite, the solution is constructed for dynamic bending of two-connected curvilinear composite plates with hinge-supported or clamped contours in a viscous medium under the action of explosive loads. The plates are hybrid, multilayered and fibrous with a symmetrical distribution of layers with respect to the middle surface. In each layer the reinforcing fibers of different materials are located in directions parallel or normal to the inner plate contour. It is shown that, depending on the load amplitude, two deformation schemes of the plate’s deformation in the form of a set of several linear surfaces separated by curved plastic hinges are possible. For each of the schemes, the equations of the dynamic deformation of the plates are obtained on the basis of the virtual power principle in combination with the d’Alembert principle. The operating conditions of these deformation mechanisms are analyzed. To simplify the calculation of double integrals over curvilinear areas with time-varying boundaries, a curvilinear orthogonal coordinate system associated with the equation of the inner contour of the plates is introduced. The analysis of the dynamic deformation is carried out for the plates under the action of arbitrary loads of explosive type. The location of the linear surfaces and the plastic hinges separating them are determined at the moment of the beginning of the plate movement depending on the value of the applied load at the initial time. The limit load, final deflections and time of deformation of the plates are determined. Numerical examples are given for a two-connected plate with an internal contour in the form of an ellipse with different variants of reinforcement, viscous foundation and supporting of the contours under the condition of the same consumption of reinforcing fibers of constant thickness. The number of parameters of the developed mathematical model allows for a wide range to change the structure of reinforcement plates, volume content of reinforcing fibers, layer thickness and physical characteristics of the composite, viscous foundation, also the geometric shape of double-connected plates and conditions of support of the contours.

Pages: 616-633 Elibrary

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

## The modulus of elasticity of polymer matrix for nanocomposites polyurethane/carbon nanotube

The mechanism of reinforcement (enhancement of elastic modulus) of nanocomposites with elastomeric matrix by carbon nanotubes was considered. It has been shown that interaction of matrix polymer and carbon nanotubes leads to formation of interfacial regions having high modulus of elasticity. This effect defines the enhancement of modulus of elasticity of polymer matrix, modified by action of nanofiller in comparison with the same characteristic for matrix polymer. This circumstance leads to reduction of real reinforcement degree of nanocomposites polymer/carbon nanotube in comparison with nominal one. The usage of real values of reinforcement degree of nanocomposites allows its correct description within the frameworks of the very various models: mixtures rule, micromechanical and percolation ones. The estimation of real values of elastic modulus for nanofiller aggregates and interfacial regions demonstrated their large (by several orders of magnitude) difference from nominal characteristics of these parameters for initial nanotubes and matrix polymer. The stiffness of polymer matrix exercises decisive influence on real value of elastic modulus of carbon nanotubes in polymer matrix of nanocomposite. This circumstance leads to approximately the same ratio of elastic moduli of carbon nanotubes aggregates for nanocomposites with elastomeric and glassy polymer matrix. The necessity of using dynamical model (with application of real characteristics of polymer matrix and carbon nanotubes) for correct description of structure and properties of nanocomposites polymer/carbon nanotube is general conclusion of the present work, whereas static models, using nominal values of the indicated characteristics, given tactless results.

Pages: 634-643 Elibrary DOI: 10.33113/mkmk.ras.2018.24.04.634_643.08

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**Larin N.V.**

## Diffraction of the cylindrical sound wave on the continuous inhomogeneous thermoelastic spherical shell

The diffraction of a cylindrical harmonic sound wave emitted by an infinitely long linear source is considered on an isotropic thermoelastic spherical shell of random thickness. The physical and mechanical characteristics of the shell material are described by continuous functions of the radial coordinate. It is assumed that the shell surfaces are bordered by non-viscous heat-conducting liquids, which are mostly different. The desired velocity potentials of acoustic and thermal waves outside and inside the shell cavity are the solutions of the Helmholtz equations, which satisfy the radiation conditions at infinity and the boundedness condition. The displacement of particles and temperature change in the thermoelastic shell are described by the system of equations of the linear coupled dynamic problem of the inhomogeneous isotropic body thermoelasticity. To simplify this system of equations, two new unknown functions are introduced. The functions are related by certain relations with the angular components of the displacement vector. The radial component of the displacement vector, the two new introduced functions and the change of the body temperature are in the form of expansions into series in spherical harmonics with unknown coefficients, which depend on the radial coordinate. Taking these expansions into account, the system of equations for describing thermomechanical perturbations in the shell reduces to a system of linear ordinary differential equations of the second order. Conditions for an ideal thermomechanical contact are satisfied on the outer and inner surfaces of the shell. Expressions for the coefficients of potential functions and boundary conditions for a system of differential equations are found from the boundary conditions. The obtained boundary value problem is solved by a spline-collocation method with the help of the cubic B -splines device. Analytic expressions describing the wave fields outside and inside the cavity of the shell are obtained. The results of the frequency and angular dependences calculations of the scattered sound field amplitude in the far zone are presented. We showed the noticeable difference in the characteristics of the sound scattering, caused by both different laws of the inhomogeneity of the shell material and the thermoelasticity of its material.

Pages: 644-659 Elibrary

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**Mikhalev A.I., Semenov N.A.**

## Electrorheological elastomer material

In this paper, was described cold cured vulcanizates were made based on microdimensional unmodified titanium dioxide and polydimethylsiloxane rubber. Cross-linked elastomeric composites with an isotropic and anisotropically distributed filler were made from these vulcanizates. In these composites, the effect of a change in the elastic properties in an applied electric field was considered. Anisotropic, with particles oriented in an applied electric field, and isotropically filled samples were studied. A method for orienting particles in the matrix during vulcanization was shown. The features of the electrorheological response of anisotropically filled and isotropically filled composites are studied. The effect of organosilane on the mechanical properties of the electrorheological elastomer is also considered. As a result, it was determined that an elastomer filled with ER particles, both isotropic and anisotropic, changes its rheological properties in an applied electric field. Anisotropic ERE exhibits a greater electrorheological effect than isotropic, since the value of storage modulus in the field increases relative to the value without field by 40%-50%, whereas in the isotropic an increase of 20% is observed. The introduction of organosilane increases the electrorheological effect for an anisotropic elastomer, while for an isotropic increase it is not visible.

Pages: 660-666 Elibrary DOI: 10.33113/mkmk.ras.2018.24.04.660_670.10

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