### №1-2018

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https://elibrary.ru/contents.asp?id=34842459

**Yankovskii A.P.**

## Comparative analysis of models of flexural deformation of reinforced metal-composite plates in conditions of steady creep

The problem of the mechanical behavior is formulated for flexible metal-reinforced composite plates in conditions of steady creeping of all phases of the materials composition. The equations, describing with varying degrees of accuracy the stress-strain state in such plates with account of their weakened resistance to the transverse shears, are obtained. The equations of the classical Kirkhoff theory, of non-classical Reissner theory and of the second variant of Timoshenko theory are appeared as special cases of these equations. The linearization of the equations is carried out on base of the method of secant modulus. For axially loaded and axially reinforced annular plates with clamped one edge and statically loaded on another edge, the simplified variant of the improved theory was developed, for which the complexity of practical realizing is comparable with the complexity of Reissner theory. Specific calculations are carried out for the bending deformation of and axially reinforced annular plates at different levels of heat loadings. It is shown that with increasing temperature the accuracy of the calculations in the framework of the traditional theories is reduced and they do not provide even 20% accuracy of determining the flexibility of such structures. It is found that the reinforcement in the directions of the main averaged stress and strain rate is not always rational from the point of view of reducing the flexibility plates in conditions of steady creep. It is shown that the traditional theory of classical and nonclassical can lead to the wrong solution of problems of optimal and rational design of flexible reinforced thin-walled structural elements.

Pages: 3-28 doi.org/10.25590/mkmk.ras.2018.24.01.003_028.01

**Karpov E.V.**

## About the destruction of spheroplastics by compression and torsion

Syntactic foams combining low density with high strength and a number of other useful properties are widely used in aerospace and shipbuilding industries. The mechanism of their destruction depends on the type of thermal-force loading due to the complex microstructure. In this paper we present an experimental study of the deformation and fracture of spheroplastics under various conditions of complex loading. The spheroplastic was an epoxy-diane syntactic foam with glass microballoons as a filler. The thickness of the glass microballoon walls was 1 μm, the average diameter was about 20 μm, the volume content in the composite was 60%. Loading was a different combination of joint and consistent compression and torsion. We also investigated the material destruction at a fixed position of the loading system clamps as a result of the aftereffect. The tests were carried out at various temperatures. Temperature affects the type of destruction of the matrix, the degree of filler particles mobility during the deformation of the composite and the state of the interphase boundaries. Analysis of loading trajectories, relaxation curves and fracture surfaces under various loading regimes and temperatures is carried out. On the basis of the analysis it was concluded that the spheroplastic under complex loading demonstrates the behavior peculiar to granular materials in which the movement of particles is impeded by adhesion with a binder. Characteristic features of granular materials are manifested with increasing compliance of the matrix and the accumulation of microstructural defects that weaken the bond between the filler particles.

Pages: 29-45 doi.org/10.25590/mkmk.ras.2018.24.01.029_045.02

**Kalmykov P.N., Lapichev N.V., Mikhaylov I.A., Myagkov N.N., Salnikov A.V., Shumikhin T.A.**

## Experimental study of fragmentation of spherical aluminum rojectiles on continuous and mesh bumpers at velocities up to 7km/s

The paper presents the results of experimental studies of fragmentation of 6.35 mm spherical aluminum projectile impacting a thin continuous aluminum plates or a steel mesh bumpers at velocities of 6 … 7 km/s. The acceleration of the projectile was achieved by a two stage light gas gun. The impact took place under vacuum condition. We used continuous and mesh bumpers are of lower and higher areal density. In each pair of bumpers of different types the areal densities were similar. For the registration of after-impact fragment cloud we used thick aluminum witness-plates situated at some distance from targets normally to the shot-line. The post-mortem examination of the morphological features of the witness-plates allowed the comparative description of the fragment cloud generated at the impact. The digital processing of the pictures of the witness-plates damages provided estimate of the volumes of craters and the sizes of deeply eroded areas. The impact on the continuous bumper of the lower areal density produced the damages in the forms of chains of small craters. Peripheral ends of some chains were split like a fork – the morphological feature of the distribution of material in the ejecta cone generated at the impact of an aluminum projectile on a thin aluminum plate. On the other hand, the experiment with mesh bumper of the similar areal density produced thread like thin craters in the form of closed contours – never observed earlier effect. The mesh with higher weight showed the highest degree of the projectile fragmentation producing the number of small fragments several times higher the in the experiments with other bumpers of lower or similar weights. The integral distribution of witness-plate crater volumes was derived for all undertaken experiments. The data analysis was performed in the frames of Weibull distribution and the comparative analysis was performed among bumpers of lower and higher weights. The distribution of kinetic energy among the fragments in the after-impact cloud was derived.

Pages: 46-69 doi.org/10.25590/mkmk.ras.2018.24.01.046_069.03

**Ivanova O.A.**

## The model of a supplied beam with viscoelastic internal interactions

Free oscillations of a one-dimensional Cosserat continuum model are considered. The model is built on the base of Ilyuishin’s mechanical modeling approach. It consists of a beam supplied by rigid massive inclusions periodically placed along the longitudinal line of the beam. Those inclusions are connected with their nearest neighbors by belt drives. We consider bending-tension motion of this construction in one plane. Model behavior in bending and tension motions of the supporting beam and relative rotation motions of the inclusions is elastic. Model behavior in the moment action of inclusions on supporting beam elements is viscoelastic (Kelvin-Voigt model is used). The linearization of the such model motion equations is made for the case of small departures from undeformed configuration. The problem of free oscillations is considered for the linearized model with boundary conditions as follows: pinning of the beam’s edges and absence of moment actions on the end inclusions. The fundamental difference between the system of equations for this model and for the one with fully elastic behavior is mentioned. The general solution of the problem of free oscillations is examined on the assumption with the special form of solution. The “antenna type” construction with known stress-strain properties is taken as an example. The computational solution of the problem of free oscillation is obtained for such construction. It is found that for each oscillation mode there exist exactly two forms of motion. The rate of decay became apparent to depend on viscosity value and oscillation mode. Some graphs are given to demonstrate the dependence.

Pages: 70-81 doi.org/10.25590/mkmk.ras.2018.24.01.070_081.04

**Saganov E.B.**

## Account of heterogeneous hardening of shape memory alloy in the problem of direct martensite transformation in torque rod with circular cross section

The work is devoted to the study of the effect of non-uniform hardening of a representative volume of a shape memory alloy (SMA) on the results of modeling the phenomenon of direct martensitic transformation in a rod of circular cross section from SMA, which proceeds at a constant torque. The problem is considered in the framework of the model of non-linear of straining of SMA in phase and structure transitions. When solving the problem, the hypothesis of flat sections for full deformations is valid. The temperature distribution over the cross-section of the rod from SMA during its cooling is considered homogeneous. In the course of the work, a comparison was made between the results obtained for different models of accounting for the martensite fraction of the representative volume of the SMA participating in the structural transition. The process of development of martensitic elements when cooling the rod from SMA is not considered in this paper. The problem is considered in a single-coupled thermomechanical formulation. It is believed that the cooling process is slow enough. The isolation of the latent heat of the phase transition and the dissipative effects are compensated by heat exchange with the surrounding medium. The problem under consideration is reduced to a system of differential equations, differently formulated in different parts of the region under consideration, the moving boundaries between which are not known in advance and are in the process of solving the problem. Therefore, this system cannot be integrated by standard methods. Proceeding from the elastic solution of the problem of torsion of rods of circular cross section, the maximum stress level is noted on the outer surface of the rod. Accordingly, the phase transition starts from the outer layers. In the general case, in the process of cooling, three regions can be distinguished in the cross section of the SMA rod. Near the outer layers, the phase transition has already been completed. The material is in the martensite state and deforms either elastically or by the mechanism of structural transformation. Near the axis of rotation, the phase transition has not yet begun. The material of the rod is in the austenitic phase state and deforms elastically. In the intermediate zone of the rod, a phase occurs, and under certain conditions, structural transitions occur. A dimensionless temperature parameter is used as the process parameter. In the course of the solution of the problem, the dependences of the dimensionless twist from the dimensionless temperature parameter, as well as the stresses over the cross-section of the rod for different cooling stages are obtained.

Pages: 82-92 doi.org/10.25590/mkmk.ras.2018.24.01.082_092.05

**Rybakov L.S.**

## Variational principles of lagrange and castigliano in theory of a plane regular truss with orthogonal structure

The linear theory of a plane regular truss built by means of the variational principles of Lagrange and Castigliano. According to glueing method, the truss was split into nodes and rods. Elastic analysis of rods and geometric conditions of the conjugation of their with the nodes showed that the stress-strain state of the truss is described by the nodal displacements, the total elongations of the rods and internal initial forces in them. All these unknown values are functions of two integer parameters, used for numbering the nodes and rods. The result of element-wise elastic analysis was the relationships connecting the total elongation of the rods with nodal displacements and initial forces. The remaining defining equations of the theory are derived from the variational principles of Lagrange and Castigliano, based on the discrete analogue of the calculus of variations. Its functionals are formed by sums and depend on functions of discrete arguments From the variational principle of Lagrange the static equations are obtained and a formulation of the boundary value problem in nodal displacements is given. The general solution of the static equations is represented to within two functions of integer parameters called force functions. Pointing to the redundancy of elastic systems, they play the same role as stress functions in the mechanics of elastic bodies. Using force functions, the compatibility equations for the total elongation of the rods are derived from the Castigliano variational principle and formulations of the boundary value problem in the initial forces and in the force functions is given.

Pages: 93-110 doi.org/10.25590/mkmk.ras.2018.24.01.093_110.06

**Potetyunko O.A., Vatulyan A.O.**

## Reconstruction of rheological parameters of inhomogeneous plate fixing

In this paper we study the oscillations of a circular viscoelastic plate of variable stiffness with different support conditions at the boundary, including in the presence of viscoelastic bonds (in the elastic case the problem was investigated by the authors earlier). In the case of steady-state vibrations, viscoelastic bonds in the boundary conditions are characterized by two complex coefficients that depend on the frequency of the oscillations. Three auxiliary problems that do not contain these coefficients were numerically formulated on the base of Lagrange’s variational principle. The problems were solved numerically using the Ritz method. The influence of the number of coordinate functions on the accuracy of the constructed solution and its dependence on the frequency of oscillations was investigated. The solution is sought in the form of a linear combination of the three constructed solutions. Satisfaction with the boundary conditions makes it possible to establish the fractional-rational structure of the solution depending on the coefficients in the boundary conditions. The problem of reconstructing the parameters of viscoelastic bonds based on the known (measured) deflection in a set of points at a fixed frequency was also solved. A system of nonlinear algebraic equations is constructed to find the required coupling coefficients, each of which defines a conditional hyperbolic dependence in the complex space. A method for selecting a single solution is presented. Computational experiments for various segments of coefficients change (weak support, rigid support) are presented. The ranges of the most successful reconstruction are founded. The effect of neglecting the viscoelasticity of the support on the reconstruction of the load amplitude was estimated.

Pages: 111-121 doi.org/10.25590/mkmk.ras.2018.24.01.111_121.07

**Kriven G.I., Lurie S.A., Lykosova E.D., Rabinskiy L.N.**

## Stress state in structural elements of modified fibrous composite materials with viscous fibers

The stress state of modified composite materials reinforced with whiskered fibers is investigated for the case of uniaxial tension. The modified composite material includes fibers that are a multiphase material, so the composite material composes of three phases: а) base fiber, б) bristled interphase layer is grown on top of the surface of fiber, в) matrix. The stress state in each phase of such a fiber composite material is studied. Two types of fiber composite material consisting of a matrix are considered. The first type is a basic carbon fiber T-650 coated with an interphase layer, where the interphase layer contain carbon nanotubes and epoxy matrix. The second type is a basic carbon fiber IM7 coated with an interphase layer, where the interphase layer contain zinc oxide nanowires and epoxy matrix. The stress state in each of the phases of the modified fibrous composite is examined to evaluate the strength and to compare the strength characteristics of the modified composite with the strength characteristics of the classical composite material consisting of carbon fibers and an epoxy matrix. The study was conducted in several stages. First, effective properties of interphase layers formed by a matrix filled with whiskers or nanotubes were determined. Interphase layers have cylindrical isotropy. As a result, the fibrous composite is considered as a composite system with a transversely isotropic property in each phase: fiber and interphase layer are transversely isotropic, matrix is isotropic. Secondly, radial, axial and circumferential stresses were investigated for each of the phases of two composite materials. The results of the studies showed that in case of uniaxial tension, the strength of the composite is controlled not by the strength of the fiber or the matrix, but is controlled by the strength of the viscous interphase layer.

Pages: 122-144 doi.org/10.25590/mkmk.ras.2018.24.01.122_144.08