SCIENCE & EDUCATION

This paper presents a model to predict a nonlinear strain of the carbon laminate; the model is based on the relations between the theory of laminated plates and the non-linear approximation of deformation curve of unidirectional layer at the shear in the layer plane. The explicit expressions of stiffness and compliance matrices were obtained via multiplying the matrices that correspond to the elastic characteristics by the matrices, considering the non-linear properties of the laminate. The paper suggests an approximation option for the non-linear properties of the layer at the shear using an exponential function. Some considerations on damage accumulation in carbon laminates were made.

In design of thin-walled structures made of fiber reinforced composites it is necessary to provide a reliable determination of resistance to load-bearing exposures. Model representations of damage accumulation were described in this paper; it was show that an adequate description is possible with a use of hereditary-type relations. Approaches allowing us to perform nonlinear strain estimation were proposed. Elastic properties of a unidirectional layer and stress-strain curves were used as initial data. Matrix algorithms for constructing constitutive equations for unidirectional and layered composites were developed. Proposed approaches could be used in strength optimization of fiber reinforced materials and in numerical methods of calculating and forecasting mechanical behavior of composite materials and their elements.

An analytical model of plastic deformation of a conical shell with the transformation of internal surface into outer one was developed with a use of the kinematic method. The shell material was assumed to be perfectly plastic. The theory of thin shells and the kinematic theorem of limit equilibrium were utilized in this work. Both geometric and physical nonlinearities were taken into account. Dependences for calculating radius of curvature of the intensive deformation zones, value of chain ring deformation and values of the deforming force as a function of axial displacement were determined. Analysis showed the possibility of using a conical shell to absorb energy with high efficiency. Obtained results could be used for calculation and selection of optimal parameters of the energy-absorbing elements in shock absorbers.

A mathematical model of deformation of flexible armored materials based on aramid fabric under high-speed effects was proposed. The model takes into account the following peculiarities of deformation parameters of composite materials of the specified class: possibility of deformation without destruction at finite deformations, a considerable difference between stress-deformation diagrams under tension and under compression, dependence of these diagrams on a loading rate, pseudo-plastic features of materials caused by pulling out threads from fabric and/or a plastic character of deformation, anisotropy of nonlinear-elastic and visco-plastic properties and other effects. A problem statement was formulated for dynamic deformation of flexible armored materials. In order to solve the problem in a two-dimensional case the method of band-adaptive grids was applied. An example of numerical solution of the problem was presented for a high-speed action of a striker onto a flexible armored material; some effects of deformation of materials of the specified class were analyzed.

Using a multi-scale homogenization method, a three-level model was developed for ceramic composite materials based on the reaction-bonded SiC. Numerical simulation was performed with the use of the finite-element method for solving a set of local problems over the periodicity cells of three structural levels. Tensor fields of stress concentration were computed in matrices and fillers. A new strength criterion of a matrix and fillers at multi-axis stress state was applied; this criterion takes into account a significant difference (more than ten times) between strength features of ceramics under tension and compression. A model that takes into account a scale effect of the strength of ceramic composites was proposed. Numerical investigation of sequential microstructural destruction of ceramic composites up to the final fracture was conducted. Computations demonstrated that at the presence of a polydisperse structure of ceramics, a change in concentrations of large-scale fractions plays less considerable role than in the case of small particles fraction.

In order to justify a reliable application domain for the calculation dependence which allows one to determine the effective heat conductivity coefficient of transversely isotropic unidirectional fiber composite in the plane perpendicular to the fibers; a mathematical model of heat transfer in representative elements of this composite’s structure was created. An ordered arrangement of parallel fibers was considered for the case when their centers coincide with the nodes of cross sectional plane grid with square cells or cells in the form of equilateral triangles. Quantitative analysis of developed mathematical model was performed using the finite element method with controlled integrated computational error. Presented results are important for assessing the temperature condition and efficiency of heat-stressed structures made of unidirectional fiber composite.

Development of a method for calculating relaxation of a contact between a fuel element and a spacer grid cell is a necessary condition for solving complex deformation problems of fuel assembly under intensive temperature and neutron fields. Due to relaxation of contact interaction the fuel element can slide in the spacer grid cell; this results in decreasing stiffness of overall assembly and its deformation. A method for calculating relaxation of contact interaction between a fuel element and a spacer grid cell based on the finite element method and the Schwarz alternating method was proposed in this work. Calculation of relaxation with the use of the proposed method will allow to estimate operation time of a fuel element till its sliding, and also to determine the disability criterion of fuel assembly.

A method of stability investigation and stabilization of equilibrium of systems with geometrical constraints was elaborated and applied to real mechatronic device that is GBB1005 Ball & Beam. For constructing mathematical model Shulgin's equations with redundant coordinates were utilized. In order to perform stability investigation it’s necessary to combine Shulgin’s equations with kinematic constraint equations, derived by differentiating geometrical constraint equations with respect to time. First approximation equations in the neighborhood of equilibrium position have zero roots; the number of these roots is equal to the number of constraints. Asymptotic stability equilibrium in mechanical system with redundant coordinates is possible, despite formal reduction to Lyapunov's special case. This work presents a more complete non-linear model of the mechanical part of GBB1005 Ball & Beam bench. One more equilibrium position of the system was found during the investigation of complete non-linear cons. At the presence of one geometrical constraint between two coordinates there are two options for selecting a redundant coordinate. It was shown that the choice of linear control subsystem depends on the choice of redundant coordinate

Several various methods could be applied for description of mass-inertia properties of the finite element; each method leads to a different mass matrix. In particular, one could use the method of lumped masses, various shape functions both matching with (consistent) or distinct from shape functions for a stiffness matrix. Other approaches are also possible. The purpose of this work was to analyze some non-trivial approaches to determination of a mass matrix, such as non-standard shape functions, oscillation modes of a bar element, a method of combining different mass matrices with weighting factors. The article compares five different mass matrices by the example of a bar finite element and identifies inherent errors of these matrices obtained during determination of eigenfrequencies of the fixed bar. A system of equations for the finite element method for all mass matrices was solved analytically in the trigonometric series defined at the nodal points. It was shown that calculation of eigenfrequency based on inconsistent mass matrices could provide higher accuracy in comparison with diagonal and consistent matrices.

A mathematical multi-level model for calculating thermo-elastic properties of textile composites with alumino-chromic phosphate matrices under high temperatures was developed. This model takes into account physical-chemical transformations occurring in an alumino-chromic phosphate binder and glass fibers under a high temperature. Comparison of calculation results with experimental data was carried out; it was shown that the developed model allowed to predict a complicated nonlinear character of changes in elastic properties of composites under heating up to 1600 К. The developed model could be used for prediction of elastic properties of composites with alumino-chromic phosphate matrices under complicated heating modes.

The self-consistency method was used to estimate the components of a thermal conductivity tensor of unidirectional fiber composite, which is transversally isotropic with respect to an axis arranged parallel to the fibers. In this method it is possible to estimate effective values of parameters for the composite as a whole by averaging the perturbed distributions of the parameters over the elements of the composite structure and equating averaged results to zero. A perturbed temperature field and heat flux density distribution over the fibers and composite matrix were calculated on the basis of the developed mathematical models of heat interaction between these elements of the structure and a transversely isotropic homogeneous material which has required coefficients of thermal conductivity. For validation of the final calculated dependences, bilateral estimates based on dual variation formulation of the stationary problem of heat conduction in an inhomogeneous solid body were used. The calculated dependencies can be used for forecasting effective thermal conductivity of unidirectional fiber composites.

The Runge–Kutta discontinuous Galerkin method for an ideal MHD system of equations on the unstructured mesh for axisymmetric cases was considered. An algorithm of divergence-free magnetic field reconstruction in cylindrical coordinates was given. This algorithm allows one to obtain physically appropriate computation results with a high order of accuracy. A mathematical simulation of magneto-rotational instability in the shell of an accreting protostar was carried out. The obtained results were illustrated and discussed. These results showed that the magneto-rotational instability leads to formation of large-scale structures and causes transfer of angular momentum to the circumference of a plasma cloud.

Dynamics of an ideal incompressible fluid filling a cylindrical vessel divided by an elastic membrane with an allowance for surface tension forces was investigated in this work. Boundary conditions for surface tension at the wetting angle of 90 degrees were formulated; normal oscillations were determined. The Bubnov-Galerkin method was used to obtain an approximate frequency equation in an axially symmetric case (plane problem). Dependence of the natural frequency of oscillations on the Bond number which characterises the strength of gravity was analysed. A critical value of the natural oscillation frequency was also determined.

A mathematical model was developed for plane aerostatic bearings with porous deboosters; this model allows to determine the main performance characteristics such as damping stiffness of load capacity in linear and angular directions and a consumable air flow. Experimental determination of permeability inhomogeneity for porous graphite deboosters was conducted. Influence of permeability variation on characteristics of an aerostatic bearing was evaluated by means of calculations with maximum and minimum values of the permeability index. This influence appeared to be smaller with a constant aerostatic gap than with constant load. On the basis of the obtained results, recommendations on manufacturing plane aerostatic bearings were given.

In this article the authors propose a finite-element method for calculating non-stationary heat-mass-transfer 3D problems in heat-shielding structures made of composite materials with consideration for thermo-decomposition and filtration of gases in pinholes. Specialized software that implements the numerical finite-element method for solving the problem of internal heat-mass-transfer in 3D elements of structures made of composite thermo-destructive materials was developed. In this paper the authors also present results of testing the developed method and software that illustrate working capacity of this method and its applicability to solving investigation problems of non-stationary heating modes of heat-shielding structures.

Artificial cavitation occurs in a hydraulic line as a result of gas feeding either into a separation zone behind bluff bodies, or directly from the wall jet in the liquid flow line. The resultant attached cavity is often unstable, so the system is set for a state which is characterized by oscillations of pressure and flow phases. This process is greatly enhanced with the proviso that the cavity is closed by hydraulic resistance established downstream of the gas injection site. A characteristic abrupt increase in the frequency of oscillation with a monotonic increase in the flow rate, and the same sharp decrease in frequency at lower flow rates could be identified as special features of the process in some working modes. The authors propose an explanation of the experimentally identified hysteretic nature of development of self-oscillations in the hydraulic lines with a limited artificial gas cavity at the output.

Results of the investigation of a new phenomenon in revolving of a rigid rotor in elastic supports with four pendular self-balancers were presented in this article. The rotor revolves with a working speed but pendulums move with a rotation frequency which is equal to one of two critical rotor’s velocities; these velocities are conditional upon, respectively, linear or angular parameters of motion. An experimental apparatus used for obtaining the dynamical phenomenon was described; data on physical parameters of the system were also provided. A current hypothesis of emergent interactions as the fundamental of creating a mathematical model of the processes of locking was proposed. A procedure of creating a mathematical model for a system with eight degrees of freedom was also proposed. Properties of dynamical links between partial systems which could be set only for stated forms of self-organization of motion of a group of freely suspended pendulums were discussed.

The method of self-consistency was applied to obtaining an estimate of the effective value of thermal conductivity of a composite with ball inclusions. This method was based on averaging of parameters of a perturbed temperature field in elements of a composite structure. The perturbed temperature field in inclusions and ball particles of the composite matrix was found by solving the steady-state heat conduction problem formulated on the basis of a constructed mathematical model of thermal interaction of these elements with a homogeneous material with an unknown coefficient of thermal conductivity. Quantitative analysis of the derived calculated relation was carried out to a sufficiently wide range of the defining parameters to verify reliability of the obtained estimate. The calculation results with respect to this relation are located within the interval between the lower and upper estimates which could be obtained from both a singular approximation of the theory of random functions and the dual variation principle of Hashin - Shtrikman.

Tensor non-linearity as a phenomenon brings together many well-known effects inherent in the specified materials, including grey cast irons study of which can be found in works by M. I. Leonov and his disciples. Processing of test results and description of properties of such media, with a very great difference in limiting deformations, is a complicated problem. In this work it was shown that the best way to solve this problem is to use V.V. Novozhilov’s theory about connection of stress and strain deviators with specified experimental results; these results allowed to improve the method of determination of material functions within the range of all ultimate deformations of simple tests. It was possible to achieve the latter with the use of “extrapolation” of initial data up to the ultimate deformation of compression. The model created by this method reflects the effect of dilatancy in a generalized sense more accurately, revealing the essence of tensor nonlinearity. It was also shown that the accumulated deformation of loosening can be chosen as a criterion of strength.

A radio telescope which is ten meters in diameter was placed on the operational orbit of the artificial Earth satellite on July 18, 2011. This telescope was mounted at spacecraft “Spectr-R” and designed for consistent radio interferometer observations of remote sources together with Earth stations. Success of this scientific project significantly depends on the accuracy of the calculated orbit of the space radio telescope. Astrometric observations are one of many information sources that were used for adjustment of the spacecraft’s orbit. Due to considerable evolution of the orbit observations were taken by means of target indication. Besides the position on the celestial sphere, a predicted value of apparent brightness is also used to identify the object. In case of “Spectr-R” this brightness can vary essentially during the flight. A model for calculation of apparent brightness of the spacecraft “Spectr-R” is presented in this work. Key parameters of this model were obtained from the analysis of telemetric and photometric data.