### №4-2019

Full versions of the articles

https://elibrary.ru/contents.asp?id=41670550

**Khokhlov Andrew V.**

## On the ability of the physically linear viscoelasticity theory to simulate sigmoid shape of stress dependence on strain rate

We study analytically properties of the stress-strain curves family generated by the Boltzmann-Volterra linear viscoelasticity constitutive equation with an arbitrary relaxation modulus under uni-axial loadings at constant strain rates. It is proved (for any decreasing relaxation modulus) that every stress-strain curve is an increasing and convex-up function of strain (without any extremum or inflection points), that stress-strain curves rise up as stress rate increase and that the stress-strain curves family converges to limit curve as stress rate tends to zero or to infinity (i.e. to equilibrium and the instantaneous stress-strain curve). We derived and analyzed the general expression for strain rate sensitivity index of stress-strain curves as the function of strain and strain rate. We found out that the strain rate sensitivity index depends only on the single argument that is the ratio of strain to strain rate. So defined function of one real variable is termed “the strain rate sensitivity function” and it may be regarded as a material function. The explicit integral expression for relaxation modulus via the strain rate sensitivity function is derived. It enables one to restore relaxation modulus assuming a strain rate sensitivity function is given. We proved that the strain rate sensitivity value is confined in the interval from zero to unity (the upper bound of strain rate sensitivity for pseudoplastic media) whatever strain and strain rate are. We found out that the linear theory can reproduce increasing or decreasing or non-monotone dependences of strain rate sensitivity on strain rate (for any fixed strain) and it can provide existence of local maximum. The analysis carried out let us to conclude that the linear viscoelasticity theory (supplied with common relaxation function which are non-exotic from any point of view) is able to produce high values of strain rate sensitivity index close to unity and to provide existence of the strain rate sensitivity index maximum with respect to strain rate. Thus, it is able to simulate qualitatively existence of flexure point on log-log graph of stress dependence on strain rate and its sigmoid shape which is one of the most distinctive features of superplastic deformation regime observed in numerous materials tests.

Pages: 453-472 Elibrary

**Krasnov L.L., Muiznieks K.I., Plume E.Z., Zhigun V.I.**

## Simple and reliable methods for determining the shear moduli of structural materials

The method of an experimental determination of shear moduli of composites accepted as standards ASTM D5379/D5379M-12 and GOST R799-2015 has been considered. Its essential failings have been noted. It has been shown that a question of a correct selection of a method for an estimation of shear moduli of constructional materials, which can allow to determine their values reliably and with acceptable for practice accuracy is not solved until the present. Some other known methods of determination of shear modulus in the plain of the plate have been examined, in particular: a method of testing of tensile or compression of the three patterns carved from an orthotropic material in the directions to the main axis of its elastic symmetry and at an angle 45 to them; quadratic plates torsion test method carried out according to three-point loading scheme; and a method of testing of prismatic patterns on the four-point bending. The detailed analysis of borders of their acceptability and validity of values of determined characteristic has been realized using a wide range of constructional materials. Their worthiness and weaknesses have been indicated. It has been shown that during the usage of the first mentioned method, there can be a variant of determination of the shear modulus, which can be based on the values of and , which greatly simplifies the process of finding it. An easy device construction for the implementing the second the mentioned method has been offered; it helps make step loading and evaluation of deflections. Its usage allows obtain stable and reproduced characteristics without exploitation of any additional equipment. The modernization of the equipment used for fulfilling the third reviewed method has been made. It considerably simplified the precise setting of a pattern and gave an opportunity to realize an examination of a slim (1,5 mm and more) patterns without losing their stability in the transverse direction. The comparison of shear modulus determined by three different methods on the same constructive materials has been fulfilled. Their quite good coherence in all reviewed situations has been identified.

Pages: 473-491 Elibrary

**Erofeev V.I., Leonteva A.V., Shekoyan A.V.**

## Propagation of plane longitudinal waves in a material with point defects

In this paper, we study the propagation of plane longitudinal waves in an infinite medium with point defects located in an nonstationary inhomogeneous temperature field. The problem is considered in a self-consistent formulation, taking into account both the influence of the acoustic wave on the formation and movement of defects, and the influence of defects on the propagation features of the acoustic wave.It is shown that in the absence of heat diffusion, the system of equations reduces to a nonlinear evolution equation, which is a generalization of the Korteweg – de Vries – Burgers equation. By the method of truncated decompositions, an exact solution of the evolution equation in the form of a stationary shock wave with a monotonic decrease has been found. It is noted that dissipative effects due to the presence of defects prevail over the dispersion associated with the migration of defects in the medium. The influence of the initial temperature and type of defects on the main parameters of a stationary wave is studied: velocity, amplitude and front width. Nonlinear waves propagate faster in media with vacancies than in media with interstitials. An increase in the initial temperature leads to an increase in the velocity of the stationary wave if the defects are interstices and to a decrease if defects are vacancies. For harmonic waves, it is shown that the presence of defects in the medium promotes the appearance of frequency-dependent dissipation and dispersion. At low frequencies close to zero, wave attenuation is practically absent, and they propagate at a constant speed close to unity, which does not depend on the type of defects or on their presence. At high frequencies, the waves also propagate at a constant speed, which depends on the type of defects. Harmonic waves have a greater length and speed in media with interstitial than in media with vacancies. The influence of the diffusion parameter on the propagation of a harmonic wave is investigated.

Pages: 492-508 Elibrary

**Arkhipov M.Y., Filina E.K., Golubev E.S., Lyakhovec A.O., Smirnov A.V.**

## The effect of the variation in physical and mechanical cfrp properties on the primary mirror performance of the millimetron space observatory

High-modulus carbon fiber reinforced plastic (CFRP) is one of the most widely used materials for manufacturing of precision dimensionally stable structures for the space telescope design. For instance, it is used for the design of the primary mirror panels of the Millimetron space observatory. To ensure unique characteristics of the primary mirror, extremely high requirements for the accuracy of the reflecting surface of the panels, as well as for their thermal stability in operational conditions at ultra-low temperatures (up to 4.5 K) are set. It is possible to meet these requirements due to the fact that the selected high-modulus CFRP is characterized by a combination of low coefficient of linear thermal expansion with high specific stiffness and low density. However, despite the unique physical, mechanical and thermal properties of the material, there are number of factors caused by both CFRP nature and technological features that can significantly decrease the accuracy of the reflecting surface and the thermal stability of the primary mirror panels. The variation in physical and mechanical properties of the ply is particularly distinguished among these factors. The paper presents a study of the effect of variation in physical and mechanical properties of the ply on the main parameters of the thermal stability of the primary mirror panels (root-mean square of the reflecting surface and the focal length) of the Millimetron space observatory carried out on the basis of numerical engineering analysis via the finite element method. The panel design is described, the methods of the development and verification of its mathematical model, the features of postprocessing the results of temperature deformations analysis are presented. The results of the numerical analysis allow to conclude that the primary mirror panels of the Millimetron space observatory have a high thermal stability and the requirements for the accuracy of the reflecting surface of the panels under operational conditions are met.

Pages: 509-521 Elibrary

**Rekov A.M., Vichuzhanin D.I.**

## Strain distribution in the neck of a flat sample of titanium vt1-00 under uniaxial tension

The components of linear and shear elastic-plastic deformations in the symmetric neck of a flat sample under uniaxial tension are determined by the digital image correlation method. The material of the specimen is the titanium grade VT1-00 with hexagonal tightly Packed crystal lattice ( phase). The specimens were extracted from sheet metal that had undergone recrystallization annealing in vacuum. The structure of the surface layer was determined by the standard method. For this material, the main feature of the mesostructure of the first level (polycrystal grain) is the presence of a large number of annealing twins. The proportion of grains with is approximately twenty to thirty percent for the entire grain array. Modern high-precision equipment, such as Instron 8801 testing machine and Strain Master digital optic system are used in experiments. The fields of displacement vectors of the sample surface areas in the neck region are determined. Irregularities of relief arising on the surface of the sample during the tension testing were taken as reference targets. In each moment of time, the linear local mesodeformations of the first kind and the value of distortion (at the level of the grain of polycrystalline specimen) was determined by using the field of the components of the displacement vectors. The deformations is calculated on virtual lines parallel to the specimen axis. The distance between the lines is taken to be half a millimeter. An approximate evaluation of the deformation value in the direction perpendicular to the specimen plane is performed by means of the incompressibility condition. The kinetics of longitudinal and transverse deformations is determined along with the tensile axis of the sample. The coefficients of mutual correlation between the parameters of the strain state are calculated. The distribution densities of linear strain and distortion tensor components are determined. The density distribution is used to determine the probability of occurrence of large values of deformations in the neck area. These probabilities are equal to the ratio of the number of grains of a polycrystal with a deformation greater than some fixed value to the total number of grains in the neck area. The work was carried out within the framework of the State Assignment of the Institute of Mechanical Engineering of the Ural branch of the Russian Academy of Sciences.

Pages: 522-530 Elibrary

**Baimova J.A., Safina L.R.**

## Deformation behavior of the graphen-nickel composite obtained at final temperatures

Composites, obtained by combining two or more materials often exhibit a number of unique properties. For last decades, composites based on carbon structures and metal nanoparticles are of great interest. In this work, the deformation behavior of the graphene-nickel composite at temperatures close to 0 K and elevated temperatures was studied by the molecular dynamics simulation. Modeling was carried out using a simple pairwise interatomic Morse potential. To obtain a composite based on crumpled graphene and nickel nanoparticles, hydrostatic compression is used at temperatures from 0 to 2000 K. In order to evaluate the strength of the obtained material, hydrostatic tension is applied to the structure. It is shown that hydrostatic compression at a temperature close to 0 K does not lead to the formation of a composite. Chemical bonds do not form between neighboring graphene flakes, and after stretching the resulting material returns to its original state. The deformation of the structure at 2000 K contributes to the formation of covalent bonds between adjacent structural elements and the formation of a single composite structure.

Pages: 531-542 Elibrary

**Gaganova N.V.**

## Shape memory alloys constitutive model extention for considering development of the martensitic elements during phase and structural transformations

The article considers a constitutive model for shape memory alloys (SMA) which allows to take into account the differences between phase and structural transformation. The model reflects the fact that hardening effect is typical for structural transformation, but not for phase transformation. Deformation due to structural transformation is described with the use of loading surface by analogue of the plasticity theory with isotropic hardening. The model describes both phase and structural mechanisms of inelastic deformation, and influence of the first mechanism on the second. The deformed state is determined by one parameter, which can be changed by phase or structural deformation. Inelastic deformation due to structural transformation in the active process is subject to the associated flow rule. The differential condition for the active process and structural deformation is formulated. Tensor increment of the structural deformation is required to be codirectional with the external normal to the loading surface, hardening parameter associated with the structural transformation correspondingly is required to be positive. The article considers the extension of this model to the constitutive equations that allow to take into account development of martensitic elements during phase and structural transformation. It was shown that the model allowed to describe cross-hardening and oriented transformation. Several cases of proportional loading for increasing, decreasing and constant stresses were considered. Deformation plots and loading surface plots were provided for each case. The results for the different material functions determining development of the martensitic elements were compared. If analytical solution was not possible, the Runge-Kutta method was used to solve differential equations for deformation depending on the volume fraction of martensite. It was shown that in the case of constant or linearly increasing stresses, the results for the considered material functions coincided. In the case of decreasing stresses, the deformation values at the end of the process were higher for the material functions taking into account development of the martensitic elements.

Pages: 543-562 Elibrary

**Saganov E.B.**

## Solution of the problem of a thick-walled cylinder under pressure from shape meory taking into account of tension-compression asymetry

The work is devoted to numerical modeling of the process of еxpansion and compression of a thick-walled cylindrical shell made of a shape memory alloy (SMA) in the martensitic inelasticity mode. The processes of expansion and compression occur under the influence of monotonically increasing internal or external pressures. The problem is considered within the framework of the model of nonlinear deformation of SMA in phase and structural transformations. The resulting solution takes into account tension-compression asymmetry of these alloys. The solution is obtained for cases of plane-stress and plane-strain states. Numerical modeling was performed in the Simulia Abaqus using user material technology. 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 distribution and compression of a thick-walled cylindrical shell was simulated in a three-dimensionally space-based formulation, taking into account the symmetry of the problem. In the framework of the work, stress diagrams were obtained over the shell cross section for various pressure values. It was established that during loading the stresses across the cross section change nonmonotonically, and the stress distribution itself has a nonlinear dependence on the radius. For both cases of pressure, the parameter of the type of stress state has an inhomogeneous distribution over the cross section of the shell. The dependence of the displacements of points on the inner and outer shell surfaces from the value of the applied pressure is also obtained. The results obtained during the work can be successfully used in the design of thermomechanical couplings from SMA.

Pages: 563-573 Elibrary

**Tikhomirova K.A.**

## Comparison of methods for calculating phase and structural transformation strains in sma structures on example of a beam bending problem

In this article, three methods for calculating the phase and structural transformation strains of shape memory alloy structures are considered and their performance is compared using as an example the problem of cantilever beam deflection. All methods are based on the hypothesis of equivalence of phase and structural transformation strains as regards the further deformation behavior of a material at the macroscopic level. Phase deformation is understood to be the deformation of oriented martensite caused by cooling the austenitic phase under load, and structural deformation is the deformation of oriented martensite caused by the isothermal reorientation of chaotic martensite. The first method involves the construction of three-dimensional constitutive equations of continuous medium mechanics and their implementation by the finite element method. The constitutive equations include two material functions: the direct transformation and martensitic inelasticity diagrams – the dependence of phase and structural transformation strains on the stress responsible for their initiation. The second method is applicable to structures whose deformation is characterized by one kinematic and one force parameter: for example, beam deflection under the action of the applied force. This method uses the structural diagrams of direct transformation and martensitic inelasticity – the dependences of phase and structural components of the kinematic parameter on the force parameter, which allows making calculations in the one-dimensional formulation. The third method can be applied only for calculating the bending of beams and plates. It involves breaking the beam into several layers, each of them experiencing only normal forces, while transverse and shear forces being neglected. The phase and structural transformation strain in each layer is calculated using the constitutive equations of the first method, but in the one-dimensional case. In the context of the beam bending problem, it has been shown that all methods give similar results. The computational efficiency of each method is estimated.

Pages: 574-594 Elibrary

Zhavoronok Sergey I.

## On the use of various equations of the n order plate theory in problems of normal wave dispersion in an elastic layer

The dispersion of normal waves in a plane elastic layer is considered. The approximate solution of this problem is obtained on the background of various formulations of the quasi-3D plate theory of N order. The plate model is based on the Lagrangian formalism of analytical dynamics of constrained continuum systems; it is defined within the configuration space with the set of field variables, the density of Lagrangian, and the constraint equations following from the boundary conditions shifted from the faces onto the base plane. The general variational formulation of the extended theory of heterogeneous anisotropic plates allows one to satisfy the boundary conditions exactly, at the same time it is covariant and allows one to use different base functions such as orthogonal polynomials or finite functions corresponding to the finite element discretization of a plate across its thickness. The equations of dynamics for an isotropic transversally heterogeneous plate are derived by the Lagrange multiplier method, and the dynamic equation with eliminated multipliers are considered; these equations are analogous to the Voronets equations in the analytical dynamics of constrained discrete systems. It is shown that the dispersion problem based on the extended plate theory leads to a singular generalized eigenvalue problem. The locking frequencies for propagating modes are computed, and the solutions based on the extended plate theory and the constraint-free one are computed; it is shown that accounting for the constraints allows one to reduce the locking effect. The solutions given by the elementary theory based on the Legendre polynomials and on the piecewise linear finite element basis (e. g. the spectral element solution) are compared; it is shown that the solution based on the orthogonal polynomials leads to faster convergence to the exact solution of Rayleigh-Lamb as compared with the spectral element solution using linear shape functions.

Pages: 595-613 Elibrary