International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 1288

ISSN 2229-5518

Software and research of fibrous composite materials elastic-plastic deformation

Askhad M. Polatov, Nodira A. Nodirjanova

Abstract— The paper describes the computer simulation of unidirectional fiber composite materials elastic-plastic deformation. Developed set of tools to automate the process of new fiber composite materials and structures designing with predetermined mechanical properties. Provides technology research process of elastic-plastic deformation of materials. Described the structure and functioning of specialized software. Computational experiment was carried out for investigate of materials strain state. Composite materials elastic-plastic analysis results are presented.

Index Terms— computer model, software, computer experiment, fiber composite, elastic-plastic state, toughness.

—————————— ——————————


ow in various areas of equipment and construction the elements of designs made of modern composite materi- als are widely used. The unidirectional fibrous compo-
sites or transversal and isotropic materials are of special inter- est. Automation of design of designs requires development of the settlement models and methods considering anisotropy of material and a configuration. The problem becomes compli- cated if material possesses elastic-plastic properties.
The integrated approach to the solution of the tasks con- nected with definition of rational structure of material is nec- essary for the effective accounting of advantages of construc- tional materials. Collaboration of fiber and a matrix gives the effect equivalent to creation of new material which properties differ from properties of its components. Due to the aforesaid, development of computer modeling of process of deformation of elements of the designs made of composite materials is very actually. The problem gains special importance at an assess- ment of durability of composite elements of designs in such areas as automotive industry, aircraft industry, astronautics, power, mechanical engineering, etc. and promotes develop- ment of researches in the field of the anisotropic theory of plasticity.
Creation of specialized software for research of elastic- plastic processes of deformation of constructional materials represents special relevance and, at the same time, is the theo- retical and applied problem having important economic value. Great scientific interest is represented by development of ef- fective systems of automation of design of new composite ma- terials and an assessment of reliability of the developed ele- ments of designs.
In computing experiments the numerical model by means of which gain new knowledge of the modeled object is used. For granting the convenient interface to the user at the de- scription of real process and carrying out computing experi- ments, it is necessary to develop computing algorithms on the basis of which the program complex is under construction. Carrying out computing experiments gives the chance to au- tomate process of design of composite materials with in ad- vance set mechanical properties, to investigate influence of a volume ratio and mechanical parameters of fiber and a matrix on design durability. Essential feature of technology of com- puter modeling and computing experiment is possibility of
carrying out a series of calculations for determination of nec- essary mechanical and geometrical parameters of the projected composite materials. The demand of researches is character- ized by that for providing strength characteristics and wide- spread introduction of constructional materials in production, it is necessary to automate process of their design.
The technique of computer modeling applied at the solu- tion of applied tasks is defined by creation of the model re- flecting real process and the computing experiment revealing the acceptable process functioning parameters. The essential contribution to development of numerical modeling was made by Samarsky A. A. [1]. The triad "model-algorithm-program" is offered them, the methodology of computing experiment and technology of computer modeling are developed. In paper of Konovalov A. N. and Yanenko N. N., is considered the modular principle of creation of software packages [2]. Algo- rithmic approach to the solution of applied tasks is offered in researches of Kabulov V. K. and his pupils [3]. Such approach differs in high extent of formalization at the solution of a wide class of tasks and automation of process of the decision. Wide use in practice of various composite materials promoted de- velopment of researches in the field of the anisotropic theory of plasticity. For the description of process of elastic-plastic deformation of materials various versions of the theory of plasticity based on a method of averaging at which composite material will be replaced homogeneous anisotropic environ- ment [4-6] are offered. Recently for the solution of problems of deformation of composite materials the method of finite ele- ments [7] is used. For carrying out engineering calculations there is now a number of computing programs and systems to which it is possible to carry software packages of LIRA, FRONT, COSMOS/M, ANSYS, NASTRAN and others.
Despite the achieved success, a problem of development of
computer modeling, computing algorithms and software for
the solution of problems of elastic-plastic deformation of con-
structional materials it is impossible to consider complete. The
insufficient attention is paid to the questions connected with
increase of efficiency of computing algorithms, specialized
software for automation of process of carrying out computing
experiments at design of new composite materials and designs
with in advance set mechanical properties. In an incomplete
measure impact on durability of designs of structural features,

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such as anisotropy of material, the volume content of fiber in materials, and also constructional not uniformity is studied.

σ ij

= σ~(δ

δ i 3δ

j 3 ) + σ


i 3δ j 3 +

Research of the new direction in a solution of the problem of automation of design of constructional materials assumes development and modification of a wide range of methods, algorithms and specialized program complexes for designing


u + u

u qu



of fibrous composites which scientific and methodical bases


where: u , u and

pu , qu - stress and strain tensor intensity

promote development of the existing technologies.
(respectively plane isotropy and isotropy transversal axis):


Pu =

1 P P = 2

2 ij ij 2

(σ 11 - σ 22 )

+ 4σ 2 ,

The elastic-plastic environment which represents the non- uniform continuous material consisting from two components is investigated: the reinforcing elements and a matrix (or bind-

pu =

1 2

pij pij =

2 2

(ε 11 - ε 22 )

+ 4ε 2 ,

ing) which ensures collaboration of the reinforcing elements. It is known that fibrous material and the transversal and iso- tropic environment are equivalent concepts. Thus replacement of the non-uniform environment with the anisotropic envi- ronment adequate to it with effective mechanical parameters

Q = 1 Q Q =

u 2 ij , ij

q = 1 q q =

u 2 ij ij

σ 2 + σ 2 ,

13 23

ε 2 + ε 2 .

13 23

gives the chance to consider inhomogeneity of composite ma-
The stress tensor ratio:
terials. In this regard, at the solution of a problem of elastic- plastic deformation of fibrous composites the theory of small


= σ ij

+ σ~(δ

i 3δ j 3

δ ij

)+ σ


i 3δ j 3


i 3δ j 3


+ σ 3 jδ i 3 ),

elastic-plastic deformations for the transversal and isotropic environment offered the prof. Pobedrya B. E. [8] is applied. Calculation of effective characteristics of fibrous composites is carried out on the basis of the expressions received by asymp- totic a method that gives the chance to consider the radial in- teraction of components caused by distinction of coefficients of Poisson of a matrix and fiber [9, 10]. Elastic-plastic calcula- tion is carried out on the basis of iterative process of a method of elastic decisions of Ilyushin A.A. For creation of the allow- ing system of the equations the method of finite elements in movements is used. The decision of system of the equations is

Qij = σ i 3δ j 3 + σ 3 j δ i 3 − 2σ 33δ i 3δ j 3 , σ = (σ 11 + σ 22 ) 2 .

In disclosing these ratios look like:

P11 = (σ 11 σ 22 ) 2 , P22 = (σ 22 σ 11 ) 2 , P12 = P21 = σ 12 ,

Q13 = Q31 = σ 13 , Q23 = Q32 = σ 23 .

Similarly prescribed ratio of strain tensor:


ij q ij i 3 j 3 33 i 3 j 3 ij ij



carried out by method of square roots, taking into account symmetric and tape structure of a matrix of coefficients.


= ε ij

+ q (δ


i 3δ j 3

δ ij

)+ ε


i 3δ j 3


i 3δ j 3

+ ε 3 j δ i 3 ),

~ = ε + ε

In reinforced composites investigation, when stiffness of


= ε i 3δ 3 j + ε 3 j δ 3i − 2ε 33δ i 3δ j 3 , q

11 22 .

reinforcing elements significantly exceeds the stiffness of bind- ing agents, simplified strain theory of plasticity could be used. It allows applying theory of small elastic-plastic strain for so- lution of specific applied tasks. Simplification is based on as- sumption that the simple stretching in the axis direction of the composite’s transversal isotropy and in perpendicular direc- tion to it, plastic strains do not occur.
For transversely isotropic solids the ratio between the
It is assumed that the transverse isotropy axis coincides with the axis OZ.
Relationship between stresses and strains intensity is repre- sented as:

Pu = 2λ4 (1 − p ( pu )) pu ,

Qu = 2λ5 (1 − χ (qu ))qu ,

stresses and strains is presented in the form of the stress tensor
decomposition on the spherical and deviatory parts:

p ( pu )

and χ (qu )

- function of plasticity which value is

zero at elastic zone.


Askhad M. Polatov, Ph.D., Institute of Mechanics, Tashkent.

For simplified transversely isotropic plasticity theory the rela- tionship between stresses and strains is given by the relations:

Senior lecturer, Department of Mechanical-mathematics, National Univer-

σ~ = (λ

+ λ )~ + λ ε ,

sity of Uzbekistan, Tashkent, Uzbekistan. Tel.: +998903715556 Tashkent.

4 7 q

5 33


σ = λ

~ + λ ε ,

Nodira A. Nodirjanova, Department of Mechanical-mathematical, National

University of Uzbekistan, Tashkent, Uzbekistan.

33 5q

3 33


P = P( p); Q = Q(q),


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P = 

2λ 7 p, if

p < p *


parameters forecasting of the projected fibrous composite ma- terials and durability of designs’ elements.

2 λ 7 p * +2λ7′ ( p p*), if

p > p *

Q = 

2λ 9 q,

if q < q *


4.1 Preprocessor

2 λ 9 q * +2λ9′ (q q*), if

q > q *

The technology of representation of area of a difficult con-

p*, q* - appropriate limit of elastic strain. Introducing the notations:

figuration is developed for automation of process of creation
of a finite element grid by means of association or removal of
"elementary" subareas [13]. Definition according to which the
area is called "elementary" if there is an algorithm of creation

λ' 

p * 

λ' 

q *  ,

of its finite element grid is for this purpose entered. The finite

α = 1 -

7 1 -

, β = 1 -


9 1 - 


element grid of area is described by a set

λ 7 

  λ 9  

expressions for stressed state components in plasticity zone could be written as:

σ 11 = (λ4 + 2λ7 )ε11 + λ2ε 22 + λ3ε 33 λ7α (ε11 ε 22 ),

σ 22 = λ4ε11 + (λ4 + 2λ7 )ε 22 + λ3ε 33 λ7α (ε 22 ε 11 ),

σ 33 = λ5 (ε 11 + ε 22 ) + λ3ε 33 ,

σ 12 = 2λ7 ε12 − 2λ7 (1 − α )ε 12 ,

σ 23 = 2λ9ε 23 − 2λ9 (1 − β )ε 23 ,

Ω = { N, M, MK, MN },

where N – number of knots; M – quantity of finite elements; MK – the massif of coordinates of knots; MN – an array of numbers of knots on finite elements. Further the ratio which allows by means of association or removal of elementary sub- areas is given to form a finite element grid of area of a difficult configuration:

σ 31 = 2λ9ε 31 − 2λ9 (1 − β )ε 31 .

k 1

Ω = I

k 2


Coefficients of λi are associated with mechanical parameters
of a transversely isotropic material in the following relation-
where I


i =1



j =1

i and j

the corresponding elementary subare-


λ3 λ E=(E1(1ν)ν/)l/, lλ, 4λ

=EE(ν(ν +kkνν))//[[((11 +ν ) / ll]],, λ 5 = EEνν//l,l,

as, k1 – number of the subareas which are subject to associa-
Are for this purpose developed: computing algorithms of for-

= − =


λ7 = G = E /[2(1 +ν )], λ9

+ 22 + λ =

= G′, l = 1 −ν − 2ν 2 k , k = E / E′.

tion, k2 – quantity of the deleted subareas.
mation of a finite element grid of elementary subareas; associ- ations and removals of subareas; definitions of the initial front and streamlining of numbers of knots on the basis of the mod- ified frontal method allowing to minimize width of a tape of


As a rule, many modern packages comprise a preprocessor (the program which is carrying out creation of settlement model, preparation of data for further calculations), the pro- cessor (the program which is carrying out calculations) and the postprocessor (the program which is carrying out visuali- zation of calculations) [11, 12]. The purpose of this work is creation of tools for automation of process of design of new fibrous composite materials and designs with in advance set mechanical properties. The automated systems are developed for achievement of this purpose:
– creation of a finite element grid of areas (preprocessor);
– solutions of a problem of elastic-plastic deformation of com-
posites (processor);
– visualization of results of calculation (postprocessor).
Such structure allows making computing experiments on reduction of time and material inputs at design of new com- posite materials and designs, to investigate influence of con- structional features of material on design durability, to submit
recommendations about increase of the bearing ability and to decrease in a material capacity of elements of designs.
The technology of calculation, computing algorithms and specialized software forms, in total, the concept of structural
nonzero coefficients of the allowing system of the equations.

4.2 Processor

Process automation construction and the decision of sys- tem of the allowing equations of a finite element methods and realization of iterative process of a method of elastic decisions of Ilyushin A.A. [14]. Algorithms are developed: creation of coefficients of a matrix of rigidity of finite elements; for- mations of the allowing system of the equations on the basis of the principle of line-by-line preparation of data for each knot separately. It captures the essence of summation and provides creation of tape system of the algebraic equations of a high order taking into account symmetry of its coefficients; deci- sions of system of the equations by the modified method of square roots taking into account symmetric and tape structure of a matrix of coefficients.
As in transformations of a method operation of multiplica-
tion of a matrix by a vector is generally used, the correspond- ing algorithm is developed for a case when only coefficients of a tape of the lower triangular matrix are set. These coefficients settle down in a rectangular matrix of Sij with sizes, where n an order of system of the equations, the l width of a tape of nonzero coefficients, including diagonal elements. And diago-

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nal elements of an initial matrix settle down on the last of l- ohm Sij matrix column. In this case for multiplication of a ma- trix of Sij by a vector of xj the ratio is used (1):

5.1 Finite element model

Automation of process of creation of a finite element grid of a three-dimensional design of a prismatic form of a difficult


yi = si , q xr

j =1

i − 1, 1 ≤ i ≤ l


+ s j ,i +l j x j , (1)

j =i

l + j − i, 1 ≤ i ≤ l
configuration is carried out in the APKEM program module. Visualization of results is carried out with use of opportunities of library of the program OpenGL [15] interface. By means of the index of a mouse on the screen of the monitor the design projection is drawn. Three-dimensional representation of a
p = 
l − 1, else
 j, 1 ≤ i ≤ l
 q = 
,  j, else  ,
 m = i + l − 1, 1 ≤ i ≤ n − l + 1
design is formed by the operation of expression applied to a
projection surface. For work of the user on the screen the working area (fig. 1.a) is formed.
As the beginning of coordinates of working area the point
serves in its top left corner. In the top right corner the toolbar
r =    
i − l + j, else ,
n, else  .
(fig. 1.b) is presented, and in right lower – the actual coordi- nates of the index of a mouse are displayed. The main tools used for generation of finite element representation of a de-
The given ratio allows at realization of a method of square
roots, with use only of coefficients of a tape of the lower trian-
gular matrix and diagonal.

4.3 Postprocessor

The algorithms allowing displaying a picture of the intense deformed condition of the studied object on the monitor screen are developed for visualization of resultant parameters. As values of movements are small in comparison with the de- sign sizes, in algorithm not real movements, but their values multiplied by the correcting k coefficient are used: (u' , v' , w' ) = k (u , v , w ) . This coefficient is selected the user depending on the solved task. To provide evident visual- ization, values of coordinates of knots are also multiplied by the correcting multiplier. Compliance between the parameter and color of filling is defined from the following ratio (2):

(2), (3),

where pmin , pmax – respectively, the minimum and maximum value of parameter, C – the machine-dependent function which is linearly displaying a numerical piece [-1; +1] in space of flowers from dark blue to the dark red. Then process of gradient filling begins. Only thus dependence between values of parameter and color is set by a ratio (3), where n – number of isolines, c1 , …, cn – values of flowers by which section areas will be painted over. The remained area of section will be painted over in the c* color. For convenience of the user color is set in the form of numbers from -100 to +100.


The ARPEK specialized program complex is developed for carrying out computing experiment in the environment of Delphi. The software has modular structure data exchange between modules is carried out through configuration files and files of data.
sign are used as follows: 1–4 - for drawing of border and fixing of basic tops (fig. 2.a); 5–8 - for finite element representation of elementary subareas; 9 (+) - for consecutive association ("sew- ing together") of elementary subareas.

a b

Fig. 1. Workspace of the user on the monitor and a tool kit
The tool 10 (3D) is used to formation of a three- dimensional finite element grid of a design on the basis of its projection. Thus the index of a mouse fixes one of tops which stretches further on the demanded distance. And, at the set distance from an initial surface the parallel trace is formed and three-dimensional representation of a design is formed (fig.
2.b, c). For visualization of a design the tool 11 (GPH) is used. At its activization the window in which left part the general view of a finite element grid of a design is arranged, and in the right part - the tool for comprehensive viewing (fig. 2.d) opens. The tool 12 (MNK) becomes more active for saving of information on finite element representation of a design.

a b

c d

Fig. 2. Stages of formation of a finite element grid

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5.2 Software functioning

Functioning of settlement modules of the ARPEK software is carried out as follows. Further settlement modules of the NERPEK program module are carried out. Calculation of ef- fective mechanical parameters of material is carried out by means of the module EFFECT. Formation of the allowing sys- tem of the equations of MKE is carried out in the RAM10 module. The method of square roots modified for systems of the equations with symmetric and tape structure is applied to the decision of system of the equations. Process of the decision consists of two stages: on the first - the RAM12 module carries out calculations according to algorithm direct, on the second - the RAM13 module - reverse motion of a method of the deci- sion. The vector of nodal movements is as a result formed. In the course of work the RAM11 module carries out calculation of values a component of the intense deformed state who reg- isters in the output file of the module – PARAMS. At the solu- tion of physically nonlinear tasks the PLASN module for spec- ification of the elastic-plastic decision on the basis of iterative process of a method of elastic decisions of A. A. Ilyushin is carried out. Zones of plastic deformations are defined on the basis of Mizes's criterion. At the exit the module writes down resultant values of an elastic-plastic state in the PARAMS file.

5.3 Visulaization

For graphic interpretation of results of calculation the pro- gram module of visualization of TASVIR is used. It allows: to visualize a picture of distribution of values of the intense de- formed state in the set sections; to execute gradient filling with use of red color for positive value of parameter and blue – for negative; to visualize isolines which values are set by the user; to draw diagram of parameters on the borders of section ad- joining axes; to correct and display pictures of deformation of designs; to execute combination of a finite element grid with diagram and gradient filling (only 10 modes of display of one section); to remove all values of the stress-strain state compo- nents for a control point; to keep the received images in the graphic file.
We will describe parameters by means of which task it is possible to receive a picture of distribution of the stress-strain state of the studied design. The file of data includes addresses of files with parameters of a finite element grid and the stress- strain state; total number of final elements and nodal points. The configuration file consists of coefficients of correction of movements and the sizes of a design; values component of movements and tension; intensity of deformations and ten- sion; values of isolines with corresponding values of tempera- ture of color; variable setup of the interface; loading parameter stress-strain state component; type of a grid and design; pa- rameter of a form of display of a design; video card parame- ters.
On an entrance the TASVIR module reads out data of a fi-
nite element grid from the DISKR1.DAT file, nodal values of
the stress-strain state from the PARAMS.TXT file and parame-
ters from the configuration CONF.TXT file. The module of
visualization uses the INIT module for line-by-line analysis of
the configuration file. Color display of distribution of values of the stress-strain state parameters according to inquiries of the user is result of operation of the module.
To remove all values of the stress-strain state components for a control point; to keep the received images in the graphic file.


6.1 Problem One

Problem of elastic-plastic uniaxial tension (Рzz = 300
MPa) of rectangular plate with concentrators in the form of
cracks are being solved. Boron/aluminum is used as fiber
composite. Effective mechanical parameters for bo- ron/aluminum are as follows:

Е=160*103 MPa, μ=0.32, Е'=260*103 MPa, μ'=0.254, G'=

51*103 MPa, G=E/(2*(1+ μ)).

Problem of uniaxial tension plate with a horizontal iso- lated linear crack in the center with length l = 0.1cm is being considered. Intensity values distribution of strains qu and рu is shown in Fig. 3. Maximum intensity values of strains qu are concentrated in the vicinity of the crack tip, but they do not reach ultimate tensile strength (Fig. 3.a). Strain intensity dis- tribution investigation shows that the plasticity zones are con- centrated in crack area (Fig. 3.b).

a b

Fig.3. Intensity values distribution of qu and pu
Further, uniaxial tension problem of plate with horizontal rectilinear cracks (l = 0.05 cm), located on plate’s side edges is considered. Intensity values distribution of strain qu is shown in Fig. 4.a. Distortion of construction sides and disclosure of crack edges are observed under tension. Increased values of qu concentrated in area of crack tips and spread vertically. Elas- tic-plastic strains - pu are formed in the vicinities of the crack tips (Fig. 4.b).

a b

Fig.4. Distribution of qu and pu values
Component values of stress - strain state in crack tips under uniaxial tension of plate with isolated crack and of plate with lateral cracks are shown in Table 1.

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р u

Р u, [MPa]

σ xx, [MPa]

σ zz, [MPa]











6.1 Problem Two

In this paragraph stress reducing process by changing of contour shape with minimal distortion stress are being stud- ied. Improvements in stresses distribution and increased con- structional strength
In the present paper elastic-plastic stress-strain state of fi-
brous boron/aluminum plates is being investigated. It is
stretched uniaxial in direction of fiber. Circular hole was cut off in the center of plate for constructional purposes. Rectan- gular plate’s dimensions are: height - 1 cm, width - 0.5 cm, thickness - 0.1 cm. Boron fiber volume fraction - 35%, hole ra- dius R = 0.05 cm, external loading Pzz = 950 MPa.
Analysis of solution results are given below. As it is known,
distribution of stresses in the plate in isolated holes presence is considerably distorted. Increased stress is observed in vicinity of hole (Fig.5.a). Second hole addition to existing hole is as- sumed (Figs.5.b-d).

а b c d

Fig.5. Distribution of strain intensity values
Computational experiment is carried out for investigating of two vertically positioned holes influence. It turns out that ad- ditional hole also causes stress increase in surrounding area. However, it is known that holes interference reduces overall stress. The values pu for distance between centers of holes h =
0.2 cm decrease in external points to 7.7%, and in internal - to
26.7% (Fig.5.b). It is interesting to note that the elastic problem
values are, accordingly, 6.7% and 32.7%. Increasing stresses in
this case are less than in case with isolated hole (Tables 2-3).



Single stress concentrator is formed by two vertically ar- ranged holes. With distance increasing of holes from each oth- er at h = 0.3 and 0.4 cm their interference disappears (Figs.5.c- d).
This phenomenon could be explained using the power flow
idea. External forces create a flow that spreads along the con-
struction. Pressure line (power flow) is rejected by the second
hole. Influence of hole after rejection of passing power flow cannot grow anymore.
Stress components τzx values distribution is given in Fig.6. Maximum τzx values concentrated on holes’ sides at an angle from π/6 to π/4 relative to the horizontal diametric section. In the vicinities of isolated holes τzx has negative values (Fig.6.a). Two vertical holes form both negative and positive zones (Figs.6.b-c).

a b c

Fig.6. Stress components values τzx distribution


The developed in work computer model, computing algo- rithms and a specialized program complex of the solution of problems of elastic-plastic deformation of constructional materi- als allow to automate process of design of earlier inaccessible, essentially new elements of designs with in advance set mechani- cal properties and a configuration. The computing algorithm and the program module of creation of a finite element grid of designs are developed. The projection of a three-dimensional design is drawn by means of the index of a mouse on the computer moni- tor. Visualization of results is carried out by means of use of op- portunities of library of the program OpenGL interface. The computing algorithm and the program module of visualization of results of calculation allowing to carry out are developed: gradi- ent filling; visualization of isolines which values are set the user; creation of an diagram of values of parameters on section bor- ders; combination of a finite element grid with schedules and

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gradient filling; a conclusion of values of resultants a component in a control point. Carrying out computing experiment allowed to investigate the deformed condition of the unidirectional compo- site material D16 (boron aluminum) and to confirm the regulari- ties connected with influence of the volume content of fiber in a composite.


The author is grateful to the leadership of National University of Uzbekistan for the rendered all-round material and moral help and support by development of algorithm and software on the basis of which the results are received, which analysis is resulted in the given job. Results of the paper were checked and have confirmation in Institute of hydrodynamics (prof. E.V. Karpov, Russia, Novosibirsk) and Kharkov aviation insti- tute (prof. A.N. Shupikov, Ukraine, Kharkov).


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