by theme:

"Mathematical model developing to stain-damage state of rubberclothering bands for mines' conveyors using information about it's macrodamages by punching"

INTRODUCTION

        The purpose of work development of mathematical model of an intense - tapes deformation condition of mine conveyors in view of macrodamages of their designs by breakdown.
         In the given scientific work the basic mathematical dependence of an intense - deformation condition of tapes with rubber and cloth has been removed in view of the stretching pressure arising in a tape at shift, and also dependence between length of the crack arising in a tape and a pressure about a crack, and as in the speed of distribution of a crack in it is established. The formula showing time of destruction of a tape is deduced, at various variants of breakdown of its design. Also in the given work mathematical models which take into account a various condition of a conveyor tape during its operation are developed. With use of the practical data of tapes destruction have been developed a theoretical quality monitoring behind processes of formation and development of explosive efforts in the damaged tapes that promotes increase in service life of a tape approximately on (10-15) %. In the given work also are investigated, with use of the developed mathematical models, the newest tapes with rubber and cloth of type 1.2, 1.2М, 1.2Ш, 1.2ШМ, 1ШТС on developments and formations of dangerous concentration of breakdowns in the given samples in view of wear process of those during their operation on mine conveyors.

1.THE REVIEW OF A QUESTION CONDITION AND STATEMENT OF THE WORK PURPOSE

                              1.1 Introductory remarks

        In connection with wide conveyors using the needs of conveyor tapes continuously grows, but because of low service life of tapes much of time spent for their replacement and the effect from using conveyors in mines or shafts definitely reduced. One of decrease factor in service life of tapes are punctures and breakdowns of conveyor tapes which on (10-20) % reduce service life of them. Models and ways of the decision about this question do not exist yet in practice though there is a certain quantity of theories about development of cracks in firm materials and plastic, but under the contents those researches do not have any answers in such homogeneous environments which tape of rubber and cloth is.

                              1.2 Theme actuality

        Scientific bases of a perfection theory of designs and calculation methods for tapes' conveyors were given in many works by researchers such as ph.V.J.Barabanov, ph.V.S.Volotnovskiy, ph.E.N.Vysochin, V.A.Zuev, ph.D.S.Monastyrskiy, G.V.Prisjazan, ph.L.N.Eppel, P.V.Jakovlev, doc.I.G.Shtokman and many other scientists as in our country and abroad.
        The organizations that work under the given question are: the Zaporozhye national technical university, the Donetsk national technical university, th open company " College of mines " the Dnepropetrovsk city, open society " RТG-RUBBER ", combine " Uralasbet ", the Ural polytechnical university, etc.
        All from the above-named scientists and the organizations mentioned questions of destruction of conveyor tapes because an accumulation of concentration of breakdowns in those while in service. So, for example, in works ph. I.G.Shtokman with all completeness research opportunities proves processes of increase of breakdowns and the reasons caused those; in works by ph.L.I.Eppelja describes processes of wear process the worker, non-working linings, boards tapes during its operation that conducts to reduction of a tape section on its width; in works by ph.V.N.Poturaeva describes developments of a crack in homogeneous environment, but only by the example of thin plates from rubber, that the full is not capable to describe development of a crack in a tape which rubber makes the certain interest in its structural share; in works by ph.D.S.Monastyrskii with all completeness describes pressure working in tapes during their operation, - but in one of works not was from the mathematical side process of occurrence and development of cracks in tapes from those damages by breakdown is described.
        Directions on research of a tape destruction at its local damage by breakdown, can lead to increase of reliability of a tape during its operation, and also to decrease in expenses for repair and replacement of a tape. Not study of the given theme results in failure of work of conveyors with the greatest possible concentration of damages by breakdown, to destruction of a homogeneous environment of a conveyor tape about the arisen breakdown and to premature replacement for all of tapes.
        On the basis of above-stated it is possible to draw a conclusion, that the theme of the given work is actual and conceptually new.

                              1.3 The purpose and a problem of work

        The purpose of the given work is development of mathematical model of an intense - deformation tapes condition for mine conveyors in view of damages of its design by breakdown.
        For achievement of the above-stated purpose the following primary goals are:

• research, use and comparison of all theoretical knowledge available on the given case in point;

• development of mathematical model it is intense - deformed tapes conditions for mine conveyors;

• analysis of development of dynamic processes in tapes at their breakdown;

• revealing laws of accumulation of pressure at breakdown of a tape during its operation;

• forecasting time of destruction of a tape depending on the form of breakdown, a place of its occurrence and a cycle tapes pressure;

• use in mathematical models of an intense - deformation tapes condition of operational factors which take into account term of operation, processes working and non-working linings, processes by go down boards of a tape, the tendency of decrease in time of resistance of tapes to breakdowns;

• research of influence of occurrence bit from height of falling of a material on a conveyor tape at passage of loading items;

• development of rational methods of calculation of operation of conveyors with the purpose of increase of an overall performance of conveyor transport of the mining enterprises.

2.BASES OF DESTRUCTION THEORIES FOR RUBBER

                              2.1 The basic physicomechanical characteristics of rubber

        In a basis of elasticity of rubber feature of molecular structure, characteristic for the majority of polymers lays. Modern sights at structure of polymers show, that their molecules represent the long flexible strings located chaotically as the confusing ball. Such form of a molecule causes flexibility of molecular circuits and highly elastic properties of materials. Macromolecules of polymer develop as the separate packs located chaotically at amorphous structure and are ordered at crystal. Such representation about " under molecular structure " rubbers is well coordinated with the experimental data received at studying of their viscoelastic properties.
        If the certain force elastic deformation will be observed is enclosed to polymeric system until pressure arising in a material will not counterbalance working forces. However full balance is not established and eventually at the fixed size of deformation of a pressure are reduced. This phenomenon of spontaneous change of a pressure at constant deformation refers to as a relaxation and underlies such phenomena, as creep, next and intrinsic losses of energy in a material. All these phenomena unite under the general name of " elastic imperfections " real firm body.
        For the description of an intense condition izotrop materials to which concerns also rubbers, the theory of elasticity assumes existence of two independent elastic constants of a pressures, expressed by the force, the area of the cross-section deformed section falling unit , elastic deformation, length of a sample determined by increase l to its initial length.
        In view of highly elastic properties of mechanical characteristics of rubber can be classified as equilibrium, having a place at the established, standard condition, and kinetic, concerning directly to action the relaxation processes.
        Believe, that at equilibrium modes of deformation during action of force in rubber the basic have time to come at the end of relaxation processes. In this case the equilibrium pressure is expressed through equilibrium deformation where Е- the equilibrium module of elasticity which size does not depend on a kind of deformation and has one or too value at compression and a stretching of samples.
        In practice equilibrium the module defines or at small speeds pressure (0,0002N/s), or at significant (some hours) endurance under loading when relaxation processes have time to come at the end and do not render appreciable influence on its size.
        Besides equilibrium values use as well their instant sizes corresponding to absolutely elastic condition of materials:

        Real sizes Е0 and G0 G0 for concrete marks of rubber find at enough high speeds presure (shock loading, a blast wave, etc.) or at low temperatures, using a principle of temperature-time superposition.
        At calculation of rubber systems enter also concept of the volumetric module:

К=Р/(V'/V),

Between modules Junga, shift and volumetric there is a connection of type:

Е=2G(1+m);
Е=3К(1-2m),

where m - puasson's factor determined by the attitude relative.
        The factor calculated thus m дThe factor calculated thus µ for rubber does not remain to constants, increasing at compression and decreasing at a stretching. Therefore use the differential form of record in view of change of volume, applying for this purpose known equation by Puasson of a kind. Structural changes at repeated loadings are most precisely shown in strongly filled rubbers of which existence of two types of a hysteresis is typical: molecular and from filling.
        Structural changes at repeated loadings are most precisely shown in strongly filled rubbers of which existence of two types of a hysteresis is typical: molecular and from filling. This hysteresis depends on temperature and speed of the deformation appendix loadings. At small speeds has time to take place molecular regroup and the material shows typical highly elastic properties that is expressed in frequency and temperature dependences strong characteristics. At high speeds recombination of molecules have not time to take place and rubber behaves as an elastic body. The molecular hysteresis is most full shown in unfilled rubbers that is more considered, and rubber fillings show poorly molecular hysteresis, the basic hysteresis is the hysteresis caused by destruction and recombination carbon connections between particles structure inside [2].

                              2.2 Calculation of long durability of tape rubber linings

        A plenty of works is devoted to calculation of long durability RTP. We shall consider one direction of calculation which can be counted rather perspective. Methods of the theory of elasticity is usually limited a pressure to a time condition of a firm body and does not mention its molecular structure which renders the big influence on stongly properties of a considered body. Therefore the most correct should be counted such settlement model which would take into account a complex of constructive and physicomechanical properties of object. The similar model has been created in A.I.Chudkovskogo's works [3,4,5]. The approach advanced by it concerns construction of the statistical theory of destruction of macrobodies in view of external influence of character, properties of a material and the constructive sizes and the form. According to this approach the firm body is considered as statistical ensemble of material points, each of represents some thermodynamic system, possessing all properties of a real body. For such a case the problem of the description of the phenomenon of destruction for a firm body is reduced to the following:

• the problem of local destruction of a body in the beginning is solved, the description of which mechanical behaviour is carried out with the help of the analysis of origin processes and development of various microdefects in a material with the set structure and properties;

• then the problem of macrodestruction of a body is solved on the basis of the analysis of behaviour of static ensemble of material points which properties are defined by the decision of the first problem or are known from experience.

        Thermodynamic descriptions of local destruction does not demand detailed knowledge of molecular mechanisms of the phenomenon of destruction. In a basis of such thermodynamic representations the marked M.Bornom analogy between the phenomenon of destruction of bodies and fusion of crystals lays. Basing on experimental data, we receive a hypothesis that effort of local destruction which is characterized by the certain change of group of tolerance Н, is achievement of density entropy some critical level S dependent from Н':

S(t*)=S*Н',

       This effort of destruction is represented also as:

wher Sе, Si- accordingly speed of an external stream entropy, connected to an exchange of energy and substance with an environment and speeds of generation entropy inside system; S - - a difference of a critical level of density entropy and density entropy in an initial condition.
        This condition of destruction allows to describe the initial destruction of a body caused by action of various mechanical-chemical factors, including chemical reactions, radiating damage, etc.
        Basin on concept of local destruction, the theory of statistical macrodestruction is created, believing, that position of points mainly random factors and therefore the surface of destruction Д is deformed by casual image. Thus the probability of local destruction is represented as

where С - a forming constant;
       V- the charactristic volume of a real material, which effective properties define properties of a point of a continuous body.
        The probability of that local destruction will take place in all points of a surface and symbolically enters the name as Р=Р(0). For calculation of destruction probability for a body on some surface the approximate method according to which as a condition of destruction of a sample formation on one of possible surfaces area of destruction Д* of the critical sizes and forms is accepted, i.e. a critical zone of destruction was used. Thus probability of that local destruction to be carried out in all points of zone Д* is defined by a condition:

        In case surface Д* degenerates in a point, this condition of destruction passes in Vejb's condition for ideally fragile kind of destruction. An equilibrium crack critical are also long is a special case of a critical zone.
        The mathematical model offered of macrodestruction of a firm body for today is difficult for realizing for the decision of specific targets in view of complexity of the device and insufficiency of the experimental information.
        Experimental mechanization of destruction of rubbers with the help of the microstructural analysis shows studying, that process of destruction begins from the very beginning of using, i.e. long before occurrence of the first seen macrocracks and is connected to origin and growth of microdefects.
        Filling of the last results system in a unstable condition and process of destruction comes to the end with macrobreak of a sample. Course of last stage is short-term and consequently the durability of products from polymers practically is completely defined by duration of a stage of accumulation of microdefects.
        For an illustration told we shall consider process of origin and development of the centers of destruction in firm polymers on example СН2 - communications(connections). The primary radical - СН2 - СН2 enters interaction with the next molecule and damages it. The damaged macromolecule is broken off, the new radical which at first breaks off hydrogen from the next macromolecule, etc. (pic.2.1, and, c) is formed. In result in a place of origin of a primary radical there is a destroyed microarea (pic. 2.1, d). Such local centers of destruction can be considered as local macrocracks. For the further researches it is obviously necessary to emphasize that fact, that as a result of detailed studying formation of submicrocrack communication between the sizes both their concentration and break of polymer is established. Submicrocrack arises at once the certain size which remains practically constant down to break of a sample. At the same time concentration continuously accrues, reaching some limiting size at which there is a macrobreak.

Picture 2.1-Circuit of submicrocrack formation

        This process of destruction previous to formation of a macrocrack, it is possible to characterize as accumulation of microdefects, absent-minded on all volume of a deformable body. Therefore at the description of such process quite natural the approach and consecutive application of methods of mechanics of the continuous environment is represented.
        At the description of many mechanical and physical phenomena the continuous environment is the basic mathematical model of a real body. From the geometrical point of view the continuous environment is smooth variety, i.e. some coherent topological space, a vicinity of which each point can geometricly be displayed on Evclid's three-dimensional space. On such variety fields density of weight S, temperatures, pressure tenzo Тб, deformation tenzo Тe which connects the metrics of variety during the set moment with the metrics of a "initial" condition, etc., all those parameters which are described completely with a thermodynamic condition of a macrocell of a researched real body. And a full set of parameters of a condition will be coordinates of space which forms every possible "instant" thermodynamic conditions of this element. The choice of parameters of a condition is in essence the first and an important point at construction of models of real firm ph. Assumptions at research:

• the researched material of a sample in an initial condition is accepted homogeneous, isotrop and capable to irreversible deformations. The considered macrosample will represent set of material points, each of which is allocated some properties of an initial body;

• process of accumulation and development of macrodefects can be interpreted as formation in originally homogeneous and psotrop material of inclusions with new, generally anisotropic properties. It is possible to motivate the same approach that in the phenomenon of destruction frequently the important role is played with dot defects (areas containing time, microcracks, etc.). Therefore, for example, the elastic pliability of inclusions should be not less an elastic pliability of the basic material;

• in a material anisotropic inclusions of various orientation, but with identical, constant elastic properties develop. Volumetric concentration of such inclusions grows in process of development of process of destruction.

        The "characteristic" volume, following Hilla [6], is understood as volume of compose a material, which contains sufficient number of inclusions to consider its macroscopical and homogeneous. The concept "macroscopical" homogeneous volume can be defined as follows - if on a surface of such volume to set loadings or displacement which in a homogeneous material (a material without inclusions) formed the homogeneous intense and deformed condition, length of a wave fluctuation fields of tenzo pressure and deformations tenzo it should unused be small in comparison with the linear sizes of this volume.
        Concerning some fixed system of orthogonal axes in case of full anisotropic, elastic properties it is possible to set orientation of separate inclusions Euler's by three corners.
        As the characteristic damages function S(t) which sets concentration of inclusions of various orientations in characteristic volume is considered.
        General concentration of damages at the moment of time t:

where p0 - general concentration of microdamages to an initial condition;
       p(t)- the certain value of microdamages concentration at the moment of time t.
        Critical damage level which achievement under the set external conditions corresponds to transition of system in mobile side:

where Е-the module of elasticity of the initial not deformed material, for rubber Е = (0,8-2) (kg / mm side) in tapes with rubber and cloth [7].

3.PRESSURE IN TAPES WITH RUBBER AND CLOTH. THEIR MATHEMATICAL MODELLING

                              3.1 A problem flat of the theory of a tape elasticity for conveyor

        We determine efforts and deformations of a multilayered material with the purpose of search of ways of improvement of operational characteristics of tapes. A power element of a tape is the fabric skeleton consisting of lines of layers of the rubberized fabric (picture 3.1). Here the fabric is urming material, and rubber - binding. An arrangement of strings of a fabric with rubberized layers rather difficultly. However it is possible to distinguish limiting type of structure in which the fabric and rubber form as though two parallel a plate. The element cut out from the rubberized lining and having the sizes х, y, z, is shown in picture 3.2.

Picture 3.1- Circuit of a sample tape

        At the decision of a problem it is accepted the following assumptions[9]:

• we consider a fabric as unline an elastic anisotropic material, and rubber - as linearly elastic izotrop element;

• we count, that in case of a flat problem components pressure tenzo zz, хz, yz and components deformation tenzo zz, хz, yz addresses in a zero;

• neglecting local effects, pressure and deformations in a fabric and rubber it is accepted to constants on section and slowly varying on length;

• fabric skeleton as continuous homogeneous body is considered;

• the effects connected to interlacings of a fabric, neglect.

Picture 3.2-Element of the rubberized lining with a designation of working pressure on its surface

        Following the general theory of environments reinforcing , we shall supply the parameters concerning to a fabric, an index "T", and the parameters concerning to rubber, an index "P". Then average pressure in a skeleton:

where- factor of reinforcing;
       ft- the area of a fabric in section уz;
       fр- the area of rubber in the same section.

        For strains the following expressions are received:

where, - tape's modals ,.

         The equivalent effort in a tape skeleton can be found

where -Puasson's factors.
        In a tape also operates shifting efforts which reach the maximal value on a drum of the conveyor at its rounding. Shift efforts are defined under the formula:

where- it is possible to take on critical durability of a tape;
       В - width of a tape, sm;
       р - explosive efforts of width of a lining of 1 sm, кN/sm;
       m - factor of safety factor [17].
        It is received, that

                              3.2 Kinematical distributions of cracks to rubbers

        Aspects of distribution of cracks were in detail considered for thin polymeric flats [6]. Here these materials are resulted for definition of settlement factors and revealing of the general laws of rubbers destruction and tapes from intensive mechanical influence.
        Research of growth of cracks were spent on samples from rubber on a basis rubber and natural rubber. Tests have shown, that development of a crack in the beginning occurs very slowly, in process of increase in length of a crack speed grows and at achievement of the certain length there is its sharp increase. The maximal speed of distribution of a crack usually on 2-3 order exceeds average speed at a slow stage of development.
        Studying kinetic the growth of defects initiated by a shaving cut, have shown, that development of a crack occurs in two stages distinguished in the speeds and topography of surfaces of destruction. The first stage - slow development of cracks - is characterized by a rough surface of break and time of development t1; the second stage - fast division of a sample into two parts - is characterized by a smooth surface of destruction and time of development of a crack t2.
        In picture 3.3 monograms of destruction of samples of tested rubbers [2] are given. At the first stage of destructions of samples of tested rubbers destruction occured more slowly development of a crack (picture 3.3, 1-5). At the certain length of a crack speed of destruction has reached the maximal value and occured practically instant (approximately for 0,02 sec) division of samples into two parts (picture3.3, section 6).
        General time of destruction of samples t:

T=t0+ t1 +t2

where t0 - time necessary for formation of a crack;
      t1 - a slow stage of development of a crack;
      t2 - a fast stage of development of a crack.

Picture 3.3 - The destruction of samples of rubbers with a preliminary cut

        If to present change of speed of distribution of a crack on its length in half-logarithmic coordinates (picture 3.3) at a stage of slow development this dependence represents the direct line which is taking place under some corner to an axis .

Picture 3.4-Schematic dependence of growth rate of a crack in dependence on its length :
lш-length of a rough zone; l2-length of a smooth zone; v0-initial speed of distribution of a crack; с-speed of a crack on the second site of destruction

        At achievement of the certain length of a crack there is a spasmodic transition to a fast stage of development, and t1>>t2.
        The analysis of the received experimental data allows to draw a conclusion on presence exponentional dependences of growth rate of a crack on its length at the first stage of destruction, i.e.

        From picture 3.4 durability can be presented tapes as:

        Generally the durability of samples can be expressed by a parity:

where b and В - the constants dependent on a material.
        Using Maceavely theory [7] according to which the end of a crack in a material has characteristic radius, it is possible to count up a pressure sтр about top of a crack under the formula:

where р - length of a crack, sm;
      J - some fictitious radius of a crack, sm.
        Using the formulas it is possible to receive, that

        Time of distribution of a crack in a sample:

where L0 - initial length of a crack.
        In tapes the cracks not dangerous to it since the pressure about their radiuses of distribution does not exceed explosive effort of a tape can be formed. The crack is not dangerous, if

using [8].
        Thus the pressure in a tape is distributed after an exhibitor aside decrease from initial radius of a crack, picture 3.5 [9].

Picture 3.5- Concentration of a pressure about an aperture not dangerous to break of a tape

        As the crack in tapee is distributed in two parties from conditionally taken the middle of its formation then and time of distribution of a crack in two parties with use of the formula for a rectilinear site of a tape without taking into account shifting efforts, it is possible to present pressure as:

        In a tape in one cross-section section some breakdowns then a pressure in a tape is relative by n breakdown can settle down:

at this break the effort in a tape is equal on :

where - the sum of lengths of cracks on width of a tape in one direction distributions;
      n - number of breakdowns in one direction of width of a tape.
      As the boundary condition is possible to take a pressure in a tape equal to a pressure of durability of a tape in view of factor of safety factor or to determine with the help rubberchacking gauges [10].

THE REPORT OF THE RECEIVED RESULTS

        The basic results of the given work were:
  1)   a conclusion of the basic mathematical dependence of pressure change in tapes at their damage by breakdown, and also time which is required for full destruction of a tape at breakdown. The given dependences look like exhibitors which shows distribution of a pressure and time of distribution of a crack to tapes as a result of breakdown in function from length of the given crack in a tape;
  2)  research a tape 2УБКНЛ-65 and calculation the critical size of a crack in it. It is established, that at breakdown of a tape in through equal 0,1 sm-the critical size of a crack is equaled by diameter of a circle of breakdown 0,8 sm, and at diameter of a circle of breakdown equal 0,5 sm-accordingly 4 sm;
  3)  at a tape there are two stages of destruction. The first when increase of a pressure in a tape about an aperture of breakdown changes on exponentional to the law on length of distribution of the crack, the given period evidently represents a rough surface of a zone of break; at the given stage speed of formation of a crack is not great. The second when occurs instant destruction a zone of a tape damage, at the given stage the pressure in a tape remains almost constant, and speed of distribution of a crack in 5-6 times exceeds speed of the given process of destruction at the first stage; the given stage of destruction of a tape is characterized by a flat smooth surface of destruction of a tape.

THE LIST OF THE USED LITERATURE

  1.Шахмейстер Л.Г., Солод Г.И. Подземные конвейерные установки.- "Недра", М., 1976,с.431
  2.Потураев В.Н. Резина в горном деле.- К., 1974, с.151
  3.Ультман В.Е., Чебаков В.М., Чудновский А.И. К вопросу о разрушении пространственно-структурных полимеров.- "Механика полимеров", 1972, №4, с.525-529.
  4.Чудновский А.И. О неизотермической деформации вязкоупругой среды.- ПМФТ ("Прикладная механика и техническая физика"), 1966, №3, с.84.
  5.Чудновский А.И. О разрушении макротел.- В кн.:Исследования по упругости и пластичности, Изд. ЛГУ, 1973, №9.
  6.Хилл Р. Упругие свойства составных тел.- Механика, М., "Мир", 1965, №5, (Сб. переводов, США).
  7.Кожушко Г.Г. Механика деформации и прогнозирования ресурса резинотканевых лент конвейеров горно-рудных предприятий- Д.,1992, с.34
  8.Спиваковский А.О., Дьячков В.К. Транспортирующие машины: Учеб. пособие для машиностроительных вузов.- 3-е изд., перераб.- М.: Машиностроение, 1983,с.487
  9.Монастырский Д.Ш. Механика процессов сборки резинотканевых конвейерных лент- Л.,1989,с.105
  10.Андреев А.В. Исследование и расчет конвейерных лент и приводов. Углетехиздат, 1959
  11.Яковлев П.В. Расчет прорезиненных лент с учетом напряжений изгиба. Материалы научно-технического совещания по ленточным конвейерам. ЦИТИ угля, Госгортехиздат, 1962
  12.Спиваковский А.О., Потапов М.Г., Котов М.А. Карьерный конвейерный транспорт. "Недра", 1965
  13.Биличенко Н.Я., Высочин Е.М., Завгородний Е.Х. Эксплуатационные режимы ленточных конвейеров. Гостехиздат, УССР, 1964
  14.Keller A. Und Blassi W. Betrachtungen und Berechnungsgrundlagen fur Fordergurte. Bergbautechnik. 1955, Helf 7, 1956, Helf 4.
  15.Чуканов В.И. Теоретические исследования изгиба ленты при набегании ее на барабан. Сб. "Транспорт горных предприятий". МИРГЭМ, 1963
  16.Штокман И.Г., Тимошкин В.А., Красиловский Л.С и др. Новая двухэлементная разборная тяговая цепь ЦДР для подземных скребковых конвейеров. "Уголь Украины", 1962, №2
  17.Григорьев В.Н. и др. Транспортные машины и комплексы подземных разработок- "Недра", М., 1976, с.400
  18.Прочность, устойчивость, колебания. Справочник, т.1, М.,"Машиностроение", 1968,с.575
  19.Перис П., Эрфган Ф. Критический анализ законов распространения трещин- "Техническая механика", М., "Мир", 1963, № 4
  20.Поляков Н.С., Штокман И.Г. Основы теории и расчета рудничных транспортных установок- "Госгортехиздат", М., 1962, с.490
  21.Иосилович Г.Б. и др. Прикладная механика- "Машиностроение", М., 1985, с.575
  22.Волотковский В.С. и др. Износ и долговечность конвейерных лент- "Недра", М., 1976, с.176